TSTP Solution File: BOO003-1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : BOO003-1 : 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 : n007.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 18:11:15 EDT 2023
% Result : Unsatisfiable 0.20s 0.74s
% 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
% 0.07/0.13 % Problem : BOO003-1 : TPTP v8.1.2. Released v1.0.0.
% 0.07/0.14 % 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 : n007.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 : Sun Aug 27 08:32:42 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.20/0.74 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.20/0.74
% 0.20/0.74 % SZS status Unsatisfiable
% 0.20/0.74
% 0.20/0.76 % SZS output start Proof
% 0.20/0.76 Take the following subset of the input axioms:
% 0.20/0.76 fof(additive_identity2, axiom, ![X]: sum(X, additive_identity, X)).
% 0.20/0.76 fof(additive_inverse2, axiom, ![X2]: sum(X2, inverse(X2), multiplicative_identity)).
% 0.20/0.76 fof(closure_of_addition, axiom, ![Y, X2]: sum(X2, Y, add(X2, Y))).
% 0.20/0.76 fof(closure_of_multiplication, axiom, ![X2, Y2]: product(X2, Y2, multiply(X2, Y2))).
% 0.20/0.76 fof(distributivity2, axiom, ![Z, V1, V2, V3, V4, X2, Y2]: (~product(X2, Y2, V1) | (~product(X2, Z, V2) | (~sum(Y2, Z, V3) | (~sum(V1, V2, V4) | product(X2, V3, V4)))))).
% 0.20/0.76 fof(distributivity5, axiom, ![X2, Y2, Z2, V1_2, V2_2, V3_2, V4_2]: (~sum(X2, Y2, V1_2) | (~sum(X2, Z2, V2_2) | (~product(Y2, Z2, V3_2) | (~sum(X2, V3_2, V4_2) | product(V1_2, V2_2, V4_2)))))).
% 0.20/0.76 fof(multiplication_is_well_defined, axiom, ![U, V, X2, Y2]: (~product(X2, Y2, U) | (~product(X2, Y2, V) | U=V))).
% 0.20/0.76 fof(multiplicative_identity1, axiom, ![X2]: product(multiplicative_identity, X2, X2)).
% 0.20/0.76 fof(multiplicative_identity2, axiom, ![X2]: product(X2, multiplicative_identity, X2)).
% 0.20/0.76 fof(prove_both_equalities, negated_conjecture, ~product(x, x, x)).
% 0.20/0.76
% 0.20/0.76 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.76 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.76 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.76 fresh(y, y, x1...xn) = u
% 0.20/0.76 C => fresh(s, t, x1...xn) = v
% 0.20/0.76 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.76 variables of u and v.
% 0.20/0.76 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.76 input problem has no model of domain size 1).
% 0.20/0.76
% 0.20/0.76 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.76
% 0.20/0.76 Axiom 1 (additive_identity2): sum(X, additive_identity, X) = true.
% 0.20/0.76 Axiom 2 (multiplicative_identity2): product(X, multiplicative_identity, X) = true.
% 0.20/0.76 Axiom 3 (multiplicative_identity1): product(multiplicative_identity, X, X) = true.
% 0.20/0.76 Axiom 4 (multiplication_is_well_defined): fresh(X, X, Y, Z) = Z.
% 0.20/0.76 Axiom 5 (additive_inverse2): sum(X, inverse(X), multiplicative_identity) = true.
% 0.20/0.76 Axiom 6 (distributivity2): fresh34(X, X, Y, Z, W) = true.
% 0.20/0.76 Axiom 7 (distributivity5): fresh22(X, X, Y, Z, W) = true.
% 0.20/0.76 Axiom 8 (closure_of_addition): sum(X, Y, add(X, Y)) = true.
% 0.20/0.76 Axiom 9 (closure_of_multiplication): product(X, Y, multiply(X, Y)) = true.
% 0.20/0.76 Axiom 10 (multiplication_is_well_defined): fresh2(X, X, Y, Z, W, V) = W.
% 0.20/0.76 Axiom 11 (distributivity2): fresh32(X, X, Y, Z, W, V, U) = product(Y, V, U).
% 0.20/0.76 Axiom 12 (distributivity5): fresh20(X, X, Y, Z, W, V, U) = product(W, V, U).
% 0.20/0.76 Axiom 13 (distributivity5): fresh21(X, X, Y, Z, W, V, U, T) = fresh22(sum(Y, Z, W), true, W, U, T).
% 0.20/0.76 Axiom 14 (distributivity2): fresh33(X, X, Y, Z, W, V, U, T, S) = fresh34(sum(Z, V, T), true, Y, T, S).
% 0.20/0.76 Axiom 15 (multiplication_is_well_defined): fresh2(product(X, Y, Z), true, X, Y, W, Z) = fresh(product(X, Y, W), true, W, Z).
% 0.20/0.76 Axiom 16 (distributivity2): fresh31(X, X, Y, Z, W, V, U, T, S) = fresh32(sum(W, U, S), true, Y, Z, V, T, S).
% 0.20/0.76 Axiom 17 (distributivity5): fresh19(X, X, Y, Z, W, V, U, T, S) = fresh20(sum(Y, V, U), true, Y, Z, W, U, S).
% 0.20/0.76 Axiom 18 (distributivity2): fresh31(product(X, Y, Z), true, X, W, V, Y, Z, U, T) = fresh33(product(X, W, V), true, X, W, V, Y, Z, U, T).
% 0.20/0.76 Axiom 19 (distributivity5): fresh19(product(X, Y, Z), true, W, X, V, Y, U, Z, T) = fresh21(sum(W, Z, T), true, W, X, V, Y, U, T).
% 0.20/0.76
% 0.20/0.76 Lemma 20: fresh(product(X, multiplicative_identity, Y), true, Y, X) = Y.
% 0.20/0.76 Proof:
% 0.20/0.76 fresh(product(X, multiplicative_identity, Y), true, Y, X)
% 0.20/0.76 = { by axiom 15 (multiplication_is_well_defined) R->L }
% 0.20/0.76 fresh2(product(X, multiplicative_identity, X), true, X, multiplicative_identity, Y, X)
% 0.20/0.76 = { by axiom 2 (multiplicative_identity2) }
% 0.20/0.76 fresh2(true, true, X, multiplicative_identity, Y, X)
% 0.20/0.76 = { by axiom 10 (multiplication_is_well_defined) }
% 0.20/0.77 Y
% 0.20/0.77
% 0.20/0.77 Goal 1 (prove_both_equalities): product(x, x, x) = true.
% 0.20/0.77 Proof:
% 0.20/0.77 product(x, x, x)
% 0.20/0.77 = { by axiom 12 (distributivity5) R->L }
% 0.20/0.77 fresh20(true, true, x, additive_identity, x, x, x)
% 0.20/0.77 = { by axiom 1 (additive_identity2) R->L }
% 0.20/0.77 fresh20(sum(x, additive_identity, x), true, x, additive_identity, x, x, x)
% 0.20/0.77 = { by axiom 17 (distributivity5) R->L }
% 0.20/0.77 fresh19(true, true, x, additive_identity, x, additive_identity, x, additive_identity, x)
% 0.20/0.77 = { by axiom 9 (closure_of_multiplication) R->L }
% 0.20/0.77 fresh19(product(additive_identity, additive_identity, multiply(additive_identity, additive_identity)), true, x, additive_identity, x, additive_identity, x, additive_identity, x)
% 0.20/0.77 = { by lemma 20 R->L }
% 0.20/0.77 fresh19(product(additive_identity, additive_identity, fresh(product(additive_identity, multiplicative_identity, multiply(additive_identity, additive_identity)), true, multiply(additive_identity, additive_identity), additive_identity)), true, x, additive_identity, x, additive_identity, x, additive_identity, x)
% 0.20/0.77 = { by lemma 20 R->L }
% 0.20/0.77 fresh19(product(additive_identity, additive_identity, fresh(product(additive_identity, fresh(product(add(additive_identity, multiplicative_identity), multiplicative_identity, multiplicative_identity), true, multiplicative_identity, add(additive_identity, multiplicative_identity)), multiply(additive_identity, additive_identity)), true, multiply(additive_identity, additive_identity), additive_identity)), true, x, additive_identity, x, additive_identity, x, additive_identity, x)
% 0.20/0.77 = { by axiom 12 (distributivity5) R->L }
% 0.20/0.77 fresh19(product(additive_identity, additive_identity, fresh(product(additive_identity, fresh(fresh20(true, true, additive_identity, multiplicative_identity, add(additive_identity, multiplicative_identity), multiplicative_identity, multiplicative_identity), true, multiplicative_identity, add(additive_identity, multiplicative_identity)), multiply(additive_identity, additive_identity)), true, multiply(additive_identity, additive_identity), additive_identity)), true, x, additive_identity, x, additive_identity, x, additive_identity, x)
% 0.20/0.77 = { by axiom 5 (additive_inverse2) R->L }
% 0.20/0.77 fresh19(product(additive_identity, additive_identity, fresh(product(additive_identity, fresh(fresh20(sum(additive_identity, inverse(additive_identity), multiplicative_identity), true, additive_identity, multiplicative_identity, add(additive_identity, multiplicative_identity), multiplicative_identity, multiplicative_identity), true, multiplicative_identity, add(additive_identity, multiplicative_identity)), multiply(additive_identity, additive_identity)), true, multiply(additive_identity, additive_identity), additive_identity)), true, x, additive_identity, x, additive_identity, x, additive_identity, x)
% 0.20/0.77 = { by axiom 17 (distributivity5) R->L }
% 0.20/0.77 fresh19(product(additive_identity, additive_identity, fresh(product(additive_identity, fresh(fresh19(true, true, additive_identity, multiplicative_identity, add(additive_identity, multiplicative_identity), inverse(additive_identity), multiplicative_identity, inverse(additive_identity), multiplicative_identity), true, multiplicative_identity, add(additive_identity, multiplicative_identity)), multiply(additive_identity, additive_identity)), true, multiply(additive_identity, additive_identity), additive_identity)), true, x, additive_identity, x, additive_identity, x, additive_identity, x)
% 0.20/0.77 = { by axiom 3 (multiplicative_identity1) R->L }
% 0.20/0.77 fresh19(product(additive_identity, additive_identity, fresh(product(additive_identity, fresh(fresh19(product(multiplicative_identity, inverse(additive_identity), inverse(additive_identity)), true, additive_identity, multiplicative_identity, add(additive_identity, multiplicative_identity), inverse(additive_identity), multiplicative_identity, inverse(additive_identity), multiplicative_identity), true, multiplicative_identity, add(additive_identity, multiplicative_identity)), multiply(additive_identity, additive_identity)), true, multiply(additive_identity, additive_identity), additive_identity)), true, x, additive_identity, x, additive_identity, x, additive_identity, x)
% 0.20/0.77 = { by axiom 19 (distributivity5) }
% 0.20/0.77 fresh19(product(additive_identity, additive_identity, fresh(product(additive_identity, fresh(fresh21(sum(additive_identity, inverse(additive_identity), multiplicative_identity), true, additive_identity, multiplicative_identity, add(additive_identity, multiplicative_identity), inverse(additive_identity), multiplicative_identity, multiplicative_identity), true, multiplicative_identity, add(additive_identity, multiplicative_identity)), multiply(additive_identity, additive_identity)), true, multiply(additive_identity, additive_identity), additive_identity)), true, x, additive_identity, x, additive_identity, x, additive_identity, x)
% 0.20/0.77 = { by axiom 5 (additive_inverse2) }
% 0.20/0.77 fresh19(product(additive_identity, additive_identity, fresh(product(additive_identity, fresh(fresh21(true, true, additive_identity, multiplicative_identity, add(additive_identity, multiplicative_identity), inverse(additive_identity), multiplicative_identity, multiplicative_identity), true, multiplicative_identity, add(additive_identity, multiplicative_identity)), multiply(additive_identity, additive_identity)), true, multiply(additive_identity, additive_identity), additive_identity)), true, x, additive_identity, x, additive_identity, x, additive_identity, x)
% 0.20/0.77 = { by axiom 13 (distributivity5) }
% 0.20/0.77 fresh19(product(additive_identity, additive_identity, fresh(product(additive_identity, fresh(fresh22(sum(additive_identity, multiplicative_identity, add(additive_identity, multiplicative_identity)), true, add(additive_identity, multiplicative_identity), multiplicative_identity, multiplicative_identity), true, multiplicative_identity, add(additive_identity, multiplicative_identity)), multiply(additive_identity, additive_identity)), true, multiply(additive_identity, additive_identity), additive_identity)), true, x, additive_identity, x, additive_identity, x, additive_identity, x)
% 0.20/0.77 = { by axiom 8 (closure_of_addition) }
% 0.20/0.77 fresh19(product(additive_identity, additive_identity, fresh(product(additive_identity, fresh(fresh22(true, true, add(additive_identity, multiplicative_identity), multiplicative_identity, multiplicative_identity), true, multiplicative_identity, add(additive_identity, multiplicative_identity)), multiply(additive_identity, additive_identity)), true, multiply(additive_identity, additive_identity), additive_identity)), true, x, additive_identity, x, additive_identity, x, additive_identity, x)
% 0.20/0.77 = { by axiom 7 (distributivity5) }
% 0.20/0.77 fresh19(product(additive_identity, additive_identity, fresh(product(additive_identity, fresh(true, true, multiplicative_identity, add(additive_identity, multiplicative_identity)), multiply(additive_identity, additive_identity)), true, multiply(additive_identity, additive_identity), additive_identity)), true, x, additive_identity, x, additive_identity, x, additive_identity, x)
% 0.20/0.77 = { by axiom 4 (multiplication_is_well_defined) }
% 0.20/0.77 fresh19(product(additive_identity, additive_identity, fresh(product(additive_identity, add(additive_identity, multiplicative_identity), multiply(additive_identity, additive_identity)), true, multiply(additive_identity, additive_identity), additive_identity)), true, x, additive_identity, x, additive_identity, x, additive_identity, x)
% 0.20/0.77 = { by axiom 11 (distributivity2) R->L }
% 0.20/0.77 fresh19(product(additive_identity, additive_identity, fresh(fresh32(true, true, additive_identity, additive_identity, multiplicative_identity, add(additive_identity, multiplicative_identity), multiply(additive_identity, additive_identity)), true, multiply(additive_identity, additive_identity), additive_identity)), true, x, additive_identity, x, additive_identity, x, additive_identity, x)
% 0.20/0.77 = { by axiom 1 (additive_identity2) R->L }
% 0.20/0.77 fresh19(product(additive_identity, additive_identity, fresh(fresh32(sum(multiply(additive_identity, additive_identity), additive_identity, multiply(additive_identity, additive_identity)), true, additive_identity, additive_identity, multiplicative_identity, add(additive_identity, multiplicative_identity), multiply(additive_identity, additive_identity)), true, multiply(additive_identity, additive_identity), additive_identity)), true, x, additive_identity, x, additive_identity, x, additive_identity, x)
% 0.20/0.77 = { by axiom 16 (distributivity2) R->L }
% 0.20/0.77 fresh19(product(additive_identity, additive_identity, fresh(fresh31(true, true, additive_identity, additive_identity, multiply(additive_identity, additive_identity), multiplicative_identity, additive_identity, add(additive_identity, multiplicative_identity), multiply(additive_identity, additive_identity)), true, multiply(additive_identity, additive_identity), additive_identity)), true, x, additive_identity, x, additive_identity, x, additive_identity, x)
% 0.20/0.77 = { by axiom 2 (multiplicative_identity2) R->L }
% 0.20/0.77 fresh19(product(additive_identity, additive_identity, fresh(fresh31(product(additive_identity, multiplicative_identity, additive_identity), true, additive_identity, additive_identity, multiply(additive_identity, additive_identity), multiplicative_identity, additive_identity, add(additive_identity, multiplicative_identity), multiply(additive_identity, additive_identity)), true, multiply(additive_identity, additive_identity), additive_identity)), true, x, additive_identity, x, additive_identity, x, additive_identity, x)
% 0.20/0.77 = { by axiom 18 (distributivity2) }
% 0.20/0.77 fresh19(product(additive_identity, additive_identity, fresh(fresh33(product(additive_identity, additive_identity, multiply(additive_identity, additive_identity)), true, additive_identity, additive_identity, multiply(additive_identity, additive_identity), multiplicative_identity, additive_identity, add(additive_identity, multiplicative_identity), multiply(additive_identity, additive_identity)), true, multiply(additive_identity, additive_identity), additive_identity)), true, x, additive_identity, x, additive_identity, x, additive_identity, x)
% 0.20/0.77 = { by axiom 9 (closure_of_multiplication) }
% 0.20/0.77 fresh19(product(additive_identity, additive_identity, fresh(fresh33(true, true, additive_identity, additive_identity, multiply(additive_identity, additive_identity), multiplicative_identity, additive_identity, add(additive_identity, multiplicative_identity), multiply(additive_identity, additive_identity)), true, multiply(additive_identity, additive_identity), additive_identity)), true, x, additive_identity, x, additive_identity, x, additive_identity, x)
% 0.20/0.77 = { by axiom 14 (distributivity2) }
% 0.20/0.77 fresh19(product(additive_identity, additive_identity, fresh(fresh34(sum(additive_identity, multiplicative_identity, add(additive_identity, multiplicative_identity)), true, additive_identity, add(additive_identity, multiplicative_identity), multiply(additive_identity, additive_identity)), true, multiply(additive_identity, additive_identity), additive_identity)), true, x, additive_identity, x, additive_identity, x, additive_identity, x)
% 0.20/0.77 = { by axiom 8 (closure_of_addition) }
% 0.20/0.77 fresh19(product(additive_identity, additive_identity, fresh(fresh34(true, true, additive_identity, add(additive_identity, multiplicative_identity), multiply(additive_identity, additive_identity)), true, multiply(additive_identity, additive_identity), additive_identity)), true, x, additive_identity, x, additive_identity, x, additive_identity, x)
% 0.20/0.77 = { by axiom 6 (distributivity2) }
% 0.20/0.77 fresh19(product(additive_identity, additive_identity, fresh(true, true, multiply(additive_identity, additive_identity), additive_identity)), true, x, additive_identity, x, additive_identity, x, additive_identity, x)
% 0.20/0.77 = { by axiom 4 (multiplication_is_well_defined) }
% 0.20/0.77 fresh19(product(additive_identity, additive_identity, additive_identity), true, x, additive_identity, x, additive_identity, x, additive_identity, x)
% 0.20/0.77 = { by axiom 19 (distributivity5) }
% 0.20/0.77 fresh21(sum(x, additive_identity, x), true, x, additive_identity, x, additive_identity, x, x)
% 0.20/0.77 = { by axiom 1 (additive_identity2) }
% 0.20/0.77 fresh21(true, true, x, additive_identity, x, additive_identity, x, x)
% 0.20/0.77 = { by axiom 13 (distributivity5) }
% 0.20/0.77 fresh22(sum(x, additive_identity, x), true, x, x, x)
% 0.20/0.77 = { by axiom 1 (additive_identity2) }
% 0.20/0.77 fresh22(true, true, x, x, x)
% 0.20/0.77 = { by axiom 7 (distributivity5) }
% 0.20/0.77 true
% 0.20/0.77 % SZS output end Proof
% 0.20/0.77
% 0.20/0.77 RESULT: Unsatisfiable (the axioms are contradictory).
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