TSTP Solution File: REL011+2 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL011+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 : n028.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 13:43:52 EDT 2023
% Result : Theorem 0.20s 0.43s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL011+2 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n028.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 : Fri Aug 25 19:18:36 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.20/0.43 Command-line arguments: --ground-connectedness --complete-subsets
% 0.20/0.43
% 0.20/0.43 % SZS status Theorem
% 0.20/0.43
% 0.20/0.45 % SZS output start Proof
% 0.20/0.45 Axiom 1 (converse_idempotence): converse(converse(X)) = X.
% 0.20/0.45 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 0.20/0.45 Axiom 3 (composition_identity): composition(X, one) = X.
% 0.20/0.45 Axiom 4 (def_top): top = join(X, complement(X)).
% 0.20/0.45 Axiom 5 (def_zero): zero = meet(X, complement(X)).
% 0.20/0.45 Axiom 6 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 0.20/0.45 Axiom 7 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 0.20/0.45 Axiom 8 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 0.20/0.45 Axiom 9 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 0.20/0.45 Axiom 10 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 0.20/0.45 Axiom 11 (goals): meet(x0, composition(converse(x1), x2)) = zero.
% 0.20/0.45 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 0.20/0.45 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 0.20/0.45 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 0.20/0.45 Axiom 15 (modular_law_1): join(meet(composition(X, Y), Z), meet(composition(X, meet(Y, composition(converse(X), Z))), Z)) = meet(composition(X, meet(Y, composition(converse(X), Z))), Z).
% 0.20/0.45
% 0.20/0.45 Lemma 16: complement(top) = zero.
% 0.20/0.45 Proof:
% 0.20/0.45 complement(top)
% 0.20/0.45 = { by axiom 4 (def_top) }
% 0.20/0.45 complement(join(complement(X), complement(complement(X))))
% 0.20/0.45 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 0.20/0.45 meet(X, complement(X))
% 0.20/0.45 = { by axiom 5 (def_zero) R->L }
% 0.20/0.45 zero
% 0.20/0.45
% 0.20/0.45 Lemma 17: join(X, join(Y, complement(X))) = join(Y, top).
% 0.20/0.45 Proof:
% 0.20/0.45 join(X, join(Y, complement(X)))
% 0.20/0.45 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 0.20/0.45 join(X, join(complement(X), Y))
% 0.20/0.45 = { by axiom 7 (maddux2_join_associativity) }
% 0.20/0.45 join(join(X, complement(X)), Y)
% 0.20/0.45 = { by axiom 4 (def_top) R->L }
% 0.20/0.45 join(top, Y)
% 0.20/0.45 = { by axiom 2 (maddux1_join_commutativity) }
% 0.20/0.45 join(Y, top)
% 0.20/0.45
% 0.20/0.45 Lemma 18: composition(converse(one), X) = X.
% 0.20/0.45 Proof:
% 0.20/0.45 composition(converse(one), X)
% 0.20/0.45 = { by axiom 1 (converse_idempotence) R->L }
% 0.20/0.45 composition(converse(one), converse(converse(X)))
% 0.20/0.45 = { by axiom 8 (converse_multiplicativity) R->L }
% 0.20/0.45 converse(composition(converse(X), one))
% 0.20/0.45 = { by axiom 3 (composition_identity) }
% 0.20/0.45 converse(converse(X))
% 0.20/0.45 = { by axiom 1 (converse_idempotence) }
% 0.20/0.45 X
% 0.20/0.45
% 0.20/0.45 Lemma 19: composition(one, X) = X.
% 0.20/0.45 Proof:
% 0.20/0.45 composition(one, X)
% 0.20/0.45 = { by lemma 18 R->L }
% 0.20/0.45 composition(converse(one), composition(one, X))
% 0.20/0.45 = { by axiom 9 (composition_associativity) }
% 0.20/0.45 composition(composition(converse(one), one), X)
% 0.20/0.45 = { by axiom 3 (composition_identity) }
% 0.20/0.45 composition(converse(one), X)
% 0.20/0.45 = { by lemma 18 }
% 0.20/0.45 X
% 0.20/0.45
% 0.20/0.45 Lemma 20: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 0.20/0.45 Proof:
% 0.20/0.45 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 0.20/0.45 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 0.20/0.45 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 0.20/0.45 = { by axiom 13 (converse_cancellativity) }
% 0.20/0.45 complement(X)
% 0.20/0.45
% 0.20/0.45 Lemma 21: join(complement(X), complement(X)) = complement(X).
% 0.20/0.45 Proof:
% 0.20/0.45 join(complement(X), complement(X))
% 0.20/0.45 = { by lemma 18 R->L }
% 0.20/0.45 join(complement(X), composition(converse(one), complement(X)))
% 0.20/0.45 = { by lemma 19 R->L }
% 0.20/0.45 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 0.20/0.45 = { by lemma 20 }
% 0.20/0.45 complement(X)
% 0.20/0.45
% 0.20/0.45 Lemma 22: join(top, complement(X)) = top.
% 0.20/0.45 Proof:
% 0.20/0.45 join(top, complement(X))
% 0.20/0.45 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 0.20/0.45 join(complement(X), top)
% 0.20/0.45 = { by lemma 17 R->L }
% 0.20/0.45 join(X, join(complement(X), complement(X)))
% 0.20/0.45 = { by lemma 21 }
% 0.20/0.45 join(X, complement(X))
% 0.20/0.45 = { by axiom 4 (def_top) R->L }
% 0.20/0.45 top
% 0.20/0.45
% 0.20/0.45 Lemma 23: join(Y, top) = join(X, top).
% 0.20/0.45 Proof:
% 0.20/0.45 join(Y, top)
% 0.20/0.45 = { by lemma 22 R->L }
% 0.20/0.45 join(Y, join(top, complement(Y)))
% 0.20/0.45 = { by lemma 17 }
% 0.20/0.45 join(top, top)
% 0.20/0.45 = { by lemma 17 R->L }
% 0.20/0.45 join(X, join(top, complement(X)))
% 0.20/0.45 = { by lemma 22 }
% 0.20/0.45 join(X, top)
% 0.20/0.45
% 0.20/0.45 Lemma 24: join(X, top) = top.
% 0.20/0.45 Proof:
% 0.20/0.46 join(X, top)
% 0.20/0.46 = { by lemma 23 }
% 0.20/0.46 join(join(zero, zero), top)
% 0.20/0.46 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 0.20/0.46 join(top, join(zero, zero))
% 0.20/0.46 = { by lemma 16 R->L }
% 0.20/0.46 join(top, join(zero, complement(top)))
% 0.20/0.46 = { by lemma 16 R->L }
% 0.20/0.46 join(top, join(complement(top), complement(top)))
% 0.20/0.46 = { by lemma 21 }
% 0.20/0.46 join(top, complement(top))
% 0.20/0.46 = { by axiom 4 (def_top) R->L }
% 0.20/0.46 top
% 0.20/0.46
% 0.20/0.46 Lemma 25: join(X, join(complement(X), Y)) = top.
% 0.20/0.46 Proof:
% 0.20/0.46 join(X, join(complement(X), Y))
% 0.20/0.46 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 0.20/0.46 join(X, join(Y, complement(X)))
% 0.20/0.46 = { by lemma 17 }
% 0.20/0.46 join(Y, top)
% 0.20/0.46 = { by lemma 23 R->L }
% 0.20/0.46 join(Z, top)
% 0.20/0.46 = { by lemma 24 }
% 0.20/0.46 top
% 0.20/0.46
% 0.20/0.46 Lemma 26: join(X, converse(top)) = top.
% 0.20/0.46 Proof:
% 0.20/0.46 join(X, converse(top))
% 0.20/0.46 = { by axiom 4 (def_top) }
% 0.20/0.46 join(X, converse(join(converse(complement(X)), complement(converse(complement(X))))))
% 0.20/0.46 = { by axiom 6 (converse_additivity) }
% 0.20/0.46 join(X, join(converse(converse(complement(X))), converse(complement(converse(complement(X))))))
% 0.20/0.46 = { by axiom 1 (converse_idempotence) }
% 0.20/0.46 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 0.20/0.46 = { by lemma 25 }
% 0.20/0.46 top
% 0.20/0.46
% 0.20/0.46 Lemma 27: converse(top) = top.
% 0.20/0.46 Proof:
% 0.20/0.46 converse(top)
% 0.20/0.46 = { by lemma 24 R->L }
% 0.20/0.46 converse(join(X, top))
% 0.20/0.46 = { by axiom 6 (converse_additivity) }
% 0.20/0.46 join(converse(X), converse(top))
% 0.20/0.46 = { by lemma 26 }
% 0.20/0.46 top
% 0.20/0.46
% 0.20/0.46 Lemma 28: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 0.20/0.46 Proof:
% 0.20/0.46 join(meet(X, Y), complement(join(complement(X), Y)))
% 0.20/0.46 = { by axiom 10 (maddux4_definiton_of_meet) }
% 0.20/0.46 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 0.20/0.46 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L }
% 0.20/0.46 X
% 0.20/0.46
% 0.20/0.46 Lemma 29: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 0.20/0.46 Proof:
% 0.20/0.46 join(zero, join(X, complement(complement(Y))))
% 0.20/0.46 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 0.20/0.46 join(zero, join(complement(complement(Y)), X))
% 0.20/0.46 = { by lemma 21 R->L }
% 0.20/0.46 join(zero, join(complement(join(complement(Y), complement(Y))), X))
% 0.20/0.46 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 0.20/0.46 join(zero, join(meet(Y, Y), X))
% 0.20/0.46 = { by axiom 7 (maddux2_join_associativity) }
% 0.20/0.46 join(join(zero, meet(Y, Y)), X)
% 0.20/0.46 = { by axiom 10 (maddux4_definiton_of_meet) }
% 0.20/0.46 join(join(zero, complement(join(complement(Y), complement(Y)))), X)
% 0.20/0.46 = { by axiom 5 (def_zero) }
% 0.20/0.46 join(join(meet(Y, complement(Y)), complement(join(complement(Y), complement(Y)))), X)
% 0.20/0.46 = { by lemma 28 }
% 0.20/0.46 join(Y, X)
% 0.20/0.46 = { by axiom 2 (maddux1_join_commutativity) }
% 0.20/0.46 join(X, Y)
% 0.20/0.46
% 0.20/0.46 Lemma 30: join(zero, complement(complement(X))) = X.
% 0.20/0.46 Proof:
% 0.20/0.46 join(zero, complement(complement(X)))
% 0.20/0.46 = { by axiom 5 (def_zero) }
% 0.20/0.46 join(meet(X, complement(X)), complement(complement(X)))
% 0.20/0.46 = { by lemma 21 R->L }
% 0.20/0.46 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 0.20/0.46 = { by lemma 28 }
% 0.20/0.46 X
% 0.20/0.46
% 0.20/0.46 Lemma 31: join(X, zero) = X.
% 0.20/0.46 Proof:
% 0.20/0.46 join(X, zero)
% 0.20/0.46 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 0.20/0.46 join(zero, X)
% 0.20/0.46 = { by lemma 29 R->L }
% 0.20/0.46 join(zero, join(zero, complement(complement(X))))
% 0.20/0.46 = { by lemma 30 R->L }
% 0.20/0.46 join(zero, join(zero, join(zero, complement(complement(complement(complement(X)))))))
% 0.20/0.46 = { by lemma 21 R->L }
% 0.20/0.46 join(zero, join(zero, join(zero, join(complement(complement(complement(complement(X)))), complement(complement(complement(complement(X))))))))
% 0.20/0.46 = { by lemma 29 }
% 0.20/0.46 join(zero, join(zero, join(complement(complement(complement(complement(X)))), complement(complement(X)))))
% 0.20/0.46 = { by axiom 2 (maddux1_join_commutativity) }
% 0.20/0.46 join(zero, join(zero, join(complement(complement(X)), complement(complement(complement(complement(X)))))))
% 0.20/0.46 = { by lemma 29 }
% 0.20/0.46 join(zero, join(complement(complement(X)), complement(complement(X))))
% 0.20/0.46 = { by lemma 21 }
% 0.20/0.46 join(zero, complement(complement(X)))
% 0.20/0.46 = { by lemma 30 }
% 0.20/0.46 X
% 0.20/0.46
% 0.20/0.46 Lemma 32: join(top, X) = top.
% 0.20/0.46 Proof:
% 0.20/0.46 join(top, X)
% 0.20/0.46 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 0.20/0.46 join(X, top)
% 0.20/0.46 = { by lemma 23 R->L }
% 0.20/0.46 join(Y, top)
% 0.20/0.46 = { by lemma 24 }
% 0.20/0.46 top
% 0.20/0.46
% 0.20/0.46 Lemma 33: join(zero, X) = X.
% 0.20/0.46 Proof:
% 0.20/0.46 join(zero, X)
% 0.20/0.46 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 0.20/0.46 join(X, zero)
% 0.20/0.46 = { by lemma 31 }
% 0.20/0.46 X
% 0.20/0.46
% 0.20/0.46 Lemma 34: meet(X, zero) = zero.
% 0.20/0.46 Proof:
% 0.20/0.46 meet(X, zero)
% 0.20/0.46 = { by axiom 10 (maddux4_definiton_of_meet) }
% 0.20/0.46 complement(join(complement(X), complement(zero)))
% 0.20/0.46 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 0.20/0.46 complement(join(complement(zero), complement(X)))
% 0.20/0.46 = { by lemma 29 R->L }
% 0.20/0.46 complement(join(zero, join(complement(zero), complement(complement(complement(X))))))
% 0.20/0.46 = { by lemma 25 }
% 0.20/0.46 complement(top)
% 0.20/0.46 = { by lemma 16 }
% 0.20/0.46 zero
% 0.20/0.46
% 0.20/0.46 Lemma 35: meet(Y, X) = meet(X, Y).
% 0.20/0.46 Proof:
% 0.20/0.46 meet(Y, X)
% 0.20/0.46 = { by axiom 10 (maddux4_definiton_of_meet) }
% 0.20/0.46 complement(join(complement(Y), complement(X)))
% 0.20/0.46 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 0.20/0.46 complement(join(complement(X), complement(Y)))
% 0.20/0.46 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 0.20/0.46 meet(X, Y)
% 0.20/0.46
% 0.20/0.46 Lemma 36: composition(top, zero) = zero.
% 0.20/0.46 Proof:
% 0.20/0.46 composition(top, zero)
% 0.20/0.46 = { by lemma 27 R->L }
% 0.20/0.46 composition(converse(top), zero)
% 0.20/0.46 = { by lemma 33 R->L }
% 0.20/0.46 join(zero, composition(converse(top), zero))
% 0.20/0.46 = { by lemma 16 R->L }
% 0.20/0.46 join(complement(top), composition(converse(top), zero))
% 0.20/0.46 = { by lemma 16 R->L }
% 0.20/0.46 join(complement(top), composition(converse(top), complement(top)))
% 0.20/0.46 = { by lemma 32 R->L }
% 0.20/0.46 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 0.20/0.46 = { by lemma 27 R->L }
% 0.20/0.46 join(complement(top), composition(converse(top), complement(join(top, composition(converse(top), top)))))
% 0.20/0.46 = { by lemma 19 R->L }
% 0.20/0.46 join(complement(top), composition(converse(top), complement(join(composition(one, top), composition(converse(top), top)))))
% 0.20/0.46 = { by axiom 12 (composition_distributivity) R->L }
% 0.20/0.46 join(complement(top), composition(converse(top), complement(composition(join(one, converse(top)), top))))
% 0.20/0.46 = { by lemma 26 }
% 0.20/0.46 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 0.20/0.46 = { by lemma 20 }
% 0.20/0.46 complement(top)
% 0.20/0.46 = { by lemma 16 }
% 0.20/0.46 zero
% 0.20/0.46
% 0.20/0.46 Lemma 37: composition(X, zero) = zero.
% 0.20/0.46 Proof:
% 0.20/0.46 composition(X, zero)
% 0.20/0.46 = { by lemma 33 R->L }
% 0.20/0.46 join(zero, composition(X, zero))
% 0.20/0.46 = { by lemma 36 R->L }
% 0.20/0.46 join(composition(top, zero), composition(X, zero))
% 0.20/0.46 = { by axiom 12 (composition_distributivity) R->L }
% 0.20/0.46 composition(join(top, X), zero)
% 0.20/0.46 = { by lemma 32 }
% 0.20/0.46 composition(top, zero)
% 0.20/0.46 = { by lemma 36 }
% 0.20/0.46 zero
% 0.20/0.46
% 0.20/0.46 Goal 1 (goals_1): meet(composition(x1, x0), x2) = zero.
% 0.20/0.46 Proof:
% 0.20/0.46 meet(composition(x1, x0), x2)
% 0.20/0.46 = { by lemma 31 R->L }
% 0.20/0.46 join(meet(composition(x1, x0), x2), zero)
% 0.20/0.46 = { by lemma 34 R->L }
% 0.20/0.46 join(meet(composition(x1, x0), x2), meet(x2, zero))
% 0.20/0.46 = { by lemma 37 R->L }
% 0.20/0.46 join(meet(composition(x1, x0), x2), meet(x2, composition(x1, zero)))
% 0.20/0.46 = { by lemma 35 }
% 0.20/0.46 join(meet(composition(x1, x0), x2), meet(composition(x1, zero), x2))
% 0.20/0.46 = { by axiom 11 (goals) R->L }
% 0.20/0.46 join(meet(composition(x1, x0), x2), meet(composition(x1, meet(x0, composition(converse(x1), x2))), x2))
% 0.20/0.46 = { by axiom 15 (modular_law_1) }
% 0.20/0.46 meet(composition(x1, meet(x0, composition(converse(x1), x2))), x2)
% 0.20/0.46 = { by axiom 11 (goals) }
% 0.20/0.46 meet(composition(x1, zero), x2)
% 0.20/0.46 = { by lemma 35 R->L }
% 0.20/0.46 meet(x2, composition(x1, zero))
% 0.20/0.46 = { by lemma 37 }
% 0.20/0.46 meet(x2, zero)
% 0.20/0.46 = { by lemma 34 }
% 0.20/0.46 zero
% 0.20/0.46 % SZS output end Proof
% 0.20/0.46
% 0.20/0.46 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------