%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : HEN006-3 : TPTP v9.3.1. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n005.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 29 10:18:57 AM UTC 2026
% Result : Unsatisfiable 0.12s 0.45s
% Output : Proof 0.45s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : HEN006-3 : TPTP v9.3.1. Released v1.0.0.
% 0.00/0.03 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.08/0.35 % Computer : n005.cluster.edu
% 0.08/0.35 % Model : x86_64 x86_64
% 0.08/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.08/0.35 % Memory : 8046.5625MB
% 0.08/0.35 % OS : Linux 6.8.0-71-generic
% 0.08/0.35 % CPULimit : 300
% 0.08/0.35 % WCLimit : 300
% 0.08/0.35 % DateTime : Sun Sep 27 10:50:37 UTC 2026
% 0.08/0.35 % CPUTime :
% 0.08/0.35 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.12/0.45 Command-line arguments: --no-flatten-goal
% 0.12/0.45
% 0.12/0.45 % SZS status Unsatisfiable
% 0.12/0.45
% 0.45/0.48 % SZS output start Proof
% 0.45/0.48 Axiom 1 (zero_is_smallest): less_equal(zero, X) = true.
% 0.45/0.48 Axiom 2 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 0.45/0.48 Axiom 3 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 0.45/0.48 Axiom 4 (quotient_smaller_than_numerator): less_equal(divide(X, Y), X) = true.
% 0.45/0.48 Axiom 5 (a_divide_b_LE_d): less_equal(divide(a, b), d) = true.
% 0.45/0.48 Axiom 6 (quotient_less_equal2): ifeq2(divide(X, Y), zero, less_equal(X, Y), true) = true.
% 0.45/0.48 Axiom 7 (quotient_less_equal1): ifeq(less_equal(X, Y), true, divide(X, Y), zero) = zero.
% 0.45/0.48 Axiom 8 (less_equal_and_equal): ifeq(less_equal(X, Y), true, ifeq(less_equal(Y, X), true, Y, X), X) = X.
% 0.45/0.48 Axiom 9 (quotient_property): less_equal(divide(divide(X, Y), divide(Z, Y)), divide(divide(X, Z), Y)) = true.
% 0.45/0.48
% 0.45/0.48 Lemma 10: divide(zero, X) = zero.
% 0.45/0.48 Proof:
% 0.45/0.48 divide(zero, X)
% 0.45/0.48 = { by axiom 3 (ifeq_axiom) R->L }
% 0.45/0.48 ifeq(true, true, divide(zero, X), zero)
% 0.45/0.48 = { by axiom 1 (zero_is_smallest) R->L }
% 0.45/0.48 ifeq(less_equal(zero, X), true, divide(zero, X), zero)
% 0.45/0.48 = { by axiom 7 (quotient_less_equal1) }
% 0.45/0.48 zero
% 0.45/0.48
% 0.45/0.48 Lemma 11: divide(divide(X, Y), X) = zero.
% 0.45/0.48 Proof:
% 0.45/0.48 divide(divide(X, Y), X)
% 0.45/0.48 = { by axiom 3 (ifeq_axiom) R->L }
% 0.45/0.48 ifeq(true, true, divide(divide(X, Y), X), zero)
% 0.45/0.48 = { by axiom 4 (quotient_smaller_than_numerator) R->L }
% 0.45/0.48 ifeq(less_equal(divide(X, Y), X), true, divide(divide(X, Y), X), zero)
% 0.45/0.48 = { by axiom 7 (quotient_less_equal1) }
% 0.45/0.48 zero
% 0.45/0.48
% 0.45/0.48 Lemma 12: divide(X, zero) = X.
% 0.45/0.48 Proof:
% 0.45/0.48 divide(X, zero)
% 0.45/0.48 = { by axiom 3 (ifeq_axiom) R->L }
% 0.45/0.48 ifeq(true, true, divide(X, zero), X)
% 0.45/0.48 = { by axiom 4 (quotient_smaller_than_numerator) R->L }
% 0.45/0.48 ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X)
% 0.45/0.48 = { by axiom 3 (ifeq_axiom) R->L }
% 0.45/0.48 ifeq(true, true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.48 = { by axiom 6 (quotient_less_equal2) R->L }
% 0.45/0.48 ifeq(ifeq2(divide(X, divide(X, zero)), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.48 = { by axiom 8 (less_equal_and_equal) R->L }
% 0.45/0.48 ifeq(ifeq2(ifeq(less_equal(divide(X, divide(X, zero)), zero), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.48 = { by axiom 2 (ifeq_axiom) R->L }
% 0.45/0.48 ifeq(ifeq2(ifeq(ifeq2(zero, zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.48 = { by axiom 3 (ifeq_axiom) R->L }
% 0.45/0.48 ifeq(ifeq2(ifeq(ifeq2(ifeq(true, true, zero, divide(divide(X, divide(X, zero)), zero)), zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.48 = { by axiom 1 (zero_is_smallest) R->L }
% 0.45/0.48 ifeq(ifeq2(ifeq(ifeq2(ifeq(less_equal(zero, divide(divide(X, divide(X, zero)), zero)), true, zero, divide(divide(X, divide(X, zero)), zero)), zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.48 = { by axiom 3 (ifeq_axiom) R->L }
% 0.45/0.48 ifeq(ifeq2(ifeq(ifeq2(ifeq(true, true, ifeq(less_equal(zero, divide(divide(X, divide(X, zero)), zero)), true, zero, divide(divide(X, divide(X, zero)), zero)), divide(divide(X, divide(X, zero)), zero)), zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.48 = { by axiom 9 (quotient_property) R->L }
% 0.45/0.48 ifeq(ifeq2(ifeq(ifeq2(ifeq(less_equal(divide(divide(X, divide(X, zero)), divide(zero, divide(X, zero))), divide(divide(X, zero), divide(X, zero))), true, ifeq(less_equal(zero, divide(divide(X, divide(X, zero)), zero)), true, zero, divide(divide(X, divide(X, zero)), zero)), divide(divide(X, divide(X, zero)), zero)), zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.48 = { by lemma 10 }
% 0.45/0.48 ifeq(ifeq2(ifeq(ifeq2(ifeq(less_equal(divide(divide(X, divide(X, zero)), zero), divide(divide(X, zero), divide(X, zero))), true, ifeq(less_equal(zero, divide(divide(X, divide(X, zero)), zero)), true, zero, divide(divide(X, divide(X, zero)), zero)), divide(divide(X, divide(X, zero)), zero)), zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.48 = { by axiom 8 (less_equal_and_equal) R->L }
% 0.45/0.48 ifeq(ifeq2(ifeq(ifeq2(ifeq(less_equal(divide(divide(X, divide(X, zero)), zero), ifeq(less_equal(divide(divide(X, zero), divide(X, zero)), zero), true, ifeq(less_equal(zero, divide(divide(X, zero), divide(X, zero))), true, zero, divide(divide(X, zero), divide(X, zero))), divide(divide(X, zero), divide(X, zero)))), true, ifeq(less_equal(zero, divide(divide(X, divide(X, zero)), zero)), true, zero, divide(divide(X, divide(X, zero)), zero)), divide(divide(X, divide(X, zero)), zero)), zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.48 = { by axiom 2 (ifeq_axiom) R->L }
% 0.45/0.48 ifeq(ifeq2(ifeq(ifeq2(ifeq(less_equal(divide(divide(X, divide(X, zero)), zero), ifeq(ifeq2(zero, zero, less_equal(divide(divide(X, zero), divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(divide(X, zero), divide(X, zero))), true, zero, divide(divide(X, zero), divide(X, zero))), divide(divide(X, zero), divide(X, zero)))), true, ifeq(less_equal(zero, divide(divide(X, divide(X, zero)), zero)), true, zero, divide(divide(X, divide(X, zero)), zero)), divide(divide(X, divide(X, zero)), zero)), zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.48 = { by axiom 3 (ifeq_axiom) R->L }
% 0.45/0.48 ifeq(ifeq2(ifeq(ifeq2(ifeq(less_equal(divide(divide(X, divide(X, zero)), zero), ifeq(ifeq2(ifeq(true, true, zero, divide(divide(divide(X, zero), divide(X, zero)), divide(zero, divide(X, zero)))), zero, less_equal(divide(divide(X, zero), divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(divide(X, zero), divide(X, zero))), true, zero, divide(divide(X, zero), divide(X, zero))), divide(divide(X, zero), divide(X, zero)))), true, ifeq(less_equal(zero, divide(divide(X, divide(X, zero)), zero)), true, zero, divide(divide(X, divide(X, zero)), zero)), divide(divide(X, divide(X, zero)), zero)), zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.48 = { by axiom 1 (zero_is_smallest) R->L }
% 0.45/0.49 ifeq(ifeq2(ifeq(ifeq2(ifeq(less_equal(divide(divide(X, divide(X, zero)), zero), ifeq(ifeq2(ifeq(less_equal(zero, divide(divide(divide(X, zero), divide(X, zero)), divide(zero, divide(X, zero)))), true, zero, divide(divide(divide(X, zero), divide(X, zero)), divide(zero, divide(X, zero)))), zero, less_equal(divide(divide(X, zero), divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(divide(X, zero), divide(X, zero))), true, zero, divide(divide(X, zero), divide(X, zero))), divide(divide(X, zero), divide(X, zero)))), true, ifeq(less_equal(zero, divide(divide(X, divide(X, zero)), zero)), true, zero, divide(divide(X, divide(X, zero)), zero)), divide(divide(X, divide(X, zero)), zero)), zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.49 = { by axiom 3 (ifeq_axiom) R->L }
% 0.45/0.49 ifeq(ifeq2(ifeq(ifeq2(ifeq(less_equal(divide(divide(X, divide(X, zero)), zero), ifeq(ifeq2(ifeq(true, true, ifeq(less_equal(zero, divide(divide(divide(X, zero), divide(X, zero)), divide(zero, divide(X, zero)))), true, zero, divide(divide(divide(X, zero), divide(X, zero)), divide(zero, divide(X, zero)))), divide(divide(divide(X, zero), divide(X, zero)), divide(zero, divide(X, zero)))), zero, less_equal(divide(divide(X, zero), divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(divide(X, zero), divide(X, zero))), true, zero, divide(divide(X, zero), divide(X, zero))), divide(divide(X, zero), divide(X, zero)))), true, ifeq(less_equal(zero, divide(divide(X, divide(X, zero)), zero)), true, zero, divide(divide(X, divide(X, zero)), zero)), divide(divide(X, divide(X, zero)), zero)), zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.49 = { by axiom 9 (quotient_property) R->L }
% 0.45/0.49 ifeq(ifeq2(ifeq(ifeq2(ifeq(less_equal(divide(divide(X, divide(X, zero)), zero), ifeq(ifeq2(ifeq(less_equal(divide(divide(divide(X, zero), divide(X, zero)), divide(zero, divide(X, zero))), divide(divide(divide(X, zero), zero), divide(X, zero))), true, ifeq(less_equal(zero, divide(divide(divide(X, zero), divide(X, zero)), divide(zero, divide(X, zero)))), true, zero, divide(divide(divide(X, zero), divide(X, zero)), divide(zero, divide(X, zero)))), divide(divide(divide(X, zero), divide(X, zero)), divide(zero, divide(X, zero)))), zero, less_equal(divide(divide(X, zero), divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(divide(X, zero), divide(X, zero))), true, zero, divide(divide(X, zero), divide(X, zero))), divide(divide(X, zero), divide(X, zero)))), true, ifeq(less_equal(zero, divide(divide(X, divide(X, zero)), zero)), true, zero, divide(divide(X, divide(X, zero)), zero)), divide(divide(X, divide(X, zero)), zero)), zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.49 = { by lemma 11 }
% 0.45/0.49 ifeq(ifeq2(ifeq(ifeq2(ifeq(less_equal(divide(divide(X, divide(X, zero)), zero), ifeq(ifeq2(ifeq(less_equal(divide(divide(divide(X, zero), divide(X, zero)), divide(zero, divide(X, zero))), zero), true, ifeq(less_equal(zero, divide(divide(divide(X, zero), divide(X, zero)), divide(zero, divide(X, zero)))), true, zero, divide(divide(divide(X, zero), divide(X, zero)), divide(zero, divide(X, zero)))), divide(divide(divide(X, zero), divide(X, zero)), divide(zero, divide(X, zero)))), zero, less_equal(divide(divide(X, zero), divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(divide(X, zero), divide(X, zero))), true, zero, divide(divide(X, zero), divide(X, zero))), divide(divide(X, zero), divide(X, zero)))), true, ifeq(less_equal(zero, divide(divide(X, divide(X, zero)), zero)), true, zero, divide(divide(X, divide(X, zero)), zero)), divide(divide(X, divide(X, zero)), zero)), zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.49 = { by axiom 8 (less_equal_and_equal) }
% 0.45/0.49 ifeq(ifeq2(ifeq(ifeq2(ifeq(less_equal(divide(divide(X, divide(X, zero)), zero), ifeq(ifeq2(divide(divide(divide(X, zero), divide(X, zero)), divide(zero, divide(X, zero))), zero, less_equal(divide(divide(X, zero), divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(divide(X, zero), divide(X, zero))), true, zero, divide(divide(X, zero), divide(X, zero))), divide(divide(X, zero), divide(X, zero)))), true, ifeq(less_equal(zero, divide(divide(X, divide(X, zero)), zero)), true, zero, divide(divide(X, divide(X, zero)), zero)), divide(divide(X, divide(X, zero)), zero)), zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.49 = { by lemma 10 }
% 0.45/0.49 ifeq(ifeq2(ifeq(ifeq2(ifeq(less_equal(divide(divide(X, divide(X, zero)), zero), ifeq(ifeq2(divide(divide(divide(X, zero), divide(X, zero)), zero), zero, less_equal(divide(divide(X, zero), divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(divide(X, zero), divide(X, zero))), true, zero, divide(divide(X, zero), divide(X, zero))), divide(divide(X, zero), divide(X, zero)))), true, ifeq(less_equal(zero, divide(divide(X, divide(X, zero)), zero)), true, zero, divide(divide(X, divide(X, zero)), zero)), divide(divide(X, divide(X, zero)), zero)), zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.49 = { by axiom 6 (quotient_less_equal2) }
% 0.45/0.49 ifeq(ifeq2(ifeq(ifeq2(ifeq(less_equal(divide(divide(X, divide(X, zero)), zero), ifeq(true, true, ifeq(less_equal(zero, divide(divide(X, zero), divide(X, zero))), true, zero, divide(divide(X, zero), divide(X, zero))), divide(divide(X, zero), divide(X, zero)))), true, ifeq(less_equal(zero, divide(divide(X, divide(X, zero)), zero)), true, zero, divide(divide(X, divide(X, zero)), zero)), divide(divide(X, divide(X, zero)), zero)), zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.49 = { by axiom 3 (ifeq_axiom) }
% 0.45/0.49 ifeq(ifeq2(ifeq(ifeq2(ifeq(less_equal(divide(divide(X, divide(X, zero)), zero), ifeq(less_equal(zero, divide(divide(X, zero), divide(X, zero))), true, zero, divide(divide(X, zero), divide(X, zero)))), true, ifeq(less_equal(zero, divide(divide(X, divide(X, zero)), zero)), true, zero, divide(divide(X, divide(X, zero)), zero)), divide(divide(X, divide(X, zero)), zero)), zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.49 = { by axiom 1 (zero_is_smallest) }
% 0.45/0.49 ifeq(ifeq2(ifeq(ifeq2(ifeq(less_equal(divide(divide(X, divide(X, zero)), zero), ifeq(true, true, zero, divide(divide(X, zero), divide(X, zero)))), true, ifeq(less_equal(zero, divide(divide(X, divide(X, zero)), zero)), true, zero, divide(divide(X, divide(X, zero)), zero)), divide(divide(X, divide(X, zero)), zero)), zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.49 = { by axiom 3 (ifeq_axiom) }
% 0.45/0.49 ifeq(ifeq2(ifeq(ifeq2(ifeq(less_equal(divide(divide(X, divide(X, zero)), zero), zero), true, ifeq(less_equal(zero, divide(divide(X, divide(X, zero)), zero)), true, zero, divide(divide(X, divide(X, zero)), zero)), divide(divide(X, divide(X, zero)), zero)), zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.49 = { by axiom 8 (less_equal_and_equal) }
% 0.45/0.49 ifeq(ifeq2(ifeq(ifeq2(divide(divide(X, divide(X, zero)), zero), zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.49 = { by axiom 6 (quotient_less_equal2) }
% 0.45/0.49 ifeq(ifeq2(ifeq(true, true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.49 = { by axiom 3 (ifeq_axiom) }
% 0.45/0.49 ifeq(ifeq2(ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.49 = { by axiom 1 (zero_is_smallest) }
% 0.45/0.49 ifeq(ifeq2(ifeq(true, true, zero, divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.49 = { by axiom 3 (ifeq_axiom) }
% 0.45/0.49 ifeq(ifeq2(zero, zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.49 = { by axiom 2 (ifeq_axiom) }
% 0.45/0.49 ifeq(less_equal(X, divide(X, zero)), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 0.45/0.49 = { by axiom 8 (less_equal_and_equal) }
% 0.45/0.49 X
% 0.45/0.49
% 0.45/0.49 Goal 1 (prove_a_divide_d_LE_b): less_equal(divide(a, d), b) = true.
% 0.45/0.49 Proof:
% 0.45/0.49 less_equal(divide(a, d), b)
% 0.45/0.49 = { by axiom 2 (ifeq_axiom) R->L }
% 0.45/0.49 ifeq2(zero, zero, less_equal(divide(a, d), b), true)
% 0.45/0.49 = { by axiom 7 (quotient_less_equal1) R->L }
% 0.45/0.49 ifeq2(ifeq(less_equal(divide(divide(divide(a, d), b), divide(divide(b, d), b)), divide(divide(divide(a, d), divide(b, d)), b)), true, divide(divide(divide(divide(a, d), b), divide(divide(b, d), b)), divide(divide(divide(a, d), divide(b, d)), b)), zero), zero, less_equal(divide(a, d), b), true)
% 0.45/0.49 = { by axiom 9 (quotient_property) }
% 0.45/0.49 ifeq2(ifeq(true, true, divide(divide(divide(divide(a, d), b), divide(divide(b, d), b)), divide(divide(divide(a, d), divide(b, d)), b)), zero), zero, less_equal(divide(a, d), b), true)
% 0.45/0.49 = { by axiom 3 (ifeq_axiom) }
% 0.45/0.49 ifeq2(divide(divide(divide(divide(a, d), b), divide(divide(b, d), b)), divide(divide(divide(a, d), divide(b, d)), b)), zero, less_equal(divide(a, d), b), true)
% 0.45/0.49 = { by lemma 11 }
% 0.45/0.49 ifeq2(divide(divide(divide(divide(a, d), b), zero), divide(divide(divide(a, d), divide(b, d)), b)), zero, less_equal(divide(a, d), b), true)
% 0.45/0.49 = { by lemma 12 }
% 0.45/0.49 ifeq2(divide(divide(divide(a, d), b), divide(divide(divide(a, d), divide(b, d)), b)), zero, less_equal(divide(a, d), b), true)
% 0.45/0.49 = { by axiom 8 (less_equal_and_equal) R->L }
% 0.45/0.49 ifeq2(divide(divide(divide(a, d), b), divide(ifeq(less_equal(divide(divide(a, d), divide(b, d)), zero), true, ifeq(less_equal(zero, divide(divide(a, d), divide(b, d))), true, zero, divide(divide(a, d), divide(b, d))), divide(divide(a, d), divide(b, d))), b)), zero, less_equal(divide(a, d), b), true)
% 0.45/0.49 = { by axiom 7 (quotient_less_equal1) R->L }
% 0.45/0.49 ifeq2(divide(divide(divide(a, d), b), divide(ifeq(less_equal(divide(divide(a, d), divide(b, d)), ifeq(less_equal(divide(a, b), d), true, divide(divide(a, b), d), zero)), true, ifeq(less_equal(zero, divide(divide(a, d), divide(b, d))), true, zero, divide(divide(a, d), divide(b, d))), divide(divide(a, d), divide(b, d))), b)), zero, less_equal(divide(a, d), b), true)
% 0.45/0.49 = { by axiom 5 (a_divide_b_LE_d) }
% 0.45/0.49 ifeq2(divide(divide(divide(a, d), b), divide(ifeq(less_equal(divide(divide(a, d), divide(b, d)), ifeq(true, true, divide(divide(a, b), d), zero)), true, ifeq(less_equal(zero, divide(divide(a, d), divide(b, d))), true, zero, divide(divide(a, d), divide(b, d))), divide(divide(a, d), divide(b, d))), b)), zero, less_equal(divide(a, d), b), true)
% 0.45/0.49 = { by axiom 3 (ifeq_axiom) }
% 0.45/0.49 ifeq2(divide(divide(divide(a, d), b), divide(ifeq(less_equal(divide(divide(a, d), divide(b, d)), divide(divide(a, b), d)), true, ifeq(less_equal(zero, divide(divide(a, d), divide(b, d))), true, zero, divide(divide(a, d), divide(b, d))), divide(divide(a, d), divide(b, d))), b)), zero, less_equal(divide(a, d), b), true)
% 0.45/0.49 = { by axiom 9 (quotient_property) }
% 0.45/0.49 ifeq2(divide(divide(divide(a, d), b), divide(ifeq(true, true, ifeq(less_equal(zero, divide(divide(a, d), divide(b, d))), true, zero, divide(divide(a, d), divide(b, d))), divide(divide(a, d), divide(b, d))), b)), zero, less_equal(divide(a, d), b), true)
% 0.45/0.49 = { by axiom 3 (ifeq_axiom) }
% 0.45/0.49 ifeq2(divide(divide(divide(a, d), b), divide(ifeq(less_equal(zero, divide(divide(a, d), divide(b, d))), true, zero, divide(divide(a, d), divide(b, d))), b)), zero, less_equal(divide(a, d), b), true)
% 0.45/0.49 = { by axiom 1 (zero_is_smallest) }
% 0.45/0.49 ifeq2(divide(divide(divide(a, d), b), divide(ifeq(true, true, zero, divide(divide(a, d), divide(b, d))), b)), zero, less_equal(divide(a, d), b), true)
% 0.45/0.49 = { by axiom 3 (ifeq_axiom) }
% 0.45/0.49 ifeq2(divide(divide(divide(a, d), b), divide(zero, b)), zero, less_equal(divide(a, d), b), true)
% 0.45/0.49 = { by lemma 10 }
% 0.45/0.49 ifeq2(divide(divide(divide(a, d), b), zero), zero, less_equal(divide(a, d), b), true)
% 0.45/0.49 = { by lemma 12 }
% 0.45/0.49 ifeq2(divide(divide(a, d), b), zero, less_equal(divide(a, d), b), true)
% 0.45/0.49 = { by axiom 6 (quotient_less_equal2) }
% 0.45/0.49 true
% 0.45/0.49 % SZS output end Proof
% 0.45/0.49
% 0.45/0.49 RESULT: Unsatisfiable (the axioms are contradictory).
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