TSTP Solution File: REL035+2 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL035+2 : TPTP v8.1.2. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n029.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 13:44:21 EDT 2023
% Result : Theorem 37.69s 5.18s
% Output : Proof 38.10s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL035+2 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n029.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Aug 25 22:20:09 EDT 2023
% 0.13/0.34 % CPUTime :
% 37.69/5.18 Command-line arguments: --ground-connectedness --complete-subsets
% 37.69/5.18
% 37.69/5.18 % SZS status Theorem
% 37.69/5.18
% 38.10/5.24 % SZS output start Proof
% 38.10/5.24 Axiom 1 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 38.10/5.24 Axiom 2 (converse_idempotence): converse(converse(X)) = X.
% 38.10/5.24 Axiom 3 (composition_identity): composition(X, one) = X.
% 38.10/5.24 Axiom 4 (goals): composition(x0, top) = x0.
% 38.10/5.24 Axiom 5 (def_top): top = join(X, complement(X)).
% 38.10/5.24 Axiom 6 (def_zero): zero = meet(X, complement(X)).
% 38.10/5.24 Axiom 7 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 38.10/5.24 Axiom 8 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 38.10/5.24 Axiom 9 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 38.10/5.24 Axiom 10 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 38.10/5.24 Axiom 11 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 38.10/5.24 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 38.10/5.24 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 38.10/5.24 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 38.10/5.24 Axiom 15 (modular_law_1): join(meet(composition(X, Y), Z), meet(composition(X, meet(Y, composition(converse(X), Z))), Z)) = meet(composition(X, meet(Y, composition(converse(X), Z))), Z).
% 38.10/5.24 Axiom 16 (modular_law_2): join(meet(composition(X, Y), Z), meet(composition(meet(X, composition(Z, converse(Y))), Y), Z)) = meet(composition(meet(X, composition(Z, converse(Y))), Y), Z).
% 38.10/5.24
% 38.10/5.24 Lemma 17: complement(top) = zero.
% 38.10/5.24 Proof:
% 38.10/5.24 complement(top)
% 38.10/5.24 = { by axiom 5 (def_top) }
% 38.10/5.24 complement(join(complement(X), complement(complement(X))))
% 38.10/5.24 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 38.10/5.24 meet(X, complement(X))
% 38.10/5.24 = { by axiom 6 (def_zero) R->L }
% 38.10/5.24 zero
% 38.10/5.24
% 38.10/5.24 Lemma 18: join(X, join(Y, complement(X))) = join(Y, top).
% 38.10/5.24 Proof:
% 38.10/5.24 join(X, join(Y, complement(X)))
% 38.10/5.24 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 38.10/5.24 join(X, join(complement(X), Y))
% 38.10/5.24 = { by axiom 8 (maddux2_join_associativity) }
% 38.10/5.24 join(join(X, complement(X)), Y)
% 38.10/5.24 = { by axiom 5 (def_top) R->L }
% 38.10/5.24 join(top, Y)
% 38.10/5.24 = { by axiom 1 (maddux1_join_commutativity) }
% 38.10/5.24 join(Y, top)
% 38.10/5.24
% 38.10/5.24 Lemma 19: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 38.10/5.24 Proof:
% 38.10/5.24 converse(composition(converse(X), Y))
% 38.10/5.24 = { by axiom 9 (converse_multiplicativity) }
% 38.10/5.24 composition(converse(Y), converse(converse(X)))
% 38.10/5.24 = { by axiom 2 (converse_idempotence) }
% 38.10/5.24 composition(converse(Y), X)
% 38.10/5.24
% 38.10/5.24 Lemma 20: composition(converse(one), X) = X.
% 38.10/5.24 Proof:
% 38.10/5.24 composition(converse(one), X)
% 38.10/5.24 = { by lemma 19 R->L }
% 38.10/5.24 converse(composition(converse(X), one))
% 38.10/5.24 = { by axiom 3 (composition_identity) }
% 38.10/5.24 converse(converse(X))
% 38.10/5.24 = { by axiom 2 (converse_idempotence) }
% 38.10/5.24 X
% 38.10/5.24
% 38.10/5.24 Lemma 21: composition(one, X) = X.
% 38.10/5.24 Proof:
% 38.10/5.24 composition(one, X)
% 38.10/5.24 = { by lemma 20 R->L }
% 38.10/5.24 composition(converse(one), composition(one, X))
% 38.10/5.24 = { by axiom 10 (composition_associativity) }
% 38.10/5.24 composition(composition(converse(one), one), X)
% 38.10/5.24 = { by axiom 3 (composition_identity) }
% 38.10/5.24 composition(converse(one), X)
% 38.10/5.24 = { by lemma 20 }
% 38.10/5.24 X
% 38.10/5.24
% 38.10/5.24 Lemma 22: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 38.10/5.24 Proof:
% 38.10/5.24 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 38.10/5.24 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 38.10/5.24 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 38.10/5.24 = { by axiom 13 (converse_cancellativity) }
% 38.10/5.24 complement(X)
% 38.10/5.24
% 38.10/5.24 Lemma 23: join(complement(X), complement(X)) = complement(X).
% 38.10/5.24 Proof:
% 38.10/5.24 join(complement(X), complement(X))
% 38.10/5.24 = { by lemma 20 R->L }
% 38.10/5.24 join(complement(X), composition(converse(one), complement(X)))
% 38.10/5.24 = { by lemma 21 R->L }
% 38.10/5.24 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 38.10/5.24 = { by lemma 22 }
% 38.10/5.24 complement(X)
% 38.10/5.24
% 38.10/5.24 Lemma 24: join(top, complement(X)) = top.
% 38.10/5.24 Proof:
% 38.10/5.24 join(top, complement(X))
% 38.10/5.24 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 38.10/5.24 join(complement(X), top)
% 38.10/5.24 = { by lemma 18 R->L }
% 38.10/5.24 join(X, join(complement(X), complement(X)))
% 38.10/5.24 = { by lemma 23 }
% 38.10/5.24 join(X, complement(X))
% 38.10/5.24 = { by axiom 5 (def_top) R->L }
% 38.10/5.24 top
% 38.10/5.24
% 38.10/5.24 Lemma 25: join(Y, top) = join(X, top).
% 38.10/5.24 Proof:
% 38.10/5.24 join(Y, top)
% 38.10/5.24 = { by lemma 24 R->L }
% 38.10/5.24 join(Y, join(top, complement(Y)))
% 38.10/5.24 = { by lemma 18 }
% 38.10/5.24 join(top, top)
% 38.10/5.24 = { by lemma 18 R->L }
% 38.10/5.24 join(X, join(top, complement(X)))
% 38.10/5.24 = { by lemma 24 }
% 38.10/5.24 join(X, top)
% 38.10/5.24
% 38.10/5.24 Lemma 26: join(X, top) = top.
% 38.10/5.24 Proof:
% 38.10/5.24 join(X, top)
% 38.10/5.24 = { by lemma 25 }
% 38.10/5.24 join(zero, top)
% 38.10/5.24 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 38.10/5.24 join(top, zero)
% 38.10/5.24 = { by lemma 17 R->L }
% 38.10/5.24 join(top, complement(top))
% 38.10/5.24 = { by axiom 5 (def_top) R->L }
% 38.10/5.24 top
% 38.10/5.24
% 38.10/5.24 Lemma 27: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 38.10/5.24 Proof:
% 38.10/5.24 converse(join(X, converse(Y)))
% 38.10/5.24 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 38.10/5.24 converse(join(converse(Y), X))
% 38.10/5.24 = { by axiom 7 (converse_additivity) }
% 38.10/5.24 join(converse(converse(Y)), converse(X))
% 38.10/5.24 = { by axiom 2 (converse_idempotence) }
% 38.10/5.24 join(Y, converse(X))
% 38.10/5.24
% 38.10/5.24 Lemma 28: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 38.10/5.24 Proof:
% 38.10/5.24 converse(join(converse(X), Y))
% 38.10/5.24 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 38.10/5.24 converse(join(Y, converse(X)))
% 38.10/5.24 = { by lemma 27 }
% 38.10/5.24 join(X, converse(Y))
% 38.10/5.24
% 38.10/5.24 Lemma 29: join(X, converse(complement(converse(X)))) = converse(top).
% 38.10/5.24 Proof:
% 38.10/5.24 join(X, converse(complement(converse(X))))
% 38.10/5.24 = { by lemma 28 R->L }
% 38.10/5.24 converse(join(converse(X), complement(converse(X))))
% 38.10/5.24 = { by axiom 5 (def_top) R->L }
% 38.10/5.24 converse(top)
% 38.10/5.24
% 38.10/5.24 Lemma 30: join(X, join(complement(X), Y)) = top.
% 38.10/5.24 Proof:
% 38.10/5.24 join(X, join(complement(X), Y))
% 38.10/5.24 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 38.10/5.24 join(X, join(Y, complement(X)))
% 38.10/5.24 = { by lemma 18 }
% 38.10/5.24 join(Y, top)
% 38.10/5.24 = { by lemma 25 R->L }
% 38.10/5.24 join(Z, top)
% 38.10/5.24 = { by lemma 26 }
% 38.10/5.24 top
% 38.10/5.24
% 38.10/5.24 Lemma 31: join(X, converse(top)) = top.
% 38.10/5.24 Proof:
% 38.10/5.24 join(X, converse(top))
% 38.10/5.24 = { by lemma 29 R->L }
% 38.10/5.24 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 38.10/5.24 = { by lemma 30 }
% 38.10/5.24 top
% 38.10/5.24
% 38.10/5.24 Lemma 32: converse(top) = top.
% 38.10/5.24 Proof:
% 38.10/5.24 converse(top)
% 38.10/5.24 = { by lemma 26 R->L }
% 38.10/5.24 converse(join(X, top))
% 38.10/5.24 = { by axiom 7 (converse_additivity) }
% 38.10/5.24 join(converse(X), converse(top))
% 38.10/5.24 = { by lemma 31 }
% 38.10/5.24 top
% 38.10/5.24
% 38.10/5.24 Lemma 33: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 38.10/5.24 Proof:
% 38.10/5.24 join(meet(X, Y), complement(join(complement(X), Y)))
% 38.10/5.24 = { by axiom 11 (maddux4_definiton_of_meet) }
% 38.10/5.24 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 38.10/5.24 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L }
% 38.10/5.24 X
% 38.10/5.24
% 38.10/5.24 Lemma 34: join(zero, meet(X, X)) = X.
% 38.10/5.24 Proof:
% 38.10/5.24 join(zero, meet(X, X))
% 38.10/5.24 = { by axiom 11 (maddux4_definiton_of_meet) }
% 38.10/5.24 join(zero, complement(join(complement(X), complement(X))))
% 38.10/5.24 = { by axiom 6 (def_zero) }
% 38.10/5.24 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 38.10/5.24 = { by lemma 33 }
% 38.10/5.24 X
% 38.10/5.24
% 38.10/5.24 Lemma 35: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 38.10/5.24 Proof:
% 38.10/5.24 join(zero, join(X, complement(complement(Y))))
% 38.10/5.24 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 38.10/5.24 join(zero, join(complement(complement(Y)), X))
% 38.10/5.24 = { by lemma 23 R->L }
% 38.10/5.24 join(zero, join(complement(join(complement(Y), complement(Y))), X))
% 38.10/5.24 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 38.10/5.24 join(zero, join(meet(Y, Y), X))
% 38.10/5.24 = { by axiom 8 (maddux2_join_associativity) }
% 38.10/5.24 join(join(zero, meet(Y, Y)), X)
% 38.10/5.24 = { by lemma 34 }
% 38.10/5.24 join(Y, X)
% 38.10/5.24 = { by axiom 1 (maddux1_join_commutativity) }
% 38.10/5.24 join(X, Y)
% 38.10/5.24
% 38.10/5.24 Lemma 36: join(zero, complement(complement(X))) = X.
% 38.10/5.24 Proof:
% 38.10/5.24 join(zero, complement(complement(X)))
% 38.10/5.24 = { by axiom 6 (def_zero) }
% 38.10/5.24 join(meet(X, complement(X)), complement(complement(X)))
% 38.10/5.24 = { by lemma 23 R->L }
% 38.10/5.24 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 38.10/5.24 = { by lemma 33 }
% 38.10/5.24 X
% 38.10/5.24
% 38.10/5.24 Lemma 37: join(X, zero) = join(X, X).
% 38.10/5.24 Proof:
% 38.10/5.24 join(X, zero)
% 38.10/5.24 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 38.10/5.24 join(zero, X)
% 38.10/5.24 = { by lemma 36 R->L }
% 38.10/5.24 join(zero, join(zero, complement(complement(X))))
% 38.10/5.24 = { by lemma 23 R->L }
% 38.10/5.24 join(zero, join(zero, join(complement(complement(X)), complement(complement(X)))))
% 38.10/5.24 = { by lemma 35 }
% 38.10/5.24 join(zero, join(complement(complement(X)), X))
% 38.10/5.24 = { by axiom 1 (maddux1_join_commutativity) }
% 38.10/5.24 join(zero, join(X, complement(complement(X))))
% 38.10/5.24 = { by lemma 35 }
% 38.10/5.24 join(X, X)
% 38.10/5.24
% 38.10/5.24 Lemma 38: join(zero, complement(X)) = complement(X).
% 38.10/5.24 Proof:
% 38.10/5.24 join(zero, complement(X))
% 38.10/5.24 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 38.10/5.24 join(complement(X), zero)
% 38.10/5.24 = { by lemma 37 }
% 38.10/5.24 join(complement(X), complement(X))
% 38.10/5.24 = { by lemma 23 }
% 38.10/5.24 complement(X)
% 38.10/5.24
% 38.10/5.24 Lemma 39: join(X, zero) = X.
% 38.10/5.24 Proof:
% 38.10/5.24 join(X, zero)
% 38.10/5.24 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 38.10/5.24 join(zero, X)
% 38.10/5.24 = { by lemma 35 R->L }
% 38.10/5.24 join(zero, join(zero, complement(complement(X))))
% 38.10/5.24 = { by lemma 38 }
% 38.10/5.24 join(zero, complement(complement(X)))
% 38.10/5.24 = { by lemma 36 }
% 38.10/5.24 X
% 38.10/5.24
% 38.10/5.24 Lemma 40: join(zero, X) = X.
% 38.10/5.24 Proof:
% 38.10/5.24 join(zero, X)
% 38.10/5.24 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 38.10/5.24 join(X, zero)
% 38.10/5.24 = { by lemma 39 }
% 38.10/5.24 X
% 38.10/5.24
% 38.10/5.24 Lemma 41: meet(Y, X) = meet(X, Y).
% 38.10/5.24 Proof:
% 38.10/5.24 meet(Y, X)
% 38.10/5.24 = { by axiom 11 (maddux4_definiton_of_meet) }
% 38.10/5.24 complement(join(complement(Y), complement(X)))
% 38.10/5.24 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 38.10/5.24 complement(join(complement(X), complement(Y)))
% 38.10/5.24 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 38.10/5.24 meet(X, Y)
% 38.10/5.24
% 38.10/5.24 Lemma 42: complement(join(zero, complement(X))) = meet(X, top).
% 38.10/5.24 Proof:
% 38.10/5.24 complement(join(zero, complement(X)))
% 38.10/5.24 = { by lemma 17 R->L }
% 38.10/5.24 complement(join(complement(top), complement(X)))
% 38.10/5.24 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 38.10/5.24 meet(top, X)
% 38.10/5.24 = { by lemma 41 R->L }
% 38.10/5.24 meet(X, top)
% 38.10/5.24
% 38.10/5.24 Lemma 43: join(X, complement(zero)) = top.
% 38.10/5.24 Proof:
% 38.10/5.24 join(X, complement(zero))
% 38.10/5.24 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 38.10/5.24 join(complement(zero), X)
% 38.10/5.24 = { by lemma 35 R->L }
% 38.10/5.24 join(zero, join(complement(zero), complement(complement(X))))
% 38.10/5.24 = { by lemma 30 }
% 38.10/5.24 top
% 38.10/5.24
% 38.10/5.24 Lemma 44: meet(X, zero) = zero.
% 38.10/5.24 Proof:
% 38.10/5.24 meet(X, zero)
% 38.10/5.24 = { by axiom 11 (maddux4_definiton_of_meet) }
% 38.10/5.24 complement(join(complement(X), complement(zero)))
% 38.10/5.24 = { by lemma 43 }
% 38.10/5.24 complement(top)
% 38.10/5.24 = { by lemma 17 }
% 38.10/5.24 zero
% 38.10/5.24
% 38.10/5.24 Lemma 45: join(meet(X, Y), meet(X, complement(Y))) = X.
% 38.10/5.24 Proof:
% 38.10/5.24 join(meet(X, Y), meet(X, complement(Y)))
% 38.10/5.24 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 38.10/5.24 join(meet(X, complement(Y)), meet(X, Y))
% 38.10/5.24 = { by axiom 11 (maddux4_definiton_of_meet) }
% 38.10/5.24 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 38.10/5.24 = { by lemma 33 }
% 38.10/5.24 X
% 38.10/5.24
% 38.10/5.24 Lemma 46: meet(X, top) = X.
% 38.10/5.24 Proof:
% 38.10/5.24 meet(X, top)
% 38.10/5.25 = { by lemma 42 R->L }
% 38.10/5.25 complement(join(zero, complement(X)))
% 38.10/5.25 = { by lemma 38 R->L }
% 38.10/5.25 join(zero, complement(join(zero, complement(X))))
% 38.10/5.25 = { by lemma 42 }
% 38.10/5.25 join(zero, meet(X, top))
% 38.10/5.25 = { by lemma 43 R->L }
% 38.10/5.25 join(zero, meet(X, join(complement(zero), complement(zero))))
% 38.10/5.25 = { by lemma 23 }
% 38.10/5.25 join(zero, meet(X, complement(zero)))
% 38.10/5.25 = { by lemma 44 R->L }
% 38.10/5.25 join(meet(X, zero), meet(X, complement(zero)))
% 38.10/5.25 = { by lemma 45 }
% 38.10/5.25 X
% 38.10/5.25
% 38.10/5.25 Lemma 47: join(meet(X, Y), meet(X, Y)) = meet(X, Y).
% 38.10/5.25 Proof:
% 38.10/5.25 join(meet(X, Y), meet(X, Y))
% 38.10/5.25 = { by lemma 41 }
% 38.10/5.25 join(meet(Y, X), meet(X, Y))
% 38.10/5.25 = { by lemma 41 }
% 38.10/5.25 join(meet(Y, X), meet(Y, X))
% 38.10/5.25 = { by axiom 11 (maddux4_definiton_of_meet) }
% 38.10/5.25 join(meet(Y, X), complement(join(complement(Y), complement(X))))
% 38.10/5.25 = { by axiom 11 (maddux4_definiton_of_meet) }
% 38.10/5.25 join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X))))
% 38.10/5.25 = { by lemma 23 }
% 38.10/5.25 complement(join(complement(Y), complement(X)))
% 38.10/5.25 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 38.10/5.25 meet(Y, X)
% 38.10/5.25 = { by lemma 41 R->L }
% 38.10/5.25 meet(X, Y)
% 38.10/5.25
% 38.10/5.25 Lemma 48: converse(zero) = zero.
% 38.10/5.25 Proof:
% 38.10/5.25 converse(zero)
% 38.10/5.25 = { by lemma 40 R->L }
% 38.10/5.25 join(zero, converse(zero))
% 38.10/5.25 = { by lemma 28 R->L }
% 38.10/5.25 converse(join(converse(zero), zero))
% 38.10/5.25 = { by lemma 37 }
% 38.10/5.25 converse(join(converse(zero), converse(zero)))
% 38.10/5.25 = { by lemma 27 }
% 38.10/5.25 join(zero, converse(converse(zero)))
% 38.10/5.25 = { by axiom 2 (converse_idempotence) }
% 38.10/5.25 join(zero, zero)
% 38.10/5.25 = { by lemma 46 R->L }
% 38.10/5.25 join(zero, meet(zero, top))
% 38.10/5.25 = { by lemma 46 R->L }
% 38.10/5.25 join(meet(zero, top), meet(zero, top))
% 38.10/5.25 = { by lemma 47 }
% 38.10/5.25 meet(zero, top)
% 38.10/5.25 = { by lemma 46 }
% 38.10/5.25 zero
% 38.10/5.25
% 38.10/5.25 Lemma 49: complement(complement(X)) = X.
% 38.10/5.25 Proof:
% 38.10/5.25 complement(complement(X))
% 38.10/5.25 = { by lemma 38 R->L }
% 38.10/5.25 join(zero, complement(complement(X)))
% 38.10/5.25 = { by lemma 36 }
% 38.10/5.25 X
% 38.10/5.25
% 38.10/5.25 Lemma 50: meet(zero, X) = zero.
% 38.10/5.25 Proof:
% 38.10/5.25 meet(zero, X)
% 38.10/5.25 = { by lemma 41 }
% 38.10/5.25 meet(X, zero)
% 38.10/5.25 = { by lemma 44 }
% 38.10/5.25 zero
% 38.10/5.25
% 38.10/5.25 Lemma 51: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 38.10/5.25 Proof:
% 38.10/5.25 meet(X, join(complement(Y), complement(Z)))
% 38.10/5.25 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 38.10/5.25 meet(X, join(complement(Z), complement(Y)))
% 38.10/5.25 = { by lemma 41 }
% 38.10/5.25 meet(join(complement(Z), complement(Y)), X)
% 38.10/5.25 = { by axiom 11 (maddux4_definiton_of_meet) }
% 38.10/5.25 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 38.10/5.25 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 38.10/5.25 complement(join(meet(Z, Y), complement(X)))
% 38.10/5.25 = { by axiom 1 (maddux1_join_commutativity) }
% 38.10/5.25 complement(join(complement(X), meet(Z, Y)))
% 38.10/5.25 = { by lemma 41 R->L }
% 38.10/5.25 complement(join(complement(X), meet(Y, Z)))
% 38.10/5.25
% 38.10/5.25 Lemma 52: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 38.10/5.25 Proof:
% 38.10/5.25 complement(join(X, complement(Y)))
% 38.10/5.25 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 38.10/5.25 complement(join(complement(Y), X))
% 38.10/5.25 = { by lemma 46 R->L }
% 38.10/5.25 complement(join(complement(Y), meet(X, top)))
% 38.10/5.25 = { by lemma 41 R->L }
% 38.10/5.25 complement(join(complement(Y), meet(top, X)))
% 38.10/5.25 = { by lemma 51 R->L }
% 38.10/5.25 meet(Y, join(complement(top), complement(X)))
% 38.10/5.25 = { by lemma 17 }
% 38.10/5.25 meet(Y, join(zero, complement(X)))
% 38.10/5.25 = { by lemma 38 }
% 38.10/5.25 meet(Y, complement(X))
% 38.10/5.25
% 38.10/5.25 Lemma 53: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 38.10/5.25 Proof:
% 38.10/5.25 complement(meet(X, complement(Y)))
% 38.10/5.25 = { by lemma 40 R->L }
% 38.10/5.25 complement(join(zero, meet(X, complement(Y))))
% 38.10/5.25 = { by lemma 52 R->L }
% 38.10/5.25 complement(join(zero, complement(join(Y, complement(X)))))
% 38.10/5.25 = { by lemma 42 }
% 38.10/5.25 meet(join(Y, complement(X)), top)
% 38.10/5.25 = { by lemma 46 }
% 38.10/5.25 join(Y, complement(X))
% 38.10/5.25
% 38.10/5.25 Lemma 54: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 38.10/5.25 Proof:
% 38.10/5.25 complement(join(complement(X), Y))
% 38.10/5.25 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 38.10/5.25 complement(join(Y, complement(X)))
% 38.10/5.25 = { by lemma 52 }
% 38.10/5.25 meet(X, complement(Y))
% 38.10/5.25
% 38.10/5.25 Lemma 55: meet(X, meet(Y, complement(X))) = zero.
% 38.10/5.25 Proof:
% 38.10/5.25 meet(X, meet(Y, complement(X)))
% 38.10/5.25 = { by lemma 41 }
% 38.10/5.25 meet(X, meet(complement(X), Y))
% 38.10/5.25 = { by axiom 11 (maddux4_definiton_of_meet) }
% 38.10/5.25 complement(join(complement(X), complement(meet(complement(X), Y))))
% 38.10/5.25 = { by lemma 47 R->L }
% 38.10/5.25 complement(join(complement(X), complement(join(meet(complement(X), Y), meet(complement(X), Y)))))
% 38.10/5.25 = { by axiom 11 (maddux4_definiton_of_meet) }
% 38.10/5.25 complement(join(complement(X), complement(join(complement(join(complement(complement(X)), complement(Y))), meet(complement(X), Y)))))
% 38.10/5.25 = { by lemma 38 R->L }
% 38.10/5.25 complement(join(complement(X), join(zero, complement(join(complement(join(complement(complement(X)), complement(Y))), meet(complement(X), Y))))))
% 38.10/5.25 = { by lemma 51 R->L }
% 38.10/5.25 complement(join(complement(X), join(zero, meet(join(complement(complement(X)), complement(Y)), join(complement(complement(X)), complement(Y))))))
% 38.10/5.25 = { by lemma 34 }
% 38.10/5.25 complement(join(complement(X), join(complement(complement(X)), complement(Y))))
% 38.10/5.25 = { by axiom 1 (maddux1_join_commutativity) }
% 38.10/5.25 complement(join(complement(X), join(complement(Y), complement(complement(X)))))
% 38.10/5.25 = { by lemma 18 }
% 38.10/5.25 complement(join(complement(Y), top))
% 38.10/5.25 = { by lemma 26 }
% 38.10/5.25 complement(top)
% 38.10/5.25 = { by lemma 17 }
% 38.10/5.25 zero
% 38.10/5.25
% 38.10/5.25 Lemma 56: meet(X, join(X, complement(Y))) = X.
% 38.10/5.25 Proof:
% 38.10/5.25 meet(X, join(X, complement(Y)))
% 38.10/5.25 = { by lemma 53 R->L }
% 38.10/5.25 meet(X, complement(meet(Y, complement(X))))
% 38.10/5.25 = { by lemma 54 R->L }
% 38.10/5.25 complement(join(complement(X), meet(Y, complement(X))))
% 38.10/5.25 = { by lemma 38 R->L }
% 38.10/5.25 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 38.10/5.25 = { by lemma 55 R->L }
% 38.10/5.25 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 38.10/5.25 = { by lemma 33 }
% 38.10/5.25 X
% 38.10/5.25
% 38.10/5.25 Lemma 57: meet(X, join(X, Y)) = X.
% 38.10/5.25 Proof:
% 38.10/5.25 meet(X, join(X, Y))
% 38.10/5.25 = { by lemma 46 R->L }
% 38.10/5.25 meet(X, join(X, meet(Y, top)))
% 38.10/5.25 = { by lemma 42 R->L }
% 38.10/5.25 meet(X, join(X, complement(join(zero, complement(Y)))))
% 38.10/5.25 = { by lemma 56 }
% 38.10/5.25 X
% 38.10/5.25
% 38.10/5.25 Lemma 58: join(zero, composition(converse(x0), complement(x0))) = zero.
% 38.10/5.25 Proof:
% 38.10/5.25 join(zero, composition(converse(x0), complement(x0)))
% 38.10/5.25 = { by lemma 17 R->L }
% 38.10/5.25 join(complement(top), composition(converse(x0), complement(x0)))
% 38.10/5.25 = { by axiom 4 (goals) R->L }
% 38.10/5.25 join(complement(top), composition(converse(x0), complement(composition(x0, top))))
% 38.10/5.25 = { by lemma 22 }
% 38.10/5.25 complement(top)
% 38.10/5.25 = { by lemma 17 }
% 38.10/5.25 zero
% 38.10/5.25
% 38.10/5.25 Lemma 59: composition(converse(x0), complement(x0)) = zero.
% 38.10/5.25 Proof:
% 38.10/5.25 composition(converse(x0), complement(x0))
% 38.10/5.25 = { by lemma 57 R->L }
% 38.10/5.25 meet(composition(converse(x0), complement(x0)), join(composition(converse(x0), complement(x0)), zero))
% 38.10/5.25 = { by axiom 1 (maddux1_join_commutativity) }
% 38.10/5.25 meet(composition(converse(x0), complement(x0)), join(zero, composition(converse(x0), complement(x0))))
% 38.10/5.25 = { by lemma 58 }
% 38.10/5.25 meet(composition(converse(x0), complement(x0)), zero)
% 38.10/5.25 = { by lemma 44 }
% 38.10/5.25 zero
% 38.10/5.25
% 38.10/5.25 Lemma 60: composition(top, zero) = zero.
% 38.10/5.25 Proof:
% 38.10/5.25 composition(top, zero)
% 38.10/5.25 = { by lemma 59 R->L }
% 38.10/5.25 composition(top, composition(converse(x0), complement(x0)))
% 38.10/5.25 = { by lemma 32 R->L }
% 38.10/5.25 composition(converse(top), composition(converse(x0), complement(x0)))
% 38.10/5.25 = { by axiom 10 (composition_associativity) }
% 38.10/5.25 composition(composition(converse(top), converse(x0)), complement(x0))
% 38.10/5.25 = { by axiom 9 (converse_multiplicativity) R->L }
% 38.10/5.25 composition(converse(composition(x0, top)), complement(x0))
% 38.10/5.25 = { by axiom 4 (goals) }
% 38.10/5.25 composition(converse(x0), complement(x0))
% 38.10/5.25 = { by lemma 59 }
% 38.10/5.25 zero
% 38.10/5.25
% 38.10/5.25 Lemma 61: composition(X, zero) = zero.
% 38.10/5.25 Proof:
% 38.10/5.25 composition(X, zero)
% 38.10/5.25 = { by lemma 40 R->L }
% 38.10/5.25 join(zero, composition(X, zero))
% 38.10/5.25 = { by lemma 60 R->L }
% 38.10/5.25 join(composition(top, zero), composition(X, zero))
% 38.10/5.25 = { by axiom 12 (composition_distributivity) R->L }
% 38.10/5.25 composition(join(top, X), zero)
% 38.10/5.25 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 38.10/5.25 composition(join(X, top), zero)
% 38.10/5.25 = { by lemma 25 R->L }
% 38.10/5.25 composition(join(Y, top), zero)
% 38.10/5.25 = { by lemma 26 }
% 38.10/5.25 composition(top, zero)
% 38.10/5.25 = { by lemma 60 }
% 38.10/5.25 zero
% 38.10/5.25
% 38.10/5.25 Lemma 62: composition(zero, X) = zero.
% 38.10/5.25 Proof:
% 38.10/5.25 composition(zero, X)
% 38.10/5.25 = { by lemma 48 R->L }
% 38.10/5.25 composition(converse(zero), X)
% 38.10/5.25 = { by lemma 19 R->L }
% 38.10/5.25 converse(composition(converse(X), zero))
% 38.10/5.25 = { by lemma 61 }
% 38.10/5.25 converse(zero)
% 38.10/5.25 = { by lemma 48 }
% 38.10/5.25 zero
% 38.10/5.25
% 38.10/5.25 Lemma 63: meet(one, composition(converse(complement(X)), X)) = zero.
% 38.10/5.25 Proof:
% 38.10/5.25 meet(one, composition(converse(complement(X)), X))
% 38.10/5.25 = { by lemma 41 }
% 38.10/5.25 meet(composition(converse(complement(X)), X), one)
% 38.10/5.25 = { by lemma 49 R->L }
% 38.10/5.25 meet(composition(converse(complement(X)), X), complement(complement(one)))
% 38.10/5.25 = { by lemma 22 R->L }
% 38.10/5.25 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(join(zero, complement(X))), complement(composition(join(zero, complement(X)), one))))))
% 38.10/5.25 = { by axiom 3 (composition_identity) }
% 38.10/5.25 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(join(zero, complement(X))), complement(join(zero, complement(X)))))))
% 38.10/5.25 = { by lemma 42 }
% 38.10/5.25 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(join(zero, complement(X))), meet(X, top)))))
% 38.10/5.25 = { by lemma 38 }
% 38.10/5.25 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(complement(X)), meet(X, top)))))
% 38.10/5.25 = { by lemma 46 }
% 38.10/5.25 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(complement(X)), X))))
% 38.10/5.25 = { by lemma 54 }
% 38.10/5.25 meet(composition(converse(complement(X)), X), meet(one, complement(composition(converse(complement(X)), X))))
% 38.10/5.25 = { by lemma 55 }
% 38.10/5.25 zero
% 38.10/5.25
% 38.10/5.25 Lemma 64: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 38.10/5.25 Proof:
% 38.10/5.25 join(meet(X, Y), meet(Y, complement(X)))
% 38.10/5.25 = { by lemma 41 }
% 38.10/5.25 join(meet(Y, X), meet(Y, complement(X)))
% 38.10/5.25 = { by lemma 45 }
% 38.10/5.25 Y
% 38.10/5.25
% 38.10/5.25 Lemma 65: converse(complement(X)) = complement(converse(X)).
% 38.10/5.25 Proof:
% 38.10/5.25 converse(complement(X))
% 38.10/5.25 = { by lemma 38 R->L }
% 38.10/5.25 converse(join(zero, complement(X)))
% 38.10/5.25 = { by lemma 33 R->L }
% 38.10/5.25 converse(join(meet(join(zero, complement(X)), complement(converse(complement(converse(complement(join(zero, complement(X)))))))), complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
% 38.10/5.25 = { by lemma 54 R->L }
% 38.10/5.25 converse(join(complement(join(complement(join(zero, complement(X))), converse(complement(converse(complement(join(zero, complement(X)))))))), complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
% 38.10/5.25 = { by lemma 29 }
% 38.10/5.25 converse(join(complement(converse(top)), complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
% 38.10/5.25 = { by lemma 32 }
% 38.10/5.25 converse(join(complement(top), complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
% 38.10/5.25 = { by lemma 17 }
% 38.10/5.25 converse(join(zero, complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
% 38.10/5.25 = { by lemma 38 }
% 38.10/5.25 converse(complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X))))))))))
% 38.10/5.25 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 38.10/5.25 converse(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))))
% 38.10/5.25 = { by lemma 39 R->L }
% 38.10/5.25 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), zero))
% 38.10/5.25 = { by lemma 50 R->L }
% 38.10/5.25 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), meet(zero, converse(complement(converse(complement(join(zero, complement(X)))))))))
% 38.10/5.25 = { by lemma 62 R->L }
% 38.10/5.25 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), meet(composition(zero, complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X)))))))))
% 38.10/5.26 = { by lemma 63 R->L }
% 38.10/5.26 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), meet(composition(meet(one, composition(converse(complement(converse(complement(join(zero, complement(X)))))), converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X)))))))))
% 38.10/5.26 = { by axiom 16 (modular_law_2) R->L }
% 38.10/5.26 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), join(meet(composition(one, complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))), meet(composition(meet(one, composition(converse(complement(converse(complement(join(zero, complement(X)))))), converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))))))
% 38.10/5.26 = { by lemma 63 }
% 38.10/5.26 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), join(meet(composition(one, complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))), meet(composition(zero, complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))))))
% 38.10/5.26 = { by lemma 21 }
% 38.10/5.26 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), join(meet(complement(join(zero, complement(X))), converse(complement(converse(complement(join(zero, complement(X))))))), meet(composition(zero, complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))))))
% 38.10/5.26 = { by lemma 62 }
% 38.10/5.26 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), join(meet(complement(join(zero, complement(X))), converse(complement(converse(complement(join(zero, complement(X))))))), meet(zero, converse(complement(converse(complement(join(zero, complement(X))))))))))
% 38.10/5.26 = { by lemma 50 }
% 38.10/5.26 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), join(meet(complement(join(zero, complement(X))), converse(complement(converse(complement(join(zero, complement(X))))))), zero)))
% 38.10/5.26 = { by lemma 39 }
% 38.10/5.26 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), meet(complement(join(zero, complement(X))), converse(complement(converse(complement(join(zero, complement(X)))))))))
% 38.10/5.26 = { by lemma 41 }
% 38.10/5.26 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), meet(converse(complement(converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X))))))
% 38.10/5.26 = { by lemma 64 }
% 38.10/5.26 converse(converse(complement(converse(complement(join(zero, complement(X)))))))
% 38.10/5.26 = { by axiom 2 (converse_idempotence) }
% 38.10/5.26 complement(converse(complement(join(zero, complement(X)))))
% 38.10/5.26 = { by lemma 42 }
% 38.10/5.26 complement(converse(meet(X, top)))
% 38.10/5.26 = { by lemma 46 }
% 38.10/5.26 complement(converse(X))
% 38.10/5.26
% 38.10/5.26 Lemma 66: join(X, meet(X, Y)) = X.
% 38.10/5.26 Proof:
% 38.10/5.26 join(X, meet(X, Y))
% 38.10/5.26 = { by axiom 11 (maddux4_definiton_of_meet) }
% 38.10/5.26 join(X, complement(join(complement(X), complement(Y))))
% 38.10/5.26 = { by lemma 53 R->L }
% 38.10/5.26 complement(meet(join(complement(X), complement(Y)), complement(X)))
% 38.10/5.26 = { by lemma 41 R->L }
% 38.10/5.26 complement(meet(complement(X), join(complement(X), complement(Y))))
% 38.10/5.26 = { by lemma 56 }
% 38.10/5.26 complement(complement(X))
% 38.10/5.26 = { by lemma 49 }
% 38.10/5.26 X
% 38.10/5.26
% 38.10/5.26 Lemma 67: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 38.10/5.26 Proof:
% 38.10/5.26 converse(composition(X, converse(Y)))
% 38.10/5.26 = { by axiom 9 (converse_multiplicativity) }
% 38.10/5.26 composition(converse(converse(Y)), converse(X))
% 38.10/5.26 = { by axiom 2 (converse_idempotence) }
% 38.10/5.26 composition(Y, converse(X))
% 38.10/5.26
% 38.10/5.26 Lemma 68: composition(converse(X), complement(composition(X, top))) = zero.
% 38.10/5.26 Proof:
% 38.10/5.26 composition(converse(X), complement(composition(X, top)))
% 38.10/5.26 = { by lemma 40 R->L }
% 38.10/5.26 join(zero, composition(converse(X), complement(composition(X, top))))
% 38.10/5.26 = { by lemma 17 R->L }
% 38.10/5.26 join(complement(top), composition(converse(X), complement(composition(X, top))))
% 38.10/5.26 = { by lemma 22 }
% 38.10/5.26 complement(top)
% 38.10/5.26 = { by lemma 17 }
% 38.10/5.26 zero
% 38.10/5.26
% 38.10/5.26 Lemma 69: meet(complement(x0), composition(x0, X)) = zero.
% 38.10/5.26 Proof:
% 38.10/5.26 meet(complement(x0), composition(x0, X))
% 38.10/5.26 = { by lemma 41 }
% 38.10/5.26 meet(composition(x0, X), complement(x0))
% 38.10/5.26 = { by axiom 4 (goals) R->L }
% 38.10/5.26 meet(composition(x0, X), complement(composition(x0, top)))
% 38.10/5.26 = { by lemma 39 R->L }
% 38.10/5.26 join(meet(composition(x0, X), complement(composition(x0, top))), zero)
% 38.10/5.26 = { by lemma 50 R->L }
% 38.10/5.26 join(meet(composition(x0, X), complement(composition(x0, top))), meet(zero, complement(composition(x0, top))))
% 38.10/5.26 = { by lemma 61 R->L }
% 38.10/5.26 join(meet(composition(x0, X), complement(composition(x0, top))), meet(composition(x0, zero), complement(composition(x0, top))))
% 38.10/5.26 = { by lemma 44 R->L }
% 38.10/5.26 join(meet(composition(x0, X), complement(composition(x0, top))), meet(composition(x0, meet(X, zero)), complement(composition(x0, top))))
% 38.10/5.26 = { by lemma 68 R->L }
% 38.10/5.26 join(meet(composition(x0, X), complement(composition(x0, top))), meet(composition(x0, meet(X, composition(converse(x0), complement(composition(x0, top))))), complement(composition(x0, top))))
% 38.10/5.26 = { by axiom 15 (modular_law_1) }
% 38.10/5.26 meet(composition(x0, meet(X, composition(converse(x0), complement(composition(x0, top))))), complement(composition(x0, top)))
% 38.10/5.26 = { by lemma 68 }
% 38.10/5.26 meet(composition(x0, meet(X, zero)), complement(composition(x0, top)))
% 38.10/5.26 = { by lemma 44 }
% 38.10/5.26 meet(composition(x0, zero), complement(composition(x0, top)))
% 38.10/5.26 = { by lemma 61 }
% 38.10/5.26 meet(zero, complement(composition(x0, top)))
% 38.10/5.26 = { by lemma 50 }
% 38.10/5.26 zero
% 38.10/5.26
% 38.10/5.26 Lemma 70: composition(complement(converse(x0)), meet(x0, X)) = zero.
% 38.10/5.26 Proof:
% 38.10/5.26 composition(complement(converse(x0)), meet(x0, X))
% 38.10/5.26 = { by lemma 65 R->L }
% 38.10/5.26 composition(converse(complement(x0)), meet(x0, X))
% 38.10/5.26 = { by lemma 19 R->L }
% 38.10/5.26 converse(composition(converse(meet(x0, X)), complement(x0)))
% 38.10/5.26 = { by lemma 39 R->L }
% 38.10/5.26 converse(join(composition(converse(meet(x0, X)), complement(x0)), zero))
% 38.10/5.26 = { by lemma 58 R->L }
% 38.10/5.26 converse(join(composition(converse(meet(x0, X)), complement(x0)), join(zero, composition(converse(x0), complement(x0)))))
% 38.10/5.26 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 38.10/5.26 converse(join(composition(converse(meet(x0, X)), complement(x0)), join(composition(converse(x0), complement(x0)), zero)))
% 38.10/5.26 = { by axiom 8 (maddux2_join_associativity) }
% 38.10/5.26 converse(join(join(composition(converse(meet(x0, X)), complement(x0)), composition(converse(x0), complement(x0))), zero))
% 38.10/5.26 = { by axiom 12 (composition_distributivity) R->L }
% 38.10/5.26 converse(join(composition(join(converse(meet(x0, X)), converse(x0)), complement(x0)), zero))
% 38.10/5.26 = { by axiom 1 (maddux1_join_commutativity) }
% 38.10/5.26 converse(join(zero, composition(join(converse(meet(x0, X)), converse(x0)), complement(x0))))
% 38.10/5.26 = { by lemma 40 }
% 38.10/5.26 converse(composition(join(converse(meet(x0, X)), converse(x0)), complement(x0)))
% 38.10/5.26 = { by axiom 1 (maddux1_join_commutativity) }
% 38.10/5.26 converse(composition(join(converse(x0), converse(meet(x0, X))), complement(x0)))
% 38.10/5.26 = { by axiom 7 (converse_additivity) R->L }
% 38.10/5.26 converse(composition(converse(join(x0, meet(x0, X))), complement(x0)))
% 38.10/5.26 = { by axiom 1 (maddux1_join_commutativity) }
% 38.10/5.26 converse(composition(converse(join(meet(x0, X), x0)), complement(x0)))
% 38.10/5.26 = { by lemma 19 }
% 38.10/5.26 composition(converse(complement(x0)), join(meet(x0, X), x0))
% 38.10/5.26 = { by lemma 65 }
% 38.10/5.26 composition(complement(converse(x0)), join(meet(x0, X), x0))
% 38.10/5.26 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 38.10/5.26 composition(complement(converse(x0)), join(x0, meet(x0, X)))
% 38.10/5.26 = { by lemma 66 }
% 38.10/5.26 composition(complement(converse(x0)), x0)
% 38.10/5.26 = { by lemma 49 R->L }
% 38.10/5.26 composition(complement(converse(x0)), complement(complement(x0)))
% 38.10/5.26 = { by lemma 65 R->L }
% 38.10/5.26 composition(converse(complement(x0)), complement(complement(x0)))
% 38.10/5.26 = { by lemma 40 R->L }
% 38.10/5.26 join(zero, composition(converse(complement(x0)), complement(complement(x0))))
% 38.10/5.26 = { by lemma 17 R->L }
% 38.10/5.26 join(complement(top), composition(converse(complement(x0)), complement(complement(x0))))
% 38.10/5.26 = { by lemma 57 R->L }
% 38.10/5.26 join(complement(top), composition(converse(complement(x0)), complement(meet(complement(x0), join(complement(x0), composition(complement(x0), top))))))
% 38.10/5.26 = { by lemma 32 R->L }
% 38.10/5.26 join(complement(top), composition(converse(complement(x0)), complement(meet(complement(x0), join(complement(x0), composition(complement(x0), converse(top)))))))
% 38.10/5.26 = { by lemma 67 R->L }
% 38.10/5.26 join(complement(top), composition(converse(complement(x0)), complement(meet(complement(x0), join(complement(x0), converse(composition(top, converse(complement(x0)))))))))
% 38.10/5.26 = { by lemma 28 R->L }
% 38.10/5.26 join(complement(top), composition(converse(complement(x0)), complement(meet(complement(x0), converse(join(converse(complement(x0)), composition(top, converse(complement(x0)))))))))
% 38.10/5.26 = { by lemma 32 R->L }
% 38.10/5.26 join(complement(top), composition(converse(complement(x0)), complement(meet(complement(x0), converse(join(converse(complement(x0)), composition(converse(top), converse(complement(x0)))))))))
% 38.10/5.26 = { by lemma 21 R->L }
% 38.10/5.26 join(complement(top), composition(converse(complement(x0)), complement(meet(complement(x0), converse(join(composition(one, converse(complement(x0))), composition(converse(top), converse(complement(x0)))))))))
% 38.10/5.26 = { by axiom 12 (composition_distributivity) R->L }
% 38.10/5.26 join(complement(top), composition(converse(complement(x0)), complement(meet(complement(x0), converse(composition(join(one, converse(top)), converse(complement(x0))))))))
% 38.10/5.26 = { by lemma 31 }
% 38.10/5.26 join(complement(top), composition(converse(complement(x0)), complement(meet(complement(x0), converse(composition(top, converse(complement(x0))))))))
% 38.10/5.26 = { by lemma 67 }
% 38.10/5.26 join(complement(top), composition(converse(complement(x0)), complement(meet(complement(x0), composition(complement(x0), converse(top))))))
% 38.10/5.26 = { by lemma 32 }
% 38.10/5.26 join(complement(top), composition(converse(complement(x0)), complement(meet(complement(x0), composition(complement(x0), top)))))
% 38.10/5.26 = { by lemma 41 }
% 38.10/5.26 join(complement(top), composition(converse(complement(x0)), complement(meet(composition(complement(x0), top), complement(x0)))))
% 38.10/5.26 = { by lemma 40 R->L }
% 38.10/5.26 join(complement(top), composition(converse(complement(x0)), complement(join(zero, meet(composition(complement(x0), top), complement(x0))))))
% 38.10/5.26 = { by lemma 50 R->L }
% 38.10/5.26 join(complement(top), composition(converse(complement(x0)), complement(join(meet(zero, x0), meet(composition(complement(x0), top), complement(x0))))))
% 38.10/5.26 = { by lemma 62 R->L }
% 38.10/5.26 join(complement(top), composition(converse(complement(x0)), complement(join(meet(composition(zero, top), x0), meet(composition(complement(x0), top), complement(x0))))))
% 38.10/5.26 = { by lemma 69 R->L }
% 38.10/5.26 join(complement(top), composition(converse(complement(x0)), complement(join(meet(composition(meet(complement(x0), composition(x0, converse(top))), top), x0), meet(composition(complement(x0), top), complement(x0))))))
% 38.10/5.26 = { by axiom 16 (modular_law_2) R->L }
% 38.10/5.26 join(complement(top), composition(converse(complement(x0)), complement(join(join(meet(composition(complement(x0), top), x0), meet(composition(meet(complement(x0), composition(x0, converse(top))), top), x0)), meet(composition(complement(x0), top), complement(x0))))))
% 38.10/5.26 = { by lemma 69 }
% 38.10/5.26 join(complement(top), composition(converse(complement(x0)), complement(join(join(meet(composition(complement(x0), top), x0), meet(composition(zero, top), x0)), meet(composition(complement(x0), top), complement(x0))))))
% 38.10/5.26 = { by lemma 62 }
% 38.10/5.26 join(complement(top), composition(converse(complement(x0)), complement(join(join(meet(composition(complement(x0), top), x0), meet(zero, x0)), meet(composition(complement(x0), top), complement(x0))))))
% 38.10/5.26 = { by lemma 50 }
% 38.10/5.26 join(complement(top), composition(converse(complement(x0)), complement(join(join(meet(composition(complement(x0), top), x0), zero), meet(composition(complement(x0), top), complement(x0))))))
% 38.10/5.26 = { by lemma 39 }
% 38.10/5.26 join(complement(top), composition(converse(complement(x0)), complement(join(meet(composition(complement(x0), top), x0), meet(composition(complement(x0), top), complement(x0))))))
% 38.10/5.26 = { by lemma 41 R->L }
% 38.10/5.26 join(complement(top), composition(converse(complement(x0)), complement(join(meet(x0, composition(complement(x0), top)), meet(composition(complement(x0), top), complement(x0))))))
% 38.10/5.26 = { by lemma 64 }
% 38.10/5.26 join(complement(top), composition(converse(complement(x0)), complement(composition(complement(x0), top))))
% 38.10/5.26 = { by lemma 22 }
% 38.10/5.26 complement(top)
% 38.10/5.26 = { by lemma 17 }
% 38.10/5.26 zero
% 38.10/5.26
% 38.10/5.26 Goal 1 (goals_1): composition(meet(x1, converse(x0)), meet(x0, x2)) = composition(x1, meet(x0, x2)).
% 38.10/5.26 Proof:
% 38.10/5.26 composition(meet(x1, converse(x0)), meet(x0, x2))
% 38.10/5.26 = { by lemma 39 R->L }
% 38.10/5.26 join(composition(meet(x1, converse(x0)), meet(x0, x2)), zero)
% 38.10/5.26 = { by lemma 70 R->L }
% 38.10/5.26 join(composition(meet(x1, converse(x0)), meet(x0, x2)), composition(complement(converse(x0)), meet(x0, x2)))
% 38.10/5.26 = { by axiom 12 (composition_distributivity) R->L }
% 38.10/5.26 composition(join(meet(x1, converse(x0)), complement(converse(x0))), meet(x0, x2))
% 38.10/5.26 = { by lemma 66 R->L }
% 38.10/5.26 composition(join(meet(x1, converse(x0)), join(complement(converse(x0)), meet(complement(converse(x0)), x1))), meet(x0, x2))
% 38.10/5.26 = { by lemma 41 R->L }
% 38.10/5.26 composition(join(meet(x1, converse(x0)), join(complement(converse(x0)), meet(x1, complement(converse(x0))))), meet(x0, x2))
% 38.10/5.26 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 38.10/5.26 composition(join(meet(x1, converse(x0)), join(meet(x1, complement(converse(x0))), complement(converse(x0)))), meet(x0, x2))
% 38.10/5.26 = { by axiom 8 (maddux2_join_associativity) }
% 38.10/5.26 composition(join(join(meet(x1, converse(x0)), meet(x1, complement(converse(x0)))), complement(converse(x0))), meet(x0, x2))
% 38.10/5.26 = { by lemma 45 }
% 38.10/5.26 composition(join(x1, complement(converse(x0))), meet(x0, x2))
% 38.10/5.26 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 38.10/5.26 composition(join(complement(converse(x0)), x1), meet(x0, x2))
% 38.10/5.26 = { by axiom 12 (composition_distributivity) }
% 38.10/5.26 join(composition(complement(converse(x0)), meet(x0, x2)), composition(x1, meet(x0, x2)))
% 38.10/5.26 = { by lemma 70 }
% 38.10/5.26 join(zero, composition(x1, meet(x0, x2)))
% 38.10/5.26 = { by lemma 40 }
% 38.10/5.26 composition(x1, meet(x0, x2))
% 38.10/5.26 % SZS output end Proof
% 38.10/5.26
% 38.10/5.26 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------