TSTP Solution File: REL011+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL011+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 : n011.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 13:43:51 EDT 2023
% Result : Theorem 7.56s 1.41s
% Output : Proof 8.51s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : REL011+1 : TPTP v8.1.2. Released v4.0.0.
% 0.11/0.13 % 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 : n011.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 19:19:38 EDT 2023
% 0.13/0.34 % CPUTime :
% 7.56/1.41 Command-line arguments: --flatten
% 7.56/1.41
% 7.56/1.41 % SZS status Theorem
% 7.56/1.41
% 8.51/1.45 % SZS output start Proof
% 8.51/1.45 Axiom 1 (converse_idempotence): converse(converse(X)) = X.
% 8.51/1.45 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 8.51/1.45 Axiom 3 (composition_identity): composition(X, one) = X.
% 8.51/1.45 Axiom 4 (def_zero): zero = meet(X, complement(X)).
% 8.51/1.45 Axiom 5 (def_top): top = join(X, complement(X)).
% 8.51/1.45 Axiom 6 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 8.51/1.45 Axiom 7 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 8.51/1.45 Axiom 8 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 8.51/1.45 Axiom 9 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 8.51/1.45 Axiom 10 (goals): meet(x0, composition(converse(x1), x2)) = zero.
% 8.51/1.45 Axiom 11 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 8.51/1.45 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 8.51/1.45 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 8.51/1.45 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 8.51/1.45
% 8.51/1.45 Lemma 15: join(X, join(Y, complement(X))) = join(Y, top).
% 8.51/1.45 Proof:
% 8.51/1.45 join(X, join(Y, complement(X)))
% 8.51/1.45 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 8.51/1.45 join(X, join(complement(X), Y))
% 8.51/1.45 = { by axiom 7 (maddux2_join_associativity) }
% 8.51/1.45 join(join(X, complement(X)), Y)
% 8.51/1.45 = { by axiom 5 (def_top) R->L }
% 8.51/1.45 join(top, Y)
% 8.51/1.45 = { by axiom 2 (maddux1_join_commutativity) }
% 8.51/1.45 join(Y, top)
% 8.51/1.45
% 8.51/1.45 Lemma 16: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 8.51/1.45 Proof:
% 8.51/1.45 converse(composition(converse(X), Y))
% 8.51/1.45 = { by axiom 8 (converse_multiplicativity) }
% 8.51/1.45 composition(converse(Y), converse(converse(X)))
% 8.51/1.45 = { by axiom 1 (converse_idempotence) }
% 8.51/1.45 composition(converse(Y), X)
% 8.51/1.45
% 8.51/1.45 Lemma 17: composition(converse(one), X) = X.
% 8.51/1.45 Proof:
% 8.51/1.45 composition(converse(one), X)
% 8.51/1.45 = { by lemma 16 R->L }
% 8.51/1.45 converse(composition(converse(X), one))
% 8.51/1.45 = { by axiom 3 (composition_identity) }
% 8.51/1.45 converse(converse(X))
% 8.51/1.45 = { by axiom 1 (converse_idempotence) }
% 8.51/1.45 X
% 8.51/1.45
% 8.51/1.45 Lemma 18: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 8.51/1.45 Proof:
% 8.51/1.45 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 8.51/1.45 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 8.51/1.45 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 8.51/1.45 = { by axiom 13 (converse_cancellativity) }
% 8.51/1.45 complement(X)
% 8.51/1.45
% 8.51/1.45 Lemma 19: join(complement(X), complement(X)) = complement(X).
% 8.51/1.45 Proof:
% 8.51/1.45 join(complement(X), complement(X))
% 8.51/1.45 = { by lemma 17 R->L }
% 8.51/1.45 join(complement(X), composition(converse(one), complement(X)))
% 8.51/1.45 = { by lemma 17 R->L }
% 8.51/1.45 join(complement(X), composition(converse(one), complement(composition(converse(one), X))))
% 8.51/1.45 = { by axiom 3 (composition_identity) R->L }
% 8.51/1.45 join(complement(X), composition(converse(one), complement(composition(composition(converse(one), one), X))))
% 8.51/1.45 = { by axiom 9 (composition_associativity) R->L }
% 8.51/1.45 join(complement(X), composition(converse(one), complement(composition(converse(one), composition(one, X)))))
% 8.51/1.45 = { by lemma 17 }
% 8.51/1.45 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 8.51/1.45 = { by lemma 18 }
% 8.51/1.45 complement(X)
% 8.51/1.45
% 8.51/1.45 Lemma 20: join(top, complement(X)) = top.
% 8.51/1.45 Proof:
% 8.51/1.45 join(top, complement(X))
% 8.51/1.45 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 8.51/1.45 join(complement(X), top)
% 8.51/1.45 = { by lemma 15 R->L }
% 8.51/1.45 join(X, join(complement(X), complement(X)))
% 8.51/1.45 = { by lemma 19 }
% 8.51/1.45 join(X, complement(X))
% 8.51/1.45 = { by axiom 5 (def_top) R->L }
% 8.51/1.45 top
% 8.51/1.45
% 8.51/1.45 Lemma 21: meet(x0, composition(converse(x1), x2)) = complement(top).
% 8.51/1.45 Proof:
% 8.51/1.45 meet(x0, composition(converse(x1), x2))
% 8.51/1.45 = { by axiom 10 (goals) }
% 8.51/1.45 zero
% 8.51/1.45 = { by axiom 4 (def_zero) }
% 8.51/1.45 meet(X, complement(X))
% 8.51/1.45 = { by axiom 11 (maddux4_definiton_of_meet) }
% 8.51/1.45 complement(join(complement(X), complement(complement(X))))
% 8.51/1.45 = { by axiom 5 (def_top) R->L }
% 8.51/1.45 complement(top)
% 8.51/1.45
% 8.51/1.45 Lemma 22: join(X, top) = top.
% 8.51/1.45 Proof:
% 8.51/1.45 join(X, top)
% 8.51/1.45 = { by lemma 20 R->L }
% 8.51/1.45 join(X, join(top, complement(X)))
% 8.51/1.45 = { by lemma 15 }
% 8.51/1.45 join(top, top)
% 8.51/1.45 = { by lemma 15 R->L }
% 8.51/1.45 join(join(meet(x0, composition(converse(x1), x2)), meet(x0, composition(converse(x1), x2))), join(top, complement(join(meet(x0, composition(converse(x1), x2)), meet(x0, composition(converse(x1), x2))))))
% 8.51/1.45 = { by lemma 20 }
% 8.51/1.45 join(join(meet(x0, composition(converse(x1), x2)), meet(x0, composition(converse(x1), x2))), top)
% 8.51/1.45 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 8.51/1.45 join(top, join(meet(x0, composition(converse(x1), x2)), meet(x0, composition(converse(x1), x2))))
% 8.51/1.45 = { by lemma 21 }
% 8.51/1.45 join(top, join(meet(x0, composition(converse(x1), x2)), complement(top)))
% 8.51/1.45 = { by lemma 21 }
% 8.51/1.45 join(top, join(complement(top), complement(top)))
% 8.51/1.45 = { by lemma 19 }
% 8.51/1.45 join(top, complement(top))
% 8.51/1.45 = { by axiom 5 (def_top) R->L }
% 8.51/1.45 top
% 8.51/1.45
% 8.51/1.45 Lemma 23: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 8.51/1.45 Proof:
% 8.51/1.45 join(meet(X, Y), complement(join(complement(X), Y)))
% 8.51/1.45 = { by axiom 11 (maddux4_definiton_of_meet) }
% 8.51/1.45 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 8.51/1.45 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L }
% 8.51/1.45 X
% 8.51/1.45
% 8.51/1.45 Lemma 24: join(meet(x0, composition(converse(x1), x2)), meet(X, X)) = X.
% 8.51/1.45 Proof:
% 8.51/1.45 join(meet(x0, composition(converse(x1), x2)), meet(X, X))
% 8.51/1.45 = { by axiom 11 (maddux4_definiton_of_meet) }
% 8.51/1.45 join(meet(x0, composition(converse(x1), x2)), complement(join(complement(X), complement(X))))
% 8.51/1.45 = { by axiom 10 (goals) }
% 8.51/1.45 join(zero, complement(join(complement(X), complement(X))))
% 8.51/1.45 = { by axiom 4 (def_zero) }
% 8.51/1.45 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 8.51/1.45 = { by lemma 23 }
% 8.51/1.45 X
% 8.51/1.45
% 8.51/1.45 Lemma 25: complement(complement(X)) = meet(X, X).
% 8.51/1.45 Proof:
% 8.51/1.45 complement(complement(X))
% 8.51/1.45 = { by lemma 19 R->L }
% 8.51/1.45 complement(join(complement(X), complement(X)))
% 8.51/1.45 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 8.51/1.45 meet(X, X)
% 8.51/1.45
% 8.51/1.45 Lemma 26: meet(Y, X) = meet(X, Y).
% 8.51/1.45 Proof:
% 8.51/1.45 meet(Y, X)
% 8.51/1.45 = { by axiom 11 (maddux4_definiton_of_meet) }
% 8.51/1.45 complement(join(complement(Y), complement(X)))
% 8.51/1.45 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 8.51/1.45 complement(join(complement(X), complement(Y)))
% 8.51/1.45 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 8.51/1.45 meet(X, Y)
% 8.51/1.45
% 8.51/1.45 Lemma 27: complement(join(meet(x0, composition(converse(x1), x2)), complement(X))) = meet(X, top).
% 8.51/1.45 Proof:
% 8.51/1.45 complement(join(meet(x0, composition(converse(x1), x2)), complement(X)))
% 8.51/1.45 = { by lemma 21 }
% 8.51/1.45 complement(join(complement(top), complement(X)))
% 8.51/1.45 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 8.51/1.45 meet(top, X)
% 8.51/1.45 = { by lemma 26 R->L }
% 8.51/1.45 meet(X, top)
% 8.51/1.45
% 8.51/1.45 Lemma 28: join(X, complement(meet(x0, composition(converse(x1), x2)))) = top.
% 8.51/1.45 Proof:
% 8.51/1.45 join(X, complement(meet(x0, composition(converse(x1), x2))))
% 8.51/1.45 = { by lemma 24 R->L }
% 8.51/1.45 join(join(meet(x0, composition(converse(x1), x2)), meet(X, X)), complement(meet(x0, composition(converse(x1), x2))))
% 8.51/1.45 = { by axiom 7 (maddux2_join_associativity) R->L }
% 8.51/1.45 join(meet(x0, composition(converse(x1), x2)), join(meet(X, X), complement(meet(x0, composition(converse(x1), x2)))))
% 8.51/1.45 = { by lemma 15 }
% 8.51/1.45 join(meet(X, X), top)
% 8.51/1.45 = { by lemma 22 }
% 8.51/1.45 top
% 8.51/1.45
% 8.51/1.45 Lemma 29: join(meet(X, Y), meet(X, complement(Y))) = X.
% 8.51/1.45 Proof:
% 8.51/1.45 join(meet(X, Y), meet(X, complement(Y)))
% 8.51/1.45 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 8.51/1.45 join(meet(X, complement(Y)), meet(X, Y))
% 8.51/1.45 = { by axiom 11 (maddux4_definiton_of_meet) }
% 8.51/1.45 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 8.51/1.45 = { by lemma 23 }
% 8.51/1.45 X
% 8.51/1.45
% 8.51/1.45 Lemma 30: join(meet(x0, composition(converse(x1), x2)), meet(X, top)) = X.
% 8.51/1.45 Proof:
% 8.51/1.45 join(meet(x0, composition(converse(x1), x2)), meet(X, top))
% 8.51/1.45 = { by lemma 28 R->L }
% 8.51/1.45 join(meet(x0, composition(converse(x1), x2)), meet(X, join(complement(meet(x0, composition(converse(x1), x2))), complement(meet(x0, composition(converse(x1), x2))))))
% 8.51/1.45 = { by lemma 19 }
% 8.51/1.45 join(meet(x0, composition(converse(x1), x2)), meet(X, complement(meet(x0, composition(converse(x1), x2)))))
% 8.51/1.45 = { by lemma 21 }
% 8.51/1.45 join(complement(top), meet(X, complement(meet(x0, composition(converse(x1), x2)))))
% 8.51/1.45 = { by lemma 28 R->L }
% 8.51/1.45 join(complement(join(complement(X), complement(meet(x0, composition(converse(x1), x2))))), meet(X, complement(meet(x0, composition(converse(x1), x2)))))
% 8.51/1.45 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 8.51/1.46 join(meet(X, meet(x0, composition(converse(x1), x2))), meet(X, complement(meet(x0, composition(converse(x1), x2)))))
% 8.51/1.46 = { by lemma 29 }
% 8.51/1.46 X
% 8.51/1.46
% 8.51/1.46 Lemma 31: join(meet(x0, composition(converse(x1), x2)), complement(X)) = complement(X).
% 8.51/1.46 Proof:
% 8.51/1.46 join(meet(x0, composition(converse(x1), x2)), complement(X))
% 8.51/1.46 = { by lemma 24 R->L }
% 8.51/1.46 join(meet(x0, composition(converse(x1), x2)), complement(join(meet(x0, composition(converse(x1), x2)), meet(X, X))))
% 8.51/1.46 = { by lemma 25 R->L }
% 8.51/1.46 join(meet(x0, composition(converse(x1), x2)), complement(join(meet(x0, composition(converse(x1), x2)), complement(complement(X)))))
% 8.51/1.46 = { by lemma 27 }
% 8.51/1.46 join(meet(x0, composition(converse(x1), x2)), meet(complement(X), top))
% 8.51/1.46 = { by lemma 30 }
% 8.51/1.46 complement(X)
% 8.51/1.46
% 8.51/1.46 Lemma 32: complement(complement(X)) = X.
% 8.51/1.46 Proof:
% 8.51/1.46 complement(complement(X))
% 8.51/1.46 = { by lemma 31 R->L }
% 8.51/1.46 join(meet(x0, composition(converse(x1), x2)), complement(complement(X)))
% 8.51/1.46 = { by lemma 25 }
% 8.51/1.46 join(meet(x0, composition(converse(x1), x2)), meet(X, X))
% 8.51/1.46 = { by lemma 24 }
% 8.51/1.46 X
% 8.51/1.46
% 8.51/1.46 Lemma 33: meet(X, X) = X.
% 8.51/1.46 Proof:
% 8.51/1.46 meet(X, X)
% 8.51/1.46 = { by lemma 25 R->L }
% 8.51/1.46 complement(complement(X))
% 8.51/1.46 = { by lemma 32 }
% 8.51/1.46 X
% 8.51/1.46
% 8.51/1.46 Lemma 34: meet(X, top) = X.
% 8.51/1.46 Proof:
% 8.51/1.46 meet(X, top)
% 8.51/1.46 = { by lemma 27 R->L }
% 8.51/1.46 complement(join(meet(x0, composition(converse(x1), x2)), complement(X)))
% 8.51/1.46 = { by lemma 31 R->L }
% 8.51/1.46 join(meet(x0, composition(converse(x1), x2)), complement(join(meet(x0, composition(converse(x1), x2)), complement(X))))
% 8.51/1.46 = { by lemma 27 }
% 8.51/1.46 join(meet(x0, composition(converse(x1), x2)), meet(X, top))
% 8.51/1.46 = { by lemma 30 }
% 8.51/1.46 X
% 8.51/1.46
% 8.51/1.46 Lemma 35: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
% 8.51/1.46 Proof:
% 8.51/1.46 complement(join(complement(X), meet(Y, Z)))
% 8.51/1.46 = { by lemma 26 }
% 8.51/1.46 complement(join(complement(X), meet(Z, Y)))
% 8.51/1.46 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 8.51/1.46 complement(join(meet(Z, Y), complement(X)))
% 8.51/1.46 = { by axiom 11 (maddux4_definiton_of_meet) }
% 8.51/1.46 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 8.51/1.46 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 8.51/1.46 meet(join(complement(Z), complement(Y)), X)
% 8.51/1.46 = { by lemma 26 R->L }
% 8.51/1.46 meet(X, join(complement(Z), complement(Y)))
% 8.51/1.46 = { by axiom 2 (maddux1_join_commutativity) }
% 8.51/1.46 meet(X, join(complement(Y), complement(Z)))
% 8.51/1.46
% 8.51/1.46 Lemma 36: complement(join(meet(x0, composition(converse(x1), x2)), meet(X, Y))) = join(complement(X), complement(Y)).
% 8.51/1.46 Proof:
% 8.51/1.46 complement(join(meet(x0, composition(converse(x1), x2)), meet(X, Y)))
% 8.51/1.46 = { by lemma 26 }
% 8.51/1.46 complement(join(meet(x0, composition(converse(x1), x2)), meet(Y, X)))
% 8.51/1.46 = { by lemma 21 }
% 8.51/1.46 complement(join(complement(top), meet(Y, X)))
% 8.51/1.46 = { by lemma 35 }
% 8.51/1.46 meet(top, join(complement(Y), complement(X)))
% 8.51/1.46 = { by lemma 26 }
% 8.51/1.46 meet(join(complement(Y), complement(X)), top)
% 8.51/1.46 = { by lemma 34 }
% 8.51/1.46 join(complement(Y), complement(X))
% 8.51/1.46 = { by axiom 2 (maddux1_join_commutativity) }
% 8.51/1.46 join(complement(X), complement(Y))
% 8.51/1.46
% 8.51/1.46 Lemma 37: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 8.51/1.46 Proof:
% 8.51/1.46 join(complement(X), complement(Y))
% 8.51/1.46 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 8.51/1.46 join(complement(Y), complement(X))
% 8.51/1.46 = { by lemma 36 R->L }
% 8.51/1.46 complement(join(meet(x0, composition(converse(x1), x2)), meet(Y, X)))
% 8.51/1.46 = { by lemma 32 R->L }
% 8.51/1.46 complement(join(meet(x0, composition(converse(x1), x2)), complement(complement(meet(Y, X)))))
% 8.51/1.46 = { by lemma 25 }
% 8.51/1.46 complement(join(meet(x0, composition(converse(x1), x2)), meet(meet(Y, X), meet(Y, X))))
% 8.51/1.46 = { by lemma 24 }
% 8.51/1.46 complement(meet(Y, X))
% 8.51/1.46 = { by lemma 26 R->L }
% 8.51/1.46 complement(meet(X, Y))
% 8.51/1.46
% 8.51/1.46 Lemma 38: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 8.51/1.46 Proof:
% 8.51/1.46 complement(meet(X, complement(Y)))
% 8.51/1.46 = { by lemma 26 }
% 8.51/1.46 complement(meet(complement(Y), X))
% 8.51/1.46 = { by lemma 31 R->L }
% 8.51/1.46 complement(meet(join(meet(x0, composition(converse(x1), x2)), complement(Y)), X))
% 8.51/1.46 = { by lemma 37 R->L }
% 8.51/1.46 join(complement(join(meet(x0, composition(converse(x1), x2)), complement(Y))), complement(X))
% 8.51/1.46 = { by lemma 27 }
% 8.51/1.46 join(meet(Y, top), complement(X))
% 8.51/1.46 = { by lemma 34 }
% 8.51/1.46 join(Y, complement(X))
% 8.51/1.46
% 8.51/1.46 Lemma 39: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 8.51/1.46 Proof:
% 8.51/1.46 complement(join(X, complement(Y)))
% 8.51/1.46 = { by lemma 38 R->L }
% 8.51/1.46 complement(complement(meet(Y, complement(X))))
% 8.51/1.46 = { by lemma 25 }
% 8.51/1.46 meet(meet(Y, complement(X)), meet(Y, complement(X)))
% 8.51/1.46 = { by lemma 33 }
% 8.51/1.46 meet(Y, complement(X))
% 8.51/1.46
% 8.51/1.46 Lemma 40: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 8.51/1.46 Proof:
% 8.51/1.46 converse(join(X, converse(Y)))
% 8.51/1.46 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 8.51/1.46 converse(join(converse(Y), X))
% 8.51/1.46 = { by axiom 6 (converse_additivity) }
% 8.51/1.46 join(converse(converse(Y)), converse(X))
% 8.51/1.46 = { by axiom 1 (converse_idempotence) }
% 8.51/1.46 join(Y, converse(X))
% 8.51/1.46
% 8.51/1.46 Lemma 41: join(X, complement(meet(X, Y))) = top.
% 8.51/1.46 Proof:
% 8.51/1.46 join(X, complement(meet(X, Y)))
% 8.51/1.46 = { by lemma 26 }
% 8.51/1.46 join(X, complement(meet(Y, X)))
% 8.51/1.46 = { by lemma 37 R->L }
% 8.51/1.46 join(X, join(complement(Y), complement(X)))
% 8.51/1.46 = { by lemma 15 }
% 8.51/1.46 join(complement(Y), top)
% 8.51/1.46 = { by lemma 22 }
% 8.51/1.46 top
% 8.51/1.46
% 8.51/1.46 Lemma 42: meet(x0, composition(converse(x1), x2)) = meet(X, meet(Y, complement(X))).
% 8.51/1.46 Proof:
% 8.51/1.46 meet(x0, composition(converse(x1), x2))
% 8.51/1.46 = { by lemma 21 }
% 8.51/1.46 complement(top)
% 8.51/1.46 = { by lemma 41 R->L }
% 8.51/1.46 complement(join(complement(X), complement(meet(complement(X), Y))))
% 8.51/1.46 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 8.51/1.46 meet(X, meet(complement(X), Y))
% 8.51/1.46 = { by lemma 26 R->L }
% 8.51/1.46 meet(X, meet(Y, complement(X)))
% 8.51/1.46
% 8.51/1.46 Lemma 43: join(complement(X), composition(Y, complement(composition(converse(Y), X)))) = complement(X).
% 8.51/1.46 Proof:
% 8.51/1.46 join(complement(X), composition(Y, complement(composition(converse(Y), X))))
% 8.51/1.46 = { by axiom 1 (converse_idempotence) R->L }
% 8.51/1.46 join(complement(X), composition(converse(converse(Y)), complement(composition(converse(Y), X))))
% 8.51/1.46 = { by lemma 18 }
% 8.51/1.46 complement(X)
% 8.51/1.46
% 8.51/1.46 Goal 1 (goals_1): meet(composition(x1, x0), x2) = zero.
% 8.51/1.46 Proof:
% 8.51/1.46 meet(composition(x1, x0), x2)
% 8.51/1.46 = { by lemma 23 R->L }
% 8.51/1.46 join(meet(meet(composition(x1, x0), x2), complement(x2)), complement(join(complement(meet(composition(x1, x0), x2)), complement(x2))))
% 8.51/1.46 = { by lemma 39 R->L }
% 8.51/1.46 join(complement(join(x2, complement(meet(composition(x1, x0), x2)))), complement(join(complement(meet(composition(x1, x0), x2)), complement(x2))))
% 8.51/1.46 = { by lemma 26 }
% 8.51/1.46 join(complement(join(x2, complement(meet(x2, composition(x1, x0))))), complement(join(complement(meet(composition(x1, x0), x2)), complement(x2))))
% 8.51/1.46 = { by lemma 41 }
% 8.51/1.46 join(complement(top), complement(join(complement(meet(composition(x1, x0), x2)), complement(x2))))
% 8.51/1.46 = { by lemma 21 R->L }
% 8.51/1.46 join(meet(x0, composition(converse(x1), x2)), complement(join(complement(meet(composition(x1, x0), x2)), complement(x2))))
% 8.51/1.46 = { by lemma 31 }
% 8.51/1.46 complement(join(complement(meet(composition(x1, x0), x2)), complement(x2)))
% 8.51/1.46 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 8.51/1.46 meet(meet(composition(x1, x0), x2), x2)
% 8.51/1.46 = { by lemma 32 R->L }
% 8.51/1.46 meet(meet(composition(x1, x0), x2), complement(complement(x2)))
% 8.51/1.46 = { by lemma 43 R->L }
% 8.51/1.46 meet(meet(composition(x1, x0), x2), complement(join(complement(x2), composition(x1, complement(composition(converse(x1), x2))))))
% 8.51/1.46 = { by lemma 29 R->L }
% 8.51/1.46 meet(meet(composition(x1, x0), x2), complement(join(complement(x2), composition(x1, complement(join(meet(composition(converse(x1), x2), x0), meet(composition(converse(x1), x2), complement(x0))))))))
% 8.51/1.46 = { by lemma 26 }
% 8.51/1.46 meet(meet(composition(x1, x0), x2), complement(join(complement(x2), composition(x1, complement(join(meet(x0, composition(converse(x1), x2)), meet(composition(converse(x1), x2), complement(x0))))))))
% 8.51/1.46 = { by lemma 36 }
% 8.51/1.46 meet(meet(composition(x1, x0), x2), complement(join(complement(x2), composition(x1, join(complement(composition(converse(x1), x2)), complement(complement(x0)))))))
% 8.51/1.46 = { by lemma 32 }
% 8.51/1.46 meet(meet(composition(x1, x0), x2), complement(join(complement(x2), composition(x1, join(complement(composition(converse(x1), x2)), x0)))))
% 8.51/1.46 = { by axiom 1 (converse_idempotence) R->L }
% 8.51/1.46 meet(meet(composition(x1, x0), x2), complement(join(complement(x2), composition(x1, join(complement(composition(converse(x1), x2)), converse(converse(x0)))))))
% 8.51/1.46 = { by lemma 40 R->L }
% 8.51/1.46 meet(meet(composition(x1, x0), x2), complement(join(complement(x2), composition(x1, converse(join(converse(x0), converse(complement(composition(converse(x1), x2)))))))))
% 8.51/1.46 = { by axiom 1 (converse_idempotence) R->L }
% 8.51/1.46 meet(meet(composition(x1, x0), x2), complement(join(complement(x2), composition(converse(converse(x1)), converse(join(converse(x0), converse(complement(composition(converse(x1), x2)))))))))
% 8.51/1.46 = { by axiom 8 (converse_multiplicativity) R->L }
% 8.51/1.46 meet(meet(composition(x1, x0), x2), complement(join(complement(x2), converse(composition(join(converse(x0), converse(complement(composition(converse(x1), x2)))), converse(x1))))))
% 8.51/1.46 = { by axiom 12 (composition_distributivity) }
% 8.51/1.46 meet(meet(composition(x1, x0), x2), complement(join(complement(x2), converse(join(composition(converse(x0), converse(x1)), composition(converse(complement(composition(converse(x1), x2))), converse(x1)))))))
% 8.51/1.46 = { by axiom 8 (converse_multiplicativity) R->L }
% 8.51/1.46 meet(meet(composition(x1, x0), x2), complement(join(complement(x2), converse(join(converse(composition(x1, x0)), composition(converse(complement(composition(converse(x1), x2))), converse(x1)))))))
% 8.51/1.46 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 8.51/1.46 meet(meet(composition(x1, x0), x2), complement(join(complement(x2), converse(join(composition(converse(complement(composition(converse(x1), x2))), converse(x1)), converse(composition(x1, x0)))))))
% 8.51/1.46 = { by lemma 40 }
% 8.51/1.46 meet(meet(composition(x1, x0), x2), complement(join(complement(x2), join(composition(x1, x0), converse(composition(converse(complement(composition(converse(x1), x2))), converse(x1)))))))
% 8.51/1.46 = { by lemma 16 }
% 8.51/1.46 meet(meet(composition(x1, x0), x2), complement(join(complement(x2), join(composition(x1, x0), composition(converse(converse(x1)), complement(composition(converse(x1), x2)))))))
% 8.51/1.46 = { by axiom 1 (converse_idempotence) }
% 8.51/1.46 meet(meet(composition(x1, x0), x2), complement(join(complement(x2), join(composition(x1, x0), composition(x1, complement(composition(converse(x1), x2)))))))
% 8.51/1.46 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 8.51/1.46 meet(meet(composition(x1, x0), x2), complement(join(complement(x2), join(composition(x1, complement(composition(converse(x1), x2))), composition(x1, x0)))))
% 8.51/1.46 = { by axiom 7 (maddux2_join_associativity) }
% 8.51/1.46 meet(meet(composition(x1, x0), x2), complement(join(join(complement(x2), composition(x1, complement(composition(converse(x1), x2)))), composition(x1, x0))))
% 8.51/1.47 = { by lemma 43 }
% 8.51/1.47 meet(meet(composition(x1, x0), x2), complement(join(complement(x2), composition(x1, x0))))
% 8.51/1.47 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 8.51/1.47 meet(meet(composition(x1, x0), x2), complement(join(composition(x1, x0), complement(x2))))
% 8.51/1.47 = { by lemma 29 R->L }
% 8.51/1.47 meet(meet(composition(x1, x0), x2), complement(join(join(meet(composition(x1, x0), x2), meet(composition(x1, x0), complement(x2))), complement(x2))))
% 8.51/1.47 = { by axiom 7 (maddux2_join_associativity) R->L }
% 8.51/1.47 meet(meet(composition(x1, x0), x2), complement(join(meet(composition(x1, x0), x2), join(meet(composition(x1, x0), complement(x2)), complement(x2)))))
% 8.51/1.47 = { by axiom 2 (maddux1_join_commutativity) }
% 8.51/1.47 meet(meet(composition(x1, x0), x2), complement(join(meet(composition(x1, x0), x2), join(complement(x2), meet(composition(x1, x0), complement(x2))))))
% 8.51/1.47 = { by lemma 26 }
% 8.51/1.47 meet(meet(composition(x1, x0), x2), complement(join(meet(composition(x1, x0), x2), join(complement(x2), meet(complement(x2), composition(x1, x0))))))
% 8.51/1.47 = { by axiom 11 (maddux4_definiton_of_meet) }
% 8.51/1.47 meet(meet(composition(x1, x0), x2), complement(join(meet(composition(x1, x0), x2), join(complement(x2), complement(join(complement(complement(x2)), complement(composition(x1, x0))))))))
% 8.51/1.47 = { by lemma 38 R->L }
% 8.51/1.47 meet(meet(composition(x1, x0), x2), complement(join(meet(composition(x1, x0), x2), complement(meet(join(complement(complement(x2)), complement(composition(x1, x0))), complement(complement(x2)))))))
% 8.51/1.47 = { by lemma 26 R->L }
% 8.51/1.47 meet(meet(composition(x1, x0), x2), complement(join(meet(composition(x1, x0), x2), complement(meet(complement(complement(x2)), join(complement(complement(x2)), complement(composition(x1, x0))))))))
% 8.51/1.47 = { by lemma 38 R->L }
% 8.51/1.47 meet(meet(composition(x1, x0), x2), complement(join(meet(composition(x1, x0), x2), complement(meet(complement(complement(x2)), complement(meet(composition(x1, x0), complement(complement(complement(x2))))))))))
% 8.51/1.47 = { by lemma 37 R->L }
% 8.51/1.47 meet(meet(composition(x1, x0), x2), complement(join(meet(composition(x1, x0), x2), complement(meet(complement(complement(x2)), join(complement(composition(x1, x0)), complement(complement(complement(complement(x2))))))))))
% 8.51/1.47 = { by lemma 35 R->L }
% 8.51/1.47 meet(meet(composition(x1, x0), x2), complement(join(meet(composition(x1, x0), x2), complement(complement(join(complement(complement(complement(x2))), meet(composition(x1, x0), complement(complement(complement(x2))))))))))
% 8.51/1.47 = { by lemma 31 R->L }
% 8.51/1.47 meet(meet(composition(x1, x0), x2), complement(join(meet(composition(x1, x0), x2), complement(join(meet(x0, composition(converse(x1), x2)), complement(join(complement(complement(complement(x2))), meet(composition(x1, x0), complement(complement(complement(x2)))))))))))
% 8.51/1.47 = { by lemma 42 }
% 8.51/1.47 meet(meet(composition(x1, x0), x2), complement(join(meet(composition(x1, x0), x2), complement(join(meet(complement(complement(x2)), meet(composition(x1, x0), complement(complement(complement(x2))))), complement(join(complement(complement(complement(x2))), meet(composition(x1, x0), complement(complement(complement(x2)))))))))))
% 8.51/1.47 = { by lemma 23 }
% 8.51/1.47 meet(meet(composition(x1, x0), x2), complement(join(meet(composition(x1, x0), x2), complement(complement(complement(x2))))))
% 8.51/1.47 = { by lemma 32 }
% 8.51/1.47 meet(meet(composition(x1, x0), x2), complement(join(meet(composition(x1, x0), x2), complement(x2))))
% 8.51/1.47 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 8.51/1.47 meet(meet(composition(x1, x0), x2), complement(join(complement(x2), meet(composition(x1, x0), x2))))
% 8.51/1.47 = { by lemma 33 R->L }
% 8.51/1.47 meet(meet(composition(x1, x0), x2), complement(join(complement(x2), meet(meet(composition(x1, x0), x2), meet(composition(x1, x0), x2)))))
% 8.51/1.47 = { by lemma 25 R->L }
% 8.51/1.47 meet(meet(composition(x1, x0), x2), complement(join(complement(x2), complement(complement(meet(composition(x1, x0), x2))))))
% 8.51/1.47 = { by lemma 39 }
% 8.51/1.47 meet(meet(composition(x1, x0), x2), meet(complement(meet(composition(x1, x0), x2)), complement(complement(x2))))
% 8.51/1.47 = { by lemma 26 R->L }
% 8.51/1.47 meet(meet(composition(x1, x0), x2), meet(complement(complement(x2)), complement(meet(composition(x1, x0), x2))))
% 8.51/1.47 = { by lemma 42 R->L }
% 8.51/1.47 meet(x0, composition(converse(x1), x2))
% 8.51/1.47 = { by axiom 10 (goals) }
% 8.51/1.47 zero
% 8.51/1.47 % SZS output end Proof
% 8.51/1.47
% 8.51/1.47 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------