TSTP Solution File: KLE016+2 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE016+2 : 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 : n027.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 46.03s 6.35s
% Output : Proof 47.13s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : KLE016+2 : 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
% 0.13/0.34 % Computer : n027.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 12:11:08 EDT 2023
% 0.13/0.35 % CPUTime :
% 46.03/6.35 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 46.03/6.35
% 46.03/6.35 % SZS status Theorem
% 46.03/6.35
% 46.96/6.41 % SZS output start Proof
% 46.96/6.41 Take the following subset of the input axioms:
% 46.96/6.41 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 46.96/6.41 fof(additive_commutativity, axiom, ![A2, B2]: addition(A2, B2)=addition(B2, A2)).
% 46.96/6.41 fof(additive_idempotence, axiom, ![A2]: addition(A2, A2)=A2).
% 46.96/6.41 fof(additive_identity, axiom, ![A2]: addition(A2, zero)=A2).
% 46.96/6.41 fof(goals, conjecture, ![X0, X1]: ((test(X1) & test(X0)) => c(multiplication(X0, X1))=addition(c(X0), c(X1)))).
% 46.96/6.41 fof(left_distributivity, axiom, ![A2, B2, C2]: multiplication(addition(A2, B2), C2)=addition(multiplication(A2, C2), multiplication(B2, C2))).
% 46.96/6.41 fof(multiplicative_associativity, axiom, ![A2, B2, C2]: multiplication(A2, multiplication(B2, C2))=multiplication(multiplication(A2, B2), C2)).
% 46.96/6.41 fof(multiplicative_left_identity, axiom, ![A2]: multiplication(one, A2)=A2).
% 46.96/6.41 fof(multiplicative_right_identity, axiom, ![A2]: multiplication(A2, one)=A2).
% 46.96/6.41 fof(right_annihilation, axiom, ![A2]: multiplication(A2, zero)=zero).
% 46.96/6.41 fof(right_distributivity, axiom, ![A2, B2, C2]: multiplication(A2, addition(B2, C2))=addition(multiplication(A2, B2), multiplication(A2, C2))).
% 46.96/6.41 fof(test_1, axiom, ![X0_2]: (test(X0_2) <=> ?[X1_2]: complement(X1_2, X0_2))).
% 46.96/6.41 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)))).
% 46.96/6.41 fof(test_3, axiom, ![X0_2, X1_2]: (test(X0_2) => (c(X0_2)=X1_2 <=> complement(X0_2, X1_2)))).
% 46.96/6.41
% 46.96/6.41 Now clausify the problem and encode Horn clauses using encoding 3 of
% 46.96/6.41 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 46.96/6.41 We repeatedly replace C & s=t => u=v by the two clauses:
% 46.96/6.41 fresh(y, y, x1...xn) = u
% 46.96/6.41 C => fresh(s, t, x1...xn) = v
% 46.96/6.41 where fresh is a fresh function symbol and x1..xn are the free
% 46.96/6.41 variables of u and v.
% 46.96/6.41 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 46.96/6.41 input problem has no model of domain size 1).
% 46.96/6.41
% 46.96/6.41 The encoding turns the above axioms into the following unit equations and goals:
% 46.96/6.41
% 46.96/6.41 Axiom 1 (goals): test(x1) = true.
% 46.96/6.41 Axiom 2 (goals_1): test(x0) = true.
% 46.96/6.41 Axiom 3 (right_annihilation): multiplication(X, zero) = zero.
% 46.96/6.41 Axiom 4 (multiplicative_right_identity): multiplication(X, one) = X.
% 46.96/6.41 Axiom 5 (multiplicative_left_identity): multiplication(one, X) = X.
% 46.96/6.41 Axiom 6 (additive_idempotence): addition(X, X) = X.
% 46.96/6.41 Axiom 7 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 46.96/6.41 Axiom 8 (additive_identity): addition(X, zero) = X.
% 46.96/6.41 Axiom 9 (test_1): fresh12(X, X, Y) = true.
% 46.96/6.41 Axiom 10 (test_1_1): fresh10(X, X, Y) = true.
% 46.96/6.41 Axiom 11 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 46.96/6.41 Axiom 12 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 46.96/6.41 Axiom 13 (test_3_1): fresh(X, X, Y, Z) = Z.
% 46.96/6.41 Axiom 14 (test_2): fresh14(X, X, Y, Z) = true.
% 46.96/6.41 Axiom 15 (test_1): fresh12(test(X), true, X) = complement(x1_2(X), X).
% 46.96/6.41 Axiom 16 (test_2): fresh9(X, X, Y, Z) = complement(Z, Y).
% 46.96/6.41 Axiom 17 (test_2_1): fresh8(X, X, Y, Z) = one.
% 46.96/6.41 Axiom 18 (test_2_2): fresh7(X, X, Y, Z) = zero.
% 46.96/6.41 Axiom 19 (test_2_3): fresh6(X, X, Y, Z) = zero.
% 46.96/6.41 Axiom 20 (test_3): fresh5(X, X, Y, Z) = complement(Y, Z).
% 46.96/6.41 Axiom 21 (test_3): fresh4(X, X, Y, Z) = true.
% 46.96/6.41 Axiom 22 (test_3_1): fresh3(X, X, Y, Z) = c(Y).
% 46.96/6.41 Axiom 23 (test_1_1): fresh10(complement(X, Y), true, Y) = test(Y).
% 46.96/6.41 Axiom 24 (test_3): fresh5(test(X), true, X, Y) = fresh4(c(X), Y, X, Y).
% 46.96/6.41 Axiom 25 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 46.96/6.41 Axiom 26 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 46.96/6.41 Axiom 27 (test_2): fresh13(X, X, Y, Z) = fresh14(addition(Y, Z), one, Y, Z).
% 46.96/6.41 Axiom 28 (test_2): fresh13(multiplication(X, Y), zero, Y, X) = fresh9(multiplication(Y, X), zero, Y, X).
% 46.96/6.41 Axiom 29 (test_2_1): fresh8(complement(X, Y), true, Y, X) = addition(Y, X).
% 46.96/6.41 Axiom 30 (test_2_2): fresh7(complement(X, Y), true, Y, X) = multiplication(Y, X).
% 46.96/6.41 Axiom 31 (test_2_3): fresh6(complement(X, Y), true, Y, X) = multiplication(X, Y).
% 46.96/6.41 Axiom 32 (test_3_1): fresh3(complement(X, Y), true, X, Y) = fresh(test(X), true, X, Y).
% 46.96/6.41
% 46.96/6.41 Lemma 33: complement(x1_2(x1), x1) = true.
% 46.96/6.41 Proof:
% 46.96/6.41 complement(x1_2(x1), x1)
% 46.96/6.41 = { by axiom 15 (test_1) R->L }
% 46.96/6.41 fresh12(test(x1), true, x1)
% 46.96/6.41 = { by axiom 1 (goals) }
% 46.96/6.41 fresh12(true, true, x1)
% 46.96/6.41 = { by axiom 9 (test_1) }
% 46.96/6.41 true
% 46.96/6.41
% 46.96/6.41 Lemma 34: multiplication(x1_2(x1), x1) = zero.
% 46.96/6.41 Proof:
% 46.96/6.41 multiplication(x1_2(x1), x1)
% 46.96/6.41 = { by axiom 31 (test_2_3) R->L }
% 46.96/6.41 fresh6(complement(x1_2(x1), x1), true, x1, x1_2(x1))
% 46.96/6.41 = { by lemma 33 }
% 46.96/6.41 fresh6(true, true, x1, x1_2(x1))
% 46.96/6.41 = { by axiom 19 (test_2_3) }
% 46.96/6.41 zero
% 46.96/6.41
% 46.96/6.41 Lemma 35: multiplication(x1, x1_2(x1)) = zero.
% 46.96/6.41 Proof:
% 46.96/6.41 multiplication(x1, x1_2(x1))
% 46.96/6.41 = { by axiom 30 (test_2_2) R->L }
% 46.96/6.41 fresh7(complement(x1_2(x1), x1), true, x1, x1_2(x1))
% 46.96/6.41 = { by lemma 33 }
% 46.96/6.41 fresh7(true, true, x1, x1_2(x1))
% 46.96/6.41 = { by axiom 18 (test_2_2) }
% 46.96/6.41 zero
% 46.96/6.41
% 46.96/6.41 Lemma 36: fresh14(addition(X, Y), one, Y, X) = fresh13(Z, Z, Y, X).
% 46.96/6.41 Proof:
% 46.96/6.41 fresh14(addition(X, Y), one, Y, X)
% 46.96/6.41 = { by axiom 7 (additive_commutativity) R->L }
% 46.96/6.41 fresh14(addition(Y, X), one, Y, X)
% 46.96/6.41 = { by axiom 27 (test_2) R->L }
% 46.96/6.41 fresh13(Z, Z, Y, X)
% 46.96/6.41
% 46.96/6.41 Lemma 37: addition(x1, x1_2(x1)) = one.
% 46.96/6.41 Proof:
% 46.96/6.41 addition(x1, x1_2(x1))
% 46.96/6.41 = { by axiom 29 (test_2_1) R->L }
% 46.96/6.41 fresh8(complement(x1_2(x1), x1), true, x1, x1_2(x1))
% 46.96/6.41 = { by lemma 33 }
% 46.96/6.41 fresh8(true, true, x1, x1_2(x1))
% 46.96/6.41 = { by axiom 17 (test_2_1) }
% 46.96/6.42 one
% 46.96/6.42
% 46.96/6.42 Lemma 38: x1_2(x1) = c(x1).
% 46.96/6.42 Proof:
% 46.96/6.42 x1_2(x1)
% 46.96/6.42 = { by axiom 13 (test_3_1) R->L }
% 46.96/6.42 fresh(true, true, x1, x1_2(x1))
% 46.96/6.42 = { by axiom 1 (goals) R->L }
% 46.96/6.42 fresh(test(x1), true, x1, x1_2(x1))
% 46.96/6.42 = { by axiom 32 (test_3_1) R->L }
% 46.96/6.42 fresh3(complement(x1, x1_2(x1)), true, x1, x1_2(x1))
% 46.96/6.42 = { by axiom 16 (test_2) R->L }
% 46.96/6.42 fresh3(fresh9(zero, zero, x1_2(x1), x1), true, x1, x1_2(x1))
% 46.96/6.42 = { by lemma 34 R->L }
% 46.96/6.42 fresh3(fresh9(multiplication(x1_2(x1), x1), zero, x1_2(x1), x1), true, x1, x1_2(x1))
% 46.96/6.42 = { by axiom 28 (test_2) R->L }
% 46.96/6.42 fresh3(fresh13(multiplication(x1, x1_2(x1)), zero, x1_2(x1), x1), true, x1, x1_2(x1))
% 46.96/6.42 = { by lemma 35 }
% 46.96/6.42 fresh3(fresh13(zero, zero, x1_2(x1), x1), true, x1, x1_2(x1))
% 46.96/6.42 = { by lemma 36 R->L }
% 46.96/6.42 fresh3(fresh14(addition(x1, x1_2(x1)), one, x1_2(x1), x1), true, x1, x1_2(x1))
% 46.96/6.42 = { by lemma 37 }
% 46.96/6.42 fresh3(fresh14(one, one, x1_2(x1), x1), true, x1, x1_2(x1))
% 46.96/6.42 = { by axiom 14 (test_2) }
% 46.96/6.42 fresh3(true, true, x1, x1_2(x1))
% 46.96/6.42 = { by axiom 22 (test_3_1) }
% 46.96/6.42 c(x1)
% 46.96/6.42
% 46.96/6.42 Lemma 39: complement(x1_2(x0), x0) = true.
% 46.96/6.42 Proof:
% 46.96/6.42 complement(x1_2(x0), x0)
% 46.96/6.42 = { by axiom 15 (test_1) R->L }
% 46.96/6.42 fresh12(test(x0), true, x0)
% 46.96/6.42 = { by axiom 2 (goals_1) }
% 46.96/6.42 fresh12(true, true, x0)
% 46.96/6.42 = { by axiom 9 (test_1) }
% 46.96/6.42 true
% 46.96/6.42
% 46.96/6.42 Lemma 40: multiplication(x1_2(x0), x0) = zero.
% 46.96/6.42 Proof:
% 46.96/6.42 multiplication(x1_2(x0), x0)
% 46.96/6.42 = { by axiom 31 (test_2_3) R->L }
% 46.96/6.42 fresh6(complement(x1_2(x0), x0), true, x0, x1_2(x0))
% 46.96/6.42 = { by lemma 39 }
% 46.96/6.42 fresh6(true, true, x0, x1_2(x0))
% 46.96/6.42 = { by axiom 19 (test_2_3) }
% 46.96/6.42 zero
% 46.96/6.42
% 46.96/6.42 Lemma 41: multiplication(x0, x1_2(x0)) = zero.
% 46.96/6.42 Proof:
% 46.96/6.42 multiplication(x0, x1_2(x0))
% 46.96/6.42 = { by axiom 30 (test_2_2) R->L }
% 46.96/6.42 fresh7(complement(x1_2(x0), x0), true, x0, x1_2(x0))
% 46.96/6.42 = { by lemma 39 }
% 46.96/6.42 fresh7(true, true, x0, x1_2(x0))
% 46.96/6.42 = { by axiom 18 (test_2_2) }
% 46.96/6.42 zero
% 46.96/6.42
% 46.96/6.42 Lemma 42: addition(x0, x1_2(x0)) = one.
% 46.96/6.42 Proof:
% 46.96/6.42 addition(x0, x1_2(x0))
% 46.96/6.42 = { by axiom 29 (test_2_1) R->L }
% 46.96/6.42 fresh8(complement(x1_2(x0), x0), true, x0, x1_2(x0))
% 46.96/6.42 = { by lemma 39 }
% 46.96/6.42 fresh8(true, true, x0, x1_2(x0))
% 46.96/6.42 = { by axiom 17 (test_2_1) }
% 46.96/6.42 one
% 46.96/6.42
% 46.96/6.42 Lemma 43: x1_2(x0) = c(x0).
% 46.96/6.42 Proof:
% 46.96/6.42 x1_2(x0)
% 46.96/6.42 = { by axiom 13 (test_3_1) R->L }
% 46.96/6.42 fresh(true, true, x0, x1_2(x0))
% 46.96/6.42 = { by axiom 2 (goals_1) R->L }
% 46.96/6.42 fresh(test(x0), true, x0, x1_2(x0))
% 46.96/6.42 = { by axiom 32 (test_3_1) R->L }
% 46.96/6.42 fresh3(complement(x0, x1_2(x0)), true, x0, x1_2(x0))
% 46.96/6.42 = { by axiom 16 (test_2) R->L }
% 46.96/6.42 fresh3(fresh9(zero, zero, x1_2(x0), x0), true, x0, x1_2(x0))
% 46.96/6.42 = { by lemma 40 R->L }
% 46.96/6.42 fresh3(fresh9(multiplication(x1_2(x0), x0), zero, x1_2(x0), x0), true, x0, x1_2(x0))
% 46.96/6.42 = { by axiom 28 (test_2) R->L }
% 46.96/6.42 fresh3(fresh13(multiplication(x0, x1_2(x0)), zero, x1_2(x0), x0), true, x0, x1_2(x0))
% 46.96/6.42 = { by lemma 41 }
% 46.96/6.42 fresh3(fresh13(zero, zero, x1_2(x0), x0), true, x0, x1_2(x0))
% 46.96/6.42 = { by lemma 36 R->L }
% 46.96/6.42 fresh3(fresh14(addition(x0, x1_2(x0)), one, x1_2(x0), x0), true, x0, x1_2(x0))
% 46.96/6.42 = { by lemma 42 }
% 46.96/6.42 fresh3(fresh14(one, one, x1_2(x0), x0), true, x0, x1_2(x0))
% 46.96/6.42 = { by axiom 14 (test_2) }
% 46.96/6.42 fresh3(true, true, x0, x1_2(x0))
% 46.96/6.42 = { by axiom 22 (test_3_1) }
% 46.96/6.42 c(x0)
% 46.96/6.42
% 46.96/6.42 Lemma 44: addition(zero, X) = X.
% 46.96/6.42 Proof:
% 46.96/6.42 addition(zero, X)
% 46.96/6.42 = { by axiom 7 (additive_commutativity) R->L }
% 46.96/6.42 addition(X, zero)
% 46.96/6.42 = { by axiom 8 (additive_identity) }
% 46.96/6.42 X
% 46.96/6.42
% 46.96/6.42 Lemma 45: addition(X, addition(X, Y)) = addition(X, Y).
% 46.96/6.42 Proof:
% 46.96/6.42 addition(X, addition(X, Y))
% 46.96/6.42 = { by axiom 12 (additive_associativity) }
% 46.96/6.42 addition(addition(X, X), Y)
% 46.96/6.42 = { by axiom 6 (additive_idempotence) }
% 46.96/6.42 addition(X, Y)
% 46.96/6.42
% 46.96/6.42 Lemma 46: addition(x1, one) = one.
% 46.96/6.42 Proof:
% 46.96/6.42 addition(x1, one)
% 46.96/6.42 = { by lemma 37 R->L }
% 46.96/6.42 addition(x1, addition(x1, x1_2(x1)))
% 46.96/6.42 = { by lemma 45 }
% 46.96/6.42 addition(x1, x1_2(x1))
% 46.96/6.42 = { by lemma 37 }
% 46.96/6.42 one
% 46.96/6.42
% 46.96/6.42 Lemma 47: addition(x0, one) = one.
% 46.96/6.42 Proof:
% 46.96/6.42 addition(x0, one)
% 46.96/6.42 = { by lemma 42 R->L }
% 46.96/6.42 addition(x0, addition(x0, x1_2(x0)))
% 46.96/6.42 = { by lemma 45 }
% 46.96/6.42 addition(x0, x1_2(x0))
% 46.96/6.42 = { by lemma 42 }
% 46.96/6.42 one
% 46.96/6.42
% 46.96/6.42 Lemma 48: complement(x0, c(x0)) = true.
% 46.96/6.42 Proof:
% 46.96/6.42 complement(x0, c(x0))
% 46.96/6.42 = { by axiom 20 (test_3) R->L }
% 46.96/6.42 fresh5(true, true, x0, c(x0))
% 46.96/6.42 = { by axiom 2 (goals_1) R->L }
% 46.96/6.42 fresh5(test(x0), true, x0, c(x0))
% 46.96/6.42 = { by axiom 24 (test_3) }
% 46.96/6.42 fresh4(c(x0), c(x0), x0, c(x0))
% 46.96/6.42 = { by axiom 21 (test_3) }
% 46.96/6.42 true
% 46.96/6.42
% 46.96/6.42 Lemma 49: addition(x0, c(x0)) = one.
% 46.96/6.42 Proof:
% 46.96/6.42 addition(x0, c(x0))
% 46.96/6.42 = { by axiom 7 (additive_commutativity) R->L }
% 46.96/6.42 addition(c(x0), x0)
% 46.96/6.42 = { by axiom 29 (test_2_1) R->L }
% 46.96/6.42 fresh8(complement(x0, c(x0)), true, c(x0), x0)
% 46.96/6.42 = { by lemma 48 }
% 46.96/6.42 fresh8(true, true, c(x0), x0)
% 46.96/6.42 = { by axiom 17 (test_2_1) }
% 46.96/6.42 one
% 46.96/6.42
% 46.96/6.42 Lemma 50: addition(X, multiplication(Y, X)) = multiplication(addition(Y, one), X).
% 46.96/6.42 Proof:
% 46.96/6.42 addition(X, multiplication(Y, X))
% 46.96/6.42 = { by axiom 5 (multiplicative_left_identity) R->L }
% 46.96/6.42 addition(multiplication(one, X), multiplication(Y, X))
% 46.96/6.42 = { by axiom 26 (left_distributivity) R->L }
% 46.96/6.42 multiplication(addition(one, Y), X)
% 46.96/6.42 = { by axiom 7 (additive_commutativity) }
% 46.96/6.42 multiplication(addition(Y, one), X)
% 46.96/6.42
% 46.96/6.42 Lemma 51: multiplication(x0, c(x0)) = zero.
% 46.96/6.42 Proof:
% 46.96/6.42 multiplication(x0, c(x0))
% 46.96/6.42 = { by axiom 31 (test_2_3) R->L }
% 46.96/6.42 fresh6(complement(x0, c(x0)), true, c(x0), x0)
% 46.96/6.42 = { by lemma 48 }
% 46.96/6.42 fresh6(true, true, c(x0), x0)
% 46.96/6.42 = { by axiom 19 (test_2_3) }
% 46.96/6.42 zero
% 46.96/6.42
% 46.96/6.42 Lemma 52: addition(X, multiplication(X, Y)) = multiplication(X, addition(Y, one)).
% 46.96/6.42 Proof:
% 46.96/6.42 addition(X, multiplication(X, Y))
% 46.96/6.42 = { by axiom 4 (multiplicative_right_identity) R->L }
% 46.96/6.42 addition(multiplication(X, one), multiplication(X, Y))
% 46.96/6.42 = { by axiom 25 (right_distributivity) R->L }
% 46.96/6.42 multiplication(X, addition(one, Y))
% 46.96/6.42 = { by axiom 7 (additive_commutativity) }
% 46.96/6.42 multiplication(X, addition(Y, one))
% 46.96/6.42
% 46.96/6.42 Lemma 53: addition(X, multiplication(addition(Y, X), x1)) = addition(X, multiplication(Y, x1)).
% 46.96/6.42 Proof:
% 46.96/6.42 addition(X, multiplication(addition(Y, X), x1))
% 46.96/6.42 = { by axiom 26 (left_distributivity) }
% 46.96/6.42 addition(X, addition(multiplication(Y, x1), multiplication(X, x1)))
% 46.96/6.42 = { by axiom 7 (additive_commutativity) R->L }
% 46.96/6.42 addition(X, addition(multiplication(X, x1), multiplication(Y, x1)))
% 46.96/6.42 = { by axiom 12 (additive_associativity) }
% 46.96/6.42 addition(addition(X, multiplication(X, x1)), multiplication(Y, x1))
% 46.96/6.42 = { by lemma 52 }
% 46.96/6.42 addition(multiplication(X, addition(x1, one)), multiplication(Y, x1))
% 46.96/6.42 = { by lemma 46 }
% 46.96/6.42 addition(multiplication(X, one), multiplication(Y, x1))
% 46.96/6.42 = { by axiom 4 (multiplicative_right_identity) }
% 46.96/6.42 addition(X, multiplication(Y, x1))
% 46.96/6.42
% 46.96/6.42 Lemma 54: addition(multiplication(X, multiplication(Y, Z)), multiplication(W, Z)) = multiplication(addition(W, multiplication(X, Y)), Z).
% 46.96/6.42 Proof:
% 46.96/6.42 addition(multiplication(X, multiplication(Y, Z)), multiplication(W, Z))
% 46.96/6.42 = { by axiom 11 (multiplicative_associativity) }
% 46.96/6.42 addition(multiplication(multiplication(X, Y), Z), multiplication(W, Z))
% 46.96/6.42 = { by axiom 26 (left_distributivity) R->L }
% 46.96/6.42 multiplication(addition(multiplication(X, Y), W), Z)
% 46.96/6.42 = { by axiom 7 (additive_commutativity) }
% 46.96/6.42 multiplication(addition(W, multiplication(X, Y)), Z)
% 46.96/6.42
% 46.96/6.42 Lemma 55: multiplication(x0, x1) = multiplication(x1, x0).
% 46.96/6.42 Proof:
% 46.96/6.42 multiplication(x0, x1)
% 46.96/6.42 = { by axiom 4 (multiplicative_right_identity) R->L }
% 46.96/6.42 multiplication(multiplication(x0, x1), one)
% 46.96/6.42 = { by lemma 49 R->L }
% 46.96/6.42 multiplication(multiplication(x0, x1), addition(x0, c(x0)))
% 46.96/6.42 = { by axiom 7 (additive_commutativity) R->L }
% 46.96/6.42 multiplication(multiplication(x0, x1), addition(c(x0), x0))
% 46.96/6.42 = { by axiom 25 (right_distributivity) }
% 46.96/6.42 addition(multiplication(multiplication(x0, x1), c(x0)), multiplication(multiplication(x0, x1), x0))
% 46.96/6.42 = { by lemma 44 R->L }
% 46.96/6.42 addition(addition(zero, multiplication(multiplication(x0, x1), c(x0))), multiplication(multiplication(x0, x1), x0))
% 46.96/6.42 = { by lemma 41 R->L }
% 46.96/6.42 addition(addition(multiplication(x0, x1_2(x0)), multiplication(multiplication(x0, x1), c(x0))), multiplication(multiplication(x0, x1), x0))
% 46.96/6.42 = { by axiom 7 (additive_commutativity) R->L }
% 46.96/6.42 addition(addition(multiplication(multiplication(x0, x1), c(x0)), multiplication(x0, x1_2(x0))), multiplication(multiplication(x0, x1), x0))
% 46.96/6.42 = { by axiom 11 (multiplicative_associativity) R->L }
% 46.96/6.42 addition(addition(multiplication(x0, multiplication(x1, c(x0))), multiplication(x0, x1_2(x0))), multiplication(multiplication(x0, x1), x0))
% 46.96/6.42 = { by axiom 25 (right_distributivity) R->L }
% 46.96/6.42 addition(multiplication(x0, addition(multiplication(x1, c(x0)), x1_2(x0))), multiplication(multiplication(x0, x1), x0))
% 46.96/6.42 = { by axiom 7 (additive_commutativity) }
% 46.96/6.42 addition(multiplication(x0, addition(x1_2(x0), multiplication(x1, c(x0)))), multiplication(multiplication(x0, x1), x0))
% 46.96/6.42 = { by lemma 43 }
% 46.96/6.42 addition(multiplication(x0, addition(c(x0), multiplication(x1, c(x0)))), multiplication(multiplication(x0, x1), x0))
% 46.96/6.42 = { by lemma 50 }
% 46.96/6.42 addition(multiplication(x0, multiplication(addition(x1, one), c(x0))), multiplication(multiplication(x0, x1), x0))
% 46.96/6.42 = { by lemma 46 }
% 46.96/6.42 addition(multiplication(x0, multiplication(one, c(x0))), multiplication(multiplication(x0, x1), x0))
% 46.96/6.42 = { by axiom 5 (multiplicative_left_identity) }
% 46.96/6.42 addition(multiplication(x0, c(x0)), multiplication(multiplication(x0, x1), x0))
% 46.96/6.42 = { by lemma 51 }
% 46.96/6.42 addition(zero, multiplication(multiplication(x0, x1), x0))
% 46.96/6.42 = { by axiom 18 (test_2_2) R->L }
% 46.96/6.42 addition(fresh7(true, true, c(x0), x0), multiplication(multiplication(x0, x1), x0))
% 46.96/6.42 = { by lemma 48 R->L }
% 46.96/6.42 addition(fresh7(complement(x0, c(x0)), true, c(x0), x0), multiplication(multiplication(x0, x1), x0))
% 46.96/6.42 = { by axiom 30 (test_2_2) }
% 46.96/6.42 addition(multiplication(c(x0), x0), multiplication(multiplication(x0, x1), x0))
% 46.96/6.42 = { by axiom 26 (left_distributivity) R->L }
% 46.96/6.42 multiplication(addition(c(x0), multiplication(x0, x1)), x0)
% 46.96/6.42 = { by lemma 53 R->L }
% 46.96/6.42 multiplication(addition(c(x0), multiplication(addition(x0, c(x0)), x1)), x0)
% 46.96/6.42 = { by lemma 43 R->L }
% 46.96/6.42 multiplication(addition(x1_2(x0), multiplication(addition(x0, c(x0)), x1)), x0)
% 46.96/6.42 = { by lemma 54 R->L }
% 46.96/6.42 addition(multiplication(addition(x0, c(x0)), multiplication(x1, x0)), multiplication(x1_2(x0), x0))
% 46.96/6.42 = { by lemma 40 }
% 46.96/6.42 addition(multiplication(addition(x0, c(x0)), multiplication(x1, x0)), zero)
% 46.96/6.42 = { by axiom 8 (additive_identity) }
% 46.96/6.42 multiplication(addition(x0, c(x0)), multiplication(x1, x0))
% 46.96/6.42 = { by lemma 49 }
% 46.96/6.42 multiplication(one, multiplication(x1, x0))
% 46.96/6.42 = { by axiom 5 (multiplicative_left_identity) }
% 46.96/6.42 multiplication(x1, x0)
% 46.96/6.42
% 46.96/6.42 Lemma 56: complement(x1, c(x1)) = true.
% 46.96/6.42 Proof:
% 46.96/6.42 complement(x1, c(x1))
% 46.96/6.42 = { by axiom 20 (test_3) R->L }
% 46.96/6.42 fresh5(true, true, x1, c(x1))
% 46.96/6.42 = { by axiom 1 (goals) R->L }
% 46.96/6.42 fresh5(test(x1), true, x1, c(x1))
% 46.96/6.42 = { by axiom 24 (test_3) }
% 46.96/6.42 fresh4(c(x1), c(x1), x1, c(x1))
% 46.96/6.42 = { by axiom 21 (test_3) }
% 46.96/6.42 true
% 46.96/6.42
% 46.96/6.42 Lemma 57: addition(x1, c(x1)) = one.
% 46.96/6.42 Proof:
% 46.96/6.42 addition(x1, c(x1))
% 46.96/6.42 = { by axiom 7 (additive_commutativity) R->L }
% 46.96/6.42 addition(c(x1), x1)
% 46.96/6.42 = { by axiom 29 (test_2_1) R->L }
% 46.96/6.42 fresh8(complement(x1, c(x1)), true, c(x1), x1)
% 46.96/6.42 = { by lemma 56 }
% 46.96/6.42 fresh8(true, true, c(x1), x1)
% 46.96/6.42 = { by axiom 17 (test_2_1) }
% 46.96/6.42 one
% 46.96/6.42
% 46.96/6.42 Lemma 58: multiplication(x1, c(x1)) = zero.
% 46.96/6.42 Proof:
% 46.96/6.42 multiplication(x1, c(x1))
% 46.96/6.42 = { by axiom 31 (test_2_3) R->L }
% 46.96/6.42 fresh6(complement(x1, c(x1)), true, c(x1), x1)
% 46.96/6.42 = { by lemma 56 }
% 46.96/6.42 fresh6(true, true, c(x1), x1)
% 46.96/6.42 = { by axiom 19 (test_2_3) }
% 46.96/6.42 zero
% 46.96/6.42
% 46.96/6.42 Lemma 59: multiplication(c(x1), x1) = zero.
% 46.96/6.42 Proof:
% 46.96/6.42 multiplication(c(x1), x1)
% 46.96/6.42 = { by axiom 30 (test_2_2) R->L }
% 46.96/6.42 fresh7(complement(x1, c(x1)), true, c(x1), x1)
% 46.96/6.42 = { by lemma 56 }
% 46.96/6.42 fresh7(true, true, c(x1), x1)
% 46.96/6.42 = { by axiom 18 (test_2_2) }
% 46.96/6.43 zero
% 46.96/6.43
% 46.96/6.43 Lemma 60: addition(Y, addition(Z, X)) = addition(X, addition(Y, Z)).
% 46.96/6.43 Proof:
% 46.96/6.43 addition(Y, addition(Z, X))
% 46.96/6.43 = { by axiom 7 (additive_commutativity) R->L }
% 46.96/6.43 addition(addition(Z, X), Y)
% 46.96/6.43 = { by axiom 7 (additive_commutativity) }
% 46.96/6.43 addition(addition(X, Z), Y)
% 46.96/6.43 = { by axiom 12 (additive_associativity) R->L }
% 46.96/6.43 addition(X, addition(Z, Y))
% 46.96/6.43 = { by axiom 7 (additive_commutativity) }
% 46.96/6.43 addition(X, addition(Y, Z))
% 46.96/6.43
% 46.96/6.43 Lemma 61: addition(Z, addition(Y, X)) = addition(X, addition(Y, Z)).
% 46.96/6.43 Proof:
% 46.96/6.43 addition(Z, addition(Y, X))
% 46.96/6.43 = { by lemma 60 }
% 46.96/6.43 addition(X, addition(Z, Y))
% 46.96/6.43 = { by axiom 7 (additive_commutativity) }
% 46.96/6.43 addition(X, addition(Y, Z))
% 46.96/6.43
% 46.96/6.43 Lemma 62: addition(addition(X, Y), Y) = addition(X, Y).
% 46.96/6.43 Proof:
% 46.96/6.43 addition(addition(X, Y), Y)
% 46.96/6.43 = { by axiom 7 (additive_commutativity) R->L }
% 46.96/6.43 addition(Y, addition(X, Y))
% 46.96/6.43 = { by axiom 7 (additive_commutativity) }
% 46.96/6.43 addition(Y, addition(Y, X))
% 46.96/6.43 = { by lemma 45 }
% 46.96/6.43 addition(Y, X)
% 46.96/6.43 = { by axiom 7 (additive_commutativity) R->L }
% 47.13/6.43 addition(X, Y)
% 47.13/6.43
% 47.13/6.43 Lemma 63: addition(addition(c(x0), c(x1)), multiplication(x0, x1)) = one.
% 47.13/6.43 Proof:
% 47.13/6.43 addition(addition(c(x0), c(x1)), multiplication(x0, x1))
% 47.13/6.43 = { by axiom 7 (additive_commutativity) R->L }
% 47.13/6.43 addition(multiplication(x0, x1), addition(c(x0), c(x1)))
% 47.13/6.43 = { by lemma 61 }
% 47.13/6.43 addition(c(x1), addition(c(x0), multiplication(x0, x1)))
% 47.13/6.43 = { by lemma 53 R->L }
% 47.13/6.43 addition(c(x1), addition(c(x0), multiplication(addition(x0, c(x0)), x1)))
% 47.13/6.43 = { by lemma 61 R->L }
% 47.13/6.43 addition(multiplication(addition(x0, c(x0)), x1), addition(c(x0), c(x1)))
% 47.13/6.43 = { by lemma 49 }
% 47.13/6.43 addition(multiplication(one, x1), addition(c(x0), c(x1)))
% 47.13/6.43 = { by axiom 5 (multiplicative_left_identity) }
% 47.13/6.43 addition(x1, addition(c(x0), c(x1)))
% 47.13/6.43 = { by axiom 7 (additive_commutativity) R->L }
% 47.13/6.43 addition(addition(c(x0), c(x1)), x1)
% 47.13/6.43 = { by lemma 62 R->L }
% 47.13/6.43 addition(addition(addition(c(x0), c(x1)), c(x1)), x1)
% 47.13/6.43 = { by axiom 12 (additive_associativity) R->L }
% 47.13/6.43 addition(addition(c(x0), c(x1)), addition(c(x1), x1))
% 47.13/6.43 = { by axiom 7 (additive_commutativity) }
% 47.13/6.43 addition(addition(c(x0), c(x1)), addition(x1, c(x1)))
% 47.13/6.43 = { by lemma 57 }
% 47.13/6.43 addition(addition(c(x0), c(x1)), one)
% 47.13/6.43 = { by lemma 49 R->L }
% 47.13/6.43 addition(addition(c(x0), c(x1)), addition(x0, c(x0)))
% 47.13/6.43 = { by axiom 7 (additive_commutativity) R->L }
% 47.13/6.43 addition(addition(c(x0), c(x1)), addition(c(x0), x0))
% 47.13/6.43 = { by axiom 12 (additive_associativity) }
% 47.13/6.43 addition(addition(addition(c(x0), c(x1)), c(x0)), x0)
% 47.13/6.43 = { by axiom 7 (additive_commutativity) R->L }
% 47.13/6.43 addition(addition(c(x0), addition(c(x0), c(x1))), x0)
% 47.13/6.43 = { by lemma 60 }
% 47.13/6.43 addition(addition(c(x1), addition(c(x0), c(x0))), x0)
% 47.13/6.43 = { by axiom 6 (additive_idempotence) }
% 47.13/6.43 addition(addition(c(x1), c(x0)), x0)
% 47.13/6.43 = { by axiom 7 (additive_commutativity) R->L }
% 47.13/6.43 addition(addition(c(x0), c(x1)), x0)
% 47.13/6.43 = { by axiom 7 (additive_commutativity) }
% 47.13/6.43 addition(x0, addition(c(x0), c(x1)))
% 47.13/6.43 = { by lemma 60 }
% 47.13/6.43 addition(c(x1), addition(x0, c(x0)))
% 47.13/6.43 = { by lemma 49 }
% 47.13/6.43 addition(c(x1), one)
% 47.13/6.43 = { by lemma 57 R->L }
% 47.13/6.43 addition(c(x1), addition(x1, c(x1)))
% 47.13/6.43 = { by lemma 45 R->L }
% 47.13/6.43 addition(c(x1), addition(x1, addition(x1, c(x1))))
% 47.13/6.43 = { by axiom 7 (additive_commutativity) R->L }
% 47.13/6.43 addition(c(x1), addition(x1, addition(c(x1), x1)))
% 47.13/6.43 = { by axiom 12 (additive_associativity) }
% 47.13/6.43 addition(addition(c(x1), x1), addition(c(x1), x1))
% 47.13/6.43 = { by axiom 6 (additive_idempotence) }
% 47.13/6.43 addition(c(x1), x1)
% 47.13/6.43 = { by axiom 7 (additive_commutativity) }
% 47.13/6.43 addition(x1, c(x1))
% 47.13/6.43 = { by lemma 57 }
% 47.13/6.43 one
% 47.13/6.43
% 47.13/6.43 Lemma 64: multiplication(addition(c(x0), c(x1)), multiplication(x0, x1)) = zero.
% 47.13/6.43 Proof:
% 47.13/6.43 multiplication(addition(c(x0), c(x1)), multiplication(x0, x1))
% 47.13/6.43 = { by axiom 7 (additive_commutativity) }
% 47.13/6.43 multiplication(addition(c(x1), c(x0)), multiplication(x0, x1))
% 47.13/6.43 = { by lemma 43 R->L }
% 47.13/6.43 multiplication(addition(c(x1), x1_2(x0)), multiplication(x0, x1))
% 47.13/6.43 = { by lemma 44 R->L }
% 47.13/6.43 addition(zero, multiplication(addition(c(x1), x1_2(x0)), multiplication(x0, x1)))
% 47.13/6.43 = { by lemma 34 R->L }
% 47.13/6.43 addition(multiplication(x1_2(x1), x1), multiplication(addition(c(x1), x1_2(x0)), multiplication(x0, x1)))
% 47.13/6.43 = { by axiom 7 (additive_commutativity) R->L }
% 47.13/6.43 addition(multiplication(addition(c(x1), x1_2(x0)), multiplication(x0, x1)), multiplication(x1_2(x1), x1))
% 47.13/6.43 = { by lemma 54 }
% 47.13/6.43 multiplication(addition(x1_2(x1), multiplication(addition(c(x1), x1_2(x0)), x0)), x1)
% 47.13/6.43 = { by lemma 38 }
% 47.13/6.43 multiplication(addition(c(x1), multiplication(addition(c(x1), x1_2(x0)), x0)), x1)
% 47.13/6.43 = { by axiom 7 (additive_commutativity) R->L }
% 47.13/6.43 multiplication(addition(c(x1), multiplication(addition(x1_2(x0), c(x1)), x0)), x1)
% 47.13/6.43 = { by axiom 26 (left_distributivity) }
% 47.13/6.43 multiplication(addition(c(x1), addition(multiplication(x1_2(x0), x0), multiplication(c(x1), x0))), x1)
% 47.13/6.43 = { by lemma 40 }
% 47.13/6.43 multiplication(addition(c(x1), addition(zero, multiplication(c(x1), x0))), x1)
% 47.13/6.43 = { by lemma 44 }
% 47.13/6.43 multiplication(addition(c(x1), multiplication(c(x1), x0)), x1)
% 47.13/6.43 = { by lemma 52 }
% 47.13/6.43 multiplication(multiplication(c(x1), addition(x0, one)), x1)
% 47.13/6.43 = { by axiom 11 (multiplicative_associativity) R->L }
% 47.13/6.43 multiplication(c(x1), multiplication(addition(x0, one), x1))
% 47.13/6.43 = { by lemma 47 }
% 47.13/6.43 multiplication(c(x1), multiplication(one, x1))
% 47.13/6.43 = { by axiom 5 (multiplicative_left_identity) }
% 47.13/6.43 multiplication(c(x1), x1)
% 47.13/6.43 = { by lemma 59 }
% 47.13/6.43 zero
% 47.13/6.43
% 47.13/6.43 Lemma 65: complement(multiplication(x0, x1), c(multiplication(x0, x1))) = true.
% 47.13/6.43 Proof:
% 47.13/6.43 complement(multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by axiom 20 (test_3) R->L }
% 47.13/6.43 fresh5(true, true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by axiom 10 (test_1_1) R->L }
% 47.13/6.43 fresh5(fresh10(true, true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by axiom 14 (test_2) R->L }
% 47.13/6.43 fresh5(fresh10(fresh14(one, one, multiplication(x0, x1), addition(c(x0), c(x1))), true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by lemma 63 R->L }
% 47.13/6.43 fresh5(fresh10(fresh14(addition(addition(c(x0), c(x1)), multiplication(x0, x1)), one, multiplication(x0, x1), addition(c(x0), c(x1))), true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by lemma 36 }
% 47.13/6.43 fresh5(fresh10(fresh13(zero, zero, multiplication(x0, x1), addition(c(x0), c(x1))), true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by lemma 64 R->L }
% 47.13/6.43 fresh5(fresh10(fresh13(multiplication(addition(c(x0), c(x1)), multiplication(x0, x1)), zero, multiplication(x0, x1), addition(c(x0), c(x1))), true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by lemma 55 }
% 47.13/6.43 fresh5(fresh10(fresh13(multiplication(addition(c(x0), c(x1)), multiplication(x0, x1)), zero, multiplication(x1, x0), addition(c(x0), c(x1))), true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by lemma 55 }
% 47.13/6.43 fresh5(fresh10(fresh13(multiplication(addition(c(x0), c(x1)), multiplication(x1, x0)), zero, multiplication(x1, x0), addition(c(x0), c(x1))), true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by axiom 28 (test_2) }
% 47.13/6.43 fresh5(fresh10(fresh9(multiplication(multiplication(x1, x0), addition(c(x0), c(x1))), zero, multiplication(x1, x0), addition(c(x0), c(x1))), true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by axiom 11 (multiplicative_associativity) R->L }
% 47.13/6.43 fresh5(fresh10(fresh9(multiplication(x1, multiplication(x0, addition(c(x0), c(x1)))), zero, multiplication(x1, x0), addition(c(x0), c(x1))), true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by axiom 25 (right_distributivity) }
% 47.13/6.43 fresh5(fresh10(fresh9(multiplication(x1, addition(multiplication(x0, c(x0)), multiplication(x0, c(x1)))), zero, multiplication(x1, x0), addition(c(x0), c(x1))), true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by lemma 51 }
% 47.13/6.43 fresh5(fresh10(fresh9(multiplication(x1, addition(zero, multiplication(x0, c(x1)))), zero, multiplication(x1, x0), addition(c(x0), c(x1))), true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by lemma 44 }
% 47.13/6.43 fresh5(fresh10(fresh9(multiplication(x1, multiplication(x0, c(x1))), zero, multiplication(x1, x0), addition(c(x0), c(x1))), true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by lemma 38 R->L }
% 47.13/6.43 fresh5(fresh10(fresh9(multiplication(x1, multiplication(x0, x1_2(x1))), zero, multiplication(x1, x0), addition(c(x0), c(x1))), true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by axiom 8 (additive_identity) R->L }
% 47.13/6.43 fresh5(fresh10(fresh9(addition(multiplication(x1, multiplication(x0, x1_2(x1))), zero), zero, multiplication(x1, x0), addition(c(x0), c(x1))), true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by lemma 35 R->L }
% 47.13/6.43 fresh5(fresh10(fresh9(addition(multiplication(x1, multiplication(x0, x1_2(x1))), multiplication(x1, x1_2(x1))), zero, multiplication(x1, x0), addition(c(x0), c(x1))), true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by lemma 54 }
% 47.13/6.43 fresh5(fresh10(fresh9(multiplication(addition(x1, multiplication(x1, x0)), x1_2(x1)), zero, multiplication(x1, x0), addition(c(x0), c(x1))), true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by lemma 38 }
% 47.13/6.43 fresh5(fresh10(fresh9(multiplication(addition(x1, multiplication(x1, x0)), c(x1)), zero, multiplication(x1, x0), addition(c(x0), c(x1))), true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by lemma 55 R->L }
% 47.13/6.43 fresh5(fresh10(fresh9(multiplication(addition(x1, multiplication(x0, x1)), c(x1)), zero, multiplication(x1, x0), addition(c(x0), c(x1))), true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by axiom 26 (left_distributivity) }
% 47.13/6.43 fresh5(fresh10(fresh9(addition(multiplication(x1, c(x1)), multiplication(multiplication(x0, x1), c(x1))), zero, multiplication(x1, x0), addition(c(x0), c(x1))), true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by lemma 58 }
% 47.13/6.43 fresh5(fresh10(fresh9(addition(zero, multiplication(multiplication(x0, x1), c(x1))), zero, multiplication(x1, x0), addition(c(x0), c(x1))), true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by lemma 44 }
% 47.13/6.43 fresh5(fresh10(fresh9(multiplication(multiplication(x0, x1), c(x1)), zero, multiplication(x1, x0), addition(c(x0), c(x1))), true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by axiom 11 (multiplicative_associativity) R->L }
% 47.13/6.43 fresh5(fresh10(fresh9(multiplication(x0, multiplication(x1, c(x1))), zero, multiplication(x1, x0), addition(c(x0), c(x1))), true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by lemma 58 }
% 47.13/6.43 fresh5(fresh10(fresh9(multiplication(x0, zero), zero, multiplication(x1, x0), addition(c(x0), c(x1))), true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by axiom 3 (right_annihilation) }
% 47.13/6.43 fresh5(fresh10(fresh9(zero, zero, multiplication(x1, x0), addition(c(x0), c(x1))), true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by axiom 16 (test_2) }
% 47.13/6.43 fresh5(fresh10(complement(addition(c(x0), c(x1)), multiplication(x1, x0)), true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by lemma 55 R->L }
% 47.13/6.43 fresh5(fresh10(complement(addition(c(x0), c(x1)), multiplication(x0, x1)), true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by axiom 23 (test_1_1) }
% 47.13/6.43 fresh5(test(multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by axiom 24 (test_3) }
% 47.13/6.43 fresh4(c(multiplication(x0, x1)), c(multiplication(x0, x1)), multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by axiom 21 (test_3) }
% 47.13/6.43 true
% 47.13/6.43
% 47.13/6.43 Lemma 66: addition(multiplication(x0, x1), c(multiplication(x0, x1))) = one.
% 47.13/6.43 Proof:
% 47.13/6.43 addition(multiplication(x0, x1), c(multiplication(x0, x1)))
% 47.13/6.43 = { by axiom 7 (additive_commutativity) R->L }
% 47.13/6.43 addition(c(multiplication(x0, x1)), multiplication(x0, x1))
% 47.13/6.43 = { by axiom 29 (test_2_1) R->L }
% 47.13/6.43 fresh8(complement(multiplication(x0, x1), c(multiplication(x0, x1))), true, c(multiplication(x0, x1)), multiplication(x0, x1))
% 47.13/6.43 = { by lemma 65 }
% 47.13/6.43 fresh8(true, true, c(multiplication(x0, x1)), multiplication(x0, x1))
% 47.13/6.43 = { by axiom 17 (test_2_1) }
% 47.13/6.43 one
% 47.13/6.43
% 47.13/6.43 Goal 1 (goals_2): c(multiplication(x0, x1)) = addition(c(x0), c(x1)).
% 47.13/6.43 Proof:
% 47.13/6.43 c(multiplication(x0, x1))
% 47.13/6.43 = { by axiom 5 (multiplicative_left_identity) R->L }
% 47.13/6.43 multiplication(one, c(multiplication(x0, x1)))
% 47.13/6.43 = { by lemma 63 R->L }
% 47.13/6.43 multiplication(addition(addition(c(x0), c(x1)), multiplication(x0, x1)), c(multiplication(x0, x1)))
% 47.13/6.43 = { by axiom 7 (additive_commutativity) R->L }
% 47.13/6.43 multiplication(addition(multiplication(x0, x1), addition(c(x0), c(x1))), c(multiplication(x0, x1)))
% 47.13/6.43 = { by axiom 26 (left_distributivity) }
% 47.13/6.43 addition(multiplication(multiplication(x0, x1), c(multiplication(x0, x1))), multiplication(addition(c(x0), c(x1)), c(multiplication(x0, x1))))
% 47.13/6.43 = { by axiom 31 (test_2_3) R->L }
% 47.13/6.44 addition(fresh6(complement(multiplication(x0, x1), c(multiplication(x0, x1))), true, c(multiplication(x0, x1)), multiplication(x0, x1)), multiplication(addition(c(x0), c(x1)), c(multiplication(x0, x1))))
% 47.13/6.44 = { by lemma 65 }
% 47.13/6.44 addition(fresh6(true, true, c(multiplication(x0, x1)), multiplication(x0, x1)), multiplication(addition(c(x0), c(x1)), c(multiplication(x0, x1))))
% 47.13/6.44 = { by axiom 19 (test_2_3) }
% 47.13/6.44 addition(zero, multiplication(addition(c(x0), c(x1)), c(multiplication(x0, x1))))
% 47.13/6.44 = { by lemma 64 R->L }
% 47.13/6.44 addition(multiplication(addition(c(x0), c(x1)), multiplication(x0, x1)), multiplication(addition(c(x0), c(x1)), c(multiplication(x0, x1))))
% 47.13/6.44 = { by lemma 62 R->L }
% 47.13/6.44 addition(multiplication(addition(c(x0), c(x1)), multiplication(x0, x1)), multiplication(addition(addition(c(x0), c(x1)), c(x1)), c(multiplication(x0, x1))))
% 47.13/6.44 = { by axiom 7 (additive_commutativity) R->L }
% 47.13/6.44 addition(multiplication(addition(c(x0), c(x1)), multiplication(x0, x1)), multiplication(addition(c(x1), addition(c(x0), c(x1))), c(multiplication(x0, x1))))
% 47.13/6.44 = { by axiom 26 (left_distributivity) }
% 47.13/6.44 addition(multiplication(addition(c(x0), c(x1)), multiplication(x0, x1)), addition(multiplication(c(x1), c(multiplication(x0, x1))), multiplication(addition(c(x0), c(x1)), c(multiplication(x0, x1)))))
% 47.13/6.44 = { by axiom 7 (additive_commutativity) R->L }
% 47.13/6.44 addition(multiplication(addition(c(x0), c(x1)), multiplication(x0, x1)), addition(multiplication(addition(c(x0), c(x1)), c(multiplication(x0, x1))), multiplication(c(x1), c(multiplication(x0, x1)))))
% 47.13/6.44 = { by axiom 12 (additive_associativity) }
% 47.13/6.44 addition(addition(multiplication(addition(c(x0), c(x1)), multiplication(x0, x1)), multiplication(addition(c(x0), c(x1)), c(multiplication(x0, x1)))), multiplication(c(x1), c(multiplication(x0, x1))))
% 47.13/6.44 = { by axiom 25 (right_distributivity) R->L }
% 47.13/6.44 addition(multiplication(addition(c(x0), c(x1)), addition(multiplication(x0, x1), c(multiplication(x0, x1)))), multiplication(c(x1), c(multiplication(x0, x1))))
% 47.13/6.44 = { by axiom 7 (additive_commutativity) }
% 47.13/6.44 addition(multiplication(c(x1), c(multiplication(x0, x1))), multiplication(addition(c(x0), c(x1)), addition(multiplication(x0, x1), c(multiplication(x0, x1)))))
% 47.13/6.44 = { by lemma 66 }
% 47.13/6.44 addition(multiplication(c(x1), c(multiplication(x0, x1))), multiplication(addition(c(x0), c(x1)), one))
% 47.13/6.44 = { by axiom 4 (multiplicative_right_identity) }
% 47.13/6.44 addition(multiplication(c(x1), c(multiplication(x0, x1))), addition(c(x0), c(x1)))
% 47.13/6.44 = { by lemma 44 R->L }
% 47.13/6.44 addition(addition(zero, multiplication(c(x1), c(multiplication(x0, x1)))), addition(c(x0), c(x1)))
% 47.13/6.44 = { by lemma 59 R->L }
% 47.13/6.44 addition(addition(multiplication(c(x1), x1), multiplication(c(x1), c(multiplication(x0, x1)))), addition(c(x0), c(x1)))
% 47.13/6.44 = { by axiom 25 (right_distributivity) R->L }
% 47.13/6.44 addition(multiplication(c(x1), addition(x1, c(multiplication(x0, x1)))), addition(c(x0), c(x1)))
% 47.13/6.44 = { by axiom 5 (multiplicative_left_identity) R->L }
% 47.13/6.44 addition(multiplication(c(x1), addition(multiplication(one, x1), c(multiplication(x0, x1)))), addition(c(x0), c(x1)))
% 47.13/6.44 = { by lemma 47 R->L }
% 47.13/6.44 addition(multiplication(c(x1), addition(multiplication(addition(x0, one), x1), c(multiplication(x0, x1)))), addition(c(x0), c(x1)))
% 47.13/6.44 = { by lemma 50 R->L }
% 47.13/6.44 addition(multiplication(c(x1), addition(addition(x1, multiplication(x0, x1)), c(multiplication(x0, x1)))), addition(c(x0), c(x1)))
% 47.13/6.44 = { by axiom 12 (additive_associativity) R->L }
% 47.13/6.44 addition(multiplication(c(x1), addition(x1, addition(multiplication(x0, x1), c(multiplication(x0, x1))))), addition(c(x0), c(x1)))
% 47.13/6.44 = { by lemma 66 }
% 47.13/6.44 addition(multiplication(c(x1), addition(x1, one)), addition(c(x0), c(x1)))
% 47.13/6.44 = { by lemma 46 }
% 47.13/6.44 addition(multiplication(c(x1), one), addition(c(x0), c(x1)))
% 47.13/6.44 = { by axiom 4 (multiplicative_right_identity) }
% 47.13/6.44 addition(c(x1), addition(c(x0), c(x1)))
% 47.13/6.44 = { by axiom 7 (additive_commutativity) }
% 47.13/6.44 addition(addition(c(x0), c(x1)), c(x1))
% 47.13/6.44 = { by lemma 62 }
% 47.13/6.44 addition(c(x0), c(x1))
% 47.13/6.44 % SZS output end Proof
% 47.13/6.44
% 47.13/6.44 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------