TSTP Solution File: HEN007-3 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : HEN007-3 : 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 : n004.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 01:56:57 EDT 2023
% Result : Unsatisfiable 0.19s 0.41s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : HEN007-3 : 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 : n004.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 13:30:53 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.19/0.41 Command-line arguments: --ground-connectedness --complete-subsets
% 0.19/0.41
% 0.19/0.41 % SZS status Unsatisfiable
% 0.19/0.41
% 0.19/0.43 % SZS output start Proof
% 0.19/0.43 Take the following subset of the input axioms:
% 0.19/0.43 fof(a_LE_b, hypothesis, less_equal(a, b)).
% 0.19/0.43 fof(less_equal_and_equal, axiom, ![X, Y]: (~less_equal(X, Y) | (~less_equal(Y, X) | X=Y))).
% 0.19/0.43 fof(prove_c_divide_b_LE_c_divide_a, negated_conjecture, ~less_equal(divide(c, b), divide(c, a))).
% 0.19/0.43 fof(quotient_less_equal1, axiom, ![X2, Y2]: (~less_equal(X2, Y2) | divide(X2, Y2)=zero)).
% 0.19/0.43 fof(quotient_less_equal2, axiom, ![X2, Y2]: (divide(X2, Y2)!=zero | less_equal(X2, Y2))).
% 0.19/0.43 fof(quotient_property, axiom, ![Z, X2, Y2]: less_equal(divide(divide(X2, Z), divide(Y2, Z)), divide(divide(X2, Y2), Z))).
% 0.19/0.43 fof(quotient_smaller_than_numerator, axiom, ![X2, Y2]: less_equal(divide(X2, Y2), X2)).
% 0.19/0.43 fof(zero_is_smallest, axiom, ![X2]: less_equal(zero, X2)).
% 0.19/0.43
% 0.19/0.43 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.43 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.43 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.43 fresh(y, y, x1...xn) = u
% 0.19/0.43 C => fresh(s, t, x1...xn) = v
% 0.19/0.44 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.44 variables of u and v.
% 0.19/0.44 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.44 input problem has no model of domain size 1).
% 0.19/0.44
% 0.19/0.44 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.44
% 0.19/0.44 Axiom 1 (zero_is_smallest): less_equal(zero, X) = true.
% 0.19/0.44 Axiom 2 (a_LE_b): less_equal(a, b) = true.
% 0.19/0.44 Axiom 3 (less_equal_and_equal): fresh(X, X, Y, Z) = Z.
% 0.19/0.44 Axiom 4 (quotient_less_equal2): fresh4(X, X, Y, Z) = true.
% 0.19/0.44 Axiom 5 (quotient_less_equal1): fresh3(X, X, Y, Z) = zero.
% 0.19/0.44 Axiom 6 (less_equal_and_equal): fresh2(X, X, Y, Z) = Y.
% 0.19/0.44 Axiom 7 (quotient_smaller_than_numerator): less_equal(divide(X, Y), X) = true.
% 0.19/0.44 Axiom 8 (quotient_less_equal2): fresh4(divide(X, Y), zero, X, Y) = less_equal(X, Y).
% 0.19/0.44 Axiom 9 (quotient_less_equal1): fresh3(less_equal(X, Y), true, X, Y) = divide(X, Y).
% 0.19/0.44 Axiom 10 (less_equal_and_equal): fresh2(less_equal(X, Y), true, Y, X) = fresh(less_equal(Y, X), true, Y, X).
% 0.19/0.44 Axiom 11 (quotient_property): less_equal(divide(divide(X, Y), divide(Z, Y)), divide(divide(X, Z), Y)) = true.
% 0.19/0.44
% 0.19/0.44 Lemma 12: divide(zero, X) = zero.
% 0.19/0.44 Proof:
% 0.19/0.44 divide(zero, X)
% 0.19/0.44 = { by axiom 9 (quotient_less_equal1) R->L }
% 0.19/0.44 fresh3(less_equal(zero, X), true, zero, X)
% 0.19/0.44 = { by axiom 1 (zero_is_smallest) }
% 0.19/0.44 fresh3(true, true, zero, X)
% 0.19/0.44 = { by axiom 5 (quotient_less_equal1) }
% 0.19/0.44 zero
% 0.19/0.44
% 0.19/0.44 Goal 1 (prove_c_divide_b_LE_c_divide_a): less_equal(divide(c, b), divide(c, a)) = true.
% 0.19/0.44 Proof:
% 0.19/0.44 less_equal(divide(c, b), divide(c, a))
% 0.19/0.44 = { by axiom 3 (less_equal_and_equal) R->L }
% 0.19/0.44 less_equal(fresh(true, true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.44 = { by axiom 4 (quotient_less_equal2) R->L }
% 0.19/0.44 less_equal(fresh(fresh4(zero, zero, divide(divide(c, a), b), divide(c, b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.44 = { by axiom 6 (less_equal_and_equal) R->L }
% 0.19/0.44 less_equal(fresh(fresh4(fresh2(true, true, zero, divide(divide(divide(c, a), b), divide(c, b))), zero, divide(divide(c, a), b), divide(c, b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.44 = { by axiom 11 (quotient_property) R->L }
% 0.19/0.44 less_equal(fresh(fresh4(fresh2(less_equal(divide(divide(divide(c, a), b), divide(c, b)), divide(divide(divide(c, a), c), b)), true, zero, divide(divide(divide(c, a), b), divide(c, b))), zero, divide(divide(c, a), b), divide(c, b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.44 = { by axiom 9 (quotient_less_equal1) R->L }
% 0.19/0.44 less_equal(fresh(fresh4(fresh2(less_equal(divide(divide(divide(c, a), b), divide(c, b)), divide(fresh3(less_equal(divide(c, a), c), true, divide(c, a), c), b)), true, zero, divide(divide(divide(c, a), b), divide(c, b))), zero, divide(divide(c, a), b), divide(c, b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.44 = { by axiom 7 (quotient_smaller_than_numerator) }
% 0.19/0.44 less_equal(fresh(fresh4(fresh2(less_equal(divide(divide(divide(c, a), b), divide(c, b)), divide(fresh3(true, true, divide(c, a), c), b)), true, zero, divide(divide(divide(c, a), b), divide(c, b))), zero, divide(divide(c, a), b), divide(c, b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.44 = { by axiom 5 (quotient_less_equal1) }
% 0.19/0.44 less_equal(fresh(fresh4(fresh2(less_equal(divide(divide(divide(c, a), b), divide(c, b)), divide(zero, b)), true, zero, divide(divide(divide(c, a), b), divide(c, b))), zero, divide(divide(c, a), b), divide(c, b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.44 = { by lemma 12 }
% 0.19/0.44 less_equal(fresh(fresh4(fresh2(less_equal(divide(divide(divide(c, a), b), divide(c, b)), zero), true, zero, divide(divide(divide(c, a), b), divide(c, b))), zero, divide(divide(c, a), b), divide(c, b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.44 = { by axiom 10 (less_equal_and_equal) }
% 0.19/0.44 less_equal(fresh(fresh4(fresh(less_equal(zero, divide(divide(divide(c, a), b), divide(c, b))), true, zero, divide(divide(divide(c, a), b), divide(c, b))), zero, divide(divide(c, a), b), divide(c, b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.44 = { by axiom 1 (zero_is_smallest) }
% 0.19/0.44 less_equal(fresh(fresh4(fresh(true, true, zero, divide(divide(divide(c, a), b), divide(c, b))), zero, divide(divide(c, a), b), divide(c, b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.44 = { by axiom 3 (less_equal_and_equal) }
% 0.19/0.44 less_equal(fresh(fresh4(divide(divide(divide(c, a), b), divide(c, b)), zero, divide(divide(c, a), b), divide(c, b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.44 = { by axiom 8 (quotient_less_equal2) }
% 0.19/0.44 less_equal(fresh(less_equal(divide(divide(c, a), b), divide(c, b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.44 = { by axiom 10 (less_equal_and_equal) R->L }
% 0.19/0.44 less_equal(fresh2(less_equal(divide(c, b), divide(divide(c, a), b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.44 = { by axiom 8 (quotient_less_equal2) R->L }
% 0.19/0.44 less_equal(fresh2(fresh4(divide(divide(c, b), divide(divide(c, a), b)), zero, divide(c, b), divide(divide(c, a), b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.44 = { by axiom 3 (less_equal_and_equal) R->L }
% 0.19/0.44 less_equal(fresh2(fresh4(fresh(true, true, zero, divide(divide(c, b), divide(divide(c, a), b))), zero, divide(c, b), divide(divide(c, a), b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.44 = { by axiom 1 (zero_is_smallest) R->L }
% 0.19/0.44 less_equal(fresh2(fresh4(fresh(less_equal(zero, divide(divide(c, b), divide(divide(c, a), b))), true, zero, divide(divide(c, b), divide(divide(c, a), b))), zero, divide(c, b), divide(divide(c, a), b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.44 = { by axiom 10 (less_equal_and_equal) R->L }
% 0.19/0.44 less_equal(fresh2(fresh4(fresh2(less_equal(divide(divide(c, b), divide(divide(c, a), b)), zero), true, zero, divide(divide(c, b), divide(divide(c, a), b))), zero, divide(c, b), divide(divide(c, a), b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.44 = { by axiom 8 (quotient_less_equal2) R->L }
% 0.19/0.44 less_equal(fresh2(fresh4(fresh2(fresh4(divide(divide(divide(c, b), divide(divide(c, a), b)), zero), zero, divide(divide(c, b), divide(divide(c, a), b)), zero), true, zero, divide(divide(c, b), divide(divide(c, a), b))), zero, divide(c, b), divide(divide(c, a), b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.44 = { by axiom 3 (less_equal_and_equal) R->L }
% 0.19/0.44 less_equal(fresh2(fresh4(fresh2(fresh4(fresh(true, true, zero, divide(divide(divide(c, b), divide(divide(c, a), b)), zero)), zero, divide(divide(c, b), divide(divide(c, a), b)), zero), true, zero, divide(divide(c, b), divide(divide(c, a), b))), zero, divide(c, b), divide(divide(c, a), b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.44 = { by axiom 1 (zero_is_smallest) R->L }
% 0.19/0.44 less_equal(fresh2(fresh4(fresh2(fresh4(fresh(less_equal(zero, divide(divide(divide(c, b), divide(divide(c, a), b)), zero)), true, zero, divide(divide(divide(c, b), divide(divide(c, a), b)), zero)), zero, divide(divide(c, b), divide(divide(c, a), b)), zero), true, zero, divide(divide(c, b), divide(divide(c, a), b))), zero, divide(c, b), divide(divide(c, a), b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.44 = { by axiom 10 (less_equal_and_equal) R->L }
% 0.19/0.44 less_equal(fresh2(fresh4(fresh2(fresh4(fresh2(less_equal(divide(divide(divide(c, b), divide(divide(c, a), b)), zero), zero), true, zero, divide(divide(divide(c, b), divide(divide(c, a), b)), zero)), zero, divide(divide(c, b), divide(divide(c, a), b)), zero), true, zero, divide(divide(c, b), divide(divide(c, a), b))), zero, divide(c, b), divide(divide(c, a), b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.44 = { by lemma 12 R->L }
% 0.19/0.44 less_equal(fresh2(fresh4(fresh2(fresh4(fresh2(less_equal(divide(divide(divide(c, b), divide(divide(c, a), b)), divide(zero, divide(divide(c, a), b))), zero), true, zero, divide(divide(divide(c, b), divide(divide(c, a), b)), zero)), zero, divide(divide(c, b), divide(divide(c, a), b)), zero), true, zero, divide(divide(c, b), divide(divide(c, a), b))), zero, divide(c, b), divide(divide(c, a), b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.45 = { by axiom 5 (quotient_less_equal1) R->L }
% 0.19/0.45 less_equal(fresh2(fresh4(fresh2(fresh4(fresh2(less_equal(divide(divide(divide(c, b), divide(divide(c, a), b)), divide(zero, divide(divide(c, a), b))), fresh3(true, true, divide(divide(c, b), zero), divide(divide(c, a), b))), true, zero, divide(divide(divide(c, b), divide(divide(c, a), b)), zero)), zero, divide(divide(c, b), divide(divide(c, a), b)), zero), true, zero, divide(divide(c, b), divide(divide(c, a), b))), zero, divide(c, b), divide(divide(c, a), b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.45 = { by axiom 11 (quotient_property) R->L }
% 0.19/0.45 less_equal(fresh2(fresh4(fresh2(fresh4(fresh2(less_equal(divide(divide(divide(c, b), divide(divide(c, a), b)), divide(zero, divide(divide(c, a), b))), fresh3(less_equal(divide(divide(c, b), divide(a, b)), divide(divide(c, a), b)), true, divide(divide(c, b), zero), divide(divide(c, a), b))), true, zero, divide(divide(divide(c, b), divide(divide(c, a), b)), zero)), zero, divide(divide(c, b), divide(divide(c, a), b)), zero), true, zero, divide(divide(c, b), divide(divide(c, a), b))), zero, divide(c, b), divide(divide(c, a), b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.45 = { by axiom 9 (quotient_less_equal1) R->L }
% 0.19/0.45 less_equal(fresh2(fresh4(fresh2(fresh4(fresh2(less_equal(divide(divide(divide(c, b), divide(divide(c, a), b)), divide(zero, divide(divide(c, a), b))), fresh3(less_equal(divide(divide(c, b), fresh3(less_equal(a, b), true, a, b)), divide(divide(c, a), b)), true, divide(divide(c, b), zero), divide(divide(c, a), b))), true, zero, divide(divide(divide(c, b), divide(divide(c, a), b)), zero)), zero, divide(divide(c, b), divide(divide(c, a), b)), zero), true, zero, divide(divide(c, b), divide(divide(c, a), b))), zero, divide(c, b), divide(divide(c, a), b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.45 = { by axiom 2 (a_LE_b) }
% 0.19/0.45 less_equal(fresh2(fresh4(fresh2(fresh4(fresh2(less_equal(divide(divide(divide(c, b), divide(divide(c, a), b)), divide(zero, divide(divide(c, a), b))), fresh3(less_equal(divide(divide(c, b), fresh3(true, true, a, b)), divide(divide(c, a), b)), true, divide(divide(c, b), zero), divide(divide(c, a), b))), true, zero, divide(divide(divide(c, b), divide(divide(c, a), b)), zero)), zero, divide(divide(c, b), divide(divide(c, a), b)), zero), true, zero, divide(divide(c, b), divide(divide(c, a), b))), zero, divide(c, b), divide(divide(c, a), b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.45 = { by axiom 5 (quotient_less_equal1) }
% 0.19/0.45 less_equal(fresh2(fresh4(fresh2(fresh4(fresh2(less_equal(divide(divide(divide(c, b), divide(divide(c, a), b)), divide(zero, divide(divide(c, a), b))), fresh3(less_equal(divide(divide(c, b), zero), divide(divide(c, a), b)), true, divide(divide(c, b), zero), divide(divide(c, a), b))), true, zero, divide(divide(divide(c, b), divide(divide(c, a), b)), zero)), zero, divide(divide(c, b), divide(divide(c, a), b)), zero), true, zero, divide(divide(c, b), divide(divide(c, a), b))), zero, divide(c, b), divide(divide(c, a), b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.45 = { by axiom 9 (quotient_less_equal1) }
% 0.19/0.45 less_equal(fresh2(fresh4(fresh2(fresh4(fresh2(less_equal(divide(divide(divide(c, b), divide(divide(c, a), b)), divide(zero, divide(divide(c, a), b))), divide(divide(divide(c, b), zero), divide(divide(c, a), b))), true, zero, divide(divide(divide(c, b), divide(divide(c, a), b)), zero)), zero, divide(divide(c, b), divide(divide(c, a), b)), zero), true, zero, divide(divide(c, b), divide(divide(c, a), b))), zero, divide(c, b), divide(divide(c, a), b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.45 = { by axiom 11 (quotient_property) }
% 0.19/0.45 less_equal(fresh2(fresh4(fresh2(fresh4(fresh2(true, true, zero, divide(divide(divide(c, b), divide(divide(c, a), b)), zero)), zero, divide(divide(c, b), divide(divide(c, a), b)), zero), true, zero, divide(divide(c, b), divide(divide(c, a), b))), zero, divide(c, b), divide(divide(c, a), b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.45 = { by axiom 6 (less_equal_and_equal) }
% 0.19/0.45 less_equal(fresh2(fresh4(fresh2(fresh4(zero, zero, divide(divide(c, b), divide(divide(c, a), b)), zero), true, zero, divide(divide(c, b), divide(divide(c, a), b))), zero, divide(c, b), divide(divide(c, a), b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.45 = { by axiom 4 (quotient_less_equal2) }
% 0.19/0.45 less_equal(fresh2(fresh4(fresh2(true, true, zero, divide(divide(c, b), divide(divide(c, a), b))), zero, divide(c, b), divide(divide(c, a), b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.45 = { by axiom 6 (less_equal_and_equal) }
% 0.19/0.45 less_equal(fresh2(fresh4(zero, zero, divide(c, b), divide(divide(c, a), b)), true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.45 = { by axiom 4 (quotient_less_equal2) }
% 0.19/0.45 less_equal(fresh2(true, true, divide(divide(c, a), b), divide(c, b)), divide(c, a))
% 0.19/0.45 = { by axiom 6 (less_equal_and_equal) }
% 0.19/0.45 less_equal(divide(divide(c, a), b), divide(c, a))
% 0.19/0.45 = { by axiom 7 (quotient_smaller_than_numerator) }
% 0.19/0.45 true
% 0.19/0.45 % SZS output end Proof
% 0.19/0.45
% 0.19/0.45 RESULT: Unsatisfiable (the axioms are contradictory).
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