TSTP Solution File: KLE011+2 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE011+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 : n012.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 05:35:27 EDT 2023
% Result : Theorem 0.19s 0.60s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : KLE011+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 : n012.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Tue Aug 29 11:23:54 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.19/0.60 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.19/0.60
% 0.19/0.60 % SZS status Theorem
% 0.19/0.60
% 0.19/0.63 % SZS output start Proof
% 0.19/0.63 Take the following subset of the input axioms:
% 0.19/0.63 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 0.19/0.63 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 0.19/0.63 fof(additive_idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 0.19/0.63 fof(additive_identity, axiom, ![A3]: addition(A3, zero)=A3).
% 0.19/0.63 fof(goals, conjecture, ![X0, X1]: ((test(X1) & test(X0)) => (leq(one, addition(addition(multiplication(addition(X1, c(X1)), X0), multiplication(addition(X0, c(X0)), X1)), multiplication(c(X0), c(X1)))) & leq(addition(addition(multiplication(addition(X1, c(X1)), X0), multiplication(addition(X0, c(X0)), X1)), multiplication(c(X0), c(X1))), one)))).
% 0.19/0.63 fof(left_distributivity, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 0.19/0.63 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 0.19/0.63 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 0.19/0.63 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 0.19/0.63 fof(right_distributivity, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 0.19/0.63 fof(test_1, axiom, ![X0_2]: (test(X0_2) <=> ?[X1_2]: complement(X1_2, X0_2))).
% 0.19/0.63 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)))).
% 0.19/0.63 fof(test_3, axiom, ![X0_2, X1_2]: (test(X0_2) => (c(X0_2)=X1_2 <=> complement(X0_2, X1_2)))).
% 0.19/0.63
% 0.19/0.63 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.63 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.63 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.63 fresh(y, y, x1...xn) = u
% 0.19/0.63 C => fresh(s, t, x1...xn) = v
% 0.19/0.63 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.63 variables of u and v.
% 0.19/0.63 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.63 input problem has no model of domain size 1).
% 0.19/0.63
% 0.19/0.63 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.63
% 0.19/0.63 Axiom 1 (goals): test(x1) = true.
% 0.19/0.63 Axiom 2 (goals_1): test(x0) = true.
% 0.19/0.63 Axiom 3 (additive_idempotence): addition(X, X) = X.
% 0.19/0.63 Axiom 4 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.19/0.63 Axiom 5 (additive_identity): addition(X, zero) = X.
% 0.19/0.63 Axiom 6 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.19/0.63 Axiom 7 (multiplicative_left_identity): multiplication(one, X) = X.
% 0.19/0.63 Axiom 8 (test_1): fresh12(X, X, Y) = true.
% 0.19/0.63 Axiom 9 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 0.19/0.63 Axiom 10 (test_1): fresh12(test(X), true, X) = complement(x1_2(X), X).
% 0.19/0.63 Axiom 11 (order): fresh11(X, X, Y, Z) = true.
% 0.19/0.63 Axiom 12 (test_2_1): fresh8(X, X, Y, Z) = one.
% 0.19/0.63 Axiom 13 (test_2_3): fresh6(X, X, Y, Z) = zero.
% 0.19/0.63 Axiom 14 (test_3): fresh5(X, X, Y, Z) = complement(Y, Z).
% 0.19/0.63 Axiom 15 (test_3): fresh4(X, X, Y, Z) = true.
% 0.19/0.63 Axiom 16 (test_3): fresh5(test(X), true, X, Y) = fresh4(c(X), Y, X, Y).
% 0.19/0.63 Axiom 17 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 0.19/0.63 Axiom 18 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 0.19/0.63 Axiom 19 (order): fresh11(addition(X, Y), Y, X, Y) = leq(X, Y).
% 0.19/0.63 Axiom 20 (test_2_1): fresh8(complement(X, Y), true, Y, X) = addition(Y, X).
% 0.19/0.63 Axiom 21 (test_2_3): fresh6(complement(X, Y), true, Y, X) = multiplication(X, Y).
% 0.19/0.63
% 0.19/0.63 Lemma 22: complement(x1, c(x1)) = true.
% 0.19/0.63 Proof:
% 0.19/0.63 complement(x1, c(x1))
% 0.19/0.63 = { by axiom 14 (test_3) R->L }
% 0.19/0.63 fresh5(true, true, x1, c(x1))
% 0.19/0.63 = { by axiom 1 (goals) R->L }
% 0.19/0.63 fresh5(test(x1), true, x1, c(x1))
% 0.19/0.63 = { by axiom 16 (test_3) }
% 0.19/0.63 fresh4(c(x1), c(x1), x1, c(x1))
% 0.19/0.63 = { by axiom 15 (test_3) }
% 0.19/0.63 true
% 0.19/0.63
% 0.19/0.63 Lemma 23: addition(x1, c(x1)) = one.
% 0.19/0.63 Proof:
% 0.19/0.63 addition(x1, c(x1))
% 0.19/0.63 = { by axiom 4 (additive_commutativity) R->L }
% 0.19/0.63 addition(c(x1), x1)
% 0.19/0.63 = { by axiom 20 (test_2_1) R->L }
% 0.19/0.63 fresh8(complement(x1, c(x1)), true, c(x1), x1)
% 0.19/0.63 = { by lemma 22 }
% 0.19/0.63 fresh8(true, true, c(x1), x1)
% 0.19/0.63 = { by axiom 12 (test_2_1) }
% 0.19/0.63 one
% 0.19/0.63
% 0.19/0.63 Lemma 24: addition(X, addition(X, Y)) = addition(X, Y).
% 0.19/0.63 Proof:
% 0.19/0.63 addition(X, addition(X, Y))
% 0.19/0.63 = { by axiom 9 (additive_associativity) }
% 0.19/0.63 addition(addition(X, X), Y)
% 0.19/0.63 = { by axiom 3 (additive_idempotence) }
% 0.19/0.63 addition(X, Y)
% 0.19/0.63
% 0.19/0.63 Lemma 25: addition(one, c(x1)) = one.
% 0.19/0.63 Proof:
% 0.19/0.63 addition(one, c(x1))
% 0.19/0.63 = { by axiom 4 (additive_commutativity) R->L }
% 0.19/0.63 addition(c(x1), one)
% 0.19/0.63 = { by lemma 23 R->L }
% 0.19/0.63 addition(c(x1), addition(x1, c(x1)))
% 0.19/0.63 = { by axiom 4 (additive_commutativity) R->L }
% 0.19/0.63 addition(c(x1), addition(c(x1), x1))
% 0.19/0.63 = { by lemma 24 }
% 0.19/0.63 addition(c(x1), x1)
% 0.19/0.63 = { by axiom 4 (additive_commutativity) }
% 0.19/0.63 addition(x1, c(x1))
% 0.19/0.63 = { by lemma 23 }
% 0.19/0.63 one
% 0.19/0.63
% 0.19/0.63 Lemma 26: addition(x0, c(x0)) = one.
% 0.19/0.63 Proof:
% 0.19/0.63 addition(x0, c(x0))
% 0.19/0.63 = { by axiom 4 (additive_commutativity) R->L }
% 0.19/0.63 addition(c(x0), x0)
% 0.19/0.63 = { by axiom 20 (test_2_1) R->L }
% 0.19/0.63 fresh8(complement(x0, c(x0)), true, c(x0), x0)
% 0.19/0.63 = { by axiom 14 (test_3) R->L }
% 0.19/0.63 fresh8(fresh5(true, true, x0, c(x0)), true, c(x0), x0)
% 0.19/0.63 = { by axiom 2 (goals_1) R->L }
% 0.19/0.63 fresh8(fresh5(test(x0), true, x0, c(x0)), true, c(x0), x0)
% 0.19/0.63 = { by axiom 16 (test_3) }
% 0.19/0.63 fresh8(fresh4(c(x0), c(x0), x0, c(x0)), true, c(x0), x0)
% 0.19/0.63 = { by axiom 15 (test_3) }
% 0.19/0.63 fresh8(true, true, c(x0), x0)
% 0.19/0.63 = { by axiom 12 (test_2_1) }
% 0.19/0.63 one
% 0.19/0.63
% 0.19/0.63 Lemma 27: addition(x0, x1_2(x0)) = one.
% 0.19/0.63 Proof:
% 0.19/0.63 addition(x0, x1_2(x0))
% 0.19/0.63 = { by axiom 20 (test_2_1) R->L }
% 0.19/0.63 fresh8(complement(x1_2(x0), x0), true, x0, x1_2(x0))
% 0.19/0.63 = { by axiom 10 (test_1) R->L }
% 0.19/0.63 fresh8(fresh12(test(x0), true, x0), true, x0, x1_2(x0))
% 0.19/0.63 = { by axiom 2 (goals_1) }
% 0.19/0.63 fresh8(fresh12(true, true, x0), true, x0, x1_2(x0))
% 0.19/0.63 = { by axiom 8 (test_1) }
% 0.19/0.63 fresh8(true, true, x0, x1_2(x0))
% 0.19/0.63 = { by axiom 12 (test_2_1) }
% 0.19/0.63 one
% 0.19/0.63
% 0.19/0.63 Lemma 28: addition(Y, addition(X, Z)) = addition(X, addition(Y, Z)).
% 0.19/0.63 Proof:
% 0.19/0.63 addition(Y, addition(X, Z))
% 0.19/0.63 = { by axiom 4 (additive_commutativity) R->L }
% 0.19/0.63 addition(addition(X, Z), Y)
% 0.19/0.63 = { by axiom 9 (additive_associativity) R->L }
% 0.19/0.63 addition(X, addition(Z, Y))
% 0.19/0.63 = { by axiom 4 (additive_commutativity) }
% 0.19/0.63 addition(X, addition(Y, Z))
% 0.19/0.63
% 0.19/0.63 Lemma 29: addition(addition(X, Y), X) = addition(X, Y).
% 0.19/0.63 Proof:
% 0.19/0.63 addition(addition(X, Y), X)
% 0.19/0.63 = { by axiom 4 (additive_commutativity) R->L }
% 0.19/0.63 addition(X, addition(X, Y))
% 0.19/0.63 = { by lemma 24 }
% 0.19/0.63 addition(X, Y)
% 0.19/0.63
% 0.19/0.63 Lemma 30: multiplication(addition(x1, c(x1)), X) = X.
% 0.19/0.63 Proof:
% 0.19/0.63 multiplication(addition(x1, c(x1)), X)
% 0.19/0.63 = { by lemma 23 }
% 0.19/0.63 multiplication(one, X)
% 0.19/0.63 = { by axiom 7 (multiplicative_left_identity) }
% 0.19/0.63 X
% 0.19/0.63
% 0.19/0.63 Lemma 31: multiplication(addition(x0, c(x0)), X) = X.
% 0.19/0.63 Proof:
% 0.19/0.63 multiplication(addition(x0, c(x0)), X)
% 0.19/0.63 = { by lemma 26 }
% 0.19/0.63 multiplication(one, X)
% 0.19/0.63 = { by axiom 7 (multiplicative_left_identity) }
% 0.19/0.63 X
% 0.19/0.63
% 0.19/0.63 Lemma 32: addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), c(x1)) = one.
% 0.19/0.63 Proof:
% 0.19/0.63 addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), c(x1))
% 0.19/0.63 = { by axiom 9 (additive_associativity) R->L }
% 0.19/0.63 addition(multiplication(addition(x1, c(x1)), x0), addition(multiplication(addition(x0, c(x0)), x1), c(x1)))
% 0.19/0.63 = { by lemma 30 }
% 0.19/0.63 addition(x0, addition(multiplication(addition(x0, c(x0)), x1), c(x1)))
% 0.19/0.63 = { by lemma 31 }
% 0.19/0.63 addition(x0, addition(x1, c(x1)))
% 0.19/0.63 = { by lemma 28 R->L }
% 0.19/0.63 addition(x1, addition(x0, c(x1)))
% 0.19/0.63 = { by axiom 4 (additive_commutativity) R->L }
% 0.19/0.63 addition(x1, addition(c(x1), x0))
% 0.19/0.63 = { by axiom 9 (additive_associativity) }
% 0.19/0.63 addition(addition(x1, c(x1)), x0)
% 0.19/0.63 = { by lemma 23 }
% 0.19/0.63 addition(one, x0)
% 0.19/0.63 = { by axiom 4 (additive_commutativity) }
% 0.19/0.63 addition(x0, one)
% 0.19/0.64 = { by lemma 27 R->L }
% 0.19/0.64 addition(x0, addition(x0, x1_2(x0)))
% 0.19/0.64 = { by lemma 24 }
% 0.19/0.64 addition(x0, x1_2(x0))
% 0.19/0.64 = { by lemma 27 }
% 0.19/0.64 one
% 0.19/0.64
% 0.19/0.64 Goal 1 (goals_2): tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))))) = tuple(true, true).
% 0.19/0.64 Proof:
% 0.19/0.64 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1)))))
% 0.19/0.64 = { by lemma 29 R->L }
% 0.19/0.64 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)))))
% 0.19/0.64 = { by axiom 6 (multiplicative_right_identity) R->L }
% 0.19/0.64 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), multiplication(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), one))))
% 0.19/0.64 = { by lemma 25 R->L }
% 0.19/0.64 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), multiplication(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), addition(one, c(x1))))))
% 0.19/0.64 = { by axiom 17 (right_distributivity) }
% 0.19/0.64 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), addition(multiplication(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), one), multiplication(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), c(x1))))))
% 0.19/0.64 = { by axiom 6 (multiplicative_right_identity) }
% 0.19/0.64 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), c(x1))))))
% 0.19/0.64 = { by axiom 9 (additive_associativity) }
% 0.19/0.64 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(addition(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1))), multiplication(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), c(x1)))))
% 0.19/0.64 = { by lemma 29 }
% 0.19/0.64 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), multiplication(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), c(x1)))))
% 0.19/0.64 = { by axiom 4 (additive_commutativity) }
% 0.19/0.64 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(multiplication(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), c(x1)), addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))))))
% 0.19/0.64 = { by lemma 31 }
% 0.19/0.64 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(multiplication(addition(multiplication(addition(x1, c(x1)), x0), x1), c(x1)), addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))))))
% 0.19/0.64 = { by lemma 30 }
% 0.19/0.64 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(multiplication(addition(x0, x1), c(x1)), addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))))))
% 0.19/0.64 = { by axiom 4 (additive_commutativity) R->L }
% 0.19/0.64 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(multiplication(addition(x1, x0), c(x1)), addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))))))
% 0.19/0.64 = { by axiom 18 (left_distributivity) }
% 0.19/0.64 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(addition(multiplication(x1, c(x1)), multiplication(x0, c(x1))), addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))))))
% 0.19/0.64 = { by axiom 21 (test_2_3) R->L }
% 0.19/0.64 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(addition(fresh6(complement(x1, c(x1)), true, c(x1), x1), multiplication(x0, c(x1))), addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))))))
% 0.19/0.64 = { by lemma 22 }
% 0.19/0.64 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(addition(fresh6(true, true, c(x1), x1), multiplication(x0, c(x1))), addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))))))
% 0.19/0.64 = { by axiom 13 (test_2_3) }
% 0.19/0.64 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(addition(zero, multiplication(x0, c(x1))), addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))))))
% 0.19/0.64 = { by axiom 4 (additive_commutativity) R->L }
% 0.19/0.64 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(addition(multiplication(x0, c(x1)), zero), addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))))))
% 0.19/0.64 = { by axiom 5 (additive_identity) }
% 0.19/0.64 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(multiplication(x0, c(x1)), addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))))))
% 0.19/0.64 = { by lemma 28 }
% 0.19/0.64 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), addition(multiplication(x0, c(x1)), multiplication(c(x0), c(x1))))))
% 0.19/0.64 = { by axiom 4 (additive_commutativity) }
% 0.19/0.64 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), addition(multiplication(c(x0), c(x1)), multiplication(x0, c(x1))))))
% 0.19/0.64 = { by axiom 18 (left_distributivity) R->L }
% 0.19/0.64 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(addition(c(x0), x0), c(x1)))))
% 0.19/0.64 = { by axiom 4 (additive_commutativity) }
% 0.19/0.64 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(addition(x0, c(x0)), c(x1)))))
% 0.19/0.64 = { by lemma 26 }
% 0.19/0.64 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(one, c(x1)))))
% 0.19/0.64 = { by axiom 7 (multiplicative_left_identity) }
% 0.19/0.64 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), c(x1))))
% 0.19/0.64 = { by lemma 32 }
% 0.19/0.64 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), leq(one, one))
% 0.19/0.64 = { by axiom 19 (order) R->L }
% 0.19/0.64 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), fresh11(addition(one, one), one, one, one))
% 0.19/0.64 = { by axiom 3 (additive_idempotence) }
% 0.19/0.64 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), fresh11(one, one, one, one))
% 0.19/0.65 = { by axiom 11 (order) }
% 0.19/0.65 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), true)
% 0.19/0.65 = { by lemma 32 R->L }
% 0.19/0.65 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), c(x1))), true)
% 0.19/0.65 = { by lemma 24 R->L }
% 0.19/0.65 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), c(x1)))), true)
% 0.19/0.65 = { by lemma 32 }
% 0.19/0.65 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), one)), true)
% 0.19/0.65 = { by lemma 26 R->L }
% 0.19/0.65 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), addition(x0, c(x0)))), true)
% 0.19/0.65 = { by axiom 6 (multiplicative_right_identity) R->L }
% 0.19/0.65 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), addition(x0, multiplication(c(x0), one)))), true)
% 0.19/0.65 = { by lemma 25 R->L }
% 0.19/0.65 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), addition(x0, multiplication(c(x0), addition(one, c(x1)))))), true)
% 0.19/0.65 = { by axiom 4 (additive_commutativity) R->L }
% 0.19/0.65 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), addition(x0, multiplication(c(x0), addition(c(x1), one))))), true)
% 0.19/0.65 = { by axiom 17 (right_distributivity) }
% 0.19/0.65 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), addition(x0, addition(multiplication(c(x0), c(x1)), multiplication(c(x0), one))))), true)
% 0.19/0.65 = { by axiom 6 (multiplicative_right_identity) }
% 0.19/0.65 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), addition(x0, addition(multiplication(c(x0), c(x1)), c(x0))))), true)
% 0.19/0.65 = { by axiom 4 (additive_commutativity) R->L }
% 0.19/0.65 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), addition(x0, addition(c(x0), multiplication(c(x0), c(x1)))))), true)
% 0.19/0.65 = { by axiom 9 (additive_associativity) }
% 0.19/0.65 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), addition(addition(x0, c(x0)), multiplication(c(x0), c(x1))))), true)
% 0.19/0.65 = { by lemma 26 }
% 0.19/0.65 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), addition(one, multiplication(c(x0), c(x1))))), true)
% 0.19/0.65 = { by lemma 28 R->L }
% 0.19/0.65 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), addition(one, addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))))), true)
% 0.19/0.65 = { by axiom 4 (additive_commutativity) R->L }
% 0.19/0.65 tuple(leq(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), addition(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one)), true)
% 0.19/0.65 = { by axiom 19 (order) R->L }
% 0.19/0.65 tuple(fresh11(addition(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), addition(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one)), addition(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), addition(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one)), true)
% 0.19/0.65 = { by lemma 24 }
% 0.19/0.65 tuple(fresh11(addition(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), addition(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one), addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), addition(addition(addition(multiplication(addition(x1, c(x1)), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(c(x0), c(x1))), one)), true)
% 0.19/0.65 = { by axiom 11 (order) }
% 0.19/0.66 tuple(true, true)
% 0.19/0.66 % SZS output end Proof
% 0.19/0.66
% 0.19/0.66 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------