TSTP Solution File: KLE014+3 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE014+3 : 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 : n014.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 05:35:28 EDT 2023
% Result : Theorem 17.91s 2.73s
% Output : Proof 18.71s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : KLE014+3 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.35 % Computer : n014.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 : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Tue Aug 29 11:49:30 EDT 2023
% 0.14/0.35 % CPUTime :
% 17.91/2.73 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 17.91/2.73
% 17.91/2.73 % SZS status Theorem
% 17.91/2.73
% 18.61/2.78 % SZS output start Proof
% 18.61/2.78 Take the following subset of the input axioms:
% 18.61/2.78 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 18.61/2.78 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 18.61/2.78 fof(additive_idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 18.61/2.78 fof(additive_identity, axiom, ![A3]: addition(A3, zero)=A3).
% 18.61/2.78 fof(goals, conjecture, ![X0, X1]: ((test(X1) & test(X0)) => (leq(c(addition(X0, X1)), multiplication(c(X0), c(X1))) & leq(multiplication(c(X0), c(X1)), c(addition(X0, X1)))))).
% 18.61/2.78 fof(left_annihilation, axiom, ![A3]: multiplication(zero, A3)=zero).
% 18.61/2.78 fof(left_distributivity, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 18.61/2.78 fof(multiplicative_associativity, axiom, ![A3, B2, C2]: multiplication(A3, multiplication(B2, C2))=multiplication(multiplication(A3, B2), C2)).
% 18.61/2.78 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 18.61/2.78 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 18.61/2.78 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 18.61/2.78 fof(right_annihilation, axiom, ![A3]: multiplication(A3, zero)=zero).
% 18.61/2.78 fof(right_distributivity, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 18.61/2.78 fof(test_1, axiom, ![X0_2]: (test(X0_2) <=> ?[X1_2]: complement(X1_2, X0_2))).
% 18.61/2.78 fof(test_2, axiom, ![X0_2, X1_2]: (complement(X1_2, X0_2) <=> (multiplication(X0_2, X1_2)=zero & (multiplication(X1_2, X0_2)=zero & addition(X0_2, X1_2)=one)))).
% 18.61/2.78 fof(test_3, axiom, ![X0_2, X1_2]: (test(X0_2) => (c(X0_2)=X1_2 <=> complement(X0_2, X1_2)))).
% 18.61/2.78
% 18.61/2.78 Now clausify the problem and encode Horn clauses using encoding 3 of
% 18.61/2.78 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 18.61/2.78 We repeatedly replace C & s=t => u=v by the two clauses:
% 18.61/2.78 fresh(y, y, x1...xn) = u
% 18.61/2.78 C => fresh(s, t, x1...xn) = v
% 18.61/2.78 where fresh is a fresh function symbol and x1..xn are the free
% 18.61/2.78 variables of u and v.
% 18.61/2.78 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 18.71/2.79 input problem has no model of domain size 1).
% 18.71/2.79
% 18.71/2.79 The encoding turns the above axioms into the following unit equations and goals:
% 18.71/2.79
% 18.71/2.79 Axiom 1 (goals): test(x1) = true.
% 18.71/2.79 Axiom 2 (goals_1): test(x0) = true.
% 18.71/2.79 Axiom 3 (right_annihilation): multiplication(X, zero) = zero.
% 18.71/2.79 Axiom 4 (multiplicative_right_identity): multiplication(X, one) = X.
% 18.71/2.79 Axiom 5 (left_annihilation): multiplication(zero, X) = zero.
% 18.71/2.79 Axiom 6 (multiplicative_left_identity): multiplication(one, X) = X.
% 18.71/2.79 Axiom 7 (additive_idempotence): addition(X, X) = X.
% 18.71/2.79 Axiom 8 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 18.71/2.79 Axiom 9 (additive_identity): addition(X, zero) = X.
% 18.71/2.79 Axiom 10 (test_1): fresh12(X, X, Y) = true.
% 18.71/2.79 Axiom 11 (test_1_1): fresh10(X, X, Y) = true.
% 18.71/2.79 Axiom 12 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 18.71/2.79 Axiom 13 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 18.71/2.79 Axiom 14 (test_2): fresh14(X, X, Y, Z) = true.
% 18.71/2.79 Axiom 15 (test_1): fresh12(test(X), true, X) = complement(x1_2(X), X).
% 18.71/2.79 Axiom 16 (order): fresh11(X, X, Y, Z) = true.
% 18.71/2.79 Axiom 17 (test_2): fresh9(X, X, Y, Z) = complement(Z, Y).
% 18.71/2.79 Axiom 18 (test_2_1): fresh8(X, X, Y, Z) = one.
% 18.71/2.79 Axiom 19 (test_2_2): fresh7(X, X, Y, Z) = zero.
% 18.71/2.79 Axiom 20 (test_2_3): fresh6(X, X, Y, Z) = zero.
% 18.71/2.79 Axiom 21 (test_3): fresh5(X, X, Y, Z) = complement(Y, Z).
% 18.71/2.79 Axiom 22 (test_3): fresh4(X, X, Y, Z) = true.
% 18.71/2.79 Axiom 23 (test_1_1): fresh10(complement(X, Y), true, Y) = test(Y).
% 18.71/2.79 Axiom 24 (test_3): fresh5(test(X), true, X, Y) = fresh4(c(X), Y, X, Y).
% 18.71/2.79 Axiom 25 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 18.71/2.79 Axiom 26 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 18.71/2.79 Axiom 27 (test_2): fresh13(X, X, Y, Z) = fresh14(addition(Y, Z), one, Y, Z).
% 18.71/2.79 Axiom 28 (order): fresh11(addition(X, Y), Y, X, Y) = leq(X, Y).
% 18.71/2.79 Axiom 29 (test_2): fresh13(multiplication(X, Y), zero, Y, X) = fresh9(multiplication(Y, X), zero, Y, X).
% 18.71/2.79 Axiom 30 (test_2_1): fresh8(complement(X, Y), true, Y, X) = addition(Y, X).
% 18.71/2.79 Axiom 31 (test_2_2): fresh7(complement(X, Y), true, Y, X) = multiplication(Y, X).
% 18.71/2.79 Axiom 32 (test_2_3): fresh6(complement(X, Y), true, Y, X) = multiplication(X, Y).
% 18.71/2.79
% 18.71/2.79 Lemma 33: addition(zero, X) = X.
% 18.71/2.79 Proof:
% 18.71/2.79 addition(zero, X)
% 18.71/2.79 = { by axiom 8 (additive_commutativity) R->L }
% 18.71/2.79 addition(X, zero)
% 18.71/2.79 = { by axiom 9 (additive_identity) }
% 18.71/2.79 X
% 18.71/2.79
% 18.71/2.79 Lemma 34: complement(x1, c(x1)) = true.
% 18.71/2.79 Proof:
% 18.71/2.79 complement(x1, c(x1))
% 18.71/2.79 = { by axiom 21 (test_3) R->L }
% 18.71/2.79 fresh5(true, true, x1, c(x1))
% 18.71/2.79 = { by axiom 1 (goals) R->L }
% 18.71/2.79 fresh5(test(x1), true, x1, c(x1))
% 18.71/2.79 = { by axiom 24 (test_3) }
% 18.71/2.79 fresh4(c(x1), c(x1), x1, c(x1))
% 18.71/2.79 = { by axiom 22 (test_3) }
% 18.71/2.79 true
% 18.71/2.79
% 18.71/2.79 Lemma 35: complement(x0, c(x0)) = true.
% 18.71/2.79 Proof:
% 18.71/2.79 complement(x0, c(x0))
% 18.71/2.79 = { by axiom 21 (test_3) R->L }
% 18.71/2.79 fresh5(true, true, x0, c(x0))
% 18.71/2.79 = { by axiom 2 (goals_1) R->L }
% 18.71/2.79 fresh5(test(x0), true, x0, c(x0))
% 18.71/2.79 = { by axiom 24 (test_3) }
% 18.71/2.79 fresh4(c(x0), c(x0), x0, c(x0))
% 18.71/2.79 = { by axiom 22 (test_3) }
% 18.71/2.79 true
% 18.71/2.79
% 18.71/2.79 Lemma 36: addition(x0, c(x0)) = one.
% 18.71/2.79 Proof:
% 18.71/2.79 addition(x0, c(x0))
% 18.71/2.79 = { by axiom 8 (additive_commutativity) R->L }
% 18.71/2.79 addition(c(x0), x0)
% 18.71/2.79 = { by axiom 30 (test_2_1) R->L }
% 18.71/2.79 fresh8(complement(x0, c(x0)), true, c(x0), x0)
% 18.71/2.79 = { by lemma 35 }
% 18.71/2.79 fresh8(true, true, c(x0), x0)
% 18.71/2.79 = { by axiom 18 (test_2_1) }
% 18.71/2.79 one
% 18.71/2.79
% 18.71/2.79 Lemma 37: addition(X, addition(X, Y)) = addition(X, Y).
% 18.71/2.79 Proof:
% 18.71/2.79 addition(X, addition(X, Y))
% 18.71/2.79 = { by axiom 13 (additive_associativity) }
% 18.71/2.79 addition(addition(X, X), Y)
% 18.71/2.79 = { by axiom 7 (additive_idempotence) }
% 18.71/2.79 addition(X, Y)
% 18.71/2.79
% 18.71/2.79 Lemma 38: addition(X, addition(Y, X)) = addition(X, Y).
% 18.71/2.79 Proof:
% 18.71/2.79 addition(X, addition(Y, X))
% 18.71/2.79 = { by axiom 8 (additive_commutativity) R->L }
% 18.71/2.79 addition(X, addition(X, Y))
% 18.71/2.79 = { by lemma 37 }
% 18.71/2.79 addition(X, Y)
% 18.71/2.79
% 18.71/2.79 Lemma 39: addition(one, c(x0)) = one.
% 18.71/2.79 Proof:
% 18.71/2.79 addition(one, c(x0))
% 18.71/2.79 = { by axiom 8 (additive_commutativity) R->L }
% 18.71/2.79 addition(c(x0), one)
% 18.71/2.79 = { by lemma 36 R->L }
% 18.71/2.79 addition(c(x0), addition(x0, c(x0)))
% 18.71/2.79 = { by lemma 38 }
% 18.71/2.79 addition(c(x0), x0)
% 18.71/2.79 = { by axiom 8 (additive_commutativity) }
% 18.71/2.79 addition(x0, c(x0))
% 18.71/2.79 = { by lemma 36 }
% 18.71/2.79 one
% 18.71/2.79
% 18.71/2.79 Lemma 40: addition(x1, c(x1)) = one.
% 18.71/2.79 Proof:
% 18.71/2.79 addition(x1, c(x1))
% 18.71/2.79 = { by axiom 8 (additive_commutativity) R->L }
% 18.71/2.79 addition(c(x1), x1)
% 18.71/2.79 = { by axiom 30 (test_2_1) R->L }
% 18.71/2.79 fresh8(complement(x1, c(x1)), true, c(x1), x1)
% 18.71/2.79 = { by lemma 34 }
% 18.71/2.79 fresh8(true, true, c(x1), x1)
% 18.71/2.79 = { by axiom 18 (test_2_1) }
% 18.71/2.79 one
% 18.71/2.79
% 18.71/2.79 Lemma 41: addition(x1, x1_2(x1)) = one.
% 18.71/2.79 Proof:
% 18.71/2.79 addition(x1, x1_2(x1))
% 18.71/2.79 = { by axiom 30 (test_2_1) R->L }
% 18.71/2.79 fresh8(complement(x1_2(x1), x1), true, x1, x1_2(x1))
% 18.71/2.79 = { by axiom 15 (test_1) R->L }
% 18.71/2.79 fresh8(fresh12(test(x1), true, x1), true, x1, x1_2(x1))
% 18.71/2.79 = { by axiom 1 (goals) }
% 18.71/2.79 fresh8(fresh12(true, true, x1), true, x1, x1_2(x1))
% 18.71/2.79 = { by axiom 10 (test_1) }
% 18.71/2.79 fresh8(true, true, x1, x1_2(x1))
% 18.71/2.79 = { by axiom 18 (test_2_1) }
% 18.71/2.79 one
% 18.71/2.79
% 18.71/2.79 Lemma 42: multiplication(x1, c(x1)) = zero.
% 18.71/2.79 Proof:
% 18.71/2.79 multiplication(x1, c(x1))
% 18.71/2.79 = { by axiom 32 (test_2_3) R->L }
% 18.71/2.79 fresh6(complement(x1, c(x1)), true, c(x1), x1)
% 18.71/2.79 = { by lemma 34 }
% 18.71/2.79 fresh6(true, true, c(x1), x1)
% 18.71/2.79 = { by axiom 20 (test_2_3) }
% 18.71/2.79 zero
% 18.71/2.79
% 18.71/2.79 Lemma 43: multiplication(c(x1), x1) = zero.
% 18.71/2.79 Proof:
% 18.71/2.79 multiplication(c(x1), x1)
% 18.71/2.79 = { by axiom 31 (test_2_2) R->L }
% 18.71/2.79 fresh7(complement(x1, c(x1)), true, c(x1), x1)
% 18.71/2.79 = { by lemma 34 }
% 18.71/2.79 fresh7(true, true, c(x1), x1)
% 18.71/2.79 = { by axiom 19 (test_2_2) }
% 18.71/2.79 zero
% 18.71/2.79
% 18.71/2.79 Lemma 44: multiplication(c(x0), x0) = zero.
% 18.71/2.79 Proof:
% 18.71/2.79 multiplication(c(x0), x0)
% 18.71/2.79 = { by axiom 31 (test_2_2) R->L }
% 18.71/2.79 fresh7(complement(x0, c(x0)), true, c(x0), x0)
% 18.71/2.79 = { by lemma 35 }
% 18.71/2.79 fresh7(true, true, c(x0), x0)
% 18.71/2.79 = { by axiom 19 (test_2_2) }
% 18.71/2.79 zero
% 18.71/2.79
% 18.71/2.79 Lemma 45: addition(X, addition(Y, Z)) = addition(Y, addition(X, Z)).
% 18.71/2.79 Proof:
% 18.71/2.79 addition(X, addition(Y, Z))
% 18.71/2.79 = { by axiom 8 (additive_commutativity) R->L }
% 18.71/2.79 addition(addition(Y, Z), X)
% 18.71/2.79 = { by axiom 13 (additive_associativity) R->L }
% 18.71/2.79 addition(Y, addition(Z, X))
% 18.71/2.79 = { by axiom 8 (additive_commutativity) }
% 18.71/2.79 addition(Y, addition(X, Z))
% 18.71/2.79
% 18.71/2.79 Lemma 46: addition(Y, addition(Z, X)) = addition(X, addition(Y, Z)).
% 18.71/2.79 Proof:
% 18.71/2.79 addition(Y, addition(Z, X))
% 18.71/2.79 = { by axiom 8 (additive_commutativity) R->L }
% 18.71/2.79 addition(addition(Z, X), Y)
% 18.71/2.79 = { by axiom 8 (additive_commutativity) }
% 18.71/2.79 addition(addition(X, Z), Y)
% 18.71/2.79 = { by axiom 13 (additive_associativity) R->L }
% 18.71/2.79 addition(X, addition(Z, Y))
% 18.71/2.79 = { by axiom 8 (additive_commutativity) }
% 18.71/2.79 addition(X, addition(Y, Z))
% 18.71/2.79
% 18.71/2.79 Lemma 47: addition(multiplication(X, Y), multiplication(X, Z)) = multiplication(X, addition(Z, Y)).
% 18.71/2.79 Proof:
% 18.71/2.79 addition(multiplication(X, Y), multiplication(X, Z))
% 18.71/2.79 = { by axiom 25 (right_distributivity) R->L }
% 18.71/2.79 multiplication(X, addition(Y, Z))
% 18.71/2.79 = { by axiom 8 (additive_commutativity) }
% 18.71/2.79 multiplication(X, addition(Z, Y))
% 18.71/2.79
% 18.71/2.79 Lemma 48: addition(multiplication(X, c(x1)), X) = X.
% 18.71/2.79 Proof:
% 18.71/2.79 addition(multiplication(X, c(x1)), X)
% 18.71/2.79 = { by axiom 4 (multiplicative_right_identity) R->L }
% 18.71/2.79 addition(multiplication(X, c(x1)), multiplication(X, one))
% 18.71/2.79 = { by lemma 47 }
% 18.71/2.79 multiplication(X, addition(one, c(x1)))
% 18.71/2.79 = { by axiom 8 (additive_commutativity) R->L }
% 18.71/2.79 multiplication(X, addition(c(x1), one))
% 18.71/2.79 = { by lemma 40 R->L }
% 18.71/2.79 multiplication(X, addition(c(x1), addition(x1, c(x1))))
% 18.71/2.79 = { by lemma 38 }
% 18.71/2.79 multiplication(X, addition(c(x1), x1))
% 18.71/2.79 = { by axiom 8 (additive_commutativity) }
% 18.71/2.79 multiplication(X, addition(x1, c(x1)))
% 18.71/2.79 = { by lemma 40 }
% 18.71/2.79 multiplication(X, one)
% 18.71/2.79 = { by axiom 4 (multiplicative_right_identity) }
% 18.71/2.79 X
% 18.71/2.79
% 18.71/2.79 Lemma 49: multiplication(multiplication(c(x0), c(x1)), x0) = zero.
% 18.71/2.79 Proof:
% 18.71/2.79 multiplication(multiplication(c(x0), c(x1)), x0)
% 18.71/2.79 = { by lemma 33 R->L }
% 18.71/2.79 addition(zero, multiplication(multiplication(c(x0), c(x1)), x0))
% 18.71/2.79 = { by lemma 44 R->L }
% 18.71/2.79 addition(multiplication(c(x0), x0), multiplication(multiplication(c(x0), c(x1)), x0))
% 18.71/2.79 = { by axiom 26 (left_distributivity) R->L }
% 18.71/2.79 multiplication(addition(c(x0), multiplication(c(x0), c(x1))), x0)
% 18.71/2.79 = { by axiom 8 (additive_commutativity) }
% 18.71/2.79 multiplication(addition(multiplication(c(x0), c(x1)), c(x0)), x0)
% 18.71/2.79 = { by lemma 48 }
% 18.71/2.79 multiplication(c(x0), x0)
% 18.71/2.79 = { by lemma 44 }
% 18.71/2.79 zero
% 18.71/2.79
% 18.71/2.79 Lemma 50: addition(addition(x0, x1), multiplication(c(x0), c(x1))) = one.
% 18.71/2.79 Proof:
% 18.71/2.79 addition(addition(x0, x1), multiplication(c(x0), c(x1)))
% 18.71/2.79 = { by axiom 8 (additive_commutativity) R->L }
% 18.71/2.79 addition(multiplication(c(x0), c(x1)), addition(x0, x1))
% 18.71/2.79 = { by axiom 6 (multiplicative_left_identity) R->L }
% 18.71/2.79 addition(multiplication(c(x0), c(x1)), multiplication(one, addition(x0, x1)))
% 18.71/2.79 = { by lemma 39 R->L }
% 18.71/2.79 addition(multiplication(c(x0), c(x1)), multiplication(addition(one, c(x0)), addition(x0, x1)))
% 18.71/2.79 = { by axiom 26 (left_distributivity) }
% 18.71/2.79 addition(multiplication(c(x0), c(x1)), addition(multiplication(one, addition(x0, x1)), multiplication(c(x0), addition(x0, x1))))
% 18.71/2.79 = { by axiom 6 (multiplicative_left_identity) }
% 18.71/2.79 addition(multiplication(c(x0), c(x1)), addition(addition(x0, x1), multiplication(c(x0), addition(x0, x1))))
% 18.71/2.79 = { by axiom 25 (right_distributivity) }
% 18.71/2.79 addition(multiplication(c(x0), c(x1)), addition(addition(x0, x1), addition(multiplication(c(x0), x0), multiplication(c(x0), x1))))
% 18.71/2.79 = { by lemma 44 }
% 18.71/2.79 addition(multiplication(c(x0), c(x1)), addition(addition(x0, x1), addition(zero, multiplication(c(x0), x1))))
% 18.71/2.79 = { by lemma 33 }
% 18.71/2.79 addition(multiplication(c(x0), c(x1)), addition(addition(x0, x1), multiplication(c(x0), x1)))
% 18.71/2.79 = { by lemma 45 R->L }
% 18.71/2.79 addition(addition(x0, x1), addition(multiplication(c(x0), c(x1)), multiplication(c(x0), x1)))
% 18.71/2.79 = { by lemma 47 }
% 18.71/2.79 addition(addition(x0, x1), multiplication(c(x0), addition(x1, c(x1))))
% 18.71/2.79 = { by lemma 40 }
% 18.71/2.79 addition(addition(x0, x1), multiplication(c(x0), one))
% 18.71/2.79 = { by axiom 4 (multiplicative_right_identity) }
% 18.71/2.79 addition(addition(x0, x1), c(x0))
% 18.71/2.79 = { by axiom 8 (additive_commutativity) R->L }
% 18.71/2.79 addition(c(x0), addition(x0, x1))
% 18.71/2.79 = { by lemma 46 }
% 18.71/2.79 addition(x1, addition(c(x0), x0))
% 18.71/2.79 = { by axiom 8 (additive_commutativity) }
% 18.71/2.79 addition(x1, addition(x0, c(x0)))
% 18.71/2.79 = { by lemma 36 }
% 18.71/2.79 addition(x1, one)
% 18.71/2.79 = { by lemma 41 R->L }
% 18.71/2.79 addition(x1, addition(x1, x1_2(x1)))
% 18.71/2.79 = { by lemma 37 }
% 18.71/2.79 addition(x1, x1_2(x1))
% 18.71/2.79 = { by lemma 41 }
% 18.71/2.79 one
% 18.71/2.79
% 18.71/2.79 Lemma 51: multiplication(addition(x0, x1), multiplication(c(x0), c(x1))) = zero.
% 18.71/2.79 Proof:
% 18.71/2.79 multiplication(addition(x0, x1), multiplication(c(x0), c(x1)))
% 18.71/2.79 = { by axiom 26 (left_distributivity) }
% 18.71/2.79 addition(multiplication(x0, multiplication(c(x0), c(x1))), multiplication(x1, multiplication(c(x0), c(x1))))
% 18.71/2.79 = { by axiom 12 (multiplicative_associativity) }
% 18.71/2.79 addition(multiplication(multiplication(x0, c(x0)), c(x1)), multiplication(x1, multiplication(c(x0), c(x1))))
% 18.71/2.79 = { by axiom 32 (test_2_3) R->L }
% 18.71/2.79 addition(multiplication(fresh6(complement(x0, c(x0)), true, c(x0), x0), c(x1)), multiplication(x1, multiplication(c(x0), c(x1))))
% 18.71/2.79 = { by lemma 35 }
% 18.71/2.79 addition(multiplication(fresh6(true, true, c(x0), x0), c(x1)), multiplication(x1, multiplication(c(x0), c(x1))))
% 18.71/2.79 = { by axiom 20 (test_2_3) }
% 18.71/2.79 addition(multiplication(zero, c(x1)), multiplication(x1, multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by axiom 5 (left_annihilation) }
% 18.71/2.80 addition(zero, multiplication(x1, multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by lemma 42 R->L }
% 18.71/2.80 addition(multiplication(x1, c(x1)), multiplication(x1, multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by axiom 25 (right_distributivity) R->L }
% 18.71/2.80 multiplication(x1, addition(c(x1), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by axiom 8 (additive_commutativity) }
% 18.71/2.80 multiplication(x1, addition(multiplication(c(x0), c(x1)), c(x1)))
% 18.71/2.80 = { by axiom 6 (multiplicative_left_identity) R->L }
% 18.71/2.80 multiplication(x1, addition(multiplication(c(x0), c(x1)), multiplication(one, c(x1))))
% 18.71/2.80 = { by axiom 26 (left_distributivity) R->L }
% 18.71/2.80 multiplication(x1, multiplication(addition(c(x0), one), c(x1)))
% 18.71/2.80 = { by axiom 8 (additive_commutativity) }
% 18.71/2.80 multiplication(x1, multiplication(addition(one, c(x0)), c(x1)))
% 18.71/2.80 = { by lemma 39 }
% 18.71/2.80 multiplication(x1, multiplication(one, c(x1)))
% 18.71/2.80 = { by axiom 6 (multiplicative_left_identity) }
% 18.71/2.80 multiplication(x1, c(x1))
% 18.71/2.80 = { by lemma 42 }
% 18.71/2.80 zero
% 18.71/2.80
% 18.71/2.80 Lemma 52: complement(addition(x0, x1), c(addition(x0, x1))) = true.
% 18.71/2.80 Proof:
% 18.71/2.80 complement(addition(x0, x1), c(addition(x0, x1)))
% 18.71/2.80 = { by axiom 21 (test_3) R->L }
% 18.71/2.80 fresh5(true, true, addition(x0, x1), c(addition(x0, x1)))
% 18.71/2.80 = { by axiom 11 (test_1_1) R->L }
% 18.71/2.80 fresh5(fresh10(true, true, addition(x0, x1)), true, addition(x0, x1), c(addition(x0, x1)))
% 18.71/2.80 = { by axiom 14 (test_2) R->L }
% 18.71/2.80 fresh5(fresh10(fresh14(one, one, addition(x0, x1), multiplication(c(x0), c(x1))), true, addition(x0, x1)), true, addition(x0, x1), c(addition(x0, x1)))
% 18.71/2.80 = { by lemma 50 R->L }
% 18.71/2.80 fresh5(fresh10(fresh14(addition(addition(x0, x1), multiplication(c(x0), c(x1))), one, addition(x0, x1), multiplication(c(x0), c(x1))), true, addition(x0, x1)), true, addition(x0, x1), c(addition(x0, x1)))
% 18.71/2.80 = { by axiom 27 (test_2) R->L }
% 18.71/2.80 fresh5(fresh10(fresh13(zero, zero, addition(x0, x1), multiplication(c(x0), c(x1))), true, addition(x0, x1)), true, addition(x0, x1), c(addition(x0, x1)))
% 18.71/2.80 = { by lemma 49 R->L }
% 18.71/2.80 fresh5(fresh10(fresh13(multiplication(multiplication(c(x0), c(x1)), x0), zero, addition(x0, x1), multiplication(c(x0), c(x1))), true, addition(x0, x1)), true, addition(x0, x1), c(addition(x0, x1)))
% 18.71/2.80 = { by axiom 12 (multiplicative_associativity) R->L }
% 18.71/2.80 fresh5(fresh10(fresh13(multiplication(c(x0), multiplication(c(x1), x0)), zero, addition(x0, x1), multiplication(c(x0), c(x1))), true, addition(x0, x1)), true, addition(x0, x1), c(addition(x0, x1)))
% 18.71/2.80 = { by lemma 33 R->L }
% 18.71/2.80 fresh5(fresh10(fresh13(multiplication(c(x0), addition(zero, multiplication(c(x1), x0))), zero, addition(x0, x1), multiplication(c(x0), c(x1))), true, addition(x0, x1)), true, addition(x0, x1), c(addition(x0, x1)))
% 18.71/2.80 = { by lemma 43 R->L }
% 18.71/2.80 fresh5(fresh10(fresh13(multiplication(c(x0), addition(multiplication(c(x1), x1), multiplication(c(x1), x0))), zero, addition(x0, x1), multiplication(c(x0), c(x1))), true, addition(x0, x1)), true, addition(x0, x1), c(addition(x0, x1)))
% 18.71/2.80 = { by axiom 25 (right_distributivity) R->L }
% 18.71/2.80 fresh5(fresh10(fresh13(multiplication(c(x0), multiplication(c(x1), addition(x1, x0))), zero, addition(x0, x1), multiplication(c(x0), c(x1))), true, addition(x0, x1)), true, addition(x0, x1), c(addition(x0, x1)))
% 18.71/2.80 = { by axiom 8 (additive_commutativity) }
% 18.71/2.80 fresh5(fresh10(fresh13(multiplication(c(x0), multiplication(c(x1), addition(x0, x1))), zero, addition(x0, x1), multiplication(c(x0), c(x1))), true, addition(x0, x1)), true, addition(x0, x1), c(addition(x0, x1)))
% 18.71/2.80 = { by axiom 12 (multiplicative_associativity) }
% 18.71/2.80 fresh5(fresh10(fresh13(multiplication(multiplication(c(x0), c(x1)), addition(x0, x1)), zero, addition(x0, x1), multiplication(c(x0), c(x1))), true, addition(x0, x1)), true, addition(x0, x1), c(addition(x0, x1)))
% 18.71/2.80 = { by axiom 29 (test_2) }
% 18.71/2.80 fresh5(fresh10(fresh9(multiplication(addition(x0, x1), multiplication(c(x0), c(x1))), zero, addition(x0, x1), multiplication(c(x0), c(x1))), true, addition(x0, x1)), true, addition(x0, x1), c(addition(x0, x1)))
% 18.71/2.80 = { by lemma 51 }
% 18.71/2.80 fresh5(fresh10(fresh9(zero, zero, addition(x0, x1), multiplication(c(x0), c(x1))), true, addition(x0, x1)), true, addition(x0, x1), c(addition(x0, x1)))
% 18.71/2.80 = { by axiom 17 (test_2) }
% 18.71/2.80 fresh5(fresh10(complement(multiplication(c(x0), c(x1)), addition(x0, x1)), true, addition(x0, x1)), true, addition(x0, x1), c(addition(x0, x1)))
% 18.71/2.80 = { by axiom 23 (test_1_1) }
% 18.71/2.80 fresh5(test(addition(x0, x1)), true, addition(x0, x1), c(addition(x0, x1)))
% 18.71/2.80 = { by axiom 24 (test_3) }
% 18.71/2.80 fresh4(c(addition(x0, x1)), c(addition(x0, x1)), addition(x0, x1), c(addition(x0, x1)))
% 18.71/2.80 = { by axiom 22 (test_3) }
% 18.71/2.80 true
% 18.71/2.80
% 18.71/2.80 Lemma 53: addition(addition(x0, x1), c(addition(x0, x1))) = one.
% 18.71/2.80 Proof:
% 18.71/2.80 addition(addition(x0, x1), c(addition(x0, x1)))
% 18.71/2.80 = { by axiom 8 (additive_commutativity) R->L }
% 18.71/2.80 addition(c(addition(x0, x1)), addition(x0, x1))
% 18.71/2.80 = { by axiom 30 (test_2_1) R->L }
% 18.71/2.80 fresh8(complement(addition(x0, x1), c(addition(x0, x1))), true, c(addition(x0, x1)), addition(x0, x1))
% 18.71/2.80 = { by lemma 52 }
% 18.71/2.80 fresh8(true, true, c(addition(x0, x1)), addition(x0, x1))
% 18.71/2.80 = { by axiom 18 (test_2_1) }
% 18.71/2.80 one
% 18.71/2.80
% 18.71/2.80 Goal 1 (goals_2): tuple(leq(multiplication(c(x0), c(x1)), c(addition(x0, x1))), leq(c(addition(x0, x1)), multiplication(c(x0), c(x1)))) = tuple(true, true).
% 18.71/2.80 Proof:
% 18.71/2.80 tuple(leq(multiplication(c(x0), c(x1)), c(addition(x0, x1))), leq(c(addition(x0, x1)), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by axiom 6 (multiplicative_left_identity) R->L }
% 18.71/2.80 tuple(leq(multiplication(one, multiplication(c(x0), c(x1))), c(addition(x0, x1))), leq(c(addition(x0, x1)), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by lemma 53 R->L }
% 18.71/2.80 tuple(leq(multiplication(addition(addition(x0, x1), c(addition(x0, x1))), multiplication(c(x0), c(x1))), c(addition(x0, x1))), leq(c(addition(x0, x1)), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by axiom 26 (left_distributivity) }
% 18.71/2.80 tuple(leq(addition(multiplication(addition(x0, x1), multiplication(c(x0), c(x1))), multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1)))), c(addition(x0, x1))), leq(c(addition(x0, x1)), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by lemma 51 }
% 18.71/2.80 tuple(leq(addition(zero, multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1)))), c(addition(x0, x1))), leq(c(addition(x0, x1)), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by lemma 33 }
% 18.71/2.80 tuple(leq(multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1))), c(addition(x0, x1))), leq(c(addition(x0, x1)), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by axiom 7 (additive_idempotence) R->L }
% 18.71/2.80 tuple(leq(multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1))), addition(c(addition(x0, x1)), c(addition(x0, x1)))), leq(c(addition(x0, x1)), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by axiom 4 (multiplicative_right_identity) R->L }
% 18.71/2.80 tuple(leq(multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1))), addition(multiplication(c(addition(x0, x1)), one), c(addition(x0, x1)))), leq(c(addition(x0, x1)), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by lemma 36 R->L }
% 18.71/2.80 tuple(leq(multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1))), addition(multiplication(c(addition(x0, x1)), addition(x0, c(x0))), c(addition(x0, x1)))), leq(c(addition(x0, x1)), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by axiom 8 (additive_commutativity) R->L }
% 18.71/2.80 tuple(leq(multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1))), addition(multiplication(c(addition(x0, x1)), addition(c(x0), x0)), c(addition(x0, x1)))), leq(c(addition(x0, x1)), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by lemma 48 R->L }
% 18.71/2.80 tuple(leq(multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1))), addition(multiplication(c(addition(x0, x1)), addition(addition(multiplication(c(x0), c(x1)), c(x0)), x0)), c(addition(x0, x1)))), leq(c(addition(x0, x1)), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by axiom 13 (additive_associativity) R->L }
% 18.71/2.80 tuple(leq(multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1))), addition(multiplication(c(addition(x0, x1)), addition(multiplication(c(x0), c(x1)), addition(c(x0), x0))), c(addition(x0, x1)))), leq(c(addition(x0, x1)), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by axiom 8 (additive_commutativity) }
% 18.71/2.80 tuple(leq(multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1))), addition(multiplication(c(addition(x0, x1)), addition(multiplication(c(x0), c(x1)), addition(x0, c(x0)))), c(addition(x0, x1)))), leq(c(addition(x0, x1)), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by lemma 36 }
% 18.71/2.80 tuple(leq(multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1))), addition(multiplication(c(addition(x0, x1)), addition(multiplication(c(x0), c(x1)), one)), c(addition(x0, x1)))), leq(c(addition(x0, x1)), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by axiom 8 (additive_commutativity) R->L }
% 18.71/2.80 tuple(leq(multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1))), addition(multiplication(c(addition(x0, x1)), addition(one, multiplication(c(x0), c(x1)))), c(addition(x0, x1)))), leq(c(addition(x0, x1)), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by axiom 25 (right_distributivity) }
% 18.71/2.80 tuple(leq(multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1))), addition(addition(multiplication(c(addition(x0, x1)), one), multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1)))), c(addition(x0, x1)))), leq(c(addition(x0, x1)), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by axiom 4 (multiplicative_right_identity) }
% 18.71/2.80 tuple(leq(multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1))), addition(addition(c(addition(x0, x1)), multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1)))), c(addition(x0, x1)))), leq(c(addition(x0, x1)), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by axiom 13 (additive_associativity) R->L }
% 18.71/2.80 tuple(leq(multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1))), addition(c(addition(x0, x1)), addition(multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1))), c(addition(x0, x1))))), leq(c(addition(x0, x1)), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by lemma 45 R->L }
% 18.71/2.80 tuple(leq(multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1))), addition(multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1))), addition(c(addition(x0, x1)), c(addition(x0, x1))))), leq(c(addition(x0, x1)), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by axiom 28 (order) R->L }
% 18.71/2.80 tuple(fresh11(addition(multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1))), addition(multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1))), addition(c(addition(x0, x1)), c(addition(x0, x1))))), addition(multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1))), addition(c(addition(x0, x1)), c(addition(x0, x1)))), multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1))), addition(multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1))), addition(c(addition(x0, x1)), c(addition(x0, x1))))), leq(c(addition(x0, x1)), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by lemma 37 }
% 18.71/2.80 tuple(fresh11(addition(multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1))), addition(c(addition(x0, x1)), c(addition(x0, x1)))), addition(multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1))), addition(c(addition(x0, x1)), c(addition(x0, x1)))), multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1))), addition(multiplication(c(addition(x0, x1)), multiplication(c(x0), c(x1))), addition(c(addition(x0, x1)), c(addition(x0, x1))))), leq(c(addition(x0, x1)), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by axiom 16 (order) }
% 18.71/2.80 tuple(true, leq(c(addition(x0, x1)), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by axiom 6 (multiplicative_left_identity) R->L }
% 18.71/2.80 tuple(true, leq(multiplication(one, c(addition(x0, x1))), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by lemma 50 R->L }
% 18.71/2.80 tuple(true, leq(multiplication(addition(addition(x0, x1), multiplication(c(x0), c(x1))), c(addition(x0, x1))), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by axiom 8 (additive_commutativity) R->L }
% 18.71/2.80 tuple(true, leq(multiplication(addition(multiplication(c(x0), c(x1)), addition(x0, x1)), c(addition(x0, x1))), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by axiom 26 (left_distributivity) }
% 18.71/2.80 tuple(true, leq(addition(multiplication(multiplication(c(x0), c(x1)), c(addition(x0, x1))), multiplication(addition(x0, x1), c(addition(x0, x1)))), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by lemma 33 R->L }
% 18.71/2.80 tuple(true, leq(addition(addition(zero, multiplication(multiplication(c(x0), c(x1)), c(addition(x0, x1)))), multiplication(addition(x0, x1), c(addition(x0, x1)))), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by lemma 49 R->L }
% 18.71/2.80 tuple(true, leq(addition(addition(multiplication(multiplication(c(x0), c(x1)), x0), multiplication(multiplication(c(x0), c(x1)), c(addition(x0, x1)))), multiplication(addition(x0, x1), c(addition(x0, x1)))), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by axiom 25 (right_distributivity) R->L }
% 18.71/2.80 tuple(true, leq(addition(multiplication(multiplication(c(x0), c(x1)), addition(x0, c(addition(x0, x1)))), multiplication(addition(x0, x1), c(addition(x0, x1)))), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by axiom 8 (additive_commutativity) }
% 18.71/2.80 tuple(true, leq(addition(multiplication(multiplication(c(x0), c(x1)), addition(c(addition(x0, x1)), x0)), multiplication(addition(x0, x1), c(addition(x0, x1)))), multiplication(c(x0), c(x1))))
% 18.71/2.80 = { by lemma 33 R->L }
% 18.71/2.80 tuple(true, leq(addition(addition(zero, multiplication(multiplication(c(x0), c(x1)), addition(c(addition(x0, x1)), x0))), multiplication(addition(x0, x1), c(addition(x0, x1)))), multiplication(c(x0), c(x1))))
% 18.71/2.81 = { by axiom 3 (right_annihilation) R->L }
% 18.71/2.81 tuple(true, leq(addition(addition(multiplication(c(x0), zero), multiplication(multiplication(c(x0), c(x1)), addition(c(addition(x0, x1)), x0))), multiplication(addition(x0, x1), c(addition(x0, x1)))), multiplication(c(x0), c(x1))))
% 18.71/2.81 = { by lemma 43 R->L }
% 18.71/2.81 tuple(true, leq(addition(addition(multiplication(c(x0), multiplication(c(x1), x1)), multiplication(multiplication(c(x0), c(x1)), addition(c(addition(x0, x1)), x0))), multiplication(addition(x0, x1), c(addition(x0, x1)))), multiplication(c(x0), c(x1))))
% 18.71/2.81 = { by axiom 12 (multiplicative_associativity) }
% 18.71/2.81 tuple(true, leq(addition(addition(multiplication(multiplication(c(x0), c(x1)), x1), multiplication(multiplication(c(x0), c(x1)), addition(c(addition(x0, x1)), x0))), multiplication(addition(x0, x1), c(addition(x0, x1)))), multiplication(c(x0), c(x1))))
% 18.71/2.81 = { by axiom 25 (right_distributivity) R->L }
% 18.71/2.81 tuple(true, leq(addition(multiplication(multiplication(c(x0), c(x1)), addition(x1, addition(c(addition(x0, x1)), x0))), multiplication(addition(x0, x1), c(addition(x0, x1)))), multiplication(c(x0), c(x1))))
% 18.71/2.81 = { by lemma 46 R->L }
% 18.71/2.81 tuple(true, leq(addition(multiplication(multiplication(c(x0), c(x1)), addition(c(addition(x0, x1)), addition(x0, x1))), multiplication(addition(x0, x1), c(addition(x0, x1)))), multiplication(c(x0), c(x1))))
% 18.71/2.81 = { by axiom 8 (additive_commutativity) }
% 18.71/2.81 tuple(true, leq(addition(multiplication(multiplication(c(x0), c(x1)), addition(addition(x0, x1), c(addition(x0, x1)))), multiplication(addition(x0, x1), c(addition(x0, x1)))), multiplication(c(x0), c(x1))))
% 18.71/2.81 = { by lemma 53 }
% 18.71/2.81 tuple(true, leq(addition(multiplication(multiplication(c(x0), c(x1)), one), multiplication(addition(x0, x1), c(addition(x0, x1)))), multiplication(c(x0), c(x1))))
% 18.71/2.81 = { by axiom 4 (multiplicative_right_identity) }
% 18.71/2.81 tuple(true, leq(addition(multiplication(c(x0), c(x1)), multiplication(addition(x0, x1), c(addition(x0, x1)))), multiplication(c(x0), c(x1))))
% 18.71/2.81 = { by axiom 32 (test_2_3) R->L }
% 18.71/2.81 tuple(true, leq(addition(multiplication(c(x0), c(x1)), fresh6(complement(addition(x0, x1), c(addition(x0, x1))), true, c(addition(x0, x1)), addition(x0, x1))), multiplication(c(x0), c(x1))))
% 18.71/2.81 = { by lemma 52 }
% 18.71/2.81 tuple(true, leq(addition(multiplication(c(x0), c(x1)), fresh6(true, true, c(addition(x0, x1)), addition(x0, x1))), multiplication(c(x0), c(x1))))
% 18.71/2.81 = { by axiom 20 (test_2_3) }
% 18.71/2.81 tuple(true, leq(addition(multiplication(c(x0), c(x1)), zero), multiplication(c(x0), c(x1))))
% 18.71/2.81 = { by axiom 9 (additive_identity) }
% 18.71/2.81 tuple(true, leq(multiplication(c(x0), c(x1)), multiplication(c(x0), c(x1))))
% 18.71/2.81 = { by axiom 28 (order) R->L }
% 18.71/2.81 tuple(true, fresh11(addition(multiplication(c(x0), c(x1)), multiplication(c(x0), c(x1))), multiplication(c(x0), c(x1)), multiplication(c(x0), c(x1)), multiplication(c(x0), c(x1))))
% 18.71/2.81 = { by axiom 7 (additive_idempotence) }
% 18.71/2.81 tuple(true, fresh11(multiplication(c(x0), c(x1)), multiplication(c(x0), c(x1)), multiplication(c(x0), c(x1)), multiplication(c(x0), c(x1))))
% 18.71/2.81 = { by axiom 16 (order) }
% 18.71/2.81 tuple(true, true)
% 18.71/2.81 % SZS output end Proof
% 18.71/2.81
% 18.71/2.81 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------