TSTP Solution File: REL018+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL018+1 : 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 13:43:59 EDT 2023
% Result : Theorem 0.23s 0.51s
% Output : Proof 0.23s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.15 % Problem : REL018+1 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.16 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.16/0.37 % Computer : n008.cluster.edu
% 0.16/0.37 % Model : x86_64 x86_64
% 0.16/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.37 % Memory : 8042.1875MB
% 0.16/0.37 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.37 % CPULimit : 300
% 0.16/0.37 % WCLimit : 300
% 0.16/0.37 % DateTime : Fri Aug 25 19:29:32 EDT 2023
% 0.16/0.37 % CPUTime :
% 0.23/0.51 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.23/0.51
% 0.23/0.51 % SZS status Theorem
% 0.23/0.51
% 0.23/0.53 % SZS output start Proof
% 0.23/0.53 Axiom 1 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 0.23/0.53 Axiom 2 (composition_identity): composition(X, one) = X.
% 0.23/0.53 Axiom 3 (goals): composition(x0, top) = x0.
% 0.23/0.53 Axiom 4 (converse_idempotence): converse(converse(X)) = X.
% 0.23/0.53 Axiom 5 (def_top): top = join(X, complement(X)).
% 0.23/0.53 Axiom 6 (def_zero): zero = meet(X, complement(X)).
% 0.23/0.53 Axiom 7 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 0.23/0.53 Axiom 8 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 0.23/0.53 Axiom 9 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 0.23/0.53 Axiom 10 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 0.23/0.53 Axiom 11 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 0.23/0.53 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 0.23/0.53 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 0.23/0.53 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 0.23/0.53
% 0.23/0.53 Lemma 15: complement(top) = zero.
% 0.23/0.53 Proof:
% 0.23/0.53 complement(top)
% 0.23/0.53 = { by axiom 5 (def_top) }
% 0.23/0.53 complement(join(complement(X), complement(complement(X))))
% 0.23/0.53 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 0.23/0.53 meet(X, complement(X))
% 0.23/0.53 = { by axiom 6 (def_zero) R->L }
% 0.23/0.53 zero
% 0.23/0.53
% 0.23/0.53 Lemma 16: join(X, join(Y, complement(X))) = join(Y, top).
% 0.23/0.53 Proof:
% 0.23/0.53 join(X, join(Y, complement(X)))
% 0.23/0.53 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 0.23/0.53 join(X, join(complement(X), Y))
% 0.23/0.53 = { by axiom 8 (maddux2_join_associativity) }
% 0.23/0.53 join(join(X, complement(X)), Y)
% 0.23/0.53 = { by axiom 5 (def_top) R->L }
% 0.23/0.53 join(top, Y)
% 0.23/0.53 = { by axiom 1 (maddux1_join_commutativity) }
% 0.23/0.53 join(Y, top)
% 0.23/0.53
% 0.23/0.53 Lemma 17: join(X, join(complement(X), Y)) = join(Y, top).
% 0.23/0.53 Proof:
% 0.23/0.53 join(X, join(complement(X), Y))
% 0.23/0.53 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 0.23/0.53 join(X, join(Y, complement(X)))
% 0.23/0.53 = { by lemma 16 }
% 0.23/0.53 join(Y, top)
% 0.23/0.53
% 0.23/0.53 Lemma 18: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 0.23/0.53 Proof:
% 0.23/0.53 converse(composition(converse(X), Y))
% 0.23/0.53 = { by axiom 9 (converse_multiplicativity) }
% 0.23/0.53 composition(converse(Y), converse(converse(X)))
% 0.23/0.53 = { by axiom 4 (converse_idempotence) }
% 0.23/0.53 composition(converse(Y), X)
% 0.23/0.53
% 0.23/0.53 Lemma 19: composition(converse(one), X) = X.
% 0.23/0.53 Proof:
% 0.23/0.53 composition(converse(one), X)
% 0.23/0.53 = { by lemma 18 R->L }
% 0.23/0.53 converse(composition(converse(X), one))
% 0.23/0.53 = { by axiom 2 (composition_identity) }
% 0.23/0.53 converse(converse(X))
% 0.23/0.53 = { by axiom 4 (converse_idempotence) }
% 0.23/0.53 X
% 0.23/0.53
% 0.23/0.53 Lemma 20: composition(one, X) = X.
% 0.23/0.53 Proof:
% 0.23/0.53 composition(one, X)
% 0.23/0.53 = { by lemma 19 R->L }
% 0.23/0.53 composition(converse(one), composition(one, X))
% 0.23/0.53 = { by axiom 10 (composition_associativity) }
% 0.23/0.53 composition(composition(converse(one), one), X)
% 0.23/0.53 = { by axiom 2 (composition_identity) }
% 0.23/0.53 composition(converse(one), X)
% 0.23/0.53 = { by lemma 19 }
% 0.23/0.53 X
% 0.23/0.53
% 0.23/0.53 Lemma 21: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 0.23/0.53 Proof:
% 0.23/0.53 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 0.23/0.53 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 0.23/0.53 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 0.23/0.53 = { by axiom 13 (converse_cancellativity) }
% 0.23/0.53 complement(X)
% 0.23/0.53
% 0.23/0.53 Lemma 22: join(complement(X), complement(X)) = complement(X).
% 0.23/0.53 Proof:
% 0.23/0.53 join(complement(X), complement(X))
% 0.23/0.53 = { by lemma 19 R->L }
% 0.23/0.53 join(complement(X), composition(converse(one), complement(X)))
% 0.23/0.53 = { by lemma 20 R->L }
% 0.23/0.53 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 0.23/0.53 = { by lemma 21 }
% 0.23/0.53 complement(X)
% 0.23/0.53
% 0.23/0.53 Lemma 23: join(X, top) = top.
% 0.23/0.53 Proof:
% 0.23/0.53 join(X, top)
% 0.23/0.53 = { by axiom 5 (def_top) }
% 0.23/0.53 join(X, join(complement(X), complement(complement(X))))
% 0.23/0.53 = { by lemma 17 }
% 0.23/0.53 join(complement(complement(X)), top)
% 0.23/0.53 = { by lemma 16 R->L }
% 0.23/0.53 join(complement(X), join(complement(complement(X)), complement(complement(X))))
% 0.23/0.53 = { by lemma 22 }
% 0.23/0.53 join(complement(X), complement(complement(X)))
% 0.23/0.53 = { by axiom 5 (def_top) R->L }
% 0.23/0.53 top
% 0.23/0.53
% 0.23/0.53 Lemma 24: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 0.23/0.53 Proof:
% 0.23/0.53 join(meet(X, Y), complement(join(complement(X), Y)))
% 0.23/0.53 = { by axiom 11 (maddux4_definiton_of_meet) }
% 0.23/0.53 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 0.23/0.53 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L }
% 0.23/0.53 X
% 0.23/0.53
% 0.23/0.53 Lemma 25: join(zero, zero) = zero.
% 0.23/0.53 Proof:
% 0.23/0.53 join(zero, zero)
% 0.23/0.53 = { by lemma 15 R->L }
% 0.23/0.53 join(zero, complement(top))
% 0.23/0.54 = { by lemma 15 R->L }
% 0.23/0.54 join(complement(top), complement(top))
% 0.23/0.54 = { by lemma 22 }
% 0.23/0.54 complement(top)
% 0.23/0.54 = { by lemma 15 }
% 0.23/0.54 zero
% 0.23/0.54
% 0.23/0.54 Lemma 26: join(zero, join(zero, X)) = join(X, zero).
% 0.23/0.54 Proof:
% 0.23/0.54 join(zero, join(zero, X))
% 0.23/0.54 = { by axiom 8 (maddux2_join_associativity) }
% 0.23/0.54 join(join(zero, zero), X)
% 0.23/0.54 = { by lemma 25 }
% 0.23/0.54 join(zero, X)
% 0.23/0.54 = { by axiom 1 (maddux1_join_commutativity) }
% 0.23/0.54 join(X, zero)
% 0.23/0.54
% 0.23/0.54 Lemma 27: join(X, zero) = X.
% 0.23/0.54 Proof:
% 0.23/0.54 join(X, zero)
% 0.23/0.54 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 0.23/0.54 join(zero, X)
% 0.23/0.54 = { by lemma 24 R->L }
% 0.23/0.54 join(zero, join(meet(X, complement(X)), complement(join(complement(X), complement(X)))))
% 0.23/0.54 = { by axiom 6 (def_zero) R->L }
% 0.23/0.54 join(zero, join(zero, complement(join(complement(X), complement(X)))))
% 0.23/0.54 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 0.23/0.54 join(zero, join(zero, meet(X, X)))
% 0.23/0.54 = { by lemma 26 }
% 0.23/0.54 join(meet(X, X), zero)
% 0.23/0.54 = { by axiom 11 (maddux4_definiton_of_meet) }
% 0.23/0.54 join(complement(join(complement(X), complement(X))), zero)
% 0.23/0.54 = { by lemma 22 }
% 0.23/0.54 join(complement(complement(X)), zero)
% 0.23/0.54 = { by axiom 1 (maddux1_join_commutativity) }
% 0.23/0.54 join(zero, complement(complement(X)))
% 0.23/0.54 = { by axiom 6 (def_zero) }
% 0.23/0.54 join(meet(X, complement(X)), complement(complement(X)))
% 0.23/0.54 = { by lemma 22 R->L }
% 0.23/0.54 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 0.23/0.54 = { by lemma 24 }
% 0.23/0.54 X
% 0.23/0.54
% 0.23/0.54 Lemma 28: join(zero, X) = X.
% 0.23/0.54 Proof:
% 0.23/0.54 join(zero, X)
% 0.23/0.54 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 0.23/0.54 join(X, zero)
% 0.23/0.54 = { by lemma 27 }
% 0.23/0.54 X
% 0.23/0.54
% 0.23/0.54 Lemma 29: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 0.23/0.54 Proof:
% 0.23/0.54 converse(join(X, converse(Y)))
% 0.23/0.54 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 0.23/0.54 converse(join(converse(Y), X))
% 0.23/0.54 = { by axiom 7 (converse_additivity) }
% 0.23/0.54 join(converse(converse(Y)), converse(X))
% 0.23/0.54 = { by axiom 4 (converse_idempotence) }
% 0.23/0.54 join(Y, converse(X))
% 0.23/0.54
% 0.23/0.54 Lemma 30: join(X, converse(top)) = converse(top).
% 0.23/0.54 Proof:
% 0.23/0.54 join(X, converse(top))
% 0.23/0.54 = { by lemma 29 R->L }
% 0.23/0.54 converse(join(top, converse(X)))
% 0.23/0.54 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 0.23/0.54 converse(join(converse(X), top))
% 0.23/0.54 = { by lemma 23 }
% 0.23/0.54 converse(top)
% 0.23/0.54
% 0.23/0.54 Lemma 31: converse(composition(top, X)) = composition(converse(X), top).
% 0.23/0.54 Proof:
% 0.23/0.54 converse(composition(top, X))
% 0.23/0.54 = { by axiom 9 (converse_multiplicativity) }
% 0.23/0.54 composition(converse(X), converse(top))
% 0.23/0.54 = { by lemma 30 R->L }
% 0.23/0.54 composition(converse(X), join(Y, converse(top)))
% 0.23/0.54 = { by lemma 30 R->L }
% 0.23/0.54 composition(converse(X), join(Y, join(complement(Y), converse(top))))
% 0.23/0.54 = { by lemma 17 }
% 0.23/0.54 composition(converse(X), join(converse(top), top))
% 0.23/0.54 = { by lemma 23 }
% 0.23/0.54 composition(converse(X), top)
% 0.23/0.54
% 0.23/0.54 Lemma 32: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 0.23/0.54 Proof:
% 0.23/0.54 converse(join(converse(X), Y))
% 0.23/0.54 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 0.23/0.54 converse(join(Y, converse(X)))
% 0.23/0.54 = { by lemma 29 }
% 0.23/0.54 join(X, converse(Y))
% 0.23/0.54
% 0.23/0.54 Goal 1 (goals_1): composition(complement(x0), top) = complement(x0).
% 0.23/0.54 Proof:
% 0.23/0.54 composition(complement(x0), top)
% 0.23/0.54 = { by axiom 4 (converse_idempotence) R->L }
% 0.23/0.54 composition(converse(converse(complement(x0))), top)
% 0.23/0.54 = { by lemma 31 R->L }
% 0.23/0.54 converse(composition(top, converse(complement(x0))))
% 0.23/0.54 = { by lemma 23 R->L }
% 0.23/0.54 converse(composition(join(one, top), converse(complement(x0))))
% 0.23/0.54 = { by axiom 12 (composition_distributivity) }
% 0.23/0.54 converse(join(composition(one, converse(complement(x0))), composition(top, converse(complement(x0)))))
% 0.23/0.54 = { by lemma 20 }
% 0.23/0.54 converse(join(converse(complement(x0)), composition(top, converse(complement(x0)))))
% 0.23/0.54 = { by lemma 32 }
% 0.23/0.54 join(complement(x0), converse(composition(top, converse(complement(x0)))))
% 0.23/0.54 = { by lemma 31 }
% 0.23/0.54 join(complement(x0), composition(converse(converse(complement(x0))), top))
% 0.23/0.54 = { by axiom 4 (converse_idempotence) }
% 0.23/0.54 join(complement(x0), composition(complement(x0), top))
% 0.23/0.54 = { by axiom 5 (def_top) }
% 0.23/0.54 join(complement(x0), composition(complement(x0), join(zero, complement(zero))))
% 0.23/0.54 = { by lemma 28 }
% 0.23/0.54 join(complement(x0), composition(complement(x0), complement(zero)))
% 0.23/0.54 = { by axiom 4 (converse_idempotence) R->L }
% 0.23/0.54 join(complement(x0), composition(converse(converse(complement(x0))), complement(zero)))
% 0.23/0.54 = { by lemma 25 R->L }
% 0.23/0.54 join(complement(x0), composition(converse(converse(complement(x0))), complement(join(zero, zero))))
% 0.23/0.54 = { by axiom 4 (converse_idempotence) R->L }
% 0.23/0.54 join(complement(x0), composition(converse(converse(complement(x0))), complement(join(zero, converse(converse(zero))))))
% 0.23/0.54 = { by lemma 27 R->L }
% 0.23/0.54 join(complement(x0), composition(converse(converse(complement(x0))), complement(join(zero, converse(join(converse(zero), zero))))))
% 0.23/0.54 = { by lemma 32 }
% 0.23/0.54 join(complement(x0), composition(converse(converse(complement(x0))), complement(join(zero, join(zero, converse(zero))))))
% 0.23/0.54 = { by lemma 26 }
% 0.23/0.54 join(complement(x0), composition(converse(converse(complement(x0))), complement(join(converse(zero), zero))))
% 0.23/0.54 = { by lemma 27 }
% 0.23/0.54 join(complement(x0), composition(converse(converse(complement(x0))), complement(converse(zero))))
% 0.23/0.54 = { by lemma 15 R->L }
% 0.23/0.54 join(complement(x0), composition(converse(converse(complement(x0))), complement(converse(complement(top)))))
% 0.23/0.54 = { by lemma 21 R->L }
% 0.23/0.54 join(complement(x0), composition(converse(converse(complement(x0))), complement(converse(join(complement(top), composition(converse(x0), complement(composition(x0, top))))))))
% 0.23/0.54 = { by axiom 3 (goals) }
% 0.23/0.54 join(complement(x0), composition(converse(converse(complement(x0))), complement(converse(join(complement(top), composition(converse(x0), complement(x0)))))))
% 0.23/0.54 = { by lemma 15 }
% 0.23/0.54 join(complement(x0), composition(converse(converse(complement(x0))), complement(converse(join(zero, composition(converse(x0), complement(x0)))))))
% 0.23/0.54 = { by lemma 28 }
% 0.23/0.54 join(complement(x0), composition(converse(converse(complement(x0))), complement(converse(composition(converse(x0), complement(x0))))))
% 0.23/0.54 = { by lemma 18 }
% 0.23/0.54 join(complement(x0), composition(converse(converse(complement(x0))), complement(composition(converse(complement(x0)), x0))))
% 0.23/0.54 = { by lemma 21 }
% 0.23/0.54 complement(x0)
% 0.23/0.54 % SZS output end Proof
% 0.23/0.54
% 0.23/0.54 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------