0.08/0.13 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.08/0.13 % Command : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn 0.14/0.35 % Computer : n008.cluster.edu 0.14/0.35 % Model : x86_64 x86_64 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.14/0.35 % Memory : 8042.1875MB 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64 0.14/0.35 % CPULimit : 960 0.14/0.35 % WCLimit : 120 0.14/0.35 % DateTime : Thu Jul 2 08:33:30 EDT 2020 0.14/0.35 % CPUTime : 3.85/0.88 % SZS status Theorem 3.85/0.88 3.85/0.88 % SZS output start Proof 3.85/0.88 Take the following subset of the input axioms: 3.85/0.89 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)). 3.85/0.89 fof(additive_commutativity, axiom, ![A, B]: addition(B, A)=addition(A, B)). 3.85/0.89 fof(additive_idempotence, axiom, ![A]: A=addition(A, A)). 3.85/0.89 fof(additive_identity, axiom, ![A]: addition(A, zero)=A). 3.85/0.89 fof(goals, conjecture, ![X0, X1]: (multiplication(X0, X1)=multiplication(X1, X0) <= (test(X1) & test(X0)))). 3.85/0.89 fof(left_distributivity, axiom, ![A, B, C]: addition(multiplication(A, C), multiplication(B, C))=multiplication(addition(A, B), C)). 3.85/0.89 fof(multiplicative_associativity, axiom, ![A, B, C]: multiplication(A, multiplication(B, C))=multiplication(multiplication(A, B), C)). 3.85/0.89 fof(multiplicative_left_identity, axiom, ![A]: multiplication(one, A)=A). 3.85/0.89 fof(multiplicative_right_identity, axiom, ![A]: multiplication(A, one)=A). 3.85/0.89 fof(right_distributivity, axiom, ![A, B, C]: addition(multiplication(A, B), multiplication(A, C))=multiplication(A, addition(B, C))). 3.85/0.89 fof(test_1, axiom, ![X0]: (test(X0) <=> ?[X1]: complement(X1, X0))). 3.85/0.89 fof(test_2, axiom, ![X0, X1]: (complement(X1, X0) <=> (zero=multiplication(X0, X1) & (zero=multiplication(X1, X0) & addition(X0, X1)=one)))). 3.85/0.89 fof(test_3, axiom, ![X0, X1]: ((complement(X0, X1) <=> X1=c(X0)) <= test(X0))). 3.85/0.89 3.85/0.89 Now clausify the problem and encode Horn clauses using encoding 3 of 3.85/0.89 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 3.85/0.89 We repeatedly replace C & s=t => u=v by the two clauses: 3.85/0.89 fresh(y, y, x1...xn) = u 3.85/0.89 C => fresh(s, t, x1...xn) = v 3.85/0.89 where fresh is a fresh function symbol and x1..xn are the free 3.85/0.89 variables of u and v. 3.85/0.89 A predicate p(X) is encoded as p(X)=true (this is sound, because the 3.85/0.89 input problem has no model of domain size 1). 3.85/0.89 3.85/0.89 The encoding turns the above axioms into the following unit equations and goals: 3.85/0.89 3.85/0.89 Axiom 1 (test_1): fresh12(X, X, Y) = true. 3.85/0.89 Axiom 2 (test_1_1): fresh10(X, X, Y) = true. 3.85/0.89 Axiom 3 (test_2_1): fresh8(X, X, Y, Z) = one. 3.85/0.89 Axiom 4 (test_2_2): fresh7(X, X, Y, Z) = zero. 3.85/0.89 Axiom 5 (test_2_3): fresh6(X, X, Y, Z) = zero. 3.85/0.89 Axiom 6 (test_3): fresh5(X, X, Y, Z) = complement(Y, Z). 3.85/0.89 Axiom 7 (test_3): fresh4(X, X, Y, Z) = true. 3.85/0.89 Axiom 8 (test_3_1): fresh(X, X, Y, Z) = Z. 3.85/0.89 Axiom 9 (test_3_1): fresh3(X, X, Y, Z) = c(Y). 3.85/0.89 Axiom 10 (left_distributivity): addition(multiplication(X, Y), multiplication(Z, Y)) = multiplication(addition(X, Z), Y). 3.85/0.89 Axiom 11 (multiplicative_right_identity): multiplication(X, one) = X. 3.85/0.89 Axiom 12 (additive_commutativity): addition(X, Y) = addition(Y, X). 3.85/0.89 Axiom 13 (multiplicative_left_identity): multiplication(one, X) = X. 3.85/0.89 Axiom 14 (right_distributivity): addition(multiplication(X, Y), multiplication(X, Z)) = multiplication(X, addition(Y, Z)). 3.85/0.89 Axiom 15 (additive_idempotence): X = addition(X, X). 3.85/0.89 Axiom 16 (additive_identity): addition(X, zero) = X. 3.85/0.89 Axiom 17 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z). 3.85/0.89 Axiom 18 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z). 3.85/0.89 Axiom 19 (test_1_1): fresh10(complement(X, Y), true, Y) = test(Y). 3.85/0.89 Axiom 20 (test_1): fresh12(test(X), true, X) = complement(sK3_test_1_X1(X), X). 3.85/0.89 Axiom 21 (test_2_3): fresh6(complement(X, Y), true, Y, X) = multiplication(X, Y). 3.85/0.89 Axiom 22 (test_2_2): fresh7(complement(X, Y), true, Y, X) = multiplication(Y, X). 3.85/0.89 Axiom 23 (test_2_1): fresh8(complement(X, Y), true, Y, X) = addition(Y, X). 3.85/0.89 Axiom 24 (test_3_1): fresh3(complement(X, Y), true, X, Y) = fresh(test(X), true, X, Y). 3.85/0.89 Axiom 25 (test_3): fresh5(test(X), true, X, Y) = fresh4(Y, c(X), X, Y). 3.85/0.89 Axiom 26 (goals): test(sK2_goals_X1) = true. 3.85/0.89 Axiom 27 (goals_1): test(sK1_goals_X0) = true. 3.85/0.89 3.85/0.89 Lemma 28: fresh5(test(X), true, X, c(X)) = true. 3.85/0.89 Proof: 3.85/0.89 fresh5(test(X), true, X, c(X)) 3.85/0.89 = { by axiom 25 (test_3) } 3.85/0.89 fresh4(c(X), c(X), X, c(X)) 3.85/0.89 = { by axiom 7 (test_3) } 3.85/0.89 true 3.85/0.89 3.85/0.89 Lemma 29: complement(sK1_goals_X0, c(sK1_goals_X0)) = true. 3.85/0.89 Proof: 3.85/0.89 complement(sK1_goals_X0, c(sK1_goals_X0)) 3.85/0.89 = { by axiom 6 (test_3) } 3.85/0.89 fresh5(true, true, sK1_goals_X0, c(sK1_goals_X0)) 3.85/0.89 = { by axiom 27 (goals_1) } 3.85/0.89 fresh5(test(sK1_goals_X0), true, sK1_goals_X0, c(sK1_goals_X0)) 3.85/0.89 = { by lemma 28 } 3.85/0.89 true 3.85/0.89 3.85/0.89 Lemma 30: multiplication(sK1_goals_X0, c(sK1_goals_X0)) = zero. 3.85/0.89 Proof: 3.85/0.89 multiplication(sK1_goals_X0, c(sK1_goals_X0)) 3.85/0.89 = { by axiom 21 (test_2_3) } 3.85/0.89 fresh6(complement(sK1_goals_X0, c(sK1_goals_X0)), true, c(sK1_goals_X0), sK1_goals_X0) 3.85/0.89 = { by lemma 29 } 3.85/0.89 fresh6(true, true, c(sK1_goals_X0), sK1_goals_X0) 3.85/0.89 = { by axiom 5 (test_2_3) } 3.85/0.89 zero 3.85/0.89 3.85/0.89 Lemma 31: addition(zero, X) = X. 3.85/0.89 Proof: 3.85/0.89 addition(zero, X) 3.85/0.89 = { by axiom 12 (additive_commutativity) } 3.85/0.89 addition(X, zero) 3.85/0.89 = { by axiom 16 (additive_identity) } 3.85/0.89 X 3.85/0.89 3.85/0.89 Lemma 32: multiplication(sK1_goals_X0, addition(X, c(sK1_goals_X0))) = multiplication(sK1_goals_X0, X). 3.85/0.89 Proof: 3.85/0.89 multiplication(sK1_goals_X0, addition(X, c(sK1_goals_X0))) 3.85/0.89 = { by axiom 12 (additive_commutativity) } 3.85/0.89 multiplication(sK1_goals_X0, addition(c(sK1_goals_X0), X)) 3.85/0.89 = { by axiom 14 (right_distributivity) } 3.85/0.89 addition(multiplication(sK1_goals_X0, c(sK1_goals_X0)), multiplication(sK1_goals_X0, X)) 3.85/0.89 = { by lemma 30 } 3.85/0.89 addition(zero, multiplication(sK1_goals_X0, X)) 3.85/0.89 = { by lemma 31 } 3.85/0.89 multiplication(sK1_goals_X0, X) 3.85/0.89 3.85/0.89 Lemma 33: complement(sK3_test_1_X1(sK1_goals_X0), sK1_goals_X0) = true. 3.85/0.89 Proof: 3.85/0.89 complement(sK3_test_1_X1(sK1_goals_X0), sK1_goals_X0) 3.85/0.89 = { by axiom 20 (test_1) } 3.85/0.89 fresh12(test(sK1_goals_X0), true, sK1_goals_X0) 3.85/0.89 = { by axiom 27 (goals_1) } 3.85/0.89 fresh12(true, true, sK1_goals_X0) 3.85/0.89 = { by axiom 1 (test_1) } 3.85/0.90 true 3.85/0.90 3.85/0.90 Lemma 34: sK3_test_1_X1(sK1_goals_X0) = c(sK1_goals_X0). 3.85/0.90 Proof: 3.85/0.90 sK3_test_1_X1(sK1_goals_X0) 3.85/0.90 = { by axiom 11 (multiplicative_right_identity) } 3.85/0.90 multiplication(sK3_test_1_X1(sK1_goals_X0), one) 3.85/0.90 = { by axiom 3 (test_2_1) } 3.85/0.90 multiplication(sK3_test_1_X1(sK1_goals_X0), fresh8(true, true, c(sK1_goals_X0), sK1_goals_X0)) 3.85/0.90 = { by lemma 29 } 3.85/0.90 multiplication(sK3_test_1_X1(sK1_goals_X0), fresh8(complement(sK1_goals_X0, c(sK1_goals_X0)), true, c(sK1_goals_X0), sK1_goals_X0)) 3.85/0.90 = { by axiom 23 (test_2_1) } 3.85/0.90 multiplication(sK3_test_1_X1(sK1_goals_X0), addition(c(sK1_goals_X0), sK1_goals_X0)) 3.85/0.90 = { by axiom 12 (additive_commutativity) } 3.85/0.90 multiplication(sK3_test_1_X1(sK1_goals_X0), addition(sK1_goals_X0, c(sK1_goals_X0))) 3.85/0.90 = { by axiom 14 (right_distributivity) } 3.85/0.90 addition(multiplication(sK3_test_1_X1(sK1_goals_X0), sK1_goals_X0), multiplication(sK3_test_1_X1(sK1_goals_X0), c(sK1_goals_X0))) 3.85/0.90 = { by axiom 21 (test_2_3) } 3.85/0.90 addition(fresh6(complement(sK3_test_1_X1(sK1_goals_X0), sK1_goals_X0), true, sK1_goals_X0, sK3_test_1_X1(sK1_goals_X0)), multiplication(sK3_test_1_X1(sK1_goals_X0), c(sK1_goals_X0))) 3.85/0.90 = { by lemma 33 } 3.85/0.90 addition(fresh6(true, true, sK1_goals_X0, sK3_test_1_X1(sK1_goals_X0)), multiplication(sK3_test_1_X1(sK1_goals_X0), c(sK1_goals_X0))) 3.85/0.90 = { by axiom 5 (test_2_3) } 3.85/0.90 addition(zero, multiplication(sK3_test_1_X1(sK1_goals_X0), c(sK1_goals_X0))) 3.85/0.90 = { by lemma 30 } 3.85/0.90 addition(multiplication(sK1_goals_X0, c(sK1_goals_X0)), multiplication(sK3_test_1_X1(sK1_goals_X0), c(sK1_goals_X0))) 3.85/0.90 = { by axiom 10 (left_distributivity) } 3.85/0.90 multiplication(addition(sK1_goals_X0, sK3_test_1_X1(sK1_goals_X0)), c(sK1_goals_X0)) 3.85/0.90 = { by axiom 23 (test_2_1) } 3.85/0.90 multiplication(fresh8(complement(sK3_test_1_X1(sK1_goals_X0), sK1_goals_X0), true, sK1_goals_X0, sK3_test_1_X1(sK1_goals_X0)), c(sK1_goals_X0)) 3.85/0.90 = { by lemma 33 } 3.85/0.90 multiplication(fresh8(true, true, sK1_goals_X0, sK3_test_1_X1(sK1_goals_X0)), c(sK1_goals_X0)) 3.85/0.90 = { by axiom 3 (test_2_1) } 3.85/0.90 multiplication(one, c(sK1_goals_X0)) 3.85/0.90 = { by axiom 13 (multiplicative_left_identity) } 3.85/0.90 c(sK1_goals_X0) 3.85/0.90 3.85/0.90 Lemma 35: test(c(sK1_goals_X0)) = true. 3.85/0.90 Proof: 3.85/0.90 test(c(sK1_goals_X0)) 3.85/0.90 = { by axiom 19 (test_1_1) } 3.85/0.90 fresh10(complement(sK1_goals_X0, c(sK1_goals_X0)), true, c(sK1_goals_X0)) 3.85/0.90 = { by lemma 29 } 3.85/0.90 fresh10(true, true, c(sK1_goals_X0)) 3.85/0.90 = { by axiom 2 (test_1_1) } 3.85/0.90 true 3.85/0.90 3.85/0.90 Lemma 36: c(c(sK1_goals_X0)) = sK1_goals_X0. 3.85/0.90 Proof: 3.85/0.90 c(c(sK1_goals_X0)) 3.85/0.90 = { by lemma 34 } 3.85/0.90 c(sK3_test_1_X1(sK1_goals_X0)) 3.85/0.90 = { by axiom 9 (test_3_1) } 3.85/0.90 fresh3(true, true, sK3_test_1_X1(sK1_goals_X0), sK1_goals_X0) 3.85/0.90 = { by lemma 33 } 3.85/0.90 fresh3(complement(sK3_test_1_X1(sK1_goals_X0), sK1_goals_X0), true, sK3_test_1_X1(sK1_goals_X0), sK1_goals_X0) 3.85/0.90 = { by axiom 24 (test_3_1) } 3.85/0.90 fresh(test(sK3_test_1_X1(sK1_goals_X0)), true, sK3_test_1_X1(sK1_goals_X0), sK1_goals_X0) 3.85/0.90 = { by lemma 34 } 3.85/0.90 fresh(test(c(sK1_goals_X0)), true, sK3_test_1_X1(sK1_goals_X0), sK1_goals_X0) 3.85/0.90 = { by lemma 35 } 3.85/0.90 fresh(true, true, sK3_test_1_X1(sK1_goals_X0), sK1_goals_X0) 3.85/0.90 = { by lemma 34 } 3.85/0.90 fresh(true, true, c(sK1_goals_X0), sK1_goals_X0) 3.85/0.90 = { by axiom 8 (test_3_1) } 3.85/0.90 sK1_goals_X0 3.85/0.90 3.85/0.90 Lemma 37: addition(sK2_goals_X1, sK3_test_1_X1(sK2_goals_X1)) = one. 3.85/0.90 Proof: 3.85/0.90 addition(sK2_goals_X1, sK3_test_1_X1(sK2_goals_X1)) 3.85/0.90 = { by axiom 23 (test_2_1) } 3.85/0.90 fresh8(complement(sK3_test_1_X1(sK2_goals_X1), sK2_goals_X1), true, sK2_goals_X1, sK3_test_1_X1(sK2_goals_X1)) 3.85/0.90 = { by axiom 20 (test_1) } 3.85/0.90 fresh8(fresh12(test(sK2_goals_X1), true, sK2_goals_X1), true, sK2_goals_X1, sK3_test_1_X1(sK2_goals_X1)) 3.85/0.90 = { by axiom 26 (goals) } 3.85/0.90 fresh8(fresh12(true, true, sK2_goals_X1), true, sK2_goals_X1, sK3_test_1_X1(sK2_goals_X1)) 3.85/0.90 = { by axiom 1 (test_1) } 3.85/0.90 fresh8(true, true, sK2_goals_X1, sK3_test_1_X1(sK2_goals_X1)) 3.85/0.90 = { by axiom 3 (test_2_1) } 3.85/0.90 one 3.85/0.90 3.85/0.90 Lemma 38: addition(one, sK2_goals_X1) = one. 3.85/0.90 Proof: 3.85/0.90 addition(one, sK2_goals_X1) 3.85/0.90 = { by axiom 12 (additive_commutativity) } 3.85/0.90 addition(sK2_goals_X1, one) 3.85/0.90 = { by lemma 37 } 3.85/0.90 addition(sK2_goals_X1, addition(sK2_goals_X1, sK3_test_1_X1(sK2_goals_X1))) 3.85/0.90 = { by axiom 17 (additive_associativity) } 3.85/0.90 addition(addition(sK2_goals_X1, sK2_goals_X1), sK3_test_1_X1(sK2_goals_X1)) 3.85/0.90 = { by axiom 15 (additive_idempotence) } 3.85/0.90 addition(sK2_goals_X1, sK3_test_1_X1(sK2_goals_X1)) 3.85/0.90 = { by lemma 37 } 3.85/0.90 one 3.85/0.90 3.85/0.90 Lemma 39: addition(c(sK1_goals_X0), c(c(sK1_goals_X0))) = one. 3.85/0.90 Proof: 3.85/0.90 addition(c(sK1_goals_X0), c(c(sK1_goals_X0))) 3.85/0.90 = { by axiom 12 (additive_commutativity) } 3.85/0.90 addition(c(c(sK1_goals_X0)), c(sK1_goals_X0)) 3.85/0.90 = { by axiom 23 (test_2_1) } 3.85/0.90 fresh8(complement(c(sK1_goals_X0), c(c(sK1_goals_X0))), true, c(c(sK1_goals_X0)), c(sK1_goals_X0)) 3.85/0.90 = { by axiom 6 (test_3) } 3.85/0.90 fresh8(fresh5(true, true, c(sK1_goals_X0), c(c(sK1_goals_X0))), true, c(c(sK1_goals_X0)), c(sK1_goals_X0)) 3.85/0.90 = { by lemma 35 } 3.85/0.90 fresh8(fresh5(test(c(sK1_goals_X0)), true, c(sK1_goals_X0), c(c(sK1_goals_X0))), true, c(c(sK1_goals_X0)), c(sK1_goals_X0)) 3.85/0.90 = { by lemma 28 } 3.85/0.90 fresh8(true, true, c(c(sK1_goals_X0)), c(sK1_goals_X0)) 3.85/0.90 = { by axiom 3 (test_2_1) } 3.85/0.90 one 3.85/0.90 3.85/0.90 Lemma 40: multiplication(addition(X, c(sK1_goals_X0)), sK1_goals_X0) = multiplication(X, sK1_goals_X0). 3.85/0.90 Proof: 3.85/0.90 multiplication(addition(X, c(sK1_goals_X0)), sK1_goals_X0) 3.85/0.90 = { by axiom 12 (additive_commutativity) } 3.85/0.90 multiplication(addition(c(sK1_goals_X0), X), sK1_goals_X0) 3.85/0.90 = { by axiom 10 (left_distributivity) } 3.85/0.90 addition(multiplication(c(sK1_goals_X0), sK1_goals_X0), multiplication(X, sK1_goals_X0)) 3.85/0.90 = { by axiom 22 (test_2_2) } 3.85/0.90 addition(fresh7(complement(sK1_goals_X0, c(sK1_goals_X0)), true, c(sK1_goals_X0), sK1_goals_X0), multiplication(X, sK1_goals_X0)) 3.85/0.90 = { by lemma 29 } 3.85/0.90 addition(fresh7(true, true, c(sK1_goals_X0), sK1_goals_X0), multiplication(X, sK1_goals_X0)) 3.85/0.90 = { by axiom 4 (test_2_2) } 3.85/0.90 addition(zero, multiplication(X, sK1_goals_X0)) 3.85/0.90 = { by lemma 31 } 3.85/0.91 multiplication(X, sK1_goals_X0) 3.85/0.91 3.85/0.91 Goal 1 (goals_2): multiplication(sK1_goals_X0, sK2_goals_X1) = multiplication(sK2_goals_X1, sK1_goals_X0). 3.85/0.91 Proof: 3.85/0.91 multiplication(sK1_goals_X0, sK2_goals_X1) 3.85/0.91 = { by lemma 32 } 3.85/0.91 multiplication(sK1_goals_X0, addition(sK2_goals_X1, c(sK1_goals_X0))) 3.85/0.91 = { by axiom 12 (additive_commutativity) } 3.85/0.91 multiplication(sK1_goals_X0, addition(c(sK1_goals_X0), sK2_goals_X1)) 3.85/0.91 = { by axiom 11 (multiplicative_right_identity) } 3.85/0.91 multiplication(sK1_goals_X0, addition(c(sK1_goals_X0), multiplication(sK2_goals_X1, one))) 3.85/0.91 = { by lemma 39 } 3.85/0.91 multiplication(sK1_goals_X0, addition(c(sK1_goals_X0), multiplication(sK2_goals_X1, addition(c(sK1_goals_X0), c(c(sK1_goals_X0)))))) 3.85/0.91 = { by axiom 12 (additive_commutativity) } 3.85/0.91 multiplication(sK1_goals_X0, addition(c(sK1_goals_X0), multiplication(sK2_goals_X1, addition(c(c(sK1_goals_X0)), c(sK1_goals_X0))))) 3.85/0.91 = { by axiom 14 (right_distributivity) } 3.85/0.91 multiplication(sK1_goals_X0, addition(c(sK1_goals_X0), addition(multiplication(sK2_goals_X1, c(c(sK1_goals_X0))), multiplication(sK2_goals_X1, c(sK1_goals_X0))))) 3.85/0.91 = { by axiom 12 (additive_commutativity) } 3.85/0.91 multiplication(sK1_goals_X0, addition(c(sK1_goals_X0), addition(multiplication(sK2_goals_X1, c(sK1_goals_X0)), multiplication(sK2_goals_X1, c(c(sK1_goals_X0)))))) 3.85/0.91 = { by axiom 17 (additive_associativity) } 3.85/0.91 multiplication(sK1_goals_X0, addition(addition(c(sK1_goals_X0), multiplication(sK2_goals_X1, c(sK1_goals_X0))), multiplication(sK2_goals_X1, c(c(sK1_goals_X0))))) 3.85/0.91 = { by axiom 13 (multiplicative_left_identity) } 3.85/0.91 multiplication(sK1_goals_X0, addition(addition(multiplication(one, c(sK1_goals_X0)), multiplication(sK2_goals_X1, c(sK1_goals_X0))), multiplication(sK2_goals_X1, c(c(sK1_goals_X0))))) 3.85/0.91 = { by axiom 10 (left_distributivity) } 3.85/0.91 multiplication(sK1_goals_X0, addition(multiplication(addition(one, sK2_goals_X1), c(sK1_goals_X0)), multiplication(sK2_goals_X1, c(c(sK1_goals_X0))))) 3.85/0.91 = { by lemma 38 } 3.85/0.91 multiplication(sK1_goals_X0, addition(multiplication(one, c(sK1_goals_X0)), multiplication(sK2_goals_X1, c(c(sK1_goals_X0))))) 3.85/0.91 = { by axiom 13 (multiplicative_left_identity) } 3.85/0.91 multiplication(sK1_goals_X0, addition(c(sK1_goals_X0), multiplication(sK2_goals_X1, c(c(sK1_goals_X0))))) 3.85/0.91 = { by lemma 36 } 3.85/0.91 multiplication(sK1_goals_X0, addition(c(sK1_goals_X0), multiplication(sK2_goals_X1, sK1_goals_X0))) 3.85/0.91 = { by axiom 12 (additive_commutativity) } 3.85/0.91 multiplication(sK1_goals_X0, addition(multiplication(sK2_goals_X1, sK1_goals_X0), c(sK1_goals_X0))) 3.85/0.91 = { by lemma 32 } 3.85/0.91 multiplication(sK1_goals_X0, multiplication(sK2_goals_X1, sK1_goals_X0)) 3.85/0.91 = { by axiom 18 (multiplicative_associativity) } 3.85/0.91 multiplication(multiplication(sK1_goals_X0, sK2_goals_X1), sK1_goals_X0) 3.85/0.91 = { by lemma 40 } 3.85/0.91 multiplication(addition(multiplication(sK1_goals_X0, sK2_goals_X1), c(sK1_goals_X0)), sK1_goals_X0) 3.85/0.91 = { by axiom 12 (additive_commutativity) } 3.85/0.91 multiplication(addition(c(sK1_goals_X0), multiplication(sK1_goals_X0, sK2_goals_X1)), sK1_goals_X0) 3.85/0.91 = { by axiom 11 (multiplicative_right_identity) } 3.85/0.91 multiplication(addition(multiplication(c(sK1_goals_X0), one), multiplication(sK1_goals_X0, sK2_goals_X1)), sK1_goals_X0) 3.85/0.91 = { by lemma 38 } 3.85/0.91 multiplication(addition(multiplication(c(sK1_goals_X0), addition(one, sK2_goals_X1)), multiplication(sK1_goals_X0, sK2_goals_X1)), sK1_goals_X0) 3.85/0.91 = { by axiom 14 (right_distributivity) } 3.85/0.91 multiplication(addition(addition(multiplication(c(sK1_goals_X0), one), multiplication(c(sK1_goals_X0), sK2_goals_X1)), multiplication(sK1_goals_X0, sK2_goals_X1)), sK1_goals_X0) 3.85/0.91 = { by axiom 11 (multiplicative_right_identity) } 3.85/0.91 multiplication(addition(addition(c(sK1_goals_X0), multiplication(c(sK1_goals_X0), sK2_goals_X1)), multiplication(sK1_goals_X0, sK2_goals_X1)), sK1_goals_X0) 3.85/0.91 = { by lemma 36 } 3.85/0.91 multiplication(addition(addition(c(sK1_goals_X0), multiplication(c(sK1_goals_X0), sK2_goals_X1)), multiplication(c(c(sK1_goals_X0)), sK2_goals_X1)), sK1_goals_X0) 3.85/0.91 = { by axiom 17 (additive_associativity) } 3.85/0.91 multiplication(addition(c(sK1_goals_X0), addition(multiplication(c(sK1_goals_X0), sK2_goals_X1), multiplication(c(c(sK1_goals_X0)), sK2_goals_X1))), sK1_goals_X0) 3.85/0.91 = { by axiom 12 (additive_commutativity) } 3.85/0.91 multiplication(addition(c(sK1_goals_X0), addition(multiplication(c(c(sK1_goals_X0)), sK2_goals_X1), multiplication(c(sK1_goals_X0), sK2_goals_X1))), sK1_goals_X0) 3.85/0.91 = { by axiom 10 (left_distributivity) } 3.85/0.91 multiplication(addition(c(sK1_goals_X0), multiplication(addition(c(c(sK1_goals_X0)), c(sK1_goals_X0)), sK2_goals_X1)), sK1_goals_X0) 3.85/0.91 = { by axiom 12 (additive_commutativity) } 3.85/0.91 multiplication(addition(c(sK1_goals_X0), multiplication(addition(c(sK1_goals_X0), c(c(sK1_goals_X0))), sK2_goals_X1)), sK1_goals_X0) 3.85/0.91 = { by lemma 39 } 3.85/0.91 multiplication(addition(c(sK1_goals_X0), multiplication(one, sK2_goals_X1)), sK1_goals_X0) 3.85/0.91 = { by axiom 13 (multiplicative_left_identity) } 3.85/0.91 multiplication(addition(c(sK1_goals_X0), sK2_goals_X1), sK1_goals_X0) 3.85/0.91 = { by axiom 12 (additive_commutativity) } 3.85/0.91 multiplication(addition(sK2_goals_X1, c(sK1_goals_X0)), sK1_goals_X0) 3.85/0.91 = { by lemma 40 } 3.85/0.91 multiplication(sK2_goals_X1, sK1_goals_X0) 3.85/0.91 % SZS output end Proof 3.85/0.91 3.85/0.91 RESULT: Theorem (the conjecture is true). 3.85/0.92 EOF