TSTP Solution File: REL009+2 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL009+2 : TPTP v8.1.2. Released v4.0.0.
% 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 13:43:50 EDT 2023
% Result : Theorem 0.22s 0.76s
% Output : Proof 0.22s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.14 % Problem : REL009+2 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.15 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.17/0.36 % Computer : n008.cluster.edu
% 0.17/0.36 % Model : x86_64 x86_64
% 0.17/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.36 % Memory : 8042.1875MB
% 0.17/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.36 % CPULimit : 300
% 0.17/0.36 % WCLimit : 300
% 0.17/0.36 % DateTime : Fri Aug 25 20:39:32 EDT 2023
% 0.17/0.36 % CPUTime :
% 0.22/0.76 Command-line arguments: --no-flatten-goal
% 0.22/0.76
% 0.22/0.76 % SZS status Theorem
% 0.22/0.76
% 0.22/0.76 % SZS output start Proof
% 0.22/0.76 Take the following subset of the input axioms:
% 0.22/0.77 fof(composition_distributivity, axiom, ![X0, X1, X2]: composition(join(X0, X1), X2)=join(composition(X0, X2), composition(X1, X2))).
% 0.22/0.77 fof(converse_additivity, axiom, ![X0_2, X1_2]: converse(join(X0_2, X1_2))=join(converse(X0_2), converse(X1_2))).
% 0.22/0.77 fof(converse_idempotence, axiom, ![X0_2]: converse(converse(X0_2))=X0_2).
% 0.22/0.77 fof(converse_multiplicativity, axiom, ![X0_2, X1_2]: converse(composition(X0_2, X1_2))=composition(converse(X1_2), converse(X0_2))).
% 0.22/0.77 fof(goals, conjecture, ![X0_2, X1_2, X2_2]: (join(X0_2, X1_2)=X1_2 => (join(composition(X0_2, X2_2), composition(X1_2, X2_2))=composition(X1_2, X2_2) & join(composition(X2_2, X0_2), composition(X2_2, X1_2))=composition(X2_2, X1_2)))).
% 0.22/0.77 fof(maddux1_join_commutativity, axiom, ![X0_2, X1_2]: join(X0_2, X1_2)=join(X1_2, X0_2)).
% 0.22/0.77
% 0.22/0.78 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.22/0.78 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.22/0.78 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.22/0.78 fresh(y, y, x1...xn) = u
% 0.22/0.78 C => fresh(s, t, x1...xn) = v
% 0.22/0.78 where fresh is a fresh function symbol and x1..xn are the free
% 0.22/0.78 variables of u and v.
% 0.22/0.78 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.22/0.78 input problem has no model of domain size 1).
% 0.22/0.78
% 0.22/0.78 The encoding turns the above axioms into the following unit equations and goals:
% 0.22/0.78
% 0.22/0.78 Axiom 1 (converse_idempotence): converse(converse(X)) = X.
% 0.22/0.78 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 0.22/0.78 Axiom 3 (goals): join(x0, x1) = x1.
% 0.22/0.78 Axiom 4 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 0.22/0.78 Axiom 5 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 0.22/0.78 Axiom 6 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 0.22/0.78
% 0.22/0.78 Goal 1 (goals_1): tuple(join(composition(x0, x2), composition(x1, x2)), join(composition(x2, x0), composition(x2, x1))) = tuple(composition(x1, x2), composition(x2, x1)).
% 0.22/0.78 Proof:
% 0.22/0.78 tuple(join(composition(x0, x2), composition(x1, x2)), join(composition(x2, x0), composition(x2, x1)))
% 0.22/0.78 = { by axiom 6 (composition_distributivity) R->L }
% 0.22/0.78 tuple(composition(join(x0, x1), x2), join(composition(x2, x0), composition(x2, x1)))
% 0.22/0.78 = { by axiom 3 (goals) }
% 0.22/0.78 tuple(composition(x1, x2), join(composition(x2, x0), composition(x2, x1)))
% 0.22/0.78 = { by axiom 1 (converse_idempotence) R->L }
% 0.22/0.78 tuple(composition(x1, x2), join(composition(x2, x0), composition(x2, converse(converse(x1)))))
% 0.22/0.78 = { by axiom 1 (converse_idempotence) R->L }
% 0.22/0.78 tuple(composition(x1, x2), converse(converse(join(composition(x2, x0), composition(x2, converse(converse(x1)))))))
% 0.22/0.78 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 0.22/0.78 tuple(composition(x1, x2), converse(converse(join(composition(x2, converse(converse(x1))), composition(x2, x0)))))
% 0.22/0.78 = { by axiom 4 (converse_additivity) }
% 0.22/0.78 tuple(composition(x1, x2), converse(join(converse(composition(x2, converse(converse(x1)))), converse(composition(x2, x0)))))
% 0.22/0.78 = { by axiom 5 (converse_multiplicativity) }
% 0.22/0.78 tuple(composition(x1, x2), converse(join(composition(converse(converse(converse(x1))), converse(x2)), converse(composition(x2, x0)))))
% 0.22/0.78 = { by axiom 1 (converse_idempotence) }
% 0.22/0.78 tuple(composition(x1, x2), converse(join(composition(converse(x1), converse(x2)), converse(composition(x2, x0)))))
% 0.22/0.78 = { by axiom 2 (maddux1_join_commutativity) }
% 0.22/0.78 tuple(composition(x1, x2), converse(join(converse(composition(x2, x0)), composition(converse(x1), converse(x2)))))
% 0.22/0.78 = { by axiom 5 (converse_multiplicativity) }
% 0.22/0.78 tuple(composition(x1, x2), converse(join(composition(converse(x0), converse(x2)), composition(converse(x1), converse(x2)))))
% 0.22/0.78 = { by axiom 6 (composition_distributivity) R->L }
% 0.22/0.78 tuple(composition(x1, x2), converse(composition(join(converse(x0), converse(x1)), converse(x2))))
% 0.22/0.78 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 0.22/0.78 tuple(composition(x1, x2), converse(composition(join(converse(x1), converse(x0)), converse(x2))))
% 0.22/0.78 = { by axiom 1 (converse_idempotence) R->L }
% 0.22/0.78 tuple(composition(x1, x2), converse(composition(join(converse(converse(converse(x1))), converse(x0)), converse(x2))))
% 0.22/0.78 = { by axiom 4 (converse_additivity) R->L }
% 0.22/0.78 tuple(composition(x1, x2), converse(composition(converse(join(converse(converse(x1)), x0)), converse(x2))))
% 0.22/0.78 = { by axiom 2 (maddux1_join_commutativity) }
% 0.22/0.78 tuple(composition(x1, x2), converse(composition(converse(join(x0, converse(converse(x1)))), converse(x2))))
% 0.22/0.78 = { by axiom 5 (converse_multiplicativity) R->L }
% 0.22/0.78 tuple(composition(x1, x2), converse(converse(composition(x2, join(x0, converse(converse(x1)))))))
% 0.22/0.78 = { by axiom 1 (converse_idempotence) }
% 0.22/0.78 tuple(composition(x1, x2), composition(x2, join(x0, converse(converse(x1)))))
% 0.22/0.78 = { by axiom 1 (converse_idempotence) }
% 0.22/0.78 tuple(composition(x1, x2), composition(x2, join(x0, x1)))
% 0.22/0.78 = { by axiom 3 (goals) }
% 0.22/0.78 tuple(composition(x1, x2), composition(x2, x1))
% 0.22/0.78 % SZS output end Proof
% 0.22/0.78
% 0.22/0.78 RESULT: Theorem (the conjecture is true).
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