TSTP Solution File: REL036+1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL036+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 : n005.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 13:44:22 EDT 2023
% Result : Theorem 15.99s 2.45s
% Output : Proof 15.99s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.13 % Problem : REL036+1 : TPTP v8.1.2. Released v4.0.0.
% 0.12/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.35 % Computer : n005.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % 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 19:49:38 EDT 2023
% 0.13/0.35 % CPUTime :
% 15.99/2.45 Command-line arguments: --flatten
% 15.99/2.45
% 15.99/2.45 % SZS status Theorem
% 15.99/2.45
% 15.99/2.50 % SZS output start Proof
% 15.99/2.50 Axiom 1 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 15.99/2.50 Axiom 2 (converse_idempotence): converse(converse(X)) = X.
% 15.99/2.50 Axiom 3 (composition_identity): composition(X, one) = X.
% 15.99/2.50 Axiom 4 (goals): composition(x0, top) = x0.
% 15.99/2.50 Axiom 5 (def_zero): zero = meet(X, complement(X)).
% 15.99/2.50 Axiom 6 (def_top): top = join(X, complement(X)).
% 15.99/2.50 Axiom 7 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 15.99/2.50 Axiom 8 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 15.99/2.50 Axiom 9 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 15.99/2.50 Axiom 10 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 15.99/2.50 Axiom 11 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 15.99/2.50 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 15.99/2.50 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 15.99/2.50 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 15.99/2.50
% 15.99/2.50 Lemma 15: complement(top) = zero.
% 15.99/2.50 Proof:
% 15.99/2.50 complement(top)
% 15.99/2.50 = { by axiom 6 (def_top) }
% 15.99/2.50 complement(join(complement(X), complement(complement(X))))
% 15.99/2.50 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 15.99/2.50 meet(X, complement(X))
% 15.99/2.50 = { by axiom 5 (def_zero) R->L }
% 15.99/2.50 zero
% 15.99/2.50
% 15.99/2.50 Lemma 16: join(X, join(Y, complement(X))) = join(Y, top).
% 15.99/2.50 Proof:
% 15.99/2.50 join(X, join(Y, complement(X)))
% 15.99/2.50 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 15.99/2.50 join(X, join(complement(X), Y))
% 15.99/2.50 = { by axiom 8 (maddux2_join_associativity) }
% 15.99/2.50 join(join(X, complement(X)), Y)
% 15.99/2.50 = { by axiom 6 (def_top) R->L }
% 15.99/2.50 join(top, Y)
% 15.99/2.50 = { by axiom 1 (maddux1_join_commutativity) }
% 15.99/2.50 join(Y, top)
% 15.99/2.50
% 15.99/2.50 Lemma 17: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 15.99/2.50 Proof:
% 15.99/2.50 converse(composition(converse(X), Y))
% 15.99/2.50 = { by axiom 9 (converse_multiplicativity) }
% 15.99/2.50 composition(converse(Y), converse(converse(X)))
% 15.99/2.50 = { by axiom 2 (converse_idempotence) }
% 15.99/2.50 composition(converse(Y), X)
% 15.99/2.50
% 15.99/2.50 Lemma 18: composition(converse(one), X) = X.
% 15.99/2.50 Proof:
% 15.99/2.50 composition(converse(one), X)
% 15.99/2.50 = { by lemma 17 R->L }
% 15.99/2.50 converse(composition(converse(X), one))
% 15.99/2.50 = { by axiom 3 (composition_identity) }
% 15.99/2.50 converse(converse(X))
% 15.99/2.50 = { by axiom 2 (converse_idempotence) }
% 15.99/2.50 X
% 15.99/2.50
% 15.99/2.50 Lemma 19: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 15.99/2.50 Proof:
% 15.99/2.50 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 15.99/2.50 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 15.99/2.50 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 15.99/2.50 = { by axiom 13 (converse_cancellativity) }
% 15.99/2.50 complement(X)
% 15.99/2.50
% 15.99/2.50 Lemma 20: join(complement(X), complement(X)) = complement(X).
% 15.99/2.50 Proof:
% 15.99/2.50 join(complement(X), complement(X))
% 15.99/2.50 = { by lemma 18 R->L }
% 15.99/2.50 join(complement(X), composition(converse(one), complement(X)))
% 15.99/2.50 = { by lemma 18 R->L }
% 15.99/2.50 join(complement(X), composition(converse(one), complement(composition(converse(one), X))))
% 15.99/2.50 = { by axiom 3 (composition_identity) R->L }
% 15.99/2.50 join(complement(X), composition(converse(one), complement(composition(composition(converse(one), one), X))))
% 15.99/2.50 = { by axiom 10 (composition_associativity) R->L }
% 15.99/2.50 join(complement(X), composition(converse(one), complement(composition(converse(one), composition(one, X)))))
% 15.99/2.50 = { by lemma 18 }
% 15.99/2.50 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 15.99/2.50 = { by lemma 19 }
% 15.99/2.50 complement(X)
% 15.99/2.50
% 15.99/2.50 Lemma 21: join(top, complement(X)) = top.
% 15.99/2.50 Proof:
% 15.99/2.50 join(top, complement(X))
% 15.99/2.50 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 15.99/2.50 join(complement(X), top)
% 15.99/2.50 = { by lemma 16 R->L }
% 15.99/2.50 join(X, join(complement(X), complement(X)))
% 15.99/2.50 = { by lemma 20 }
% 15.99/2.50 join(X, complement(X))
% 15.99/2.50 = { by axiom 6 (def_top) R->L }
% 15.99/2.50 top
% 15.99/2.50
% 15.99/2.50 Lemma 22: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 15.99/2.50 Proof:
% 15.99/2.50 join(meet(X, Y), complement(join(complement(X), Y)))
% 15.99/2.50 = { by axiom 11 (maddux4_definiton_of_meet) }
% 15.99/2.50 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 15.99/2.50 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L }
% 15.99/2.50 X
% 15.99/2.50
% 15.99/2.50 Lemma 23: join(zero, meet(X, X)) = X.
% 15.99/2.50 Proof:
% 15.99/2.50 join(zero, meet(X, X))
% 15.99/2.50 = { by axiom 11 (maddux4_definiton_of_meet) }
% 15.99/2.50 join(zero, complement(join(complement(X), complement(X))))
% 15.99/2.50 = { by axiom 5 (def_zero) }
% 15.99/2.50 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 15.99/2.50 = { by lemma 22 }
% 15.99/2.50 X
% 15.99/2.50
% 15.99/2.50 Lemma 24: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 15.99/2.50 Proof:
% 15.99/2.50 join(zero, join(X, complement(complement(Y))))
% 15.99/2.50 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 15.99/2.50 join(zero, join(complement(complement(Y)), X))
% 15.99/2.50 = { by lemma 20 R->L }
% 15.99/2.50 join(zero, join(complement(join(complement(Y), complement(Y))), X))
% 15.99/2.50 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 15.99/2.50 join(zero, join(meet(Y, Y), X))
% 15.99/2.50 = { by axiom 8 (maddux2_join_associativity) }
% 15.99/2.50 join(join(zero, meet(Y, Y)), X)
% 15.99/2.50 = { by lemma 23 }
% 15.99/2.50 join(Y, X)
% 15.99/2.50 = { by axiom 1 (maddux1_join_commutativity) }
% 15.99/2.50 join(X, Y)
% 15.99/2.50
% 15.99/2.50 Lemma 25: join(zero, complement(complement(X))) = X.
% 15.99/2.50 Proof:
% 15.99/2.50 join(zero, complement(complement(X)))
% 15.99/2.50 = { by axiom 5 (def_zero) }
% 15.99/2.50 join(meet(X, complement(X)), complement(complement(X)))
% 15.99/2.50 = { by lemma 20 R->L }
% 15.99/2.50 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 15.99/2.50 = { by lemma 22 }
% 15.99/2.50 X
% 15.99/2.50
% 15.99/2.50 Lemma 26: join(zero, complement(X)) = complement(X).
% 15.99/2.50 Proof:
% 15.99/2.50 join(zero, complement(X))
% 15.99/2.50 = { by lemma 25 R->L }
% 15.99/2.50 join(zero, join(zero, complement(complement(complement(X)))))
% 15.99/2.50 = { by lemma 20 R->L }
% 15.99/2.50 join(zero, join(zero, join(complement(complement(complement(X))), complement(complement(complement(X))))))
% 15.99/2.50 = { by lemma 24 }
% 15.99/2.50 join(zero, join(complement(complement(complement(X))), complement(X)))
% 15.99/2.50 = { by axiom 1 (maddux1_join_commutativity) }
% 15.99/2.50 join(zero, join(complement(X), complement(complement(complement(X)))))
% 15.99/2.50 = { by lemma 24 }
% 15.99/2.50 join(complement(X), complement(X))
% 15.99/2.50 = { by lemma 20 }
% 15.99/2.50 complement(X)
% 15.99/2.50
% 15.99/2.50 Lemma 27: join(X, zero) = X.
% 15.99/2.50 Proof:
% 15.99/2.50 join(X, zero)
% 15.99/2.50 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 15.99/2.50 join(zero, X)
% 15.99/2.50 = { by lemma 24 R->L }
% 15.99/2.50 join(zero, join(zero, complement(complement(X))))
% 15.99/2.50 = { by lemma 26 }
% 15.99/2.50 join(zero, complement(complement(X)))
% 15.99/2.51 = { by lemma 25 }
% 15.99/2.51 X
% 15.99/2.51
% 15.99/2.51 Lemma 28: join(zero, X) = X.
% 15.99/2.51 Proof:
% 15.99/2.51 join(zero, X)
% 15.99/2.51 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 15.99/2.51 join(X, zero)
% 15.99/2.51 = { by lemma 27 }
% 15.99/2.51 X
% 15.99/2.51
% 15.99/2.51 Lemma 29: join(X, top) = top.
% 15.99/2.51 Proof:
% 15.99/2.51 join(X, top)
% 15.99/2.51 = { by lemma 21 R->L }
% 15.99/2.51 join(X, join(top, complement(X)))
% 15.99/2.51 = { by lemma 16 }
% 15.99/2.51 join(top, top)
% 15.99/2.51 = { by lemma 16 R->L }
% 15.99/2.51 join(zero, join(top, complement(zero)))
% 15.99/2.51 = { by lemma 21 }
% 15.99/2.51 join(zero, top)
% 15.99/2.51 = { by lemma 28 }
% 15.99/2.51 top
% 15.99/2.51
% 15.99/2.51 Lemma 30: meet(Y, X) = meet(X, Y).
% 15.99/2.51 Proof:
% 15.99/2.51 meet(Y, X)
% 15.99/2.51 = { by axiom 11 (maddux4_definiton_of_meet) }
% 15.99/2.51 complement(join(complement(Y), complement(X)))
% 15.99/2.51 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 15.99/2.51 complement(join(complement(X), complement(Y)))
% 15.99/2.51 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 15.99/2.51 meet(X, Y)
% 15.99/2.51
% 15.99/2.51 Lemma 31: complement(join(zero, complement(X))) = meet(X, top).
% 15.99/2.51 Proof:
% 15.99/2.51 complement(join(zero, complement(X)))
% 15.99/2.51 = { by lemma 15 R->L }
% 15.99/2.51 complement(join(complement(top), complement(X)))
% 15.99/2.51 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 15.99/2.51 meet(top, X)
% 15.99/2.51 = { by lemma 30 R->L }
% 15.99/2.51 meet(X, top)
% 15.99/2.51
% 15.99/2.51 Lemma 32: join(X, complement(zero)) = top.
% 15.99/2.51 Proof:
% 15.99/2.51 join(X, complement(zero))
% 15.99/2.51 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 15.99/2.51 join(complement(zero), X)
% 15.99/2.51 = { by lemma 24 R->L }
% 15.99/2.51 join(zero, join(complement(zero), complement(complement(X))))
% 15.99/2.51 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 15.99/2.51 join(zero, join(complement(complement(X)), complement(zero)))
% 15.99/2.51 = { by lemma 16 }
% 15.99/2.51 join(complement(complement(X)), top)
% 15.99/2.51 = { by lemma 29 }
% 15.99/2.51 top
% 15.99/2.51
% 15.99/2.51 Lemma 33: meet(X, zero) = zero.
% 15.99/2.51 Proof:
% 15.99/2.51 meet(X, zero)
% 15.99/2.51 = { by axiom 11 (maddux4_definiton_of_meet) }
% 15.99/2.51 complement(join(complement(X), complement(zero)))
% 15.99/2.51 = { by lemma 32 }
% 15.99/2.51 complement(top)
% 15.99/2.51 = { by lemma 15 }
% 15.99/2.51 zero
% 15.99/2.51
% 15.99/2.51 Lemma 34: join(meet(X, Y), meet(X, complement(Y))) = X.
% 15.99/2.51 Proof:
% 15.99/2.51 join(meet(X, Y), meet(X, complement(Y)))
% 15.99/2.51 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 15.99/2.51 join(meet(X, complement(Y)), meet(X, Y))
% 15.99/2.51 = { by axiom 11 (maddux4_definiton_of_meet) }
% 15.99/2.51 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 15.99/2.51 = { by lemma 22 }
% 15.99/2.51 X
% 15.99/2.51
% 15.99/2.51 Lemma 35: meet(X, top) = X.
% 15.99/2.51 Proof:
% 15.99/2.51 meet(X, top)
% 15.99/2.51 = { by lemma 31 R->L }
% 15.99/2.51 complement(join(zero, complement(X)))
% 15.99/2.51 = { by lemma 26 R->L }
% 15.99/2.51 join(zero, complement(join(zero, complement(X))))
% 15.99/2.51 = { by lemma 31 }
% 15.99/2.51 join(zero, meet(X, top))
% 15.99/2.51 = { by lemma 32 R->L }
% 15.99/2.51 join(zero, meet(X, join(complement(zero), complement(zero))))
% 15.99/2.51 = { by lemma 20 }
% 15.99/2.51 join(zero, meet(X, complement(zero)))
% 15.99/2.51 = { by lemma 33 R->L }
% 15.99/2.51 join(meet(X, zero), meet(X, complement(zero)))
% 15.99/2.51 = { by lemma 34 }
% 15.99/2.51 X
% 15.99/2.51
% 15.99/2.51 Lemma 36: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 15.99/2.51 Proof:
% 15.99/2.51 meet(X, join(complement(Y), complement(Z)))
% 15.99/2.51 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 15.99/2.51 meet(X, join(complement(Z), complement(Y)))
% 15.99/2.51 = { by lemma 30 }
% 15.99/2.51 meet(join(complement(Z), complement(Y)), X)
% 15.99/2.51 = { by axiom 11 (maddux4_definiton_of_meet) }
% 15.99/2.51 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 15.99/2.51 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 15.99/2.51 complement(join(meet(Z, Y), complement(X)))
% 15.99/2.51 = { by axiom 1 (maddux1_join_commutativity) }
% 15.99/2.51 complement(join(complement(X), meet(Z, Y)))
% 15.99/2.51 = { by lemma 30 R->L }
% 15.99/2.51 complement(join(complement(X), meet(Y, Z)))
% 15.99/2.51
% 15.99/2.51 Lemma 37: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 15.99/2.51 Proof:
% 15.99/2.51 complement(join(X, complement(Y)))
% 15.99/2.51 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 15.99/2.51 complement(join(complement(Y), X))
% 15.99/2.51 = { by lemma 35 R->L }
% 15.99/2.51 complement(join(complement(Y), meet(X, top)))
% 15.99/2.51 = { by lemma 30 R->L }
% 15.99/2.51 complement(join(complement(Y), meet(top, X)))
% 15.99/2.51 = { by lemma 36 R->L }
% 15.99/2.51 meet(Y, join(complement(top), complement(X)))
% 15.99/2.51 = { by lemma 15 }
% 15.99/2.51 meet(Y, join(zero, complement(X)))
% 15.99/2.51 = { by lemma 26 }
% 15.99/2.51 meet(Y, complement(X))
% 15.99/2.51
% 15.99/2.51 Lemma 38: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 15.99/2.51 Proof:
% 15.99/2.51 complement(meet(X, complement(Y)))
% 15.99/2.51 = { by lemma 28 R->L }
% 15.99/2.51 complement(join(zero, meet(X, complement(Y))))
% 15.99/2.51 = { by lemma 37 R->L }
% 15.99/2.51 complement(join(zero, complement(join(Y, complement(X)))))
% 15.99/2.51 = { by lemma 31 }
% 15.99/2.51 meet(join(Y, complement(X)), top)
% 15.99/2.51 = { by lemma 35 }
% 15.99/2.51 join(Y, complement(X))
% 15.99/2.51
% 15.99/2.51 Lemma 39: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 15.99/2.51 Proof:
% 15.99/2.51 complement(join(complement(X), Y))
% 15.99/2.51 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 15.99/2.51 complement(join(Y, complement(X)))
% 15.99/2.51 = { by lemma 37 }
% 15.99/2.51 meet(X, complement(Y))
% 15.99/2.51
% 15.99/2.51 Lemma 40: join(X, complement(meet(X, Y))) = top.
% 15.99/2.51 Proof:
% 15.99/2.51 join(X, complement(meet(X, Y)))
% 15.99/2.51 = { by lemma 30 }
% 15.99/2.51 join(X, complement(meet(Y, X)))
% 15.99/2.51 = { by axiom 11 (maddux4_definiton_of_meet) }
% 15.99/2.51 join(X, complement(complement(join(complement(Y), complement(X)))))
% 15.99/2.51 = { by lemma 20 R->L }
% 15.99/2.51 join(X, complement(join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X))))))
% 15.99/2.51 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 15.99/2.51 join(X, complement(join(meet(Y, X), complement(join(complement(Y), complement(X))))))
% 15.99/2.51 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 15.99/2.51 join(X, complement(join(meet(Y, X), meet(Y, X))))
% 15.99/2.51 = { by lemma 30 R->L }
% 15.99/2.51 join(X, complement(join(meet(Y, X), meet(X, Y))))
% 15.99/2.51 = { by lemma 30 R->L }
% 15.99/2.51 join(X, complement(join(meet(X, Y), meet(X, Y))))
% 15.99/2.51 = { by axiom 11 (maddux4_definiton_of_meet) }
% 15.99/2.51 join(X, complement(join(complement(join(complement(X), complement(Y))), meet(X, Y))))
% 15.99/2.51 = { by lemma 26 R->L }
% 15.99/2.51 join(X, join(zero, complement(join(complement(join(complement(X), complement(Y))), meet(X, Y)))))
% 15.99/2.51 = { by lemma 36 R->L }
% 15.99/2.51 join(X, join(zero, meet(join(complement(X), complement(Y)), join(complement(X), complement(Y)))))
% 15.99/2.51 = { by lemma 23 }
% 15.99/2.51 join(X, join(complement(X), complement(Y)))
% 15.99/2.51 = { by axiom 1 (maddux1_join_commutativity) }
% 15.99/2.51 join(X, join(complement(Y), complement(X)))
% 15.99/2.51 = { by lemma 16 }
% 15.99/2.51 join(complement(Y), top)
% 15.99/2.51 = { by lemma 29 }
% 15.99/2.51 top
% 15.99/2.51
% 15.99/2.51 Lemma 41: meet(X, join(X, complement(Y))) = X.
% 15.99/2.51 Proof:
% 15.99/2.51 meet(X, join(X, complement(Y)))
% 15.99/2.51 = { by lemma 38 R->L }
% 15.99/2.51 meet(X, complement(meet(Y, complement(X))))
% 15.99/2.51 = { by lemma 39 R->L }
% 15.99/2.51 complement(join(complement(X), meet(Y, complement(X))))
% 15.99/2.51 = { by lemma 26 R->L }
% 15.99/2.51 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 15.99/2.51 = { by lemma 15 R->L }
% 15.99/2.51 join(complement(top), complement(join(complement(X), meet(Y, complement(X)))))
% 15.99/2.51 = { by lemma 40 R->L }
% 15.99/2.51 join(complement(join(complement(X), complement(meet(complement(X), Y)))), complement(join(complement(X), meet(Y, complement(X)))))
% 15.99/2.51 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 15.99/2.51 join(meet(X, meet(complement(X), Y)), complement(join(complement(X), meet(Y, complement(X)))))
% 15.99/2.51 = { by lemma 30 R->L }
% 15.99/2.51 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 15.99/2.51 = { by lemma 22 }
% 15.99/2.51 X
% 15.99/2.51
% 15.99/2.51 Lemma 42: join(X, meet(X, Y)) = X.
% 15.99/2.51 Proof:
% 15.99/2.51 join(X, meet(X, Y))
% 15.99/2.51 = { by axiom 11 (maddux4_definiton_of_meet) }
% 15.99/2.51 join(X, complement(join(complement(X), complement(Y))))
% 15.99/2.51 = { by lemma 38 R->L }
% 15.99/2.51 complement(meet(join(complement(X), complement(Y)), complement(X)))
% 15.99/2.51 = { by lemma 30 R->L }
% 15.99/2.51 complement(meet(complement(X), join(complement(X), complement(Y))))
% 15.99/2.51 = { by lemma 41 }
% 15.99/2.51 complement(complement(X))
% 15.99/2.51 = { by lemma 26 R->L }
% 15.99/2.51 join(zero, complement(complement(X)))
% 15.99/2.51 = { by lemma 25 }
% 15.99/2.51 X
% 15.99/2.51
% 15.99/2.51 Lemma 43: join(meet(X, Y), Y) = Y.
% 15.99/2.51 Proof:
% 15.99/2.51 join(meet(X, Y), Y)
% 15.99/2.51 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 15.99/2.51 join(Y, meet(X, Y))
% 15.99/2.51 = { by lemma 30 R->L }
% 15.99/2.51 join(Y, meet(Y, X))
% 15.99/2.51 = { by lemma 42 }
% 15.99/2.51 Y
% 15.99/2.51
% 15.99/2.51 Lemma 44: meet(X, join(X, Y)) = X.
% 15.99/2.51 Proof:
% 15.99/2.51 meet(X, join(X, Y))
% 15.99/2.51 = { by lemma 35 R->L }
% 15.99/2.51 meet(X, join(X, meet(Y, top)))
% 15.99/2.51 = { by lemma 31 R->L }
% 15.99/2.51 meet(X, join(X, complement(join(zero, complement(Y)))))
% 15.99/2.51 = { by lemma 41 }
% 15.99/2.51 X
% 15.99/2.51
% 15.99/2.51 Lemma 45: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 15.99/2.51 Proof:
% 15.99/2.51 meet(complement(X), complement(Y))
% 15.99/2.51 = { by lemma 30 }
% 15.99/2.51 meet(complement(Y), complement(X))
% 15.99/2.51 = { by lemma 26 R->L }
% 15.99/2.51 meet(join(zero, complement(Y)), complement(X))
% 15.99/2.51 = { by lemma 37 R->L }
% 15.99/2.51 complement(join(X, complement(join(zero, complement(Y)))))
% 15.99/2.51 = { by lemma 31 }
% 15.99/2.51 complement(join(X, meet(Y, top)))
% 15.99/2.51 = { by lemma 35 }
% 15.99/2.51 complement(join(X, Y))
% 15.99/2.51
% 15.99/2.51 Lemma 46: meet(complement(Z), meet(Y, X)) = meet(X, meet(Y, complement(Z))).
% 15.99/2.51 Proof:
% 15.99/2.51 meet(complement(Z), meet(Y, X))
% 15.99/2.51 = { by lemma 30 }
% 15.99/2.51 meet(complement(Z), meet(X, Y))
% 15.99/2.51 = { by lemma 30 }
% 15.99/2.51 meet(meet(X, Y), complement(Z))
% 15.99/2.51 = { by axiom 11 (maddux4_definiton_of_meet) }
% 15.99/2.51 meet(complement(join(complement(X), complement(Y))), complement(Z))
% 15.99/2.51 = { by lemma 45 }
% 15.99/2.51 complement(join(join(complement(X), complement(Y)), Z))
% 15.99/2.51 = { by axiom 8 (maddux2_join_associativity) R->L }
% 15.99/2.51 complement(join(complement(X), join(complement(Y), Z)))
% 15.99/2.51 = { by lemma 39 }
% 15.99/2.51 meet(X, complement(join(complement(Y), Z)))
% 15.99/2.51 = { by lemma 39 }
% 15.99/2.51 meet(X, meet(Y, complement(Z)))
% 15.99/2.51
% 15.99/2.51 Lemma 47: meet(Y, meet(Z, X)) = meet(X, meet(Y, Z)).
% 15.99/2.51 Proof:
% 15.99/2.51 meet(Y, meet(Z, X))
% 15.99/2.51 = { by lemma 35 R->L }
% 15.99/2.51 meet(meet(Y, top), meet(Z, X))
% 15.99/2.51 = { by lemma 31 R->L }
% 15.99/2.51 meet(complement(join(zero, complement(Y))), meet(Z, X))
% 15.99/2.51 = { by lemma 46 }
% 15.99/2.51 meet(X, meet(Z, complement(join(zero, complement(Y)))))
% 15.99/2.51 = { by lemma 31 }
% 15.99/2.51 meet(X, meet(Z, meet(Y, top)))
% 15.99/2.51 = { by lemma 35 }
% 15.99/2.51 meet(X, meet(Z, Y))
% 15.99/2.51 = { by lemma 30 R->L }
% 15.99/2.51 meet(X, meet(Y, Z))
% 15.99/2.51
% 15.99/2.51 Lemma 48: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 15.99/2.51 Proof:
% 15.99/2.51 meet(Y, meet(X, Z))
% 15.99/2.51 = { by lemma 30 }
% 15.99/2.51 meet(Y, meet(Z, X))
% 15.99/2.51 = { by lemma 47 }
% 15.99/2.51 meet(X, meet(Y, Z))
% 15.99/2.51
% 15.99/2.51 Lemma 49: join(composition(X, Y), composition(X, Z)) = composition(X, join(Y, Z)).
% 15.99/2.51 Proof:
% 15.99/2.51 join(composition(X, Y), composition(X, Z))
% 15.99/2.51 = { by axiom 2 (converse_idempotence) R->L }
% 15.99/2.51 join(composition(X, Y), composition(converse(converse(X)), Z))
% 15.99/2.51 = { by axiom 2 (converse_idempotence) R->L }
% 15.99/2.51 join(converse(converse(composition(X, Y))), composition(converse(converse(X)), Z))
% 15.99/2.51 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 15.99/2.51 join(composition(converse(converse(X)), Z), converse(converse(composition(X, Y))))
% 15.99/2.52 = { by lemma 17 R->L }
% 15.99/2.52 join(converse(composition(converse(Z), converse(X))), converse(converse(composition(X, Y))))
% 15.99/2.52 = { by axiom 7 (converse_additivity) R->L }
% 15.99/2.52 converse(join(composition(converse(Z), converse(X)), converse(composition(X, Y))))
% 15.99/2.52 = { by axiom 1 (maddux1_join_commutativity) }
% 15.99/2.52 converse(join(converse(composition(X, Y)), composition(converse(Z), converse(X))))
% 15.99/2.52 = { by axiom 9 (converse_multiplicativity) }
% 15.99/2.52 converse(join(composition(converse(Y), converse(X)), composition(converse(Z), converse(X))))
% 15.99/2.52 = { by axiom 12 (composition_distributivity) R->L }
% 15.99/2.52 converse(composition(join(converse(Y), converse(Z)), converse(X)))
% 15.99/2.52 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 15.99/2.52 converse(composition(join(converse(Z), converse(Y)), converse(X)))
% 15.99/2.52 = { by axiom 2 (converse_idempotence) R->L }
% 15.99/2.52 converse(composition(join(converse(converse(converse(Z))), converse(Y)), converse(X)))
% 15.99/2.52 = { by axiom 7 (converse_additivity) R->L }
% 15.99/2.52 converse(composition(converse(join(converse(converse(Z)), Y)), converse(X)))
% 15.99/2.52 = { by axiom 1 (maddux1_join_commutativity) }
% 15.99/2.52 converse(composition(converse(join(Y, converse(converse(Z)))), converse(X)))
% 15.99/2.52 = { by axiom 9 (converse_multiplicativity) R->L }
% 15.99/2.52 converse(converse(composition(X, join(Y, converse(converse(Z))))))
% 15.99/2.52 = { by axiom 2 (converse_idempotence) }
% 15.99/2.52 composition(X, join(Y, converse(converse(Z))))
% 15.99/2.52 = { by axiom 2 (converse_idempotence) }
% 15.99/2.52 composition(X, join(Y, Z))
% 15.99/2.52
% 15.99/2.52 Lemma 50: join(composition(X, Z), composition(X, Y)) = composition(X, join(Y, Z)).
% 15.99/2.52 Proof:
% 15.99/2.52 join(composition(X, Z), composition(X, Y))
% 15.99/2.52 = { by lemma 49 }
% 15.99/2.52 composition(X, join(Z, Y))
% 15.99/2.52 = { by axiom 1 (maddux1_join_commutativity) }
% 15.99/2.52 composition(X, join(Y, Z))
% 15.99/2.52
% 15.99/2.52 Lemma 51: meet(complement(X), join(Y, complement(Z))) = complement(join(X, meet(Z, complement(Y)))).
% 15.99/2.52 Proof:
% 15.99/2.52 meet(complement(X), join(Y, complement(Z)))
% 15.99/2.52 = { by lemma 30 }
% 15.99/2.52 meet(join(Y, complement(Z)), complement(X))
% 15.99/2.52 = { by lemma 37 R->L }
% 15.99/2.52 complement(join(X, complement(join(Y, complement(Z)))))
% 15.99/2.52 = { by lemma 37 }
% 15.99/2.52 complement(join(X, meet(Z, complement(Y))))
% 15.99/2.52
% 15.99/2.52 Lemma 52: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 15.99/2.52 Proof:
% 15.99/2.52 join(meet(X, Y), meet(Y, complement(X)))
% 15.99/2.52 = { by lemma 30 }
% 15.99/2.52 join(meet(Y, X), meet(Y, complement(X)))
% 15.99/2.52 = { by lemma 34 }
% 15.99/2.52 Y
% 15.99/2.52
% 15.99/2.52 Lemma 53: join(zero, composition(converse(x0), complement(composition(x0, top)))) = zero.
% 15.99/2.52 Proof:
% 15.99/2.52 join(zero, composition(converse(x0), complement(composition(x0, top))))
% 15.99/2.52 = { by axiom 4 (goals) R->L }
% 15.99/2.52 join(zero, composition(converse(composition(x0, top)), complement(composition(x0, top))))
% 15.99/2.52 = { by lemma 15 R->L }
% 15.99/2.52 join(complement(top), composition(converse(composition(x0, top)), complement(composition(x0, top))))
% 15.99/2.52 = { by axiom 4 (goals) R->L }
% 15.99/2.52 join(complement(top), composition(converse(composition(x0, top)), complement(composition(composition(x0, top), top))))
% 15.99/2.52 = { by lemma 19 }
% 15.99/2.52 complement(top)
% 15.99/2.52 = { by lemma 15 }
% 15.99/2.52 zero
% 15.99/2.52
% 15.99/2.52 Lemma 54: composition(meet(X, converse(x0)), complement(composition(x0, top))) = zero.
% 15.99/2.52 Proof:
% 15.99/2.52 composition(meet(X, converse(x0)), complement(composition(x0, top)))
% 15.99/2.52 = { by lemma 27 R->L }
% 15.99/2.52 join(composition(meet(X, converse(x0)), complement(composition(x0, top))), zero)
% 15.99/2.52 = { by lemma 53 R->L }
% 15.99/2.52 join(composition(meet(X, converse(x0)), complement(composition(x0, top))), join(zero, composition(converse(x0), complement(composition(x0, top)))))
% 15.99/2.52 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 15.99/2.52 join(composition(meet(X, converse(x0)), complement(composition(x0, top))), join(composition(converse(x0), complement(composition(x0, top))), zero))
% 15.99/2.52 = { by axiom 8 (maddux2_join_associativity) }
% 15.99/2.52 join(join(composition(meet(X, converse(x0)), complement(composition(x0, top))), composition(converse(x0), complement(composition(x0, top)))), zero)
% 15.99/2.52 = { by axiom 12 (composition_distributivity) R->L }
% 15.99/2.52 join(composition(join(meet(X, converse(x0)), converse(x0)), complement(composition(x0, top))), zero)
% 15.99/2.52 = { by axiom 1 (maddux1_join_commutativity) }
% 15.99/2.52 join(zero, composition(join(meet(X, converse(x0)), converse(x0)), complement(composition(x0, top))))
% 15.99/2.52 = { by lemma 28 }
% 15.99/2.52 composition(join(meet(X, converse(x0)), converse(x0)), complement(composition(x0, top)))
% 15.99/2.52 = { by lemma 43 }
% 15.99/2.52 composition(converse(x0), complement(composition(x0, top)))
% 15.99/2.52 = { by lemma 44 R->L }
% 15.99/2.52 meet(composition(converse(x0), complement(composition(x0, top))), join(composition(converse(x0), complement(composition(x0, top))), zero))
% 15.99/2.52 = { by axiom 1 (maddux1_join_commutativity) }
% 15.99/2.52 meet(composition(converse(x0), complement(composition(x0, top))), join(zero, composition(converse(x0), complement(composition(x0, top)))))
% 15.99/2.52 = { by lemma 53 }
% 15.99/2.52 meet(composition(converse(x0), complement(composition(x0, top))), zero)
% 15.99/2.52 = { by lemma 33 }
% 15.99/2.52 zero
% 15.99/2.52
% 15.99/2.52 Lemma 55: join(meet(x0, X), meet(composition(x0, top), complement(X))) = composition(x0, top).
% 15.99/2.52 Proof:
% 15.99/2.52 join(meet(x0, X), meet(composition(x0, top), complement(X)))
% 15.99/2.52 = { by axiom 4 (goals) }
% 15.99/2.52 join(meet(x0, X), meet(x0, complement(X)))
% 15.99/2.52 = { by lemma 34 }
% 15.99/2.52 x0
% 15.99/2.52 = { by axiom 4 (goals) R->L }
% 15.99/2.52 composition(x0, top)
% 15.99/2.52
% 15.99/2.52 Goal 1 (goals_1): join(composition(meet(x1, converse(x0)), x2), composition(meet(x1, converse(x0)), meet(x0, x2))) = composition(meet(x1, converse(x0)), meet(x0, x2)).
% 15.99/2.52 Proof:
% 15.99/2.52 join(composition(meet(x1, converse(x0)), x2), composition(meet(x1, converse(x0)), meet(x0, x2)))
% 15.99/2.52 = { by lemma 27 R->L }
% 15.99/2.52 join(join(composition(meet(x1, converse(x0)), x2), composition(meet(x1, converse(x0)), meet(x0, x2))), zero)
% 15.99/2.52 = { by lemma 54 R->L }
% 15.99/2.52 join(join(composition(meet(x1, converse(x0)), x2), composition(meet(x1, converse(x0)), meet(x0, x2))), composition(meet(x1, converse(x0)), complement(composition(x0, top))))
% 15.99/2.52 = { by lemma 50 }
% 15.99/2.52 join(composition(meet(x1, converse(x0)), join(meet(x0, x2), x2)), composition(meet(x1, converse(x0)), complement(composition(x0, top))))
% 15.99/2.52 = { by lemma 43 }
% 15.99/2.52 join(composition(meet(x1, converse(x0)), x2), composition(meet(x1, converse(x0)), complement(composition(x0, top))))
% 15.99/2.52 = { by lemma 50 }
% 15.99/2.52 composition(meet(x1, converse(x0)), join(complement(composition(x0, top)), x2))
% 15.99/2.52 = { by axiom 1 (maddux1_join_commutativity) }
% 15.99/2.52 composition(meet(x1, converse(x0)), join(x2, complement(composition(x0, top))))
% 15.99/2.52 = { by lemma 52 R->L }
% 15.99/2.52 composition(meet(x1, converse(x0)), join(meet(composition(x0, top), join(x2, complement(composition(x0, top)))), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.52 = { by lemma 30 }
% 15.99/2.52 composition(meet(x1, converse(x0)), join(meet(join(x2, complement(composition(x0, top))), composition(x0, top)), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.52 = { by lemma 55 R->L }
% 15.99/2.52 composition(meet(x1, converse(x0)), join(meet(join(x2, complement(composition(x0, top))), join(meet(x0, x2), meet(composition(x0, top), complement(x2)))), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.52 = { by lemma 38 R->L }
% 15.99/2.52 composition(meet(x1, converse(x0)), join(meet(complement(meet(composition(x0, top), complement(x2))), join(meet(x0, x2), meet(composition(x0, top), complement(x2)))), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.52 = { by lemma 30 }
% 15.99/2.52 composition(meet(x1, converse(x0)), join(meet(join(meet(x0, x2), meet(composition(x0, top), complement(x2))), complement(meet(composition(x0, top), complement(x2)))), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.52 = { by lemma 37 R->L }
% 15.99/2.52 composition(meet(x1, converse(x0)), join(complement(join(meet(composition(x0, top), complement(x2)), complement(join(meet(x0, x2), meet(composition(x0, top), complement(x2)))))), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.52 = { by lemma 45 R->L }
% 15.99/2.52 composition(meet(x1, converse(x0)), join(complement(join(meet(composition(x0, top), complement(x2)), meet(complement(meet(x0, x2)), complement(meet(composition(x0, top), complement(x2)))))), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.52 = { by lemma 52 R->L }
% 15.99/2.52 composition(meet(x1, converse(x0)), join(join(meet(meet(x0, x2), complement(join(meet(composition(x0, top), complement(x2)), meet(complement(meet(x0, x2)), complement(meet(composition(x0, top), complement(x2))))))), meet(complement(join(meet(composition(x0, top), complement(x2)), meet(complement(meet(x0, x2)), complement(meet(composition(x0, top), complement(x2)))))), complement(meet(x0, x2)))), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.52 = { by lemma 30 R->L }
% 15.99/2.52 composition(meet(x1, converse(x0)), join(join(meet(meet(x0, x2), complement(join(meet(composition(x0, top), complement(x2)), meet(complement(meet(x0, x2)), complement(meet(composition(x0, top), complement(x2))))))), meet(complement(meet(x0, x2)), complement(join(meet(composition(x0, top), complement(x2)), meet(complement(meet(x0, x2)), complement(meet(composition(x0, top), complement(x2)))))))), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.52 = { by lemma 51 R->L }
% 15.99/2.52 composition(meet(x1, converse(x0)), join(join(meet(meet(x0, x2), complement(join(meet(composition(x0, top), complement(x2)), meet(complement(meet(x0, x2)), complement(meet(composition(x0, top), complement(x2))))))), meet(complement(meet(x0, x2)), meet(complement(meet(composition(x0, top), complement(x2))), join(meet(composition(x0, top), complement(x2)), complement(complement(meet(x0, x2))))))), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.52 = { by lemma 48 }
% 15.99/2.52 composition(meet(x1, converse(x0)), join(join(meet(meet(x0, x2), complement(join(meet(composition(x0, top), complement(x2)), meet(complement(meet(x0, x2)), complement(meet(composition(x0, top), complement(x2))))))), meet(complement(meet(composition(x0, top), complement(x2))), meet(complement(meet(x0, x2)), join(meet(composition(x0, top), complement(x2)), complement(complement(meet(x0, x2))))))), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.52 = { by lemma 46 }
% 15.99/2.52 composition(meet(x1, converse(x0)), join(join(meet(meet(x0, x2), complement(join(meet(composition(x0, top), complement(x2)), meet(complement(meet(x0, x2)), complement(meet(composition(x0, top), complement(x2))))))), meet(join(meet(composition(x0, top), complement(x2)), complement(complement(meet(x0, x2)))), meet(complement(meet(x0, x2)), complement(meet(composition(x0, top), complement(x2)))))), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.52 = { by lemma 37 R->L }
% 15.99/2.52 composition(meet(x1, converse(x0)), join(join(meet(meet(x0, x2), complement(join(meet(composition(x0, top), complement(x2)), meet(complement(meet(x0, x2)), complement(meet(composition(x0, top), complement(x2))))))), meet(join(meet(composition(x0, top), complement(x2)), complement(complement(meet(x0, x2)))), complement(join(meet(composition(x0, top), complement(x2)), complement(complement(meet(x0, x2))))))), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.52 = { by axiom 5 (def_zero) R->L }
% 15.99/2.52 composition(meet(x1, converse(x0)), join(join(meet(meet(x0, x2), complement(join(meet(composition(x0, top), complement(x2)), meet(complement(meet(x0, x2)), complement(meet(composition(x0, top), complement(x2))))))), zero), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.52 = { by lemma 27 }
% 15.99/2.53 composition(meet(x1, converse(x0)), join(meet(meet(x0, x2), complement(join(meet(composition(x0, top), complement(x2)), meet(complement(meet(x0, x2)), complement(meet(composition(x0, top), complement(x2))))))), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.53 = { by lemma 45 }
% 15.99/2.53 composition(meet(x1, converse(x0)), join(meet(meet(x0, x2), complement(join(meet(composition(x0, top), complement(x2)), complement(join(meet(x0, x2), meet(composition(x0, top), complement(x2))))))), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.53 = { by lemma 37 }
% 15.99/2.53 composition(meet(x1, converse(x0)), join(meet(meet(x0, x2), meet(join(meet(x0, x2), meet(composition(x0, top), complement(x2))), complement(meet(composition(x0, top), complement(x2))))), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.53 = { by lemma 30 R->L }
% 15.99/2.53 composition(meet(x1, converse(x0)), join(meet(meet(x0, x2), meet(complement(meet(composition(x0, top), complement(x2))), join(meet(x0, x2), meet(composition(x0, top), complement(x2))))), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.53 = { by lemma 55 }
% 15.99/2.53 composition(meet(x1, converse(x0)), join(meet(meet(x0, x2), meet(complement(meet(composition(x0, top), complement(x2))), composition(x0, top))), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.53 = { by lemma 48 }
% 15.99/2.53 composition(meet(x1, converse(x0)), join(meet(complement(meet(composition(x0, top), complement(x2))), meet(meet(x0, x2), composition(x0, top))), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.53 = { by axiom 11 (maddux4_definiton_of_meet) }
% 15.99/2.53 composition(meet(x1, converse(x0)), join(meet(complement(meet(composition(x0, top), complement(x2))), complement(join(complement(meet(x0, x2)), complement(composition(x0, top))))), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.53 = { by lemma 26 R->L }
% 15.99/2.53 composition(meet(x1, converse(x0)), join(meet(complement(meet(composition(x0, top), complement(x2))), join(zero, complement(join(complement(meet(x0, x2)), complement(composition(x0, top)))))), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.53 = { by lemma 15 R->L }
% 15.99/2.53 composition(meet(x1, converse(x0)), join(meet(complement(meet(composition(x0, top), complement(x2))), join(complement(top), complement(join(complement(meet(x0, x2)), complement(composition(x0, top)))))), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.53 = { by lemma 40 R->L }
% 15.99/2.53 composition(meet(x1, converse(x0)), join(meet(complement(meet(composition(x0, top), complement(x2))), join(complement(join(x0, complement(meet(x0, x2)))), complement(join(complement(meet(x0, x2)), complement(composition(x0, top)))))), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.53 = { by axiom 4 (goals) R->L }
% 15.99/2.53 composition(meet(x1, converse(x0)), join(meet(complement(meet(composition(x0, top), complement(x2))), join(complement(join(composition(x0, top), complement(meet(x0, x2)))), complement(join(complement(meet(x0, x2)), complement(composition(x0, top)))))), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.53 = { by lemma 37 }
% 15.99/2.53 composition(meet(x1, converse(x0)), join(meet(complement(meet(composition(x0, top), complement(x2))), join(meet(meet(x0, x2), complement(composition(x0, top))), complement(join(complement(meet(x0, x2)), complement(composition(x0, top)))))), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.53 = { by lemma 22 }
% 15.99/2.53 composition(meet(x1, converse(x0)), join(meet(complement(meet(composition(x0, top), complement(x2))), meet(x0, x2)), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.53 = { by lemma 38 }
% 15.99/2.53 composition(meet(x1, converse(x0)), join(meet(join(x2, complement(composition(x0, top))), meet(x0, x2)), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.53 = { by axiom 4 (goals) R->L }
% 15.99/2.53 composition(meet(x1, converse(x0)), join(meet(join(x2, complement(composition(x0, top))), meet(composition(x0, top), x2)), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.53 = { by lemma 47 R->L }
% 15.99/2.53 composition(meet(x1, converse(x0)), join(meet(composition(x0, top), meet(x2, join(x2, complement(composition(x0, top))))), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.53 = { by lemma 44 }
% 15.99/2.53 composition(meet(x1, converse(x0)), join(meet(composition(x0, top), x2), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.53 = { by axiom 4 (goals) }
% 15.99/2.53 composition(meet(x1, converse(x0)), join(meet(x0, x2), meet(join(x2, complement(composition(x0, top))), complement(composition(x0, top)))))
% 15.99/2.53 = { by lemma 30 }
% 15.99/2.53 composition(meet(x1, converse(x0)), join(meet(x0, x2), meet(complement(composition(x0, top)), join(x2, complement(composition(x0, top))))))
% 15.99/2.53 = { by lemma 51 }
% 15.99/2.53 composition(meet(x1, converse(x0)), join(meet(x0, x2), complement(join(composition(x0, top), meet(composition(x0, top), complement(x2))))))
% 15.99/2.53 = { by lemma 42 }
% 15.99/2.53 composition(meet(x1, converse(x0)), join(meet(x0, x2), complement(composition(x0, top))))
% 15.99/2.53 = { by lemma 49 R->L }
% 15.99/2.53 join(composition(meet(x1, converse(x0)), meet(x0, x2)), composition(meet(x1, converse(x0)), complement(composition(x0, top))))
% 15.99/2.53 = { by lemma 54 }
% 15.99/2.53 join(composition(meet(x1, converse(x0)), meet(x0, x2)), zero)
% 15.99/2.53 = { by lemma 27 }
% 15.99/2.53 composition(meet(x1, converse(x0)), meet(x0, x2))
% 15.99/2.53 % SZS output end Proof
% 15.99/2.53
% 15.99/2.53 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------