TSTP Solution File: REL041+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL041+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 : n003.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 13:44:26 EDT 2023
% Result : Theorem 31.72s 4.44s
% Output : Proof 31.72s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.12 % Problem : REL041+1 : TPTP v8.1.2. Released v4.0.0.
% 0.10/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.34 % Computer : n003.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Fri Aug 25 18:58:08 EDT 2023
% 0.14/0.34 % CPUTime :
% 31.72/4.44 Command-line arguments: --flatten
% 31.72/4.44
% 31.72/4.44 % SZS status Theorem
% 31.72/4.44
% 31.72/4.57 % SZS output start Proof
% 31.72/4.57 Axiom 1 (converse_idempotence): converse(converse(X)) = X.
% 31.72/4.58 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 31.72/4.58 Axiom 3 (composition_identity): composition(X, one) = X.
% 31.72/4.58 Axiom 4 (def_zero): zero = meet(X, complement(X)).
% 31.72/4.58 Axiom 5 (def_top): top = join(X, complement(X)).
% 31.72/4.58 Axiom 6 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 31.72/4.58 Axiom 7 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 31.72/4.58 Axiom 8 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 31.72/4.58 Axiom 9 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 31.72/4.58 Axiom 10 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 31.72/4.58 Axiom 11 (goals): join(composition(converse(x0), x0), one) = one.
% 31.72/4.58 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 31.72/4.58 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 31.72/4.58 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 31.72/4.58
% 31.72/4.58 Lemma 15: complement(top) = zero.
% 31.72/4.58 Proof:
% 31.72/4.58 complement(top)
% 31.72/4.58 = { by axiom 5 (def_top) }
% 31.72/4.58 complement(join(complement(X), complement(complement(X))))
% 31.72/4.58 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 31.72/4.58 meet(X, complement(X))
% 31.72/4.58 = { by axiom 4 (def_zero) R->L }
% 31.72/4.58 zero
% 31.72/4.58
% 31.72/4.58 Lemma 16: join(X, join(Y, complement(X))) = join(Y, top).
% 31.72/4.58 Proof:
% 31.72/4.58 join(X, join(Y, complement(X)))
% 31.72/4.58 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 31.72/4.58 join(X, join(complement(X), Y))
% 31.72/4.58 = { by axiom 7 (maddux2_join_associativity) }
% 31.72/4.58 join(join(X, complement(X)), Y)
% 31.72/4.58 = { by axiom 5 (def_top) R->L }
% 31.72/4.58 join(top, Y)
% 31.72/4.58 = { by axiom 2 (maddux1_join_commutativity) }
% 31.72/4.58 join(Y, top)
% 31.72/4.58
% 31.72/4.58 Lemma 17: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 31.72/4.58 Proof:
% 31.72/4.58 converse(composition(converse(X), Y))
% 31.72/4.58 = { by axiom 8 (converse_multiplicativity) }
% 31.72/4.58 composition(converse(Y), converse(converse(X)))
% 31.72/4.58 = { by axiom 1 (converse_idempotence) }
% 31.72/4.58 composition(converse(Y), X)
% 31.72/4.58
% 31.72/4.58 Lemma 18: composition(converse(join(composition(converse(x0), x0), one)), X) = X.
% 31.72/4.58 Proof:
% 31.72/4.58 composition(converse(join(composition(converse(x0), x0), one)), X)
% 31.72/4.58 = { by axiom 11 (goals) }
% 31.72/4.58 composition(converse(one), X)
% 31.72/4.58 = { by lemma 17 R->L }
% 31.72/4.58 converse(composition(converse(X), one))
% 31.72/4.58 = { by axiom 3 (composition_identity) }
% 31.72/4.58 converse(converse(X))
% 31.72/4.58 = { by axiom 1 (converse_idempotence) }
% 31.72/4.58 X
% 31.72/4.58
% 31.72/4.58 Lemma 19: composition(join(composition(converse(x0), x0), one), X) = X.
% 31.72/4.58 Proof:
% 31.72/4.58 composition(join(composition(converse(x0), x0), one), X)
% 31.72/4.58 = { by lemma 18 R->L }
% 31.72/4.58 composition(converse(join(composition(converse(x0), x0), one)), composition(join(composition(converse(x0), x0), one), X))
% 31.72/4.58 = { by axiom 11 (goals) }
% 31.72/4.58 composition(converse(join(composition(converse(x0), x0), one)), composition(one, X))
% 31.72/4.58 = { by axiom 9 (composition_associativity) }
% 31.72/4.58 composition(composition(converse(join(composition(converse(x0), x0), one)), one), X)
% 31.72/4.58 = { by axiom 3 (composition_identity) }
% 31.72/4.58 composition(converse(join(composition(converse(x0), x0), one)), X)
% 31.72/4.58 = { by lemma 18 }
% 31.72/4.58 X
% 31.72/4.58
% 31.72/4.58 Lemma 20: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 31.72/4.58 Proof:
% 31.72/4.58 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 31.72/4.58 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 31.72/4.58 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 31.72/4.58 = { by axiom 13 (converse_cancellativity) }
% 31.72/4.58 complement(X)
% 31.72/4.58
% 31.72/4.58 Lemma 21: join(complement(X), complement(X)) = complement(X).
% 31.72/4.58 Proof:
% 31.72/4.58 join(complement(X), complement(X))
% 31.72/4.58 = { by lemma 18 R->L }
% 31.72/4.58 join(complement(X), composition(converse(join(composition(converse(x0), x0), one)), complement(X)))
% 31.72/4.58 = { by lemma 19 R->L }
% 31.72/4.58 join(complement(X), composition(converse(join(composition(converse(x0), x0), one)), complement(composition(join(composition(converse(x0), x0), one), X))))
% 31.72/4.58 = { by lemma 20 }
% 31.72/4.58 complement(X)
% 31.72/4.58
% 31.72/4.58 Lemma 22: join(top, complement(X)) = top.
% 31.72/4.58 Proof:
% 31.72/4.58 join(top, complement(X))
% 31.72/4.58 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 31.72/4.58 join(complement(X), top)
% 31.72/4.58 = { by lemma 16 R->L }
% 31.72/4.58 join(X, join(complement(X), complement(X)))
% 31.72/4.58 = { by lemma 21 }
% 31.72/4.58 join(X, complement(X))
% 31.72/4.58 = { by axiom 5 (def_top) R->L }
% 31.72/4.58 top
% 31.72/4.58
% 31.72/4.58 Lemma 23: join(Y, top) = join(X, top).
% 31.72/4.58 Proof:
% 31.72/4.58 join(Y, top)
% 31.72/4.58 = { by lemma 22 R->L }
% 31.72/4.58 join(Y, join(top, complement(Y)))
% 31.72/4.58 = { by lemma 16 }
% 31.72/4.58 join(top, top)
% 31.72/4.58 = { by lemma 16 R->L }
% 31.72/4.58 join(X, join(top, complement(X)))
% 31.72/4.58 = { by lemma 22 }
% 31.72/4.58 join(X, top)
% 31.72/4.58
% 31.72/4.58 Lemma 24: join(X, top) = top.
% 31.72/4.58 Proof:
% 31.72/4.58 join(X, top)
% 31.72/4.58 = { by lemma 23 }
% 31.72/4.58 join(complement(Y), top)
% 31.72/4.58 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 31.72/4.58 join(top, complement(Y))
% 31.72/4.58 = { by lemma 22 }
% 31.72/4.58 top
% 31.72/4.58
% 31.72/4.58 Lemma 25: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 31.72/4.58 Proof:
% 31.72/4.58 converse(join(X, converse(Y)))
% 31.72/4.58 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 31.72/4.58 converse(join(converse(Y), X))
% 31.72/4.58 = { by axiom 6 (converse_additivity) }
% 31.72/4.58 join(converse(converse(Y)), converse(X))
% 31.72/4.58 = { by axiom 1 (converse_idempotence) }
% 31.72/4.58 join(Y, converse(X))
% 31.72/4.58
% 31.72/4.58 Lemma 26: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 31.72/4.58 Proof:
% 31.72/4.58 converse(join(converse(X), Y))
% 31.72/4.58 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 31.72/4.58 converse(join(Y, converse(X)))
% 31.72/4.58 = { by lemma 25 }
% 31.72/4.58 join(X, converse(Y))
% 31.72/4.58
% 31.72/4.58 Lemma 27: join(X, join(complement(X), Y)) = top.
% 31.72/4.58 Proof:
% 31.72/4.58 join(X, join(complement(X), Y))
% 31.72/4.58 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 31.72/4.58 join(X, join(Y, complement(X)))
% 31.72/4.58 = { by lemma 16 }
% 31.72/4.58 join(Y, top)
% 31.72/4.58 = { by lemma 23 R->L }
% 31.72/4.58 join(Z, top)
% 31.72/4.58 = { by lemma 24 }
% 31.72/4.58 top
% 31.72/4.58
% 31.72/4.58 Lemma 28: converse(top) = top.
% 31.72/4.58 Proof:
% 31.72/4.58 converse(top)
% 31.72/4.58 = { by lemma 24 R->L }
% 31.72/4.58 converse(join(X, top))
% 31.72/4.58 = { by axiom 6 (converse_additivity) }
% 31.72/4.58 join(converse(X), converse(top))
% 31.72/4.58 = { by axiom 5 (def_top) }
% 31.72/4.58 join(converse(X), converse(join(converse(complement(converse(X))), complement(converse(complement(converse(X)))))))
% 31.72/4.58 = { by lemma 26 }
% 31.72/4.58 join(converse(X), join(complement(converse(X)), converse(complement(converse(complement(converse(X)))))))
% 31.72/4.58 = { by lemma 27 }
% 31.72/4.58 top
% 31.72/4.58
% 31.72/4.58 Lemma 29: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 31.72/4.58 Proof:
% 31.72/4.58 join(meet(X, Y), complement(join(complement(X), Y)))
% 31.72/4.58 = { by axiom 10 (maddux4_definiton_of_meet) }
% 31.72/4.58 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 31.72/4.58 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L }
% 31.72/4.58 X
% 31.72/4.58
% 31.72/4.58 Lemma 30: join(zero, meet(X, X)) = X.
% 31.72/4.58 Proof:
% 31.72/4.58 join(zero, meet(X, X))
% 31.72/4.58 = { by axiom 10 (maddux4_definiton_of_meet) }
% 31.72/4.58 join(zero, complement(join(complement(X), complement(X))))
% 31.72/4.58 = { by axiom 4 (def_zero) }
% 31.72/4.58 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 31.72/4.58 = { by lemma 29 }
% 31.72/4.58 X
% 31.72/4.58
% 31.72/4.58 Lemma 31: complement(complement(X)) = meet(X, X).
% 31.72/4.58 Proof:
% 31.72/4.58 complement(complement(X))
% 31.72/4.58 = { by lemma 21 R->L }
% 31.72/4.58 complement(join(complement(X), complement(X)))
% 31.72/4.58 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 31.72/4.58 meet(X, X)
% 31.72/4.58
% 31.72/4.58 Lemma 32: meet(Y, X) = meet(X, Y).
% 31.72/4.58 Proof:
% 31.72/4.58 meet(Y, X)
% 31.72/4.58 = { by axiom 10 (maddux4_definiton_of_meet) }
% 31.72/4.58 complement(join(complement(Y), complement(X)))
% 31.72/4.58 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 31.72/4.58 complement(join(complement(X), complement(Y)))
% 31.72/4.58 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 31.72/4.58 meet(X, Y)
% 31.72/4.58
% 31.72/4.58 Lemma 33: complement(join(zero, complement(X))) = meet(X, top).
% 31.72/4.58 Proof:
% 31.72/4.58 complement(join(zero, complement(X)))
% 31.72/4.58 = { by lemma 15 R->L }
% 31.72/4.58 complement(join(complement(top), complement(X)))
% 31.72/4.58 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 31.72/4.58 meet(top, X)
% 31.72/4.58 = { by lemma 32 R->L }
% 31.72/4.58 meet(X, top)
% 31.72/4.58
% 31.72/4.58 Lemma 34: join(X, complement(zero)) = top.
% 31.72/4.58 Proof:
% 31.72/4.58 join(X, complement(zero))
% 31.72/4.58 = { by lemma 30 R->L }
% 31.72/4.58 join(join(zero, meet(X, X)), complement(zero))
% 31.72/4.58 = { by axiom 7 (maddux2_join_associativity) R->L }
% 31.72/4.58 join(zero, join(meet(X, X), complement(zero)))
% 31.72/4.58 = { by axiom 2 (maddux1_join_commutativity) }
% 31.72/4.58 join(zero, join(complement(zero), meet(X, X)))
% 31.72/4.58 = { by lemma 27 }
% 31.72/4.58 top
% 31.72/4.58
% 31.72/4.58 Lemma 35: join(meet(X, Y), meet(X, complement(Y))) = X.
% 31.72/4.58 Proof:
% 31.72/4.58 join(meet(X, Y), meet(X, complement(Y)))
% 31.72/4.58 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 31.72/4.58 join(meet(X, complement(Y)), meet(X, Y))
% 31.72/4.58 = { by axiom 10 (maddux4_definiton_of_meet) }
% 31.72/4.58 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 31.72/4.58 = { by lemma 29 }
% 31.72/4.58 X
% 31.72/4.58
% 31.72/4.58 Lemma 36: join(zero, meet(X, top)) = X.
% 31.72/4.58 Proof:
% 31.72/4.58 join(zero, meet(X, top))
% 31.72/4.58 = { by lemma 34 R->L }
% 31.72/4.58 join(zero, meet(X, join(complement(zero), complement(zero))))
% 31.72/4.58 = { by lemma 21 }
% 31.72/4.59 join(zero, meet(X, complement(zero)))
% 31.72/4.59 = { by lemma 15 R->L }
% 31.72/4.59 join(complement(top), meet(X, complement(zero)))
% 31.72/4.59 = { by lemma 34 R->L }
% 31.72/4.59 join(complement(join(complement(X), complement(zero))), meet(X, complement(zero)))
% 31.72/4.59 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 31.72/4.59 join(meet(X, zero), meet(X, complement(zero)))
% 31.72/4.59 = { by lemma 35 }
% 31.72/4.59 X
% 31.72/4.59
% 31.72/4.59 Lemma 37: join(zero, complement(X)) = complement(X).
% 31.72/4.59 Proof:
% 31.72/4.59 join(zero, complement(X))
% 31.72/4.59 = { by lemma 30 R->L }
% 31.72/4.59 join(zero, complement(join(zero, meet(X, X))))
% 31.72/4.59 = { by lemma 31 R->L }
% 31.72/4.59 join(zero, complement(join(zero, complement(complement(X)))))
% 31.72/4.59 = { by lemma 33 }
% 31.72/4.59 join(zero, meet(complement(X), top))
% 31.72/4.59 = { by lemma 36 }
% 31.72/4.59 complement(X)
% 31.72/4.59
% 31.72/4.59 Lemma 38: complement(complement(X)) = X.
% 31.72/4.59 Proof:
% 31.72/4.59 complement(complement(X))
% 31.72/4.59 = { by lemma 37 R->L }
% 31.72/4.59 join(zero, complement(complement(X)))
% 31.72/4.59 = { by lemma 31 }
% 31.72/4.59 join(zero, meet(X, X))
% 31.72/4.59 = { by lemma 30 }
% 31.72/4.59 X
% 31.72/4.59
% 31.72/4.59 Lemma 39: join(X, zero) = X.
% 31.72/4.59 Proof:
% 31.72/4.59 join(X, zero)
% 31.72/4.59 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 31.72/4.59 join(zero, X)
% 31.72/4.59 = { by lemma 38 R->L }
% 31.72/4.59 join(zero, complement(complement(X)))
% 31.72/4.59 = { by lemma 31 }
% 31.72/4.59 join(zero, meet(X, X))
% 31.72/4.59 = { by lemma 30 }
% 31.72/4.59 X
% 31.72/4.59
% 31.72/4.59 Lemma 40: join(zero, X) = X.
% 31.72/4.59 Proof:
% 31.72/4.59 join(zero, X)
% 31.72/4.59 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 31.72/4.59 join(X, zero)
% 31.72/4.59 = { by lemma 39 }
% 31.72/4.59 X
% 31.72/4.59
% 31.72/4.59 Lemma 41: meet(X, X) = X.
% 31.72/4.59 Proof:
% 31.72/4.59 meet(X, X)
% 31.72/4.59 = { by lemma 31 R->L }
% 31.72/4.59 complement(complement(X))
% 31.72/4.59 = { by lemma 38 }
% 31.72/4.59 X
% 31.72/4.59
% 31.72/4.59 Lemma 42: meet(X, top) = X.
% 31.72/4.59 Proof:
% 31.72/4.59 meet(X, top)
% 31.72/4.59 = { by lemma 33 R->L }
% 31.72/4.59 complement(join(zero, complement(X)))
% 31.72/4.59 = { by lemma 37 R->L }
% 31.72/4.59 join(zero, complement(join(zero, complement(X))))
% 31.72/4.59 = { by lemma 33 }
% 31.72/4.59 join(zero, meet(X, top))
% 31.72/4.59 = { by lemma 36 }
% 31.72/4.59 X
% 31.72/4.59
% 31.72/4.59 Lemma 43: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
% 31.72/4.59 Proof:
% 31.72/4.59 complement(join(complement(X), meet(Y, Z)))
% 31.72/4.59 = { by lemma 32 }
% 31.72/4.59 complement(join(complement(X), meet(Z, Y)))
% 31.72/4.59 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 31.72/4.59 complement(join(meet(Z, Y), complement(X)))
% 31.72/4.59 = { by axiom 10 (maddux4_definiton_of_meet) }
% 31.72/4.59 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 31.72/4.59 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 31.72/4.59 meet(join(complement(Z), complement(Y)), X)
% 31.72/4.59 = { by lemma 32 R->L }
% 31.72/4.59 meet(X, join(complement(Z), complement(Y)))
% 31.72/4.59 = { by axiom 2 (maddux1_join_commutativity) }
% 31.72/4.59 meet(X, join(complement(Y), complement(Z)))
% 31.72/4.59
% 31.72/4.59 Lemma 44: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 31.72/4.59 Proof:
% 31.72/4.59 join(complement(X), complement(Y))
% 31.72/4.59 = { by lemma 42 R->L }
% 31.72/4.59 meet(join(complement(X), complement(Y)), top)
% 31.72/4.59 = { by lemma 32 R->L }
% 31.72/4.59 meet(top, join(complement(X), complement(Y)))
% 31.72/4.59 = { by lemma 43 R->L }
% 31.72/4.59 complement(join(complement(top), meet(X, Y)))
% 31.72/4.59 = { by lemma 15 }
% 31.72/4.59 complement(join(zero, meet(X, Y)))
% 31.72/4.59 = { by lemma 32 R->L }
% 31.72/4.59 complement(join(zero, meet(Y, X)))
% 31.72/4.59 = { by axiom 2 (maddux1_join_commutativity) }
% 31.72/4.59 complement(join(meet(Y, X), zero))
% 31.72/4.59 = { by lemma 39 }
% 31.72/4.59 complement(meet(Y, X))
% 31.72/4.59 = { by lemma 32 R->L }
% 31.72/4.59 complement(meet(X, Y))
% 31.72/4.59
% 31.72/4.59 Lemma 45: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 31.72/4.59 Proof:
% 31.72/4.59 complement(meet(X, complement(Y)))
% 31.72/4.59 = { by lemma 32 }
% 31.72/4.59 complement(meet(complement(Y), X))
% 31.72/4.59 = { by lemma 37 R->L }
% 31.72/4.59 complement(meet(join(zero, complement(Y)), X))
% 31.72/4.59 = { by lemma 44 R->L }
% 31.72/4.59 join(complement(join(zero, complement(Y))), complement(X))
% 31.72/4.59 = { by lemma 33 }
% 31.72/4.59 join(meet(Y, top), complement(X))
% 31.72/4.59 = { by lemma 42 }
% 31.72/4.59 join(Y, complement(X))
% 31.72/4.59
% 31.72/4.59 Lemma 46: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 31.72/4.59 Proof:
% 31.72/4.59 complement(join(X, complement(Y)))
% 31.72/4.59 = { by lemma 45 R->L }
% 31.72/4.59 complement(complement(meet(Y, complement(X))))
% 31.72/4.59 = { by lemma 31 }
% 31.72/4.59 meet(meet(Y, complement(X)), meet(Y, complement(X)))
% 31.72/4.59 = { by lemma 41 }
% 31.72/4.59 meet(Y, complement(X))
% 31.72/4.59
% 31.72/4.59 Lemma 47: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 31.72/4.59 Proof:
% 31.72/4.59 complement(join(complement(X), Y))
% 31.72/4.59 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 31.72/4.59 complement(join(Y, complement(X)))
% 31.72/4.59 = { by lemma 46 }
% 31.72/4.59 meet(X, complement(Y))
% 31.72/4.59
% 31.72/4.59 Lemma 48: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 31.72/4.59 Proof:
% 31.72/4.59 complement(meet(complement(X), Y))
% 31.72/4.59 = { by lemma 32 }
% 31.72/4.59 complement(meet(Y, complement(X)))
% 31.72/4.59 = { by lemma 45 }
% 31.72/4.59 join(X, complement(Y))
% 31.72/4.59
% 31.72/4.59 Lemma 49: meet(X, meet(Y, complement(X))) = zero.
% 31.72/4.59 Proof:
% 31.72/4.59 meet(X, meet(Y, complement(X)))
% 31.72/4.59 = { by lemma 32 }
% 31.72/4.59 meet(X, meet(complement(X), Y))
% 31.72/4.59 = { by axiom 10 (maddux4_definiton_of_meet) }
% 31.72/4.59 complement(join(complement(X), complement(meet(complement(X), Y))))
% 31.72/4.59 = { by lemma 32 }
% 31.72/4.59 complement(join(complement(X), complement(meet(Y, complement(X)))))
% 31.72/4.59 = { by lemma 44 R->L }
% 31.72/4.59 complement(join(complement(X), join(complement(Y), complement(complement(X)))))
% 31.72/4.59 = { by lemma 16 }
% 31.72/4.59 complement(join(complement(Y), top))
% 31.72/4.59 = { by lemma 24 }
% 31.72/4.59 complement(top)
% 31.72/4.59 = { by lemma 15 }
% 31.72/4.59 zero
% 31.72/4.59
% 31.72/4.59 Lemma 50: meet(X, join(Y, X)) = X.
% 31.72/4.59 Proof:
% 31.72/4.59 meet(X, join(Y, X))
% 31.72/4.59 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 31.72/4.59 meet(X, join(X, Y))
% 31.72/4.59 = { by lemma 41 R->L }
% 31.72/4.59 meet(X, join(X, meet(Y, Y)))
% 31.72/4.59 = { by lemma 31 R->L }
% 31.72/4.59 meet(X, join(X, complement(complement(Y))))
% 31.72/4.59 = { by lemma 45 R->L }
% 31.72/4.59 meet(X, complement(meet(complement(Y), complement(X))))
% 31.72/4.59 = { by lemma 44 R->L }
% 31.72/4.59 meet(X, join(complement(complement(Y)), complement(complement(X))))
% 31.72/4.59 = { by lemma 43 R->L }
% 31.72/4.59 complement(join(complement(X), meet(complement(Y), complement(X))))
% 31.72/4.59 = { by lemma 37 R->L }
% 31.72/4.59 join(zero, complement(join(complement(X), meet(complement(Y), complement(X)))))
% 31.72/4.59 = { by lemma 49 R->L }
% 31.72/4.59 join(meet(X, meet(complement(Y), complement(X))), complement(join(complement(X), meet(complement(Y), complement(X)))))
% 31.72/4.59 = { by lemma 29 }
% 31.72/4.59 X
% 31.72/4.59
% 31.72/4.59 Lemma 51: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 31.72/4.59 Proof:
% 31.72/4.59 meet(complement(X), complement(Y))
% 31.72/4.59 = { by lemma 32 }
% 31.72/4.59 meet(complement(Y), complement(X))
% 31.72/4.59 = { by lemma 46 R->L }
% 31.72/4.59 complement(join(X, complement(complement(Y))))
% 31.72/4.59 = { by lemma 31 }
% 31.72/4.59 complement(join(X, meet(Y, Y)))
% 31.72/4.59 = { by lemma 41 }
% 31.72/4.59 complement(join(X, Y))
% 31.72/4.59
% 31.72/4.59 Lemma 52: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 31.72/4.59 Proof:
% 31.72/4.59 converse(composition(X, converse(Y)))
% 31.72/4.59 = { by axiom 8 (converse_multiplicativity) }
% 31.72/4.59 composition(converse(converse(Y)), converse(X))
% 31.72/4.59 = { by axiom 1 (converse_idempotence) }
% 31.72/4.59 composition(Y, converse(X))
% 31.72/4.59
% 31.72/4.59 Lemma 53: join(composition(X, complement(Y)), composition(X, Y)) = converse(composition(top, converse(X))).
% 31.72/4.59 Proof:
% 31.72/4.59 join(composition(X, complement(Y)), composition(X, Y))
% 31.72/4.59 = { by axiom 1 (converse_idempotence) R->L }
% 31.72/4.59 converse(converse(join(composition(X, complement(Y)), composition(X, Y))))
% 31.72/4.59 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 31.72/4.59 converse(converse(join(composition(X, Y), composition(X, complement(Y)))))
% 31.72/4.59 = { by axiom 1 (converse_idempotence) R->L }
% 31.72/4.59 converse(converse(join(composition(X, Y), composition(X, complement(converse(converse(Y)))))))
% 31.72/4.59 = { by axiom 1 (converse_idempotence) R->L }
% 31.72/4.59 converse(converse(join(composition(X, Y), composition(converse(converse(X)), complement(converse(converse(Y)))))))
% 31.72/4.59 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 31.72/4.59 converse(converse(join(composition(converse(converse(X)), complement(converse(converse(Y)))), composition(X, Y))))
% 31.72/4.59 = { by axiom 6 (converse_additivity) }
% 31.72/4.59 converse(join(converse(composition(converse(converse(X)), complement(converse(converse(Y))))), converse(composition(X, Y))))
% 31.72/4.59 = { by lemma 17 }
% 31.72/4.59 converse(join(composition(converse(complement(converse(converse(Y)))), converse(X)), converse(composition(X, Y))))
% 31.72/4.59 = { by axiom 2 (maddux1_join_commutativity) }
% 31.72/4.59 converse(join(converse(composition(X, Y)), composition(converse(complement(converse(converse(Y)))), converse(X))))
% 31.72/4.59 = { by axiom 8 (converse_multiplicativity) }
% 31.72/4.59 converse(join(composition(converse(Y), converse(X)), composition(converse(complement(converse(converse(Y)))), converse(X))))
% 31.72/4.59 = { by axiom 12 (composition_distributivity) R->L }
% 31.72/4.59 converse(composition(join(converse(Y), converse(complement(converse(converse(Y))))), converse(X)))
% 31.72/4.59 = { by lemma 26 R->L }
% 31.72/4.59 converse(composition(converse(join(converse(converse(Y)), complement(converse(converse(Y))))), converse(X)))
% 31.72/4.59 = { by axiom 5 (def_top) R->L }
% 31.72/4.59 converse(composition(converse(top), converse(X)))
% 31.72/4.59 = { by lemma 28 }
% 31.72/4.59 converse(composition(top, converse(X)))
% 31.72/4.59
% 31.72/4.60 Lemma 54: composition(converse(X), complement(composition(X, top))) = zero.
% 31.72/4.60 Proof:
% 31.72/4.60 composition(converse(X), complement(composition(X, top)))
% 31.72/4.60 = { by lemma 40 R->L }
% 31.72/4.60 join(zero, composition(converse(X), complement(composition(X, top))))
% 31.72/4.60 = { by lemma 15 R->L }
% 31.72/4.60 join(complement(top), composition(converse(X), complement(composition(X, top))))
% 31.72/4.60 = { by lemma 20 }
% 31.72/4.60 complement(top)
% 31.72/4.60 = { by lemma 15 }
% 31.72/4.60 zero
% 31.72/4.60
% 31.72/4.60 Lemma 55: join(composition(X, Y), composition(X, Z)) = composition(X, join(Y, Z)).
% 31.72/4.60 Proof:
% 31.72/4.60 join(composition(X, Y), composition(X, Z))
% 31.72/4.60 = { by axiom 1 (converse_idempotence) R->L }
% 31.72/4.60 join(composition(X, Y), composition(X, converse(converse(Z))))
% 31.72/4.60 = { by axiom 1 (converse_idempotence) R->L }
% 31.72/4.60 converse(converse(join(composition(X, Y), composition(X, converse(converse(Z))))))
% 31.72/4.60 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 31.72/4.60 converse(converse(join(composition(X, converse(converse(Z))), composition(X, Y))))
% 31.72/4.60 = { by axiom 6 (converse_additivity) }
% 31.72/4.60 converse(join(converse(composition(X, converse(converse(Z)))), converse(composition(X, Y))))
% 31.72/4.60 = { by lemma 52 }
% 31.72/4.60 converse(join(composition(converse(Z), converse(X)), converse(composition(X, Y))))
% 31.72/4.60 = { by axiom 2 (maddux1_join_commutativity) }
% 31.72/4.60 converse(join(converse(composition(X, Y)), composition(converse(Z), converse(X))))
% 31.72/4.60 = { by axiom 8 (converse_multiplicativity) }
% 31.72/4.60 converse(join(composition(converse(Y), converse(X)), composition(converse(Z), converse(X))))
% 31.72/4.60 = { by axiom 12 (composition_distributivity) R->L }
% 31.72/4.60 converse(composition(join(converse(Y), converse(Z)), converse(X)))
% 31.72/4.60 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 31.72/4.60 converse(composition(join(converse(Z), converse(Y)), converse(X)))
% 31.72/4.60 = { by lemma 25 R->L }
% 31.72/4.60 converse(composition(converse(join(Y, converse(converse(Z)))), converse(X)))
% 31.72/4.60 = { by axiom 8 (converse_multiplicativity) R->L }
% 31.72/4.60 converse(converse(composition(X, join(Y, converse(converse(Z))))))
% 31.72/4.60 = { by axiom 1 (converse_idempotence) }
% 31.72/4.60 composition(X, join(Y, converse(converse(Z))))
% 31.72/4.60 = { by axiom 1 (converse_idempotence) }
% 31.72/4.60 composition(X, join(Y, Z))
% 31.72/4.60
% 31.72/4.60 Lemma 56: composition(composition(converse(X), X), top) = composition(converse(X), top).
% 31.72/4.60 Proof:
% 31.72/4.60 composition(composition(converse(X), X), top)
% 31.72/4.60 = { by lemma 28 R->L }
% 31.72/4.60 composition(composition(converse(X), X), converse(top))
% 31.72/4.60 = { by lemma 17 R->L }
% 31.72/4.60 composition(converse(composition(converse(X), X)), converse(top))
% 31.72/4.60 = { by axiom 8 (converse_multiplicativity) R->L }
% 31.72/4.60 converse(composition(top, composition(converse(X), X)))
% 31.72/4.60 = { by axiom 9 (composition_associativity) }
% 31.72/4.60 converse(composition(composition(top, converse(X)), X))
% 31.72/4.60 = { by axiom 8 (converse_multiplicativity) }
% 31.72/4.60 composition(converse(X), converse(composition(top, converse(X))))
% 31.72/4.60 = { by lemma 53 R->L }
% 31.72/4.60 composition(converse(X), join(composition(X, complement(Y)), composition(X, Y)))
% 31.72/4.60 = { by lemma 40 R->L }
% 31.72/4.60 join(zero, composition(converse(X), join(composition(X, complement(Y)), composition(X, Y))))
% 31.72/4.60 = { by lemma 54 R->L }
% 31.72/4.60 join(composition(converse(X), complement(composition(X, top))), composition(converse(X), join(composition(X, complement(Y)), composition(X, Y))))
% 31.72/4.60 = { by lemma 55 }
% 31.72/4.60 composition(converse(X), join(complement(composition(X, top)), join(composition(X, complement(Y)), composition(X, Y))))
% 31.72/4.60 = { by lemma 28 R->L }
% 31.72/4.60 composition(converse(X), join(complement(composition(X, converse(top))), join(composition(X, complement(Y)), composition(X, Y))))
% 31.72/4.60 = { by lemma 52 R->L }
% 31.72/4.60 composition(converse(X), join(complement(converse(composition(top, converse(X)))), join(composition(X, complement(Y)), composition(X, Y))))
% 31.72/4.60 = { by lemma 53 R->L }
% 31.72/4.60 composition(converse(X), join(complement(join(composition(X, complement(Y)), composition(X, Y))), join(composition(X, complement(Y)), composition(X, Y))))
% 31.72/4.60 = { by axiom 2 (maddux1_join_commutativity) }
% 31.72/4.60 composition(converse(X), join(join(composition(X, complement(Y)), composition(X, Y)), complement(join(composition(X, complement(Y)), composition(X, Y)))))
% 31.72/4.60 = { by axiom 5 (def_top) R->L }
% 31.72/4.60 composition(converse(X), top)
% 31.72/4.60
% 31.72/4.60 Goal 1 (goals_1): meet(composition(x0, x1), composition(x0, complement(x1))) = zero.
% 31.72/4.60 Proof:
% 31.72/4.60 meet(composition(x0, x1), composition(x0, complement(x1)))
% 31.72/4.60 = { by lemma 32 R->L }
% 31.72/4.60 meet(composition(x0, complement(x1)), composition(x0, x1))
% 31.72/4.60 = { by lemma 38 R->L }
% 31.72/4.60 meet(composition(x0, complement(x1)), complement(complement(composition(x0, x1))))
% 31.72/4.60 = { by lemma 20 R->L }
% 31.72/4.60 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(converse(converse(x0)), complement(composition(converse(x0), composition(x0, x1)))))))
% 31.72/4.60 = { by axiom 9 (composition_associativity) }
% 31.72/4.60 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(converse(converse(x0)), complement(composition(composition(converse(x0), x0), x1))))))
% 31.72/4.60 = { by axiom 1 (converse_idempotence) }
% 31.72/4.60 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(composition(composition(converse(x0), x0), x1))))))
% 31.72/4.60 = { by lemma 35 R->L }
% 31.72/4.60 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, join(meet(complement(composition(composition(converse(x0), x0), x1)), x1), meet(complement(composition(composition(converse(x0), x0), x1)), complement(x1)))))))
% 31.72/4.60 = { by lemma 32 R->L }
% 31.72/4.60 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, join(meet(x1, complement(composition(composition(converse(x0), x0), x1))), meet(complement(composition(composition(converse(x0), x0), x1)), complement(x1)))))))
% 31.72/4.60 = { by lemma 32 R->L }
% 31.72/4.60 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, join(meet(x1, complement(composition(composition(converse(x0), x0), x1))), meet(complement(x1), complement(composition(composition(converse(x0), x0), x1))))))))
% 31.72/4.60 = { by lemma 47 R->L }
% 31.72/4.60 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, join(complement(join(complement(x1), composition(composition(converse(x0), x0), x1))), meet(complement(x1), complement(composition(composition(converse(x0), x0), x1))))))))
% 31.72/4.60 = { by axiom 9 (composition_associativity) R->L }
% 31.72/4.60 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, join(complement(join(complement(x1), composition(converse(x0), composition(x0, x1)))), meet(complement(x1), complement(composition(composition(converse(x0), x0), x1))))))))
% 31.72/4.61 = { by lemma 20 R->L }
% 31.72/4.61 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, join(complement(join(join(complement(x1), composition(converse(x0), complement(composition(x0, x1)))), composition(converse(x0), composition(x0, x1)))), meet(complement(x1), complement(composition(composition(converse(x0), x0), x1))))))))
% 31.72/4.61 = { by axiom 7 (maddux2_join_associativity) R->L }
% 31.72/4.61 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, join(complement(join(complement(x1), join(composition(converse(x0), complement(composition(x0, x1))), composition(converse(x0), composition(x0, x1))))), meet(complement(x1), complement(composition(composition(converse(x0), x0), x1))))))))
% 31.72/4.61 = { by axiom 2 (maddux1_join_commutativity) }
% 31.72/4.61 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, join(complement(join(complement(x1), join(composition(converse(x0), composition(x0, x1)), composition(converse(x0), complement(composition(x0, x1)))))), meet(complement(x1), complement(composition(composition(converse(x0), x0), x1))))))))
% 31.72/4.61 = { by lemma 55 }
% 31.72/4.61 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, join(complement(join(complement(x1), composition(converse(x0), join(composition(x0, x1), complement(composition(x0, x1)))))), meet(complement(x1), complement(composition(composition(converse(x0), x0), x1))))))))
% 31.72/4.61 = { by axiom 5 (def_top) R->L }
% 31.72/4.61 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, join(complement(join(complement(x1), composition(converse(x0), top))), meet(complement(x1), complement(composition(composition(converse(x0), x0), x1))))))))
% 31.72/4.61 = { by lemma 56 R->L }
% 31.72/4.61 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, join(complement(join(complement(x1), composition(composition(converse(x0), x0), top))), meet(complement(x1), complement(composition(composition(converse(x0), x0), x1))))))))
% 31.72/4.61 = { by lemma 47 }
% 31.72/4.61 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, join(meet(x1, complement(composition(composition(converse(x0), x0), top))), meet(complement(x1), complement(composition(composition(converse(x0), x0), x1))))))))
% 31.72/4.61 = { by lemma 51 }
% 31.72/4.61 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, join(meet(x1, complement(composition(composition(converse(x0), x0), top))), complement(join(x1, composition(composition(converse(x0), x0), x1))))))))
% 31.72/4.61 = { by lemma 46 R->L }
% 31.72/4.61 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, join(complement(join(composition(composition(converse(x0), x0), top), complement(x1))), complement(join(x1, composition(composition(converse(x0), x0), x1))))))))
% 31.72/4.61 = { by lemma 44 }
% 31.72/4.61 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(meet(join(composition(composition(converse(x0), x0), top), complement(x1)), join(x1, composition(composition(converse(x0), x0), x1))))))))
% 31.72/4.61 = { by lemma 32 R->L }
% 31.72/4.61 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(meet(join(x1, composition(composition(converse(x0), x0), x1)), join(composition(composition(converse(x0), x0), top), complement(x1))))))))
% 31.72/4.61 = { by lemma 19 R->L }
% 31.72/4.61 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(meet(join(composition(join(composition(converse(x0), x0), one), x1), composition(composition(converse(x0), x0), x1)), join(composition(composition(converse(x0), x0), top), complement(x1))))))))
% 31.72/4.61 = { by axiom 12 (composition_distributivity) R->L }
% 31.72/4.61 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(meet(composition(join(join(composition(converse(x0), x0), one), composition(converse(x0), x0)), x1), join(composition(composition(converse(x0), x0), top), complement(x1))))))))
% 31.72/4.61 = { by axiom 11 (goals) }
% 31.72/4.61 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(meet(composition(join(one, composition(converse(x0), x0)), x1), join(composition(composition(converse(x0), x0), top), complement(x1))))))))
% 31.72/4.61 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 31.72/4.61 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(meet(composition(join(composition(converse(x0), x0), one), x1), join(composition(composition(converse(x0), x0), top), complement(x1))))))))
% 31.72/4.61 = { by lemma 19 }
% 31.72/4.61 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(meet(x1, join(composition(composition(converse(x0), x0), top), complement(x1))))))))
% 31.72/4.61 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 31.72/4.61 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(meet(x1, join(complement(x1), composition(composition(converse(x0), x0), top))))))))
% 31.72/4.61 = { by lemma 50 R->L }
% 31.72/4.61 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(meet(meet(x1, join(complement(x1), composition(composition(converse(x0), x0), top))), join(composition(composition(converse(x0), x0), top), meet(x1, join(complement(x1), composition(composition(converse(x0), x0), top))))))))))
% 31.72/4.61 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 31.72/4.61 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(meet(meet(x1, join(complement(x1), composition(composition(converse(x0), x0), top))), join(composition(composition(converse(x0), x0), top), meet(x1, join(composition(composition(converse(x0), x0), top), complement(x1))))))))))
% 31.72/4.61 = { by lemma 48 R->L }
% 31.72/4.61 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(meet(meet(x1, join(complement(x1), composition(composition(converse(x0), x0), top))), join(composition(composition(converse(x0), x0), top), meet(x1, complement(meet(complement(composition(composition(converse(x0), x0), top)), x1))))))))))
% 31.72/4.61 = { by lemma 47 R->L }
% 31.72/4.61 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(meet(meet(x1, join(complement(x1), composition(composition(converse(x0), x0), top))), join(composition(composition(converse(x0), x0), top), complement(join(complement(x1), meet(complement(composition(composition(converse(x0), x0), top)), x1))))))))))
% 31.72/4.61 = { by lemma 48 R->L }
% 31.72/4.61 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(meet(meet(x1, join(complement(x1), composition(composition(converse(x0), x0), top))), complement(meet(complement(composition(composition(converse(x0), x0), top)), join(complement(x1), meet(complement(composition(composition(converse(x0), x0), top)), x1))))))))))
% 31.72/4.62 = { by lemma 39 R->L }
% 31.72/4.62 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(meet(meet(x1, join(complement(x1), composition(composition(converse(x0), x0), top))), complement(join(meet(complement(composition(composition(converse(x0), x0), top)), join(complement(x1), meet(complement(composition(composition(converse(x0), x0), top)), x1))), zero))))))))
% 31.72/4.62 = { by lemma 15 R->L }
% 31.72/4.62 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(meet(meet(x1, join(complement(x1), composition(composition(converse(x0), x0), top))), complement(join(meet(complement(composition(composition(converse(x0), x0), top)), join(complement(x1), meet(complement(composition(composition(converse(x0), x0), top)), x1))), complement(top)))))))))
% 31.72/4.62 = { by axiom 5 (def_top) }
% 31.72/4.62 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(meet(meet(x1, join(complement(x1), composition(composition(converse(x0), x0), top))), complement(join(meet(complement(composition(composition(converse(x0), x0), top)), join(complement(x1), meet(complement(composition(composition(converse(x0), x0), top)), x1))), complement(join(join(complement(complement(composition(composition(converse(x0), x0), top))), complement(x1)), complement(join(complement(complement(composition(composition(converse(x0), x0), top))), complement(x1)))))))))))))
% 31.72/4.62 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 31.72/4.62 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(meet(meet(x1, join(complement(x1), composition(composition(converse(x0), x0), top))), complement(join(meet(complement(composition(composition(converse(x0), x0), top)), join(complement(x1), meet(complement(composition(composition(converse(x0), x0), top)), x1))), complement(join(join(complement(complement(composition(composition(converse(x0), x0), top))), complement(x1)), meet(complement(composition(composition(converse(x0), x0), top)), x1)))))))))))
% 31.72/4.62 = { by axiom 7 (maddux2_join_associativity) R->L }
% 31.72/4.62 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(meet(meet(x1, join(complement(x1), composition(composition(converse(x0), x0), top))), complement(join(meet(complement(composition(composition(converse(x0), x0), top)), join(complement(x1), meet(complement(composition(composition(converse(x0), x0), top)), x1))), complement(join(complement(complement(composition(composition(converse(x0), x0), top))), join(complement(x1), meet(complement(composition(composition(converse(x0), x0), top)), x1))))))))))))
% 31.72/4.62 = { by lemma 29 }
% 31.72/4.62 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(meet(meet(x1, join(complement(x1), composition(composition(converse(x0), x0), top))), complement(complement(composition(composition(converse(x0), x0), top)))))))))
% 31.72/4.62 = { by lemma 38 }
% 31.72/4.62 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(meet(meet(x1, join(complement(x1), composition(composition(converse(x0), x0), top))), composition(composition(converse(x0), x0), top)))))))
% 31.72/4.62 = { by lemma 32 }
% 31.72/4.62 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(meet(composition(composition(converse(x0), x0), top), meet(x1, join(complement(x1), composition(composition(converse(x0), x0), top)))))))))
% 31.72/4.62 = { by lemma 41 R->L }
% 31.72/4.62 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(meet(meet(composition(composition(converse(x0), x0), top), meet(x1, join(complement(x1), composition(composition(converse(x0), x0), top)))), meet(composition(composition(converse(x0), x0), top), meet(x1, join(complement(x1), composition(composition(converse(x0), x0), top))))))))))
% 31.72/4.62 = { by lemma 31 R->L }
% 31.72/4.62 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(complement(complement(meet(composition(composition(converse(x0), x0), top), meet(x1, join(complement(x1), composition(composition(converse(x0), x0), top)))))))))))
% 31.72/4.62 = { by lemma 32 }
% 31.72/4.62 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(complement(complement(meet(composition(composition(converse(x0), x0), top), meet(join(complement(x1), composition(composition(converse(x0), x0), top)), x1)))))))))
% 31.72/4.62 = { by lemma 44 R->L }
% 31.72/4.62 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(complement(join(complement(composition(composition(converse(x0), x0), top)), complement(meet(join(complement(x1), composition(composition(converse(x0), x0), top)), x1)))))))))
% 31.72/4.62 = { by lemma 44 R->L }
% 31.72/4.62 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(complement(join(complement(composition(composition(converse(x0), x0), top)), join(complement(join(complement(x1), composition(composition(converse(x0), x0), top))), complement(x1)))))))))
% 31.72/4.62 = { by axiom 7 (maddux2_join_associativity) }
% 31.72/4.62 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(complement(join(join(complement(composition(composition(converse(x0), x0), top)), complement(join(complement(x1), composition(composition(converse(x0), x0), top)))), complement(x1))))))))
% 31.72/4.62 = { by lemma 45 R->L }
% 31.72/4.62 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(complement(complement(meet(x1, complement(join(complement(composition(composition(converse(x0), x0), top)), complement(join(complement(x1), composition(composition(converse(x0), x0), top)))))))))))))
% 31.72/4.62 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 31.72/4.62 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(complement(complement(meet(x1, meet(composition(composition(converse(x0), x0), top), join(complement(x1), composition(composition(converse(x0), x0), top)))))))))))
% 31.72/4.62 = { by lemma 32 R->L }
% 31.72/4.62 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(complement(complement(meet(x1, meet(join(complement(x1), composition(composition(converse(x0), x0), top)), composition(composition(converse(x0), x0), top))))))))))
% 31.72/4.62 = { by lemma 38 }
% 31.72/4.62 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(meet(x1, meet(join(complement(x1), composition(composition(converse(x0), x0), top)), composition(composition(converse(x0), x0), top))))))))
% 31.72/4.62 = { by lemma 32 R->L }
% 31.72/4.63 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(meet(x1, meet(composition(composition(converse(x0), x0), top), join(complement(x1), composition(composition(converse(x0), x0), top)))))))))
% 31.72/4.63 = { by lemma 50 }
% 31.72/4.63 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(meet(x1, composition(composition(converse(x0), x0), top)))))))
% 31.72/4.63 = { by lemma 44 R->L }
% 31.72/4.63 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, join(complement(x1), complement(composition(composition(converse(x0), x0), top)))))))
% 31.72/4.63 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 31.72/4.63 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, join(complement(composition(composition(converse(x0), x0), top)), complement(x1))))))
% 31.72/4.63 = { by lemma 56 }
% 31.72/4.63 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, join(complement(composition(converse(x0), top)), complement(x1))))))
% 31.72/4.63 = { by lemma 55 R->L }
% 31.72/4.63 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), join(composition(x0, complement(composition(converse(x0), top))), composition(x0, complement(x1))))))
% 31.72/4.63 = { by axiom 1 (converse_idempotence) R->L }
% 31.72/4.63 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), join(composition(converse(converse(x0)), complement(composition(converse(x0), top))), composition(x0, complement(x1))))))
% 31.72/4.63 = { by lemma 54 }
% 31.72/4.63 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), join(zero, composition(x0, complement(x1))))))
% 31.72/4.63 = { by lemma 40 }
% 31.72/4.63 meet(composition(x0, complement(x1)), complement(join(complement(composition(x0, x1)), composition(x0, complement(x1)))))
% 31.72/4.63 = { by axiom 2 (maddux1_join_commutativity) }
% 31.72/4.63 meet(composition(x0, complement(x1)), complement(join(composition(x0, complement(x1)), complement(composition(x0, x1)))))
% 31.72/4.63 = { by lemma 51 R->L }
% 31.72/4.63 meet(composition(x0, complement(x1)), meet(complement(composition(x0, complement(x1))), complement(complement(composition(x0, x1)))))
% 31.72/4.63 = { by lemma 32 }
% 31.72/4.63 meet(composition(x0, complement(x1)), meet(complement(complement(composition(x0, x1))), complement(composition(x0, complement(x1)))))
% 31.72/4.63 = { by lemma 49 }
% 31.72/4.63 zero
% 31.72/4.63 % SZS output end Proof
% 31.72/4.63
% 31.72/4.63 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------