TSTP Solution File: HEN009-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : HEN009-1 : 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 : n013.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 01:57:00 EDT 2023
% Result : Unsatisfiable 28.72s 4.21s
% Output : Proof 29.77s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : HEN009-1 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.33 % Computer : n013.cluster.edu
% 0.14/0.33 % Model : x86_64 x86_64
% 0.14/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.33 % Memory : 8042.1875MB
% 0.14/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.33 % CPULimit : 300
% 0.14/0.33 % WCLimit : 300
% 0.14/0.33 % DateTime : Thu Aug 24 13:25:47 EDT 2023
% 0.14/0.33 % CPUTime :
% 28.72/4.21 Command-line arguments: --ground-connectedness --complete-subsets
% 28.72/4.21
% 28.72/4.21 % SZS status Unsatisfiable
% 28.72/4.21
% 29.77/4.27 % SZS output start Proof
% 29.77/4.27 Take the following subset of the input axioms:
% 29.77/4.27 fof(closure, axiom, ![X, Y]: quotient(X, Y, divide(X, Y))).
% 29.77/4.27 fof(divisor_existence, axiom, ![Z, X2, Y2]: (~quotient(X2, Y2, Z) | less_equal(Z, X2))).
% 29.77/4.27 fof(identity_divide_a, hypothesis, quotient(identity, a, idQa)).
% 29.77/4.27 fof(identity_divide_idQ_idQa, hypothesis, quotient(identity, idQ_idQa, idQ__idQ_idQa)).
% 29.77/4.27 fof(identity_divide_idQa, hypothesis, quotient(identity, idQa, idQ_idQa)).
% 29.77/4.27 fof(identity_is_largest, axiom, ![X2]: less_equal(X2, identity)).
% 29.77/4.27 fof(less_equal_and_equal, axiom, ![X2, Y2]: (~less_equal(X2, Y2) | (~less_equal(Y2, X2) | X2=Y2))).
% 29.77/4.27 fof(less_equal_quotient, axiom, ![X2, Y2]: (~quotient(X2, Y2, zero) | less_equal(X2, Y2))).
% 29.77/4.27 fof(prove_one_inversion_equals_three, negated_conjecture, idQa!=idQ__idQ_idQa).
% 29.77/4.27 fof(quotient_less_equal, axiom, ![X2, Y2]: (~less_equal(X2, Y2) | quotient(X2, Y2, zero))).
% 29.77/4.27 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))))))).
% 29.77/4.27 fof(well_defined, axiom, ![W, X2, Y2, Z2]: (~quotient(X2, Y2, Z2) | (~quotient(X2, Y2, W) | Z2=W))).
% 29.77/4.27 fof(zero_is_smallest, axiom, ![X2]: less_equal(zero, X2)).
% 29.77/4.27
% 29.77/4.27 Now clausify the problem and encode Horn clauses using encoding 3 of
% 29.77/4.27 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 29.77/4.27 We repeatedly replace C & s=t => u=v by the two clauses:
% 29.77/4.27 fresh(y, y, x1...xn) = u
% 29.77/4.27 C => fresh(s, t, x1...xn) = v
% 29.77/4.27 where fresh is a fresh function symbol and x1..xn are the free
% 29.77/4.27 variables of u and v.
% 29.77/4.27 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 29.77/4.27 input problem has no model of domain size 1).
% 29.77/4.27
% 29.77/4.27 The encoding turns the above axioms into the following unit equations and goals:
% 29.77/4.27
% 29.77/4.27 Axiom 1 (identity_is_largest): less_equal(X, identity) = true.
% 29.77/4.27 Axiom 2 (zero_is_smallest): less_equal(zero, X) = true.
% 29.77/4.27 Axiom 3 (identity_divide_a): quotient(identity, a, idQa) = true.
% 29.77/4.27 Axiom 4 (identity_divide_idQ_idQa): quotient(identity, idQ_idQa, idQ__idQ_idQa) = true.
% 29.77/4.27 Axiom 5 (identity_divide_idQa): quotient(identity, idQa, idQ_idQa) = true.
% 29.77/4.27 Axiom 6 (well_defined): fresh(X, X, Y, Z) = Z.
% 29.77/4.27 Axiom 7 (quotient_property): fresh12(X, X, Y, Z) = true.
% 29.77/4.27 Axiom 8 (less_equal_quotient): fresh7(X, X, Y, Z) = true.
% 29.77/4.27 Axiom 9 (divisor_existence): fresh6(X, X, Y, Z) = true.
% 29.77/4.27 Axiom 10 (quotient_less_equal): fresh5(X, X, Y, Z) = true.
% 29.77/4.27 Axiom 11 (less_equal_and_equal): fresh4(X, X, Y, Z) = Y.
% 29.77/4.27 Axiom 12 (less_equal_and_equal): fresh3(X, X, Y, Z) = Z.
% 29.77/4.27 Axiom 13 (closure): quotient(X, Y, divide(X, Y)) = true.
% 29.77/4.27 Axiom 14 (quotient_less_equal): fresh5(less_equal(X, Y), true, X, Y) = quotient(X, Y, zero).
% 29.77/4.27 Axiom 15 (less_equal_and_equal): fresh4(less_equal(X, Y), true, Y, X) = fresh3(less_equal(Y, X), true, Y, X).
% 29.77/4.27 Axiom 16 (well_defined): fresh2(X, X, Y, Z, W, V) = W.
% 29.77/4.27 Axiom 17 (quotient_property): fresh10(X, X, Y, Z, W, V, U) = less_equal(V, U).
% 29.77/4.27 Axiom 18 (less_equal_quotient): fresh7(quotient(X, Y, zero), true, X, Y) = less_equal(X, Y).
% 29.77/4.27 Axiom 19 (divisor_existence): fresh6(quotient(X, Y, Z), true, X, Z) = less_equal(Z, X).
% 29.77/4.27 Axiom 20 (quotient_property): fresh11(X, X, Y, Z, W, V, U, T, S) = fresh12(quotient(Y, Z, W), true, T, S).
% 29.77/4.27 Axiom 21 (well_defined): fresh2(quotient(X, Y, Z), true, X, Y, W, Z) = fresh(quotient(X, Y, W), true, W, Z).
% 29.77/4.27 Axiom 22 (quotient_property): fresh9(X, X, Y, Z, W, V, U, T, S, X2) = fresh10(quotient(Y, V, T), true, Y, Z, W, S, X2).
% 29.77/4.27 Axiom 23 (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).
% 29.77/4.27 Axiom 24 (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).
% 29.77/4.27
% 29.77/4.27 Lemma 25: fresh9(X, X, Y, Z, W, V, U, divide(Y, V), T, S) = less_equal(T, S).
% 29.77/4.27 Proof:
% 29.77/4.27 fresh9(X, X, Y, Z, W, V, U, divide(Y, V), T, S)
% 29.77/4.27 = { by axiom 22 (quotient_property) }
% 29.77/4.27 fresh10(quotient(Y, V, divide(Y, V)), true, Y, Z, W, T, S)
% 29.77/4.27 = { by axiom 13 (closure) }
% 29.77/4.27 fresh10(true, true, Y, Z, W, T, S)
% 29.77/4.27 = { by axiom 17 (quotient_property) }
% 29.77/4.27 less_equal(T, S)
% 29.77/4.27
% 29.77/4.27 Lemma 26: fresh8(Y2, Y2, Y, Z, W, V, U, T, S, X2) = fresh8(X, X, Y, Z, W, V, U, T, S, X2).
% 29.77/4.27 Proof:
% 29.77/4.27 fresh8(Y2, Y2, Y, Z, W, V, U, T, S, X2)
% 29.77/4.27 = { by axiom 23 (quotient_property) }
% 29.77/4.27 fresh11(quotient(Z, V, U), true, Y, Z, W, V, T, S, X2)
% 29.77/4.27 = { by axiom 23 (quotient_property) R->L }
% 29.77/4.27 fresh8(X, X, Y, Z, W, V, U, T, S, X2)
% 29.77/4.27
% 29.77/4.27 Lemma 27: fresh9(Y2, Y2, Y, Z, W, V, Z2, T, S, X2) = fresh9(X, X, Y, Z, W, V, U, T, S, X2).
% 29.77/4.27 Proof:
% 29.77/4.27 fresh9(Y2, Y2, Y, Z, W, V, Z2, T, S, X2)
% 29.77/4.27 = { by axiom 22 (quotient_property) }
% 29.77/4.27 fresh10(quotient(Y, V, T), true, Y, Z, W, S, X2)
% 29.77/4.27 = { by axiom 22 (quotient_property) R->L }
% 29.77/4.27 fresh9(X, X, Y, Z, W, V, U, T, S, X2)
% 29.77/4.27
% 29.77/4.27 Lemma 28: fresh8(X, X, Y, Z, W, V, U, T, divide(T, U), divide(W, V)) = fresh9(S, S, Y, Z, W, V, X2, T, divide(T, U), divide(W, V)).
% 29.77/4.27 Proof:
% 29.77/4.27 fresh8(X, X, Y, Z, W, V, U, T, divide(T, U), divide(W, V))
% 29.77/4.27 = { by lemma 26 }
% 29.77/4.27 fresh8(true, true, Y, Z, W, V, U, T, divide(T, U), divide(W, V))
% 29.77/4.27 = { by axiom 13 (closure) R->L }
% 29.77/4.27 fresh8(quotient(T, U, divide(T, U)), true, Y, Z, W, V, U, T, divide(T, U), divide(W, V))
% 29.77/4.27 = { by axiom 24 (quotient_property) }
% 29.77/4.27 fresh9(quotient(W, V, divide(W, V)), true, Y, Z, W, V, U, T, divide(T, U), divide(W, V))
% 29.77/4.27 = { by axiom 13 (closure) }
% 29.77/4.27 fresh9(true, true, Y, Z, W, V, U, T, divide(T, U), divide(W, V))
% 29.77/4.27 = { by lemma 27 R->L }
% 29.77/4.27 fresh9(S, S, Y, Z, W, V, X2, T, divide(T, U), divide(W, V))
% 29.77/4.27
% 29.77/4.27 Lemma 29: quotient(zero, X, zero) = true.
% 29.77/4.27 Proof:
% 29.77/4.27 quotient(zero, X, zero)
% 29.77/4.27 = { by axiom 14 (quotient_less_equal) R->L }
% 29.77/4.27 fresh5(less_equal(zero, X), true, zero, X)
% 29.77/4.27 = { by axiom 2 (zero_is_smallest) }
% 29.77/4.27 fresh5(true, true, zero, X)
% 29.77/4.27 = { by axiom 10 (quotient_less_equal) }
% 29.77/4.27 true
% 29.77/4.27
% 29.77/4.27 Lemma 30: less_equal(divide(divide(X, Y), zero), divide(divide(X, zero), Y)) = true.
% 29.77/4.27 Proof:
% 29.77/4.27 less_equal(divide(divide(X, Y), zero), divide(divide(X, zero), Y))
% 29.77/4.27 = { by lemma 25 R->L }
% 29.77/4.27 fresh9(Z, Z, X, zero, divide(X, zero), Y, W, divide(X, Y), divide(divide(X, Y), zero), divide(divide(X, zero), Y))
% 29.77/4.27 = { by lemma 28 R->L }
% 29.77/4.27 fresh8(V, V, X, zero, divide(X, zero), Y, zero, divide(X, Y), divide(divide(X, Y), zero), divide(divide(X, zero), Y))
% 29.77/4.27 = { by axiom 23 (quotient_property) }
% 29.77/4.27 fresh11(quotient(zero, Y, zero), true, X, zero, divide(X, zero), Y, divide(X, Y), divide(divide(X, Y), zero), divide(divide(X, zero), Y))
% 29.77/4.27 = { by lemma 29 }
% 29.77/4.27 fresh11(true, true, X, zero, divide(X, zero), Y, divide(X, Y), divide(divide(X, Y), zero), divide(divide(X, zero), Y))
% 29.77/4.27 = { by axiom 20 (quotient_property) }
% 29.77/4.27 fresh12(quotient(X, zero, divide(X, zero)), true, divide(divide(X, Y), zero), divide(divide(X, zero), Y))
% 29.77/4.27 = { by axiom 13 (closure) }
% 29.77/4.27 fresh12(true, true, divide(divide(X, Y), zero), divide(divide(X, zero), Y))
% 29.77/4.27 = { by axiom 7 (quotient_property) }
% 29.77/4.27 true
% 29.77/4.27
% 29.77/4.27 Lemma 31: less_equal(divide(X, Y), X) = true.
% 29.77/4.27 Proof:
% 29.77/4.27 less_equal(divide(X, Y), X)
% 29.77/4.27 = { by axiom 19 (divisor_existence) R->L }
% 29.77/4.27 fresh6(quotient(X, Y, divide(X, Y)), true, X, divide(X, Y))
% 29.77/4.27 = { by axiom 13 (closure) }
% 29.77/4.27 fresh6(true, true, X, divide(X, Y))
% 29.77/4.27 = { by axiom 9 (divisor_existence) }
% 29.77/4.27 true
% 29.77/4.27
% 29.77/4.27 Lemma 32: quotient(divide(X, Y), X, zero) = true.
% 29.77/4.27 Proof:
% 29.77/4.27 quotient(divide(X, Y), X, zero)
% 29.77/4.27 = { by axiom 14 (quotient_less_equal) R->L }
% 29.77/4.27 fresh5(less_equal(divide(X, Y), X), true, divide(X, Y), X)
% 29.77/4.27 = { by lemma 31 }
% 29.77/4.27 fresh5(true, true, divide(X, Y), X)
% 29.77/4.27 = { by axiom 10 (quotient_less_equal) }
% 29.77/4.27 true
% 29.77/4.27
% 29.77/4.27 Lemma 33: divide(divide(X, Y), X) = zero.
% 29.77/4.27 Proof:
% 29.77/4.27 divide(divide(X, Y), X)
% 29.77/4.27 = { by axiom 6 (well_defined) R->L }
% 29.77/4.27 fresh(true, true, zero, divide(divide(X, Y), X))
% 29.77/4.27 = { by lemma 32 R->L }
% 29.77/4.27 fresh(quotient(divide(X, Y), X, zero), true, zero, divide(divide(X, Y), X))
% 29.77/4.27 = { by axiom 21 (well_defined) R->L }
% 29.77/4.27 fresh2(quotient(divide(X, Y), X, divide(divide(X, Y), X)), true, divide(X, Y), X, zero, divide(divide(X, Y), X))
% 29.77/4.27 = { by axiom 13 (closure) }
% 29.77/4.27 fresh2(true, true, divide(X, Y), X, zero, divide(divide(X, Y), X))
% 29.77/4.27 = { by axiom 16 (well_defined) }
% 29.77/4.27 zero
% 29.77/4.27
% 29.77/4.27 Lemma 34: divide(X, X) = zero.
% 29.77/4.27 Proof:
% 29.77/4.27 divide(X, X)
% 29.77/4.27 = { by axiom 11 (less_equal_and_equal) R->L }
% 29.77/4.27 fresh4(true, true, divide(X, X), zero)
% 29.77/4.27 = { by axiom 2 (zero_is_smallest) R->L }
% 29.77/4.27 fresh4(less_equal(zero, divide(X, X)), true, divide(X, X), zero)
% 29.77/4.27 = { by axiom 15 (less_equal_and_equal) }
% 29.77/4.27 fresh3(less_equal(divide(X, X), zero), true, divide(X, X), zero)
% 29.77/4.27 = { by axiom 18 (less_equal_quotient) R->L }
% 29.77/4.27 fresh3(fresh7(quotient(divide(X, X), zero, zero), true, divide(X, X), zero), true, divide(X, X), zero)
% 29.77/4.27 = { by axiom 12 (less_equal_and_equal) R->L }
% 29.77/4.27 fresh3(fresh7(quotient(divide(X, X), zero, fresh3(true, true, divide(divide(X, X), zero), zero)), true, divide(X, X), zero), true, divide(X, X), zero)
% 29.77/4.27 = { by lemma 30 R->L }
% 29.77/4.28 fresh3(fresh7(quotient(divide(X, X), zero, fresh3(less_equal(divide(divide(X, X), zero), divide(divide(X, zero), X)), true, divide(divide(X, X), zero), zero)), true, divide(X, X), zero), true, divide(X, X), zero)
% 29.77/4.28 = { by lemma 33 }
% 29.77/4.28 fresh3(fresh7(quotient(divide(X, X), zero, fresh3(less_equal(divide(divide(X, X), zero), zero), true, divide(divide(X, X), zero), zero)), true, divide(X, X), zero), true, divide(X, X), zero)
% 29.77/4.28 = { by axiom 15 (less_equal_and_equal) R->L }
% 29.77/4.28 fresh3(fresh7(quotient(divide(X, X), zero, fresh4(less_equal(zero, divide(divide(X, X), zero)), true, divide(divide(X, X), zero), zero)), true, divide(X, X), zero), true, divide(X, X), zero)
% 29.77/4.28 = { by axiom 2 (zero_is_smallest) }
% 29.77/4.28 fresh3(fresh7(quotient(divide(X, X), zero, fresh4(true, true, divide(divide(X, X), zero), zero)), true, divide(X, X), zero), true, divide(X, X), zero)
% 29.77/4.28 = { by axiom 11 (less_equal_and_equal) }
% 29.77/4.28 fresh3(fresh7(quotient(divide(X, X), zero, divide(divide(X, X), zero)), true, divide(X, X), zero), true, divide(X, X), zero)
% 29.77/4.28 = { by axiom 13 (closure) }
% 29.77/4.28 fresh3(fresh7(true, true, divide(X, X), zero), true, divide(X, X), zero)
% 29.77/4.28 = { by axiom 8 (less_equal_quotient) }
% 29.77/4.28 fresh3(true, true, divide(X, X), zero)
% 29.77/4.28 = { by axiom 12 (less_equal_and_equal) }
% 29.77/4.28 zero
% 29.77/4.28
% 29.77/4.28 Lemma 35: divide(X, zero) = X.
% 29.77/4.28 Proof:
% 29.77/4.28 divide(X, zero)
% 29.77/4.28 = { by axiom 12 (less_equal_and_equal) R->L }
% 29.77/4.28 fresh3(true, true, X, divide(X, zero))
% 29.77/4.28 = { by axiom 8 (less_equal_quotient) R->L }
% 29.77/4.28 fresh3(fresh7(true, true, X, divide(X, zero)), true, X, divide(X, zero))
% 29.77/4.28 = { by axiom 13 (closure) R->L }
% 29.77/4.28 fresh3(fresh7(quotient(X, divide(X, zero), divide(X, divide(X, zero))), true, X, divide(X, zero)), true, X, divide(X, zero))
% 29.77/4.28 = { by axiom 11 (less_equal_and_equal) R->L }
% 29.77/4.28 fresh3(fresh7(quotient(X, divide(X, zero), fresh4(true, true, divide(X, divide(X, zero)), zero)), true, X, divide(X, zero)), true, X, divide(X, zero))
% 29.77/4.28 = { by axiom 2 (zero_is_smallest) R->L }
% 29.77/4.28 fresh3(fresh7(quotient(X, divide(X, zero), fresh4(less_equal(zero, divide(X, divide(X, zero))), true, divide(X, divide(X, zero)), zero)), true, X, divide(X, zero)), true, X, divide(X, zero))
% 29.77/4.28 = { by axiom 15 (less_equal_and_equal) }
% 29.77/4.28 fresh3(fresh7(quotient(X, divide(X, zero), fresh3(less_equal(divide(X, divide(X, zero)), zero), true, divide(X, divide(X, zero)), zero)), true, X, divide(X, zero)), true, X, divide(X, zero))
% 29.77/4.28 = { by axiom 18 (less_equal_quotient) R->L }
% 29.77/4.28 fresh3(fresh7(quotient(X, divide(X, zero), fresh3(fresh7(quotient(divide(X, divide(X, zero)), zero, zero), true, divide(X, divide(X, zero)), zero), true, divide(X, divide(X, zero)), zero)), true, X, divide(X, zero)), true, X, divide(X, zero))
% 29.77/4.28 = { by axiom 12 (less_equal_and_equal) R->L }
% 29.77/4.28 fresh3(fresh7(quotient(X, divide(X, zero), fresh3(fresh7(quotient(divide(X, divide(X, zero)), zero, fresh3(true, true, divide(divide(X, divide(X, zero)), zero), zero)), true, divide(X, divide(X, zero)), zero), true, divide(X, divide(X, zero)), zero)), true, X, divide(X, zero)), true, X, divide(X, zero))
% 29.77/4.28 = { by lemma 30 R->L }
% 29.77/4.28 fresh3(fresh7(quotient(X, divide(X, zero), fresh3(fresh7(quotient(divide(X, divide(X, zero)), zero, fresh3(less_equal(divide(divide(X, divide(X, zero)), zero), divide(divide(X, zero), divide(X, zero))), true, divide(divide(X, divide(X, zero)), zero), zero)), true, divide(X, divide(X, zero)), zero), true, divide(X, divide(X, zero)), zero)), true, X, divide(X, zero)), true, X, divide(X, zero))
% 29.77/4.28 = { by lemma 34 }
% 29.77/4.28 fresh3(fresh7(quotient(X, divide(X, zero), fresh3(fresh7(quotient(divide(X, divide(X, zero)), zero, fresh3(less_equal(divide(divide(X, divide(X, zero)), zero), zero), true, divide(divide(X, divide(X, zero)), zero), zero)), true, divide(X, divide(X, zero)), zero), true, divide(X, divide(X, zero)), zero)), true, X, divide(X, zero)), true, X, divide(X, zero))
% 29.77/4.28 = { by axiom 15 (less_equal_and_equal) R->L }
% 29.77/4.28 fresh3(fresh7(quotient(X, divide(X, zero), fresh3(fresh7(quotient(divide(X, divide(X, zero)), zero, fresh4(less_equal(zero, divide(divide(X, divide(X, zero)), zero)), true, divide(divide(X, divide(X, zero)), zero), zero)), true, divide(X, divide(X, zero)), zero), true, divide(X, divide(X, zero)), zero)), true, X, divide(X, zero)), true, X, divide(X, zero))
% 29.77/4.28 = { by axiom 2 (zero_is_smallest) }
% 29.77/4.28 fresh3(fresh7(quotient(X, divide(X, zero), fresh3(fresh7(quotient(divide(X, divide(X, zero)), zero, fresh4(true, true, divide(divide(X, divide(X, zero)), zero), zero)), true, divide(X, divide(X, zero)), zero), true, divide(X, divide(X, zero)), zero)), true, X, divide(X, zero)), true, X, divide(X, zero))
% 29.77/4.28 = { by axiom 11 (less_equal_and_equal) }
% 29.77/4.28 fresh3(fresh7(quotient(X, divide(X, zero), fresh3(fresh7(quotient(divide(X, divide(X, zero)), zero, divide(divide(X, divide(X, zero)), zero)), true, divide(X, divide(X, zero)), zero), true, divide(X, divide(X, zero)), zero)), true, X, divide(X, zero)), true, X, divide(X, zero))
% 29.77/4.28 = { by axiom 13 (closure) }
% 29.77/4.28 fresh3(fresh7(quotient(X, divide(X, zero), fresh3(fresh7(true, true, divide(X, divide(X, zero)), zero), true, divide(X, divide(X, zero)), zero)), true, X, divide(X, zero)), true, X, divide(X, zero))
% 29.77/4.28 = { by axiom 8 (less_equal_quotient) }
% 29.77/4.28 fresh3(fresh7(quotient(X, divide(X, zero), fresh3(true, true, divide(X, divide(X, zero)), zero)), true, X, divide(X, zero)), true, X, divide(X, zero))
% 29.77/4.28 = { by axiom 12 (less_equal_and_equal) }
% 29.77/4.28 fresh3(fresh7(quotient(X, divide(X, zero), zero), true, X, divide(X, zero)), true, X, divide(X, zero))
% 29.77/4.28 = { by axiom 18 (less_equal_quotient) }
% 29.77/4.28 fresh3(less_equal(X, divide(X, zero)), true, X, divide(X, zero))
% 29.77/4.28 = { by axiom 15 (less_equal_and_equal) R->L }
% 29.77/4.28 fresh4(less_equal(divide(X, zero), X), true, X, divide(X, zero))
% 29.77/4.28 = { by lemma 31 }
% 29.77/4.28 fresh4(true, true, X, divide(X, zero))
% 29.77/4.28 = { by axiom 11 (less_equal_and_equal) }
% 29.77/4.28 X
% 29.77/4.28
% 29.77/4.28 Lemma 36: divide(identity, idQ_idQa) = idQ__idQ_idQa.
% 29.77/4.28 Proof:
% 29.77/4.28 divide(identity, idQ_idQa)
% 29.77/4.28 = { by axiom 16 (well_defined) R->L }
% 29.77/4.28 fresh2(true, true, identity, idQ_idQa, divide(identity, idQ_idQa), idQ__idQ_idQa)
% 29.77/4.28 = { by axiom 4 (identity_divide_idQ_idQa) R->L }
% 29.77/4.28 fresh2(quotient(identity, idQ_idQa, idQ__idQ_idQa), true, identity, idQ_idQa, divide(identity, idQ_idQa), idQ__idQ_idQa)
% 29.77/4.28 = { by axiom 21 (well_defined) }
% 29.77/4.28 fresh(quotient(identity, idQ_idQa, divide(identity, idQ_idQa)), true, divide(identity, idQ_idQa), idQ__idQ_idQa)
% 29.77/4.28 = { by axiom 13 (closure) }
% 29.77/4.28 fresh(true, true, divide(identity, idQ_idQa), idQ__idQ_idQa)
% 29.77/4.28 = { by axiom 6 (well_defined) }
% 29.77/4.28 idQ__idQ_idQa
% 29.77/4.28
% 29.77/4.28 Lemma 37: fresh8(X, X, Y, Z, W, V, divide(Z, V), U, T, S) = fresh12(quotient(Y, Z, W), true, T, S).
% 29.77/4.28 Proof:
% 29.77/4.28 fresh8(X, X, Y, Z, W, V, divide(Z, V), U, T, S)
% 29.77/4.28 = { by axiom 23 (quotient_property) }
% 29.77/4.28 fresh11(quotient(Z, V, divide(Z, V)), true, Y, Z, W, V, U, T, S)
% 29.77/4.28 = { by axiom 13 (closure) }
% 29.77/4.28 fresh11(true, true, Y, Z, W, V, U, T, S)
% 29.77/4.28 = { by axiom 20 (quotient_property) }
% 29.77/4.28 fresh12(quotient(Y, Z, W), true, T, S)
% 29.77/4.28
% 29.77/4.28 Lemma 38: fresh9(X, X, Y, Z, W, V, U, T, divide(T, divide(Z, V)), divide(W, V)) = fresh12(quotient(Y, Z, W), true, divide(T, divide(Z, V)), divide(W, V)).
% 29.77/4.28 Proof:
% 29.77/4.28 fresh9(X, X, Y, Z, W, V, U, T, divide(T, divide(Z, V)), divide(W, V))
% 29.77/4.28 = { by lemma 28 R->L }
% 29.77/4.28 fresh8(S, S, Y, Z, W, V, divide(Z, V), T, divide(T, divide(Z, V)), divide(W, V))
% 29.77/4.28 = { by lemma 37 }
% 29.77/4.28 fresh12(quotient(Y, Z, W), true, divide(T, divide(Z, V)), divide(W, V))
% 29.77/4.28
% 29.77/4.28 Lemma 39: fresh12(quotient(X, Y, Z), true, divide(divide(X, W), divide(Y, W)), divide(Z, W)) = less_equal(divide(divide(X, W), divide(Y, W)), divide(Z, W)).
% 29.77/4.28 Proof:
% 29.77/4.28 fresh12(quotient(X, Y, Z), true, divide(divide(X, W), divide(Y, W)), divide(Z, W))
% 29.77/4.28 = { by lemma 38 R->L }
% 29.77/4.28 fresh9(V, V, X, Y, Z, W, U, divide(X, W), divide(divide(X, W), divide(Y, W)), divide(Z, W))
% 29.77/4.28 = { by lemma 25 }
% 29.77/4.28 less_equal(divide(divide(X, W), divide(Y, W)), divide(Z, W))
% 29.77/4.28
% 29.77/4.28 Lemma 40: divide(idQ_idQa, divide(a, idQa)) = zero.
% 29.77/4.28 Proof:
% 29.77/4.28 divide(idQ_idQa, divide(a, idQa))
% 29.77/4.28 = { by axiom 11 (less_equal_and_equal) R->L }
% 29.77/4.28 fresh4(true, true, divide(idQ_idQa, divide(a, idQa)), zero)
% 29.77/4.28 = { by axiom 2 (zero_is_smallest) R->L }
% 29.77/4.28 fresh4(less_equal(zero, divide(idQ_idQa, divide(a, idQa))), true, divide(idQ_idQa, divide(a, idQa)), zero)
% 29.77/4.28 = { by axiom 15 (less_equal_and_equal) }
% 29.77/4.28 fresh3(less_equal(divide(idQ_idQa, divide(a, idQa)), zero), true, divide(idQ_idQa, divide(a, idQa)), zero)
% 29.77/4.28 = { by lemma 34 R->L }
% 29.77/4.28 fresh3(less_equal(divide(idQ_idQa, divide(a, idQa)), divide(idQa, idQa)), true, divide(idQ_idQa, divide(a, idQa)), zero)
% 29.77/4.28 = { by axiom 6 (well_defined) R->L }
% 29.77/4.28 fresh3(less_equal(divide(fresh(true, true, divide(identity, idQa), idQ_idQa), divide(a, idQa)), divide(idQa, idQa)), true, divide(idQ_idQa, divide(a, idQa)), zero)
% 29.77/4.28 = { by axiom 13 (closure) R->L }
% 29.77/4.28 fresh3(less_equal(divide(fresh(quotient(identity, idQa, divide(identity, idQa)), true, divide(identity, idQa), idQ_idQa), divide(a, idQa)), divide(idQa, idQa)), true, divide(idQ_idQa, divide(a, idQa)), zero)
% 29.77/4.28 = { by axiom 21 (well_defined) R->L }
% 29.77/4.28 fresh3(less_equal(divide(fresh2(quotient(identity, idQa, idQ_idQa), true, identity, idQa, divide(identity, idQa), idQ_idQa), divide(a, idQa)), divide(idQa, idQa)), true, divide(idQ_idQa, divide(a, idQa)), zero)
% 29.77/4.28 = { by axiom 5 (identity_divide_idQa) }
% 29.77/4.28 fresh3(less_equal(divide(fresh2(true, true, identity, idQa, divide(identity, idQa), idQ_idQa), divide(a, idQa)), divide(idQa, idQa)), true, divide(idQ_idQa, divide(a, idQa)), zero)
% 29.77/4.28 = { by axiom 16 (well_defined) }
% 29.77/4.28 fresh3(less_equal(divide(divide(identity, idQa), divide(a, idQa)), divide(idQa, idQa)), true, divide(idQ_idQa, divide(a, idQa)), zero)
% 29.77/4.28 = { by lemma 39 R->L }
% 29.77/4.28 fresh3(fresh12(quotient(identity, a, idQa), true, divide(divide(identity, idQa), divide(a, idQa)), divide(idQa, idQa)), true, divide(idQ_idQa, divide(a, idQa)), zero)
% 29.77/4.28 = { by axiom 3 (identity_divide_a) }
% 29.77/4.28 fresh3(fresh12(true, true, divide(divide(identity, idQa), divide(a, idQa)), divide(idQa, idQa)), true, divide(idQ_idQa, divide(a, idQa)), zero)
% 29.77/4.28 = { by axiom 7 (quotient_property) }
% 29.77/4.28 fresh3(true, true, divide(idQ_idQa, divide(a, idQa)), zero)
% 29.77/4.28 = { by axiom 12 (less_equal_and_equal) }
% 29.77/4.28 zero
% 29.77/4.28
% 29.77/4.28 Lemma 41: 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).
% 29.77/4.28 Proof:
% 29.77/4.28 fresh8(X, X, Y, Z, zero, W, V, U, divide(U, V), zero)
% 29.77/4.28 = { by lemma 26 }
% 29.77/4.28 fresh8(true, true, Y, Z, zero, W, V, U, divide(U, V), zero)
% 29.77/4.28 = { by axiom 13 (closure) R->L }
% 29.77/4.28 fresh8(quotient(U, V, divide(U, V)), true, Y, Z, zero, W, V, U, divide(U, V), zero)
% 29.77/4.28 = { by axiom 24 (quotient_property) }
% 29.77/4.28 fresh9(quotient(zero, W, zero), true, Y, Z, zero, W, V, U, divide(U, V), zero)
% 29.77/4.28 = { by lemma 29 }
% 29.77/4.28 fresh9(true, true, Y, Z, zero, W, V, U, divide(U, V), zero)
% 29.77/4.28 = { by lemma 27 R->L }
% 29.77/4.28 fresh9(T, T, Y, Z, zero, W, S, U, divide(U, V), zero)
% 29.77/4.28
% 29.77/4.28 Lemma 42: divide(a, idQa) = idQ_idQa.
% 29.77/4.28 Proof:
% 29.77/4.28 divide(a, idQa)
% 29.77/4.28 = { by axiom 12 (less_equal_and_equal) R->L }
% 29.77/4.28 fresh3(true, true, idQ_idQa, divide(a, idQa))
% 29.77/4.28 = { by axiom 8 (less_equal_quotient) R->L }
% 29.77/4.28 fresh3(fresh7(true, true, idQ_idQa, divide(a, idQa)), true, idQ_idQa, divide(a, idQa))
% 29.77/4.28 = { by axiom 13 (closure) R->L }
% 29.77/4.28 fresh3(fresh7(quotient(idQ_idQa, divide(a, idQa), divide(idQ_idQa, divide(a, idQa))), true, idQ_idQa, divide(a, idQa)), true, idQ_idQa, divide(a, idQa))
% 29.77/4.28 = { by lemma 40 }
% 29.77/4.28 fresh3(fresh7(quotient(idQ_idQa, divide(a, idQa), zero), true, idQ_idQa, divide(a, idQa)), true, idQ_idQa, divide(a, idQa))
% 29.77/4.28 = { by axiom 18 (less_equal_quotient) }
% 29.77/4.28 fresh3(less_equal(idQ_idQa, divide(a, idQa)), true, idQ_idQa, divide(a, idQa))
% 29.77/4.28 = { by axiom 15 (less_equal_and_equal) R->L }
% 29.77/4.28 fresh4(less_equal(divide(a, idQa), idQ_idQa), true, idQ_idQa, divide(a, idQa))
% 29.77/4.28 = { by axiom 18 (less_equal_quotient) R->L }
% 29.77/4.28 fresh4(fresh7(quotient(divide(a, idQa), idQ_idQa, zero), true, divide(a, idQa), idQ_idQa), true, idQ_idQa, divide(a, idQa))
% 29.77/4.28 = { by axiom 12 (less_equal_and_equal) R->L }
% 29.77/4.28 fresh4(fresh7(quotient(divide(a, idQa), idQ_idQa, fresh3(true, true, divide(divide(a, idQa), idQ_idQa), zero)), true, divide(a, idQa), idQ_idQa), true, idQ_idQa, divide(a, idQa))
% 29.77/4.28 = { by axiom 7 (quotient_property) R->L }
% 29.77/4.28 fresh4(fresh7(quotient(divide(a, idQa), idQ_idQa, fresh3(fresh12(true, true, divide(divide(a, idQa), idQ_idQa), zero), true, divide(divide(a, idQa), idQ_idQa), zero)), true, divide(a, idQa), idQ_idQa), true, idQ_idQa, divide(a, idQa))
% 29.77/4.28 = { by axiom 10 (quotient_less_equal) R->L }
% 29.77/4.28 fresh4(fresh7(quotient(divide(a, idQa), idQ_idQa, fresh3(fresh12(fresh5(true, true, a, identity), true, divide(divide(a, idQa), idQ_idQa), zero), true, divide(divide(a, idQa), idQ_idQa), zero)), true, divide(a, idQa), idQ_idQa), true, idQ_idQa, divide(a, idQa))
% 29.77/4.28 = { by axiom 1 (identity_is_largest) R->L }
% 29.77/4.28 fresh4(fresh7(quotient(divide(a, idQa), idQ_idQa, fresh3(fresh12(fresh5(less_equal(a, identity), true, a, identity), true, divide(divide(a, idQa), idQ_idQa), zero), true, divide(divide(a, idQa), idQ_idQa), zero)), true, divide(a, idQa), idQ_idQa), true, idQ_idQa, divide(a, idQa))
% 29.77/4.28 = { by axiom 14 (quotient_less_equal) }
% 29.77/4.28 fresh4(fresh7(quotient(divide(a, idQa), idQ_idQa, fresh3(fresh12(quotient(a, identity, zero), true, divide(divide(a, idQa), idQ_idQa), zero), true, divide(divide(a, idQa), idQ_idQa), zero)), true, divide(a, idQa), idQ_idQa), true, idQ_idQa, divide(a, idQa))
% 29.77/4.28 = { by axiom 20 (quotient_property) R->L }
% 29.77/4.28 fresh4(fresh7(quotient(divide(a, idQa), idQ_idQa, fresh3(fresh11(true, true, a, identity, zero, idQa, divide(a, idQa), divide(divide(a, idQa), idQ_idQa), zero), true, divide(divide(a, idQa), idQ_idQa), zero)), true, divide(a, idQa), idQ_idQa), true, idQ_idQa, divide(a, idQa))
% 29.77/4.28 = { by axiom 5 (identity_divide_idQa) R->L }
% 29.77/4.28 fresh4(fresh7(quotient(divide(a, idQa), idQ_idQa, fresh3(fresh11(quotient(identity, idQa, idQ_idQa), true, a, identity, zero, idQa, divide(a, idQa), divide(divide(a, idQa), idQ_idQa), zero), true, divide(divide(a, idQa), idQ_idQa), zero)), true, divide(a, idQa), idQ_idQa), true, idQ_idQa, divide(a, idQa))
% 29.77/4.28 = { by axiom 23 (quotient_property) R->L }
% 29.77/4.28 fresh4(fresh7(quotient(divide(a, idQa), idQ_idQa, fresh3(fresh8(X, X, a, identity, zero, idQa, idQ_idQa, divide(a, idQa), divide(divide(a, idQa), idQ_idQa), zero), true, divide(divide(a, idQa), idQ_idQa), zero)), true, divide(a, idQa), idQ_idQa), true, idQ_idQa, divide(a, idQa))
% 29.77/4.28 = { by lemma 41 }
% 29.77/4.28 fresh4(fresh7(quotient(divide(a, idQa), idQ_idQa, fresh3(fresh9(Y, Y, a, identity, zero, idQa, Z, divide(a, idQa), divide(divide(a, idQa), idQ_idQa), zero), true, divide(divide(a, idQa), idQ_idQa), zero)), true, divide(a, idQa), idQ_idQa), true, idQ_idQa, divide(a, idQa))
% 29.77/4.28 = { by lemma 25 }
% 29.77/4.29 fresh4(fresh7(quotient(divide(a, idQa), idQ_idQa, fresh3(less_equal(divide(divide(a, idQa), idQ_idQa), zero), true, divide(divide(a, idQa), idQ_idQa), zero)), true, divide(a, idQa), idQ_idQa), true, idQ_idQa, divide(a, idQa))
% 29.77/4.29 = { by axiom 15 (less_equal_and_equal) R->L }
% 29.77/4.29 fresh4(fresh7(quotient(divide(a, idQa), idQ_idQa, fresh4(less_equal(zero, divide(divide(a, idQa), idQ_idQa)), true, divide(divide(a, idQa), idQ_idQa), zero)), true, divide(a, idQa), idQ_idQa), true, idQ_idQa, divide(a, idQa))
% 29.77/4.29 = { by axiom 2 (zero_is_smallest) }
% 29.77/4.29 fresh4(fresh7(quotient(divide(a, idQa), idQ_idQa, fresh4(true, true, divide(divide(a, idQa), idQ_idQa), zero)), true, divide(a, idQa), idQ_idQa), true, idQ_idQa, divide(a, idQa))
% 29.77/4.29 = { by axiom 11 (less_equal_and_equal) }
% 29.77/4.29 fresh4(fresh7(quotient(divide(a, idQa), idQ_idQa, divide(divide(a, idQa), idQ_idQa)), true, divide(a, idQa), idQ_idQa), true, idQ_idQa, divide(a, idQa))
% 29.77/4.29 = { by axiom 13 (closure) }
% 29.77/4.29 fresh4(fresh7(true, true, divide(a, idQa), idQ_idQa), true, idQ_idQa, divide(a, idQa))
% 29.77/4.29 = { by axiom 8 (less_equal_quotient) }
% 29.77/4.29 fresh4(true, true, idQ_idQa, divide(a, idQa))
% 29.77/4.29 = { by axiom 11 (less_equal_and_equal) }
% 29.77/4.29 idQ_idQa
% 29.77/4.29
% 29.77/4.29 Lemma 43: divide(divide(X, divide(Y, Z)), Y) = divide(X, Y).
% 29.77/4.29 Proof:
% 29.77/4.29 divide(divide(X, divide(Y, Z)), Y)
% 29.77/4.29 = { by axiom 12 (less_equal_and_equal) R->L }
% 29.77/4.29 fresh3(true, true, divide(X, Y), divide(divide(X, divide(Y, Z)), Y))
% 29.77/4.29 = { by axiom 7 (quotient_property) R->L }
% 29.77/4.29 fresh3(fresh12(true, true, divide(X, Y), divide(divide(X, divide(Y, Z)), Y)), true, divide(X, Y), divide(divide(X, divide(Y, Z)), Y))
% 29.77/4.29 = { by axiom 13 (closure) R->L }
% 29.77/4.29 fresh3(fresh12(quotient(X, divide(Y, Z), divide(X, divide(Y, Z))), true, divide(X, Y), divide(divide(X, divide(Y, Z)), Y)), true, divide(X, Y), divide(divide(X, divide(Y, Z)), Y))
% 29.77/4.29 = { by lemma 35 R->L }
% 29.77/4.29 fresh3(fresh12(quotient(X, divide(Y, Z), divide(X, divide(Y, Z))), true, divide(divide(X, Y), zero), divide(divide(X, divide(Y, Z)), Y)), true, divide(X, Y), divide(divide(X, divide(Y, Z)), Y))
% 29.77/4.29 = { by lemma 33 R->L }
% 29.77/4.29 fresh3(fresh12(quotient(X, divide(Y, Z), divide(X, divide(Y, Z))), true, divide(divide(X, Y), divide(divide(Y, Z), Y)), divide(divide(X, divide(Y, Z)), Y)), true, divide(X, Y), divide(divide(X, divide(Y, Z)), Y))
% 29.77/4.29 = { by lemma 38 R->L }
% 29.77/4.29 fresh3(fresh9(W, W, X, divide(Y, Z), divide(X, divide(Y, Z)), Y, V, divide(X, Y), divide(divide(X, Y), divide(divide(Y, Z), Y)), divide(divide(X, divide(Y, Z)), Y)), true, divide(X, Y), divide(divide(X, divide(Y, Z)), Y))
% 29.77/4.29 = { by lemma 33 }
% 29.77/4.29 fresh3(fresh9(W, W, X, divide(Y, Z), divide(X, divide(Y, Z)), Y, V, divide(X, Y), divide(divide(X, Y), zero), divide(divide(X, divide(Y, Z)), Y)), true, divide(X, Y), divide(divide(X, divide(Y, Z)), Y))
% 29.77/4.29 = { by lemma 35 }
% 29.77/4.29 fresh3(fresh9(W, W, X, divide(Y, Z), divide(X, divide(Y, Z)), Y, V, divide(X, Y), divide(X, Y), divide(divide(X, divide(Y, Z)), Y)), true, divide(X, Y), divide(divide(X, divide(Y, Z)), Y))
% 29.77/4.29 = { by lemma 25 }
% 29.77/4.29 fresh3(less_equal(divide(X, Y), divide(divide(X, divide(Y, Z)), Y)), true, divide(X, Y), divide(divide(X, divide(Y, Z)), Y))
% 29.77/4.29 = { by axiom 15 (less_equal_and_equal) R->L }
% 29.77/4.29 fresh4(less_equal(divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), true, divide(X, Y), divide(divide(X, divide(Y, Z)), Y))
% 29.77/4.29 = { by axiom 18 (less_equal_quotient) R->L }
% 29.77/4.29 fresh4(fresh7(quotient(divide(divide(X, divide(Y, Z)), Y), divide(X, Y), zero), true, divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), true, divide(X, Y), divide(divide(X, divide(Y, Z)), Y))
% 29.77/4.29 = { by axiom 12 (less_equal_and_equal) R->L }
% 29.77/4.29 fresh4(fresh7(quotient(divide(divide(X, divide(Y, Z)), Y), divide(X, Y), fresh3(true, true, divide(divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), zero)), true, divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), true, divide(X, Y), divide(divide(X, divide(Y, Z)), Y))
% 29.77/4.29 = { by axiom 7 (quotient_property) R->L }
% 29.77/4.29 fresh4(fresh7(quotient(divide(divide(X, divide(Y, Z)), Y), divide(X, Y), fresh3(fresh12(true, true, divide(divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), zero), true, divide(divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), zero)), true, divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), true, divide(X, Y), divide(divide(X, divide(Y, Z)), Y))
% 29.77/4.29 = { by lemma 32 R->L }
% 29.77/4.29 fresh4(fresh7(quotient(divide(divide(X, divide(Y, Z)), Y), divide(X, Y), fresh3(fresh12(quotient(divide(X, divide(Y, Z)), X, zero), true, divide(divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), zero), true, divide(divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), zero)), true, divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), true, divide(X, Y), divide(divide(X, divide(Y, Z)), Y))
% 29.77/4.29 = { by lemma 37 R->L }
% 29.77/4.29 fresh4(fresh7(quotient(divide(divide(X, divide(Y, Z)), Y), divide(X, Y), fresh3(fresh8(U, U, divide(X, divide(Y, Z)), X, zero, Y, divide(X, Y), divide(divide(X, divide(Y, Z)), Y), divide(divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), zero), true, divide(divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), zero)), true, divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), true, divide(X, Y), divide(divide(X, divide(Y, Z)), Y))
% 29.77/4.29 = { by lemma 41 }
% 29.77/4.29 fresh4(fresh7(quotient(divide(divide(X, divide(Y, Z)), Y), divide(X, Y), fresh3(fresh9(T, T, divide(X, divide(Y, Z)), X, zero, Y, S, divide(divide(X, divide(Y, Z)), Y), divide(divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), zero), true, divide(divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), zero)), true, divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), true, divide(X, Y), divide(divide(X, divide(Y, Z)), Y))
% 29.77/4.29 = { by lemma 25 }
% 29.77/4.29 fresh4(fresh7(quotient(divide(divide(X, divide(Y, Z)), Y), divide(X, Y), fresh3(less_equal(divide(divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), zero), true, divide(divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), zero)), true, divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), true, divide(X, Y), divide(divide(X, divide(Y, Z)), Y))
% 29.77/4.29 = { by axiom 15 (less_equal_and_equal) R->L }
% 29.77/4.29 fresh4(fresh7(quotient(divide(divide(X, divide(Y, Z)), Y), divide(X, Y), fresh4(less_equal(zero, divide(divide(divide(X, divide(Y, Z)), Y), divide(X, Y))), true, divide(divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), zero)), true, divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), true, divide(X, Y), divide(divide(X, divide(Y, Z)), Y))
% 29.77/4.29 = { by axiom 2 (zero_is_smallest) }
% 29.77/4.29 fresh4(fresh7(quotient(divide(divide(X, divide(Y, Z)), Y), divide(X, Y), fresh4(true, true, divide(divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), zero)), true, divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), true, divide(X, Y), divide(divide(X, divide(Y, Z)), Y))
% 29.77/4.29 = { by axiom 11 (less_equal_and_equal) }
% 29.77/4.29 fresh4(fresh7(quotient(divide(divide(X, divide(Y, Z)), Y), divide(X, Y), divide(divide(divide(X, divide(Y, Z)), Y), divide(X, Y))), true, divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), true, divide(X, Y), divide(divide(X, divide(Y, Z)), Y))
% 29.77/4.29 = { by axiom 13 (closure) }
% 29.77/4.29 fresh4(fresh7(true, true, divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), true, divide(X, Y), divide(divide(X, divide(Y, Z)), Y))
% 29.77/4.29 = { by axiom 8 (less_equal_quotient) }
% 29.77/4.29 fresh4(true, true, divide(X, Y), divide(divide(X, divide(Y, Z)), Y))
% 29.77/4.29 = { by axiom 11 (less_equal_and_equal) }
% 29.77/4.29 divide(X, Y)
% 29.77/4.29
% 29.77/4.29 Goal 1 (prove_one_inversion_equals_three): idQa = idQ__idQ_idQa.
% 29.77/4.29 Proof:
% 29.77/4.29 idQa
% 29.77/4.29 = { by axiom 12 (less_equal_and_equal) R->L }
% 29.77/4.29 fresh3(true, true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by axiom 8 (less_equal_quotient) R->L }
% 29.77/4.29 fresh3(fresh7(true, true, idQ__idQ_idQa, idQa), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by axiom 13 (closure) R->L }
% 29.77/4.29 fresh3(fresh7(quotient(idQ__idQ_idQa, idQa, divide(idQ__idQ_idQa, idQa)), true, idQ__idQ_idQa, idQa), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by lemma 43 R->L }
% 29.77/4.29 fresh3(fresh7(quotient(idQ__idQ_idQa, idQa, divide(divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), idQa)), true, idQ__idQ_idQa, idQa), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by axiom 11 (less_equal_and_equal) R->L }
% 29.77/4.29 fresh3(fresh7(quotient(idQ__idQ_idQa, idQa, divide(fresh4(true, true, divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), zero), idQa)), true, idQ__idQ_idQa, idQa), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by axiom 2 (zero_is_smallest) R->L }
% 29.77/4.29 fresh3(fresh7(quotient(idQ__idQ_idQa, idQa, divide(fresh4(less_equal(zero, divide(idQ__idQ_idQa, divide(idQa, idQ_idQa))), true, divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), zero), idQa)), true, idQ__idQ_idQa, idQa), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by axiom 15 (less_equal_and_equal) }
% 29.77/4.29 fresh3(fresh7(quotient(idQ__idQ_idQa, idQa, divide(fresh3(less_equal(divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), zero), true, divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), zero), idQa)), true, idQ__idQ_idQa, idQa), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by lemma 36 R->L }
% 29.77/4.29 fresh3(fresh7(quotient(idQ__idQ_idQa, idQa, divide(fresh3(less_equal(divide(divide(identity, idQ_idQa), divide(idQa, idQ_idQa)), zero), true, divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), zero), idQa)), true, idQ__idQ_idQa, idQa), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by lemma 42 R->L }
% 29.77/4.29 fresh3(fresh7(quotient(idQ__idQ_idQa, idQa, divide(fresh3(less_equal(divide(divide(identity, divide(a, idQa)), divide(idQa, idQ_idQa)), zero), true, divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), zero), idQa)), true, idQ__idQ_idQa, idQa), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by lemma 42 R->L }
% 29.77/4.29 fresh3(fresh7(quotient(idQ__idQ_idQa, idQa, divide(fresh3(less_equal(divide(divide(identity, divide(a, idQa)), divide(idQa, divide(a, idQa))), zero), true, divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), zero), idQa)), true, idQ__idQ_idQa, idQa), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by lemma 40 R->L }
% 29.77/4.29 fresh3(fresh7(quotient(idQ__idQ_idQa, idQa, divide(fresh3(less_equal(divide(divide(identity, divide(a, idQa)), divide(idQa, divide(a, idQa))), divide(idQ_idQa, divide(a, idQa))), true, divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), zero), idQa)), true, idQ__idQ_idQa, idQa), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by lemma 39 R->L }
% 29.77/4.29 fresh3(fresh7(quotient(idQ__idQ_idQa, idQa, divide(fresh3(fresh12(quotient(identity, idQa, idQ_idQa), true, divide(divide(identity, divide(a, idQa)), divide(idQa, divide(a, idQa))), divide(idQ_idQa, divide(a, idQa))), true, divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), zero), idQa)), true, idQ__idQ_idQa, idQa), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by axiom 5 (identity_divide_idQa) }
% 29.77/4.29 fresh3(fresh7(quotient(idQ__idQ_idQa, idQa, divide(fresh3(fresh12(true, true, divide(divide(identity, divide(a, idQa)), divide(idQa, divide(a, idQa))), divide(idQ_idQa, divide(a, idQa))), true, divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), zero), idQa)), true, idQ__idQ_idQa, idQa), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by axiom 7 (quotient_property) }
% 29.77/4.29 fresh3(fresh7(quotient(idQ__idQ_idQa, idQa, divide(fresh3(true, true, divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), zero), idQa)), true, idQ__idQ_idQa, idQa), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by axiom 12 (less_equal_and_equal) }
% 29.77/4.29 fresh3(fresh7(quotient(idQ__idQ_idQa, idQa, divide(zero, idQa)), true, idQ__idQ_idQa, idQa), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by axiom 12 (less_equal_and_equal) R->L }
% 29.77/4.29 fresh3(fresh7(quotient(idQ__idQ_idQa, idQa, fresh3(true, true, zero, divide(zero, idQa))), true, idQ__idQ_idQa, idQa), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by axiom 2 (zero_is_smallest) R->L }
% 29.77/4.29 fresh3(fresh7(quotient(idQ__idQ_idQa, idQa, fresh3(less_equal(zero, divide(zero, idQa)), true, zero, divide(zero, idQa))), true, idQ__idQ_idQa, idQa), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by axiom 15 (less_equal_and_equal) R->L }
% 29.77/4.29 fresh3(fresh7(quotient(idQ__idQ_idQa, idQa, fresh4(less_equal(divide(zero, idQa), zero), true, zero, divide(zero, idQa))), true, idQ__idQ_idQa, idQa), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by lemma 31 }
% 29.77/4.29 fresh3(fresh7(quotient(idQ__idQ_idQa, idQa, fresh4(true, true, zero, divide(zero, idQa))), true, idQ__idQ_idQa, idQa), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by axiom 11 (less_equal_and_equal) }
% 29.77/4.29 fresh3(fresh7(quotient(idQ__idQ_idQa, idQa, zero), true, idQ__idQ_idQa, idQa), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by axiom 18 (less_equal_quotient) }
% 29.77/4.29 fresh3(less_equal(idQ__idQ_idQa, idQa), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by axiom 15 (less_equal_and_equal) R->L }
% 29.77/4.29 fresh4(less_equal(idQa, idQ__idQ_idQa), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by lemma 36 R->L }
% 29.77/4.29 fresh4(less_equal(idQa, divide(identity, idQ_idQa)), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by lemma 42 R->L }
% 29.77/4.29 fresh4(less_equal(idQa, divide(identity, divide(a, idQa))), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by axiom 6 (well_defined) R->L }
% 29.77/4.29 fresh4(less_equal(fresh(true, true, divide(identity, a), idQa), divide(identity, divide(a, idQa))), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by axiom 13 (closure) R->L }
% 29.77/4.29 fresh4(less_equal(fresh(quotient(identity, a, divide(identity, a)), true, divide(identity, a), idQa), divide(identity, divide(a, idQa))), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by axiom 21 (well_defined) R->L }
% 29.77/4.29 fresh4(less_equal(fresh2(quotient(identity, a, idQa), true, identity, a, divide(identity, a), idQa), divide(identity, divide(a, idQa))), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by axiom 3 (identity_divide_a) }
% 29.77/4.29 fresh4(less_equal(fresh2(true, true, identity, a, divide(identity, a), idQa), divide(identity, divide(a, idQa))), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by axiom 16 (well_defined) }
% 29.77/4.29 fresh4(less_equal(divide(identity, a), divide(identity, divide(a, idQa))), true, idQ__idQ_idQa, idQa)
% 29.77/4.29 = { by lemma 43 R->L }
% 29.77/4.30 fresh4(less_equal(divide(divide(identity, divide(a, idQa)), a), divide(identity, divide(a, idQa))), true, idQ__idQ_idQa, idQa)
% 29.77/4.30 = { by lemma 31 }
% 29.77/4.30 fresh4(true, true, idQ__idQ_idQa, idQa)
% 29.77/4.30 = { by axiom 11 (less_equal_and_equal) }
% 29.77/4.30 idQ__idQ_idQa
% 29.77/4.30 % SZS output end Proof
% 29.77/4.30
% 29.77/4.30 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------