TSTP Solution File: REL037+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL037+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 : n017.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 13:44:23 EDT 2023
% Result : Theorem 17.78s 2.68s
% Output : Proof 18.54s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : REL037+1 : TPTP v8.1.2. Released v4.0.0.
% 0.07/0.13 % 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 : n017.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
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Fri Aug 25 20:20:10 EDT 2023
% 0.13/0.35 % CPUTime :
% 17.78/2.68 Command-line arguments: --ground-connectedness --complete-subsets
% 17.78/2.68
% 17.78/2.68 % SZS status Theorem
% 17.78/2.68
% 17.78/2.73 % SZS output start Proof
% 17.78/2.73 Axiom 1 (converse_idempotence): converse(converse(X)) = X.
% 17.78/2.73 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 17.78/2.73 Axiom 3 (composition_identity): composition(X, one) = X.
% 17.78/2.73 Axiom 4 (goals): composition(x0, top) = x0.
% 17.78/2.73 Axiom 5 (def_zero): zero = meet(X, complement(X)).
% 17.78/2.73 Axiom 6 (def_top): top = join(X, complement(X)).
% 17.78/2.73 Axiom 7 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 17.78/2.73 Axiom 8 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 17.78/2.73 Axiom 9 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 17.78/2.73 Axiom 10 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 17.78/2.73 Axiom 11 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 17.78/2.73 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 17.78/2.73 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 17.78/2.73 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 17.78/2.73
% 17.78/2.73 Lemma 15: complement(top) = zero.
% 17.78/2.73 Proof:
% 17.78/2.73 complement(top)
% 17.78/2.73 = { by axiom 6 (def_top) }
% 18.54/2.73 complement(join(complement(X), complement(complement(X))))
% 18.54/2.73 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 18.54/2.73 meet(X, complement(X))
% 18.54/2.73 = { by axiom 5 (def_zero) R->L }
% 18.54/2.73 zero
% 18.54/2.73
% 18.54/2.73 Lemma 16: join(X, join(Y, complement(X))) = join(Y, top).
% 18.54/2.73 Proof:
% 18.54/2.73 join(X, join(Y, complement(X)))
% 18.54/2.73 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 18.54/2.73 join(X, join(complement(X), Y))
% 18.54/2.73 = { by axiom 8 (maddux2_join_associativity) }
% 18.54/2.73 join(join(X, complement(X)), Y)
% 18.54/2.73 = { by axiom 6 (def_top) R->L }
% 18.54/2.73 join(top, Y)
% 18.54/2.73 = { by axiom 2 (maddux1_join_commutativity) }
% 18.54/2.73 join(Y, top)
% 18.54/2.73
% 18.54/2.73 Lemma 17: composition(converse(one), X) = X.
% 18.54/2.73 Proof:
% 18.54/2.73 composition(converse(one), X)
% 18.54/2.73 = { by axiom 1 (converse_idempotence) R->L }
% 18.54/2.73 composition(converse(one), converse(converse(X)))
% 18.54/2.73 = { by axiom 9 (converse_multiplicativity) R->L }
% 18.54/2.73 converse(composition(converse(X), one))
% 18.54/2.73 = { by axiom 3 (composition_identity) }
% 18.54/2.73 converse(converse(X))
% 18.54/2.73 = { by axiom 1 (converse_idempotence) }
% 18.54/2.73 X
% 18.54/2.73
% 18.54/2.73 Lemma 18: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 18.54/2.73 Proof:
% 18.54/2.73 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 18.54/2.73 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 18.54/2.73 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 18.54/2.73 = { by axiom 13 (converse_cancellativity) }
% 18.54/2.73 complement(X)
% 18.54/2.73
% 18.54/2.73 Lemma 19: join(complement(X), complement(X)) = complement(X).
% 18.54/2.73 Proof:
% 18.54/2.73 join(complement(X), complement(X))
% 18.54/2.73 = { by lemma 17 R->L }
% 18.54/2.73 join(complement(X), composition(converse(one), complement(X)))
% 18.54/2.73 = { by lemma 17 R->L }
% 18.54/2.73 join(complement(X), composition(converse(one), complement(composition(converse(one), X))))
% 18.54/2.73 = { by axiom 3 (composition_identity) R->L }
% 18.54/2.73 join(complement(X), composition(converse(one), complement(composition(composition(converse(one), one), X))))
% 18.54/2.73 = { by axiom 10 (composition_associativity) R->L }
% 18.54/2.73 join(complement(X), composition(converse(one), complement(composition(converse(one), composition(one, X)))))
% 18.54/2.73 = { by lemma 17 }
% 18.54/2.73 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 18.54/2.73 = { by lemma 18 }
% 18.54/2.73 complement(X)
% 18.54/2.73
% 18.54/2.73 Lemma 20: join(top, complement(X)) = top.
% 18.54/2.73 Proof:
% 18.54/2.73 join(top, complement(X))
% 18.54/2.73 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 18.54/2.73 join(complement(X), top)
% 18.54/2.73 = { by lemma 16 R->L }
% 18.54/2.73 join(X, join(complement(X), complement(X)))
% 18.54/2.73 = { by lemma 19 }
% 18.54/2.73 join(X, complement(X))
% 18.54/2.73 = { by axiom 6 (def_top) R->L }
% 18.54/2.73 top
% 18.54/2.73
% 18.54/2.73 Lemma 21: join(X, top) = top.
% 18.54/2.73 Proof:
% 18.54/2.73 join(X, top)
% 18.54/2.73 = { by lemma 20 R->L }
% 18.54/2.73 join(X, join(top, complement(X)))
% 18.54/2.73 = { by lemma 16 }
% 18.54/2.73 join(top, top)
% 18.54/2.73 = { by lemma 16 R->L }
% 18.54/2.73 join(zero, join(top, complement(zero)))
% 18.54/2.73 = { by lemma 20 }
% 18.54/2.73 join(zero, top)
% 18.54/2.73 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 18.54/2.73 join(top, zero)
% 18.54/2.73 = { by lemma 15 R->L }
% 18.54/2.73 join(top, complement(top))
% 18.54/2.73 = { by axiom 6 (def_top) R->L }
% 18.54/2.73 top
% 18.54/2.73
% 18.54/2.73 Lemma 22: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 18.54/2.73 Proof:
% 18.54/2.73 join(meet(X, Y), complement(join(complement(X), Y)))
% 18.54/2.73 = { by axiom 11 (maddux4_definiton_of_meet) }
% 18.54/2.73 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 18.54/2.73 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L }
% 18.54/2.73 X
% 18.54/2.73
% 18.54/2.73 Lemma 23: join(zero, meet(X, X)) = X.
% 18.54/2.73 Proof:
% 18.54/2.73 join(zero, meet(X, X))
% 18.54/2.73 = { by axiom 11 (maddux4_definiton_of_meet) }
% 18.54/2.73 join(zero, complement(join(complement(X), complement(X))))
% 18.54/2.73 = { by axiom 5 (def_zero) }
% 18.54/2.73 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 18.54/2.73 = { by lemma 22 }
% 18.54/2.73 X
% 18.54/2.73
% 18.54/2.73 Lemma 24: complement(complement(X)) = meet(X, X).
% 18.54/2.73 Proof:
% 18.54/2.73 complement(complement(X))
% 18.54/2.73 = { by lemma 19 R->L }
% 18.54/2.73 complement(join(complement(X), complement(X)))
% 18.54/2.73 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 18.54/2.73 meet(X, X)
% 18.54/2.73
% 18.54/2.73 Lemma 25: meet(Y, X) = meet(X, Y).
% 18.54/2.73 Proof:
% 18.54/2.73 meet(Y, X)
% 18.54/2.73 = { by axiom 11 (maddux4_definiton_of_meet) }
% 18.54/2.73 complement(join(complement(Y), complement(X)))
% 18.54/2.73 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 18.54/2.73 complement(join(complement(X), complement(Y)))
% 18.54/2.73 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 18.54/2.73 meet(X, Y)
% 18.54/2.73
% 18.54/2.73 Lemma 26: complement(join(zero, complement(X))) = meet(X, top).
% 18.54/2.73 Proof:
% 18.54/2.73 complement(join(zero, complement(X)))
% 18.54/2.73 = { by lemma 15 R->L }
% 18.54/2.73 complement(join(complement(top), complement(X)))
% 18.54/2.73 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 18.54/2.73 meet(top, X)
% 18.54/2.73 = { by lemma 25 R->L }
% 18.54/2.73 meet(X, top)
% 18.54/2.73
% 18.54/2.73 Lemma 27: join(X, complement(zero)) = top.
% 18.54/2.73 Proof:
% 18.54/2.73 join(X, complement(zero))
% 18.54/2.73 = { by lemma 23 R->L }
% 18.54/2.73 join(join(zero, meet(X, X)), complement(zero))
% 18.54/2.73 = { by axiom 8 (maddux2_join_associativity) R->L }
% 18.54/2.73 join(zero, join(meet(X, X), complement(zero)))
% 18.54/2.73 = { by lemma 16 }
% 18.54/2.73 join(meet(X, X), top)
% 18.54/2.73 = { by lemma 21 }
% 18.54/2.73 top
% 18.54/2.73
% 18.54/2.73 Lemma 28: meet(X, zero) = zero.
% 18.54/2.73 Proof:
% 18.54/2.73 meet(X, zero)
% 18.54/2.73 = { by axiom 11 (maddux4_definiton_of_meet) }
% 18.54/2.73 complement(join(complement(X), complement(zero)))
% 18.54/2.73 = { by lemma 27 }
% 18.54/2.73 complement(top)
% 18.54/2.73 = { by lemma 15 }
% 18.54/2.73 zero
% 18.54/2.73
% 18.54/2.73 Lemma 29: join(meet(X, Y), meet(X, complement(Y))) = X.
% 18.54/2.73 Proof:
% 18.54/2.73 join(meet(X, Y), meet(X, complement(Y)))
% 18.54/2.73 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 18.54/2.73 join(meet(X, complement(Y)), meet(X, Y))
% 18.54/2.73 = { by axiom 11 (maddux4_definiton_of_meet) }
% 18.54/2.73 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 18.54/2.73 = { by lemma 22 }
% 18.54/2.73 X
% 18.54/2.73
% 18.54/2.73 Lemma 30: join(zero, meet(X, top)) = X.
% 18.54/2.73 Proof:
% 18.54/2.73 join(zero, meet(X, top))
% 18.54/2.73 = { by lemma 27 R->L }
% 18.54/2.73 join(zero, meet(X, join(complement(zero), complement(zero))))
% 18.54/2.73 = { by lemma 19 }
% 18.54/2.73 join(zero, meet(X, complement(zero)))
% 18.54/2.73 = { by lemma 28 R->L }
% 18.54/2.73 join(meet(X, zero), meet(X, complement(zero)))
% 18.54/2.73 = { by lemma 29 }
% 18.54/2.73 X
% 18.54/2.73
% 18.54/2.73 Lemma 31: join(zero, complement(X)) = complement(X).
% 18.54/2.73 Proof:
% 18.54/2.73 join(zero, complement(X))
% 18.54/2.73 = { by lemma 23 R->L }
% 18.54/2.73 join(zero, complement(join(zero, meet(X, X))))
% 18.54/2.73 = { by lemma 24 R->L }
% 18.54/2.73 join(zero, complement(join(zero, complement(complement(X)))))
% 18.54/2.73 = { by lemma 26 }
% 18.54/2.73 join(zero, meet(complement(X), top))
% 18.54/2.73 = { by lemma 30 }
% 18.54/2.73 complement(X)
% 18.54/2.73
% 18.54/2.73 Lemma 32: complement(complement(X)) = X.
% 18.54/2.73 Proof:
% 18.54/2.73 complement(complement(X))
% 18.54/2.73 = { by lemma 31 R->L }
% 18.54/2.73 join(zero, complement(complement(X)))
% 18.54/2.73 = { by lemma 24 }
% 18.54/2.73 join(zero, meet(X, X))
% 18.54/2.73 = { by lemma 23 }
% 18.54/2.73 X
% 18.54/2.73
% 18.54/2.73 Lemma 33: join(X, zero) = X.
% 18.54/2.73 Proof:
% 18.54/2.73 join(X, zero)
% 18.54/2.73 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 18.54/2.73 join(zero, X)
% 18.54/2.73 = { by lemma 32 R->L }
% 18.54/2.73 join(zero, complement(complement(X)))
% 18.54/2.73 = { by lemma 24 }
% 18.54/2.73 join(zero, meet(X, X))
% 18.54/2.73 = { by lemma 23 }
% 18.54/2.73 X
% 18.54/2.73
% 18.54/2.73 Lemma 34: meet(X, X) = X.
% 18.54/2.73 Proof:
% 18.54/2.73 meet(X, X)
% 18.54/2.73 = { by lemma 24 R->L }
% 18.54/2.73 complement(complement(X))
% 18.54/2.73 = { by lemma 32 }
% 18.54/2.73 X
% 18.54/2.73
% 18.54/2.73 Lemma 35: meet(X, top) = X.
% 18.54/2.73 Proof:
% 18.54/2.73 meet(X, top)
% 18.54/2.73 = { by lemma 26 R->L }
% 18.54/2.73 complement(join(zero, complement(X)))
% 18.54/2.73 = { by lemma 31 R->L }
% 18.54/2.73 join(zero, complement(join(zero, complement(X))))
% 18.54/2.73 = { by lemma 26 }
% 18.54/2.73 join(zero, meet(X, top))
% 18.54/2.73 = { by lemma 30 }
% 18.54/2.73 X
% 18.54/2.73
% 18.54/2.73 Lemma 36: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
% 18.54/2.73 Proof:
% 18.54/2.73 complement(join(complement(X), meet(Y, Z)))
% 18.54/2.73 = { by lemma 25 }
% 18.54/2.73 complement(join(complement(X), meet(Z, Y)))
% 18.54/2.73 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 18.54/2.73 complement(join(meet(Z, Y), complement(X)))
% 18.54/2.73 = { by axiom 11 (maddux4_definiton_of_meet) }
% 18.54/2.73 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 18.54/2.73 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 18.54/2.73 meet(join(complement(Z), complement(Y)), X)
% 18.54/2.73 = { by lemma 25 R->L }
% 18.54/2.73 meet(X, join(complement(Z), complement(Y)))
% 18.54/2.73 = { by axiom 2 (maddux1_join_commutativity) }
% 18.54/2.73 meet(X, join(complement(Y), complement(Z)))
% 18.54/2.73
% 18.54/2.73 Lemma 37: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 18.54/2.73 Proof:
% 18.54/2.73 join(complement(X), complement(Y))
% 18.54/2.73 = { by lemma 35 R->L }
% 18.54/2.73 meet(join(complement(X), complement(Y)), top)
% 18.54/2.73 = { by lemma 25 R->L }
% 18.54/2.73 meet(top, join(complement(X), complement(Y)))
% 18.54/2.73 = { by lemma 36 R->L }
% 18.54/2.73 complement(join(complement(top), meet(X, Y)))
% 18.54/2.73 = { by lemma 15 }
% 18.54/2.73 complement(join(zero, meet(X, Y)))
% 18.54/2.73 = { by lemma 25 R->L }
% 18.54/2.73 complement(join(zero, meet(Y, X)))
% 18.54/2.73 = { by axiom 2 (maddux1_join_commutativity) }
% 18.54/2.73 complement(join(meet(Y, X), zero))
% 18.54/2.73 = { by lemma 33 }
% 18.54/2.73 complement(meet(Y, X))
% 18.54/2.73 = { by lemma 25 R->L }
% 18.54/2.73 complement(meet(X, Y))
% 18.54/2.73
% 18.54/2.73 Lemma 38: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 18.54/2.73 Proof:
% 18.54/2.73 complement(meet(X, complement(Y)))
% 18.54/2.73 = { by lemma 25 }
% 18.54/2.73 complement(meet(complement(Y), X))
% 18.54/2.73 = { by lemma 31 R->L }
% 18.54/2.73 complement(meet(join(zero, complement(Y)), X))
% 18.54/2.73 = { by lemma 37 R->L }
% 18.54/2.73 join(complement(join(zero, complement(Y))), complement(X))
% 18.54/2.73 = { by lemma 26 }
% 18.54/2.73 join(meet(Y, top), complement(X))
% 18.54/2.73 = { by lemma 35 }
% 18.54/2.73 join(Y, complement(X))
% 18.54/2.73
% 18.54/2.73 Lemma 39: meet(X, join(X, complement(Y))) = X.
% 18.54/2.73 Proof:
% 18.54/2.73 meet(X, join(X, complement(Y)))
% 18.54/2.73 = { by lemma 38 R->L }
% 18.54/2.73 meet(X, complement(meet(Y, complement(X))))
% 18.54/2.73 = { by lemma 37 R->L }
% 18.54/2.73 meet(X, join(complement(Y), complement(complement(X))))
% 18.54/2.73 = { by lemma 36 R->L }
% 18.54/2.73 complement(join(complement(X), meet(Y, complement(X))))
% 18.54/2.73 = { by lemma 31 R->L }
% 18.54/2.74 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 18.54/2.74 = { by lemma 15 R->L }
% 18.54/2.74 join(complement(top), complement(join(complement(X), meet(Y, complement(X)))))
% 18.54/2.74 = { by lemma 21 R->L }
% 18.54/2.74 join(complement(join(complement(Y), top)), complement(join(complement(X), meet(Y, complement(X)))))
% 18.54/2.74 = { by lemma 16 R->L }
% 18.54/2.74 join(complement(join(complement(X), join(complement(Y), complement(complement(X))))), complement(join(complement(X), meet(Y, complement(X)))))
% 18.54/2.74 = { by lemma 37 }
% 18.54/2.74 join(complement(join(complement(X), complement(meet(Y, complement(X))))), complement(join(complement(X), meet(Y, complement(X)))))
% 18.54/2.74 = { by lemma 25 R->L }
% 18.54/2.74 join(complement(join(complement(X), complement(meet(complement(X), Y)))), complement(join(complement(X), meet(Y, complement(X)))))
% 18.54/2.74 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 18.54/2.74 join(meet(X, meet(complement(X), Y)), complement(join(complement(X), meet(Y, complement(X)))))
% 18.54/2.74 = { by lemma 25 R->L }
% 18.54/2.74 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 18.54/2.74 = { by lemma 22 }
% 18.54/2.74 X
% 18.54/2.74
% 18.54/2.74 Lemma 40: join(meet(X, Y), Y) = Y.
% 18.54/2.74 Proof:
% 18.54/2.74 join(meet(X, Y), Y)
% 18.54/2.74 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 18.54/2.74 join(Y, meet(X, Y))
% 18.54/2.74 = { by lemma 25 R->L }
% 18.54/2.74 join(Y, meet(Y, X))
% 18.54/2.74 = { by axiom 11 (maddux4_definiton_of_meet) }
% 18.54/2.74 join(Y, complement(join(complement(Y), complement(X))))
% 18.54/2.74 = { by lemma 38 R->L }
% 18.54/2.74 complement(meet(join(complement(Y), complement(X)), complement(Y)))
% 18.54/2.74 = { by lemma 25 R->L }
% 18.54/2.74 complement(meet(complement(Y), join(complement(Y), complement(X))))
% 18.54/2.74 = { by lemma 39 }
% 18.54/2.74 complement(complement(Y))
% 18.54/2.74 = { by lemma 32 }
% 18.54/2.74 Y
% 18.54/2.74
% 18.54/2.74 Lemma 41: meet(X, join(Y, X)) = X.
% 18.54/2.74 Proof:
% 18.54/2.74 meet(X, join(Y, X))
% 18.54/2.74 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 18.54/2.74 meet(X, join(X, Y))
% 18.54/2.74 = { by lemma 34 R->L }
% 18.54/2.74 meet(X, join(X, meet(Y, Y)))
% 18.54/2.74 = { by lemma 24 R->L }
% 18.54/2.74 meet(X, join(X, complement(complement(Y))))
% 18.54/2.74 = { by lemma 39 }
% 18.54/2.74 X
% 18.54/2.74
% 18.54/2.74 Lemma 42: meet(meet(X, Y), Z) = meet(X, meet(Z, Y)).
% 18.54/2.74 Proof:
% 18.54/2.74 meet(meet(X, Y), Z)
% 18.54/2.74 = { by lemma 25 }
% 18.54/2.74 meet(Z, meet(X, Y))
% 18.54/2.74 = { by lemma 34 R->L }
% 18.54/2.74 meet(meet(Z, meet(X, Y)), meet(Z, meet(X, Y)))
% 18.54/2.74 = { by lemma 24 R->L }
% 18.54/2.74 complement(complement(meet(Z, meet(X, Y))))
% 18.54/2.74 = { by lemma 25 }
% 18.54/2.74 complement(complement(meet(Z, meet(Y, X))))
% 18.54/2.74 = { by lemma 37 R->L }
% 18.54/2.74 complement(join(complement(Z), complement(meet(Y, X))))
% 18.54/2.74 = { by lemma 37 R->L }
% 18.54/2.74 complement(join(complement(Z), join(complement(Y), complement(X))))
% 18.54/2.74 = { by axiom 8 (maddux2_join_associativity) }
% 18.54/2.74 complement(join(join(complement(Z), complement(Y)), complement(X)))
% 18.54/2.74 = { by lemma 38 R->L }
% 18.54/2.74 complement(complement(meet(X, complement(join(complement(Z), complement(Y))))))
% 18.54/2.74 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 18.54/2.74 complement(complement(meet(X, meet(Z, Y))))
% 18.54/2.74 = { by lemma 25 R->L }
% 18.54/2.74 complement(complement(meet(X, meet(Y, Z))))
% 18.54/2.74 = { by lemma 32 }
% 18.54/2.74 meet(X, meet(Y, Z))
% 18.54/2.74 = { by lemma 25 R->L }
% 18.54/2.74 meet(X, meet(Z, Y))
% 18.54/2.74
% 18.54/2.74 Lemma 43: join(zero, composition(converse(x0), complement(x0))) = zero.
% 18.54/2.74 Proof:
% 18.54/2.74 join(zero, composition(converse(x0), complement(x0)))
% 18.54/2.74 = { by lemma 15 R->L }
% 18.54/2.74 join(complement(top), composition(converse(x0), complement(x0)))
% 18.54/2.74 = { by axiom 4 (goals) R->L }
% 18.54/2.74 join(complement(top), composition(converse(x0), complement(composition(x0, top))))
% 18.54/2.74 = { by lemma 18 }
% 18.54/2.74 complement(top)
% 18.54/2.74 = { by lemma 15 }
% 18.54/2.74 zero
% 18.54/2.74
% 18.54/2.74 Lemma 44: join(composition(X, Y), composition(X, Z)) = composition(X, join(Y, Z)).
% 18.54/2.74 Proof:
% 18.54/2.74 join(composition(X, Y), composition(X, Z))
% 18.54/2.74 = { by axiom 1 (converse_idempotence) R->L }
% 18.54/2.74 join(composition(X, Y), composition(X, converse(converse(Z))))
% 18.54/2.74 = { by axiom 1 (converse_idempotence) R->L }
% 18.54/2.74 converse(converse(join(composition(X, Y), composition(X, converse(converse(Z))))))
% 18.54/2.74 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 18.54/2.74 converse(converse(join(composition(X, converse(converse(Z))), composition(X, Y))))
% 18.54/2.74 = { by axiom 7 (converse_additivity) }
% 18.54/2.74 converse(join(converse(composition(X, converse(converse(Z)))), converse(composition(X, Y))))
% 18.54/2.74 = { by axiom 9 (converse_multiplicativity) }
% 18.54/2.74 converse(join(composition(converse(converse(converse(Z))), converse(X)), converse(composition(X, Y))))
% 18.54/2.74 = { by axiom 1 (converse_idempotence) }
% 18.54/2.74 converse(join(composition(converse(Z), converse(X)), converse(composition(X, Y))))
% 18.54/2.74 = { by axiom 2 (maddux1_join_commutativity) }
% 18.54/2.74 converse(join(converse(composition(X, Y)), composition(converse(Z), converse(X))))
% 18.54/2.74 = { by axiom 9 (converse_multiplicativity) }
% 18.54/2.74 converse(join(composition(converse(Y), converse(X)), composition(converse(Z), converse(X))))
% 18.54/2.74 = { by axiom 12 (composition_distributivity) R->L }
% 18.54/2.74 converse(composition(join(converse(Y), converse(Z)), converse(X)))
% 18.54/2.74 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 18.54/2.74 converse(composition(join(converse(Z), converse(Y)), converse(X)))
% 18.54/2.74 = { by axiom 1 (converse_idempotence) R->L }
% 18.54/2.74 converse(composition(join(converse(converse(converse(Z))), converse(Y)), converse(X)))
% 18.54/2.74 = { by axiom 7 (converse_additivity) R->L }
% 18.54/2.74 converse(composition(converse(join(converse(converse(Z)), Y)), converse(X)))
% 18.54/2.74 = { by axiom 2 (maddux1_join_commutativity) }
% 18.54/2.74 converse(composition(converse(join(Y, converse(converse(Z)))), converse(X)))
% 18.54/2.74 = { by axiom 9 (converse_multiplicativity) R->L }
% 18.54/2.74 converse(converse(composition(X, join(Y, converse(converse(Z))))))
% 18.54/2.74 = { by axiom 1 (converse_idempotence) }
% 18.54/2.74 composition(X, join(Y, converse(converse(Z))))
% 18.54/2.74 = { by axiom 1 (converse_idempotence) }
% 18.54/2.74 composition(X, join(Y, Z))
% 18.54/2.74
% 18.54/2.74 Lemma 45: composition(meet(X, converse(x0)), complement(x0)) = zero.
% 18.54/2.74 Proof:
% 18.54/2.74 composition(meet(X, converse(x0)), complement(x0))
% 18.54/2.74 = { by lemma 33 R->L }
% 18.54/2.74 join(composition(meet(X, converse(x0)), complement(x0)), zero)
% 18.54/2.74 = { by lemma 43 R->L }
% 18.54/2.74 join(composition(meet(X, converse(x0)), complement(x0)), join(zero, composition(converse(x0), complement(x0))))
% 18.54/2.74 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 18.54/2.74 join(composition(meet(X, converse(x0)), complement(x0)), join(composition(converse(x0), complement(x0)), zero))
% 18.54/2.74 = { by axiom 8 (maddux2_join_associativity) }
% 18.54/2.74 join(join(composition(meet(X, converse(x0)), complement(x0)), composition(converse(x0), complement(x0))), zero)
% 18.54/2.74 = { by axiom 12 (composition_distributivity) R->L }
% 18.54/2.74 join(composition(join(meet(X, converse(x0)), converse(x0)), complement(x0)), zero)
% 18.54/2.74 = { by lemma 33 }
% 18.54/2.74 composition(join(meet(X, converse(x0)), converse(x0)), complement(x0))
% 18.54/2.74 = { by lemma 40 }
% 18.54/2.74 composition(converse(x0), complement(x0))
% 18.54/2.74 = { by lemma 41 R->L }
% 18.54/2.74 meet(composition(converse(x0), complement(x0)), join(zero, composition(converse(x0), complement(x0))))
% 18.54/2.74 = { by lemma 43 }
% 18.54/2.74 meet(composition(converse(x0), complement(x0)), zero)
% 18.54/2.74 = { by lemma 28 }
% 18.54/2.74 zero
% 18.54/2.74
% 18.54/2.74 Goal 1 (goals_1): composition(meet(x1, converse(x0)), meet(x0, x2)) = composition(meet(x1, converse(x0)), x2).
% 18.54/2.74 Proof:
% 18.54/2.74 composition(meet(x1, converse(x0)), meet(x0, x2))
% 18.54/2.74 = { by lemma 33 R->L }
% 18.54/2.74 join(composition(meet(x1, converse(x0)), meet(x0, x2)), zero)
% 18.54/2.74 = { by lemma 45 R->L }
% 18.54/2.74 join(composition(meet(x1, converse(x0)), meet(x0, x2)), composition(meet(x1, converse(x0)), complement(x0)))
% 18.54/2.74 = { by lemma 44 }
% 18.54/2.74 composition(meet(x1, converse(x0)), join(meet(x0, x2), complement(x0)))
% 18.54/2.74 = { by lemma 38 R->L }
% 18.54/2.74 composition(meet(x1, converse(x0)), complement(meet(x0, complement(meet(x0, x2)))))
% 18.54/2.74 = { by lemma 25 }
% 18.54/2.74 composition(meet(x1, converse(x0)), complement(meet(complement(meet(x0, x2)), x0)))
% 18.54/2.74 = { by lemma 29 R->L }
% 18.54/2.74 composition(meet(x1, converse(x0)), complement(meet(complement(meet(x0, x2)), join(meet(x0, x2), meet(x0, complement(x2))))))
% 18.54/2.74 = { by lemma 32 R->L }
% 18.54/2.74 composition(meet(x1, converse(x0)), complement(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2))))))))
% 18.54/2.74 = { by lemma 22 R->L }
% 18.54/2.74 composition(meet(x1, converse(x0)), complement(join(meet(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))), join(complement(join(zero, complement(meet(x0, complement(x2))))), meet(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))), join(zero, complement(meet(x0, complement(x2))))))), complement(join(complement(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2))))))), join(complement(join(zero, complement(meet(x0, complement(x2))))), meet(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))), join(zero, complement(meet(x0, complement(x2)))))))))))
% 18.54/2.74 = { by axiom 8 (maddux2_join_associativity) }
% 18.54/2.74 composition(meet(x1, converse(x0)), complement(join(meet(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))), join(complement(join(zero, complement(meet(x0, complement(x2))))), meet(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))), join(zero, complement(meet(x0, complement(x2))))))), complement(join(join(complement(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2))))))), complement(join(zero, complement(meet(x0, complement(x2)))))), meet(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))), join(zero, complement(meet(x0, complement(x2))))))))))
% 18.54/2.74 = { by axiom 11 (maddux4_definiton_of_meet) }
% 18.54/2.74 composition(meet(x1, converse(x0)), complement(join(meet(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))), join(complement(join(zero, complement(meet(x0, complement(x2))))), meet(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))), join(zero, complement(meet(x0, complement(x2))))))), complement(join(join(complement(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2))))))), complement(join(zero, complement(meet(x0, complement(x2)))))), complement(join(complement(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2))))))), complement(join(zero, complement(meet(x0, complement(x2))))))))))))
% 18.54/2.74 = { by axiom 6 (def_top) R->L }
% 18.54/2.74 composition(meet(x1, converse(x0)), complement(join(meet(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))), join(complement(join(zero, complement(meet(x0, complement(x2))))), meet(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))), join(zero, complement(meet(x0, complement(x2))))))), complement(top))))
% 18.54/2.74 = { by lemma 15 }
% 18.54/2.74 composition(meet(x1, converse(x0)), complement(join(meet(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))), join(complement(join(zero, complement(meet(x0, complement(x2))))), meet(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))), join(zero, complement(meet(x0, complement(x2))))))), zero)))
% 18.54/2.74 = { by lemma 33 }
% 18.54/2.74 composition(meet(x1, converse(x0)), complement(meet(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))), join(complement(join(zero, complement(meet(x0, complement(x2))))), meet(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))), join(zero, complement(meet(x0, complement(x2)))))))))
% 18.54/2.74 = { by lemma 26 }
% 18.54/2.74 composition(meet(x1, converse(x0)), complement(meet(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))), join(meet(meet(x0, complement(x2)), top), meet(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))), join(zero, complement(meet(x0, complement(x2)))))))))
% 18.54/2.74 = { by lemma 35 }
% 18.54/2.75 composition(meet(x1, converse(x0)), complement(meet(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))), join(meet(x0, complement(x2)), meet(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))), join(zero, complement(meet(x0, complement(x2)))))))))
% 18.54/2.75 = { by lemma 31 }
% 18.54/2.75 composition(meet(x1, converse(x0)), complement(meet(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))), join(meet(x0, complement(x2)), meet(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))), complement(meet(x0, complement(x2))))))))
% 18.54/2.75 = { by lemma 25 R->L }
% 18.54/2.75 composition(meet(x1, converse(x0)), complement(meet(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))), join(meet(x0, complement(x2)), meet(complement(meet(x0, complement(x2))), meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))))))))
% 18.54/2.75 = { by lemma 25 }
% 18.54/2.75 composition(meet(x1, converse(x0)), complement(meet(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))), join(meet(x0, complement(x2)), meet(complement(meet(x0, complement(x2))), meet(join(meet(x0, x2), complement(complement(meet(x0, complement(x2))))), complement(meet(x0, x2))))))))
% 18.54/2.75 = { by lemma 42 R->L }
% 18.54/2.75 composition(meet(x1, converse(x0)), complement(meet(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))), join(meet(x0, complement(x2)), meet(meet(complement(meet(x0, complement(x2))), complement(meet(x0, x2))), join(meet(x0, x2), complement(complement(meet(x0, complement(x2))))))))))
% 18.54/2.75 = { by lemma 38 R->L }
% 18.54/2.75 composition(meet(x1, converse(x0)), complement(meet(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))), join(meet(x0, complement(x2)), meet(meet(complement(meet(x0, complement(x2))), complement(meet(x0, x2))), complement(meet(complement(meet(x0, complement(x2))), complement(meet(x0, x2)))))))))
% 18.54/2.75 = { by axiom 5 (def_zero) R->L }
% 18.54/2.75 composition(meet(x1, converse(x0)), complement(meet(meet(complement(meet(x0, x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))), join(meet(x0, complement(x2)), zero))))
% 18.54/2.75 = { by lemma 42 }
% 18.54/2.75 composition(meet(x1, converse(x0)), complement(meet(complement(meet(x0, x2)), meet(join(meet(x0, complement(x2)), zero), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))))))
% 18.54/2.75 = { by lemma 33 }
% 18.54/2.75 composition(meet(x1, converse(x0)), complement(meet(complement(meet(x0, x2)), meet(meet(x0, complement(x2)), join(meet(x0, x2), complement(complement(meet(x0, complement(x2)))))))))
% 18.54/2.75 = { by lemma 32 }
% 18.54/2.75 composition(meet(x1, converse(x0)), complement(meet(complement(meet(x0, x2)), meet(meet(x0, complement(x2)), join(meet(x0, x2), meet(x0, complement(x2)))))))
% 18.54/2.75 = { by lemma 41 }
% 18.54/2.75 composition(meet(x1, converse(x0)), complement(meet(complement(meet(x0, x2)), meet(x0, complement(x2)))))
% 18.54/2.75 = { by lemma 25 R->L }
% 18.54/2.75 composition(meet(x1, converse(x0)), complement(meet(meet(x0, complement(x2)), complement(meet(x0, x2)))))
% 18.54/2.75 = { by lemma 42 }
% 18.54/2.75 composition(meet(x1, converse(x0)), complement(meet(x0, meet(complement(meet(x0, x2)), complement(x2)))))
% 18.54/2.75 = { by lemma 25 }
% 18.54/2.75 composition(meet(x1, converse(x0)), complement(meet(x0, meet(complement(x2), complement(meet(x0, x2))))))
% 18.54/2.75 = { by lemma 34 R->L }
% 18.54/2.75 composition(meet(x1, converse(x0)), complement(meet(x0, meet(meet(complement(x2), complement(meet(x0, x2))), meet(complement(x2), complement(meet(x0, x2)))))))
% 18.54/2.75 = { by lemma 24 R->L }
% 18.54/2.75 composition(meet(x1, converse(x0)), complement(meet(x0, complement(complement(meet(complement(x2), complement(meet(x0, x2))))))))
% 18.54/2.75 = { by lemma 38 }
% 18.54/2.75 composition(meet(x1, converse(x0)), complement(meet(x0, complement(join(meet(x0, x2), complement(complement(x2)))))))
% 18.54/2.75 = { by lemma 24 }
% 18.54/2.75 composition(meet(x1, converse(x0)), complement(meet(x0, complement(join(meet(x0, x2), meet(x2, x2))))))
% 18.54/2.75 = { by lemma 34 }
% 18.54/2.75 composition(meet(x1, converse(x0)), complement(meet(x0, complement(join(meet(x0, x2), x2)))))
% 18.54/2.75 = { by lemma 40 }
% 18.54/2.75 composition(meet(x1, converse(x0)), complement(meet(x0, complement(x2))))
% 18.54/2.75 = { by lemma 38 }
% 18.54/2.75 composition(meet(x1, converse(x0)), join(x2, complement(x0)))
% 18.54/2.75 = { by lemma 44 R->L }
% 18.54/2.75 join(composition(meet(x1, converse(x0)), x2), composition(meet(x1, converse(x0)), complement(x0)))
% 18.54/2.75 = { by lemma 45 }
% 18.54/2.75 join(composition(meet(x1, converse(x0)), x2), zero)
% 18.54/2.75 = { by lemma 33 }
% 18.54/2.75 composition(meet(x1, converse(x0)), x2)
% 18.54/2.75 % SZS output end Proof
% 18.54/2.75
% 18.54/2.75 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------