TSTP Solution File: RNG037-1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : RNG037-1 : 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 : n012.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 13:58:58 EDT 2023
% Result : Unsatisfiable 4.26s 0.92s
% Output : Proof 4.66s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : RNG037-1 : TPTP v8.1.2. Released v1.0.0.
% 0.13/0.13 % 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 : n012.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
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Sun Aug 27 02:54:25 EDT 2023
% 0.13/0.35 % CPUTime :
% 4.26/0.92 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 4.26/0.92
% 4.26/0.92 % SZS status Unsatisfiable
% 4.26/0.92
% 4.26/0.96 % SZS output start Proof
% 4.26/0.96 Take the following subset of the input axioms:
% 4.26/0.96 fof(a_inverse_times_b, hypothesis, product(a, additive_inverse(b), c)).
% 4.26/0.96 fof(a_times_b, hypothesis, product(a, b, d)).
% 4.26/0.96 fof(addition_is_well_defined, axiom, ![X, Y, U, V]: (~sum(X, Y, U) | (~sum(X, Y, V) | U=V))).
% 4.26/0.96 fof(additive_identity2, axiom, ![X2]: sum(X2, additive_identity, X2)).
% 4.26/0.96 fof(associativity_of_addition1, axiom, ![Z, W, X2, Y2, U2, V5]: (~sum(X2, Y2, U2) | (~sum(Y2, Z, V5) | (~sum(U2, Z, W) | sum(X2, V5, W))))).
% 4.26/0.96 fof(associativity_of_addition2, axiom, ![X2, Y2, U2, V5, Z2, W2]: (~sum(X2, Y2, U2) | (~sum(Y2, Z2, V5) | (~sum(X2, V5, W2) | sum(U2, Z2, W2))))).
% 4.26/0.96 fof(closure_of_addition, axiom, ![X2, Y2]: sum(X2, Y2, add(X2, Y2))).
% 4.26/0.96 fof(closure_of_multiplication, axiom, ![X2, Y2]: product(X2, Y2, multiply(X2, Y2))).
% 4.26/0.96 fof(commutativity_of_addition, axiom, ![X2, Y2, Z2]: (~sum(X2, Y2, Z2) | sum(Y2, X2, Z2))).
% 4.26/0.96 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)))))).
% 4.26/0.96 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)))))).
% 4.26/0.96 fof(left_inverse, axiom, ![X2]: sum(additive_inverse(X2), X2, additive_identity)).
% 4.26/0.96 fof(multiplication_is_well_defined, axiom, ![X2, Y2, U2, V5]: (~product(X2, Y2, U2) | (~product(X2, Y2, V5) | U2=V5))).
% 4.26/0.96 fof(prove_sum_is_additive_identity, negated_conjecture, ~sum(c, d, additive_identity)).
% 4.26/0.96 fof(right_inverse, axiom, ![X2]: sum(X2, additive_inverse(X2), additive_identity)).
% 4.26/0.96
% 4.26/0.96 Now clausify the problem and encode Horn clauses using encoding 3 of
% 4.26/0.96 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 4.26/0.96 We repeatedly replace C & s=t => u=v by the two clauses:
% 4.26/0.96 fresh(y, y, x1...xn) = u
% 4.26/0.96 C => fresh(s, t, x1...xn) = v
% 4.26/0.96 where fresh is a fresh function symbol and x1..xn are the free
% 4.26/0.96 variables of u and v.
% 4.26/0.96 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 4.26/0.96 input problem has no model of domain size 1).
% 4.26/0.96
% 4.26/0.96 The encoding turns the above axioms into the following unit equations and goals:
% 4.26/0.96
% 4.26/0.96 Axiom 1 (additive_identity2): sum(X, additive_identity, X) = true.
% 4.26/0.96 Axiom 2 (a_times_b): product(a, b, d) = true.
% 4.26/0.96 Axiom 3 (right_inverse): sum(X, additive_inverse(X), additive_identity) = true.
% 4.26/0.96 Axiom 4 (left_inverse): sum(additive_inverse(X), X, additive_identity) = true.
% 4.26/0.96 Axiom 5 (a_inverse_times_b): product(a, additive_inverse(b), c) = true.
% 4.26/0.96 Axiom 6 (multiplication_is_well_defined): fresh(X, X, Y, Z) = Z.
% 4.26/0.96 Axiom 7 (addition_is_well_defined): fresh3(X, X, Y, Z) = Z.
% 4.26/0.96 Axiom 8 (closure_of_addition): sum(X, Y, add(X, Y)) = true.
% 4.26/0.96 Axiom 9 (closure_of_multiplication): product(X, Y, multiply(X, Y)) = true.
% 4.26/0.96 Axiom 10 (associativity_of_addition1): fresh33(X, X, Y, Z, W) = true.
% 4.26/0.96 Axiom 11 (associativity_of_addition2): fresh31(X, X, Y, Z, W) = true.
% 4.26/0.96 Axiom 12 (distributivity1): fresh25(X, X, Y, Z, W) = true.
% 4.26/0.96 Axiom 13 (distributivity2): fresh21(X, X, Y, Z, W) = true.
% 4.26/0.96 Axiom 14 (commutativity_of_addition): fresh5(X, X, Y, Z, W) = true.
% 4.26/0.96 Axiom 15 (addition_is_well_defined): fresh4(X, X, Y, Z, W, V) = W.
% 4.26/0.96 Axiom 16 (multiplication_is_well_defined): fresh2(X, X, Y, Z, W, V) = W.
% 4.26/0.96 Axiom 17 (distributivity2): fresh19(X, X, Y, Z, W, V, U) = product(Y, V, U).
% 4.26/0.96 Axiom 18 (associativity_of_addition1): fresh9(X, X, Y, Z, W, V, U) = sum(Y, V, U).
% 4.26/0.96 Axiom 19 (associativity_of_addition2): fresh8(X, X, Y, Z, W, V, U) = sum(W, V, U).
% 4.26/0.96 Axiom 20 (associativity_of_addition1): fresh32(X, X, Y, Z, W, V, U, T) = fresh33(sum(Y, Z, W), true, Y, U, T).
% 4.26/0.96 Axiom 21 (associativity_of_addition2): fresh30(X, X, Y, Z, W, V, U, T) = fresh31(sum(Y, Z, W), true, W, V, T).
% 4.26/0.96 Axiom 22 (distributivity1): fresh23(X, X, Y, Z, W, V, U, T) = sum(Z, V, T).
% 4.26/0.96 Axiom 23 (commutativity_of_addition): fresh5(sum(X, Y, Z), true, X, Y, Z) = sum(Y, X, Z).
% 4.26/0.96 Axiom 24 (distributivity1): fresh24(X, X, Y, Z, W, V, U, T, S) = fresh25(sum(Z, V, T), true, W, U, S).
% 4.26/0.96 Axiom 25 (distributivity2): fresh20(X, X, Y, Z, W, V, U, T, S) = fresh21(sum(Z, V, T), true, Y, T, S).
% 4.26/0.96 Axiom 26 (addition_is_well_defined): fresh4(sum(X, Y, Z), true, X, Y, W, Z) = fresh3(sum(X, Y, W), true, W, Z).
% 4.26/0.96 Axiom 27 (multiplication_is_well_defined): fresh2(product(X, Y, Z), true, X, Y, W, Z) = fresh(product(X, Y, W), true, W, Z).
% 4.26/0.96 Axiom 28 (distributivity2): fresh18(X, X, Y, Z, W, V, U, T, S) = fresh19(sum(W, U, S), true, Y, Z, V, T, S).
% 4.26/0.96 Axiom 29 (associativity_of_addition1): fresh32(sum(X, Y, Z), true, W, V, X, Y, U, Z) = fresh9(sum(V, Y, U), true, W, V, X, U, Z).
% 4.26/0.96 Axiom 30 (associativity_of_addition2): fresh30(sum(X, Y, Z), true, W, X, V, Y, Z, U) = fresh8(sum(W, Z, U), true, W, X, V, Y, U).
% 4.26/0.96 Axiom 31 (distributivity1): fresh22(X, X, Y, Z, W, V, U, T, S) = fresh23(product(Y, Z, W), true, Z, W, V, U, T, S).
% 4.26/0.96 Axiom 32 (distributivity1): fresh22(product(X, Y, Z), true, X, W, V, U, T, Y, Z) = fresh24(product(X, U, T), true, X, W, V, U, T, Y, Z).
% 4.26/0.96 Axiom 33 (distributivity2): fresh18(product(X, Y, Z), true, X, W, V, Y, Z, U, T) = fresh20(product(X, W, V), true, X, W, V, Y, Z, U, T).
% 4.66/0.96
% 4.66/0.96 Lemma 34: fresh3(sum(X, Y, Z), true, Z, add(X, Y)) = Z.
% 4.66/0.96 Proof:
% 4.66/0.96 fresh3(sum(X, Y, Z), true, Z, add(X, Y))
% 4.66/0.96 = { by axiom 26 (addition_is_well_defined) R->L }
% 4.66/0.96 fresh4(sum(X, Y, add(X, Y)), true, X, Y, Z, add(X, Y))
% 4.66/0.96 = { by axiom 8 (closure_of_addition) }
% 4.66/0.96 fresh4(true, true, X, Y, Z, add(X, Y))
% 4.66/0.96 = { by axiom 15 (addition_is_well_defined) }
% 4.66/0.96 Z
% 4.66/0.96
% 4.66/0.96 Lemma 35: sum(X, Y, add(Y, X)) = true.
% 4.66/0.96 Proof:
% 4.66/0.96 sum(X, Y, add(Y, X))
% 4.66/0.96 = { by axiom 23 (commutativity_of_addition) R->L }
% 4.66/0.96 fresh5(sum(Y, X, add(Y, X)), true, Y, X, add(Y, X))
% 4.66/0.96 = { by axiom 8 (closure_of_addition) }
% 4.66/0.96 fresh5(true, true, Y, X, add(Y, X))
% 4.66/0.96 = { by axiom 14 (commutativity_of_addition) }
% 4.66/0.96 true
% 4.66/0.96
% 4.66/0.96 Lemma 36: add(X, Y) = add(Y, X).
% 4.66/0.96 Proof:
% 4.66/0.96 add(X, Y)
% 4.66/0.96 = { by lemma 34 R->L }
% 4.66/0.96 fresh3(sum(Y, X, add(X, Y)), true, add(X, Y), add(Y, X))
% 4.66/0.96 = { by lemma 35 }
% 4.66/0.96 fresh3(true, true, add(X, Y), add(Y, X))
% 4.66/0.96 = { by axiom 7 (addition_is_well_defined) }
% 4.66/0.96 add(Y, X)
% 4.66/0.96
% 4.66/0.96 Lemma 37: add(add(X, Y), Z) = add(X, add(Z, Y)).
% 4.66/0.96 Proof:
% 4.66/0.96 add(add(X, Y), Z)
% 4.66/0.96 = { by lemma 36 R->L }
% 4.66/0.96 add(Z, add(X, Y))
% 4.66/0.96 = { by lemma 34 R->L }
% 4.66/0.96 fresh3(sum(X, add(Z, Y), add(Z, add(X, Y))), true, add(Z, add(X, Y)), add(X, add(Z, Y)))
% 4.66/0.96 = { by lemma 36 R->L }
% 4.66/0.96 fresh3(sum(X, add(Z, Y), add(add(X, Y), Z)), true, add(Z, add(X, Y)), add(X, add(Z, Y)))
% 4.66/0.96 = { by lemma 36 R->L }
% 4.66/0.96 fresh3(sum(X, add(Y, Z), add(add(X, Y), Z)), true, add(Z, add(X, Y)), add(X, add(Z, Y)))
% 4.66/0.96 = { by axiom 18 (associativity_of_addition1) R->L }
% 4.66/0.96 fresh3(fresh9(true, true, X, Y, add(X, Y), add(Y, Z), add(add(X, Y), Z)), true, add(Z, add(X, Y)), add(X, add(Z, Y)))
% 4.66/0.96 = { by axiom 8 (closure_of_addition) R->L }
% 4.66/0.96 fresh3(fresh9(sum(Y, Z, add(Y, Z)), true, X, Y, add(X, Y), add(Y, Z), add(add(X, Y), Z)), true, add(Z, add(X, Y)), add(X, add(Z, Y)))
% 4.66/0.96 = { by axiom 29 (associativity_of_addition1) R->L }
% 4.66/0.96 fresh3(fresh32(sum(add(X, Y), Z, add(add(X, Y), Z)), true, X, Y, add(X, Y), Z, add(Y, Z), add(add(X, Y), Z)), true, add(Z, add(X, Y)), add(X, add(Z, Y)))
% 4.66/0.96 = { by axiom 8 (closure_of_addition) }
% 4.66/0.96 fresh3(fresh32(true, true, X, Y, add(X, Y), Z, add(Y, Z), add(add(X, Y), Z)), true, add(Z, add(X, Y)), add(X, add(Z, Y)))
% 4.66/0.96 = { by axiom 20 (associativity_of_addition1) }
% 4.66/0.96 fresh3(fresh33(sum(X, Y, add(X, Y)), true, X, add(Y, Z), add(add(X, Y), Z)), true, add(Z, add(X, Y)), add(X, add(Z, Y)))
% 4.66/0.96 = { by axiom 8 (closure_of_addition) }
% 4.66/0.96 fresh3(fresh33(true, true, X, add(Y, Z), add(add(X, Y), Z)), true, add(Z, add(X, Y)), add(X, add(Z, Y)))
% 4.66/0.96 = { by axiom 10 (associativity_of_addition1) }
% 4.66/0.96 fresh3(true, true, add(Z, add(X, Y)), add(X, add(Z, Y)))
% 4.66/0.96 = { by axiom 7 (addition_is_well_defined) }
% 4.66/0.96 add(X, add(Z, Y))
% 4.66/0.96
% 4.66/0.96 Lemma 38: fresh30(X, X, Y, Z, add(Y, Z), W, V, U) = true.
% 4.66/0.96 Proof:
% 4.66/0.96 fresh30(X, X, Y, Z, add(Y, Z), W, V, U)
% 4.66/0.96 = { by axiom 21 (associativity_of_addition2) }
% 4.66/0.97 fresh31(sum(Y, Z, add(Y, Z)), true, add(Y, Z), W, U)
% 4.66/0.97 = { by axiom 8 (closure_of_addition) }
% 4.66/0.97 fresh31(true, true, add(Y, Z), W, U)
% 4.66/0.97 = { by axiom 11 (associativity_of_addition2) }
% 4.66/0.97 true
% 4.66/0.97
% 4.66/0.97 Lemma 39: sum(add(X, additive_inverse(Y)), Y, X) = true.
% 4.66/0.97 Proof:
% 4.66/0.97 sum(add(X, additive_inverse(Y)), Y, X)
% 4.66/0.97 = { by axiom 19 (associativity_of_addition2) R->L }
% 4.66/0.97 fresh8(true, true, X, additive_inverse(Y), add(X, additive_inverse(Y)), Y, X)
% 4.66/0.97 = { by axiom 1 (additive_identity2) R->L }
% 4.66/0.97 fresh8(sum(X, additive_identity, X), true, X, additive_inverse(Y), add(X, additive_inverse(Y)), Y, X)
% 4.66/0.97 = { by axiom 30 (associativity_of_addition2) R->L }
% 4.66/0.97 fresh30(sum(additive_inverse(Y), Y, additive_identity), true, X, additive_inverse(Y), add(X, additive_inverse(Y)), Y, additive_identity, X)
% 4.66/0.97 = { by axiom 4 (left_inverse) }
% 4.66/0.97 fresh30(true, true, X, additive_inverse(Y), add(X, additive_inverse(Y)), Y, additive_identity, X)
% 4.66/0.97 = { by lemma 38 }
% 4.66/0.97 true
% 4.66/0.97
% 4.66/0.97 Lemma 40: add(X, add(Y, additive_inverse(X))) = Y.
% 4.66/0.97 Proof:
% 4.66/0.97 add(X, add(Y, additive_inverse(X)))
% 4.66/0.97 = { by lemma 36 R->L }
% 4.66/0.97 add(add(Y, additive_inverse(X)), X)
% 4.66/0.97 = { by axiom 7 (addition_is_well_defined) R->L }
% 4.66/0.97 fresh3(true, true, Y, add(add(Y, additive_inverse(X)), X))
% 4.66/0.97 = { by lemma 39 R->L }
% 4.66/0.97 fresh3(sum(add(Y, additive_inverse(X)), X, Y), true, Y, add(add(Y, additive_inverse(X)), X))
% 4.66/0.97 = { by lemma 34 }
% 4.66/0.97 Y
% 4.66/0.97
% 4.66/0.97 Goal 1 (prove_sum_is_additive_identity): sum(c, d, additive_identity) = true.
% 4.66/0.97 Proof:
% 4.66/0.97 sum(c, d, additive_identity)
% 4.66/0.97 = { by lemma 40 R->L }
% 4.66/0.97 sum(add(d, add(c, additive_inverse(d))), d, additive_identity)
% 4.66/0.97 = { by lemma 36 }
% 4.66/0.97 sum(add(d, add(additive_inverse(d), c)), d, additive_identity)
% 4.66/0.97 = { by lemma 37 R->L }
% 4.66/0.97 sum(add(add(d, c), additive_inverse(d)), d, additive_identity)
% 4.66/0.97 = { by lemma 34 R->L }
% 4.66/0.97 sum(add(add(d, c), additive_inverse(fresh3(sum(add(d, d), c, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.97 = { by axiom 16 (multiplication_is_well_defined) R->L }
% 4.66/0.97 sum(add(add(d, c), additive_inverse(fresh3(sum(fresh2(true, true, a, add(b, b), add(d, d), multiply(a, add(b, b))), c, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.97 = { by axiom 9 (closure_of_multiplication) R->L }
% 4.66/0.97 sum(add(add(d, c), additive_inverse(fresh3(sum(fresh2(product(a, add(b, b), multiply(a, add(b, b))), true, a, add(b, b), add(d, d), multiply(a, add(b, b))), c, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.97 = { by axiom 27 (multiplication_is_well_defined) }
% 4.66/0.97 sum(add(add(d, c), additive_inverse(fresh3(sum(fresh(product(a, add(b, b), add(d, d)), true, add(d, d), multiply(a, add(b, b))), c, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.97 = { by axiom 17 (distributivity2) R->L }
% 4.66/0.97 sum(add(add(d, c), additive_inverse(fresh3(sum(fresh(fresh19(true, true, a, b, b, add(b, b), add(d, d)), true, add(d, d), multiply(a, add(b, b))), c, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.97 = { by axiom 8 (closure_of_addition) R->L }
% 4.66/0.97 sum(add(add(d, c), additive_inverse(fresh3(sum(fresh(fresh19(sum(d, d, add(d, d)), true, a, b, b, add(b, b), add(d, d)), true, add(d, d), multiply(a, add(b, b))), c, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.97 = { by axiom 28 (distributivity2) R->L }
% 4.66/0.97 sum(add(add(d, c), additive_inverse(fresh3(sum(fresh(fresh18(true, true, a, b, d, b, d, add(b, b), add(d, d)), true, add(d, d), multiply(a, add(b, b))), c, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.97 = { by axiom 2 (a_times_b) R->L }
% 4.66/0.97 sum(add(add(d, c), additive_inverse(fresh3(sum(fresh(fresh18(product(a, b, d), true, a, b, d, b, d, add(b, b), add(d, d)), true, add(d, d), multiply(a, add(b, b))), c, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.97 = { by axiom 33 (distributivity2) }
% 4.66/0.97 sum(add(add(d, c), additive_inverse(fresh3(sum(fresh(fresh20(product(a, b, d), true, a, b, d, b, d, add(b, b), add(d, d)), true, add(d, d), multiply(a, add(b, b))), c, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.97 = { by axiom 2 (a_times_b) }
% 4.66/0.97 sum(add(add(d, c), additive_inverse(fresh3(sum(fresh(fresh20(true, true, a, b, d, b, d, add(b, b), add(d, d)), true, add(d, d), multiply(a, add(b, b))), c, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.97 = { by axiom 25 (distributivity2) }
% 4.66/0.97 sum(add(add(d, c), additive_inverse(fresh3(sum(fresh(fresh21(sum(b, b, add(b, b)), true, a, add(b, b), add(d, d)), true, add(d, d), multiply(a, add(b, b))), c, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.97 = { by axiom 8 (closure_of_addition) }
% 4.66/0.97 sum(add(add(d, c), additive_inverse(fresh3(sum(fresh(fresh21(true, true, a, add(b, b), add(d, d)), true, add(d, d), multiply(a, add(b, b))), c, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.97 = { by axiom 13 (distributivity2) }
% 4.66/0.97 sum(add(add(d, c), additive_inverse(fresh3(sum(fresh(true, true, add(d, d), multiply(a, add(b, b))), c, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.97 = { by axiom 6 (multiplication_is_well_defined) }
% 4.66/0.97 sum(add(add(d, c), additive_inverse(fresh3(sum(multiply(a, add(b, b)), c, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.97 = { by axiom 7 (addition_is_well_defined) R->L }
% 4.66/0.97 sum(add(add(d, c), additive_inverse(fresh3(sum(multiply(a, add(b, fresh3(true, true, add(b, additive_identity), b))), c, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.97 = { by axiom 8 (closure_of_addition) R->L }
% 4.66/0.97 sum(add(add(d, c), additive_inverse(fresh3(sum(multiply(a, add(b, fresh3(sum(b, additive_identity, add(b, additive_identity)), true, add(b, additive_identity), b))), c, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.97 = { by axiom 26 (addition_is_well_defined) R->L }
% 4.66/0.97 sum(add(add(d, c), additive_inverse(fresh3(sum(multiply(a, add(b, fresh4(sum(b, additive_identity, b), true, b, additive_identity, add(b, additive_identity), b))), c, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.97 = { by axiom 1 (additive_identity2) }
% 4.66/0.97 sum(add(add(d, c), additive_inverse(fresh3(sum(multiply(a, add(b, fresh4(true, true, b, additive_identity, add(b, additive_identity), b))), c, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.98 = { by axiom 15 (addition_is_well_defined) }
% 4.66/0.98 sum(add(add(d, c), additive_inverse(fresh3(sum(multiply(a, add(b, add(b, additive_identity))), c, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.98 = { by axiom 7 (addition_is_well_defined) R->L }
% 4.66/0.98 sum(add(add(d, c), additive_inverse(fresh3(sum(multiply(a, add(b, add(b, fresh3(true, true, add(additive_inverse(additive_inverse(b)), additive_inverse(b)), additive_identity)))), c, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.98 = { by lemma 35 R->L }
% 4.66/0.98 sum(add(add(d, c), additive_inverse(fresh3(sum(multiply(a, add(b, add(b, fresh3(sum(additive_inverse(b), additive_inverse(additive_inverse(b)), add(additive_inverse(additive_inverse(b)), additive_inverse(b))), true, add(additive_inverse(additive_inverse(b)), additive_inverse(b)), additive_identity)))), c, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.98 = { by axiom 26 (addition_is_well_defined) R->L }
% 4.66/0.98 sum(add(add(d, c), additive_inverse(fresh3(sum(multiply(a, add(b, add(b, fresh4(sum(additive_inverse(b), additive_inverse(additive_inverse(b)), additive_identity), true, additive_inverse(b), additive_inverse(additive_inverse(b)), add(additive_inverse(additive_inverse(b)), additive_inverse(b)), additive_identity)))), c, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.98 = { by axiom 3 (right_inverse) }
% 4.66/0.98 sum(add(add(d, c), additive_inverse(fresh3(sum(multiply(a, add(b, add(b, fresh4(true, true, additive_inverse(b), additive_inverse(additive_inverse(b)), add(additive_inverse(additive_inverse(b)), additive_inverse(b)), additive_identity)))), c, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.98 = { by axiom 15 (addition_is_well_defined) }
% 4.66/0.98 sum(add(add(d, c), additive_inverse(fresh3(sum(multiply(a, add(b, add(b, add(additive_inverse(additive_inverse(b)), additive_inverse(b))))), c, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.98 = { by lemma 40 }
% 4.66/0.98 sum(add(add(d, c), additive_inverse(fresh3(sum(multiply(a, add(b, additive_inverse(additive_inverse(b)))), c, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.98 = { by axiom 22 (distributivity1) R->L }
% 4.66/0.98 sum(add(add(d, c), additive_inverse(fresh3(fresh23(true, true, add(b, additive_inverse(additive_inverse(b))), multiply(a, add(b, additive_inverse(additive_inverse(b)))), additive_inverse(b), c, b, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.98 = { by axiom 9 (closure_of_multiplication) R->L }
% 4.66/0.98 sum(add(add(d, c), additive_inverse(fresh3(fresh23(product(a, add(b, additive_inverse(additive_inverse(b))), multiply(a, add(b, additive_inverse(additive_inverse(b))))), true, add(b, additive_inverse(additive_inverse(b))), multiply(a, add(b, additive_inverse(additive_inverse(b)))), additive_inverse(b), c, b, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.98 = { by axiom 31 (distributivity1) R->L }
% 4.66/0.98 sum(add(add(d, c), additive_inverse(fresh3(fresh22(true, true, a, add(b, additive_inverse(additive_inverse(b))), multiply(a, add(b, additive_inverse(additive_inverse(b)))), additive_inverse(b), c, b, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.98 = { by axiom 2 (a_times_b) R->L }
% 4.66/0.98 sum(add(add(d, c), additive_inverse(fresh3(fresh22(product(a, b, d), true, a, add(b, additive_inverse(additive_inverse(b))), multiply(a, add(b, additive_inverse(additive_inverse(b)))), additive_inverse(b), c, b, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.98 = { by axiom 32 (distributivity1) }
% 4.66/0.98 sum(add(add(d, c), additive_inverse(fresh3(fresh24(product(a, additive_inverse(b), c), true, a, add(b, additive_inverse(additive_inverse(b))), multiply(a, add(b, additive_inverse(additive_inverse(b)))), additive_inverse(b), c, b, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.98 = { by axiom 5 (a_inverse_times_b) }
% 4.66/0.98 sum(add(add(d, c), additive_inverse(fresh3(fresh24(true, true, a, add(b, additive_inverse(additive_inverse(b))), multiply(a, add(b, additive_inverse(additive_inverse(b)))), additive_inverse(b), c, b, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.98 = { by axiom 24 (distributivity1) }
% 4.66/0.98 sum(add(add(d, c), additive_inverse(fresh3(fresh25(sum(add(b, additive_inverse(additive_inverse(b))), additive_inverse(b), b), true, multiply(a, add(b, additive_inverse(additive_inverse(b)))), c, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.98 = { by lemma 39 }
% 4.66/0.98 sum(add(add(d, c), additive_inverse(fresh3(fresh25(true, true, multiply(a, add(b, additive_inverse(additive_inverse(b)))), c, d), true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.98 = { by axiom 12 (distributivity1) }
% 4.66/0.98 sum(add(add(d, c), additive_inverse(fresh3(true, true, d, add(add(d, d), c)))), d, additive_identity)
% 4.66/0.98 = { by axiom 7 (addition_is_well_defined) }
% 4.66/0.98 sum(add(add(d, c), additive_inverse(add(add(d, d), c))), d, additive_identity)
% 4.66/0.98 = { by lemma 37 }
% 4.66/0.98 sum(add(add(d, c), additive_inverse(add(d, add(c, d)))), d, additive_identity)
% 4.66/0.98 = { by lemma 36 }
% 4.66/0.98 sum(add(add(d, c), additive_inverse(add(d, add(d, c)))), d, additive_identity)
% 4.66/0.98 = { by lemma 36 R->L }
% 4.66/0.98 sum(add(add(d, c), additive_inverse(add(add(d, c), d))), d, additive_identity)
% 4.66/0.98 = { by lemma 36 R->L }
% 4.66/0.98 sum(add(additive_inverse(add(add(d, c), d)), add(d, c)), d, additive_identity)
% 4.66/0.98 = { by axiom 19 (associativity_of_addition2) R->L }
% 4.66/0.98 fresh8(true, true, additive_inverse(add(add(d, c), d)), add(d, c), add(additive_inverse(add(add(d, c), d)), add(d, c)), d, additive_identity)
% 4.66/0.98 = { by axiom 4 (left_inverse) R->L }
% 4.66/0.98 fresh8(sum(additive_inverse(add(add(d, c), d)), add(add(d, c), d), additive_identity), true, additive_inverse(add(add(d, c), d)), add(d, c), add(additive_inverse(add(add(d, c), d)), add(d, c)), d, additive_identity)
% 4.66/0.98 = { by axiom 30 (associativity_of_addition2) R->L }
% 4.66/0.98 fresh30(sum(add(d, c), d, add(add(d, c), d)), true, additive_inverse(add(add(d, c), d)), add(d, c), add(additive_inverse(add(add(d, c), d)), add(d, c)), d, add(add(d, c), d), additive_identity)
% 4.66/0.98 = { by axiom 8 (closure_of_addition) }
% 4.66/0.98 fresh30(true, true, additive_inverse(add(add(d, c), d)), add(d, c), add(additive_inverse(add(add(d, c), d)), add(d, c)), d, add(add(d, c), d), additive_identity)
% 4.66/0.98 = { by lemma 38 }
% 4.66/0.98 true
% 4.66/0.98 % SZS output end Proof
% 4.66/0.98
% 4.66/0.98 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------