TSTP Solution File: GRP030-1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : GRP030-1 : 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 : n032.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 01:16:41 EDT 2023
% Result : Unsatisfiable 0.09s 0.32s
% Output : Proof 0.09s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : GRP030-1 : TPTP v8.1.2. Released v1.0.0.
% 0.08/0.10 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.09/0.28 % Computer : n032.cluster.edu
% 0.09/0.28 % Model : x86_64 x86_64
% 0.09/0.28 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.28 % Memory : 8042.1875MB
% 0.09/0.28 % OS : Linux 3.10.0-693.el7.x86_64
% 0.09/0.28 % CPULimit : 300
% 0.09/0.28 % WCLimit : 300
% 0.09/0.28 % DateTime : Tue Aug 29 00:10:46 EDT 2023
% 0.09/0.29 % CPUTime :
% 0.09/0.32 Command-line arguments: --no-flatten-goal
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% 0.09/0.32 % SZS status Unsatisfiable
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% 0.09/0.33 % SZS output start Proof
% 0.09/0.33 Take the following subset of the input axioms:
% 0.09/0.33 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))))).
% 0.09/0.33 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))))).
% 0.09/0.33 fof(left_identity, hypothesis, ![A]: product(identity, A, A)).
% 0.09/0.33 fof(left_inverse, hypothesis, ![A2]: product(inverse(A2), A2, identity)).
% 0.09/0.33 fof(prove_identity_is_a_right_identity, negated_conjecture, ~product(a, identity, a)).
% 0.09/0.33 fof(total_function1, axiom, ![X2, Y2]: product(X2, Y2, multiply(X2, Y2))).
% 0.09/0.33 fof(total_function2, axiom, ![X2, Y2, Z2, W2]: (~product(X2, Y2, Z2) | (~product(X2, Y2, W2) | Z2=W2))).
% 0.09/0.33
% 0.09/0.33 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.09/0.33 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.09/0.33 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.09/0.33 fresh(y, y, x1...xn) = u
% 0.09/0.33 C => fresh(s, t, x1...xn) = v
% 0.09/0.33 where fresh is a fresh function symbol and x1..xn are the free
% 0.09/0.33 variables of u and v.
% 0.09/0.33 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.09/0.33 input problem has no model of domain size 1).
% 0.09/0.33
% 0.09/0.33 The encoding turns the above axioms into the following unit equations and goals:
% 0.09/0.33
% 0.09/0.33 Axiom 1 (left_identity): product(identity, X, X) = true.
% 0.09/0.33 Axiom 2 (total_function2): fresh(X, X, Y, Z) = Z.
% 0.09/0.33 Axiom 3 (left_inverse): product(inverse(X), X, identity) = true.
% 0.09/0.33 Axiom 4 (associativity1): fresh8(X, X, Y, Z, W) = true.
% 0.09/0.33 Axiom 5 (associativity2): fresh6(X, X, Y, Z, W) = true.
% 0.09/0.33 Axiom 6 (total_function1): product(X, Y, multiply(X, Y)) = true.
% 0.09/0.33 Axiom 7 (total_function2): fresh2(X, X, Y, Z, W, V) = W.
% 0.09/0.33 Axiom 8 (associativity1): fresh4(X, X, Y, Z, W, V, U) = product(Y, V, U).
% 0.09/0.33 Axiom 9 (associativity2): fresh3(X, X, Y, Z, W, V, U) = product(W, V, U).
% 0.09/0.33 Axiom 10 (associativity1): fresh7(X, X, Y, Z, W, V, U, T) = fresh8(product(Y, Z, W), true, Y, U, T).
% 0.09/0.33 Axiom 11 (associativity2): fresh5(X, X, Y, Z, W, V, U, T) = fresh6(product(Y, Z, W), true, W, V, T).
% 0.09/0.33 Axiom 12 (total_function2): fresh2(product(X, Y, Z), true, X, Y, W, Z) = fresh(product(X, Y, W), true, W, Z).
% 0.09/0.33 Axiom 13 (associativity1): fresh7(product(X, Y, Z), true, W, V, X, Y, U, Z) = fresh4(product(V, Y, U), true, W, V, X, U, Z).
% 0.09/0.33 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).
% 0.09/0.33
% 0.09/0.33 Lemma 15: fresh7(X, X, Y, inverse(Z), identity, W, identity, Z) = product(Y, identity, Z).
% 0.09/0.33 Proof:
% 0.09/0.33 fresh7(X, X, Y, inverse(Z), identity, W, identity, Z)
% 0.09/0.33 = { by axiom 10 (associativity1) }
% 0.09/0.33 fresh8(product(Y, inverse(Z), identity), true, Y, identity, Z)
% 0.09/0.33 = { by axiom 10 (associativity1) R->L }
% 0.09/0.33 fresh7(true, true, Y, inverse(Z), identity, Z, identity, Z)
% 0.09/0.33 = { by axiom 1 (left_identity) R->L }
% 0.09/0.33 fresh7(product(identity, Z, Z), true, Y, inverse(Z), identity, Z, identity, Z)
% 0.09/0.33 = { by axiom 13 (associativity1) }
% 0.09/0.33 fresh4(product(inverse(Z), Z, identity), true, Y, inverse(Z), identity, identity, Z)
% 0.09/0.33 = { by axiom 3 (left_inverse) }
% 0.09/0.33 fresh4(true, true, Y, inverse(Z), identity, identity, Z)
% 0.09/0.33 = { by axiom 8 (associativity1) }
% 0.09/0.33 product(Y, identity, Z)
% 0.09/0.33
% 0.09/0.33 Goal 1 (prove_identity_is_a_right_identity): product(a, identity, a) = true.
% 0.09/0.33 Proof:
% 0.09/0.33 product(a, identity, a)
% 0.09/0.33 = { by axiom 7 (total_function2) R->L }
% 0.09/0.33 product(fresh2(true, true, inverse(inverse(a)), identity, a, multiply(inverse(inverse(a)), identity)), identity, a)
% 0.09/0.33 = { by axiom 6 (total_function1) R->L }
% 0.09/0.33 product(fresh2(product(inverse(inverse(a)), identity, multiply(inverse(inverse(a)), identity)), true, inverse(inverse(a)), identity, a, multiply(inverse(inverse(a)), identity)), identity, a)
% 0.09/0.33 = { by axiom 12 (total_function2) }
% 0.09/0.33 product(fresh(product(inverse(inverse(a)), identity, a), true, a, multiply(inverse(inverse(a)), identity)), identity, a)
% 0.09/0.33 = { by lemma 15 R->L }
% 0.09/0.33 product(fresh(fresh7(X, X, inverse(inverse(a)), inverse(a), identity, Y, identity, a), true, a, multiply(inverse(inverse(a)), identity)), identity, a)
% 0.09/0.33 = { by axiom 10 (associativity1) }
% 0.09/0.33 product(fresh(fresh8(product(inverse(inverse(a)), inverse(a), identity), true, inverse(inverse(a)), identity, a), true, a, multiply(inverse(inverse(a)), identity)), identity, a)
% 0.09/0.33 = { by axiom 3 (left_inverse) }
% 0.09/0.33 product(fresh(fresh8(true, true, inverse(inverse(a)), identity, a), true, a, multiply(inverse(inverse(a)), identity)), identity, a)
% 0.09/0.33 = { by axiom 4 (associativity1) }
% 0.09/0.33 product(fresh(true, true, a, multiply(inverse(inverse(a)), identity)), identity, a)
% 0.09/0.33 = { by axiom 2 (total_function2) }
% 0.09/0.33 product(multiply(inverse(inverse(a)), identity), identity, a)
% 0.09/0.33 = { by lemma 15 R->L }
% 0.09/0.33 fresh7(Z, Z, multiply(inverse(inverse(a)), identity), inverse(a), identity, W, identity, a)
% 0.09/0.33 = { by axiom 10 (associativity1) }
% 0.09/0.33 fresh8(product(multiply(inverse(inverse(a)), identity), inverse(a), identity), true, multiply(inverse(inverse(a)), identity), identity, a)
% 0.09/0.33 = { by axiom 9 (associativity2) R->L }
% 0.09/0.33 fresh8(fresh3(true, true, inverse(inverse(a)), identity, multiply(inverse(inverse(a)), identity), inverse(a), identity), true, multiply(inverse(inverse(a)), identity), identity, a)
% 0.09/0.33 = { by axiom 3 (left_inverse) R->L }
% 0.09/0.33 fresh8(fresh3(product(inverse(inverse(a)), inverse(a), identity), true, inverse(inverse(a)), identity, multiply(inverse(inverse(a)), identity), inverse(a), identity), true, multiply(inverse(inverse(a)), identity), identity, a)
% 0.09/0.33 = { by axiom 14 (associativity2) R->L }
% 0.09/0.33 fresh8(fresh5(product(identity, inverse(a), inverse(a)), true, inverse(inverse(a)), identity, multiply(inverse(inverse(a)), identity), inverse(a), inverse(a), identity), true, multiply(inverse(inverse(a)), identity), identity, a)
% 0.09/0.33 = { by axiom 1 (left_identity) }
% 0.09/0.33 fresh8(fresh5(true, true, inverse(inverse(a)), identity, multiply(inverse(inverse(a)), identity), inverse(a), inverse(a), identity), true, multiply(inverse(inverse(a)), identity), identity, a)
% 0.09/0.33 = { by axiom 11 (associativity2) }
% 0.09/0.33 fresh8(fresh6(product(inverse(inverse(a)), identity, multiply(inverse(inverse(a)), identity)), true, multiply(inverse(inverse(a)), identity), inverse(a), identity), true, multiply(inverse(inverse(a)), identity), identity, a)
% 0.09/0.33 = { by axiom 6 (total_function1) }
% 0.09/0.33 fresh8(fresh6(true, true, multiply(inverse(inverse(a)), identity), inverse(a), identity), true, multiply(inverse(inverse(a)), identity), identity, a)
% 0.09/0.33 = { by axiom 5 (associativity2) }
% 0.09/0.33 fresh8(true, true, multiply(inverse(inverse(a)), identity), identity, a)
% 0.09/0.33 = { by axiom 4 (associativity1) }
% 0.09/0.33 true
% 0.09/0.33 % SZS output end Proof
% 0.09/0.33
% 0.09/0.33 RESULT: Unsatisfiable (the axioms are contradictory).
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