TSTP Solution File: RNG007-1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : RNG007-1 : 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 : n019.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 13:58:45 EDT 2023
% Result : Unsatisfiable 4.73s 1.02s
% Output : Proof 5.51s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : RNG007-1 : 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
% 0.13/0.34 % Computer : n019.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 : Sun Aug 27 01:48:28 EDT 2023
% 0.13/0.34 % CPUTime :
% 4.73/1.02 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 4.73/1.02
% 4.73/1.02 % SZS status Unsatisfiable
% 4.73/1.02
% 5.29/1.06 % SZS output start Proof
% 5.29/1.06 Take the following subset of the input axioms:
% 5.29/1.06 fof(addition_is_well_defined, axiom, ![X, Y, U, V]: (~sum(X, Y, U) | (~sum(X, Y, V) | U=V))).
% 5.29/1.06 fof(additive_identity1, axiom, ![X2]: sum(additive_identity, X2, X2)).
% 5.29/1.06 fof(additive_identity2, axiom, ![X2]: sum(X2, additive_identity, X2)).
% 5.29/1.06 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))))).
% 5.29/1.06 fof(closure_of_addition, axiom, ![X2, Y2]: sum(X2, Y2, add(X2, Y2))).
% 5.29/1.06 fof(closure_of_multiplication, axiom, ![X2, Y2]: product(X2, Y2, multiply(X2, Y2))).
% 5.29/1.06 fof(commutativity_of_addition, axiom, ![X2, Y2, Z2]: (~sum(X2, Y2, Z2) | sum(Y2, X2, Z2))).
% 5.29/1.06 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)))))).
% 5.29/1.06 fof(distributivity3, axiom, ![X2, Y2, Z2, V1_2, V2_2, V3_2, V4_2]: (~product(Y2, X2, V1_2) | (~product(Z2, X2, V2_2) | (~sum(Y2, Z2, V3_2) | (~product(V3_2, X2, V4_2) | sum(V1_2, V2_2, V4_2)))))).
% 5.29/1.06 fof(left_inverse, axiom, ![X2]: sum(additive_inverse(X2), X2, additive_identity)).
% 5.29/1.06 fof(prove_a_plus_a_is_id, negated_conjecture, ~sum(a, a, additive_identity)).
% 5.29/1.06 fof(right_inverse, axiom, ![X2]: sum(X2, additive_inverse(X2), additive_identity)).
% 5.29/1.06 fof(x_squared_is_x, hypothesis, ![X2]: product(X2, X2, X2)).
% 5.29/1.06
% 5.29/1.06 Now clausify the problem and encode Horn clauses using encoding 3 of
% 5.29/1.06 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 5.29/1.06 We repeatedly replace C & s=t => u=v by the two clauses:
% 5.29/1.06 fresh(y, y, x1...xn) = u
% 5.29/1.06 C => fresh(s, t, x1...xn) = v
% 5.29/1.06 where fresh is a fresh function symbol and x1..xn are the free
% 5.29/1.06 variables of u and v.
% 5.29/1.06 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 5.29/1.06 input problem has no model of domain size 1).
% 5.29/1.06
% 5.29/1.06 The encoding turns the above axioms into the following unit equations and goals:
% 5.29/1.06
% 5.29/1.06 Axiom 1 (additive_identity2): sum(X, additive_identity, X) = true.
% 5.29/1.06 Axiom 2 (additive_identity1): sum(additive_identity, X, X) = true.
% 5.29/1.06 Axiom 3 (x_squared_is_x): product(X, X, X) = true.
% 5.29/1.06 Axiom 4 (right_inverse): sum(X, additive_inverse(X), additive_identity) = true.
% 5.29/1.06 Axiom 5 (left_inverse): sum(additive_inverse(X), X, additive_identity) = true.
% 5.29/1.06 Axiom 6 (addition_is_well_defined): fresh3(X, X, Y, Z) = Z.
% 5.29/1.06 Axiom 7 (closure_of_addition): sum(X, Y, add(X, Y)) = true.
% 5.29/1.06 Axiom 8 (closure_of_multiplication): product(X, Y, multiply(X, Y)) = true.
% 5.29/1.06 Axiom 9 (associativity_of_addition1): fresh33(X, X, Y, Z, W) = true.
% 5.29/1.06 Axiom 10 (distributivity1): fresh25(X, X, Y, Z, W) = true.
% 5.29/1.06 Axiom 11 (distributivity3): fresh17(X, X, Y, Z, W) = true.
% 5.29/1.06 Axiom 12 (commutativity_of_addition): fresh5(X, X, Y, Z, W) = true.
% 5.29/1.06 Axiom 13 (addition_is_well_defined): fresh4(X, X, Y, Z, W, V) = W.
% 5.29/1.06 Axiom 14 (associativity_of_addition1): fresh9(X, X, Y, Z, W, V, U) = sum(Y, V, U).
% 5.29/1.06 Axiom 15 (associativity_of_addition1): fresh32(X, X, Y, Z, W, V, U, T) = fresh33(sum(Y, Z, W), true, Y, U, T).
% 5.29/1.06 Axiom 16 (distributivity1): fresh23(X, X, Y, Z, W, V, U, T) = sum(Z, V, T).
% 5.29/1.06 Axiom 17 (distributivity3): fresh15(X, X, Y, Z, W, V, U, T) = sum(Z, V, T).
% 5.29/1.06 Axiom 18 (commutativity_of_addition): fresh5(sum(X, Y, Z), true, X, Y, Z) = sum(Y, X, Z).
% 5.29/1.06 Axiom 19 (distributivity1): fresh24(X, X, Y, Z, W, V, U, T, S) = fresh25(sum(Z, V, T), true, W, U, S).
% 5.29/1.06 Axiom 20 (distributivity3): fresh16(X, X, Y, Z, W, V, U, T, S) = fresh17(sum(Y, V, T), true, W, U, S).
% 5.29/1.06 Axiom 21 (addition_is_well_defined): fresh4(sum(X, Y, Z), true, X, Y, W, Z) = fresh3(sum(X, Y, W), true, W, Z).
% 5.29/1.06 Axiom 22 (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).
% 5.29/1.06 Axiom 23 (distributivity1): fresh22(X, X, Y, Z, W, V, U, T, S) = fresh23(product(Y, Z, W), true, Z, W, V, U, T, S).
% 5.29/1.06 Axiom 24 (distributivity3): fresh14(X, X, Y, Z, W, V, U, T, S) = fresh15(product(Y, Z, W), true, Y, W, V, U, T, S).
% 5.29/1.06 Axiom 25 (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).
% 5.29/1.06 Axiom 26 (distributivity3): fresh14(product(X, Y, Z), true, W, Y, V, U, T, X, Z) = fresh16(product(U, Y, T), true, W, Y, V, U, T, X, Z).
% 5.29/1.06
% 5.29/1.06 Lemma 27: fresh3(sum(X, Y, Z), true, Z, add(X, Y)) = Z.
% 5.29/1.06 Proof:
% 5.29/1.06 fresh3(sum(X, Y, Z), true, Z, add(X, Y))
% 5.29/1.06 = { by axiom 21 (addition_is_well_defined) R->L }
% 5.29/1.06 fresh4(sum(X, Y, add(X, Y)), true, X, Y, Z, add(X, Y))
% 5.29/1.06 = { by axiom 7 (closure_of_addition) }
% 5.29/1.06 fresh4(true, true, X, Y, Z, add(X, Y))
% 5.29/1.06 = { by axiom 13 (addition_is_well_defined) }
% 5.29/1.06 Z
% 5.29/1.06
% 5.29/1.06 Lemma 28: sum(X, Y, add(Y, X)) = true.
% 5.29/1.06 Proof:
% 5.29/1.06 sum(X, Y, add(Y, X))
% 5.29/1.06 = { by axiom 18 (commutativity_of_addition) R->L }
% 5.29/1.06 fresh5(sum(Y, X, add(Y, X)), true, Y, X, add(Y, X))
% 5.29/1.06 = { by axiom 7 (closure_of_addition) }
% 5.29/1.06 fresh5(true, true, Y, X, add(Y, X))
% 5.29/1.06 = { by axiom 12 (commutativity_of_addition) }
% 5.29/1.06 true
% 5.29/1.06
% 5.29/1.06 Lemma 29: add(X, Y) = add(Y, X).
% 5.29/1.06 Proof:
% 5.29/1.06 add(X, Y)
% 5.29/1.06 = { by lemma 27 R->L }
% 5.29/1.06 fresh3(sum(Y, X, add(X, Y)), true, add(X, Y), add(Y, X))
% 5.29/1.06 = { by lemma 28 }
% 5.29/1.06 fresh3(true, true, add(X, Y), add(Y, X))
% 5.29/1.06 = { by axiom 6 (addition_is_well_defined) }
% 5.29/1.06 add(Y, X)
% 5.29/1.06
% 5.29/1.06 Lemma 30: fresh9(sum(X, Y, Z), true, W, X, additive_identity, Z, Y) = fresh32(V, V, W, X, additive_identity, U, Z, Y).
% 5.29/1.06 Proof:
% 5.29/1.06 fresh9(sum(X, Y, Z), true, W, X, additive_identity, Z, Y)
% 5.29/1.06 = { by axiom 22 (associativity_of_addition1) R->L }
% 5.29/1.06 fresh32(sum(additive_identity, Y, Y), true, W, X, additive_identity, Y, Z, Y)
% 5.29/1.06 = { by axiom 2 (additive_identity1) }
% 5.29/1.06 fresh32(true, true, W, X, additive_identity, Y, Z, Y)
% 5.29/1.06 = { by axiom 15 (associativity_of_addition1) }
% 5.29/1.06 fresh33(sum(W, X, additive_identity), true, W, Z, Y)
% 5.29/1.06 = { by axiom 15 (associativity_of_addition1) R->L }
% 5.29/1.06 fresh32(V, V, W, X, additive_identity, U, Z, Y)
% 5.29/1.06
% 5.29/1.06 Lemma 31: fresh3(sum(X, additive_identity, Y), true, Y, X) = Y.
% 5.29/1.06 Proof:
% 5.29/1.06 fresh3(sum(X, additive_identity, Y), true, Y, X)
% 5.29/1.06 = { by axiom 21 (addition_is_well_defined) R->L }
% 5.29/1.06 fresh4(sum(X, additive_identity, X), true, X, additive_identity, Y, X)
% 5.29/1.06 = { by axiom 1 (additive_identity2) }
% 5.29/1.06 fresh4(true, true, X, additive_identity, Y, X)
% 5.29/1.06 = { by axiom 13 (addition_is_well_defined) }
% 5.29/1.06 Y
% 5.29/1.06
% 5.29/1.06 Lemma 32: additive_inverse(additive_inverse(X)) = X.
% 5.29/1.06 Proof:
% 5.29/1.06 additive_inverse(additive_inverse(X))
% 5.29/1.06 = { by axiom 6 (addition_is_well_defined) R->L }
% 5.29/1.06 fresh3(true, true, X, additive_inverse(additive_inverse(X)))
% 5.29/1.06 = { by axiom 9 (associativity_of_addition1) R->L }
% 5.29/1.06 fresh3(fresh33(true, true, additive_inverse(additive_inverse(X)), additive_identity, X), true, X, additive_inverse(additive_inverse(X)))
% 5.29/1.06 = { by axiom 5 (left_inverse) R->L }
% 5.29/1.06 fresh3(fresh33(sum(additive_inverse(additive_inverse(X)), additive_inverse(X), additive_identity), true, additive_inverse(additive_inverse(X)), additive_identity, X), true, X, additive_inverse(additive_inverse(X)))
% 5.29/1.06 = { by axiom 15 (associativity_of_addition1) R->L }
% 5.29/1.06 fresh3(fresh32(Y, Y, additive_inverse(additive_inverse(X)), additive_inverse(X), additive_identity, Z, additive_identity, X), true, X, additive_inverse(additive_inverse(X)))
% 5.29/1.06 = { by lemma 30 R->L }
% 5.29/1.06 fresh3(fresh9(sum(additive_inverse(X), X, additive_identity), true, additive_inverse(additive_inverse(X)), additive_inverse(X), additive_identity, additive_identity, X), true, X, additive_inverse(additive_inverse(X)))
% 5.29/1.06 = { by axiom 5 (left_inverse) }
% 5.29/1.06 fresh3(fresh9(true, true, additive_inverse(additive_inverse(X)), additive_inverse(X), additive_identity, additive_identity, X), true, X, additive_inverse(additive_inverse(X)))
% 5.29/1.06 = { by axiom 14 (associativity_of_addition1) }
% 5.29/1.06 fresh3(sum(additive_inverse(additive_inverse(X)), additive_identity, X), true, X, additive_inverse(additive_inverse(X)))
% 5.29/1.06 = { by lemma 31 }
% 5.29/1.06 X
% 5.29/1.06
% 5.29/1.06 Lemma 33: add(additive_inverse(X), X) = additive_identity.
% 5.29/1.06 Proof:
% 5.29/1.06 add(additive_inverse(X), X)
% 5.29/1.07 = { by axiom 13 (addition_is_well_defined) R->L }
% 5.51/1.07 fresh4(true, true, X, additive_inverse(X), add(additive_inverse(X), X), additive_identity)
% 5.51/1.07 = { by axiom 4 (right_inverse) R->L }
% 5.51/1.07 fresh4(sum(X, additive_inverse(X), additive_identity), true, X, additive_inverse(X), add(additive_inverse(X), X), additive_identity)
% 5.51/1.07 = { by axiom 21 (addition_is_well_defined) }
% 5.51/1.07 fresh3(sum(X, additive_inverse(X), add(additive_inverse(X), X)), true, add(additive_inverse(X), X), additive_identity)
% 5.51/1.07 = { by lemma 28 }
% 5.51/1.07 fresh3(true, true, add(additive_inverse(X), X), additive_identity)
% 5.51/1.07 = { by axiom 6 (addition_is_well_defined) }
% 5.51/1.07 additive_identity
% 5.51/1.07
% 5.51/1.07 Lemma 34: add(X, add(Y, additive_inverse(X))) = Y.
% 5.51/1.07 Proof:
% 5.51/1.07 add(X, add(Y, additive_inverse(X)))
% 5.51/1.07 = { by axiom 6 (addition_is_well_defined) R->L }
% 5.51/1.07 fresh3(true, true, Y, add(X, add(Y, additive_inverse(X))))
% 5.51/1.07 = { by axiom 9 (associativity_of_addition1) R->L }
% 5.51/1.07 fresh3(fresh33(true, true, X, add(additive_inverse(X), Y), Y), true, Y, add(X, add(Y, additive_inverse(X))))
% 5.51/1.07 = { by axiom 4 (right_inverse) R->L }
% 5.51/1.07 fresh3(fresh33(sum(X, additive_inverse(X), additive_identity), true, X, add(additive_inverse(X), Y), Y), true, Y, add(X, add(Y, additive_inverse(X))))
% 5.51/1.07 = { by axiom 15 (associativity_of_addition1) R->L }
% 5.51/1.07 fresh3(fresh32(Z, Z, X, additive_inverse(X), additive_identity, W, add(additive_inverse(X), Y), Y), true, Y, add(X, add(Y, additive_inverse(X))))
% 5.51/1.07 = { by lemma 30 R->L }
% 5.51/1.07 fresh3(fresh9(sum(additive_inverse(X), Y, add(additive_inverse(X), Y)), true, X, additive_inverse(X), additive_identity, add(additive_inverse(X), Y), Y), true, Y, add(X, add(Y, additive_inverse(X))))
% 5.51/1.07 = { by axiom 7 (closure_of_addition) }
% 5.51/1.07 fresh3(fresh9(true, true, X, additive_inverse(X), additive_identity, add(additive_inverse(X), Y), Y), true, Y, add(X, add(Y, additive_inverse(X))))
% 5.51/1.07 = { by axiom 14 (associativity_of_addition1) }
% 5.51/1.07 fresh3(sum(X, add(additive_inverse(X), Y), Y), true, Y, add(X, add(Y, additive_inverse(X))))
% 5.51/1.07 = { by lemma 29 }
% 5.51/1.07 fresh3(sum(X, add(Y, additive_inverse(X)), Y), true, Y, add(X, add(Y, additive_inverse(X))))
% 5.51/1.07 = { by lemma 27 }
% 5.51/1.07 Y
% 5.51/1.07
% 5.51/1.07 Lemma 35: add(additive_inverse(X), add(X, Y)) = Y.
% 5.51/1.07 Proof:
% 5.51/1.07 add(additive_inverse(X), add(X, Y))
% 5.51/1.07 = { by lemma 29 R->L }
% 5.51/1.07 add(additive_inverse(X), add(Y, X))
% 5.51/1.07 = { by lemma 32 R->L }
% 5.51/1.07 add(additive_inverse(X), add(Y, additive_inverse(additive_inverse(X))))
% 5.51/1.07 = { by lemma 34 }
% 5.51/1.07 Y
% 5.51/1.07
% 5.51/1.07 Lemma 36: fresh22(X, X, Y, Z, multiply(Y, Z), W, V, U, T) = sum(multiply(Y, Z), V, T).
% 5.51/1.07 Proof:
% 5.51/1.07 fresh22(X, X, Y, Z, multiply(Y, Z), W, V, U, T)
% 5.51/1.07 = { by axiom 23 (distributivity1) }
% 5.51/1.07 fresh23(product(Y, Z, multiply(Y, Z)), true, Z, multiply(Y, Z), W, V, U, T)
% 5.51/1.07 = { by axiom 8 (closure_of_multiplication) }
% 5.51/1.07 fresh23(true, true, Z, multiply(Y, Z), W, V, U, T)
% 5.51/1.07 = { by axiom 16 (distributivity1) }
% 5.51/1.07 sum(multiply(Y, Z), V, T)
% 5.51/1.07
% 5.51/1.07 Lemma 37: fresh14(X, X, Y, Z, multiply(Y, Z), W, V, U, T) = sum(multiply(Y, Z), V, T).
% 5.51/1.07 Proof:
% 5.51/1.07 fresh14(X, X, Y, Z, multiply(Y, Z), W, V, U, T)
% 5.51/1.07 = { by axiom 24 (distributivity3) }
% 5.51/1.07 fresh15(product(Y, Z, multiply(Y, Z)), true, Y, multiply(Y, Z), W, V, U, T)
% 5.51/1.07 = { by axiom 8 (closure_of_multiplication) }
% 5.51/1.07 fresh15(true, true, Y, multiply(Y, Z), W, V, U, T)
% 5.51/1.07 = { by axiom 17 (distributivity3) }
% 5.51/1.07 sum(multiply(Y, Z), V, T)
% 5.51/1.07
% 5.51/1.07 Lemma 38: fresh22(product(X, Y, Z), true, X, W, V, X, X, Y, Z) = fresh25(sum(W, X, Y), true, V, X, Z).
% 5.51/1.07 Proof:
% 5.51/1.07 fresh22(product(X, Y, Z), true, X, W, V, X, X, Y, Z)
% 5.51/1.07 = { by axiom 25 (distributivity1) }
% 5.51/1.07 fresh24(product(X, X, X), true, X, W, V, X, X, Y, Z)
% 5.51/1.07 = { by axiom 3 (x_squared_is_x) }
% 5.51/1.07 fresh24(true, true, X, W, V, X, X, Y, Z)
% 5.51/1.07 = { by axiom 19 (distributivity1) }
% 5.51/1.07 fresh25(sum(W, X, Y), true, V, X, Z)
% 5.51/1.07
% 5.51/1.07 Lemma 39: fresh14(product(X, Y, Z), true, W, Y, V, Y, Y, X, Z) = fresh17(sum(W, Y, X), true, V, Y, Z).
% 5.51/1.07 Proof:
% 5.51/1.07 fresh14(product(X, Y, Z), true, W, Y, V, Y, Y, X, Z)
% 5.51/1.07 = { by axiom 26 (distributivity3) }
% 5.51/1.07 fresh16(product(Y, Y, Y), true, W, Y, V, Y, Y, X, Z)
% 5.51/1.07 = { by axiom 3 (x_squared_is_x) }
% 5.51/1.07 fresh16(true, true, W, Y, V, Y, Y, X, Z)
% 5.51/1.07 = { by axiom 20 (distributivity3) }
% 5.51/1.07 fresh17(sum(W, Y, X), true, V, Y, Z)
% 5.51/1.07
% 5.51/1.07 Goal 1 (prove_a_plus_a_is_id): sum(a, a, additive_identity) = true.
% 5.51/1.07 Proof:
% 5.51/1.07 sum(a, a, additive_identity)
% 5.51/1.07 = { by axiom 6 (addition_is_well_defined) R->L }
% 5.51/1.07 sum(fresh3(true, true, add(a, additive_identity), a), a, additive_identity)
% 5.51/1.07 = { by axiom 7 (closure_of_addition) R->L }
% 5.51/1.07 sum(fresh3(sum(a, additive_identity, add(a, additive_identity)), true, add(a, additive_identity), a), a, additive_identity)
% 5.51/1.07 = { by lemma 31 }
% 5.51/1.07 sum(add(a, additive_identity), a, additive_identity)
% 5.51/1.07 = { by lemma 27 R->L }
% 5.51/1.07 sum(add(a, fresh3(sum(multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a), additive_identity), true, additive_identity, add(multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a)))), a, additive_identity)
% 5.51/1.07 = { by lemma 36 R->L }
% 5.51/1.07 sum(add(a, fresh3(fresh22(true, true, additive_inverse(a), additive_inverse(additive_inverse(a)), multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a), additive_inverse(a), additive_identity, additive_identity), true, additive_identity, add(multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a)))), a, additive_identity)
% 5.51/1.07 = { by axiom 8 (closure_of_multiplication) R->L }
% 5.51/1.07 sum(add(a, fresh3(fresh22(product(additive_inverse(a), additive_identity, multiply(additive_inverse(a), additive_identity)), true, additive_inverse(a), additive_inverse(additive_inverse(a)), multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a), additive_inverse(a), additive_identity, additive_identity), true, additive_identity, add(multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a)))), a, additive_identity)
% 5.51/1.07 = { by lemma 35 R->L }
% 5.51/1.07 sum(add(a, fresh3(fresh22(product(additive_inverse(a), additive_identity, add(additive_inverse(additive_inverse(a)), add(additive_inverse(a), multiply(additive_inverse(a), additive_identity)))), true, additive_inverse(a), additive_inverse(additive_inverse(a)), multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a), additive_inverse(a), additive_identity, additive_identity), true, additive_identity, add(multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a)))), a, additive_identity)
% 5.51/1.07 = { by lemma 29 R->L }
% 5.51/1.07 sum(add(a, fresh3(fresh22(product(additive_inverse(a), additive_identity, add(additive_inverse(additive_inverse(a)), add(multiply(additive_inverse(a), additive_identity), additive_inverse(a)))), true, additive_inverse(a), additive_inverse(additive_inverse(a)), multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a), additive_inverse(a), additive_identity, additive_identity), true, additive_identity, add(multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a)))), a, additive_identity)
% 5.51/1.07 = { by axiom 6 (addition_is_well_defined) R->L }
% 5.51/1.07 sum(add(a, fresh3(fresh22(product(additive_inverse(a), additive_identity, add(additive_inverse(additive_inverse(a)), fresh3(true, true, additive_inverse(a), add(multiply(additive_inverse(a), additive_identity), additive_inverse(a))))), true, additive_inverse(a), additive_inverse(additive_inverse(a)), multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a), additive_inverse(a), additive_identity, additive_identity), true, additive_identity, add(multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a)))), a, additive_identity)
% 5.51/1.07 = { by axiom 10 (distributivity1) R->L }
% 5.51/1.07 sum(add(a, fresh3(fresh22(product(additive_inverse(a), additive_identity, add(additive_inverse(additive_inverse(a)), fresh3(fresh25(true, true, multiply(additive_inverse(a), additive_identity), additive_inverse(a), additive_inverse(a)), true, additive_inverse(a), add(multiply(additive_inverse(a), additive_identity), additive_inverse(a))))), true, additive_inverse(a), additive_inverse(additive_inverse(a)), multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a), additive_inverse(a), additive_identity, additive_identity), true, additive_identity, add(multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a)))), a, additive_identity)
% 5.51/1.07 = { by axiom 2 (additive_identity1) R->L }
% 5.51/1.07 sum(add(a, fresh3(fresh22(product(additive_inverse(a), additive_identity, add(additive_inverse(additive_inverse(a)), fresh3(fresh25(sum(additive_identity, additive_inverse(a), additive_inverse(a)), true, multiply(additive_inverse(a), additive_identity), additive_inverse(a), additive_inverse(a)), true, additive_inverse(a), add(multiply(additive_inverse(a), additive_identity), additive_inverse(a))))), true, additive_inverse(a), additive_inverse(additive_inverse(a)), multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a), additive_inverse(a), additive_identity, additive_identity), true, additive_identity, add(multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a)))), a, additive_identity)
% 5.51/1.07 = { by lemma 38 R->L }
% 5.51/1.07 sum(add(a, fresh3(fresh22(product(additive_inverse(a), additive_identity, add(additive_inverse(additive_inverse(a)), fresh3(fresh22(product(additive_inverse(a), additive_inverse(a), additive_inverse(a)), true, additive_inverse(a), additive_identity, multiply(additive_inverse(a), additive_identity), additive_inverse(a), additive_inverse(a), additive_inverse(a), additive_inverse(a)), true, additive_inverse(a), add(multiply(additive_inverse(a), additive_identity), additive_inverse(a))))), true, additive_inverse(a), additive_inverse(additive_inverse(a)), multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a), additive_inverse(a), additive_identity, additive_identity), true, additive_identity, add(multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a)))), a, additive_identity)
% 5.51/1.07 = { by axiom 3 (x_squared_is_x) }
% 5.51/1.07 sum(add(a, fresh3(fresh22(product(additive_inverse(a), additive_identity, add(additive_inverse(additive_inverse(a)), fresh3(fresh22(true, true, additive_inverse(a), additive_identity, multiply(additive_inverse(a), additive_identity), additive_inverse(a), additive_inverse(a), additive_inverse(a), additive_inverse(a)), true, additive_inverse(a), add(multiply(additive_inverse(a), additive_identity), additive_inverse(a))))), true, additive_inverse(a), additive_inverse(additive_inverse(a)), multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a), additive_inverse(a), additive_identity, additive_identity), true, additive_identity, add(multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a)))), a, additive_identity)
% 5.51/1.07 = { by lemma 36 }
% 5.51/1.07 sum(add(a, fresh3(fresh22(product(additive_inverse(a), additive_identity, add(additive_inverse(additive_inverse(a)), fresh3(sum(multiply(additive_inverse(a), additive_identity), additive_inverse(a), additive_inverse(a)), true, additive_inverse(a), add(multiply(additive_inverse(a), additive_identity), additive_inverse(a))))), true, additive_inverse(a), additive_inverse(additive_inverse(a)), multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a), additive_inverse(a), additive_identity, additive_identity), true, additive_identity, add(multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a)))), a, additive_identity)
% 5.51/1.07 = { by lemma 27 }
% 5.51/1.07 sum(add(a, fresh3(fresh22(product(additive_inverse(a), additive_identity, add(additive_inverse(additive_inverse(a)), additive_inverse(a))), true, additive_inverse(a), additive_inverse(additive_inverse(a)), multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a), additive_inverse(a), additive_identity, additive_identity), true, additive_identity, add(multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a)))), a, additive_identity)
% 5.51/1.07 = { by lemma 33 }
% 5.51/1.07 sum(add(a, fresh3(fresh22(product(additive_inverse(a), additive_identity, additive_identity), true, additive_inverse(a), additive_inverse(additive_inverse(a)), multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a), additive_inverse(a), additive_identity, additive_identity), true, additive_identity, add(multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a)))), a, additive_identity)
% 5.51/1.07 = { by lemma 38 }
% 5.51/1.07 sum(add(a, fresh3(fresh25(sum(additive_inverse(additive_inverse(a)), additive_inverse(a), additive_identity), true, multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a), additive_identity), true, additive_identity, add(multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a)))), a, additive_identity)
% 5.51/1.07 = { by axiom 5 (left_inverse) }
% 5.51/1.07 sum(add(a, fresh3(fresh25(true, true, multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a), additive_identity), true, additive_identity, add(multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a)))), a, additive_identity)
% 5.51/1.07 = { by axiom 10 (distributivity1) }
% 5.51/1.07 sum(add(a, fresh3(true, true, additive_identity, add(multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a)))), a, additive_identity)
% 5.51/1.07 = { by axiom 6 (addition_is_well_defined) }
% 5.51/1.07 sum(add(a, add(multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), additive_inverse(a))), a, additive_identity)
% 5.51/1.07 = { by lemma 34 }
% 5.51/1.07 sum(multiply(additive_inverse(a), additive_inverse(additive_inverse(a))), a, additive_identity)
% 5.51/1.07 = { by lemma 32 }
% 5.51/1.07 sum(multiply(additive_inverse(a), a), a, additive_identity)
% 5.51/1.07 = { by lemma 37 R->L }
% 5.51/1.07 fresh14(true, true, additive_inverse(a), a, multiply(additive_inverse(a), a), a, a, additive_identity, additive_identity)
% 5.51/1.07 = { by axiom 8 (closure_of_multiplication) R->L }
% 5.51/1.07 fresh14(product(additive_identity, a, multiply(additive_identity, a)), true, additive_inverse(a), a, multiply(additive_inverse(a), a), a, a, additive_identity, additive_identity)
% 5.51/1.07 = { by lemma 35 R->L }
% 5.51/1.07 fresh14(product(additive_identity, a, add(additive_inverse(a), add(a, multiply(additive_identity, a)))), true, additive_inverse(a), a, multiply(additive_inverse(a), a), a, a, additive_identity, additive_identity)
% 5.51/1.07 = { by lemma 29 R->L }
% 5.51/1.07 fresh14(product(additive_identity, a, add(additive_inverse(a), add(multiply(additive_identity, a), a))), true, additive_inverse(a), a, multiply(additive_inverse(a), a), a, a, additive_identity, additive_identity)
% 5.51/1.07 = { by axiom 6 (addition_is_well_defined) R->L }
% 5.51/1.07 fresh14(product(additive_identity, a, add(additive_inverse(a), fresh3(true, true, a, add(multiply(additive_identity, a), a)))), true, additive_inverse(a), a, multiply(additive_inverse(a), a), a, a, additive_identity, additive_identity)
% 5.51/1.07 = { by axiom 11 (distributivity3) R->L }
% 5.51/1.07 fresh14(product(additive_identity, a, add(additive_inverse(a), fresh3(fresh17(true, true, multiply(additive_identity, a), a, a), true, a, add(multiply(additive_identity, a), a)))), true, additive_inverse(a), a, multiply(additive_inverse(a), a), a, a, additive_identity, additive_identity)
% 5.51/1.07 = { by axiom 2 (additive_identity1) R->L }
% 5.51/1.07 fresh14(product(additive_identity, a, add(additive_inverse(a), fresh3(fresh17(sum(additive_identity, a, a), true, multiply(additive_identity, a), a, a), true, a, add(multiply(additive_identity, a), a)))), true, additive_inverse(a), a, multiply(additive_inverse(a), a), a, a, additive_identity, additive_identity)
% 5.51/1.07 = { by lemma 39 R->L }
% 5.51/1.07 fresh14(product(additive_identity, a, add(additive_inverse(a), fresh3(fresh14(product(a, a, a), true, additive_identity, a, multiply(additive_identity, a), a, a, a, a), true, a, add(multiply(additive_identity, a), a)))), true, additive_inverse(a), a, multiply(additive_inverse(a), a), a, a, additive_identity, additive_identity)
% 5.51/1.07 = { by axiom 3 (x_squared_is_x) }
% 5.51/1.07 fresh14(product(additive_identity, a, add(additive_inverse(a), fresh3(fresh14(true, true, additive_identity, a, multiply(additive_identity, a), a, a, a, a), true, a, add(multiply(additive_identity, a), a)))), true, additive_inverse(a), a, multiply(additive_inverse(a), a), a, a, additive_identity, additive_identity)
% 5.51/1.07 = { by lemma 37 }
% 5.51/1.07 fresh14(product(additive_identity, a, add(additive_inverse(a), fresh3(sum(multiply(additive_identity, a), a, a), true, a, add(multiply(additive_identity, a), a)))), true, additive_inverse(a), a, multiply(additive_inverse(a), a), a, a, additive_identity, additive_identity)
% 5.51/1.07 = { by lemma 27 }
% 5.51/1.08 fresh14(product(additive_identity, a, add(additive_inverse(a), a)), true, additive_inverse(a), a, multiply(additive_inverse(a), a), a, a, additive_identity, additive_identity)
% 5.51/1.08 = { by lemma 33 }
% 5.51/1.08 fresh14(product(additive_identity, a, additive_identity), true, additive_inverse(a), a, multiply(additive_inverse(a), a), a, a, additive_identity, additive_identity)
% 5.51/1.08 = { by lemma 39 }
% 5.51/1.08 fresh17(sum(additive_inverse(a), a, additive_identity), true, multiply(additive_inverse(a), a), a, additive_identity)
% 5.51/1.08 = { by axiom 5 (left_inverse) }
% 5.51/1.08 fresh17(true, true, multiply(additive_inverse(a), a), a, additive_identity)
% 5.51/1.08 = { by axiom 11 (distributivity3) }
% 5.51/1.08 true
% 5.51/1.08 % SZS output end Proof
% 5.51/1.08
% 5.51/1.08 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------