TSTP Solution File: GRP013-1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : GRP013-1 : 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 : n026.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 0.17s 0.39s
% Output : Proof 0.17s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : GRP013-1 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.33 % Computer : n026.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Mon Aug 28 21:21:35 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.17/0.39 Command-line arguments: --no-flatten-goal
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% 0.17/0.39 % SZS status Unsatisfiable
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% 0.17/0.40 % SZS output start Proof
% 0.17/0.40 Take the following subset of the input axioms:
% 0.17/0.40 fof(a_times_b_is_c, hypothesis, product(a, b, c)).
% 0.17/0.40 fof(inverse_a_times_inverse_b_is_d, hypothesis, product(inverse(a), inverse(b), d)).
% 0.17/0.40 fof(inverses_have_property, hypothesis, ![B, C, A2]: (~product(inverse(A2), inverse(B), C) | product(A2, C, B))).
% 0.17/0.40 fof(prove_c_times_d_is_identity, negated_conjecture, ~product(c, d, identity)).
% 0.17/0.40 fof(right_identity, axiom, ![X]: product(X, identity, X)).
% 0.17/0.40 fof(right_inverse, axiom, ![X2]: product(X2, inverse(X2), identity)).
% 0.17/0.40 fof(squareness, hypothesis, ![A]: product(A, A, identity)).
% 0.17/0.40 fof(total_function2, axiom, ![Y, Z, W, X2]: (~product(X2, Y, Z) | (~product(X2, Y, W) | Z=W))).
% 0.17/0.40
% 0.17/0.40 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.17/0.40 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.17/0.40 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.17/0.40 fresh(y, y, x1...xn) = u
% 0.17/0.40 C => fresh(s, t, x1...xn) = v
% 0.17/0.40 where fresh is a fresh function symbol and x1..xn are the free
% 0.17/0.40 variables of u and v.
% 0.17/0.40 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.17/0.40 input problem has no model of domain size 1).
% 0.17/0.40
% 0.17/0.40 The encoding turns the above axioms into the following unit equations and goals:
% 0.17/0.40
% 0.17/0.40 Axiom 1 (squareness): product(X, X, identity) = true.
% 0.17/0.40 Axiom 2 (right_identity): product(X, identity, X) = true.
% 0.17/0.40 Axiom 3 (a_times_b_is_c): product(a, b, c) = true.
% 0.17/0.40 Axiom 4 (total_function2): fresh(X, X, Y, Z) = Z.
% 0.17/0.40 Axiom 5 (right_inverse): product(X, inverse(X), identity) = true.
% 0.17/0.40 Axiom 6 (inverses_have_property): fresh3(X, X, Y, Z, W) = true.
% 0.17/0.40 Axiom 7 (inverse_a_times_inverse_b_is_d): product(inverse(a), inverse(b), d) = true.
% 0.17/0.40 Axiom 8 (total_function2): fresh2(X, X, Y, Z, W, V) = W.
% 0.17/0.40 Axiom 9 (total_function2): fresh2(product(X, Y, Z), true, X, Y, W, Z) = fresh(product(X, Y, W), true, W, Z).
% 0.17/0.40 Axiom 10 (inverses_have_property): fresh3(product(inverse(X), inverse(Y), Z), true, X, Y, Z) = product(X, Z, Y).
% 0.17/0.40
% 0.17/0.40 Lemma 11: inverse(X) = X.
% 0.17/0.40 Proof:
% 0.17/0.40 inverse(X)
% 0.17/0.40 = { by axiom 8 (total_function2) R->L }
% 0.17/0.40 fresh2(true, true, X, identity, inverse(X), X)
% 0.17/0.40 = { by axiom 2 (right_identity) R->L }
% 0.17/0.40 fresh2(product(X, identity, X), true, X, identity, inverse(X), X)
% 0.17/0.40 = { by axiom 9 (total_function2) }
% 0.17/0.40 fresh(product(X, identity, inverse(X)), true, inverse(X), X)
% 0.17/0.40 = { by axiom 10 (inverses_have_property) R->L }
% 0.17/0.40 fresh(fresh3(product(inverse(X), inverse(inverse(X)), identity), true, X, inverse(X), identity), true, inverse(X), X)
% 0.17/0.40 = { by axiom 5 (right_inverse) }
% 0.17/0.40 fresh(fresh3(true, true, X, inverse(X), identity), true, inverse(X), X)
% 0.17/0.40 = { by axiom 6 (inverses_have_property) }
% 0.17/0.40 fresh(true, true, inverse(X), X)
% 0.17/0.40 = { by axiom 4 (total_function2) }
% 0.17/0.40 X
% 0.17/0.40
% 0.17/0.40 Goal 1 (prove_c_times_d_is_identity): product(c, d, identity) = true.
% 0.17/0.40 Proof:
% 0.17/0.40 product(c, d, identity)
% 0.17/0.40 = { by axiom 8 (total_function2) R->L }
% 0.17/0.40 product(c, fresh2(true, true, a, b, d, c), identity)
% 0.17/0.40 = { by axiom 3 (a_times_b_is_c) R->L }
% 0.17/0.40 product(c, fresh2(product(a, b, c), true, a, b, d, c), identity)
% 0.17/0.40 = { by axiom 9 (total_function2) }
% 0.17/0.40 product(c, fresh(product(a, b, d), true, d, c), identity)
% 0.17/0.40 = { by lemma 11 R->L }
% 0.17/0.40 product(c, fresh(product(a, inverse(b), d), true, d, c), identity)
% 0.17/0.40 = { by lemma 11 R->L }
% 0.17/0.40 product(c, fresh(product(inverse(a), inverse(b), d), true, d, c), identity)
% 0.17/0.40 = { by axiom 7 (inverse_a_times_inverse_b_is_d) }
% 0.17/0.40 product(c, fresh(true, true, d, c), identity)
% 0.17/0.40 = { by axiom 4 (total_function2) }
% 0.17/0.40 product(c, c, identity)
% 0.17/0.40 = { by axiom 1 (squareness) }
% 0.17/0.40 true
% 0.17/0.40 % SZS output end Proof
% 0.17/0.40
% 0.17/0.40 RESULT: Unsatisfiable (the axioms are contradictory).
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