TSTP Solution File: REL004+2 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL004+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 : n008.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 13:43:44 EDT 2023
% Result : Theorem 0.22s 0.62s
% Output : Proof 0.22s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : REL004+2 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.15/0.35 % Computer : n008.cluster.edu
% 0.15/0.35 % Model : x86_64 x86_64
% 0.15/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.35 % Memory : 8042.1875MB
% 0.15/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.35 % CPULimit : 300
% 0.15/0.35 % WCLimit : 300
% 0.15/0.35 % DateTime : Fri Aug 25 19:06:47 EDT 2023
% 0.15/0.35 % CPUTime :
% 0.22/0.62 Command-line arguments: --no-flatten-goal
% 0.22/0.62
% 0.22/0.62 % SZS status Theorem
% 0.22/0.62
% 0.22/0.65 % SZS output start Proof
% 0.22/0.65 Axiom 1 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 0.22/0.65 Axiom 2 (converse_idempotence): converse(converse(X)) = X.
% 0.22/0.65 Axiom 3 (composition_identity): composition(X, one) = X.
% 0.22/0.65 Axiom 4 (def_zero): zero = meet(X, complement(X)).
% 0.22/0.65 Axiom 5 (def_top): top = join(X, complement(X)).
% 0.22/0.65 Axiom 6 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 0.22/0.65 Axiom 7 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 0.22/0.65 Axiom 8 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 0.22/0.65 Axiom 9 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 0.22/0.65 Axiom 10 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 0.22/0.65 Axiom 11 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 0.22/0.65 Axiom 12 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 0.22/0.65 Axiom 13 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 0.22/0.65 Axiom 14 (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).
% 0.22/0.65
% 0.22/0.65 Lemma 15: complement(top) = zero.
% 0.22/0.65 Proof:
% 0.22/0.65 complement(top)
% 0.22/0.65 = { by axiom 5 (def_top) }
% 0.22/0.65 complement(join(complement(X), complement(complement(X))))
% 0.22/0.65 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 0.22/0.65 meet(X, complement(X))
% 0.22/0.65 = { by axiom 4 (def_zero) R->L }
% 0.22/0.65 zero
% 0.22/0.65
% 0.22/0.65 Lemma 16: join(X, join(Y, complement(X))) = join(Y, top).
% 0.22/0.65 Proof:
% 0.22/0.65 join(X, join(Y, complement(X)))
% 0.22/0.65 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 0.22/0.65 join(X, join(complement(X), Y))
% 0.22/0.65 = { by axiom 7 (maddux2_join_associativity) }
% 0.22/0.65 join(join(X, complement(X)), Y)
% 0.22/0.65 = { by axiom 5 (def_top) R->L }
% 0.22/0.65 join(top, Y)
% 0.22/0.65 = { by axiom 1 (maddux1_join_commutativity) }
% 0.22/0.65 join(Y, top)
% 0.22/0.65
% 0.22/0.65 Lemma 17: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 0.22/0.65 Proof:
% 0.22/0.65 converse(composition(converse(X), Y))
% 0.22/0.65 = { by axiom 8 (converse_multiplicativity) }
% 0.22/0.65 composition(converse(Y), converse(converse(X)))
% 0.22/0.65 = { by axiom 2 (converse_idempotence) }
% 0.22/0.65 composition(converse(Y), X)
% 0.22/0.65
% 0.22/0.65 Lemma 18: composition(converse(one), X) = X.
% 0.22/0.65 Proof:
% 0.22/0.65 composition(converse(one), X)
% 0.22/0.65 = { by lemma 17 R->L }
% 0.22/0.65 converse(composition(converse(X), one))
% 0.22/0.65 = { by axiom 3 (composition_identity) }
% 0.22/0.65 converse(converse(X))
% 0.22/0.65 = { by axiom 2 (converse_idempotence) }
% 0.22/0.65 X
% 0.22/0.65
% 0.22/0.65 Lemma 19: composition(one, X) = X.
% 0.22/0.65 Proof:
% 0.22/0.65 composition(one, X)
% 0.22/0.65 = { by lemma 18 R->L }
% 0.22/0.65 composition(converse(one), composition(one, X))
% 0.22/0.65 = { by axiom 9 (composition_associativity) }
% 0.22/0.65 composition(composition(converse(one), one), X)
% 0.22/0.65 = { by axiom 3 (composition_identity) }
% 0.22/0.65 composition(converse(one), X)
% 0.22/0.65 = { by lemma 18 }
% 0.22/0.65 X
% 0.22/0.65
% 0.22/0.65 Lemma 20: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 0.22/0.65 Proof:
% 0.22/0.65 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 0.22/0.65 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 0.22/0.65 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 0.22/0.65 = { by axiom 12 (converse_cancellativity) }
% 0.22/0.66 complement(X)
% 0.22/0.66
% 0.22/0.66 Lemma 21: join(complement(X), complement(X)) = complement(X).
% 0.22/0.66 Proof:
% 0.22/0.66 join(complement(X), complement(X))
% 0.22/0.66 = { by lemma 18 R->L }
% 0.22/0.66 join(complement(X), composition(converse(one), complement(X)))
% 0.22/0.66 = { by lemma 19 R->L }
% 0.22/0.66 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 0.22/0.66 = { by lemma 20 }
% 0.22/0.66 complement(X)
% 0.22/0.66
% 0.22/0.66 Lemma 22: join(top, complement(X)) = top.
% 0.22/0.66 Proof:
% 0.22/0.66 join(top, complement(X))
% 0.22/0.66 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 0.22/0.66 join(complement(X), top)
% 0.22/0.66 = { by lemma 16 R->L }
% 0.22/0.66 join(X, join(complement(X), complement(X)))
% 0.22/0.66 = { by lemma 21 }
% 0.22/0.66 join(X, complement(X))
% 0.22/0.66 = { by axiom 5 (def_top) R->L }
% 0.22/0.66 top
% 0.22/0.66
% 0.22/0.66 Lemma 23: join(Y, top) = join(X, top).
% 0.22/0.66 Proof:
% 0.22/0.66 join(Y, top)
% 0.22/0.66 = { by lemma 22 R->L }
% 0.22/0.66 join(Y, join(top, complement(Y)))
% 0.22/0.66 = { by lemma 16 }
% 0.22/0.66 join(top, top)
% 0.22/0.66 = { by lemma 16 R->L }
% 0.22/0.66 join(X, join(top, complement(X)))
% 0.22/0.66 = { by lemma 22 }
% 0.22/0.66 join(X, top)
% 0.22/0.66
% 0.22/0.66 Lemma 24: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 0.22/0.66 Proof:
% 0.22/0.66 join(meet(X, Y), complement(join(complement(X), Y)))
% 0.22/0.66 = { by axiom 10 (maddux4_definiton_of_meet) }
% 0.22/0.66 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 0.22/0.66 = { by axiom 13 (maddux3_a_kind_of_de_Morgan) R->L }
% 0.22/0.66 X
% 0.22/0.66
% 0.22/0.66 Lemma 25: join(zero, meet(X, X)) = X.
% 0.22/0.66 Proof:
% 0.22/0.66 join(zero, meet(X, X))
% 0.22/0.66 = { by axiom 10 (maddux4_definiton_of_meet) }
% 0.22/0.66 join(zero, complement(join(complement(X), complement(X))))
% 0.22/0.66 = { by axiom 4 (def_zero) }
% 0.22/0.66 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 0.22/0.66 = { by lemma 24 }
% 0.22/0.66 X
% 0.22/0.66
% 0.22/0.66 Lemma 26: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 0.22/0.66 Proof:
% 0.22/0.66 join(zero, join(X, complement(complement(Y))))
% 0.22/0.66 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 0.22/0.66 join(zero, join(complement(complement(Y)), X))
% 0.22/0.66 = { by lemma 21 R->L }
% 0.22/0.66 join(zero, join(complement(join(complement(Y), complement(Y))), X))
% 0.22/0.66 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 0.22/0.66 join(zero, join(meet(Y, Y), X))
% 0.22/0.66 = { by axiom 7 (maddux2_join_associativity) }
% 0.22/0.66 join(join(zero, meet(Y, Y)), X)
% 0.22/0.66 = { by lemma 25 }
% 0.22/0.66 join(Y, X)
% 0.22/0.66 = { by axiom 1 (maddux1_join_commutativity) }
% 0.22/0.66 join(X, Y)
% 0.22/0.66
% 0.22/0.66 Lemma 27: join(zero, complement(complement(X))) = X.
% 0.22/0.66 Proof:
% 0.22/0.66 join(zero, complement(complement(X)))
% 0.22/0.66 = { by axiom 4 (def_zero) }
% 0.22/0.66 join(meet(X, complement(X)), complement(complement(X)))
% 0.22/0.66 = { by lemma 21 R->L }
% 0.22/0.66 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 0.22/0.66 = { by lemma 24 }
% 0.22/0.66 X
% 0.22/0.66
% 0.22/0.66 Lemma 28: join(X, zero) = join(X, X).
% 0.22/0.66 Proof:
% 0.22/0.66 join(X, zero)
% 0.22/0.66 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 0.22/0.66 join(zero, X)
% 0.22/0.66 = { by lemma 27 R->L }
% 0.22/0.66 join(zero, join(zero, complement(complement(X))))
% 0.22/0.66 = { by lemma 21 R->L }
% 0.22/0.66 join(zero, join(zero, join(complement(complement(X)), complement(complement(X)))))
% 0.22/0.66 = { by lemma 26 }
% 0.22/0.66 join(zero, join(complement(complement(X)), X))
% 0.22/0.66 = { by axiom 1 (maddux1_join_commutativity) }
% 0.22/0.66 join(zero, join(X, complement(complement(X))))
% 0.22/0.66 = { by lemma 26 }
% 0.22/0.66 join(X, X)
% 0.22/0.66
% 0.22/0.66 Lemma 29: join(zero, complement(X)) = complement(X).
% 0.22/0.66 Proof:
% 0.22/0.66 join(zero, complement(X))
% 0.22/0.66 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 0.22/0.66 join(complement(X), zero)
% 0.22/0.66 = { by lemma 28 }
% 0.22/0.66 join(complement(X), complement(X))
% 0.22/0.66 = { by lemma 21 }
% 0.22/0.66 complement(X)
% 0.22/0.66
% 0.22/0.66 Lemma 30: join(X, zero) = X.
% 0.22/0.66 Proof:
% 0.22/0.66 join(X, zero)
% 0.22/0.66 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 0.22/0.66 join(zero, X)
% 0.22/0.66 = { by lemma 26 R->L }
% 0.22/0.66 join(zero, join(zero, complement(complement(X))))
% 0.22/0.66 = { by lemma 29 }
% 0.22/0.66 join(zero, complement(complement(X)))
% 0.22/0.66 = { by lemma 27 }
% 0.22/0.66 X
% 0.22/0.66
% 0.22/0.66 Lemma 31: join(X, top) = top.
% 0.22/0.66 Proof:
% 0.22/0.66 join(X, top)
% 0.22/0.66 = { by lemma 23 }
% 0.22/0.66 join(join(zero, zero), top)
% 0.22/0.66 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 0.22/0.66 join(top, join(zero, zero))
% 0.22/0.66 = { by lemma 30 }
% 0.22/0.66 join(top, zero)
% 0.22/0.66 = { by lemma 30 }
% 0.22/0.66 top
% 0.22/0.66
% 0.22/0.66 Lemma 32: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 0.22/0.66 Proof:
% 0.22/0.66 converse(join(X, converse(Y)))
% 0.22/0.66 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 0.22/0.66 converse(join(converse(Y), X))
% 0.22/0.66 = { by axiom 6 (converse_additivity) }
% 0.22/0.66 join(converse(converse(Y)), converse(X))
% 0.22/0.66 = { by axiom 2 (converse_idempotence) }
% 0.22/0.66 join(Y, converse(X))
% 0.22/0.66
% 0.22/0.66 Lemma 33: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 0.22/0.66 Proof:
% 0.22/0.66 converse(join(converse(X), Y))
% 0.22/0.66 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 0.22/0.66 converse(join(Y, converse(X)))
% 0.22/0.66 = { by lemma 32 }
% 0.22/0.66 join(X, converse(Y))
% 0.22/0.66
% 0.22/0.66 Lemma 34: join(X, converse(complement(converse(X)))) = converse(top).
% 0.22/0.66 Proof:
% 0.22/0.66 join(X, converse(complement(converse(X))))
% 0.22/0.66 = { by lemma 33 R->L }
% 0.22/0.66 converse(join(converse(X), complement(converse(X))))
% 0.22/0.66 = { by axiom 5 (def_top) R->L }
% 0.22/0.66 converse(top)
% 0.22/0.66
% 0.22/0.66 Lemma 35: join(X, join(complement(X), Y)) = top.
% 0.22/0.66 Proof:
% 0.22/0.66 join(X, join(complement(X), Y))
% 0.22/0.66 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 0.22/0.66 join(X, join(Y, complement(X)))
% 0.22/0.66 = { by lemma 16 }
% 0.22/0.66 join(Y, top)
% 0.22/0.66 = { by lemma 23 R->L }
% 0.22/0.66 join(Z, top)
% 0.22/0.66 = { by lemma 31 }
% 0.22/0.66 top
% 0.22/0.66
% 0.22/0.66 Lemma 36: join(X, converse(top)) = top.
% 0.22/0.66 Proof:
% 0.22/0.66 join(X, converse(top))
% 0.22/0.66 = { by lemma 34 R->L }
% 0.22/0.66 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 0.22/0.66 = { by lemma 35 }
% 0.22/0.66 top
% 0.22/0.66
% 0.22/0.66 Lemma 37: converse(top) = top.
% 0.22/0.66 Proof:
% 0.22/0.66 converse(top)
% 0.22/0.66 = { by lemma 31 R->L }
% 0.22/0.66 converse(join(X, top))
% 0.22/0.66 = { by axiom 6 (converse_additivity) }
% 0.22/0.66 join(converse(X), converse(top))
% 0.22/0.66 = { by lemma 36 }
% 0.22/0.66 top
% 0.22/0.66
% 0.22/0.66 Lemma 38: join(zero, X) = X.
% 0.22/0.66 Proof:
% 0.22/0.66 join(zero, X)
% 0.22/0.66 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 0.22/0.66 join(X, zero)
% 0.22/0.66 = { by lemma 30 }
% 0.22/0.66 X
% 0.22/0.66
% 0.22/0.66 Lemma 39: meet(Y, X) = meet(X, Y).
% 0.22/0.66 Proof:
% 0.22/0.66 meet(Y, X)
% 0.22/0.66 = { by axiom 10 (maddux4_definiton_of_meet) }
% 0.22/0.66 complement(join(complement(Y), complement(X)))
% 0.22/0.66 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 0.22/0.66 complement(join(complement(X), complement(Y)))
% 0.22/0.66 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 0.22/0.66 meet(X, Y)
% 0.22/0.66
% 0.22/0.66 Lemma 40: complement(join(zero, complement(X))) = meet(X, top).
% 0.22/0.66 Proof:
% 0.22/0.66 complement(join(zero, complement(X)))
% 0.22/0.66 = { by lemma 15 R->L }
% 0.22/0.66 complement(join(complement(top), complement(X)))
% 0.22/0.66 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 0.22/0.66 meet(top, X)
% 0.22/0.66 = { by lemma 39 R->L }
% 0.22/0.66 meet(X, top)
% 0.22/0.66
% 0.22/0.66 Lemma 41: join(X, complement(zero)) = top.
% 0.22/0.66 Proof:
% 0.22/0.66 join(X, complement(zero))
% 0.22/0.66 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 0.22/0.66 join(complement(zero), X)
% 0.22/0.66 = { by lemma 26 R->L }
% 0.22/0.66 join(zero, join(complement(zero), complement(complement(X))))
% 0.22/0.66 = { by lemma 35 }
% 0.22/0.66 top
% 0.22/0.66
% 0.22/0.66 Lemma 42: meet(X, zero) = zero.
% 0.22/0.66 Proof:
% 0.22/0.66 meet(X, zero)
% 0.22/0.66 = { by axiom 10 (maddux4_definiton_of_meet) }
% 0.22/0.66 complement(join(complement(X), complement(zero)))
% 0.22/0.66 = { by lemma 41 }
% 0.22/0.66 complement(top)
% 0.22/0.66 = { by lemma 15 }
% 0.22/0.66 zero
% 0.22/0.66
% 0.22/0.66 Lemma 43: join(meet(X, Y), meet(X, complement(Y))) = X.
% 0.22/0.66 Proof:
% 0.22/0.66 join(meet(X, Y), meet(X, complement(Y)))
% 0.22/0.66 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 0.22/0.66 join(meet(X, complement(Y)), meet(X, Y))
% 0.22/0.66 = { by axiom 10 (maddux4_definiton_of_meet) }
% 0.22/0.66 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 0.22/0.66 = { by lemma 24 }
% 0.22/0.66 X
% 0.22/0.66
% 0.22/0.66 Lemma 44: meet(X, top) = X.
% 0.22/0.66 Proof:
% 0.22/0.66 meet(X, top)
% 0.22/0.66 = { by lemma 40 R->L }
% 0.22/0.66 complement(join(zero, complement(X)))
% 0.22/0.66 = { by lemma 29 R->L }
% 0.22/0.66 join(zero, complement(join(zero, complement(X))))
% 0.22/0.66 = { by lemma 40 }
% 0.22/0.66 join(zero, meet(X, top))
% 0.22/0.66 = { by lemma 41 R->L }
% 0.22/0.66 join(zero, meet(X, join(complement(zero), complement(zero))))
% 0.22/0.66 = { by lemma 21 }
% 0.22/0.66 join(zero, meet(X, complement(zero)))
% 0.22/0.66 = { by lemma 42 R->L }
% 0.22/0.66 join(meet(X, zero), meet(X, complement(zero)))
% 0.22/0.66 = { by lemma 43 }
% 0.22/0.66 X
% 0.22/0.66
% 0.22/0.66 Lemma 45: join(meet(X, Y), meet(X, Y)) = meet(X, Y).
% 0.22/0.66 Proof:
% 0.22/0.66 join(meet(X, Y), meet(X, Y))
% 0.22/0.66 = { by lemma 39 }
% 0.22/0.66 join(meet(Y, X), meet(X, Y))
% 0.22/0.66 = { by lemma 39 }
% 0.22/0.66 join(meet(Y, X), meet(Y, X))
% 0.22/0.66 = { by axiom 10 (maddux4_definiton_of_meet) }
% 0.22/0.66 join(meet(Y, X), complement(join(complement(Y), complement(X))))
% 0.22/0.66 = { by axiom 10 (maddux4_definiton_of_meet) }
% 0.22/0.66 join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X))))
% 0.22/0.66 = { by lemma 21 }
% 0.22/0.66 complement(join(complement(Y), complement(X)))
% 0.22/0.66 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 0.22/0.66 meet(Y, X)
% 0.22/0.66 = { by lemma 39 R->L }
% 0.22/0.66 meet(X, Y)
% 0.22/0.66
% 0.22/0.66 Lemma 46: converse(zero) = zero.
% 0.22/0.66 Proof:
% 0.22/0.66 converse(zero)
% 0.22/0.66 = { by lemma 38 R->L }
% 0.22/0.66 join(zero, converse(zero))
% 0.22/0.66 = { by lemma 33 R->L }
% 0.22/0.66 converse(join(converse(zero), zero))
% 0.22/0.66 = { by lemma 28 }
% 0.22/0.66 converse(join(converse(zero), converse(zero)))
% 0.22/0.66 = { by lemma 32 }
% 0.22/0.66 join(zero, converse(converse(zero)))
% 0.22/0.66 = { by axiom 2 (converse_idempotence) }
% 0.22/0.66 join(zero, zero)
% 0.22/0.66 = { by lemma 44 R->L }
% 0.22/0.66 join(zero, meet(zero, top))
% 0.22/0.66 = { by lemma 44 R->L }
% 0.22/0.66 join(meet(zero, top), meet(zero, top))
% 0.22/0.66 = { by lemma 45 }
% 0.22/0.66 meet(zero, top)
% 0.22/0.66 = { by lemma 44 }
% 0.22/0.66 zero
% 0.22/0.66
% 0.22/0.66 Lemma 47: join(top, X) = top.
% 0.22/0.66 Proof:
% 0.22/0.66 join(top, X)
% 0.22/0.66 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 0.22/0.66 join(X, top)
% 0.22/0.66 = { by lemma 23 R->L }
% 0.22/0.66 join(Y, top)
% 0.22/0.66 = { by lemma 31 }
% 0.22/0.66 top
% 0.22/0.66
% 0.22/0.66 Lemma 48: meet(zero, X) = zero.
% 0.22/0.66 Proof:
% 0.22/0.66 meet(zero, X)
% 0.22/0.66 = { by lemma 39 }
% 0.22/0.66 meet(X, zero)
% 0.22/0.66 = { by lemma 42 }
% 0.22/0.66 zero
% 0.22/0.66
% 0.22/0.66 Lemma 49: composition(top, zero) = zero.
% 0.22/0.66 Proof:
% 0.22/0.66 composition(top, zero)
% 0.22/0.66 = { by lemma 37 R->L }
% 0.22/0.66 composition(converse(top), zero)
% 0.22/0.66 = { by lemma 38 R->L }
% 0.22/0.66 join(zero, composition(converse(top), zero))
% 0.22/0.66 = { by lemma 15 R->L }
% 0.22/0.66 join(complement(top), composition(converse(top), zero))
% 0.22/0.66 = { by lemma 15 R->L }
% 0.22/0.66 join(complement(top), composition(converse(top), complement(top)))
% 0.22/0.66 = { by lemma 47 R->L }
% 0.22/0.66 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 0.22/0.66 = { by lemma 37 R->L }
% 0.22/0.66 join(complement(top), composition(converse(top), complement(join(top, composition(converse(top), top)))))
% 0.22/0.66 = { by lemma 19 R->L }
% 0.22/0.66 join(complement(top), composition(converse(top), complement(join(composition(one, top), composition(converse(top), top)))))
% 0.22/0.66 = { by axiom 11 (composition_distributivity) R->L }
% 0.22/0.66 join(complement(top), composition(converse(top), complement(composition(join(one, converse(top)), top))))
% 0.22/0.66 = { by lemma 36 }
% 0.22/0.66 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 0.22/0.66 = { by lemma 20 }
% 0.22/0.66 complement(top)
% 0.22/0.66 = { by lemma 15 }
% 0.22/0.66 zero
% 0.22/0.66
% 0.22/0.66 Lemma 50: composition(zero, X) = zero.
% 0.22/0.66 Proof:
% 0.22/0.66 composition(zero, X)
% 0.22/0.66 = { by lemma 46 R->L }
% 0.22/0.66 composition(converse(zero), X)
% 0.22/0.66 = { by lemma 17 R->L }
% 0.22/0.66 converse(composition(converse(X), zero))
% 0.22/0.66 = { by lemma 38 R->L }
% 0.22/0.66 converse(join(zero, composition(converse(X), zero)))
% 0.22/0.66 = { by lemma 49 R->L }
% 0.22/0.66 converse(join(composition(top, zero), composition(converse(X), zero)))
% 0.22/0.66 = { by axiom 11 (composition_distributivity) R->L }
% 0.22/0.66 converse(composition(join(top, converse(X)), zero))
% 0.22/0.66 = { by lemma 47 }
% 0.22/0.66 converse(composition(top, zero))
% 0.22/0.66 = { by lemma 49 }
% 0.22/0.66 converse(zero)
% 0.22/0.66 = { by lemma 46 }
% 0.22/0.66 zero
% 0.22/0.66
% 0.22/0.66 Lemma 51: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 0.22/0.66 Proof:
% 0.22/0.66 meet(X, join(complement(Y), complement(Z)))
% 0.22/0.66 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 0.22/0.66 meet(X, join(complement(Z), complement(Y)))
% 0.22/0.66 = { by lemma 39 }
% 0.22/0.66 meet(join(complement(Z), complement(Y)), X)
% 0.22/0.66 = { by axiom 10 (maddux4_definiton_of_meet) }
% 0.22/0.66 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 0.22/0.66 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 0.22/0.66 complement(join(meet(Z, Y), complement(X)))
% 0.22/0.66 = { by axiom 1 (maddux1_join_commutativity) }
% 0.22/0.66 complement(join(complement(X), meet(Z, Y)))
% 0.22/0.66 = { by lemma 39 R->L }
% 0.22/0.66 complement(join(complement(X), meet(Y, Z)))
% 0.22/0.66
% 0.22/0.66 Lemma 52: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 0.22/0.66 Proof:
% 0.22/0.66 complement(join(complement(X), Y))
% 0.22/0.66 = { by lemma 44 R->L }
% 0.22/0.66 complement(join(complement(X), meet(Y, top)))
% 0.22/0.66 = { by lemma 39 R->L }
% 0.22/0.66 complement(join(complement(X), meet(top, Y)))
% 0.22/0.66 = { by lemma 51 R->L }
% 0.22/0.66 meet(X, join(complement(top), complement(Y)))
% 0.22/0.66 = { by lemma 15 }
% 0.22/0.66 meet(X, join(zero, complement(Y)))
% 0.22/0.67 = { by lemma 29 }
% 0.22/0.67 meet(X, complement(Y))
% 0.22/0.67
% 0.22/0.67 Lemma 53: meet(one, composition(converse(complement(X)), X)) = zero.
% 0.22/0.67 Proof:
% 0.22/0.67 meet(one, composition(converse(complement(X)), X))
% 0.22/0.67 = { by lemma 39 }
% 0.22/0.67 meet(composition(converse(complement(X)), X), one)
% 0.22/0.67 = { by lemma 27 R->L }
% 0.22/0.67 meet(composition(converse(complement(X)), X), join(zero, complement(complement(one))))
% 0.22/0.67 = { by lemma 29 }
% 0.22/0.67 meet(composition(converse(complement(X)), X), complement(complement(one)))
% 0.22/0.67 = { by lemma 20 R->L }
% 0.22/0.67 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(join(zero, complement(X))), complement(composition(join(zero, complement(X)), one))))))
% 0.22/0.67 = { by axiom 3 (composition_identity) }
% 0.22/0.67 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(join(zero, complement(X))), complement(join(zero, complement(X)))))))
% 0.22/0.67 = { by lemma 40 }
% 0.22/0.67 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(join(zero, complement(X))), meet(X, top)))))
% 0.22/0.67 = { by lemma 29 }
% 0.22/0.67 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(complement(X)), meet(X, top)))))
% 0.22/0.67 = { by lemma 44 }
% 0.22/0.67 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(complement(X)), X))))
% 0.22/0.67 = { by lemma 52 }
% 0.22/0.67 meet(composition(converse(complement(X)), X), meet(one, complement(composition(converse(complement(X)), X))))
% 0.22/0.67 = { by lemma 39 }
% 0.22/0.67 meet(composition(converse(complement(X)), X), meet(complement(composition(converse(complement(X)), X)), one))
% 0.22/0.67 = { by axiom 10 (maddux4_definiton_of_meet) }
% 0.22/0.67 complement(join(complement(composition(converse(complement(X)), X)), complement(meet(complement(composition(converse(complement(X)), X)), one))))
% 0.22/0.67 = { by lemma 45 R->L }
% 0.22/0.67 complement(join(complement(composition(converse(complement(X)), X)), complement(join(meet(complement(composition(converse(complement(X)), X)), one), meet(complement(composition(converse(complement(X)), X)), one)))))
% 0.22/0.67 = { by axiom 10 (maddux4_definiton_of_meet) }
% 0.22/0.67 complement(join(complement(composition(converse(complement(X)), X)), complement(join(complement(join(complement(complement(composition(converse(complement(X)), X))), complement(one))), meet(complement(composition(converse(complement(X)), X)), one)))))
% 0.22/0.67 = { by lemma 29 R->L }
% 0.22/0.67 complement(join(complement(composition(converse(complement(X)), X)), join(zero, complement(join(complement(join(complement(complement(composition(converse(complement(X)), X))), complement(one))), meet(complement(composition(converse(complement(X)), X)), one))))))
% 0.22/0.67 = { by lemma 51 R->L }
% 0.22/0.67 complement(join(complement(composition(converse(complement(X)), X)), join(zero, meet(join(complement(complement(composition(converse(complement(X)), X))), complement(one)), join(complement(complement(composition(converse(complement(X)), X))), complement(one))))))
% 0.22/0.67 = { by lemma 25 }
% 0.22/0.67 complement(join(complement(composition(converse(complement(X)), X)), join(complement(complement(composition(converse(complement(X)), X))), complement(one))))
% 0.22/0.67 = { by axiom 1 (maddux1_join_commutativity) }
% 0.22/0.67 complement(join(complement(composition(converse(complement(X)), X)), join(complement(one), complement(complement(composition(converse(complement(X)), X))))))
% 0.22/0.67 = { by lemma 16 }
% 0.22/0.67 complement(join(complement(one), top))
% 0.22/0.67 = { by lemma 31 }
% 0.22/0.67 complement(top)
% 0.22/0.67 = { by lemma 15 }
% 0.22/0.67 zero
% 0.22/0.67
% 0.22/0.67 Goal 1 (goals): converse(complement(x0)) = complement(converse(x0)).
% 0.22/0.67 Proof:
% 0.22/0.67 converse(complement(x0))
% 0.22/0.67 = { by lemma 29 R->L }
% 0.22/0.67 converse(join(zero, complement(x0)))
% 0.22/0.67 = { by lemma 24 R->L }
% 0.22/0.67 converse(join(meet(join(zero, complement(x0)), complement(converse(complement(converse(complement(join(zero, complement(x0)))))))), complement(join(complement(join(zero, complement(x0))), complement(converse(complement(converse(complement(join(zero, complement(x0)))))))))))
% 0.22/0.67 = { by lemma 52 R->L }
% 0.22/0.67 converse(join(complement(join(complement(join(zero, complement(x0))), converse(complement(converse(complement(join(zero, complement(x0)))))))), complement(join(complement(join(zero, complement(x0))), complement(converse(complement(converse(complement(join(zero, complement(x0)))))))))))
% 0.22/0.67 = { by lemma 34 }
% 0.22/0.67 converse(join(complement(converse(top)), complement(join(complement(join(zero, complement(x0))), complement(converse(complement(converse(complement(join(zero, complement(x0)))))))))))
% 0.22/0.67 = { by lemma 37 }
% 0.22/0.67 converse(join(complement(top), complement(join(complement(join(zero, complement(x0))), complement(converse(complement(converse(complement(join(zero, complement(x0)))))))))))
% 0.22/0.67 = { by lemma 15 }
% 0.22/0.67 converse(join(zero, complement(join(complement(join(zero, complement(x0))), complement(converse(complement(converse(complement(join(zero, complement(x0)))))))))))
% 0.22/0.67 = { by lemma 29 }
% 0.22/0.67 converse(complement(join(complement(join(zero, complement(x0))), complement(converse(complement(converse(complement(join(zero, complement(x0))))))))))
% 0.22/0.67 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 0.22/0.67 converse(meet(join(zero, complement(x0)), converse(complement(converse(complement(join(zero, complement(x0))))))))
% 0.22/0.67 = { by lemma 30 R->L }
% 0.22/0.67 converse(join(meet(join(zero, complement(x0)), converse(complement(converse(complement(join(zero, complement(x0))))))), zero))
% 0.22/0.67 = { by lemma 48 R->L }
% 0.22/0.67 converse(join(meet(join(zero, complement(x0)), converse(complement(converse(complement(join(zero, complement(x0))))))), meet(zero, converse(complement(converse(complement(join(zero, complement(x0)))))))))
% 0.22/0.67 = { by lemma 50 R->L }
% 0.22/0.67 converse(join(meet(join(zero, complement(x0)), converse(complement(converse(complement(join(zero, complement(x0))))))), meet(composition(zero, complement(join(zero, complement(x0)))), converse(complement(converse(complement(join(zero, complement(x0)))))))))
% 0.22/0.67 = { by lemma 53 R->L }
% 0.22/0.67 converse(join(meet(join(zero, complement(x0)), converse(complement(converse(complement(join(zero, complement(x0))))))), meet(composition(meet(one, composition(converse(complement(converse(complement(join(zero, complement(x0)))))), converse(complement(join(zero, complement(x0)))))), complement(join(zero, complement(x0)))), converse(complement(converse(complement(join(zero, complement(x0)))))))))
% 0.22/0.67 = { by axiom 14 (modular_law_2) R->L }
% 0.22/0.67 converse(join(meet(join(zero, complement(x0)), converse(complement(converse(complement(join(zero, complement(x0))))))), join(meet(composition(one, complement(join(zero, complement(x0)))), converse(complement(converse(complement(join(zero, complement(x0))))))), meet(composition(meet(one, composition(converse(complement(converse(complement(join(zero, complement(x0)))))), converse(complement(join(zero, complement(x0)))))), complement(join(zero, complement(x0)))), converse(complement(converse(complement(join(zero, complement(x0))))))))))
% 0.22/0.67 = { by lemma 53 }
% 0.22/0.67 converse(join(meet(join(zero, complement(x0)), converse(complement(converse(complement(join(zero, complement(x0))))))), join(meet(composition(one, complement(join(zero, complement(x0)))), converse(complement(converse(complement(join(zero, complement(x0))))))), meet(composition(zero, complement(join(zero, complement(x0)))), converse(complement(converse(complement(join(zero, complement(x0))))))))))
% 0.22/0.67 = { by lemma 19 }
% 0.22/0.67 converse(join(meet(join(zero, complement(x0)), converse(complement(converse(complement(join(zero, complement(x0))))))), join(meet(complement(join(zero, complement(x0))), converse(complement(converse(complement(join(zero, complement(x0))))))), meet(composition(zero, complement(join(zero, complement(x0)))), converse(complement(converse(complement(join(zero, complement(x0))))))))))
% 0.22/0.67 = { by lemma 50 }
% 0.22/0.67 converse(join(meet(join(zero, complement(x0)), converse(complement(converse(complement(join(zero, complement(x0))))))), join(meet(complement(join(zero, complement(x0))), converse(complement(converse(complement(join(zero, complement(x0))))))), meet(zero, converse(complement(converse(complement(join(zero, complement(x0))))))))))
% 0.22/0.67 = { by lemma 48 }
% 0.22/0.67 converse(join(meet(join(zero, complement(x0)), converse(complement(converse(complement(join(zero, complement(x0))))))), join(meet(complement(join(zero, complement(x0))), converse(complement(converse(complement(join(zero, complement(x0))))))), zero)))
% 0.22/0.67 = { by lemma 30 }
% 0.22/0.67 converse(join(meet(join(zero, complement(x0)), converse(complement(converse(complement(join(zero, complement(x0))))))), meet(complement(join(zero, complement(x0))), converse(complement(converse(complement(join(zero, complement(x0)))))))))
% 0.22/0.67 = { by lemma 39 }
% 0.22/0.67 converse(join(meet(join(zero, complement(x0)), converse(complement(converse(complement(join(zero, complement(x0))))))), meet(converse(complement(converse(complement(join(zero, complement(x0)))))), complement(join(zero, complement(x0))))))
% 0.22/0.67 = { by lemma 39 }
% 0.22/0.67 converse(join(meet(converse(complement(converse(complement(join(zero, complement(x0)))))), join(zero, complement(x0))), meet(converse(complement(converse(complement(join(zero, complement(x0)))))), complement(join(zero, complement(x0))))))
% 0.22/0.67 = { by lemma 43 }
% 0.22/0.67 converse(converse(complement(converse(complement(join(zero, complement(x0)))))))
% 0.22/0.67 = { by axiom 2 (converse_idempotence) }
% 0.22/0.67 complement(converse(complement(join(zero, complement(x0)))))
% 0.22/0.67 = { by lemma 40 }
% 0.22/0.67 complement(converse(meet(x0, top)))
% 0.22/0.67 = { by lemma 44 }
% 0.22/0.67 complement(converse(x0))
% 0.22/0.67 % SZS output end Proof
% 0.22/0.67
% 0.22/0.67 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------