TSTP Solution File: HEN006-7 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : HEN006-7 : TPTP v8.1.2. Released v1.0.0.
% 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 : n017.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 01:56:56 EDT 2023
% Result : Unsatisfiable 10.60s 1.70s
% Output : Proof 11.05s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : HEN006-7 : TPTP v8.1.2. Released v1.0.0.
% 0.11/0.13 % 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 : n017.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 : Thu Aug 24 12:55:09 EDT 2023
% 0.13/0.34 % CPUTime :
% 10.60/1.70 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 10.60/1.70
% 10.60/1.70 % SZS status Unsatisfiable
% 10.60/1.70
% 11.05/1.75 % SZS output start Proof
% 11.05/1.75 Take the following subset of the input axioms:
% 11.05/1.75 fof(closure, axiom, ![X, Y]: quotient(X, Y, divide(X, Y))).
% 11.05/1.75 fof(divisor_existence, axiom, ![Z, X2, Y2]: (~quotient(X2, Y2, Z) | less_equal(Z, X2))).
% 11.05/1.75 fof(less_equal_and_equal, axiom, ![X2, Y2]: (~less_equal(X2, Y2) | (~less_equal(Y2, X2) | X2=Y2))).
% 11.05/1.75 fof(less_equal_quotient, axiom, ![X2, Y2]: (~quotient(X2, Y2, zero) | less_equal(X2, Y2))).
% 11.05/1.75 fof(prove_xQzLEy, negated_conjecture, ~less_equal(xQz, y)).
% 11.05/1.75 fof(quotient_less_equal, axiom, ![X2, Y2]: (~less_equal(X2, Y2) | quotient(X2, Y2, zero))).
% 11.05/1.75 fof(quotient_property, axiom, ![V1, V2, V3, V4, V5, X2, Y2, Z2]: (~quotient(X2, Y2, V1) | (~quotient(Y2, Z2, V2) | (~quotient(X2, Z2, V3) | (~quotient(V3, V2, V4) | (~quotient(V1, Z2, V5) | less_equal(V4, V5))))))).
% 11.05/1.75 fof(well_defined, axiom, ![W, X2, Y2, Z2]: (~quotient(X2, Y2, Z2) | (~quotient(X2, Y2, W) | Z2=W))).
% 11.05/1.75 fof(xQy, hypothesis, quotient(x, y, xQy)).
% 11.05/1.75 fof(xQyLEz, hypothesis, less_equal(xQy, z)).
% 11.05/1.75 fof(xQz, hypothesis, quotient(x, z, xQz)).
% 11.05/1.75 fof(x_divde_zero_is_x, axiom, ![X2]: quotient(X2, zero, X2)).
% 11.05/1.75 fof(x_divide_x_is_zero, axiom, ![X2]: quotient(X2, X2, zero)).
% 11.05/1.75 fof(zero_divide_anything_is_zero, axiom, ![X2]: quotient(zero, X2, zero)).
% 11.05/1.75 fof(zero_is_smallest, axiom, ![X2]: less_equal(zero, X2)).
% 11.05/1.75
% 11.05/1.75 Now clausify the problem and encode Horn clauses using encoding 3 of
% 11.05/1.75 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 11.05/1.75 We repeatedly replace C & s=t => u=v by the two clauses:
% 11.05/1.75 fresh(y, y, x1...xn) = u
% 11.05/1.75 C => fresh(s, t, x1...xn) = v
% 11.05/1.75 where fresh is a fresh function symbol and x1..xn are the free
% 11.05/1.75 variables of u and v.
% 11.05/1.75 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 11.05/1.75 input problem has no model of domain size 1).
% 11.05/1.75
% 11.05/1.75 The encoding turns the above axioms into the following unit equations and goals:
% 11.05/1.75
% 11.05/1.75 Axiom 1 (zero_is_smallest): less_equal(zero, X) = true.
% 11.05/1.75 Axiom 2 (xQyLEz): less_equal(xQy, z) = true.
% 11.05/1.75 Axiom 3 (x_divide_x_is_zero): quotient(X, X, zero) = true.
% 11.05/1.75 Axiom 4 (x_divde_zero_is_x): quotient(X, zero, X) = true.
% 11.05/1.76 Axiom 5 (zero_divide_anything_is_zero): quotient(zero, X, zero) = true.
% 11.05/1.76 Axiom 6 (xQy): quotient(x, y, xQy) = true.
% 11.05/1.76 Axiom 7 (xQz): quotient(x, z, xQz) = true.
% 11.05/1.76 Axiom 8 (well_defined): fresh(X, X, Y, Z) = Z.
% 11.05/1.76 Axiom 9 (quotient_property): fresh12(X, X, Y, Z) = true.
% 11.05/1.76 Axiom 10 (less_equal_quotient): fresh7(X, X, Y, Z) = true.
% 11.05/1.76 Axiom 11 (divisor_existence): fresh6(X, X, Y, Z) = true.
% 11.05/1.76 Axiom 12 (quotient_less_equal): fresh5(X, X, Y, Z) = true.
% 11.05/1.76 Axiom 13 (less_equal_and_equal): fresh4(X, X, Y, Z) = Y.
% 11.05/1.76 Axiom 14 (less_equal_and_equal): fresh3(X, X, Y, Z) = Z.
% 11.05/1.76 Axiom 15 (closure): quotient(X, Y, divide(X, Y)) = true.
% 11.05/1.76 Axiom 16 (quotient_less_equal): fresh5(less_equal(X, Y), true, X, Y) = quotient(X, Y, zero).
% 11.05/1.76 Axiom 17 (less_equal_and_equal): fresh4(less_equal(X, Y), true, Y, X) = fresh3(less_equal(Y, X), true, Y, X).
% 11.05/1.76 Axiom 18 (well_defined): fresh2(X, X, Y, Z, W, V) = W.
% 11.05/1.76 Axiom 19 (quotient_property): fresh10(X, X, Y, Z, W, V, U) = less_equal(V, U).
% 11.05/1.76 Axiom 20 (less_equal_quotient): fresh7(quotient(X, Y, zero), true, X, Y) = less_equal(X, Y).
% 11.05/1.76 Axiom 21 (divisor_existence): fresh6(quotient(X, Y, Z), true, X, Z) = less_equal(Z, X).
% 11.05/1.76 Axiom 22 (quotient_property): fresh11(X, X, Y, Z, W, V, U, T, S) = fresh12(quotient(Y, Z, W), true, T, S).
% 11.05/1.76 Axiom 23 (well_defined): fresh2(quotient(X, Y, Z), true, X, Y, W, Z) = fresh(quotient(X, Y, W), true, W, Z).
% 11.05/1.76 Axiom 24 (quotient_property): fresh9(X, X, Y, Z, W, V, U, T, S, X2) = fresh10(quotient(Y, V, T), true, Y, Z, W, S, X2).
% 11.05/1.76 Axiom 25 (quotient_property): fresh8(X, X, Y, Z, W, V, U, T, S, X2) = fresh11(quotient(Z, V, U), true, Y, Z, W, V, T, S, X2).
% 11.05/1.76 Axiom 26 (quotient_property): fresh8(quotient(X, Y, Z), true, W, V, U, T, Y, X, Z, S) = fresh9(quotient(U, T, S), true, W, V, U, T, Y, X, Z, S).
% 11.05/1.76
% 11.05/1.76 Lemma 27: divide(X, zero) = X.
% 11.05/1.76 Proof:
% 11.05/1.76 divide(X, zero)
% 11.05/1.76 = { by axiom 18 (well_defined) R->L }
% 11.05/1.76 fresh2(true, true, X, zero, divide(X, zero), X)
% 11.05/1.76 = { by axiom 4 (x_divde_zero_is_x) R->L }
% 11.05/1.76 fresh2(quotient(X, zero, X), true, X, zero, divide(X, zero), X)
% 11.05/1.76 = { by axiom 23 (well_defined) }
% 11.05/1.76 fresh(quotient(X, zero, divide(X, zero)), true, divide(X, zero), X)
% 11.05/1.76 = { by axiom 15 (closure) }
% 11.05/1.76 fresh(true, true, divide(X, zero), X)
% 11.05/1.76 = { by axiom 8 (well_defined) }
% 11.05/1.76 X
% 11.05/1.76
% 11.05/1.76 Lemma 28: divide(x, z) = xQz.
% 11.05/1.76 Proof:
% 11.05/1.76 divide(x, z)
% 11.05/1.76 = { by axiom 18 (well_defined) R->L }
% 11.05/1.76 fresh2(true, true, x, z, divide(x, z), xQz)
% 11.05/1.76 = { by axiom 7 (xQz) R->L }
% 11.05/1.76 fresh2(quotient(x, z, xQz), true, x, z, divide(x, z), xQz)
% 11.05/1.76 = { by axiom 23 (well_defined) }
% 11.05/1.76 fresh(quotient(x, z, divide(x, z)), true, divide(x, z), xQz)
% 11.05/1.76 = { by axiom 15 (closure) }
% 11.05/1.76 fresh(true, true, divide(x, z), xQz)
% 11.05/1.76 = { by axiom 8 (well_defined) }
% 11.05/1.76 xQz
% 11.05/1.76
% 11.05/1.76 Lemma 29: fresh8(X, X, Y, Z, W, V, divide(Z, V), U, T, S) = fresh12(quotient(Y, Z, W), true, T, S).
% 11.05/1.76 Proof:
% 11.05/1.76 fresh8(X, X, Y, Z, W, V, divide(Z, V), U, T, S)
% 11.05/1.76 = { by axiom 25 (quotient_property) }
% 11.05/1.76 fresh11(quotient(Z, V, divide(Z, V)), true, Y, Z, W, V, U, T, S)
% 11.05/1.76 = { by axiom 15 (closure) }
% 11.05/1.76 fresh11(true, true, Y, Z, W, V, U, T, S)
% 11.05/1.76 = { by axiom 22 (quotient_property) }
% 11.05/1.76 fresh12(quotient(Y, Z, W), true, T, S)
% 11.05/1.76
% 11.05/1.76 Lemma 30: fresh9(X, X, Y, Z, W, V, U, divide(Y, V), T, S) = less_equal(T, S).
% 11.05/1.76 Proof:
% 11.05/1.76 fresh9(X, X, Y, Z, W, V, U, divide(Y, V), T, S)
% 11.05/1.76 = { by axiom 24 (quotient_property) }
% 11.05/1.76 fresh10(quotient(Y, V, divide(Y, V)), true, Y, Z, W, T, S)
% 11.05/1.76 = { by axiom 15 (closure) }
% 11.05/1.76 fresh10(true, true, Y, Z, W, T, S)
% 11.05/1.76 = { by axiom 19 (quotient_property) }
% 11.05/1.76 less_equal(T, S)
% 11.05/1.76
% 11.05/1.76 Lemma 31: fresh12(quotient(X, Y, Z), true, divide(divide(X, Z), divide(Y, Z)), zero) = less_equal(divide(divide(X, Z), divide(Y, Z)), zero).
% 11.05/1.76 Proof:
% 11.05/1.76 fresh12(quotient(X, Y, Z), true, divide(divide(X, Z), divide(Y, Z)), zero)
% 11.05/1.76 = { by lemma 29 R->L }
% 11.05/1.76 fresh8(true, true, X, Y, Z, Z, divide(Y, Z), divide(X, Z), divide(divide(X, Z), divide(Y, Z)), zero)
% 11.05/1.76 = { by axiom 15 (closure) R->L }
% 11.05/1.76 fresh8(quotient(divide(X, Z), divide(Y, Z), divide(divide(X, Z), divide(Y, Z))), true, X, Y, Z, Z, divide(Y, Z), divide(X, Z), divide(divide(X, Z), divide(Y, Z)), zero)
% 11.05/1.76 = { by axiom 26 (quotient_property) }
% 11.05/1.76 fresh9(quotient(Z, Z, zero), true, X, Y, Z, Z, divide(Y, Z), divide(X, Z), divide(divide(X, Z), divide(Y, Z)), zero)
% 11.05/1.76 = { by axiom 3 (x_divide_x_is_zero) }
% 11.05/1.76 fresh9(true, true, X, Y, Z, Z, divide(Y, Z), divide(X, Z), divide(divide(X, Z), divide(Y, Z)), zero)
% 11.05/1.76 = { by lemma 30 }
% 11.05/1.76 less_equal(divide(divide(X, Z), divide(Y, Z)), zero)
% 11.05/1.76
% 11.05/1.76 Lemma 32: quotient(divide(X, Y), X, zero) = true.
% 11.05/1.76 Proof:
% 11.05/1.76 quotient(divide(X, Y), X, zero)
% 11.05/1.76 = { by axiom 16 (quotient_less_equal) R->L }
% 11.05/1.76 fresh5(less_equal(divide(X, Y), X), true, divide(X, Y), X)
% 11.05/1.76 = { by axiom 21 (divisor_existence) R->L }
% 11.05/1.76 fresh5(fresh6(quotient(X, Y, divide(X, Y)), true, X, divide(X, Y)), true, divide(X, Y), X)
% 11.05/1.76 = { by axiom 15 (closure) }
% 11.05/1.76 fresh5(fresh6(true, true, X, divide(X, Y)), true, divide(X, Y), X)
% 11.05/1.76 = { by axiom 11 (divisor_existence) }
% 11.05/1.76 fresh5(true, true, divide(X, Y), X)
% 11.05/1.76 = { by axiom 12 (quotient_less_equal) }
% 11.05/1.76 true
% 11.05/1.76
% 11.05/1.76 Lemma 33: fresh8(Y2, Y2, Y, Z, W, V, U, T, S, X2) = fresh8(X, X, Y, Z, W, V, U, T, S, X2).
% 11.05/1.76 Proof:
% 11.05/1.76 fresh8(Y2, Y2, Y, Z, W, V, U, T, S, X2)
% 11.05/1.76 = { by axiom 25 (quotient_property) }
% 11.05/1.76 fresh11(quotient(Z, V, U), true, Y, Z, W, V, T, S, X2)
% 11.05/1.76 = { by axiom 25 (quotient_property) R->L }
% 11.05/1.76 fresh8(X, X, Y, Z, W, V, U, T, S, X2)
% 11.05/1.76
% 11.05/1.76 Lemma 34: fresh9(quotient(X, Y, Z), true, W, V, X, Y, zero, U, U, Z) = fresh8(T, T, W, V, X, Y, zero, U, U, Z).
% 11.05/1.76 Proof:
% 11.05/1.76 fresh9(quotient(X, Y, Z), true, W, V, X, Y, zero, U, U, Z)
% 11.05/1.76 = { by axiom 26 (quotient_property) R->L }
% 11.05/1.76 fresh8(quotient(U, zero, U), true, W, V, X, Y, zero, U, U, Z)
% 11.05/1.76 = { by axiom 4 (x_divde_zero_is_x) }
% 11.05/1.76 fresh8(true, true, W, V, X, Y, zero, U, U, Z)
% 11.05/1.76 = { by lemma 33 R->L }
% 11.05/1.76 fresh8(T, T, W, V, X, Y, zero, U, U, Z)
% 11.05/1.76
% 11.05/1.76 Lemma 35: fresh8(X, X, Y, Z, zero, W, V, U, divide(U, V), zero) = fresh9(T, T, Y, Z, zero, W, S, U, divide(U, V), zero).
% 11.05/1.76 Proof:
% 11.05/1.76 fresh8(X, X, Y, Z, zero, W, V, U, divide(U, V), zero)
% 11.05/1.76 = { by lemma 33 }
% 11.05/1.76 fresh8(true, true, Y, Z, zero, W, V, U, divide(U, V), zero)
% 11.05/1.76 = { by axiom 15 (closure) R->L }
% 11.05/1.76 fresh8(quotient(U, V, divide(U, V)), true, Y, Z, zero, W, V, U, divide(U, V), zero)
% 11.05/1.76 = { by axiom 26 (quotient_property) }
% 11.05/1.76 fresh9(quotient(zero, W, zero), true, Y, Z, zero, W, V, U, divide(U, V), zero)
% 11.05/1.76 = { by axiom 5 (zero_divide_anything_is_zero) }
% 11.05/1.76 fresh9(true, true, Y, Z, zero, W, V, U, divide(U, V), zero)
% 11.05/1.76 = { by axiom 24 (quotient_property) }
% 11.05/1.76 fresh10(quotient(Y, W, U), true, Y, Z, zero, divide(U, V), zero)
% 11.05/1.76 = { by axiom 24 (quotient_property) R->L }
% 11.05/1.76 fresh9(T, T, Y, Z, zero, W, S, U, divide(U, V), zero)
% 11.05/1.76
% 11.05/1.76 Lemma 36: divide(xQz, y) = zero.
% 11.05/1.76 Proof:
% 11.05/1.76 divide(xQz, y)
% 11.05/1.76 = { by lemma 27 R->L }
% 11.05/1.76 divide(divide(xQz, y), zero)
% 11.05/1.76 = { by axiom 14 (less_equal_and_equal) R->L }
% 11.05/1.76 divide(divide(xQz, y), fresh3(true, true, divide(divide(x, xQy), y), zero))
% 11.05/1.76 = { by axiom 9 (quotient_property) R->L }
% 11.05/1.76 divide(divide(xQz, y), fresh3(fresh12(true, true, divide(divide(x, xQy), y), zero), true, divide(divide(x, xQy), y), zero))
% 11.05/1.76 = { by axiom 15 (closure) R->L }
% 11.05/1.76 divide(divide(xQz, y), fresh3(fresh12(quotient(divide(x, xQy), divide(y, xQy), divide(divide(x, xQy), divide(y, xQy))), true, divide(divide(x, xQy), y), zero), true, divide(divide(x, xQy), y), zero))
% 11.05/1.76 = { by axiom 13 (less_equal_and_equal) R->L }
% 11.05/1.76 divide(divide(xQz, y), fresh3(fresh12(quotient(divide(x, xQy), divide(y, xQy), fresh4(true, true, divide(divide(x, xQy), divide(y, xQy)), zero)), true, divide(divide(x, xQy), y), zero), true, divide(divide(x, xQy), y), zero))
% 11.05/1.76 = { by axiom 1 (zero_is_smallest) R->L }
% 11.05/1.76 divide(divide(xQz, y), fresh3(fresh12(quotient(divide(x, xQy), divide(y, xQy), fresh4(less_equal(zero, divide(divide(x, xQy), divide(y, xQy))), true, divide(divide(x, xQy), divide(y, xQy)), zero)), true, divide(divide(x, xQy), y), zero), true, divide(divide(x, xQy), y), zero))
% 11.05/1.76 = { by axiom 17 (less_equal_and_equal) }
% 11.05/1.76 divide(divide(xQz, y), fresh3(fresh12(quotient(divide(x, xQy), divide(y, xQy), fresh3(less_equal(divide(divide(x, xQy), divide(y, xQy)), zero), true, divide(divide(x, xQy), divide(y, xQy)), zero)), true, divide(divide(x, xQy), y), zero), true, divide(divide(x, xQy), y), zero))
% 11.05/1.76 = { by lemma 31 R->L }
% 11.05/1.76 divide(divide(xQz, y), fresh3(fresh12(quotient(divide(x, xQy), divide(y, xQy), fresh3(fresh12(quotient(x, y, xQy), true, divide(divide(x, xQy), divide(y, xQy)), zero), true, divide(divide(x, xQy), divide(y, xQy)), zero)), true, divide(divide(x, xQy), y), zero), true, divide(divide(x, xQy), y), zero))
% 11.05/1.76 = { by axiom 6 (xQy) }
% 11.05/1.76 divide(divide(xQz, y), fresh3(fresh12(quotient(divide(x, xQy), divide(y, xQy), fresh3(fresh12(true, true, divide(divide(x, xQy), divide(y, xQy)), zero), true, divide(divide(x, xQy), divide(y, xQy)), zero)), true, divide(divide(x, xQy), y), zero), true, divide(divide(x, xQy), y), zero))
% 11.05/1.76 = { by axiom 9 (quotient_property) }
% 11.05/1.76 divide(divide(xQz, y), fresh3(fresh12(quotient(divide(x, xQy), divide(y, xQy), fresh3(true, true, divide(divide(x, xQy), divide(y, xQy)), zero)), true, divide(divide(x, xQy), y), zero), true, divide(divide(x, xQy), y), zero))
% 11.05/1.76 = { by axiom 14 (less_equal_and_equal) }
% 11.05/1.76 divide(divide(xQz, y), fresh3(fresh12(quotient(divide(x, xQy), divide(y, xQy), zero), true, divide(divide(x, xQy), y), zero), true, divide(divide(x, xQy), y), zero))
% 11.05/1.76 = { by axiom 22 (quotient_property) R->L }
% 11.05/1.76 divide(divide(xQz, y), fresh3(fresh11(true, true, divide(x, xQy), divide(y, xQy), zero, y, divide(divide(x, xQy), y), divide(divide(x, xQy), y), zero), true, divide(divide(x, xQy), y), zero))
% 11.05/1.76 = { by lemma 32 R->L }
% 11.05/1.76 divide(divide(xQz, y), fresh3(fresh11(quotient(divide(y, xQy), y, zero), true, divide(x, xQy), divide(y, xQy), zero, y, divide(divide(x, xQy), y), divide(divide(x, xQy), y), zero), true, divide(divide(x, xQy), y), zero))
% 11.05/1.76 = { by axiom 25 (quotient_property) R->L }
% 11.05/1.76 divide(divide(xQz, y), fresh3(fresh8(X, X, divide(x, xQy), divide(y, xQy), zero, y, zero, divide(divide(x, xQy), y), divide(divide(x, xQy), y), zero), true, divide(divide(x, xQy), y), zero))
% 11.05/1.76 = { by lemma 34 R->L }
% 11.05/1.76 divide(divide(xQz, y), fresh3(fresh9(quotient(zero, y, zero), true, divide(x, xQy), divide(y, xQy), zero, y, zero, divide(divide(x, xQy), y), divide(divide(x, xQy), y), zero), true, divide(divide(x, xQy), y), zero))
% 11.05/1.76 = { by axiom 5 (zero_divide_anything_is_zero) }
% 11.05/1.76 divide(divide(xQz, y), fresh3(fresh9(true, true, divide(x, xQy), divide(y, xQy), zero, y, zero, divide(divide(x, xQy), y), divide(divide(x, xQy), y), zero), true, divide(divide(x, xQy), y), zero))
% 11.05/1.76 = { by lemma 30 }
% 11.05/1.76 divide(divide(xQz, y), fresh3(less_equal(divide(divide(x, xQy), y), zero), true, divide(divide(x, xQy), y), zero))
% 11.05/1.76 = { by axiom 17 (less_equal_and_equal) R->L }
% 11.05/1.76 divide(divide(xQz, y), fresh4(less_equal(zero, divide(divide(x, xQy), y)), true, divide(divide(x, xQy), y), zero))
% 11.05/1.76 = { by axiom 1 (zero_is_smallest) }
% 11.05/1.76 divide(divide(xQz, y), fresh4(true, true, divide(divide(x, xQy), y), zero))
% 11.05/1.76 = { by axiom 13 (less_equal_and_equal) }
% 11.05/1.76 divide(divide(xQz, y), divide(divide(x, xQy), y))
% 11.05/1.76 = { by lemma 28 R->L }
% 11.05/1.76 divide(divide(divide(x, z), y), divide(divide(x, xQy), y))
% 11.05/1.76 = { by axiom 13 (less_equal_and_equal) R->L }
% 11.05/1.76 divide(divide(fresh4(true, true, divide(x, z), divide(divide(x, xQy), z)), y), divide(divide(x, xQy), y))
% 11.05/1.76 = { by axiom 10 (less_equal_quotient) R->L }
% 11.05/1.76 divide(divide(fresh4(fresh7(true, true, divide(divide(x, xQy), z), xQz), true, divide(x, z), divide(divide(x, xQy), z)), y), divide(divide(x, xQy), y))
% 11.05/1.76 = { by axiom 15 (closure) R->L }
% 11.05/1.76 divide(divide(fresh4(fresh7(quotient(divide(divide(x, xQy), z), xQz, divide(divide(divide(x, xQy), z), xQz)), true, divide(divide(x, xQy), z), xQz), true, divide(x, z), divide(divide(x, xQy), z)), y), divide(divide(x, xQy), y))
% 11.05/1.76 = { by axiom 13 (less_equal_and_equal) R->L }
% 11.05/1.76 divide(divide(fresh4(fresh7(quotient(divide(divide(x, xQy), z), xQz, fresh4(true, true, divide(divide(divide(x, xQy), z), xQz), zero)), true, divide(divide(x, xQy), z), xQz), true, divide(x, z), divide(divide(x, xQy), z)), y), divide(divide(x, xQy), y))
% 11.05/1.76 = { by axiom 1 (zero_is_smallest) R->L }
% 11.05/1.76 divide(divide(fresh4(fresh7(quotient(divide(divide(x, xQy), z), xQz, fresh4(less_equal(zero, divide(divide(divide(x, xQy), z), xQz)), true, divide(divide(divide(x, xQy), z), xQz), zero)), true, divide(divide(x, xQy), z), xQz), true, divide(x, z), divide(divide(x, xQy), z)), y), divide(divide(x, xQy), y))
% 11.05/1.76 = { by axiom 17 (less_equal_and_equal) }
% 11.05/1.76 divide(divide(fresh4(fresh7(quotient(divide(divide(x, xQy), z), xQz, fresh3(less_equal(divide(divide(divide(x, xQy), z), xQz), zero), true, divide(divide(divide(x, xQy), z), xQz), zero)), true, divide(divide(x, xQy), z), xQz), true, divide(x, z), divide(divide(x, xQy), z)), y), divide(divide(x, xQy), y))
% 11.05/1.77 = { by lemma 30 R->L }
% 11.05/1.77 divide(divide(fresh4(fresh7(quotient(divide(divide(x, xQy), z), xQz, fresh3(fresh9(Y, Y, divide(x, xQy), x, zero, z, Z, divide(divide(x, xQy), z), divide(divide(divide(x, xQy), z), xQz), zero), true, divide(divide(divide(x, xQy), z), xQz), zero)), true, divide(divide(x, xQy), z), xQz), true, divide(x, z), divide(divide(x, xQy), z)), y), divide(divide(x, xQy), y))
% 11.05/1.77 = { by lemma 35 R->L }
% 11.05/1.77 divide(divide(fresh4(fresh7(quotient(divide(divide(x, xQy), z), xQz, fresh3(fresh8(W, W, divide(x, xQy), x, zero, z, xQz, divide(divide(x, xQy), z), divide(divide(divide(x, xQy), z), xQz), zero), true, divide(divide(divide(x, xQy), z), xQz), zero)), true, divide(divide(x, xQy), z), xQz), true, divide(x, z), divide(divide(x, xQy), z)), y), divide(divide(x, xQy), y))
% 11.05/1.77 = { by axiom 25 (quotient_property) }
% 11.05/1.77 divide(divide(fresh4(fresh7(quotient(divide(divide(x, xQy), z), xQz, fresh3(fresh11(quotient(x, z, xQz), true, divide(x, xQy), x, zero, z, divide(divide(x, xQy), z), divide(divide(divide(x, xQy), z), xQz), zero), true, divide(divide(divide(x, xQy), z), xQz), zero)), true, divide(divide(x, xQy), z), xQz), true, divide(x, z), divide(divide(x, xQy), z)), y), divide(divide(x, xQy), y))
% 11.05/1.77 = { by axiom 7 (xQz) }
% 11.05/1.77 divide(divide(fresh4(fresh7(quotient(divide(divide(x, xQy), z), xQz, fresh3(fresh11(true, true, divide(x, xQy), x, zero, z, divide(divide(x, xQy), z), divide(divide(divide(x, xQy), z), xQz), zero), true, divide(divide(divide(x, xQy), z), xQz), zero)), true, divide(divide(x, xQy), z), xQz), true, divide(x, z), divide(divide(x, xQy), z)), y), divide(divide(x, xQy), y))
% 11.05/1.77 = { by axiom 22 (quotient_property) }
% 11.05/1.77 divide(divide(fresh4(fresh7(quotient(divide(divide(x, xQy), z), xQz, fresh3(fresh12(quotient(divide(x, xQy), x, zero), true, divide(divide(divide(x, xQy), z), xQz), zero), true, divide(divide(divide(x, xQy), z), xQz), zero)), true, divide(divide(x, xQy), z), xQz), true, divide(x, z), divide(divide(x, xQy), z)), y), divide(divide(x, xQy), y))
% 11.05/1.77 = { by lemma 32 }
% 11.05/1.77 divide(divide(fresh4(fresh7(quotient(divide(divide(x, xQy), z), xQz, fresh3(fresh12(true, true, divide(divide(divide(x, xQy), z), xQz), zero), true, divide(divide(divide(x, xQy), z), xQz), zero)), true, divide(divide(x, xQy), z), xQz), true, divide(x, z), divide(divide(x, xQy), z)), y), divide(divide(x, xQy), y))
% 11.05/1.77 = { by axiom 9 (quotient_property) }
% 11.05/1.77 divide(divide(fresh4(fresh7(quotient(divide(divide(x, xQy), z), xQz, fresh3(true, true, divide(divide(divide(x, xQy), z), xQz), zero)), true, divide(divide(x, xQy), z), xQz), true, divide(x, z), divide(divide(x, xQy), z)), y), divide(divide(x, xQy), y))
% 11.05/1.77 = { by axiom 14 (less_equal_and_equal) }
% 11.05/1.77 divide(divide(fresh4(fresh7(quotient(divide(divide(x, xQy), z), xQz, zero), true, divide(divide(x, xQy), z), xQz), true, divide(x, z), divide(divide(x, xQy), z)), y), divide(divide(x, xQy), y))
% 11.05/1.77 = { by axiom 20 (less_equal_quotient) }
% 11.05/1.77 divide(divide(fresh4(less_equal(divide(divide(x, xQy), z), xQz), true, divide(x, z), divide(divide(x, xQy), z)), y), divide(divide(x, xQy), y))
% 11.05/1.77 = { by lemma 28 R->L }
% 11.05/1.77 divide(divide(fresh4(less_equal(divide(divide(x, xQy), z), divide(x, z)), true, divide(x, z), divide(divide(x, xQy), z)), y), divide(divide(x, xQy), y))
% 11.05/1.77 = { by axiom 17 (less_equal_and_equal) }
% 11.05/1.77 divide(divide(fresh3(less_equal(divide(x, z), divide(divide(x, xQy), z)), true, divide(x, z), divide(divide(x, xQy), z)), y), divide(divide(x, xQy), y))
% 11.05/1.77 = { by lemma 30 R->L }
% 11.05/1.77 divide(divide(fresh3(fresh9(true, true, x, xQy, divide(x, xQy), z, zero, divide(x, z), divide(x, z), divide(divide(x, xQy), z)), true, divide(x, z), divide(divide(x, xQy), z)), y), divide(divide(x, xQy), y))
% 11.05/1.77 = { by axiom 15 (closure) R->L }
% 11.05/1.77 divide(divide(fresh3(fresh9(quotient(divide(x, xQy), z, divide(divide(x, xQy), z)), true, x, xQy, divide(x, xQy), z, zero, divide(x, z), divide(x, z), divide(divide(x, xQy), z)), true, divide(x, z), divide(divide(x, xQy), z)), y), divide(divide(x, xQy), y))
% 11.05/1.77 = { by lemma 34 }
% 11.05/1.77 divide(divide(fresh3(fresh8(V, V, x, xQy, divide(x, xQy), z, zero, divide(x, z), divide(x, z), divide(divide(x, xQy), z)), true, divide(x, z), divide(divide(x, xQy), z)), y), divide(divide(x, xQy), y))
% 11.05/1.77 = { by axiom 25 (quotient_property) }
% 11.05/1.77 divide(divide(fresh3(fresh11(quotient(xQy, z, zero), true, x, xQy, divide(x, xQy), z, divide(x, z), divide(x, z), divide(divide(x, xQy), z)), true, divide(x, z), divide(divide(x, xQy), z)), y), divide(divide(x, xQy), y))
% 11.05/1.77 = { by axiom 16 (quotient_less_equal) R->L }
% 11.05/1.77 divide(divide(fresh3(fresh11(fresh5(less_equal(xQy, z), true, xQy, z), true, x, xQy, divide(x, xQy), z, divide(x, z), divide(x, z), divide(divide(x, xQy), z)), true, divide(x, z), divide(divide(x, xQy), z)), y), divide(divide(x, xQy), y))
% 11.05/1.77 = { by axiom 2 (xQyLEz) }
% 11.05/1.77 divide(divide(fresh3(fresh11(fresh5(true, true, xQy, z), true, x, xQy, divide(x, xQy), z, divide(x, z), divide(x, z), divide(divide(x, xQy), z)), true, divide(x, z), divide(divide(x, xQy), z)), y), divide(divide(x, xQy), y))
% 11.05/1.77 = { by axiom 12 (quotient_less_equal) }
% 11.05/1.77 divide(divide(fresh3(fresh11(true, true, x, xQy, divide(x, xQy), z, divide(x, z), divide(x, z), divide(divide(x, xQy), z)), true, divide(x, z), divide(divide(x, xQy), z)), y), divide(divide(x, xQy), y))
% 11.05/1.77 = { by axiom 22 (quotient_property) }
% 11.05/1.77 divide(divide(fresh3(fresh12(quotient(x, xQy, divide(x, xQy)), true, divide(x, z), divide(divide(x, xQy), z)), true, divide(x, z), divide(divide(x, xQy), z)), y), divide(divide(x, xQy), y))
% 11.05/1.77 = { by axiom 15 (closure) }
% 11.05/1.77 divide(divide(fresh3(fresh12(true, true, divide(x, z), divide(divide(x, xQy), z)), true, divide(x, z), divide(divide(x, xQy), z)), y), divide(divide(x, xQy), y))
% 11.05/1.77 = { by axiom 9 (quotient_property) }
% 11.05/1.77 divide(divide(fresh3(true, true, divide(x, z), divide(divide(x, xQy), z)), y), divide(divide(x, xQy), y))
% 11.05/1.77 = { by axiom 14 (less_equal_and_equal) }
% 11.05/1.77 divide(divide(divide(divide(x, xQy), z), y), divide(divide(x, xQy), y))
% 11.05/1.77 = { by axiom 13 (less_equal_and_equal) R->L }
% 11.05/1.77 fresh4(true, true, divide(divide(divide(divide(x, xQy), z), y), divide(divide(x, xQy), y)), zero)
% 11.05/1.77 = { by axiom 1 (zero_is_smallest) R->L }
% 11.05/1.77 fresh4(less_equal(zero, divide(divide(divide(divide(x, xQy), z), y), divide(divide(x, xQy), y))), true, divide(divide(divide(divide(x, xQy), z), y), divide(divide(x, xQy), y)), zero)
% 11.05/1.77 = { by axiom 17 (less_equal_and_equal) }
% 11.05/1.77 fresh3(less_equal(divide(divide(divide(divide(x, xQy), z), y), divide(divide(x, xQy), y)), zero), true, divide(divide(divide(divide(x, xQy), z), y), divide(divide(x, xQy), y)), zero)
% 11.05/1.77 = { by lemma 30 R->L }
% 11.05/1.77 fresh3(fresh9(U, U, divide(divide(x, xQy), z), divide(x, xQy), zero, y, T, divide(divide(divide(x, xQy), z), y), divide(divide(divide(divide(x, xQy), z), y), divide(divide(x, xQy), y)), zero), true, divide(divide(divide(divide(x, xQy), z), y), divide(divide(x, xQy), y)), zero)
% 11.05/1.77 = { by lemma 35 R->L }
% 11.05/1.77 fresh3(fresh8(S, S, divide(divide(x, xQy), z), divide(x, xQy), zero, y, divide(divide(x, xQy), y), divide(divide(divide(x, xQy), z), y), divide(divide(divide(divide(x, xQy), z), y), divide(divide(x, xQy), y)), zero), true, divide(divide(divide(divide(x, xQy), z), y), divide(divide(x, xQy), y)), zero)
% 11.05/1.77 = { by lemma 29 }
% 11.05/1.77 fresh3(fresh12(quotient(divide(divide(x, xQy), z), divide(x, xQy), zero), true, divide(divide(divide(divide(x, xQy), z), y), divide(divide(x, xQy), y)), zero), true, divide(divide(divide(divide(x, xQy), z), y), divide(divide(x, xQy), y)), zero)
% 11.05/1.77 = { by lemma 32 }
% 11.05/1.77 fresh3(fresh12(true, true, divide(divide(divide(divide(x, xQy), z), y), divide(divide(x, xQy), y)), zero), true, divide(divide(divide(divide(x, xQy), z), y), divide(divide(x, xQy), y)), zero)
% 11.05/1.77 = { by axiom 9 (quotient_property) }
% 11.05/1.77 fresh3(true, true, divide(divide(divide(divide(x, xQy), z), y), divide(divide(x, xQy), y)), zero)
% 11.05/1.77 = { by axiom 14 (less_equal_and_equal) }
% 11.05/1.77 zero
% 11.05/1.77
% 11.05/1.77 Goal 1 (prove_xQzLEy): less_equal(xQz, y) = true.
% 11.05/1.77 Proof:
% 11.05/1.77 less_equal(xQz, y)
% 11.05/1.77 = { by lemma 27 R->L }
% 11.05/1.77 less_equal(xQz, divide(y, zero))
% 11.05/1.77 = { by lemma 36 R->L }
% 11.05/1.77 less_equal(xQz, divide(y, divide(xQz, y)))
% 11.05/1.77 = { by lemma 27 R->L }
% 11.05/1.77 less_equal(divide(xQz, zero), divide(y, divide(xQz, y)))
% 11.05/1.77 = { by lemma 36 R->L }
% 11.05/1.77 less_equal(divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y)))
% 11.05/1.77 = { by axiom 20 (less_equal_quotient) R->L }
% 11.05/1.77 fresh7(quotient(divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y)), zero), true, divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y)))
% 11.05/1.77 = { by axiom 14 (less_equal_and_equal) R->L }
% 11.05/1.77 fresh7(quotient(divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y)), fresh3(true, true, divide(divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y))), zero)), true, divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y)))
% 11.05/1.77 = { by axiom 9 (quotient_property) R->L }
% 11.05/1.77 fresh7(quotient(divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y)), fresh3(fresh12(true, true, divide(divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y))), zero), true, divide(divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y))), zero)), true, divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y)))
% 11.05/1.77 = { by axiom 15 (closure) R->L }
% 11.05/1.77 fresh7(quotient(divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y)), fresh3(fresh12(quotient(xQz, y, divide(xQz, y)), true, divide(divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y))), zero), true, divide(divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y))), zero)), true, divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y)))
% 11.05/1.77 = { by lemma 31 }
% 11.05/1.77 fresh7(quotient(divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y)), fresh3(less_equal(divide(divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y))), zero), true, divide(divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y))), zero)), true, divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y)))
% 11.05/1.77 = { by axiom 17 (less_equal_and_equal) R->L }
% 11.05/1.77 fresh7(quotient(divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y)), fresh4(less_equal(zero, divide(divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y)))), true, divide(divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y))), zero)), true, divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y)))
% 11.05/1.77 = { by axiom 1 (zero_is_smallest) }
% 11.05/1.77 fresh7(quotient(divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y)), fresh4(true, true, divide(divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y))), zero)), true, divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y)))
% 11.05/1.77 = { by axiom 13 (less_equal_and_equal) }
% 11.05/1.77 fresh7(quotient(divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y)), divide(divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y)))), true, divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y)))
% 11.05/1.77 = { by axiom 15 (closure) }
% 11.05/1.77 fresh7(true, true, divide(xQz, divide(xQz, y)), divide(y, divide(xQz, y)))
% 11.05/1.77 = { by axiom 10 (less_equal_quotient) }
% 11.05/1.77 true
% 11.05/1.77 % SZS output end Proof
% 11.05/1.77
% 11.05/1.77 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------