TSTP Solution File: HEN009-3 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : HEN009-3 : 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 : n014.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 0.21s 0.49s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : HEN009-3 : 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
% 0.13/0.34 % Computer : n014.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
% 0.20/0.34 % CPULimit : 300
% 0.20/0.34 % WCLimit : 300
% 0.20/0.34 % DateTime : Thu Aug 24 13:22:18 EDT 2023
% 0.20/0.35 % CPUTime :
% 0.21/0.49 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.21/0.49
% 0.21/0.49 % SZS status Unsatisfiable
% 0.21/0.49
% 0.21/0.54 % SZS output start Proof
% 0.21/0.54 Take the following subset of the input axioms:
% 0.21/0.54 fof(id_divide_a_is_b, hypothesis, divide(identity, a)=b).
% 0.21/0.54 fof(id_divide_b_is_c, hypothesis, divide(identity, b)=c).
% 0.21/0.54 fof(id_divide_c_is_d, hypothesis, divide(identity, c)=d).
% 0.21/0.54 fof(identity_is_largest, axiom, ![X]: less_equal(X, identity)).
% 0.21/0.54 fof(less_equal_and_equal, axiom, ![Y, X2]: (~less_equal(X2, Y) | (~less_equal(Y, X2) | X2=Y))).
% 0.21/0.54 fof(part_of_theorem, hypothesis, divide(identity, a)!=divide(identity, divide(identity, divide(identity, a)))).
% 0.21/0.54 fof(prove_b_equals_d, negated_conjecture, b!=d).
% 0.21/0.54 fof(quotient_less_equal1, axiom, ![X2, Y2]: (~less_equal(X2, Y2) | divide(X2, Y2)=zero)).
% 0.21/0.54 fof(quotient_less_equal2, axiom, ![X2, Y2]: (divide(X2, Y2)!=zero | less_equal(X2, Y2))).
% 0.21/0.54 fof(quotient_property, axiom, ![Z, X2, Y2]: less_equal(divide(divide(X2, Z), divide(Y2, Z)), divide(divide(X2, Y2), Z))).
% 0.21/0.54 fof(quotient_smaller_than_numerator, axiom, ![X2, Y2]: less_equal(divide(X2, Y2), X2)).
% 0.21/0.54 fof(zero_is_smallest, axiom, ![X2]: less_equal(zero, X2)).
% 0.21/0.54
% 0.21/0.54 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.54 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.54 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.54 fresh(y, y, x1...xn) = u
% 0.21/0.54 C => fresh(s, t, x1...xn) = v
% 0.21/0.54 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.54 variables of u and v.
% 0.21/0.54 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.54 input problem has no model of domain size 1).
% 0.21/0.54
% 0.21/0.54 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.54
% 0.21/0.54 Axiom 1 (id_divide_a_is_b): divide(identity, a) = b.
% 0.21/0.54 Axiom 2 (id_divide_b_is_c): divide(identity, b) = c.
% 0.21/0.54 Axiom 3 (id_divide_c_is_d): divide(identity, c) = d.
% 0.21/0.54 Axiom 4 (identity_is_largest): less_equal(X, identity) = true.
% 0.21/0.54 Axiom 5 (zero_is_smallest): less_equal(zero, X) = true.
% 0.21/0.54 Axiom 6 (quotient_smaller_than_numerator): less_equal(divide(X, Y), X) = true.
% 0.21/0.54 Axiom 7 (less_equal_and_equal): fresh(X, X, Y, Z) = Z.
% 0.21/0.54 Axiom 8 (quotient_less_equal2): fresh4(X, X, Y, Z) = true.
% 0.21/0.54 Axiom 9 (quotient_less_equal1): fresh3(X, X, Y, Z) = zero.
% 0.21/0.54 Axiom 10 (less_equal_and_equal): fresh2(X, X, Y, Z) = Y.
% 0.21/0.54 Axiom 11 (quotient_less_equal2): fresh4(divide(X, Y), zero, X, Y) = less_equal(X, Y).
% 0.21/0.54 Axiom 12 (quotient_less_equal1): fresh3(less_equal(X, Y), true, X, Y) = divide(X, Y).
% 0.21/0.54 Axiom 13 (less_equal_and_equal): fresh2(less_equal(X, Y), true, Y, X) = fresh(less_equal(Y, X), true, Y, X).
% 0.21/0.54 Axiom 14 (quotient_property): less_equal(divide(divide(X, Y), divide(Z, Y)), divide(divide(X, Z), Y)) = true.
% 0.21/0.54
% 0.21/0.54 Lemma 15: divide(zero, X) = zero.
% 0.21/0.54 Proof:
% 0.21/0.54 divide(zero, X)
% 0.21/0.54 = { by axiom 12 (quotient_less_equal1) R->L }
% 0.21/0.54 fresh3(less_equal(zero, X), true, zero, X)
% 0.21/0.54 = { by axiom 5 (zero_is_smallest) }
% 0.21/0.54 fresh3(true, true, zero, X)
% 0.21/0.54 = { by axiom 9 (quotient_less_equal1) }
% 0.21/0.54 zero
% 0.21/0.54
% 0.21/0.54 Lemma 16: divide(divide(X, Y), X) = zero.
% 0.21/0.54 Proof:
% 0.21/0.54 divide(divide(X, Y), X)
% 0.21/0.54 = { by axiom 12 (quotient_less_equal1) R->L }
% 0.21/0.54 fresh3(less_equal(divide(X, Y), X), true, divide(X, Y), X)
% 0.21/0.54 = { by axiom 6 (quotient_smaller_than_numerator) }
% 0.21/0.54 fresh3(true, true, divide(X, Y), X)
% 0.21/0.54 = { by axiom 9 (quotient_less_equal1) }
% 0.21/0.54 zero
% 0.21/0.54
% 0.21/0.54 Lemma 17: divide(divide(divide(X, Y), Z), divide(X, Z)) = zero.
% 0.21/0.54 Proof:
% 0.21/0.54 divide(divide(divide(X, Y), Z), divide(X, Z))
% 0.21/0.54 = { by axiom 7 (less_equal_and_equal) R->L }
% 0.21/0.54 fresh(true, true, zero, divide(divide(divide(X, Y), Z), divide(X, Z)))
% 0.21/0.54 = { by axiom 5 (zero_is_smallest) R->L }
% 0.21/0.54 fresh(less_equal(zero, divide(divide(divide(X, Y), Z), divide(X, Z))), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z)))
% 0.21/0.54 = { by axiom 13 (less_equal_and_equal) R->L }
% 0.21/0.54 fresh2(less_equal(divide(divide(divide(X, Y), Z), divide(X, Z)), zero), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z)))
% 0.21/0.54 = { by lemma 15 R->L }
% 0.21/0.54 fresh2(less_equal(divide(divide(divide(X, Y), Z), divide(X, Z)), divide(zero, Z)), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z)))
% 0.21/0.54 = { by lemma 16 R->L }
% 0.21/0.54 fresh2(less_equal(divide(divide(divide(X, Y), Z), divide(X, Z)), divide(divide(divide(X, Y), X), Z)), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z)))
% 0.21/0.54 = { by axiom 14 (quotient_property) }
% 0.21/0.54 fresh2(true, true, zero, divide(divide(divide(X, Y), Z), divide(X, Z)))
% 0.21/0.54 = { by axiom 10 (less_equal_and_equal) }
% 0.21/0.54 zero
% 0.21/0.54
% 0.21/0.54 Lemma 18: divide(X, X) = zero.
% 0.21/0.54 Proof:
% 0.21/0.54 divide(X, X)
% 0.21/0.54 = { by axiom 7 (less_equal_and_equal) R->L }
% 0.21/0.54 fresh(true, true, zero, divide(X, X))
% 0.21/0.54 = { by axiom 5 (zero_is_smallest) R->L }
% 0.21/0.54 fresh(less_equal(zero, divide(X, X)), true, zero, divide(X, X))
% 0.21/0.54 = { by axiom 13 (less_equal_and_equal) R->L }
% 0.21/0.54 fresh2(less_equal(divide(X, X), zero), true, zero, divide(X, X))
% 0.21/0.54 = { by axiom 11 (quotient_less_equal2) R->L }
% 0.21/0.54 fresh2(fresh4(divide(divide(X, X), zero), zero, divide(X, X), zero), true, zero, divide(X, X))
% 0.21/0.54 = { by lemma 15 R->L }
% 0.21/0.54 fresh2(fresh4(divide(divide(X, X), divide(zero, X)), zero, divide(X, X), zero), true, zero, divide(X, X))
% 0.21/0.54 = { by axiom 7 (less_equal_and_equal) R->L }
% 0.21/0.54 fresh2(fresh4(fresh(true, true, zero, divide(divide(X, X), divide(zero, X))), zero, divide(X, X), zero), true, zero, divide(X, X))
% 0.21/0.54 = { by axiom 5 (zero_is_smallest) R->L }
% 0.21/0.54 fresh2(fresh4(fresh(less_equal(zero, divide(divide(X, X), divide(zero, X))), true, zero, divide(divide(X, X), divide(zero, X))), zero, divide(X, X), zero), true, zero, divide(X, X))
% 0.21/0.54 = { by axiom 13 (less_equal_and_equal) R->L }
% 0.21/0.54 fresh2(fresh4(fresh2(less_equal(divide(divide(X, X), divide(zero, X)), zero), true, zero, divide(divide(X, X), divide(zero, X))), zero, divide(X, X), zero), true, zero, divide(X, X))
% 0.21/0.54 = { by lemma 16 R->L }
% 0.21/0.54 fresh2(fresh4(fresh2(less_equal(divide(divide(X, X), divide(zero, X)), divide(divide(X, zero), X)), true, zero, divide(divide(X, X), divide(zero, X))), zero, divide(X, X), zero), true, zero, divide(X, X))
% 0.21/0.54 = { by axiom 14 (quotient_property) }
% 0.21/0.54 fresh2(fresh4(fresh2(true, true, zero, divide(divide(X, X), divide(zero, X))), zero, divide(X, X), zero), true, zero, divide(X, X))
% 0.21/0.54 = { by axiom 10 (less_equal_and_equal) }
% 0.21/0.54 fresh2(fresh4(zero, zero, divide(X, X), zero), true, zero, divide(X, X))
% 0.21/0.54 = { by axiom 8 (quotient_less_equal2) }
% 0.21/0.54 fresh2(true, true, zero, divide(X, X))
% 0.21/0.55 = { by axiom 10 (less_equal_and_equal) }
% 0.21/0.55 zero
% 0.21/0.55
% 0.21/0.55 Lemma 19: divide(d, divide(b, c)) = zero.
% 0.21/0.55 Proof:
% 0.21/0.55 divide(d, divide(b, c))
% 0.21/0.55 = { by axiom 7 (less_equal_and_equal) R->L }
% 0.21/0.55 fresh(true, true, zero, divide(d, divide(b, c)))
% 0.21/0.55 = { by axiom 5 (zero_is_smallest) R->L }
% 0.21/0.55 fresh(less_equal(zero, divide(d, divide(b, c))), true, zero, divide(d, divide(b, c)))
% 0.21/0.55 = { by axiom 13 (less_equal_and_equal) R->L }
% 0.21/0.55 fresh2(less_equal(divide(d, divide(b, c)), zero), true, zero, divide(d, divide(b, c)))
% 0.21/0.55 = { by lemma 18 R->L }
% 0.21/0.55 fresh2(less_equal(divide(d, divide(b, c)), divide(c, c)), true, zero, divide(d, divide(b, c)))
% 0.21/0.55 = { by axiom 3 (id_divide_c_is_d) R->L }
% 0.21/0.55 fresh2(less_equal(divide(divide(identity, c), divide(b, c)), divide(c, c)), true, zero, divide(d, divide(b, c)))
% 0.21/0.55 = { by axiom 2 (id_divide_b_is_c) R->L }
% 0.21/0.55 fresh2(less_equal(divide(divide(identity, c), divide(b, c)), divide(divide(identity, b), c)), true, zero, divide(d, divide(b, c)))
% 0.21/0.55 = { by axiom 14 (quotient_property) }
% 0.21/0.55 fresh2(true, true, zero, divide(d, divide(b, c)))
% 0.21/0.55 = { by axiom 10 (less_equal_and_equal) }
% 0.21/0.55 zero
% 0.21/0.55
% 0.21/0.55 Lemma 20: divide(b, c) = d.
% 0.21/0.55 Proof:
% 0.21/0.55 divide(b, c)
% 0.21/0.55 = { by axiom 10 (less_equal_and_equal) R->L }
% 0.21/0.55 fresh2(true, true, divide(b, c), d)
% 0.21/0.55 = { by axiom 8 (quotient_less_equal2) R->L }
% 0.21/0.55 fresh2(fresh4(zero, zero, d, divide(b, c)), true, divide(b, c), d)
% 0.21/0.55 = { by lemma 19 R->L }
% 0.21/0.55 fresh2(fresh4(divide(d, divide(b, c)), zero, d, divide(b, c)), true, divide(b, c), d)
% 0.21/0.55 = { by axiom 11 (quotient_less_equal2) }
% 0.21/0.55 fresh2(less_equal(d, divide(b, c)), true, divide(b, c), d)
% 0.21/0.55 = { by axiom 13 (less_equal_and_equal) }
% 0.21/0.55 fresh(less_equal(divide(b, c), d), true, divide(b, c), d)
% 0.21/0.55 = { by axiom 11 (quotient_less_equal2) R->L }
% 0.21/0.55 fresh(fresh4(divide(divide(b, c), d), zero, divide(b, c), d), true, divide(b, c), d)
% 0.21/0.55 = { by axiom 3 (id_divide_c_is_d) R->L }
% 0.21/0.55 fresh(fresh4(divide(divide(b, c), divide(identity, c)), zero, divide(b, c), d), true, divide(b, c), d)
% 0.21/0.55 = { by axiom 1 (id_divide_a_is_b) R->L }
% 0.21/0.55 fresh(fresh4(divide(divide(divide(identity, a), c), divide(identity, c)), zero, divide(b, c), d), true, divide(b, c), d)
% 0.21/0.55 = { by lemma 17 }
% 0.21/0.55 fresh(fresh4(zero, zero, divide(b, c), d), true, divide(b, c), d)
% 0.21/0.55 = { by axiom 8 (quotient_less_equal2) }
% 0.21/0.55 fresh(true, true, divide(b, c), d)
% 0.21/0.55 = { by axiom 7 (less_equal_and_equal) }
% 0.21/0.55 d
% 0.21/0.55
% 0.21/0.55 Lemma 21: divide(divide(X, Y), divide(identity, Y)) = zero.
% 0.21/0.55 Proof:
% 0.21/0.55 divide(divide(X, Y), divide(identity, Y))
% 0.21/0.55 = { by axiom 7 (less_equal_and_equal) R->L }
% 0.21/0.55 fresh(true, true, zero, divide(divide(X, Y), divide(identity, Y)))
% 0.21/0.55 = { by axiom 5 (zero_is_smallest) R->L }
% 0.21/0.55 fresh(less_equal(zero, divide(divide(X, Y), divide(identity, Y))), true, zero, divide(divide(X, Y), divide(identity, Y)))
% 0.21/0.55 = { by axiom 13 (less_equal_and_equal) R->L }
% 0.21/0.55 fresh2(less_equal(divide(divide(X, Y), divide(identity, Y)), zero), true, zero, divide(divide(X, Y), divide(identity, Y)))
% 0.21/0.55 = { by lemma 15 R->L }
% 0.21/0.55 fresh2(less_equal(divide(divide(X, Y), divide(identity, Y)), divide(zero, Y)), true, zero, divide(divide(X, Y), divide(identity, Y)))
% 0.21/0.55 = { by axiom 9 (quotient_less_equal1) R->L }
% 0.21/0.55 fresh2(less_equal(divide(divide(X, Y), divide(identity, Y)), divide(fresh3(true, true, X, identity), Y)), true, zero, divide(divide(X, Y), divide(identity, Y)))
% 0.21/0.55 = { by axiom 4 (identity_is_largest) R->L }
% 0.21/0.55 fresh2(less_equal(divide(divide(X, Y), divide(identity, Y)), divide(fresh3(less_equal(X, identity), true, X, identity), Y)), true, zero, divide(divide(X, Y), divide(identity, Y)))
% 0.21/0.55 = { by axiom 12 (quotient_less_equal1) }
% 0.21/0.55 fresh2(less_equal(divide(divide(X, Y), divide(identity, Y)), divide(divide(X, identity), Y)), true, zero, divide(divide(X, Y), divide(identity, Y)))
% 0.21/0.55 = { by axiom 14 (quotient_property) }
% 0.21/0.55 fresh2(true, true, zero, divide(divide(X, Y), divide(identity, Y)))
% 0.21/0.55 = { by axiom 10 (less_equal_and_equal) }
% 0.21/0.55 zero
% 0.21/0.55
% 0.21/0.55 Lemma 22: divide(d, a) = b.
% 0.21/0.55 Proof:
% 0.21/0.55 divide(d, a)
% 0.21/0.55 = { by axiom 10 (less_equal_and_equal) R->L }
% 0.21/0.55 fresh2(true, true, divide(d, a), b)
% 0.21/0.55 = { by axiom 8 (quotient_less_equal2) R->L }
% 0.21/0.55 fresh2(fresh4(zero, zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by axiom 10 (less_equal_and_equal) R->L }
% 0.21/0.55 fresh2(fresh4(fresh2(true, true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by axiom 14 (quotient_property) R->L }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(divide(identity, d), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by axiom 7 (less_equal_and_equal) R->L }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(fresh(true, true, divide(c, d), divide(identity, d)), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by axiom 8 (quotient_less_equal2) R->L }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(fresh(fresh4(zero, zero, divide(c, d), divide(identity, d)), true, divide(c, d), divide(identity, d)), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by lemma 17 R->L }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(fresh(fresh4(divide(divide(divide(identity, b), d), divide(identity, d)), zero, divide(c, d), divide(identity, d)), true, divide(c, d), divide(identity, d)), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by axiom 2 (id_divide_b_is_c) }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(fresh(fresh4(divide(divide(c, d), divide(identity, d)), zero, divide(c, d), divide(identity, d)), true, divide(c, d), divide(identity, d)), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by axiom 11 (quotient_less_equal2) }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(fresh(less_equal(divide(c, d), divide(identity, d)), true, divide(c, d), divide(identity, d)), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by axiom 13 (less_equal_and_equal) R->L }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(fresh2(less_equal(divide(identity, d), divide(c, d)), true, divide(c, d), divide(identity, d)), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by axiom 11 (quotient_less_equal2) R->L }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(fresh2(fresh4(divide(divide(identity, d), divide(c, d)), zero, divide(identity, d), divide(c, d)), true, divide(c, d), divide(identity, d)), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by axiom 7 (less_equal_and_equal) R->L }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(fresh2(fresh4(fresh(true, true, zero, divide(divide(identity, d), divide(c, d))), zero, divide(identity, d), divide(c, d)), true, divide(c, d), divide(identity, d)), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by axiom 5 (zero_is_smallest) R->L }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(fresh2(fresh4(fresh(less_equal(zero, divide(divide(identity, d), divide(c, d))), true, zero, divide(divide(identity, d), divide(c, d))), zero, divide(identity, d), divide(c, d)), true, divide(c, d), divide(identity, d)), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by axiom 13 (less_equal_and_equal) R->L }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(fresh2(fresh4(fresh2(less_equal(divide(divide(identity, d), divide(c, d)), zero), true, zero, divide(divide(identity, d), divide(c, d))), zero, divide(identity, d), divide(c, d)), true, divide(c, d), divide(identity, d)), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by lemma 20 R->L }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(fresh2(fresh4(fresh2(less_equal(divide(divide(identity, divide(b, c)), divide(c, d)), zero), true, zero, divide(divide(identity, d), divide(c, d))), zero, divide(identity, d), divide(c, d)), true, divide(c, d), divide(identity, d)), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by lemma 20 R->L }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(fresh2(fresh4(fresh2(less_equal(divide(divide(identity, divide(b, c)), divide(c, divide(b, c))), zero), true, zero, divide(divide(identity, d), divide(c, d))), zero, divide(identity, d), divide(c, d)), true, divide(c, d), divide(identity, d)), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by lemma 19 R->L }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(fresh2(fresh4(fresh2(less_equal(divide(divide(identity, divide(b, c)), divide(c, divide(b, c))), divide(d, divide(b, c))), true, zero, divide(divide(identity, d), divide(c, d))), zero, divide(identity, d), divide(c, d)), true, divide(c, d), divide(identity, d)), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by axiom 3 (id_divide_c_is_d) R->L }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(fresh2(fresh4(fresh2(less_equal(divide(divide(identity, divide(b, c)), divide(c, divide(b, c))), divide(divide(identity, c), divide(b, c))), true, zero, divide(divide(identity, d), divide(c, d))), zero, divide(identity, d), divide(c, d)), true, divide(c, d), divide(identity, d)), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by axiom 14 (quotient_property) }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(fresh2(fresh4(fresh2(true, true, zero, divide(divide(identity, d), divide(c, d))), zero, divide(identity, d), divide(c, d)), true, divide(c, d), divide(identity, d)), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by axiom 10 (less_equal_and_equal) }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(fresh2(fresh4(zero, zero, divide(identity, d), divide(c, d)), true, divide(c, d), divide(identity, d)), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by axiom 8 (quotient_less_equal2) }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(fresh2(true, true, divide(c, d), divide(identity, d)), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by axiom 10 (less_equal_and_equal) }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(divide(c, d), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by axiom 7 (less_equal_and_equal) R->L }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(divide(fresh(true, true, divide(a, b), c), d), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by axiom 8 (quotient_less_equal2) R->L }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(divide(fresh(fresh4(zero, zero, divide(a, b), c), true, divide(a, b), c), d), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by lemma 21 R->L }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(divide(fresh(fresh4(divide(divide(a, b), divide(identity, b)), zero, divide(a, b), c), true, divide(a, b), c), d), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by axiom 2 (id_divide_b_is_c) }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(divide(fresh(fresh4(divide(divide(a, b), c), zero, divide(a, b), c), true, divide(a, b), c), d), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by axiom 11 (quotient_less_equal2) }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(divide(fresh(less_equal(divide(a, b), c), true, divide(a, b), c), d), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by axiom 13 (less_equal_and_equal) R->L }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(divide(fresh2(less_equal(c, divide(a, b)), true, divide(a, b), c), d), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by axiom 11 (quotient_less_equal2) R->L }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(divide(fresh2(fresh4(divide(c, divide(a, b)), zero, c, divide(a, b)), true, divide(a, b), c), d), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by axiom 7 (less_equal_and_equal) R->L }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(divide(fresh2(fresh4(fresh(true, true, zero, divide(c, divide(a, b))), zero, c, divide(a, b)), true, divide(a, b), c), d), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by axiom 5 (zero_is_smallest) R->L }
% 0.21/0.55 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(divide(fresh2(fresh4(fresh(less_equal(zero, divide(c, divide(a, b))), true, zero, divide(c, divide(a, b))), zero, c, divide(a, b)), true, divide(a, b), c), d), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.55 = { by axiom 13 (less_equal_and_equal) R->L }
% 0.21/0.56 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(divide(fresh2(fresh4(fresh2(less_equal(divide(c, divide(a, b)), zero), true, zero, divide(c, divide(a, b))), zero, c, divide(a, b)), true, divide(a, b), c), d), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.56 = { by lemma 18 R->L }
% 0.21/0.56 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(divide(fresh2(fresh4(fresh2(less_equal(divide(c, divide(a, b)), divide(b, b)), true, zero, divide(c, divide(a, b))), zero, c, divide(a, b)), true, divide(a, b), c), d), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.56 = { by axiom 2 (id_divide_b_is_c) R->L }
% 0.21/0.56 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(divide(fresh2(fresh4(fresh2(less_equal(divide(divide(identity, b), divide(a, b)), divide(b, b)), true, zero, divide(c, divide(a, b))), zero, c, divide(a, b)), true, divide(a, b), c), d), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.56 = { by axiom 1 (id_divide_a_is_b) R->L }
% 0.21/0.56 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(divide(fresh2(fresh4(fresh2(less_equal(divide(divide(identity, b), divide(a, b)), divide(divide(identity, a), b)), true, zero, divide(c, divide(a, b))), zero, c, divide(a, b)), true, divide(a, b), c), d), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.56 = { by axiom 14 (quotient_property) }
% 0.21/0.56 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(divide(fresh2(fresh4(fresh2(true, true, zero, divide(c, divide(a, b))), zero, c, divide(a, b)), true, divide(a, b), c), d), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.56 = { by axiom 10 (less_equal_and_equal) }
% 0.21/0.56 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(divide(fresh2(fresh4(zero, zero, c, divide(a, b)), true, divide(a, b), c), d), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.56 = { by axiom 8 (quotient_less_equal2) }
% 0.21/0.56 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(divide(fresh2(true, true, divide(a, b), c), d), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.56 = { by axiom 10 (less_equal_and_equal) }
% 0.21/0.56 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), divide(divide(divide(a, b), d), a)), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.56 = { by axiom 7 (less_equal_and_equal) R->L }
% 0.21/0.56 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), fresh(true, true, zero, divide(divide(divide(a, b), d), a))), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.56 = { by axiom 5 (zero_is_smallest) R->L }
% 0.21/0.56 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), fresh(less_equal(zero, divide(divide(divide(a, b), d), a)), true, zero, divide(divide(divide(a, b), d), a))), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.56 = { by axiom 13 (less_equal_and_equal) R->L }
% 0.21/0.56 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), fresh2(less_equal(divide(divide(divide(a, b), d), a), zero), true, zero, divide(divide(divide(a, b), d), a))), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.56 = { by lemma 16 R->L }
% 0.21/0.56 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), fresh2(less_equal(divide(divide(divide(a, b), d), a), divide(divide(a, b), a)), true, zero, divide(divide(divide(a, b), d), a))), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.56 = { by axiom 11 (quotient_less_equal2) R->L }
% 0.21/0.56 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), fresh2(fresh4(divide(divide(divide(divide(a, b), d), a), divide(divide(a, b), a)), zero, divide(divide(divide(a, b), d), a), divide(divide(a, b), a)), true, zero, divide(divide(divide(a, b), d), a))), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.56 = { by lemma 17 }
% 0.21/0.56 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), fresh2(fresh4(zero, zero, divide(divide(divide(a, b), d), a), divide(divide(a, b), a)), true, zero, divide(divide(divide(a, b), d), a))), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.56 = { by axiom 8 (quotient_less_equal2) }
% 0.21/0.56 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), fresh2(true, true, zero, divide(divide(divide(a, b), d), a))), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.56 = { by axiom 10 (less_equal_and_equal) }
% 0.21/0.56 fresh2(fresh4(fresh2(less_equal(divide(divide(identity, a), divide(d, a)), zero), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.56 = { by axiom 1 (id_divide_a_is_b) }
% 0.21/0.56 fresh2(fresh4(fresh2(less_equal(divide(b, divide(d, a)), zero), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.56 = { by axiom 13 (less_equal_and_equal) }
% 0.21/0.56 fresh2(fresh4(fresh(less_equal(zero, divide(b, divide(d, a))), true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.56 = { by axiom 5 (zero_is_smallest) }
% 0.21/0.56 fresh2(fresh4(fresh(true, true, zero, divide(b, divide(d, a))), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.56 = { by axiom 7 (less_equal_and_equal) }
% 0.21/0.56 fresh2(fresh4(divide(b, divide(d, a)), zero, b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.56 = { by axiom 11 (quotient_less_equal2) }
% 0.21/0.56 fresh2(less_equal(b, divide(d, a)), true, divide(d, a), b)
% 0.21/0.56 = { by axiom 13 (less_equal_and_equal) }
% 0.21/0.56 fresh(less_equal(divide(d, a), b), true, divide(d, a), b)
% 0.21/0.56 = { by axiom 11 (quotient_less_equal2) R->L }
% 0.21/0.56 fresh(fresh4(divide(divide(d, a), b), zero, divide(d, a), b), true, divide(d, a), b)
% 0.21/0.56 = { by axiom 1 (id_divide_a_is_b) R->L }
% 0.21/0.56 fresh(fresh4(divide(divide(d, a), divide(identity, a)), zero, divide(d, a), b), true, divide(d, a), b)
% 0.21/0.56 = { by lemma 21 }
% 0.21/0.56 fresh(fresh4(zero, zero, divide(d, a), b), true, divide(d, a), b)
% 0.21/0.56 = { by axiom 8 (quotient_less_equal2) }
% 0.21/0.56 fresh(true, true, divide(d, a), b)
% 0.21/0.56 = { by axiom 7 (less_equal_and_equal) }
% 0.21/0.56 b
% 0.21/0.56
% 0.21/0.56 Lemma 23: d = b.
% 0.21/0.56 Proof:
% 0.21/0.56 d
% 0.21/0.56 = { by axiom 10 (less_equal_and_equal) R->L }
% 0.21/0.56 fresh2(true, true, d, divide(d, a))
% 0.21/0.56 = { by axiom 6 (quotient_smaller_than_numerator) R->L }
% 0.21/0.56 fresh2(less_equal(divide(d, a), d), true, d, divide(d, a))
% 0.21/0.56 = { by axiom 13 (less_equal_and_equal) }
% 0.21/0.56 fresh(less_equal(d, divide(d, a)), true, d, divide(d, a))
% 0.21/0.56 = { by lemma 22 }
% 0.21/0.56 fresh(less_equal(d, b), true, d, divide(d, a))
% 0.21/0.56 = { by lemma 20 R->L }
% 0.21/0.56 fresh(less_equal(divide(b, c), b), true, d, divide(d, a))
% 0.21/0.56 = { by axiom 6 (quotient_smaller_than_numerator) }
% 0.21/0.56 fresh(true, true, d, divide(d, a))
% 0.21/0.56 = { by axiom 7 (less_equal_and_equal) }
% 0.21/0.56 divide(d, a)
% 0.21/0.56 = { by lemma 22 }
% 0.21/0.56 b
% 0.21/0.56
% 0.21/0.56 Goal 1 (prove_b_equals_d): b = d.
% 0.21/0.56 Proof:
% 0.21/0.56 b
% 0.21/0.56 = { by lemma 23 R->L }
% 0.21/0.56 d
% 0.21/0.56
% 0.21/0.56 Goal 2 (part_of_theorem): divide(identity, a) = divide(identity, divide(identity, divide(identity, a))).
% 0.21/0.56 Proof:
% 0.21/0.56 divide(identity, a)
% 0.21/0.56 = { by axiom 1 (id_divide_a_is_b) }
% 0.21/0.56 b
% 0.21/0.56 = { by lemma 23 R->L }
% 0.21/0.56 d
% 0.21/0.56 = { by axiom 3 (id_divide_c_is_d) R->L }
% 0.21/0.56 divide(identity, c)
% 0.21/0.56 = { by axiom 2 (id_divide_b_is_c) R->L }
% 0.21/0.56 divide(identity, divide(identity, b))
% 0.21/0.56 = { by axiom 1 (id_divide_a_is_b) R->L }
% 0.21/0.56 divide(identity, divide(identity, divide(identity, a)))
% 0.21/0.56 % SZS output end Proof
% 0.21/0.56
% 0.21/0.56 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------