TSTP Solution File: LDA004-1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LDA004-1 : 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 : n008.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 09:04:46 EDT 2023
% Result : Unsatisfiable 197.51s 25.74s
% Output : Proof 197.85s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LDA004-1 : 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
% 0.13/0.33 % Computer : n008.cluster.edu
% 0.13/0.33 % Model : x86_64 x86_64
% 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.33 % Memory : 8042.1875MB
% 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.33 % CPULimit : 300
% 0.13/0.33 % WCLimit : 300
% 0.13/0.33 % DateTime : Sun Aug 27 01:10:17 EDT 2023
% 0.13/0.33 % CPUTime :
% 197.51/25.74 Command-line arguments: --no-flatten-goal
% 197.51/25.74
% 197.51/25.74 % SZS status Unsatisfiable
% 197.51/25.74
% 197.51/25.76 % SZS output start Proof
% 197.51/25.76 Take the following subset of the input axioms:
% 197.51/25.76 fof(a1, axiom, ![X, Y, Z]: f(X, f(Y, Z))=f(f(X, Y), f(X, Z))).
% 197.51/25.76 fof(a2, axiom, ![X2, Y2]: left(X2, f(X2, Y2))).
% 197.51/25.76 fof(a3, axiom, ![X2, Y2, Z2]: (~left(X2, Y2) | (~left(Y2, Z2) | left(X2, Z2)))).
% 197.51/25.76 fof(clause_10, axiom, a=f(f(n3, n2), u2)).
% 197.51/25.76 fof(clause_11, axiom, b=f(u1, u3)).
% 197.51/25.76 fof(clause_4, axiom, n2=f(n1, n1)).
% 197.51/25.76 fof(clause_5, axiom, n3=f(n2, n1)).
% 197.51/25.76 fof(clause_6, axiom, u=f(n2, n2)).
% 197.51/25.76 fof(clause_7, axiom, u1=f(u, n1)).
% 197.51/25.76 fof(clause_8, axiom, u2=f(u, n2)).
% 197.51/25.76 fof(clause_9, axiom, u3=f(u, n3)).
% 197.51/25.76 fof(prove_equation, negated_conjecture, ~left(a, b)).
% 197.51/25.76
% 197.51/25.76 Now clausify the problem and encode Horn clauses using encoding 3 of
% 197.51/25.76 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 197.51/25.76 We repeatedly replace C & s=t => u=v by the two clauses:
% 197.51/25.76 fresh(y, y, x1...xn) = u
% 197.51/25.76 C => fresh(s, t, x1...xn) = v
% 197.51/25.76 where fresh is a fresh function symbol and x1..xn are the free
% 197.51/25.76 variables of u and v.
% 197.51/25.76 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 197.51/25.76 input problem has no model of domain size 1).
% 197.51/25.76
% 197.51/25.76 The encoding turns the above axioms into the following unit equations and goals:
% 197.51/25.76
% 197.51/25.76 Axiom 1 (clause_11): b = f(u1, u3).
% 197.51/25.76 Axiom 2 (clause_4): n2 = f(n1, n1).
% 197.51/25.76 Axiom 3 (clause_9): u3 = f(u, n3).
% 197.51/25.76 Axiom 4 (clause_7): u1 = f(u, n1).
% 197.51/25.76 Axiom 5 (clause_8): u2 = f(u, n2).
% 197.51/25.76 Axiom 6 (clause_5): n3 = f(n2, n1).
% 197.51/25.76 Axiom 7 (clause_6): u = f(n2, n2).
% 197.51/25.76 Axiom 8 (a3): fresh2(X, X, Y, Z) = true.
% 197.51/25.76 Axiom 9 (a2): left(X, f(X, Y)) = true.
% 197.51/25.76 Axiom 10 (clause_10): a = f(f(n3, n2), u2).
% 197.51/25.77 Axiom 11 (a3): fresh(X, X, Y, Z, W) = left(Y, W).
% 197.51/25.77 Axiom 12 (a1): f(X, f(Y, Z)) = f(f(X, Y), f(X, Z)).
% 197.51/25.77 Axiom 13 (a3): fresh(left(X, Y), true, Z, X, Y) = fresh2(left(Z, X), true, Z, Y).
% 197.51/25.77
% 197.51/25.77 Lemma 14: f(n1, n2) = u.
% 197.51/25.77 Proof:
% 197.51/25.77 f(n1, n2)
% 197.51/25.77 = { by axiom 2 (clause_4) }
% 197.51/25.77 f(n1, f(n1, n1))
% 197.51/25.77 = { by axiom 12 (a1) }
% 197.51/25.77 f(f(n1, n1), f(n1, n1))
% 197.51/25.77 = { by axiom 2 (clause_4) R->L }
% 197.51/25.77 f(n2, f(n1, n1))
% 197.51/25.77 = { by axiom 2 (clause_4) R->L }
% 197.51/25.77 f(n2, n2)
% 197.51/25.77 = { by axiom 7 (clause_6) R->L }
% 197.51/25.77 u
% 197.51/25.77
% 197.51/25.77 Lemma 15: f(n2, f(n1, X)) = f(n3, f(n2, X)).
% 197.51/25.77 Proof:
% 197.51/25.77 f(n2, f(n1, X))
% 197.51/25.77 = { by axiom 12 (a1) }
% 197.51/25.77 f(f(n2, n1), f(n2, X))
% 197.51/25.77 = { by axiom 6 (clause_5) R->L }
% 197.51/25.77 f(n3, f(n2, X))
% 197.51/25.77
% 197.51/25.77 Lemma 16: f(n3, n3) = u.
% 197.51/25.77 Proof:
% 197.51/25.77 f(n3, n3)
% 197.51/25.77 = { by axiom 6 (clause_5) }
% 197.51/25.77 f(n3, f(n2, n1))
% 197.51/25.77 = { by lemma 15 R->L }
% 197.51/25.77 f(n2, f(n1, n1))
% 197.51/25.77 = { by axiom 2 (clause_4) R->L }
% 197.51/25.77 f(n2, n2)
% 197.51/25.77 = { by axiom 7 (clause_6) R->L }
% 197.51/25.77 u
% 197.51/25.77
% 197.51/25.77 Lemma 17: fresh2(left(X, Y), true, X, f(Y, Z)) = left(X, f(Y, Z)).
% 197.51/25.77 Proof:
% 197.51/25.77 fresh2(left(X, Y), true, X, f(Y, Z))
% 197.51/25.77 = { by axiom 13 (a3) R->L }
% 197.51/25.77 fresh(left(Y, f(Y, Z)), true, X, Y, f(Y, Z))
% 197.51/25.77 = { by axiom 9 (a2) }
% 197.51/25.77 fresh(true, true, X, Y, f(Y, Z))
% 197.51/25.77 = { by axiom 11 (a3) }
% 197.51/25.77 left(X, f(Y, Z))
% 197.51/25.77
% 197.51/25.77 Lemma 18: f(f(X, f(Y, Z)), f(f(X, Y), W)) = f(f(X, Y), f(f(X, Z), W)).
% 197.51/25.77 Proof:
% 197.51/25.77 f(f(X, f(Y, Z)), f(f(X, Y), W))
% 197.51/25.77 = { by axiom 12 (a1) }
% 197.51/25.77 f(f(f(X, Y), f(X, Z)), f(f(X, Y), W))
% 197.51/25.77 = { by axiom 12 (a1) R->L }
% 197.51/25.77 f(f(X, Y), f(f(X, Z), W))
% 197.51/25.77
% 197.51/25.77 Goal 1 (prove_equation): left(a, b) = true.
% 197.51/25.77 Proof:
% 197.51/25.77 left(a, b)
% 197.85/25.77 = { by axiom 1 (clause_11) }
% 197.85/25.77 left(a, f(u1, u3))
% 197.85/25.77 = { by axiom 4 (clause_7) }
% 197.85/25.77 left(a, f(f(u, n1), u3))
% 197.85/25.77 = { by axiom 3 (clause_9) }
% 197.85/25.77 left(a, f(f(u, n1), f(u, n3)))
% 197.85/25.77 = { by axiom 12 (a1) R->L }
% 197.85/25.77 left(a, f(u, f(n1, n3)))
% 197.85/25.77 = { by axiom 6 (clause_5) }
% 197.85/25.77 left(a, f(u, f(n1, f(n2, n1))))
% 197.85/25.77 = { by axiom 12 (a1) }
% 197.85/25.77 left(a, f(u, f(f(n1, n2), f(n1, n1))))
% 197.85/25.77 = { by axiom 2 (clause_4) R->L }
% 197.85/25.77 left(a, f(u, f(f(n1, n2), n2)))
% 197.85/25.77 = { by lemma 14 }
% 197.85/25.77 left(a, f(u, f(u, n2)))
% 197.85/25.77 = { by axiom 12 (a1) }
% 197.85/25.77 left(a, f(f(u, u), f(u, n2)))
% 197.85/25.77 = { by axiom 7 (clause_6) }
% 197.85/25.77 left(a, f(f(u, f(n2, n2)), f(u, n2)))
% 197.85/25.77 = { by axiom 7 (clause_6) }
% 197.85/25.77 left(a, f(f(f(n2, n2), f(n2, n2)), f(u, n2)))
% 197.85/25.77 = { by axiom 12 (a1) R->L }
% 197.85/25.77 left(a, f(f(n2, f(n2, n2)), f(u, n2)))
% 197.85/25.77 = { by axiom 7 (clause_6) R->L }
% 197.85/25.77 left(a, f(f(n2, u), f(u, n2)))
% 197.85/25.77 = { by lemma 14 R->L }
% 197.85/25.77 left(a, f(f(n2, f(n1, n2)), f(u, n2)))
% 197.85/25.77 = { by lemma 15 }
% 197.85/25.77 left(a, f(f(n3, f(n2, n2)), f(u, n2)))
% 197.85/25.77 = { by axiom 7 (clause_6) R->L }
% 197.85/25.77 left(a, f(f(n3, u), f(u, n2)))
% 197.85/25.77 = { by lemma 16 R->L }
% 197.85/25.77 left(a, f(f(n3, u), f(f(n3, n3), n2)))
% 197.85/25.77 = { by lemma 18 R->L }
% 197.85/25.77 left(a, f(f(n3, f(u, n3)), f(f(n3, u), n2)))
% 197.85/25.77 = { by axiom 3 (clause_9) R->L }
% 197.85/25.77 left(a, f(f(n3, u3), f(f(n3, u), n2)))
% 197.85/25.77 = { by lemma 18 R->L }
% 197.85/25.77 left(a, f(f(n3, f(u3, u)), f(f(n3, u3), n2)))
% 197.85/25.77 = { by lemma 17 R->L }
% 197.85/25.77 fresh2(left(a, f(n3, f(u3, u))), true, a, f(f(n3, f(u3, u)), f(f(n3, u3), n2)))
% 197.85/25.77 = { by axiom 7 (clause_6) }
% 197.85/25.77 fresh2(left(a, f(n3, f(u3, f(n2, n2)))), true, a, f(f(n3, f(u3, u)), f(f(n3, u3), n2)))
% 197.85/25.77 = { by axiom 3 (clause_9) }
% 197.85/25.77 fresh2(left(a, f(n3, f(f(u, n3), f(n2, n2)))), true, a, f(f(n3, f(u3, u)), f(f(n3, u3), n2)))
% 197.85/25.77 = { by axiom 7 (clause_6) }
% 197.85/25.77 fresh2(left(a, f(n3, f(f(f(n2, n2), n3), f(n2, n2)))), true, a, f(f(n3, f(u3, u)), f(f(n3, u3), n2)))
% 197.85/25.77 = { by axiom 6 (clause_5) }
% 197.85/25.77 fresh2(left(a, f(n3, f(f(f(n2, n2), f(n2, n1)), f(n2, n2)))), true, a, f(f(n3, f(u3, u)), f(f(n3, u3), n2)))
% 197.85/25.77 = { by axiom 12 (a1) R->L }
% 197.85/25.77 fresh2(left(a, f(n3, f(f(n2, f(n2, n1)), f(n2, n2)))), true, a, f(f(n3, f(u3, u)), f(f(n3, u3), n2)))
% 197.85/25.77 = { by axiom 6 (clause_5) R->L }
% 197.85/25.77 fresh2(left(a, f(n3, f(f(n2, n3), f(n2, n2)))), true, a, f(f(n3, f(u3, u)), f(f(n3, u3), n2)))
% 197.85/25.77 = { by axiom 12 (a1) R->L }
% 197.85/25.77 fresh2(left(a, f(n3, f(n2, f(n3, n2)))), true, a, f(f(n3, f(u3, u)), f(f(n3, u3), n2)))
% 197.85/25.77 = { by axiom 12 (a1) }
% 197.85/25.77 fresh2(left(a, f(f(n3, n2), f(n3, f(n3, n2)))), true, a, f(f(n3, f(u3, u)), f(f(n3, u3), n2)))
% 197.85/25.77 = { by axiom 12 (a1) }
% 197.85/25.77 fresh2(left(a, f(f(n3, n2), f(f(n3, n3), f(n3, n2)))), true, a, f(f(n3, f(u3, u)), f(f(n3, u3), n2)))
% 197.85/25.77 = { by lemma 16 }
% 197.85/25.77 fresh2(left(a, f(f(n3, n2), f(u, f(n3, n2)))), true, a, f(f(n3, f(u3, u)), f(f(n3, u3), n2)))
% 197.85/25.77 = { by axiom 12 (a1) }
% 197.85/25.77 fresh2(left(a, f(f(n3, n2), f(f(u, n3), f(u, n2)))), true, a, f(f(n3, f(u3, u)), f(f(n3, u3), n2)))
% 197.85/25.77 = { by axiom 3 (clause_9) R->L }
% 197.85/25.77 fresh2(left(a, f(f(n3, n2), f(u3, f(u, n2)))), true, a, f(f(n3, f(u3, u)), f(f(n3, u3), n2)))
% 197.85/25.77 = { by axiom 5 (clause_8) R->L }
% 197.85/25.77 fresh2(left(a, f(f(n3, n2), f(u3, u2))), true, a, f(f(n3, f(u3, u)), f(f(n3, u3), n2)))
% 197.85/25.77 = { by axiom 12 (a1) }
% 197.85/25.77 fresh2(left(a, f(f(f(n3, n2), u3), f(f(n3, n2), u2))), true, a, f(f(n3, f(u3, u)), f(f(n3, u3), n2)))
% 197.85/25.77 = { by axiom 3 (clause_9) }
% 197.85/25.77 fresh2(left(a, f(f(f(n3, n2), f(u, n3)), f(f(n3, n2), u2))), true, a, f(f(n3, f(u3, u)), f(f(n3, u3), n2)))
% 197.85/25.77 = { by axiom 6 (clause_5) }
% 197.85/25.77 fresh2(left(a, f(f(f(n3, n2), f(u, f(n2, n1))), f(f(n3, n2), u2))), true, a, f(f(n3, f(u3, u)), f(f(n3, u3), n2)))
% 197.85/25.77 = { by axiom 12 (a1) }
% 197.85/25.77 fresh2(left(a, f(f(f(n3, n2), f(f(u, n2), f(u, n1))), f(f(n3, n2), u2))), true, a, f(f(n3, f(u3, u)), f(f(n3, u3), n2)))
% 197.85/25.77 = { by axiom 4 (clause_7) R->L }
% 197.85/25.77 fresh2(left(a, f(f(f(n3, n2), f(f(u, n2), u1)), f(f(n3, n2), u2))), true, a, f(f(n3, f(u3, u)), f(f(n3, u3), n2)))
% 197.85/25.77 = { by axiom 5 (clause_8) R->L }
% 197.85/25.77 fresh2(left(a, f(f(f(n3, n2), f(u2, u1)), f(f(n3, n2), u2))), true, a, f(f(n3, f(u3, u)), f(f(n3, u3), n2)))
% 197.85/25.77 = { by axiom 12 (a1) }
% 197.85/25.77 fresh2(left(a, f(f(f(f(n3, n2), u2), f(f(n3, n2), u1)), f(f(n3, n2), u2))), true, a, f(f(n3, f(u3, u)), f(f(n3, u3), n2)))
% 197.85/25.77 = { by axiom 10 (clause_10) R->L }
% 197.85/25.77 fresh2(left(a, f(f(a, f(f(n3, n2), u1)), f(f(n3, n2), u2))), true, a, f(f(n3, f(u3, u)), f(f(n3, u3), n2)))
% 197.85/25.77 = { by lemma 17 R->L }
% 197.85/25.77 fresh2(fresh2(left(a, f(a, f(f(n3, n2), u1))), true, a, f(f(a, f(f(n3, n2), u1)), f(f(n3, n2), u2))), true, a, f(f(n3, f(u3, u)), f(f(n3, u3), n2)))
% 197.85/25.77 = { by axiom 9 (a2) }
% 197.85/25.77 fresh2(fresh2(true, true, a, f(f(a, f(f(n3, n2), u1)), f(f(n3, n2), u2))), true, a, f(f(n3, f(u3, u)), f(f(n3, u3), n2)))
% 197.85/25.77 = { by axiom 8 (a3) }
% 197.85/25.77 fresh2(true, true, a, f(f(n3, f(u3, u)), f(f(n3, u3), n2)))
% 197.85/25.77 = { by axiom 8 (a3) }
% 197.85/25.77 true
% 197.85/25.77 % SZS output end Proof
% 197.85/25.77
% 197.85/25.77 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------