TSTP Solution File: FLD018-3 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : FLD018-3 : TPTP v8.1.2. Bugfixed v2.1.0.
% 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 : n023.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 : Wed Aug 30 22:36:51 EDT 2023
% Result : Unsatisfiable 0.20s 0.49s
% 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
% 0.07/0.12 % Problem : FLD018-3 : TPTP v8.1.2. Bugfixed v2.1.0.
% 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 : n023.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 : Mon Aug 28 00:39:55 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.20/0.49 Command-line arguments: --kbo-weight0 --lhs-weight 5 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --goal-heuristic
% 0.20/0.49
% 0.20/0.49 % SZS status Unsatisfiable
% 0.20/0.49
% 0.20/0.50 % SZS output start Proof
% 0.20/0.50 Take the following subset of the input axioms:
% 0.20/0.50 fof(a_is_defined, hypothesis, defined(a)).
% 0.20/0.50 fof(associativity_addition_2, axiom, ![X, V, W, Y, U, Z]: (sum(U, Z, W) | (~sum(X, Y, U) | (~sum(Y, Z, V) | ~sum(X, V, W))))).
% 0.20/0.50 fof(commutativity_addition, axiom, ![X2, Y2, Z2]: (sum(Y2, X2, Z2) | ~sum(X2, Y2, Z2))).
% 0.20/0.50 fof(existence_of_identity_addition, axiom, ![X2]: (sum(additive_identity, X2, X2) | ~defined(X2))).
% 0.20/0.50 fof(existence_of_inverse_addition, axiom, ![X2]: (sum(additive_inverse(X2), X2, additive_identity) | ~defined(X2))).
% 0.20/0.50 fof(not_sum_3, negated_conjecture, ~sum(additive_identity, additive_inverse(a), additive_identity)).
% 0.20/0.50 fof(sum_2, negated_conjecture, sum(additive_identity, a, additive_identity)).
% 0.20/0.50 fof(well_definedness_of_additive_identity, axiom, defined(additive_identity)).
% 0.20/0.50 fof(well_definedness_of_additive_inverse, axiom, ![X2]: (defined(additive_inverse(X2)) | ~defined(X2))).
% 0.20/0.50
% 0.20/0.50 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.50 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.50 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.50 fresh(y, y, x1...xn) = u
% 0.20/0.50 C => fresh(s, t, x1...xn) = v
% 0.20/0.50 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.50 variables of u and v.
% 0.20/0.50 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.50 input problem has no model of domain size 1).
% 0.20/0.50
% 0.20/0.50 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.50
% 0.20/0.50 Axiom 1 (well_definedness_of_additive_identity): defined(additive_identity) = true.
% 0.20/0.50 Axiom 2 (a_is_defined): defined(a) = true.
% 0.20/0.50 Axiom 3 (sum_2): sum(additive_identity, a, additive_identity) = true.
% 0.20/0.50 Axiom 4 (existence_of_identity_addition): fresh14(X, X, Y) = true.
% 0.20/0.50 Axiom 5 (existence_of_inverse_addition): fresh12(X, X, Y) = true.
% 0.20/0.50 Axiom 6 (well_definedness_of_additive_inverse): fresh3(X, X, Y) = true.
% 0.20/0.50 Axiom 7 (existence_of_identity_addition): fresh14(defined(X), true, X) = sum(additive_identity, X, X).
% 0.20/0.50 Axiom 8 (existence_of_inverse_addition): fresh12(defined(X), true, X) = sum(additive_inverse(X), X, additive_identity).
% 0.20/0.50 Axiom 9 (well_definedness_of_additive_inverse): fresh3(defined(X), true, X) = defined(additive_inverse(X)).
% 0.20/0.50 Axiom 10 (associativity_addition_2): fresh42(X, X, Y, Z, W) = true.
% 0.20/0.50 Axiom 11 (commutativity_addition): fresh18(X, X, Y, Z, W) = true.
% 0.20/0.50 Axiom 12 (associativity_addition_2): fresh21(X, X, Y, Z, W, V, U) = sum(Y, Z, W).
% 0.20/0.50 Axiom 13 (commutativity_addition): fresh18(sum(X, Y, Z), true, Y, X, Z) = sum(Y, X, Z).
% 0.20/0.50 Axiom 14 (associativity_addition_2): fresh41(X, X, Y, Z, W, V, U, T) = fresh42(sum(V, U, Y), true, Y, Z, W).
% 0.20/0.50 Axiom 15 (associativity_addition_2): fresh41(sum(X, Y, Z), true, W, Y, V, U, X, Z) = fresh21(sum(U, Z, V), true, W, Y, V, U, X).
% 0.20/0.50
% 0.20/0.50 Lemma 16: fresh41(sum(X, Y, additive_identity), true, Z, Y, additive_identity, additive_inverse(additive_identity), X, additive_identity) = sum(Z, Y, additive_identity).
% 0.20/0.50 Proof:
% 0.20/0.50 fresh41(sum(X, Y, additive_identity), true, Z, Y, additive_identity, additive_inverse(additive_identity), X, additive_identity)
% 0.20/0.50 = { by axiom 15 (associativity_addition_2) }
% 0.20/0.50 fresh21(sum(additive_inverse(additive_identity), additive_identity, additive_identity), true, Z, Y, additive_identity, additive_inverse(additive_identity), X)
% 0.20/0.50 = { by axiom 8 (existence_of_inverse_addition) R->L }
% 0.20/0.50 fresh21(fresh12(defined(additive_identity), true, additive_identity), true, Z, Y, additive_identity, additive_inverse(additive_identity), X)
% 0.20/0.50 = { by axiom 1 (well_definedness_of_additive_identity) }
% 0.20/0.50 fresh21(fresh12(true, true, additive_identity), true, Z, Y, additive_identity, additive_inverse(additive_identity), X)
% 0.20/0.50 = { by axiom 5 (existence_of_inverse_addition) }
% 0.20/0.50 fresh21(true, true, Z, Y, additive_identity, additive_inverse(additive_identity), X)
% 0.20/0.50 = { by axiom 12 (associativity_addition_2) }
% 0.20/0.50 sum(Z, Y, additive_identity)
% 0.20/0.50
% 0.20/0.50 Goal 1 (not_sum_3): sum(additive_identity, additive_inverse(a), additive_identity) = true.
% 0.20/0.50 Proof:
% 0.20/0.50 sum(additive_identity, additive_inverse(a), additive_identity)
% 0.20/0.50 = { by lemma 16 R->L }
% 0.20/0.50 fresh41(sum(a, additive_inverse(a), additive_identity), true, additive_identity, additive_inverse(a), additive_identity, additive_inverse(additive_identity), a, additive_identity)
% 0.20/0.50 = { by axiom 13 (commutativity_addition) R->L }
% 0.20/0.50 fresh41(fresh18(sum(additive_inverse(a), a, additive_identity), true, a, additive_inverse(a), additive_identity), true, additive_identity, additive_inverse(a), additive_identity, additive_inverse(additive_identity), a, additive_identity)
% 0.20/0.50 = { by axiom 8 (existence_of_inverse_addition) R->L }
% 0.20/0.50 fresh41(fresh18(fresh12(defined(a), true, a), true, a, additive_inverse(a), additive_identity), true, additive_identity, additive_inverse(a), additive_identity, additive_inverse(additive_identity), a, additive_identity)
% 0.20/0.50 = { by axiom 2 (a_is_defined) }
% 0.20/0.50 fresh41(fresh18(fresh12(true, true, a), true, a, additive_inverse(a), additive_identity), true, additive_identity, additive_inverse(a), additive_identity, additive_inverse(additive_identity), a, additive_identity)
% 0.20/0.50 = { by axiom 5 (existence_of_inverse_addition) }
% 0.20/0.50 fresh41(fresh18(true, true, a, additive_inverse(a), additive_identity), true, additive_identity, additive_inverse(a), additive_identity, additive_inverse(additive_identity), a, additive_identity)
% 0.20/0.50 = { by axiom 11 (commutativity_addition) }
% 0.20/0.50 fresh41(true, true, additive_identity, additive_inverse(a), additive_identity, additive_inverse(additive_identity), a, additive_identity)
% 0.20/0.50 = { by axiom 14 (associativity_addition_2) }
% 0.20/0.50 fresh42(sum(additive_inverse(additive_identity), a, additive_identity), true, additive_identity, additive_inverse(a), additive_identity)
% 0.20/0.50 = { by lemma 16 R->L }
% 0.20/0.50 fresh42(fresh41(sum(additive_identity, a, additive_identity), true, additive_inverse(additive_identity), a, additive_identity, additive_inverse(additive_identity), additive_identity, additive_identity), true, additive_identity, additive_inverse(a), additive_identity)
% 0.20/0.50 = { by axiom 3 (sum_2) }
% 0.20/0.50 fresh42(fresh41(true, true, additive_inverse(additive_identity), a, additive_identity, additive_inverse(additive_identity), additive_identity, additive_identity), true, additive_identity, additive_inverse(a), additive_identity)
% 0.20/0.50 = { by axiom 14 (associativity_addition_2) }
% 0.20/0.50 fresh42(fresh42(sum(additive_inverse(additive_identity), additive_identity, additive_inverse(additive_identity)), true, additive_inverse(additive_identity), a, additive_identity), true, additive_identity, additive_inverse(a), additive_identity)
% 0.20/0.50 = { by axiom 13 (commutativity_addition) R->L }
% 0.20/0.50 fresh42(fresh42(fresh18(sum(additive_identity, additive_inverse(additive_identity), additive_inverse(additive_identity)), true, additive_inverse(additive_identity), additive_identity, additive_inverse(additive_identity)), true, additive_inverse(additive_identity), a, additive_identity), true, additive_identity, additive_inverse(a), additive_identity)
% 0.20/0.50 = { by axiom 7 (existence_of_identity_addition) R->L }
% 0.20/0.50 fresh42(fresh42(fresh18(fresh14(defined(additive_inverse(additive_identity)), true, additive_inverse(additive_identity)), true, additive_inverse(additive_identity), additive_identity, additive_inverse(additive_identity)), true, additive_inverse(additive_identity), a, additive_identity), true, additive_identity, additive_inverse(a), additive_identity)
% 0.20/0.50 = { by axiom 9 (well_definedness_of_additive_inverse) R->L }
% 0.20/0.50 fresh42(fresh42(fresh18(fresh14(fresh3(defined(additive_identity), true, additive_identity), true, additive_inverse(additive_identity)), true, additive_inverse(additive_identity), additive_identity, additive_inverse(additive_identity)), true, additive_inverse(additive_identity), a, additive_identity), true, additive_identity, additive_inverse(a), additive_identity)
% 0.20/0.50 = { by axiom 1 (well_definedness_of_additive_identity) }
% 0.20/0.50 fresh42(fresh42(fresh18(fresh14(fresh3(true, true, additive_identity), true, additive_inverse(additive_identity)), true, additive_inverse(additive_identity), additive_identity, additive_inverse(additive_identity)), true, additive_inverse(additive_identity), a, additive_identity), true, additive_identity, additive_inverse(a), additive_identity)
% 0.20/0.50 = { by axiom 6 (well_definedness_of_additive_inverse) }
% 0.20/0.50 fresh42(fresh42(fresh18(fresh14(true, true, additive_inverse(additive_identity)), true, additive_inverse(additive_identity), additive_identity, additive_inverse(additive_identity)), true, additive_inverse(additive_identity), a, additive_identity), true, additive_identity, additive_inverse(a), additive_identity)
% 0.20/0.50 = { by axiom 4 (existence_of_identity_addition) }
% 0.20/0.50 fresh42(fresh42(fresh18(true, true, additive_inverse(additive_identity), additive_identity, additive_inverse(additive_identity)), true, additive_inverse(additive_identity), a, additive_identity), true, additive_identity, additive_inverse(a), additive_identity)
% 0.20/0.50 = { by axiom 11 (commutativity_addition) }
% 0.20/0.50 fresh42(fresh42(true, true, additive_inverse(additive_identity), a, additive_identity), true, additive_identity, additive_inverse(a), additive_identity)
% 0.20/0.50 = { by axiom 10 (associativity_addition_2) }
% 0.20/0.50 fresh42(true, true, additive_identity, additive_inverse(a), additive_identity)
% 0.20/0.50 = { by axiom 10 (associativity_addition_2) }
% 0.20/0.50 true
% 0.20/0.50 % SZS output end Proof
% 0.20/0.50
% 0.20/0.50 RESULT: Unsatisfiable (the axioms are contradictory).
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