TSTP Solution File: KLE016+3 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE016+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 : n002.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 05:35:29 EDT 2023
% Result : Theorem 19.90s 2.95s
% Output : Proof 21.12s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.09/0.12 % Problem : KLE016+3 : TPTP v8.1.2. Released v4.0.0.
% 0.12/0.13 % 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 : n002.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 12:19:50 EDT 2023
% 0.13/0.34 % CPUTime :
% 19.90/2.95 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 19.90/2.95
% 19.90/2.95 % SZS status Theorem
% 19.90/2.95
% 20.74/3.00 % SZS output start Proof
% 20.74/3.00 Take the following subset of the input axioms:
% 20.74/3.00 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 20.74/3.00 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 20.74/3.00 fof(additive_idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 20.74/3.00 fof(additive_identity, axiom, ![A3]: addition(A3, zero)=A3).
% 20.74/3.00 fof(goals, conjecture, ![X0, X1]: ((test(X1) & test(X0)) => (leq(c(multiplication(X0, X1)), addition(c(X0), c(X1))) & leq(addition(c(X0), c(X1)), c(multiplication(X0, X1)))))).
% 20.74/3.00 fof(left_distributivity, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 20.74/3.00 fof(multiplicative_associativity, axiom, ![A3, B2, C2]: multiplication(A3, multiplication(B2, C2))=multiplication(multiplication(A3, B2), C2)).
% 20.74/3.00 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 20.74/3.00 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 20.74/3.00 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 20.74/3.00 fof(right_annihilation, axiom, ![A3]: multiplication(A3, zero)=zero).
% 20.74/3.01 fof(right_distributivity, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 20.74/3.01 fof(test_1, axiom, ![X0_2]: (test(X0_2) <=> ?[X1_2]: complement(X1_2, X0_2))).
% 20.74/3.01 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)))).
% 20.74/3.01 fof(test_3, axiom, ![X0_2, X1_2]: (test(X0_2) => (c(X0_2)=X1_2 <=> complement(X0_2, X1_2)))).
% 20.74/3.01
% 20.74/3.01 Now clausify the problem and encode Horn clauses using encoding 3 of
% 20.74/3.01 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 20.74/3.01 We repeatedly replace C & s=t => u=v by the two clauses:
% 20.74/3.01 fresh(y, y, x1...xn) = u
% 20.74/3.01 C => fresh(s, t, x1...xn) = v
% 20.74/3.01 where fresh is a fresh function symbol and x1..xn are the free
% 20.74/3.01 variables of u and v.
% 20.74/3.01 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 20.74/3.01 input problem has no model of domain size 1).
% 20.74/3.01
% 20.74/3.01 The encoding turns the above axioms into the following unit equations and goals:
% 20.74/3.01
% 20.74/3.01 Axiom 1 (goals): test(x1) = true.
% 20.74/3.01 Axiom 2 (goals_1): test(x0) = true.
% 20.74/3.01 Axiom 3 (right_annihilation): multiplication(X, zero) = zero.
% 20.74/3.01 Axiom 4 (multiplicative_right_identity): multiplication(X, one) = X.
% 20.74/3.01 Axiom 5 (multiplicative_left_identity): multiplication(one, X) = X.
% 20.74/3.01 Axiom 6 (additive_idempotence): addition(X, X) = X.
% 20.74/3.01 Axiom 7 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 20.74/3.01 Axiom 8 (additive_identity): addition(X, zero) = X.
% 20.74/3.01 Axiom 9 (test_1): fresh12(X, X, Y) = true.
% 20.74/3.01 Axiom 10 (test_1_1): fresh10(X, X, Y) = true.
% 20.74/3.01 Axiom 11 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 20.74/3.01 Axiom 12 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 20.74/3.01 Axiom 13 (test_3_1): fresh(X, X, Y, Z) = Z.
% 20.74/3.01 Axiom 14 (test_2): fresh14(X, X, Y, Z) = true.
% 20.74/3.01 Axiom 15 (test_1): fresh12(test(X), true, X) = complement(x1_2(X), X).
% 20.74/3.01 Axiom 16 (order): fresh11(X, X, Y, Z) = true.
% 20.74/3.01 Axiom 17 (test_2): fresh9(X, X, Y, Z) = complement(Z, Y).
% 20.74/3.01 Axiom 18 (test_2_1): fresh8(X, X, Y, Z) = one.
% 20.74/3.01 Axiom 19 (test_2_2): fresh7(X, X, Y, Z) = zero.
% 20.74/3.01 Axiom 20 (test_2_3): fresh6(X, X, Y, Z) = zero.
% 20.74/3.01 Axiom 21 (test_3): fresh5(X, X, Y, Z) = complement(Y, Z).
% 20.74/3.01 Axiom 22 (test_3): fresh4(X, X, Y, Z) = true.
% 20.74/3.01 Axiom 23 (test_3_1): fresh3(X, X, Y, Z) = c(Y).
% 20.74/3.01 Axiom 24 (test_1_1): fresh10(complement(X, Y), true, Y) = test(Y).
% 20.74/3.01 Axiom 25 (test_3): fresh5(test(X), true, X, Y) = fresh4(c(X), Y, X, Y).
% 20.74/3.01 Axiom 26 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 20.74/3.01 Axiom 27 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 20.74/3.01 Axiom 28 (test_2): fresh13(X, X, Y, Z) = fresh14(addition(Y, Z), one, Y, Z).
% 20.74/3.01 Axiom 29 (order): fresh11(addition(X, Y), Y, X, Y) = leq(X, Y).
% 20.74/3.01 Axiom 30 (test_2): fresh13(multiplication(X, Y), zero, Y, X) = fresh9(multiplication(Y, X), zero, Y, X).
% 20.74/3.01 Axiom 31 (test_2_1): fresh8(complement(X, Y), true, Y, X) = addition(Y, X).
% 20.74/3.01 Axiom 32 (test_2_2): fresh7(complement(X, Y), true, Y, X) = multiplication(Y, X).
% 20.74/3.01 Axiom 33 (test_2_3): fresh6(complement(X, Y), true, Y, X) = multiplication(X, Y).
% 20.74/3.01 Axiom 34 (test_3_1): fresh3(complement(X, Y), true, X, Y) = fresh(test(X), true, X, Y).
% 20.74/3.01
% 20.74/3.01 Lemma 35: complement(x1_2(x1), x1) = true.
% 20.74/3.01 Proof:
% 20.74/3.01 complement(x1_2(x1), x1)
% 20.74/3.01 = { by axiom 15 (test_1) R->L }
% 20.74/3.01 fresh12(test(x1), true, x1)
% 20.74/3.01 = { by axiom 1 (goals) }
% 20.74/3.01 fresh12(true, true, x1)
% 20.74/3.01 = { by axiom 9 (test_1) }
% 20.74/3.01 true
% 20.74/3.01
% 20.74/3.01 Lemma 36: multiplication(x1_2(x1), x1) = zero.
% 20.74/3.01 Proof:
% 20.74/3.01 multiplication(x1_2(x1), x1)
% 20.74/3.01 = { by axiom 33 (test_2_3) R->L }
% 20.74/3.01 fresh6(complement(x1_2(x1), x1), true, x1, x1_2(x1))
% 20.74/3.01 = { by lemma 35 }
% 20.74/3.01 fresh6(true, true, x1, x1_2(x1))
% 20.74/3.01 = { by axiom 20 (test_2_3) }
% 20.74/3.01 zero
% 20.74/3.01
% 20.74/3.01 Lemma 37: multiplication(x1, x1_2(x1)) = zero.
% 20.74/3.01 Proof:
% 20.74/3.01 multiplication(x1, x1_2(x1))
% 20.74/3.01 = { by axiom 32 (test_2_2) R->L }
% 20.74/3.01 fresh7(complement(x1_2(x1), x1), true, x1, x1_2(x1))
% 20.74/3.01 = { by lemma 35 }
% 20.74/3.01 fresh7(true, true, x1, x1_2(x1))
% 20.74/3.01 = { by axiom 19 (test_2_2) }
% 20.74/3.01 zero
% 20.74/3.01
% 20.74/3.01 Lemma 38: fresh14(addition(X, Y), one, Y, X) = fresh13(Z, Z, Y, X).
% 20.74/3.01 Proof:
% 20.74/3.01 fresh14(addition(X, Y), one, Y, X)
% 20.74/3.01 = { by axiom 7 (additive_commutativity) R->L }
% 20.74/3.01 fresh14(addition(Y, X), one, Y, X)
% 20.74/3.01 = { by axiom 28 (test_2) R->L }
% 20.74/3.01 fresh13(Z, Z, Y, X)
% 20.74/3.01
% 20.74/3.01 Lemma 39: addition(x1, x1_2(x1)) = one.
% 20.74/3.01 Proof:
% 20.74/3.01 addition(x1, x1_2(x1))
% 20.74/3.01 = { by axiom 31 (test_2_1) R->L }
% 20.74/3.01 fresh8(complement(x1_2(x1), x1), true, x1, x1_2(x1))
% 20.74/3.01 = { by lemma 35 }
% 20.74/3.01 fresh8(true, true, x1, x1_2(x1))
% 20.74/3.01 = { by axiom 18 (test_2_1) }
% 20.74/3.01 one
% 20.74/3.01
% 20.74/3.01 Lemma 40: x1_2(x1) = c(x1).
% 20.74/3.01 Proof:
% 20.74/3.01 x1_2(x1)
% 20.74/3.01 = { by axiom 13 (test_3_1) R->L }
% 20.74/3.01 fresh(true, true, x1, x1_2(x1))
% 20.74/3.01 = { by axiom 1 (goals) R->L }
% 20.74/3.01 fresh(test(x1), true, x1, x1_2(x1))
% 20.74/3.01 = { by axiom 34 (test_3_1) R->L }
% 20.74/3.01 fresh3(complement(x1, x1_2(x1)), true, x1, x1_2(x1))
% 20.74/3.01 = { by axiom 17 (test_2) R->L }
% 20.74/3.01 fresh3(fresh9(zero, zero, x1_2(x1), x1), true, x1, x1_2(x1))
% 20.74/3.01 = { by lemma 36 R->L }
% 20.74/3.01 fresh3(fresh9(multiplication(x1_2(x1), x1), zero, x1_2(x1), x1), true, x1, x1_2(x1))
% 20.74/3.01 = { by axiom 30 (test_2) R->L }
% 20.74/3.01 fresh3(fresh13(multiplication(x1, x1_2(x1)), zero, x1_2(x1), x1), true, x1, x1_2(x1))
% 20.74/3.01 = { by lemma 37 }
% 20.74/3.01 fresh3(fresh13(zero, zero, x1_2(x1), x1), true, x1, x1_2(x1))
% 20.74/3.01 = { by lemma 38 R->L }
% 20.74/3.01 fresh3(fresh14(addition(x1, x1_2(x1)), one, x1_2(x1), x1), true, x1, x1_2(x1))
% 20.74/3.01 = { by lemma 39 }
% 20.74/3.01 fresh3(fresh14(one, one, x1_2(x1), x1), true, x1, x1_2(x1))
% 20.74/3.01 = { by axiom 14 (test_2) }
% 20.74/3.01 fresh3(true, true, x1, x1_2(x1))
% 20.74/3.01 = { by axiom 23 (test_3_1) }
% 20.74/3.01 c(x1)
% 20.74/3.01
% 20.74/3.01 Lemma 41: complement(x1_2(x0), x0) = true.
% 20.74/3.01 Proof:
% 20.74/3.01 complement(x1_2(x0), x0)
% 20.74/3.01 = { by axiom 15 (test_1) R->L }
% 20.74/3.01 fresh12(test(x0), true, x0)
% 20.74/3.01 = { by axiom 2 (goals_1) }
% 20.74/3.01 fresh12(true, true, x0)
% 20.74/3.01 = { by axiom 9 (test_1) }
% 20.74/3.01 true
% 20.74/3.01
% 20.74/3.01 Lemma 42: multiplication(x1_2(x0), x0) = zero.
% 20.74/3.01 Proof:
% 20.74/3.01 multiplication(x1_2(x0), x0)
% 20.74/3.01 = { by axiom 33 (test_2_3) R->L }
% 20.74/3.01 fresh6(complement(x1_2(x0), x0), true, x0, x1_2(x0))
% 20.74/3.01 = { by lemma 41 }
% 20.74/3.01 fresh6(true, true, x0, x1_2(x0))
% 20.74/3.01 = { by axiom 20 (test_2_3) }
% 20.74/3.01 zero
% 20.74/3.01
% 20.74/3.01 Lemma 43: multiplication(x0, x1_2(x0)) = zero.
% 20.74/3.01 Proof:
% 20.74/3.01 multiplication(x0, x1_2(x0))
% 20.74/3.01 = { by axiom 32 (test_2_2) R->L }
% 20.74/3.01 fresh7(complement(x1_2(x0), x0), true, x0, x1_2(x0))
% 20.74/3.01 = { by lemma 41 }
% 20.74/3.01 fresh7(true, true, x0, x1_2(x0))
% 20.74/3.01 = { by axiom 19 (test_2_2) }
% 20.74/3.01 zero
% 20.74/3.01
% 20.74/3.01 Lemma 44: addition(x0, x1_2(x0)) = one.
% 20.74/3.01 Proof:
% 20.74/3.01 addition(x0, x1_2(x0))
% 20.74/3.01 = { by axiom 31 (test_2_1) R->L }
% 20.74/3.01 fresh8(complement(x1_2(x0), x0), true, x0, x1_2(x0))
% 20.74/3.01 = { by lemma 41 }
% 20.74/3.01 fresh8(true, true, x0, x1_2(x0))
% 20.74/3.01 = { by axiom 18 (test_2_1) }
% 20.74/3.01 one
% 20.74/3.01
% 20.74/3.01 Lemma 45: x1_2(x0) = c(x0).
% 20.74/3.01 Proof:
% 20.74/3.01 x1_2(x0)
% 20.74/3.01 = { by axiom 13 (test_3_1) R->L }
% 20.74/3.01 fresh(true, true, x0, x1_2(x0))
% 20.74/3.01 = { by axiom 2 (goals_1) R->L }
% 20.74/3.01 fresh(test(x0), true, x0, x1_2(x0))
% 20.74/3.01 = { by axiom 34 (test_3_1) R->L }
% 20.74/3.01 fresh3(complement(x0, x1_2(x0)), true, x0, x1_2(x0))
% 20.74/3.01 = { by axiom 17 (test_2) R->L }
% 20.74/3.01 fresh3(fresh9(zero, zero, x1_2(x0), x0), true, x0, x1_2(x0))
% 20.74/3.01 = { by lemma 42 R->L }
% 20.74/3.01 fresh3(fresh9(multiplication(x1_2(x0), x0), zero, x1_2(x0), x0), true, x0, x1_2(x0))
% 20.74/3.01 = { by axiom 30 (test_2) R->L }
% 20.74/3.01 fresh3(fresh13(multiplication(x0, x1_2(x0)), zero, x1_2(x0), x0), true, x0, x1_2(x0))
% 20.74/3.01 = { by lemma 43 }
% 20.74/3.01 fresh3(fresh13(zero, zero, x1_2(x0), x0), true, x0, x1_2(x0))
% 20.74/3.01 = { by lemma 38 R->L }
% 20.74/3.01 fresh3(fresh14(addition(x0, x1_2(x0)), one, x1_2(x0), x0), true, x0, x1_2(x0))
% 20.74/3.01 = { by lemma 44 }
% 20.74/3.01 fresh3(fresh14(one, one, x1_2(x0), x0), true, x0, x1_2(x0))
% 20.74/3.01 = { by axiom 14 (test_2) }
% 20.74/3.01 fresh3(true, true, x0, x1_2(x0))
% 20.74/3.01 = { by axiom 23 (test_3_1) }
% 20.74/3.01 c(x0)
% 20.74/3.01
% 20.74/3.01 Lemma 46: addition(zero, X) = X.
% 20.74/3.01 Proof:
% 20.74/3.01 addition(zero, X)
% 20.74/3.01 = { by axiom 7 (additive_commutativity) R->L }
% 20.74/3.01 addition(X, zero)
% 20.74/3.01 = { by axiom 8 (additive_identity) }
% 20.74/3.01 X
% 20.74/3.01
% 20.74/3.01 Lemma 47: addition(X, addition(X, Y)) = addition(X, Y).
% 20.74/3.01 Proof:
% 20.74/3.01 addition(X, addition(X, Y))
% 20.74/3.01 = { by axiom 12 (additive_associativity) }
% 20.74/3.01 addition(addition(X, X), Y)
% 20.74/3.01 = { by axiom 6 (additive_idempotence) }
% 20.74/3.01 addition(X, Y)
% 20.74/3.01
% 20.74/3.01 Lemma 48: addition(x1, one) = one.
% 20.74/3.01 Proof:
% 20.74/3.01 addition(x1, one)
% 20.74/3.01 = { by lemma 39 R->L }
% 20.74/3.01 addition(x1, addition(x1, x1_2(x1)))
% 20.74/3.01 = { by lemma 47 }
% 20.74/3.01 addition(x1, x1_2(x1))
% 20.74/3.01 = { by lemma 39 }
% 20.74/3.01 one
% 20.74/3.01
% 20.74/3.01 Lemma 49: addition(x0, one) = one.
% 20.74/3.01 Proof:
% 20.74/3.01 addition(x0, one)
% 20.74/3.01 = { by lemma 44 R->L }
% 20.74/3.01 addition(x0, addition(x0, x1_2(x0)))
% 20.74/3.01 = { by lemma 47 }
% 20.74/3.01 addition(x0, x1_2(x0))
% 20.74/3.01 = { by lemma 44 }
% 20.74/3.01 one
% 20.74/3.01
% 20.74/3.01 Lemma 50: complement(x0, c(x0)) = true.
% 20.74/3.01 Proof:
% 20.74/3.01 complement(x0, c(x0))
% 20.74/3.01 = { by axiom 21 (test_3) R->L }
% 20.74/3.01 fresh5(true, true, x0, c(x0))
% 20.74/3.01 = { by axiom 2 (goals_1) R->L }
% 20.74/3.01 fresh5(test(x0), true, x0, c(x0))
% 20.74/3.01 = { by axiom 25 (test_3) }
% 20.74/3.01 fresh4(c(x0), c(x0), x0, c(x0))
% 20.74/3.01 = { by axiom 22 (test_3) }
% 20.74/3.01 true
% 20.74/3.01
% 20.74/3.01 Lemma 51: addition(x0, c(x0)) = one.
% 20.74/3.01 Proof:
% 20.74/3.01 addition(x0, c(x0))
% 20.74/3.01 = { by axiom 7 (additive_commutativity) R->L }
% 20.74/3.01 addition(c(x0), x0)
% 20.74/3.01 = { by axiom 31 (test_2_1) R->L }
% 20.74/3.01 fresh8(complement(x0, c(x0)), true, c(x0), x0)
% 20.74/3.01 = { by lemma 50 }
% 20.74/3.01 fresh8(true, true, c(x0), x0)
% 20.74/3.01 = { by axiom 18 (test_2_1) }
% 20.74/3.01 one
% 20.74/3.01
% 20.74/3.01 Lemma 52: addition(X, multiplication(Y, X)) = multiplication(addition(Y, one), X).
% 20.74/3.01 Proof:
% 20.74/3.01 addition(X, multiplication(Y, X))
% 20.74/3.01 = { by axiom 5 (multiplicative_left_identity) R->L }
% 20.74/3.01 addition(multiplication(one, X), multiplication(Y, X))
% 20.74/3.01 = { by axiom 27 (left_distributivity) R->L }
% 20.74/3.01 multiplication(addition(one, Y), X)
% 20.74/3.01 = { by axiom 7 (additive_commutativity) }
% 20.74/3.01 multiplication(addition(Y, one), X)
% 20.74/3.01
% 20.74/3.01 Lemma 53: multiplication(x0, c(x0)) = zero.
% 20.74/3.01 Proof:
% 20.74/3.01 multiplication(x0, c(x0))
% 20.74/3.01 = { by axiom 33 (test_2_3) R->L }
% 20.74/3.01 fresh6(complement(x0, c(x0)), true, c(x0), x0)
% 20.74/3.01 = { by lemma 50 }
% 20.74/3.01 fresh6(true, true, c(x0), x0)
% 20.74/3.01 = { by axiom 20 (test_2_3) }
% 20.74/3.01 zero
% 20.74/3.01
% 20.74/3.01 Lemma 54: addition(X, multiplication(X, Y)) = multiplication(X, addition(Y, one)).
% 20.74/3.01 Proof:
% 20.74/3.01 addition(X, multiplication(X, Y))
% 20.74/3.01 = { by axiom 4 (multiplicative_right_identity) R->L }
% 20.74/3.01 addition(multiplication(X, one), multiplication(X, Y))
% 20.74/3.01 = { by axiom 26 (right_distributivity) R->L }
% 20.74/3.01 multiplication(X, addition(one, Y))
% 20.74/3.01 = { by axiom 7 (additive_commutativity) }
% 20.74/3.01 multiplication(X, addition(Y, one))
% 20.74/3.01
% 20.74/3.01 Lemma 55: addition(X, multiplication(addition(Y, X), x1)) = addition(X, multiplication(Y, x1)).
% 20.74/3.01 Proof:
% 20.74/3.01 addition(X, multiplication(addition(Y, X), x1))
% 20.74/3.01 = { by axiom 27 (left_distributivity) }
% 20.74/3.01 addition(X, addition(multiplication(Y, x1), multiplication(X, x1)))
% 20.74/3.01 = { by axiom 7 (additive_commutativity) R->L }
% 20.74/3.01 addition(X, addition(multiplication(X, x1), multiplication(Y, x1)))
% 20.74/3.01 = { by axiom 12 (additive_associativity) }
% 20.74/3.01 addition(addition(X, multiplication(X, x1)), multiplication(Y, x1))
% 20.74/3.01 = { by lemma 54 }
% 20.74/3.01 addition(multiplication(X, addition(x1, one)), multiplication(Y, x1))
% 20.74/3.01 = { by lemma 48 }
% 20.74/3.01 addition(multiplication(X, one), multiplication(Y, x1))
% 20.74/3.01 = { by axiom 4 (multiplicative_right_identity) }
% 20.74/3.01 addition(X, multiplication(Y, x1))
% 20.74/3.01
% 20.74/3.01 Lemma 56: addition(multiplication(X, multiplication(Y, Z)), multiplication(W, Z)) = multiplication(addition(W, multiplication(X, Y)), Z).
% 20.74/3.01 Proof:
% 20.74/3.01 addition(multiplication(X, multiplication(Y, Z)), multiplication(W, Z))
% 20.74/3.01 = { by axiom 11 (multiplicative_associativity) }
% 20.74/3.01 addition(multiplication(multiplication(X, Y), Z), multiplication(W, Z))
% 20.74/3.01 = { by axiom 27 (left_distributivity) R->L }
% 20.74/3.01 multiplication(addition(multiplication(X, Y), W), Z)
% 20.74/3.01 = { by axiom 7 (additive_commutativity) }
% 20.74/3.01 multiplication(addition(W, multiplication(X, Y)), Z)
% 20.74/3.01
% 20.74/3.01 Lemma 57: multiplication(x0, x1) = multiplication(x1, x0).
% 20.74/3.01 Proof:
% 20.74/3.01 multiplication(x0, x1)
% 20.74/3.01 = { by axiom 4 (multiplicative_right_identity) R->L }
% 20.74/3.01 multiplication(multiplication(x0, x1), one)
% 20.74/3.01 = { by lemma 51 R->L }
% 20.74/3.01 multiplication(multiplication(x0, x1), addition(x0, c(x0)))
% 20.74/3.01 = { by axiom 7 (additive_commutativity) R->L }
% 20.74/3.01 multiplication(multiplication(x0, x1), addition(c(x0), x0))
% 20.74/3.01 = { by axiom 26 (right_distributivity) }
% 20.74/3.01 addition(multiplication(multiplication(x0, x1), c(x0)), multiplication(multiplication(x0, x1), x0))
% 20.74/3.01 = { by lemma 46 R->L }
% 20.74/3.01 addition(addition(zero, multiplication(multiplication(x0, x1), c(x0))), multiplication(multiplication(x0, x1), x0))
% 20.74/3.01 = { by lemma 43 R->L }
% 20.74/3.01 addition(addition(multiplication(x0, x1_2(x0)), multiplication(multiplication(x0, x1), c(x0))), multiplication(multiplication(x0, x1), x0))
% 20.74/3.01 = { by axiom 7 (additive_commutativity) R->L }
% 20.74/3.01 addition(addition(multiplication(multiplication(x0, x1), c(x0)), multiplication(x0, x1_2(x0))), multiplication(multiplication(x0, x1), x0))
% 20.74/3.01 = { by axiom 11 (multiplicative_associativity) R->L }
% 20.74/3.01 addition(addition(multiplication(x0, multiplication(x1, c(x0))), multiplication(x0, x1_2(x0))), multiplication(multiplication(x0, x1), x0))
% 20.74/3.01 = { by axiom 26 (right_distributivity) R->L }
% 20.74/3.01 addition(multiplication(x0, addition(multiplication(x1, c(x0)), x1_2(x0))), multiplication(multiplication(x0, x1), x0))
% 20.74/3.01 = { by axiom 7 (additive_commutativity) }
% 20.74/3.01 addition(multiplication(x0, addition(x1_2(x0), multiplication(x1, c(x0)))), multiplication(multiplication(x0, x1), x0))
% 20.74/3.01 = { by lemma 45 }
% 20.74/3.01 addition(multiplication(x0, addition(c(x0), multiplication(x1, c(x0)))), multiplication(multiplication(x0, x1), x0))
% 20.74/3.01 = { by lemma 52 }
% 20.74/3.01 addition(multiplication(x0, multiplication(addition(x1, one), c(x0))), multiplication(multiplication(x0, x1), x0))
% 20.74/3.01 = { by lemma 48 }
% 20.74/3.01 addition(multiplication(x0, multiplication(one, c(x0))), multiplication(multiplication(x0, x1), x0))
% 20.74/3.01 = { by axiom 5 (multiplicative_left_identity) }
% 20.74/3.01 addition(multiplication(x0, c(x0)), multiplication(multiplication(x0, x1), x0))
% 20.74/3.01 = { by lemma 53 }
% 20.74/3.01 addition(zero, multiplication(multiplication(x0, x1), x0))
% 20.74/3.01 = { by axiom 19 (test_2_2) R->L }
% 20.74/3.01 addition(fresh7(true, true, c(x0), x0), multiplication(multiplication(x0, x1), x0))
% 20.74/3.01 = { by lemma 50 R->L }
% 20.74/3.01 addition(fresh7(complement(x0, c(x0)), true, c(x0), x0), multiplication(multiplication(x0, x1), x0))
% 20.74/3.01 = { by axiom 32 (test_2_2) }
% 20.74/3.01 addition(multiplication(c(x0), x0), multiplication(multiplication(x0, x1), x0))
% 20.74/3.01 = { by axiom 27 (left_distributivity) R->L }
% 20.74/3.01 multiplication(addition(c(x0), multiplication(x0, x1)), x0)
% 20.74/3.01 = { by lemma 55 R->L }
% 20.74/3.01 multiplication(addition(c(x0), multiplication(addition(x0, c(x0)), x1)), x0)
% 20.74/3.01 = { by lemma 45 R->L }
% 20.74/3.01 multiplication(addition(x1_2(x0), multiplication(addition(x0, c(x0)), x1)), x0)
% 20.74/3.01 = { by lemma 56 R->L }
% 20.74/3.01 addition(multiplication(addition(x0, c(x0)), multiplication(x1, x0)), multiplication(x1_2(x0), x0))
% 20.74/3.01 = { by lemma 42 }
% 20.74/3.01 addition(multiplication(addition(x0, c(x0)), multiplication(x1, x0)), zero)
% 20.74/3.01 = { by axiom 8 (additive_identity) }
% 20.74/3.02 multiplication(addition(x0, c(x0)), multiplication(x1, x0))
% 20.74/3.02 = { by lemma 51 }
% 20.74/3.02 multiplication(one, multiplication(x1, x0))
% 20.74/3.02 = { by axiom 5 (multiplicative_left_identity) }
% 20.74/3.02 multiplication(x1, x0)
% 20.74/3.02
% 20.74/3.02 Lemma 58: complement(x1, c(x1)) = true.
% 20.74/3.02 Proof:
% 20.74/3.02 complement(x1, c(x1))
% 20.74/3.02 = { by axiom 21 (test_3) R->L }
% 20.74/3.02 fresh5(true, true, x1, c(x1))
% 20.74/3.02 = { by axiom 1 (goals) R->L }
% 20.74/3.02 fresh5(test(x1), true, x1, c(x1))
% 20.74/3.02 = { by axiom 25 (test_3) }
% 20.74/3.02 fresh4(c(x1), c(x1), x1, c(x1))
% 20.74/3.02 = { by axiom 22 (test_3) }
% 20.74/3.02 true
% 20.74/3.02
% 20.74/3.02 Lemma 59: addition(x1, c(x1)) = one.
% 20.74/3.02 Proof:
% 20.74/3.02 addition(x1, c(x1))
% 20.74/3.02 = { by axiom 7 (additive_commutativity) R->L }
% 20.74/3.02 addition(c(x1), x1)
% 20.74/3.02 = { by axiom 31 (test_2_1) R->L }
% 20.74/3.02 fresh8(complement(x1, c(x1)), true, c(x1), x1)
% 21.03/3.02 = { by lemma 58 }
% 21.03/3.02 fresh8(true, true, c(x1), x1)
% 21.03/3.02 = { by axiom 18 (test_2_1) }
% 21.03/3.02 one
% 21.03/3.02
% 21.03/3.02 Lemma 60: multiplication(x1, c(x1)) = zero.
% 21.03/3.02 Proof:
% 21.03/3.02 multiplication(x1, c(x1))
% 21.03/3.02 = { by axiom 33 (test_2_3) R->L }
% 21.03/3.02 fresh6(complement(x1, c(x1)), true, c(x1), x1)
% 21.03/3.02 = { by lemma 58 }
% 21.03/3.02 fresh6(true, true, c(x1), x1)
% 21.03/3.02 = { by axiom 20 (test_2_3) }
% 21.03/3.02 zero
% 21.03/3.02
% 21.03/3.02 Lemma 61: multiplication(c(x1), x1) = zero.
% 21.03/3.02 Proof:
% 21.03/3.02 multiplication(c(x1), x1)
% 21.03/3.02 = { by axiom 32 (test_2_2) R->L }
% 21.03/3.02 fresh7(complement(x1, c(x1)), true, c(x1), x1)
% 21.03/3.02 = { by lemma 58 }
% 21.03/3.02 fresh7(true, true, c(x1), x1)
% 21.03/3.02 = { by axiom 19 (test_2_2) }
% 21.03/3.02 zero
% 21.03/3.02
% 21.03/3.02 Lemma 62: addition(Y, addition(Z, X)) = addition(X, addition(Y, Z)).
% 21.03/3.02 Proof:
% 21.03/3.02 addition(Y, addition(Z, X))
% 21.03/3.02 = { by axiom 7 (additive_commutativity) R->L }
% 21.03/3.02 addition(addition(Z, X), Y)
% 21.03/3.02 = { by axiom 7 (additive_commutativity) }
% 21.03/3.02 addition(addition(X, Z), Y)
% 21.03/3.02 = { by axiom 12 (additive_associativity) R->L }
% 21.03/3.02 addition(X, addition(Z, Y))
% 21.03/3.02 = { by axiom 7 (additive_commutativity) }
% 21.03/3.02 addition(X, addition(Y, Z))
% 21.03/3.02
% 21.03/3.02 Lemma 63: addition(Z, addition(Y, X)) = addition(X, addition(Y, Z)).
% 21.03/3.02 Proof:
% 21.03/3.02 addition(Z, addition(Y, X))
% 21.03/3.02 = { by lemma 62 }
% 21.03/3.02 addition(X, addition(Z, Y))
% 21.03/3.02 = { by axiom 7 (additive_commutativity) }
% 21.03/3.02 addition(X, addition(Y, Z))
% 21.03/3.02
% 21.03/3.02 Lemma 64: addition(addition(X, Y), Y) = addition(X, Y).
% 21.03/3.02 Proof:
% 21.03/3.02 addition(addition(X, Y), Y)
% 21.03/3.02 = { by axiom 7 (additive_commutativity) R->L }
% 21.03/3.02 addition(Y, addition(X, Y))
% 21.03/3.02 = { by axiom 7 (additive_commutativity) }
% 21.03/3.02 addition(Y, addition(Y, X))
% 21.03/3.02 = { by lemma 47 }
% 21.03/3.02 addition(Y, X)
% 21.03/3.02 = { by axiom 7 (additive_commutativity) R->L }
% 21.03/3.02 addition(X, Y)
% 21.03/3.02
% 21.03/3.02 Lemma 65: addition(addition(c(x0), c(x1)), multiplication(x0, x1)) = one.
% 21.03/3.02 Proof:
% 21.03/3.02 addition(addition(c(x0), c(x1)), multiplication(x0, x1))
% 21.03/3.02 = { by axiom 7 (additive_commutativity) R->L }
% 21.03/3.02 addition(multiplication(x0, x1), addition(c(x0), c(x1)))
% 21.03/3.02 = { by lemma 63 }
% 21.03/3.02 addition(c(x1), addition(c(x0), multiplication(x0, x1)))
% 21.03/3.02 = { by lemma 55 R->L }
% 21.03/3.02 addition(c(x1), addition(c(x0), multiplication(addition(x0, c(x0)), x1)))
% 21.03/3.02 = { by lemma 63 R->L }
% 21.03/3.02 addition(multiplication(addition(x0, c(x0)), x1), addition(c(x0), c(x1)))
% 21.03/3.02 = { by lemma 51 }
% 21.03/3.02 addition(multiplication(one, x1), addition(c(x0), c(x1)))
% 21.03/3.02 = { by axiom 5 (multiplicative_left_identity) }
% 21.03/3.02 addition(x1, addition(c(x0), c(x1)))
% 21.03/3.02 = { by axiom 7 (additive_commutativity) R->L }
% 21.03/3.02 addition(addition(c(x0), c(x1)), x1)
% 21.03/3.02 = { by lemma 64 R->L }
% 21.03/3.02 addition(addition(addition(c(x0), c(x1)), c(x1)), x1)
% 21.03/3.02 = { by axiom 12 (additive_associativity) R->L }
% 21.03/3.02 addition(addition(c(x0), c(x1)), addition(c(x1), x1))
% 21.03/3.02 = { by axiom 7 (additive_commutativity) }
% 21.03/3.02 addition(addition(c(x0), c(x1)), addition(x1, c(x1)))
% 21.03/3.02 = { by lemma 59 }
% 21.03/3.02 addition(addition(c(x0), c(x1)), one)
% 21.03/3.02 = { by lemma 51 R->L }
% 21.03/3.02 addition(addition(c(x0), c(x1)), addition(x0, c(x0)))
% 21.03/3.02 = { by axiom 7 (additive_commutativity) R->L }
% 21.03/3.02 addition(addition(c(x0), c(x1)), addition(c(x0), x0))
% 21.03/3.02 = { by axiom 12 (additive_associativity) }
% 21.03/3.02 addition(addition(addition(c(x0), c(x1)), c(x0)), x0)
% 21.03/3.02 = { by axiom 7 (additive_commutativity) R->L }
% 21.03/3.02 addition(addition(c(x0), addition(c(x0), c(x1))), x0)
% 21.03/3.02 = { by lemma 62 }
% 21.03/3.02 addition(addition(c(x1), addition(c(x0), c(x0))), x0)
% 21.03/3.02 = { by axiom 6 (additive_idempotence) }
% 21.03/3.02 addition(addition(c(x1), c(x0)), x0)
% 21.03/3.02 = { by axiom 7 (additive_commutativity) R->L }
% 21.03/3.02 addition(addition(c(x0), c(x1)), x0)
% 21.03/3.02 = { by axiom 7 (additive_commutativity) }
% 21.03/3.02 addition(x0, addition(c(x0), c(x1)))
% 21.03/3.02 = { by lemma 62 }
% 21.03/3.02 addition(c(x1), addition(x0, c(x0)))
% 21.03/3.02 = { by lemma 51 }
% 21.03/3.02 addition(c(x1), one)
% 21.03/3.02 = { by lemma 59 R->L }
% 21.03/3.02 addition(c(x1), addition(x1, c(x1)))
% 21.03/3.02 = { by lemma 47 R->L }
% 21.03/3.02 addition(c(x1), addition(x1, addition(x1, c(x1))))
% 21.03/3.02 = { by axiom 7 (additive_commutativity) R->L }
% 21.03/3.02 addition(c(x1), addition(x1, addition(c(x1), x1)))
% 21.03/3.02 = { by axiom 12 (additive_associativity) }
% 21.03/3.02 addition(addition(c(x1), x1), addition(c(x1), x1))
% 21.03/3.02 = { by axiom 6 (additive_idempotence) }
% 21.03/3.02 addition(c(x1), x1)
% 21.03/3.02 = { by axiom 7 (additive_commutativity) }
% 21.03/3.02 addition(x1, c(x1))
% 21.03/3.02 = { by lemma 59 }
% 21.03/3.02 one
% 21.03/3.02
% 21.03/3.02 Lemma 66: multiplication(addition(c(x0), c(x1)), multiplication(x0, x1)) = zero.
% 21.03/3.02 Proof:
% 21.03/3.02 multiplication(addition(c(x0), c(x1)), multiplication(x0, x1))
% 21.03/3.02 = { by axiom 7 (additive_commutativity) }
% 21.03/3.02 multiplication(addition(c(x1), c(x0)), multiplication(x0, x1))
% 21.03/3.02 = { by lemma 45 R->L }
% 21.03/3.02 multiplication(addition(c(x1), x1_2(x0)), multiplication(x0, x1))
% 21.03/3.02 = { by lemma 46 R->L }
% 21.03/3.02 addition(zero, multiplication(addition(c(x1), x1_2(x0)), multiplication(x0, x1)))
% 21.03/3.02 = { by lemma 36 R->L }
% 21.03/3.02 addition(multiplication(x1_2(x1), x1), multiplication(addition(c(x1), x1_2(x0)), multiplication(x0, x1)))
% 21.03/3.02 = { by axiom 7 (additive_commutativity) R->L }
% 21.03/3.02 addition(multiplication(addition(c(x1), x1_2(x0)), multiplication(x0, x1)), multiplication(x1_2(x1), x1))
% 21.03/3.02 = { by lemma 56 }
% 21.03/3.02 multiplication(addition(x1_2(x1), multiplication(addition(c(x1), x1_2(x0)), x0)), x1)
% 21.03/3.02 = { by lemma 40 }
% 21.03/3.02 multiplication(addition(c(x1), multiplication(addition(c(x1), x1_2(x0)), x0)), x1)
% 21.03/3.02 = { by axiom 7 (additive_commutativity) R->L }
% 21.03/3.02 multiplication(addition(c(x1), multiplication(addition(x1_2(x0), c(x1)), x0)), x1)
% 21.03/3.02 = { by axiom 27 (left_distributivity) }
% 21.03/3.02 multiplication(addition(c(x1), addition(multiplication(x1_2(x0), x0), multiplication(c(x1), x0))), x1)
% 21.03/3.02 = { by lemma 42 }
% 21.03/3.02 multiplication(addition(c(x1), addition(zero, multiplication(c(x1), x0))), x1)
% 21.03/3.02 = { by lemma 46 }
% 21.03/3.02 multiplication(addition(c(x1), multiplication(c(x1), x0)), x1)
% 21.03/3.02 = { by lemma 54 }
% 21.03/3.02 multiplication(multiplication(c(x1), addition(x0, one)), x1)
% 21.03/3.02 = { by axiom 11 (multiplicative_associativity) R->L }
% 21.03/3.02 multiplication(c(x1), multiplication(addition(x0, one), x1))
% 21.03/3.02 = { by lemma 49 }
% 21.03/3.02 multiplication(c(x1), multiplication(one, x1))
% 21.03/3.02 = { by axiom 5 (multiplicative_left_identity) }
% 21.03/3.02 multiplication(c(x1), x1)
% 21.03/3.02 = { by lemma 61 }
% 21.03/3.02 zero
% 21.03/3.02
% 21.03/3.02 Lemma 67: complement(multiplication(x0, x1), c(multiplication(x0, x1))) = true.
% 21.03/3.02 Proof:
% 21.03/3.02 complement(multiplication(x0, x1), c(multiplication(x0, x1)))
% 21.03/3.02 = { by axiom 21 (test_3) R->L }
% 21.03/3.02 fresh5(true, true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 21.03/3.02 = { by axiom 10 (test_1_1) R->L }
% 21.03/3.02 fresh5(fresh10(true, true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 21.03/3.02 = { by axiom 14 (test_2) R->L }
% 21.03/3.02 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)))
% 21.03/3.02 = { by lemma 65 R->L }
% 21.03/3.02 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)))
% 21.03/3.02 = { by lemma 38 }
% 21.03/3.02 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)))
% 21.03/3.02 = { by lemma 66 R->L }
% 21.03/3.02 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)))
% 21.03/3.02 = { by lemma 57 }
% 21.03/3.02 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)))
% 21.03/3.02 = { by lemma 57 }
% 21.03/3.02 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)))
% 21.03/3.02 = { by axiom 30 (test_2) }
% 21.03/3.02 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)))
% 21.03/3.02 = { by axiom 11 (multiplicative_associativity) R->L }
% 21.03/3.02 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)))
% 21.03/3.02 = { by axiom 26 (right_distributivity) }
% 21.03/3.02 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)))
% 21.03/3.02 = { by lemma 53 }
% 21.03/3.02 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)))
% 21.03/3.02 = { by lemma 46 }
% 21.03/3.02 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)))
% 21.03/3.02 = { by lemma 40 R->L }
% 21.03/3.02 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)))
% 21.03/3.02 = { by axiom 8 (additive_identity) R->L }
% 21.03/3.02 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)))
% 21.03/3.02 = { by lemma 37 R->L }
% 21.03/3.02 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)))
% 21.03/3.02 = { by lemma 56 }
% 21.03/3.02 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)))
% 21.03/3.02 = { by lemma 40 }
% 21.03/3.02 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)))
% 21.03/3.02 = { by lemma 57 R->L }
% 21.03/3.02 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)))
% 21.03/3.02 = { by axiom 27 (left_distributivity) }
% 21.03/3.02 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)))
% 21.03/3.02 = { by lemma 60 }
% 21.03/3.02 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)))
% 21.03/3.02 = { by lemma 46 }
% 21.03/3.02 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)))
% 21.03/3.02 = { by axiom 11 (multiplicative_associativity) R->L }
% 21.03/3.02 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)))
% 21.03/3.02 = { by lemma 60 }
% 21.03/3.02 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)))
% 21.03/3.02 = { by axiom 3 (right_annihilation) }
% 21.03/3.02 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)))
% 21.03/3.02 = { by axiom 17 (test_2) }
% 21.03/3.02 fresh5(fresh10(complement(addition(c(x0), c(x1)), multiplication(x1, x0)), true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 21.03/3.02 = { by lemma 57 R->L }
% 21.03/3.02 fresh5(fresh10(complement(addition(c(x0), c(x1)), multiplication(x0, x1)), true, multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 21.03/3.02 = { by axiom 24 (test_1_1) }
% 21.03/3.02 fresh5(test(multiplication(x0, x1)), true, multiplication(x0, x1), c(multiplication(x0, x1)))
% 21.03/3.02 = { by axiom 25 (test_3) }
% 21.03/3.02 fresh4(c(multiplication(x0, x1)), c(multiplication(x0, x1)), multiplication(x0, x1), c(multiplication(x0, x1)))
% 21.03/3.02 = { by axiom 22 (test_3) }
% 21.03/3.02 true
% 21.03/3.02
% 21.03/3.02 Lemma 68: addition(multiplication(x0, x1), c(multiplication(x0, x1))) = one.
% 21.03/3.02 Proof:
% 21.03/3.02 addition(multiplication(x0, x1), c(multiplication(x0, x1)))
% 21.03/3.02 = { by axiom 7 (additive_commutativity) R->L }
% 21.03/3.02 addition(c(multiplication(x0, x1)), multiplication(x0, x1))
% 21.03/3.02 = { by axiom 31 (test_2_1) R->L }
% 21.03/3.02 fresh8(complement(multiplication(x0, x1), c(multiplication(x0, x1))), true, c(multiplication(x0, x1)), multiplication(x0, x1))
% 21.03/3.02 = { by lemma 67 }
% 21.03/3.02 fresh8(true, true, c(multiplication(x0, x1)), multiplication(x0, x1))
% 21.03/3.02 = { by axiom 18 (test_2_1) }
% 21.03/3.02 one
% 21.03/3.02
% 21.03/3.02 Lemma 69: multiplication(addition(c(x0), c(x1)), c(multiplication(x0, x1))) = c(multiplication(x0, x1)).
% 21.03/3.02 Proof:
% 21.03/3.02 multiplication(addition(c(x0), c(x1)), c(multiplication(x0, x1)))
% 21.03/3.02 = { by lemma 46 R->L }
% 21.03/3.02 addition(zero, multiplication(addition(c(x0), c(x1)), c(multiplication(x0, x1))))
% 21.03/3.02 = { by axiom 20 (test_2_3) R->L }
% 21.03/3.02 addition(fresh6(true, true, c(multiplication(x0, x1)), multiplication(x0, x1)), multiplication(addition(c(x0), c(x1)), c(multiplication(x0, x1))))
% 21.03/3.02 = { by lemma 67 R->L }
% 21.03/3.02 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))))
% 21.03/3.02 = { by axiom 33 (test_2_3) }
% 21.03/3.02 addition(multiplication(multiplication(x0, x1), c(multiplication(x0, x1))), multiplication(addition(c(x0), c(x1)), c(multiplication(x0, x1))))
% 21.03/3.02 = { by axiom 27 (left_distributivity) R->L }
% 21.03/3.02 multiplication(addition(multiplication(x0, x1), addition(c(x0), c(x1))), c(multiplication(x0, x1)))
% 21.03/3.02 = { by axiom 7 (additive_commutativity) }
% 21.03/3.02 multiplication(addition(addition(c(x0), c(x1)), multiplication(x0, x1)), c(multiplication(x0, x1)))
% 21.03/3.03 = { by lemma 65 }
% 21.03/3.03 multiplication(one, c(multiplication(x0, x1)))
% 21.03/3.03 = { by axiom 5 (multiplicative_left_identity) }
% 21.03/3.03 c(multiplication(x0, x1))
% 21.03/3.03
% 21.03/3.03 Lemma 70: addition(addition(c(x0), c(x1)), multiplication(c(x1), c(multiplication(x0, x1)))) = multiplication(addition(c(x0), c(x1)), c(multiplication(x0, x1))).
% 21.03/3.03 Proof:
% 21.03/3.03 addition(addition(c(x0), c(x1)), multiplication(c(x1), c(multiplication(x0, x1))))
% 21.03/3.03 = { by axiom 7 (additive_commutativity) R->L }
% 21.03/3.03 addition(multiplication(c(x1), c(multiplication(x0, x1))), addition(c(x0), c(x1)))
% 21.03/3.03 = { by axiom 4 (multiplicative_right_identity) R->L }
% 21.03/3.03 addition(multiplication(c(x1), c(multiplication(x0, x1))), multiplication(addition(c(x0), c(x1)), one))
% 21.03/3.03 = { by lemma 68 R->L }
% 21.03/3.03 addition(multiplication(c(x1), c(multiplication(x0, x1))), multiplication(addition(c(x0), c(x1)), addition(multiplication(x0, x1), c(multiplication(x0, x1)))))
% 21.03/3.03 = { by axiom 7 (additive_commutativity) R->L }
% 21.03/3.03 addition(multiplication(addition(c(x0), c(x1)), addition(multiplication(x0, x1), c(multiplication(x0, x1)))), multiplication(c(x1), c(multiplication(x0, x1))))
% 21.03/3.03 = { by axiom 26 (right_distributivity) }
% 21.03/3.03 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))))
% 21.03/3.03 = { by axiom 12 (additive_associativity) R->L }
% 21.03/3.03 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)))))
% 21.03/3.03 = { by axiom 7 (additive_commutativity) }
% 21.03/3.03 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)))))
% 21.03/3.03 = { by axiom 27 (left_distributivity) R->L }
% 21.03/3.03 addition(multiplication(addition(c(x0), c(x1)), multiplication(x0, x1)), multiplication(addition(c(x1), addition(c(x0), c(x1))), c(multiplication(x0, x1))))
% 21.03/3.03 = { by axiom 7 (additive_commutativity) }
% 21.03/3.03 addition(multiplication(addition(c(x0), c(x1)), multiplication(x0, x1)), multiplication(addition(addition(c(x0), c(x1)), c(x1)), c(multiplication(x0, x1))))
% 21.03/3.03 = { by lemma 64 }
% 21.03/3.03 addition(multiplication(addition(c(x0), c(x1)), multiplication(x0, x1)), multiplication(addition(c(x0), c(x1)), c(multiplication(x0, x1))))
% 21.03/3.03 = { by lemma 66 }
% 21.03/3.03 addition(zero, multiplication(addition(c(x0), c(x1)), c(multiplication(x0, x1))))
% 21.03/3.03 = { by lemma 46 }
% 21.03/3.03 multiplication(addition(c(x0), c(x1)), c(multiplication(x0, x1)))
% 21.03/3.03
% 21.03/3.03 Goal 1 (goals_2): tuple(leq(addition(c(x0), c(x1)), c(multiplication(x0, x1))), leq(c(multiplication(x0, x1)), addition(c(x0), c(x1)))) = tuple(true, true).
% 21.03/3.03 Proof:
% 21.03/3.03 tuple(leq(addition(c(x0), c(x1)), c(multiplication(x0, x1))), leq(c(multiplication(x0, x1)), addition(c(x0), c(x1))))
% 21.03/3.03 = { by lemma 69 R->L }
% 21.03/3.03 tuple(leq(addition(c(x0), c(x1)), c(multiplication(x0, x1))), leq(multiplication(addition(c(x0), c(x1)), c(multiplication(x0, x1))), addition(c(x0), c(x1))))
% 21.03/3.03 = { by lemma 70 R->L }
% 21.03/3.03 tuple(leq(addition(c(x0), c(x1)), c(multiplication(x0, x1))), leq(addition(addition(c(x0), c(x1)), multiplication(c(x1), c(multiplication(x0, x1)))), addition(c(x0), c(x1))))
% 21.03/3.03 = { by axiom 7 (additive_commutativity) }
% 21.03/3.03 tuple(leq(addition(c(x0), c(x1)), c(multiplication(x0, x1))), leq(addition(multiplication(c(x1), c(multiplication(x0, x1))), addition(c(x0), c(x1))), addition(c(x0), c(x1))))
% 21.03/3.03 = { by lemma 46 R->L }
% 21.03/3.03 tuple(leq(addition(c(x0), c(x1)), c(multiplication(x0, x1))), leq(addition(addition(zero, multiplication(c(x1), c(multiplication(x0, x1)))), addition(c(x0), c(x1))), addition(c(x0), c(x1))))
% 21.03/3.03 = { by lemma 61 R->L }
% 21.03/3.03 tuple(leq(addition(c(x0), c(x1)), c(multiplication(x0, x1))), leq(addition(addition(multiplication(c(x1), x1), multiplication(c(x1), c(multiplication(x0, x1)))), addition(c(x0), c(x1))), addition(c(x0), c(x1))))
% 21.03/3.03 = { by axiom 26 (right_distributivity) R->L }
% 21.03/3.03 tuple(leq(addition(c(x0), c(x1)), c(multiplication(x0, x1))), leq(addition(multiplication(c(x1), addition(x1, c(multiplication(x0, x1)))), addition(c(x0), c(x1))), addition(c(x0), c(x1))))
% 21.03/3.03 = { by axiom 5 (multiplicative_left_identity) R->L }
% 21.03/3.03 tuple(leq(addition(c(x0), c(x1)), c(multiplication(x0, x1))), leq(addition(multiplication(c(x1), addition(multiplication(one, x1), c(multiplication(x0, x1)))), addition(c(x0), c(x1))), addition(c(x0), c(x1))))
% 21.03/3.03 = { by lemma 49 R->L }
% 21.03/3.03 tuple(leq(addition(c(x0), c(x1)), c(multiplication(x0, x1))), leq(addition(multiplication(c(x1), addition(multiplication(addition(x0, one), x1), c(multiplication(x0, x1)))), addition(c(x0), c(x1))), addition(c(x0), c(x1))))
% 21.03/3.03 = { by lemma 52 R->L }
% 21.03/3.03 tuple(leq(addition(c(x0), c(x1)), c(multiplication(x0, x1))), leq(addition(multiplication(c(x1), addition(addition(x1, multiplication(x0, x1)), c(multiplication(x0, x1)))), addition(c(x0), c(x1))), addition(c(x0), c(x1))))
% 21.03/3.03 = { by axiom 12 (additive_associativity) R->L }
% 21.03/3.03 tuple(leq(addition(c(x0), c(x1)), c(multiplication(x0, x1))), leq(addition(multiplication(c(x1), addition(x1, addition(multiplication(x0, x1), c(multiplication(x0, x1))))), addition(c(x0), c(x1))), addition(c(x0), c(x1))))
% 21.03/3.03 = { by lemma 68 }
% 21.03/3.03 tuple(leq(addition(c(x0), c(x1)), c(multiplication(x0, x1))), leq(addition(multiplication(c(x1), addition(x1, one)), addition(c(x0), c(x1))), addition(c(x0), c(x1))))
% 21.03/3.03 = { by lemma 48 }
% 21.03/3.03 tuple(leq(addition(c(x0), c(x1)), c(multiplication(x0, x1))), leq(addition(multiplication(c(x1), one), addition(c(x0), c(x1))), addition(c(x0), c(x1))))
% 21.03/3.03 = { by axiom 4 (multiplicative_right_identity) }
% 21.03/3.03 tuple(leq(addition(c(x0), c(x1)), c(multiplication(x0, x1))), leq(addition(c(x1), addition(c(x0), c(x1))), addition(c(x0), c(x1))))
% 21.03/3.03 = { by axiom 7 (additive_commutativity) }
% 21.03/3.03 tuple(leq(addition(c(x0), c(x1)), c(multiplication(x0, x1))), leq(addition(addition(c(x0), c(x1)), c(x1)), addition(c(x0), c(x1))))
% 21.03/3.03 = { by lemma 64 }
% 21.03/3.03 tuple(leq(addition(c(x0), c(x1)), c(multiplication(x0, x1))), leq(addition(c(x0), c(x1)), addition(c(x0), c(x1))))
% 21.03/3.03 = { by axiom 29 (order) R->L }
% 21.03/3.03 tuple(leq(addition(c(x0), c(x1)), c(multiplication(x0, x1))), fresh11(addition(addition(c(x0), c(x1)), addition(c(x0), c(x1))), addition(c(x0), c(x1)), addition(c(x0), c(x1)), addition(c(x0), c(x1))))
% 21.03/3.03 = { by axiom 6 (additive_idempotence) }
% 21.03/3.03 tuple(leq(addition(c(x0), c(x1)), c(multiplication(x0, x1))), fresh11(addition(c(x0), c(x1)), addition(c(x0), c(x1)), addition(c(x0), c(x1)), addition(c(x0), c(x1))))
% 21.03/3.03 = { by axiom 16 (order) }
% 21.03/3.03 tuple(leq(addition(c(x0), c(x1)), c(multiplication(x0, x1))), true)
% 21.03/3.03 = { by axiom 6 (additive_idempotence) R->L }
% 21.03/3.03 tuple(leq(addition(c(x0), c(x1)), addition(c(multiplication(x0, x1)), c(multiplication(x0, x1)))), true)
% 21.03/3.03 = { by lemma 69 R->L }
% 21.03/3.03 tuple(leq(addition(c(x0), c(x1)), addition(c(multiplication(x0, x1)), multiplication(addition(c(x0), c(x1)), c(multiplication(x0, x1))))), true)
% 21.03/3.03 = { by lemma 70 R->L }
% 21.03/3.03 tuple(leq(addition(c(x0), c(x1)), addition(c(multiplication(x0, x1)), addition(addition(c(x0), c(x1)), multiplication(c(x1), c(multiplication(x0, x1)))))), true)
% 21.03/3.03 = { by axiom 7 (additive_commutativity) R->L }
% 21.03/3.03 tuple(leq(addition(c(x0), c(x1)), addition(c(multiplication(x0, x1)), addition(multiplication(c(x1), c(multiplication(x0, x1))), addition(c(x0), c(x1))))), true)
% 21.03/3.03 = { by axiom 12 (additive_associativity) }
% 21.03/3.03 tuple(leq(addition(c(x0), c(x1)), addition(addition(c(multiplication(x0, x1)), multiplication(c(x1), c(multiplication(x0, x1)))), addition(c(x0), c(x1)))), true)
% 21.03/3.03 = { by axiom 7 (additive_commutativity) }
% 21.03/3.03 tuple(leq(addition(c(x0), c(x1)), addition(addition(c(x0), c(x1)), addition(c(multiplication(x0, x1)), multiplication(c(x1), c(multiplication(x0, x1)))))), true)
% 21.12/3.03 = { by axiom 29 (order) R->L }
% 21.12/3.03 tuple(fresh11(addition(addition(c(x0), c(x1)), addition(addition(c(x0), c(x1)), addition(c(multiplication(x0, x1)), multiplication(c(x1), c(multiplication(x0, x1)))))), addition(addition(c(x0), c(x1)), addition(c(multiplication(x0, x1)), multiplication(c(x1), c(multiplication(x0, x1))))), addition(c(x0), c(x1)), addition(addition(c(x0), c(x1)), addition(c(multiplication(x0, x1)), multiplication(c(x1), c(multiplication(x0, x1)))))), true)
% 21.12/3.03 = { by lemma 47 }
% 21.12/3.03 tuple(fresh11(addition(addition(c(x0), c(x1)), addition(c(multiplication(x0, x1)), multiplication(c(x1), c(multiplication(x0, x1))))), addition(addition(c(x0), c(x1)), addition(c(multiplication(x0, x1)), multiplication(c(x1), c(multiplication(x0, x1))))), addition(c(x0), c(x1)), addition(addition(c(x0), c(x1)), addition(c(multiplication(x0, x1)), multiplication(c(x1), c(multiplication(x0, x1)))))), true)
% 21.12/3.03 = { by axiom 16 (order) }
% 21.12/3.03 tuple(true, true)
% 21.12/3.03 % SZS output end Proof
% 21.12/3.03
% 21.12/3.03 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------