TSTP Solution File: HEN005-4 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : HEN005-4 : 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 : n005.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 01:56:54 EDT 2023
% Result : Unsatisfiable 0.21s 0.40s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : HEN005-4 : TPTP v8.1.2. Released v1.0.0.
% 0.12/0.13 % 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 : n005.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 13:23:53 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.21/0.40 Command-line arguments: --ground-connectedness --complete-subsets
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% 0.21/0.40 % SZS status Unsatisfiable
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% 0.21/0.40 % SZS output start Proof
% 0.21/0.40 Take the following subset of the input axioms:
% 0.21/0.40 fof(a_LE_b, hypothesis, less_equal(a, b)).
% 0.21/0.40 fof(b_LE_c, hypothesis, less_equal(b, c)).
% 0.21/0.40 fof(prove_a_LE_c, negated_conjecture, ~less_equal(a, c)).
% 0.21/0.40 fof(quotient_less_equal1, axiom, ![X, Y]: (~less_equal(X, Y) | divide(X, Y)=zero)).
% 0.21/0.40 fof(quotient_less_equal2, axiom, ![X2, Y2]: (divide(X2, Y2)!=zero | less_equal(X2, Y2))).
% 0.21/0.40 fof(quotient_property, axiom, ![Z, X2, Y2]: less_equal(divide(divide(X2, Z), divide(Y2, Z)), divide(divide(X2, Y2), Z))).
% 0.21/0.40 fof(x_divide_zero_is_x, axiom, ![X2]: divide(X2, zero)=X2).
% 0.21/0.40 fof(zero_divide_anything_is_zero, axiom, ![X2]: divide(zero, X2)=zero).
% 0.21/0.40
% 0.21/0.40 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.40 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.40 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.40 fresh(y, y, x1...xn) = u
% 0.21/0.40 C => fresh(s, t, x1...xn) = v
% 0.21/0.40 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.40 variables of u and v.
% 0.21/0.40 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.40 input problem has no model of domain size 1).
% 0.21/0.40
% 0.21/0.40 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.40
% 0.21/0.40 Axiom 1 (b_LE_c): less_equal(b, c) = true.
% 0.21/0.40 Axiom 2 (a_LE_b): less_equal(a, b) = true.
% 0.21/0.40 Axiom 3 (x_divide_zero_is_x): divide(X, zero) = X.
% 0.21/0.40 Axiom 4 (zero_divide_anything_is_zero): divide(zero, X) = zero.
% 0.21/0.40 Axiom 5 (quotient_less_equal2): fresh4(X, X, Y, Z) = true.
% 0.21/0.40 Axiom 6 (quotient_less_equal1): fresh3(X, X, Y, Z) = zero.
% 0.21/0.40 Axiom 7 (quotient_less_equal2): fresh4(divide(X, Y), zero, X, Y) = less_equal(X, Y).
% 0.21/0.40 Axiom 8 (quotient_less_equal1): fresh3(less_equal(X, Y), true, X, Y) = divide(X, Y).
% 0.21/0.41 Axiom 9 (quotient_property): less_equal(divide(divide(X, Y), divide(Z, Y)), divide(divide(X, Z), Y)) = true.
% 0.21/0.41
% 0.21/0.41 Goal 1 (prove_a_LE_c): less_equal(a, c) = true.
% 0.21/0.41 Proof:
% 0.21/0.41 less_equal(a, c)
% 0.21/0.41 = { by axiom 7 (quotient_less_equal2) R->L }
% 0.21/0.41 fresh4(divide(a, c), zero, a, c)
% 0.21/0.41 = { by axiom 3 (x_divide_zero_is_x) R->L }
% 0.21/0.41 fresh4(divide(divide(a, c), zero), zero, a, c)
% 0.21/0.41 = { by axiom 8 (quotient_less_equal1) R->L }
% 0.21/0.41 fresh4(fresh3(less_equal(divide(a, c), zero), true, divide(a, c), zero), zero, a, c)
% 0.21/0.41 = { by axiom 4 (zero_divide_anything_is_zero) R->L }
% 0.21/0.41 fresh4(fresh3(less_equal(divide(a, c), divide(zero, c)), true, divide(a, c), zero), zero, a, c)
% 0.21/0.41 = { by axiom 6 (quotient_less_equal1) R->L }
% 0.21/0.41 fresh4(fresh3(less_equal(divide(a, c), divide(fresh3(true, true, a, b), c)), true, divide(a, c), zero), zero, a, c)
% 0.21/0.41 = { by axiom 2 (a_LE_b) R->L }
% 0.21/0.41 fresh4(fresh3(less_equal(divide(a, c), divide(fresh3(less_equal(a, b), true, a, b), c)), true, divide(a, c), zero), zero, a, c)
% 0.21/0.41 = { by axiom 8 (quotient_less_equal1) }
% 0.21/0.41 fresh4(fresh3(less_equal(divide(a, c), divide(divide(a, b), c)), true, divide(a, c), zero), zero, a, c)
% 0.21/0.41 = { by axiom 3 (x_divide_zero_is_x) R->L }
% 0.21/0.41 fresh4(fresh3(less_equal(divide(divide(a, c), zero), divide(divide(a, b), c)), true, divide(a, c), zero), zero, a, c)
% 0.21/0.41 = { by axiom 6 (quotient_less_equal1) R->L }
% 0.21/0.41 fresh4(fresh3(less_equal(divide(divide(a, c), fresh3(true, true, b, c)), divide(divide(a, b), c)), true, divide(a, c), zero), zero, a, c)
% 0.21/0.41 = { by axiom 1 (b_LE_c) R->L }
% 0.21/0.41 fresh4(fresh3(less_equal(divide(divide(a, c), fresh3(less_equal(b, c), true, b, c)), divide(divide(a, b), c)), true, divide(a, c), zero), zero, a, c)
% 0.21/0.41 = { by axiom 8 (quotient_less_equal1) }
% 0.21/0.41 fresh4(fresh3(less_equal(divide(divide(a, c), divide(b, c)), divide(divide(a, b), c)), true, divide(a, c), zero), zero, a, c)
% 0.21/0.41 = { by axiom 9 (quotient_property) }
% 0.21/0.41 fresh4(fresh3(true, true, divide(a, c), zero), zero, a, c)
% 0.21/0.41 = { by axiom 6 (quotient_less_equal1) }
% 0.21/0.41 fresh4(zero, zero, a, c)
% 0.21/0.41 = { by axiom 5 (quotient_less_equal2) }
% 0.21/0.41 true
% 0.21/0.41 % SZS output end Proof
% 0.21/0.41
% 0.21/0.41 RESULT: Unsatisfiable (the axioms are contradictory).
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