TSTP Solution File: RNG001-3 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : RNG001-3 : 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 13:58:41 EDT 2023
% Result : Unsatisfiable 29.68s 4.15s
% Output : Proof 30.10s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : RNG001-3 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.12 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.11/0.31 % Computer : n032.cluster.edu
% 0.11/0.31 % Model : x86_64 x86_64
% 0.11/0.31 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.31 % Memory : 8042.1875MB
% 0.11/0.31 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.31 % CPULimit : 300
% 0.11/0.31 % WCLimit : 300
% 0.11/0.31 % DateTime : Sun Aug 27 02:01:44 EDT 2023
% 0.11/0.32 % CPUTime :
% 29.68/4.15 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 29.68/4.15
% 29.68/4.15 % SZS status Unsatisfiable
% 29.68/4.15
% 29.68/4.17 % SZS output start Proof
% 29.68/4.17 Take the following subset of the input axioms:
% 29.68/4.17 fof(additive_identity1, axiom, ![X]: sum(additive_identity, X, X)).
% 29.68/4.17 fof(additive_inverse1, axiom, ![X2]: sum(additive_inverse(X2), X2, additive_identity)).
% 29.68/4.17 fof(associativity_of_addition1, axiom, ![Y, U, Z, V, W, X2]: (~sum(X2, Y, U) | (~sum(Y, Z, V) | (~sum(U, Z, W) | sum(X2, V, W))))).
% 29.68/4.17 fof(associativity_of_addition2, axiom, ![X2, Y2, Z2, U2, V5, W2]: (~sum(X2, Y2, U2) | (~sum(Y2, Z2, V5) | (~sum(X2, V5, W2) | sum(U2, Z2, W2))))).
% 29.68/4.17 fof(closure_of_multiplication, axiom, ![X2, Y2]: product(X2, Y2, multiply(X2, Y2))).
% 29.68/4.17 fof(distributivity1, axiom, ![V1, V2, V3, V4, X2, Y2, Z2]: (~product(X2, Y2, V1) | (~product(X2, Z2, V2) | (~sum(Y2, Z2, V3) | (~product(X2, V3, V4) | sum(V1, V2, V4)))))).
% 29.68/4.17 fof(distributivity2, axiom, ![X2, Y2, Z2, V1_2, V2_2, V3_2, V4_2]: (~product(X2, Y2, V1_2) | (~product(X2, Z2, V2_2) | (~sum(Y2, Z2, V3_2) | (~sum(V1_2, V2_2, V4_2) | product(X2, V3_2, V4_2)))))).
% 29.68/4.17 fof(prove_a_times_additive_id_is_additive_id, negated_conjecture, ~product(a, additive_identity, additive_identity)).
% 29.68/4.17
% 29.68/4.17 Now clausify the problem and encode Horn clauses using encoding 3 of
% 29.68/4.17 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 29.68/4.17 We repeatedly replace C & s=t => u=v by the two clauses:
% 29.68/4.17 fresh(y, y, x1...xn) = u
% 29.68/4.17 C => fresh(s, t, x1...xn) = v
% 29.68/4.17 where fresh is a fresh function symbol and x1..xn are the free
% 29.68/4.17 variables of u and v.
% 29.68/4.17 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 29.68/4.17 input problem has no model of domain size 1).
% 29.68/4.17
% 29.68/4.17 The encoding turns the above axioms into the following unit equations and goals:
% 29.68/4.17
% 29.68/4.17 Axiom 1 (additive_identity1): sum(additive_identity, X, X) = true.
% 29.68/4.17 Axiom 2 (additive_inverse1): sum(additive_inverse(X), X, additive_identity) = true.
% 29.68/4.17 Axiom 3 (associativity_of_addition1): fresh14(X, X, Y, Z, W) = true.
% 29.68/4.17 Axiom 4 (associativity_of_addition2): fresh12(X, X, Y, Z, W) = true.
% 29.68/4.17 Axiom 5 (distributivity1): fresh10(X, X, Y, Z, W) = true.
% 29.68/4.17 Axiom 6 (distributivity2): fresh6(X, X, Y, Z, W) = true.
% 29.68/4.17 Axiom 7 (closure_of_multiplication): product(X, Y, multiply(X, Y)) = true.
% 29.68/4.17 Axiom 8 (associativity_of_addition2): fresh(X, X, Y, Z, W, V, U) = sum(W, V, U).
% 29.68/4.17 Axiom 9 (distributivity2): fresh4(X, X, Y, Z, W, V, U) = product(Y, V, U).
% 29.68/4.17 Axiom 10 (associativity_of_addition1): fresh2(X, X, Y, Z, W, V, U) = sum(Y, V, U).
% 29.68/4.17 Axiom 11 (associativity_of_addition1): fresh13(X, X, Y, Z, W, V, U, T) = fresh14(sum(Y, Z, W), true, Y, U, T).
% 29.68/4.17 Axiom 12 (associativity_of_addition2): fresh11(X, X, Y, Z, W, V, U, T) = fresh12(sum(Y, Z, W), true, W, V, T).
% 29.68/4.17 Axiom 13 (distributivity1): fresh8(X, X, Y, Z, W, V, U, T) = sum(Z, V, T).
% 29.68/4.17 Axiom 14 (distributivity1): fresh9(X, X, Y, Z, W, V, U, T, S) = fresh10(sum(Z, V, T), true, W, U, S).
% 29.68/4.17 Axiom 15 (distributivity2): fresh5(X, X, Y, Z, W, V, U, T, S) = fresh6(sum(Z, V, T), true, Y, T, S).
% 29.68/4.17 Axiom 16 (distributivity2): fresh3(X, X, Y, Z, W, V, U, T, S) = fresh4(sum(W, U, S), true, Y, Z, V, T, S).
% 29.68/4.17 Axiom 17 (associativity_of_addition1): fresh13(sum(X, Y, Z), true, W, V, X, Y, U, Z) = fresh2(sum(V, Y, U), true, W, V, X, U, Z).
% 29.68/4.17 Axiom 18 (associativity_of_addition2): fresh11(sum(X, Y, Z), true, W, X, V, Y, Z, U) = fresh(sum(W, Z, U), true, W, X, V, Y, U).
% 29.68/4.17 Axiom 19 (distributivity1): fresh7(X, X, Y, Z, W, V, U, T, S) = fresh8(product(Y, Z, W), true, Z, W, V, U, T, S).
% 29.68/4.17 Axiom 20 (distributivity1): fresh7(product(X, Y, Z), true, X, W, V, U, T, Y, Z) = fresh9(product(X, U, T), true, X, W, V, U, T, Y, Z).
% 29.68/4.17 Axiom 21 (distributivity2): fresh3(product(X, Y, Z), true, X, W, V, Y, Z, U, T) = fresh5(product(X, W, V), true, X, W, V, Y, Z, U, T).
% 29.68/4.17
% 29.68/4.17 Lemma 22: fresh13(X, X, Y, additive_inverse(Z), additive_identity, W, additive_identity, Z) = sum(Y, additive_identity, Z).
% 29.68/4.17 Proof:
% 29.68/4.17 fresh13(X, X, Y, additive_inverse(Z), additive_identity, W, additive_identity, Z)
% 29.68/4.17 = { by axiom 11 (associativity_of_addition1) }
% 29.68/4.17 fresh14(sum(Y, additive_inverse(Z), additive_identity), true, Y, additive_identity, Z)
% 29.68/4.17 = { by axiom 11 (associativity_of_addition1) R->L }
% 29.68/4.17 fresh13(true, true, Y, additive_inverse(Z), additive_identity, Z, additive_identity, Z)
% 29.68/4.17 = { by axiom 1 (additive_identity1) R->L }
% 29.68/4.17 fresh13(sum(additive_identity, Z, Z), true, Y, additive_inverse(Z), additive_identity, Z, additive_identity, Z)
% 29.68/4.17 = { by axiom 17 (associativity_of_addition1) }
% 29.68/4.17 fresh2(sum(additive_inverse(Z), Z, additive_identity), true, Y, additive_inverse(Z), additive_identity, additive_identity, Z)
% 29.68/4.17 = { by axiom 2 (additive_inverse1) }
% 29.68/4.17 fresh2(true, true, Y, additive_inverse(Z), additive_identity, additive_identity, Z)
% 29.68/4.17 = { by axiom 10 (associativity_of_addition1) }
% 29.68/4.17 sum(Y, additive_identity, Z)
% 29.68/4.17
% 29.68/4.17 Lemma 23: sum(X, additive_inverse(X), additive_identity) = true.
% 29.68/4.17 Proof:
% 29.68/4.17 sum(X, additive_inverse(X), additive_identity)
% 29.68/4.17 = { by axiom 8 (associativity_of_addition2) R->L }
% 29.68/4.17 fresh(true, true, additive_inverse(additive_inverse(X)), additive_identity, X, additive_inverse(X), additive_identity)
% 29.68/4.17 = { by axiom 2 (additive_inverse1) R->L }
% 29.68/4.17 fresh(sum(additive_inverse(additive_inverse(X)), additive_inverse(X), additive_identity), true, additive_inverse(additive_inverse(X)), additive_identity, X, additive_inverse(X), additive_identity)
% 29.68/4.17 = { by axiom 18 (associativity_of_addition2) R->L }
% 29.68/4.17 fresh11(sum(additive_identity, additive_inverse(X), additive_inverse(X)), true, additive_inverse(additive_inverse(X)), additive_identity, X, additive_inverse(X), additive_inverse(X), additive_identity)
% 29.68/4.17 = { by axiom 1 (additive_identity1) }
% 29.68/4.17 fresh11(true, true, additive_inverse(additive_inverse(X)), additive_identity, X, additive_inverse(X), additive_inverse(X), additive_identity)
% 29.68/4.17 = { by axiom 12 (associativity_of_addition2) }
% 29.68/4.17 fresh12(sum(additive_inverse(additive_inverse(X)), additive_identity, X), true, X, additive_inverse(X), additive_identity)
% 29.68/4.17 = { by lemma 22 R->L }
% 29.68/4.17 fresh12(fresh13(Y, Y, additive_inverse(additive_inverse(X)), additive_inverse(X), additive_identity, Z, additive_identity, X), true, X, additive_inverse(X), additive_identity)
% 29.68/4.17 = { by axiom 11 (associativity_of_addition1) }
% 29.68/4.17 fresh12(fresh14(sum(additive_inverse(additive_inverse(X)), additive_inverse(X), additive_identity), true, additive_inverse(additive_inverse(X)), additive_identity, X), true, X, additive_inverse(X), additive_identity)
% 29.68/4.17 = { by axiom 2 (additive_inverse1) }
% 29.68/4.17 fresh12(fresh14(true, true, additive_inverse(additive_inverse(X)), additive_identity, X), true, X, additive_inverse(X), additive_identity)
% 29.68/4.17 = { by axiom 3 (associativity_of_addition1) }
% 29.68/4.17 fresh12(true, true, X, additive_inverse(X), additive_identity)
% 29.68/4.17 = { by axiom 4 (associativity_of_addition2) }
% 29.68/4.17 true
% 29.68/4.17
% 29.68/4.17 Lemma 24: sum(X, additive_identity, X) = true.
% 29.68/4.17 Proof:
% 29.68/4.17 sum(X, additive_identity, X)
% 29.68/4.17 = { by lemma 22 R->L }
% 29.68/4.17 fresh13(Y, Y, X, additive_inverse(X), additive_identity, Z, additive_identity, X)
% 29.68/4.17 = { by axiom 11 (associativity_of_addition1) }
% 29.68/4.17 fresh14(sum(X, additive_inverse(X), additive_identity), true, X, additive_identity, X)
% 29.68/4.17 = { by lemma 23 }
% 29.68/4.17 fresh14(true, true, X, additive_identity, X)
% 29.68/4.17 = { by axiom 3 (associativity_of_addition1) }
% 29.68/4.17 true
% 29.68/4.17
% 29.68/4.17 Lemma 25: fresh10(sum(X, Y, Z), true, multiply(W, X), multiply(W, Y), multiply(W, Z)) = sum(multiply(W, X), multiply(W, Y), multiply(W, Z)).
% 29.68/4.17 Proof:
% 29.68/4.17 fresh10(sum(X, Y, Z), true, multiply(W, X), multiply(W, Y), multiply(W, Z))
% 29.68/4.17 = { by axiom 14 (distributivity1) R->L }
% 29.68/4.17 fresh9(true, true, W, X, multiply(W, X), Y, multiply(W, Y), Z, multiply(W, Z))
% 29.68/4.17 = { by axiom 7 (closure_of_multiplication) R->L }
% 29.68/4.17 fresh9(product(W, Y, multiply(W, Y)), true, W, X, multiply(W, X), Y, multiply(W, Y), Z, multiply(W, Z))
% 29.68/4.17 = { by axiom 20 (distributivity1) R->L }
% 29.68/4.17 fresh7(product(W, Z, multiply(W, Z)), true, W, X, multiply(W, X), Y, multiply(W, Y), Z, multiply(W, Z))
% 29.68/4.17 = { by axiom 7 (closure_of_multiplication) }
% 29.68/4.17 fresh7(true, true, W, X, multiply(W, X), Y, multiply(W, Y), Z, multiply(W, Z))
% 29.68/4.17 = { by axiom 19 (distributivity1) }
% 29.68/4.17 fresh8(product(W, X, multiply(W, X)), true, X, multiply(W, X), Y, multiply(W, Y), Z, multiply(W, Z))
% 29.68/4.17 = { by axiom 7 (closure_of_multiplication) }
% 29.68/4.17 fresh8(true, true, X, multiply(W, X), Y, multiply(W, Y), Z, multiply(W, Z))
% 29.68/4.17 = { by axiom 13 (distributivity1) }
% 29.68/4.17 sum(multiply(W, X), multiply(W, Y), multiply(W, Z))
% 29.68/4.17
% 29.68/4.17 Goal 1 (prove_a_times_additive_id_is_additive_id): product(a, additive_identity, additive_identity) = true.
% 29.68/4.17 Proof:
% 29.68/4.17 product(a, additive_identity, additive_identity)
% 29.68/4.17 = { by axiom 9 (distributivity2) R->L }
% 29.68/4.17 fresh4(true, true, a, additive_identity, additive_identity, additive_identity, additive_identity)
% 29.68/4.17 = { by axiom 3 (associativity_of_addition1) R->L }
% 29.68/4.17 fresh4(fresh14(true, true, multiply(a, additive_identity), multiply(a, additive_identity), additive_identity), true, a, additive_identity, additive_identity, additive_identity, additive_identity)
% 29.68/4.17 = { by axiom 5 (distributivity1) R->L }
% 29.68/4.17 fresh4(fresh14(fresh10(true, true, multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity)), true, multiply(a, additive_identity), multiply(a, additive_identity), additive_identity), true, a, additive_identity, additive_identity, additive_identity, additive_identity)
% 29.68/4.17 = { by lemma 24 R->L }
% 29.68/4.17 fresh4(fresh14(fresh10(sum(additive_identity, additive_identity, additive_identity), true, multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity)), true, multiply(a, additive_identity), multiply(a, additive_identity), additive_identity), true, a, additive_identity, additive_identity, additive_identity, additive_identity)
% 29.68/4.17 = { by lemma 25 }
% 29.68/4.17 fresh4(fresh14(sum(multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity)), true, multiply(a, additive_identity), multiply(a, additive_identity), additive_identity), true, a, additive_identity, additive_identity, additive_identity, additive_identity)
% 29.68/4.17 = { by axiom 11 (associativity_of_addition1) R->L }
% 29.68/4.17 fresh4(fresh13(true, true, multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity), additive_identity, multiply(a, additive_identity), additive_identity), true, a, additive_identity, additive_identity, additive_identity, additive_identity)
% 29.68/4.17 = { by axiom 3 (associativity_of_addition1) R->L }
% 29.68/4.17 fresh4(fresh13(fresh14(true, true, multiply(a, additive_identity), additive_identity, additive_identity), true, multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity), additive_identity, multiply(a, additive_identity), additive_identity), true, a, additive_identity, additive_identity, additive_identity, additive_identity)
% 29.68/4.17 = { by axiom 5 (distributivity1) R->L }
% 29.68/4.17 fresh4(fresh13(fresh14(fresh10(true, true, multiply(a, additive_identity), multiply(a, X), multiply(a, X)), true, multiply(a, additive_identity), additive_identity, additive_identity), true, multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity), additive_identity, multiply(a, additive_identity), additive_identity), true, a, additive_identity, additive_identity, additive_identity, additive_identity)
% 29.68/4.17 = { by axiom 1 (additive_identity1) R->L }
% 29.68/4.17 fresh4(fresh13(fresh14(fresh10(sum(additive_identity, X, X), true, multiply(a, additive_identity), multiply(a, X), multiply(a, X)), true, multiply(a, additive_identity), additive_identity, additive_identity), true, multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity), additive_identity, multiply(a, additive_identity), additive_identity), true, a, additive_identity, additive_identity, additive_identity, additive_identity)
% 29.68/4.17 = { by lemma 25 }
% 29.68/4.17 fresh4(fresh13(fresh14(sum(multiply(a, additive_identity), multiply(a, X), multiply(a, X)), true, multiply(a, additive_identity), additive_identity, additive_identity), true, multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity), additive_identity, multiply(a, additive_identity), additive_identity), true, a, additive_identity, additive_identity, additive_identity, additive_identity)
% 29.68/4.17 = { by axiom 11 (associativity_of_addition1) R->L }
% 29.68/4.17 fresh4(fresh13(fresh13(true, true, multiply(a, additive_identity), multiply(a, X), multiply(a, X), additive_inverse(multiply(a, X)), additive_identity, additive_identity), true, multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity), additive_identity, multiply(a, additive_identity), additive_identity), true, a, additive_identity, additive_identity, additive_identity, additive_identity)
% 29.68/4.17 = { by lemma 23 R->L }
% 29.68/4.18 fresh4(fresh13(fresh13(sum(multiply(a, X), additive_inverse(multiply(a, X)), additive_identity), true, multiply(a, additive_identity), multiply(a, X), multiply(a, X), additive_inverse(multiply(a, X)), additive_identity, additive_identity), true, multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity), additive_identity, multiply(a, additive_identity), additive_identity), true, a, additive_identity, additive_identity, additive_identity, additive_identity)
% 29.68/4.18 = { by axiom 17 (associativity_of_addition1) }
% 29.68/4.18 fresh4(fresh13(fresh2(sum(multiply(a, X), additive_inverse(multiply(a, X)), additive_identity), true, multiply(a, additive_identity), multiply(a, X), multiply(a, X), additive_identity, additive_identity), true, multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity), additive_identity, multiply(a, additive_identity), additive_identity), true, a, additive_identity, additive_identity, additive_identity, additive_identity)
% 29.68/4.18 = { by lemma 23 }
% 29.68/4.18 fresh4(fresh13(fresh2(true, true, multiply(a, additive_identity), multiply(a, X), multiply(a, X), additive_identity, additive_identity), true, multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity), additive_identity, multiply(a, additive_identity), additive_identity), true, a, additive_identity, additive_identity, additive_identity, additive_identity)
% 29.68/4.18 = { by axiom 10 (associativity_of_addition1) }
% 29.68/4.18 fresh4(fresh13(sum(multiply(a, additive_identity), additive_identity, additive_identity), true, multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity), additive_identity, multiply(a, additive_identity), additive_identity), true, a, additive_identity, additive_identity, additive_identity, additive_identity)
% 29.68/4.18 = { by axiom 17 (associativity_of_addition1) }
% 29.68/4.18 fresh4(fresh2(sum(multiply(a, additive_identity), additive_identity, multiply(a, additive_identity)), true, multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity), additive_identity), true, a, additive_identity, additive_identity, additive_identity, additive_identity)
% 29.68/4.18 = { by lemma 24 }
% 30.10/4.18 fresh4(fresh2(true, true, multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity), multiply(a, additive_identity), additive_identity), true, a, additive_identity, additive_identity, additive_identity, additive_identity)
% 30.10/4.18 = { by axiom 10 (associativity_of_addition1) }
% 30.10/4.18 fresh4(sum(multiply(a, additive_identity), multiply(a, additive_identity), additive_identity), true, a, additive_identity, additive_identity, additive_identity, additive_identity)
% 30.10/4.18 = { by axiom 16 (distributivity2) R->L }
% 30.10/4.18 fresh3(true, true, a, additive_identity, multiply(a, additive_identity), additive_identity, multiply(a, additive_identity), additive_identity, additive_identity)
% 30.10/4.18 = { by axiom 7 (closure_of_multiplication) R->L }
% 30.10/4.18 fresh3(product(a, additive_identity, multiply(a, additive_identity)), true, a, additive_identity, multiply(a, additive_identity), additive_identity, multiply(a, additive_identity), additive_identity, additive_identity)
% 30.10/4.18 = { by axiom 21 (distributivity2) }
% 30.10/4.18 fresh5(product(a, additive_identity, multiply(a, additive_identity)), true, a, additive_identity, multiply(a, additive_identity), additive_identity, multiply(a, additive_identity), additive_identity, additive_identity)
% 30.10/4.18 = { by axiom 7 (closure_of_multiplication) }
% 30.10/4.18 fresh5(true, true, a, additive_identity, multiply(a, additive_identity), additive_identity, multiply(a, additive_identity), additive_identity, additive_identity)
% 30.10/4.18 = { by axiom 15 (distributivity2) }
% 30.10/4.18 fresh6(sum(additive_identity, additive_identity, additive_identity), true, a, additive_identity, additive_identity)
% 30.10/4.18 = { by axiom 1 (additive_identity1) }
% 30.10/4.18 fresh6(true, true, a, additive_identity, additive_identity)
% 30.10/4.18 = { by axiom 6 (distributivity2) }
% 30.10/4.18 true
% 30.10/4.18 % SZS output end Proof
% 30.10/4.18
% 30.10/4.18 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------