TSTP Solution File: HEN009-4 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : HEN009-4 : TPTP v8.1.2. Bugfixed v1.2.1.
% 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 : n023.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 0.19s 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 : HEN009-4 : TPTP v8.1.2. Bugfixed v1.2.1.
% 0.12/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.34 % Computer : n023.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Thu Aug 24 13:46:27 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.19/0.41 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
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% 0.19/0.41 % SZS status Unsatisfiable
% 0.19/0.41
% 0.19/0.42 % SZS output start Proof
% 0.19/0.42 Take the following subset of the input axioms:
% 0.19/0.42 fof(identity_is_largest, axiom, ![X]: less_equal(X, identity)).
% 0.19/0.42 fof(less_equal_and_equal, axiom, ![Y, X2]: (~less_equal(X2, Y) | (~less_equal(Y, X2) | X2=Y))).
% 0.19/0.42 fof(property_of_divide1, axiom, ![Z, X2, Y2]: (~less_equal(divide(X2, Y2), Z) | less_equal(divide(X2, Z), Y2))).
% 0.19/0.42 fof(property_of_divide2, axiom, ![X2, Y2, Z2]: (~less_equal(X2, Y2) | less_equal(divide(Z2, Y2), divide(Z2, X2)))).
% 0.19/0.42 fof(property_of_divide3, axiom, ![X2, Y2, Z2]: (~less_equal(X2, Y2) | less_equal(divide(X2, Z2), divide(Y2, Z2)))).
% 0.19/0.42 fof(prove_this, negated_conjecture, divide(identity, a)!=divide(identity, divide(identity, divide(identity, a)))).
% 0.19/0.42
% 0.19/0.42 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.42 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.42 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.42 fresh(y, y, x1...xn) = u
% 0.19/0.42 C => fresh(s, t, x1...xn) = v
% 0.19/0.42 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.42 variables of u and v.
% 0.19/0.42 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.42 input problem has no model of domain size 1).
% 0.19/0.42
% 0.19/0.42 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.42
% 0.19/0.42 Axiom 1 (identity_is_largest): less_equal(X, identity) = true.
% 0.19/0.42 Axiom 2 (less_equal_and_equal): fresh(X, X, Y, Z) = Z.
% 0.19/0.42 Axiom 3 (less_equal_and_equal): fresh2(X, X, Y, Z) = Y.
% 0.19/0.42 Axiom 4 (property_of_divide2): fresh9(X, X, Y, Z, W) = true.
% 0.19/0.42 Axiom 5 (property_of_divide1): fresh8(X, X, Y, Z, W) = true.
% 0.19/0.42 Axiom 6 (property_of_divide3): fresh7(X, X, Y, Z, W) = true.
% 0.19/0.42 Axiom 7 (less_equal_and_equal): fresh2(less_equal(X, Y), true, Y, X) = fresh(less_equal(Y, X), true, Y, X).
% 0.19/0.42 Axiom 8 (property_of_divide2): fresh9(less_equal(X, Y), true, X, Y, Z) = less_equal(divide(Z, Y), divide(Z, X)).
% 0.19/0.42 Axiom 9 (property_of_divide3): fresh7(less_equal(X, Y), true, X, Y, Z) = less_equal(divide(X, Z), divide(Y, Z)).
% 0.19/0.42 Axiom 10 (property_of_divide1): fresh8(less_equal(divide(X, Y), Z), true, X, Y, Z) = less_equal(divide(X, Z), Y).
% 0.19/0.42
% 0.19/0.42 Lemma 11: less_equal(divide(X, divide(identity, Y)), Y) = true.
% 0.19/0.42 Proof:
% 0.19/0.42 less_equal(divide(X, divide(identity, Y)), Y)
% 0.19/0.42 = { by axiom 10 (property_of_divide1) R->L }
% 0.19/0.42 fresh8(less_equal(divide(X, Y), divide(identity, Y)), true, X, Y, divide(identity, Y))
% 0.19/0.42 = { by axiom 9 (property_of_divide3) R->L }
% 0.19/0.42 fresh8(fresh7(less_equal(X, identity), true, X, identity, Y), true, X, Y, divide(identity, Y))
% 0.19/0.42 = { by axiom 1 (identity_is_largest) }
% 0.19/0.42 fresh8(fresh7(true, true, X, identity, Y), true, X, Y, divide(identity, Y))
% 0.19/0.42 = { by axiom 6 (property_of_divide3) }
% 0.19/0.42 fresh8(true, true, X, Y, divide(identity, Y))
% 0.19/0.42 = { by axiom 5 (property_of_divide1) }
% 0.19/0.42 true
% 0.19/0.42
% 0.19/0.42 Goal 1 (prove_this): divide(identity, a) = divide(identity, divide(identity, divide(identity, a))).
% 0.19/0.42 Proof:
% 0.19/0.42 divide(identity, a)
% 0.19/0.42 = { by axiom 2 (less_equal_and_equal) R->L }
% 0.19/0.42 fresh(true, true, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.19/0.42 = { by lemma 11 R->L }
% 0.19/0.42 fresh(less_equal(divide(identity, divide(identity, divide(identity, a))), divide(identity, a)), true, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.19/0.42 = { by axiom 7 (less_equal_and_equal) R->L }
% 0.19/0.42 fresh2(less_equal(divide(identity, a), divide(identity, divide(identity, divide(identity, a)))), true, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.19/0.42 = { by axiom 8 (property_of_divide2) R->L }
% 0.19/0.42 fresh2(fresh9(less_equal(divide(identity, divide(identity, a)), a), true, divide(identity, divide(identity, a)), a, identity), true, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.19/0.42 = { by lemma 11 }
% 0.19/0.42 fresh2(fresh9(true, true, divide(identity, divide(identity, a)), a, identity), true, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.19/0.42 = { by axiom 4 (property_of_divide2) }
% 0.19/0.42 fresh2(true, true, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.19/0.42 = { by axiom 3 (less_equal_and_equal) }
% 0.19/0.42 divide(identity, divide(identity, divide(identity, a)))
% 0.19/0.42 % SZS output end Proof
% 0.19/0.42
% 0.19/0.42 RESULT: Unsatisfiable (the axioms are contradictory).
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