TSTP Solution File: KLE071+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE071+1 : TPTP v8.1.2. Released v4.0.0.
% 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 : n022.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 05:35:46 EDT 2023
% Result : Theorem 4.31s 0.94s
% Output : Proof 4.79s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : KLE071+1 : TPTP v8.1.2. Released v4.0.0.
% 0.07/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n022.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 : Tue Aug 29 11:50:53 EDT 2023
% 0.13/0.35 % CPUTime :
% 4.31/0.94 Command-line arguments: --kbo-weight0 --lhs-weight 5 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --goal-heuristic
% 4.31/0.94
% 4.31/0.94 % SZS status Theorem
% 4.31/0.94
% 4.79/0.96 % SZS output start Proof
% 4.79/0.96 Take the following subset of the input axioms:
% 4.79/0.97 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 4.79/0.97 fof(additive_commutativity, axiom, ![A2, B2]: addition(A2, B2)=addition(B2, A2)).
% 4.79/0.97 fof(domain1, axiom, ![X0]: addition(X0, multiplication(domain(X0), X0))=multiplication(domain(X0), X0)).
% 4.79/0.97 fof(domain2, axiom, ![X1, X0_2]: domain(multiplication(X0_2, X1))=domain(multiplication(X0_2, domain(X1)))).
% 4.79/0.97 fof(domain3, axiom, ![X0_2]: addition(domain(X0_2), one)=one).
% 4.79/0.97 fof(domain5, axiom, ![X0_2, X1_2]: domain(addition(X0_2, X1_2))=addition(domain(X0_2), domain(X1_2))).
% 4.79/0.97 fof(goals, conjecture, ![X2, X0_2, X1_2]: addition(domain(X0_2), multiplication(domain(X1_2), domain(X2)))=multiplication(addition(domain(X0_2), domain(X1_2)), addition(domain(X0_2), domain(X2)))).
% 4.79/0.97 fof(left_distributivity, axiom, ![A2, B2, C2]: multiplication(addition(A2, B2), C2)=addition(multiplication(A2, C2), multiplication(B2, C2))).
% 4.79/0.97 fof(multiplicative_associativity, axiom, ![A2, B2, C2]: multiplication(A2, multiplication(B2, C2))=multiplication(multiplication(A2, B2), C2)).
% 4.79/0.97 fof(multiplicative_left_identity, axiom, ![A2]: multiplication(one, A2)=A2).
% 4.79/0.97 fof(multiplicative_right_identity, axiom, ![A2]: multiplication(A2, one)=A2).
% 4.79/0.97 fof(right_distributivity, axiom, ![A2, B2, C2]: multiplication(A2, addition(B2, C2))=addition(multiplication(A2, B2), multiplication(A2, C2))).
% 4.79/0.97
% 4.79/0.97 Now clausify the problem and encode Horn clauses using encoding 3 of
% 4.79/0.97 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 4.79/0.97 We repeatedly replace C & s=t => u=v by the two clauses:
% 4.79/0.97 fresh(y, y, x1...xn) = u
% 4.79/0.97 C => fresh(s, t, x1...xn) = v
% 4.79/0.97 where fresh is a fresh function symbol and x1..xn are the free
% 4.79/0.97 variables of u and v.
% 4.79/0.97 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 4.79/0.97 input problem has no model of domain size 1).
% 4.79/0.97
% 4.79/0.97 The encoding turns the above axioms into the following unit equations and goals:
% 4.79/0.97
% 4.79/0.97 Axiom 1 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 4.79/0.97 Axiom 2 (multiplicative_right_identity): multiplication(X, one) = X.
% 4.79/0.97 Axiom 3 (multiplicative_left_identity): multiplication(one, X) = X.
% 4.79/0.97 Axiom 4 (domain3): addition(domain(X), one) = one.
% 4.79/0.97 Axiom 5 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 4.79/0.97 Axiom 6 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 4.79/0.97 Axiom 7 (domain2): domain(multiplication(X, Y)) = domain(multiplication(X, domain(Y))).
% 4.79/0.97 Axiom 8 (domain1): addition(X, multiplication(domain(X), X)) = multiplication(domain(X), X).
% 4.79/0.97 Axiom 9 (domain5): domain(addition(X, Y)) = addition(domain(X), domain(Y)).
% 4.79/0.97 Axiom 10 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 4.79/0.97 Axiom 11 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 4.79/0.97
% 4.79/0.97 Lemma 12: domain(domain(X)) = domain(X).
% 4.79/0.97 Proof:
% 4.79/0.97 domain(domain(X))
% 4.79/0.97 = { by axiom 3 (multiplicative_left_identity) R->L }
% 4.79/0.97 domain(multiplication(one, domain(X)))
% 4.79/0.97 = { by axiom 7 (domain2) R->L }
% 4.79/0.97 domain(multiplication(one, X))
% 4.79/0.97 = { by axiom 3 (multiplicative_left_identity) }
% 4.79/0.97 domain(X)
% 4.79/0.97
% 4.79/0.97 Lemma 13: addition(one, domain(X)) = one.
% 4.79/0.97 Proof:
% 4.79/0.97 addition(one, domain(X))
% 4.79/0.97 = { by axiom 1 (additive_commutativity) R->L }
% 4.79/0.97 addition(domain(X), one)
% 4.79/0.97 = { by axiom 4 (domain3) }
% 4.79/0.97 one
% 4.79/0.97
% 4.79/0.97 Lemma 14: multiplication(addition(one, Y), X) = addition(X, multiplication(Y, X)).
% 4.79/0.97 Proof:
% 4.79/0.97 multiplication(addition(one, Y), X)
% 4.79/0.97 = { by axiom 11 (left_distributivity) }
% 4.79/0.97 addition(multiplication(one, X), multiplication(Y, X))
% 4.79/0.97 = { by axiom 3 (multiplicative_left_identity) }
% 4.79/0.97 addition(X, multiplication(Y, X))
% 4.79/0.97
% 4.79/0.97 Lemma 15: multiplication(domain(X), X) = X.
% 4.79/0.97 Proof:
% 4.79/0.97 multiplication(domain(X), X)
% 4.79/0.97 = { by axiom 8 (domain1) R->L }
% 4.79/0.97 addition(X, multiplication(domain(X), X))
% 4.79/0.97 = { by lemma 14 R->L }
% 4.79/0.97 multiplication(addition(one, domain(X)), X)
% 4.79/0.97 = { by lemma 13 }
% 4.79/0.97 multiplication(one, X)
% 4.79/0.97 = { by axiom 3 (multiplicative_left_identity) }
% 4.79/0.97 X
% 4.79/0.97
% 4.79/0.97 Lemma 16: addition(domain(X), domain(Y)) = domain(addition(Y, X)).
% 4.79/0.97 Proof:
% 4.79/0.97 addition(domain(X), domain(Y))
% 4.79/0.97 = { by axiom 9 (domain5) R->L }
% 4.79/0.97 domain(addition(X, Y))
% 4.79/0.97 = { by axiom 1 (additive_commutativity) }
% 4.79/0.97 domain(addition(Y, X))
% 4.79/0.97
% 4.79/0.97 Lemma 17: addition(domain(X), domain(Y)) = domain(addition(X, domain(Y))).
% 4.79/0.97 Proof:
% 4.79/0.97 addition(domain(X), domain(Y))
% 4.79/0.97 = { by lemma 12 R->L }
% 4.79/0.97 addition(domain(X), domain(domain(Y)))
% 4.79/0.97 = { by lemma 16 }
% 4.79/0.97 domain(addition(domain(Y), X))
% 4.79/0.97 = { by axiom 1 (additive_commutativity) }
% 4.79/0.97 domain(addition(X, domain(Y)))
% 4.79/0.97
% 4.79/0.97 Lemma 18: domain(addition(domain(Y), X)) = domain(addition(X, Y)).
% 4.79/0.97 Proof:
% 4.79/0.97 domain(addition(domain(Y), X))
% 4.79/0.97 = { by axiom 9 (domain5) }
% 4.79/0.97 addition(domain(domain(Y)), domain(X))
% 4.79/0.97 = { by lemma 12 }
% 4.79/0.97 addition(domain(Y), domain(X))
% 4.79/0.97 = { by lemma 16 }
% 4.79/0.97 domain(addition(X, Y))
% 4.79/0.97
% 4.79/0.97 Lemma 19: domain(addition(domain(X), Y)) = domain(addition(X, Y)).
% 4.79/0.97 Proof:
% 4.79/0.97 domain(addition(domain(X), Y))
% 4.79/0.97 = { by lemma 18 }
% 4.79/0.97 domain(addition(Y, X))
% 4.79/0.97 = { by axiom 1 (additive_commutativity) }
% 4.79/0.97 domain(addition(X, Y))
% 4.79/0.97
% 4.79/0.97 Lemma 20: addition(X, multiplication(X, domain(Y))) = X.
% 4.79/0.97 Proof:
% 4.79/0.97 addition(X, multiplication(X, domain(Y)))
% 4.79/0.97 = { by axiom 2 (multiplicative_right_identity) R->L }
% 4.79/0.97 addition(multiplication(X, one), multiplication(X, domain(Y)))
% 4.79/0.97 = { by axiom 10 (right_distributivity) R->L }
% 4.79/0.97 multiplication(X, addition(one, domain(Y)))
% 4.79/0.97 = { by lemma 13 }
% 4.79/0.97 multiplication(X, one)
% 4.79/0.97 = { by axiom 2 (multiplicative_right_identity) }
% 4.79/0.97 X
% 4.79/0.97
% 4.79/0.97 Lemma 21: addition(X, multiplication(domain(Y), X)) = X.
% 4.79/0.97 Proof:
% 4.79/0.97 addition(X, multiplication(domain(Y), X))
% 4.79/0.97 = { by lemma 14 R->L }
% 4.79/0.97 multiplication(addition(one, domain(Y)), X)
% 4.79/0.97 = { by lemma 13 }
% 4.79/0.97 multiplication(one, X)
% 4.79/0.97 = { by axiom 3 (multiplicative_left_identity) }
% 4.79/0.97 X
% 4.79/0.97
% 4.79/0.97 Lemma 22: multiplication(domain(X), addition(X, Y)) = addition(X, multiplication(domain(X), Y)).
% 4.79/0.97 Proof:
% 4.79/0.97 multiplication(domain(X), addition(X, Y))
% 4.79/0.97 = { by axiom 10 (right_distributivity) }
% 4.79/0.97 addition(multiplication(domain(X), X), multiplication(domain(X), Y))
% 4.79/0.97 = { by lemma 15 }
% 4.79/0.97 addition(X, multiplication(domain(X), Y))
% 4.79/0.97
% 4.79/0.97 Lemma 23: multiplication(domain(multiplication(domain(X), Y)), Y) = multiplication(domain(X), Y).
% 4.79/0.97 Proof:
% 4.79/0.97 multiplication(domain(multiplication(domain(X), Y)), Y)
% 4.79/0.97 = { by lemma 21 R->L }
% 4.79/0.97 multiplication(domain(multiplication(domain(X), Y)), addition(Y, multiplication(domain(X), Y)))
% 4.79/0.97 = { by axiom 1 (additive_commutativity) R->L }
% 4.79/0.97 multiplication(domain(multiplication(domain(X), Y)), addition(multiplication(domain(X), Y), Y))
% 4.79/0.97 = { by lemma 22 }
% 4.79/0.97 addition(multiplication(domain(X), Y), multiplication(domain(multiplication(domain(X), Y)), Y))
% 4.79/0.97 = { by axiom 11 (left_distributivity) R->L }
% 4.79/0.97 multiplication(addition(domain(X), domain(multiplication(domain(X), Y))), Y)
% 4.79/0.97 = { by lemma 16 }
% 4.79/0.97 multiplication(domain(addition(multiplication(domain(X), Y), X)), Y)
% 4.79/0.97 = { by axiom 9 (domain5) }
% 4.79/0.97 multiplication(addition(domain(multiplication(domain(X), Y)), domain(X)), Y)
% 4.79/0.97 = { by axiom 7 (domain2) }
% 4.79/0.97 multiplication(addition(domain(multiplication(domain(X), domain(Y))), domain(X)), Y)
% 4.79/0.97 = { by axiom 9 (domain5) R->L }
% 4.79/0.97 multiplication(domain(addition(multiplication(domain(X), domain(Y)), X)), Y)
% 4.79/0.97 = { by axiom 1 (additive_commutativity) }
% 4.79/0.97 multiplication(domain(addition(X, multiplication(domain(X), domain(Y)))), Y)
% 4.79/0.97 = { by lemma 19 R->L }
% 4.79/0.97 multiplication(domain(addition(domain(X), multiplication(domain(X), domain(Y)))), Y)
% 4.79/0.97 = { by lemma 20 }
% 4.79/0.97 multiplication(domain(domain(X)), Y)
% 4.79/0.97 = { by lemma 12 }
% 4.79/0.97 multiplication(domain(X), Y)
% 4.79/0.97
% 4.79/0.97 Lemma 24: domain(multiplication(domain(X), domain(Y))) = multiplication(domain(X), domain(Y)).
% 4.79/0.97 Proof:
% 4.79/0.97 domain(multiplication(domain(X), domain(Y)))
% 4.79/0.97 = { by lemma 20 R->L }
% 4.79/0.97 addition(domain(multiplication(domain(X), domain(Y))), multiplication(domain(multiplication(domain(X), domain(Y))), domain(domain(Y))))
% 4.79/0.97 = { by lemma 12 R->L }
% 4.79/0.97 addition(domain(multiplication(domain(X), domain(Y))), multiplication(domain(domain(multiplication(domain(X), domain(Y)))), domain(domain(Y))))
% 4.79/0.97 = { by lemma 22 R->L }
% 4.79/0.97 multiplication(domain(domain(multiplication(domain(X), domain(Y)))), addition(domain(multiplication(domain(X), domain(Y))), domain(domain(Y))))
% 4.79/0.97 = { by axiom 9 (domain5) R->L }
% 4.79/0.97 multiplication(domain(domain(multiplication(domain(X), domain(Y)))), domain(addition(multiplication(domain(X), domain(Y)), domain(Y))))
% 4.79/0.97 = { by lemma 12 }
% 4.79/0.97 multiplication(domain(multiplication(domain(X), domain(Y))), domain(addition(multiplication(domain(X), domain(Y)), domain(Y))))
% 4.79/0.97 = { by axiom 1 (additive_commutativity) }
% 4.79/0.97 multiplication(domain(multiplication(domain(X), domain(Y))), domain(addition(domain(Y), multiplication(domain(X), domain(Y)))))
% 4.79/0.97 = { by lemma 21 }
% 4.79/0.97 multiplication(domain(multiplication(domain(X), domain(Y))), domain(domain(Y)))
% 4.79/0.97 = { by lemma 12 }
% 4.79/0.97 multiplication(domain(multiplication(domain(X), domain(Y))), domain(Y))
% 4.79/0.97 = { by lemma 23 }
% 4.79/0.97 multiplication(domain(X), domain(Y))
% 4.79/0.97
% 4.79/0.97 Lemma 25: addition(multiplication(domain(X), domain(Y)), domain(Z)) = domain(addition(Z, multiplication(domain(X), domain(Y)))).
% 4.79/0.97 Proof:
% 4.79/0.97 addition(multiplication(domain(X), domain(Y)), domain(Z))
% 4.79/0.97 = { by lemma 24 R->L }
% 4.79/0.97 addition(domain(multiplication(domain(X), domain(Y))), domain(Z))
% 4.79/0.97 = { by axiom 9 (domain5) R->L }
% 4.79/0.97 domain(addition(multiplication(domain(X), domain(Y)), Z))
% 4.79/0.97 = { by axiom 1 (additive_commutativity) }
% 4.79/0.97 domain(addition(Z, multiplication(domain(X), domain(Y))))
% 4.79/0.97
% 4.79/0.97 Goal 1 (goals): addition(domain(x0), multiplication(domain(x1), domain(x2))) = multiplication(addition(domain(x0), domain(x1)), addition(domain(x0), domain(x2))).
% 4.79/0.97 Proof:
% 4.79/0.97 addition(domain(x0), multiplication(domain(x1), domain(x2)))
% 4.79/0.97 = { by lemma 20 R->L }
% 4.79/0.97 addition(addition(domain(x0), multiplication(domain(x1), domain(x2))), multiplication(addition(domain(x0), multiplication(domain(x1), domain(x2))), domain(x2)))
% 4.79/0.97 = { by axiom 1 (additive_commutativity) R->L }
% 4.79/0.97 addition(addition(domain(x0), multiplication(domain(x1), domain(x2))), multiplication(addition(multiplication(domain(x1), domain(x2)), domain(x0)), domain(x2)))
% 4.79/0.97 = { by axiom 11 (left_distributivity) }
% 4.79/0.97 addition(addition(domain(x0), multiplication(domain(x1), domain(x2))), addition(multiplication(multiplication(domain(x1), domain(x2)), domain(x2)), multiplication(domain(x0), domain(x2))))
% 4.79/0.97 = { by lemma 24 R->L }
% 4.79/0.97 addition(addition(domain(x0), multiplication(domain(x1), domain(x2))), addition(multiplication(domain(multiplication(domain(x1), domain(x2))), domain(x2)), multiplication(domain(x0), domain(x2))))
% 4.79/0.97 = { by lemma 23 }
% 4.79/0.97 addition(addition(domain(x0), multiplication(domain(x1), domain(x2))), addition(multiplication(domain(x1), domain(x2)), multiplication(domain(x0), domain(x2))))
% 4.79/0.97 = { by axiom 11 (left_distributivity) R->L }
% 4.79/0.97 addition(addition(domain(x0), multiplication(domain(x1), domain(x2))), multiplication(addition(domain(x1), domain(x0)), domain(x2)))
% 4.79/0.97 = { by axiom 1 (additive_commutativity) }
% 4.79/0.97 addition(addition(domain(x0), multiplication(domain(x1), domain(x2))), multiplication(addition(domain(x0), domain(x1)), domain(x2)))
% 4.79/0.97 = { by axiom 1 (additive_commutativity) R->L }
% 4.79/0.97 addition(addition(multiplication(domain(x1), domain(x2)), domain(x0)), multiplication(addition(domain(x0), domain(x1)), domain(x2)))
% 4.79/0.97 = { by lemma 21 R->L }
% 4.79/0.97 addition(addition(multiplication(domain(x1), domain(x2)), addition(domain(x0), multiplication(domain(x1), domain(x0)))), multiplication(addition(domain(x0), domain(x1)), domain(x2)))
% 4.79/0.97 = { by lemma 15 R->L }
% 4.79/0.97 addition(addition(multiplication(domain(x1), domain(x2)), addition(multiplication(domain(domain(x0)), domain(x0)), multiplication(domain(x1), domain(x0)))), multiplication(addition(domain(x0), domain(x1)), domain(x2)))
% 4.79/0.97 = { by lemma 12 }
% 4.79/0.97 addition(addition(multiplication(domain(x1), domain(x2)), addition(multiplication(domain(x0), domain(x0)), multiplication(domain(x1), domain(x0)))), multiplication(addition(domain(x0), domain(x1)), domain(x2)))
% 4.79/0.97 = { by axiom 11 (left_distributivity) R->L }
% 4.79/0.97 addition(addition(multiplication(domain(x1), domain(x2)), multiplication(addition(domain(x0), domain(x1)), domain(x0))), multiplication(addition(domain(x0), domain(x1)), domain(x2)))
% 4.79/0.97 = { by lemma 21 R->L }
% 4.79/0.97 addition(addition(addition(multiplication(domain(x1), domain(x2)), multiplication(domain(x0), multiplication(domain(x1), domain(x2)))), multiplication(addition(domain(x0), domain(x1)), domain(x0))), multiplication(addition(domain(x0), domain(x1)), domain(x2)))
% 4.79/0.97 = { by lemma 21 R->L }
% 4.79/0.97 addition(addition(addition(addition(multiplication(domain(x1), domain(x2)), multiplication(domain(x1), multiplication(domain(x1), domain(x2)))), multiplication(domain(x0), multiplication(domain(x1), domain(x2)))), multiplication(addition(domain(x0), domain(x1)), domain(x0))), multiplication(addition(domain(x0), domain(x1)), domain(x2)))
% 4.79/0.97 = { by lemma 12 R->L }
% 4.79/0.97 addition(addition(addition(addition(multiplication(domain(x1), domain(x2)), multiplication(domain(domain(x1)), multiplication(domain(x1), domain(x2)))), multiplication(domain(x0), multiplication(domain(x1), domain(x2)))), multiplication(addition(domain(x0), domain(x1)), domain(x0))), multiplication(addition(domain(x0), domain(x1)), domain(x2)))
% 4.79/0.98 = { by axiom 6 (multiplicative_associativity) }
% 4.79/0.98 addition(addition(addition(addition(multiplication(domain(x1), domain(x2)), multiplication(multiplication(domain(domain(x1)), domain(x1)), domain(x2))), multiplication(domain(x0), multiplication(domain(x1), domain(x2)))), multiplication(addition(domain(x0), domain(x1)), domain(x0))), multiplication(addition(domain(x0), domain(x1)), domain(x2)))
% 4.79/0.98 = { by axiom 11 (left_distributivity) R->L }
% 4.79/0.98 addition(addition(addition(multiplication(addition(domain(x1), multiplication(domain(domain(x1)), domain(x1))), domain(x2)), multiplication(domain(x0), multiplication(domain(x1), domain(x2)))), multiplication(addition(domain(x0), domain(x1)), domain(x0))), multiplication(addition(domain(x0), domain(x1)), domain(x2)))
% 4.79/0.98 = { by axiom 8 (domain1) }
% 4.79/0.98 addition(addition(addition(multiplication(multiplication(domain(domain(x1)), domain(x1)), domain(x2)), multiplication(domain(x0), multiplication(domain(x1), domain(x2)))), multiplication(addition(domain(x0), domain(x1)), domain(x0))), multiplication(addition(domain(x0), domain(x1)), domain(x2)))
% 4.79/0.98 = { by axiom 6 (multiplicative_associativity) R->L }
% 4.79/0.98 addition(addition(addition(multiplication(domain(domain(x1)), multiplication(domain(x1), domain(x2))), multiplication(domain(x0), multiplication(domain(x1), domain(x2)))), multiplication(addition(domain(x0), domain(x1)), domain(x0))), multiplication(addition(domain(x0), domain(x1)), domain(x2)))
% 4.79/0.98 = { by lemma 12 }
% 4.79/0.98 addition(addition(addition(multiplication(domain(x1), multiplication(domain(x1), domain(x2))), multiplication(domain(x0), multiplication(domain(x1), domain(x2)))), multiplication(addition(domain(x0), domain(x1)), domain(x0))), multiplication(addition(domain(x0), domain(x1)), domain(x2)))
% 4.79/0.98 = { by axiom 11 (left_distributivity) R->L }
% 4.79/0.98 addition(addition(multiplication(addition(domain(x1), domain(x0)), multiplication(domain(x1), domain(x2))), multiplication(addition(domain(x0), domain(x1)), domain(x0))), multiplication(addition(domain(x0), domain(x1)), domain(x2)))
% 4.79/0.98 = { by axiom 1 (additive_commutativity) }
% 4.79/0.98 addition(addition(multiplication(addition(domain(x0), domain(x1)), multiplication(domain(x1), domain(x2))), multiplication(addition(domain(x0), domain(x1)), domain(x0))), multiplication(addition(domain(x0), domain(x1)), domain(x2)))
% 4.79/0.98 = { by axiom 10 (right_distributivity) R->L }
% 4.79/0.98 addition(multiplication(addition(domain(x0), domain(x1)), addition(multiplication(domain(x1), domain(x2)), domain(x0))), multiplication(addition(domain(x0), domain(x1)), domain(x2)))
% 4.79/0.98 = { by axiom 1 (additive_commutativity) }
% 4.79/0.98 addition(multiplication(addition(domain(x0), domain(x1)), addition(domain(x0), multiplication(domain(x1), domain(x2)))), multiplication(addition(domain(x0), domain(x1)), domain(x2)))
% 4.79/0.98 = { by axiom 10 (right_distributivity) R->L }
% 4.79/0.98 multiplication(addition(domain(x0), domain(x1)), addition(addition(domain(x0), multiplication(domain(x1), domain(x2))), domain(x2)))
% 4.79/0.98 = { by axiom 1 (additive_commutativity) }
% 4.79/0.98 multiplication(addition(domain(x0), domain(x1)), addition(domain(x2), addition(domain(x0), multiplication(domain(x1), domain(x2)))))
% 4.79/0.98 = { by lemma 24 R->L }
% 4.79/0.98 multiplication(addition(domain(x0), domain(x1)), addition(domain(x2), addition(domain(x0), domain(multiplication(domain(x1), domain(x2))))))
% 4.79/0.98 = { by lemma 16 }
% 4.79/0.98 multiplication(addition(domain(x0), domain(x1)), addition(domain(x2), domain(addition(multiplication(domain(x1), domain(x2)), x0))))
% 4.79/0.98 = { by lemma 18 R->L }
% 4.79/0.98 multiplication(addition(domain(x0), domain(x1)), addition(domain(x2), domain(addition(domain(x0), multiplication(domain(x1), domain(x2))))))
% 4.79/0.98 = { by lemma 16 }
% 4.79/0.98 multiplication(addition(domain(x0), domain(x1)), domain(addition(addition(domain(x0), multiplication(domain(x1), domain(x2))), x2)))
% 4.79/0.98 = { by lemma 18 R->L }
% 4.79/0.98 multiplication(addition(domain(x0), domain(x1)), domain(addition(domain(x2), addition(domain(x0), multiplication(domain(x1), domain(x2))))))
% 4.79/0.98 = { by lemma 19 }
% 4.79/0.98 multiplication(addition(domain(x0), domain(x1)), domain(addition(x2, addition(domain(x0), multiplication(domain(x1), domain(x2))))))
% 4.79/0.98 = { by axiom 5 (additive_associativity) }
% 4.79/0.98 multiplication(addition(domain(x0), domain(x1)), domain(addition(addition(x2, domain(x0)), multiplication(domain(x1), domain(x2)))))
% 4.79/0.98 = { by lemma 25 R->L }
% 4.79/0.98 multiplication(addition(domain(x0), domain(x1)), addition(multiplication(domain(x1), domain(x2)), domain(addition(x2, domain(x0)))))
% 4.79/0.98 = { by axiom 1 (additive_commutativity) R->L }
% 4.79/0.98 multiplication(addition(domain(x0), domain(x1)), addition(multiplication(domain(x1), domain(x2)), domain(addition(domain(x0), x2))))
% 4.79/0.98 = { by lemma 16 R->L }
% 4.79/0.98 multiplication(addition(domain(x0), domain(x1)), addition(multiplication(domain(x1), domain(x2)), addition(domain(x2), domain(domain(x0)))))
% 4.79/0.98 = { by lemma 12 }
% 4.79/0.98 multiplication(addition(domain(x0), domain(x1)), addition(multiplication(domain(x1), domain(x2)), addition(domain(x2), domain(x0))))
% 4.79/0.98 = { by axiom 1 (additive_commutativity) }
% 4.79/0.98 multiplication(addition(domain(x0), domain(x1)), addition(multiplication(domain(x1), domain(x2)), addition(domain(x0), domain(x2))))
% 4.79/0.98 = { by lemma 17 }
% 4.79/0.98 multiplication(addition(domain(x0), domain(x1)), addition(multiplication(domain(x1), domain(x2)), domain(addition(x0, domain(x2)))))
% 4.79/0.98 = { by lemma 25 }
% 4.79/0.98 multiplication(addition(domain(x0), domain(x1)), domain(addition(addition(x0, domain(x2)), multiplication(domain(x1), domain(x2)))))
% 4.79/0.98 = { by axiom 5 (additive_associativity) R->L }
% 4.79/0.98 multiplication(addition(domain(x0), domain(x1)), domain(addition(x0, addition(domain(x2), multiplication(domain(x1), domain(x2))))))
% 4.79/0.98 = { by lemma 21 }
% 4.79/0.98 multiplication(addition(domain(x0), domain(x1)), domain(addition(x0, domain(x2))))
% 4.79/0.98 = { by lemma 17 R->L }
% 4.79/0.98 multiplication(addition(domain(x0), domain(x1)), addition(domain(x0), domain(x2)))
% 4.79/0.98 % SZS output end Proof
% 4.79/0.98
% 4.79/0.98 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------