0.00/0.11 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.00/0.11 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof 0.11/0.32 % Computer : n029.cluster.edu 0.11/0.32 % Model : x86_64 x86_64 0.11/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.11/0.32 % Memory : 8042.1875MB 0.11/0.32 % OS : Linux 3.10.0-693.el7.x86_64 0.11/0.32 % CPULimit : 1200 0.11/0.32 % WCLimit : 120 0.11/0.32 % DateTime : Tue Jul 13 16:57:25 EDT 2021 0.11/0.32 % CPUTime : 61.78/8.11 % SZS status Theorem 61.78/8.11 62.68/8.24 % SZS output start Proof 62.68/8.24 Axiom 1 (converse_idempotence): converse(converse(X)) = X. 62.68/8.24 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X). 62.68/8.24 Axiom 3 (composition_identity): composition(X, one) = X. 62.68/8.24 Axiom 4 (def_zero): zero = meet(X, complement(X)). 62.68/8.24 Axiom 5 (def_top): join(X, complement(X)) = top. 62.68/8.24 Axiom 6 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)). 62.68/8.24 Axiom 7 (maddux2_join_associativity): join(join(X, Y), Z) = join(X, join(Y, Z)). 62.68/8.24 Axiom 8 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)). 62.68/8.24 Axiom 9 (composition_associativity): composition(composition(X, Y), Z) = composition(X, composition(Y, Z)). 62.68/8.24 Axiom 10 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))). 62.68/8.24 Axiom 11 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)). 62.68/8.24 Axiom 12 (goals): join(composition(complement(x0), x1), complement(x2)) = complement(x2). 62.68/8.24 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y). 62.68/8.24 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))). 62.68/8.24 Axiom 15 (dedekind_law): composition(meet(X, composition(Y, converse(Z))), meet(Z, composition(converse(X), Y))) = join(meet(composition(X, Z), Y), composition(meet(X, composition(Y, converse(Z))), meet(Z, composition(converse(X), Y)))). 62.68/8.24 62.68/8.24 Lemma 16: converse(composition(converse(X), Y)) = composition(converse(Y), X). 62.68/8.24 Proof: 62.68/8.24 converse(composition(converse(X), Y)) 62.68/8.24 = { by axiom 8 (converse_multiplicativity) } 62.68/8.24 composition(converse(Y), converse(converse(X))) 62.68/8.24 = { by axiom 1 (converse_idempotence) } 62.68/8.24 composition(converse(Y), X) 62.68/8.24 62.68/8.24 Lemma 17: composition(converse(one), X) = X. 62.68/8.24 Proof: 62.68/8.24 composition(converse(one), X) 62.68/8.24 = { by lemma 16 R->L } 62.68/8.24 converse(composition(converse(X), one)) 62.68/8.24 = { by axiom 3 (composition_identity) } 62.68/8.24 converse(converse(X)) 62.68/8.24 = { by axiom 1 (converse_idempotence) } 62.68/8.24 X 62.68/8.24 62.68/8.24 Lemma 18: composition(one, X) = X. 62.68/8.24 Proof: 62.68/8.24 composition(one, X) 62.68/8.24 = { by lemma 17 R->L } 62.68/8.24 composition(converse(one), composition(one, X)) 62.68/8.24 = { by axiom 9 (composition_associativity) R->L } 62.68/8.24 composition(composition(converse(one), one), X) 62.68/8.24 = { by axiom 3 (composition_identity) } 62.68/8.24 composition(converse(one), X) 62.68/8.24 = { by lemma 17 } 62.68/8.24 X 62.68/8.24 62.68/8.24 Lemma 19: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X). 62.68/8.24 Proof: 62.68/8.24 join(complement(X), composition(converse(Y), complement(composition(Y, X)))) 62.68/8.24 = { by axiom 2 (maddux1_join_commutativity) R->L } 62.68/8.24 join(composition(converse(Y), complement(composition(Y, X))), complement(X)) 62.68/8.24 = { by axiom 13 (converse_cancellativity) } 62.68/8.24 complement(X) 62.68/8.24 62.68/8.24 Lemma 20: join(complement(X), complement(X)) = complement(X). 62.68/8.24 Proof: 62.68/8.24 join(complement(X), complement(X)) 62.68/8.24 = { by lemma 17 R->L } 62.68/8.24 join(complement(X), composition(converse(one), complement(X))) 62.68/8.24 = { by lemma 18 R->L } 62.68/8.24 join(complement(X), composition(converse(one), complement(composition(one, X)))) 62.68/8.24 = { by lemma 19 } 62.68/8.24 complement(X) 62.68/8.24 62.68/8.24 Lemma 21: meet(Y, X) = meet(X, Y). 62.68/8.24 Proof: 62.68/8.24 meet(Y, X) 62.68/8.24 = { by axiom 10 (maddux4_definiton_of_meet) } 62.68/8.24 complement(join(complement(Y), complement(X))) 62.68/8.24 = { by axiom 2 (maddux1_join_commutativity) R->L } 62.68/8.24 complement(join(complement(X), complement(Y))) 62.68/8.24 = { by axiom 10 (maddux4_definiton_of_meet) R->L } 62.68/8.24 meet(X, Y) 62.68/8.24 62.68/8.24 Lemma 22: complement(join(complement(X), complement(Y))) = meet(Y, X). 62.68/8.24 Proof: 62.68/8.24 complement(join(complement(X), complement(Y))) 62.68/8.24 = { by axiom 10 (maddux4_definiton_of_meet) R->L } 62.68/8.24 meet(X, Y) 62.68/8.24 = { by lemma 21 R->L } 62.68/8.24 meet(Y, X) 62.68/8.24 62.68/8.24 Lemma 23: complement(complement(X)) = meet(X, X). 62.68/8.24 Proof: 62.68/8.24 complement(complement(X)) 62.68/8.24 = { by lemma 20 R->L } 62.68/8.24 complement(join(complement(X), complement(X))) 62.68/8.24 = { by lemma 22 } 62.68/8.24 meet(X, X) 62.68/8.24 62.68/8.24 Lemma 24: join(meet(X, Y), complement(join(complement(X), Y))) = X. 62.68/8.24 Proof: 62.68/8.24 join(meet(X, Y), complement(join(complement(X), Y))) 62.68/8.24 = { by axiom 10 (maddux4_definiton_of_meet) } 62.68/8.24 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))) 62.68/8.24 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L } 62.68/8.24 X 62.68/8.24 62.68/8.24 Lemma 25: join(zero, meet(X, X)) = X. 62.68/8.24 Proof: 62.68/8.24 join(zero, meet(X, X)) 62.68/8.24 = { by axiom 10 (maddux4_definiton_of_meet) } 62.68/8.24 join(zero, complement(join(complement(X), complement(X)))) 62.68/8.24 = { by axiom 4 (def_zero) } 62.68/8.24 join(meet(X, complement(X)), complement(join(complement(X), complement(X)))) 62.68/8.24 = { by lemma 24 } 62.68/8.24 X 62.68/8.24 62.68/8.24 Lemma 26: complement(top) = zero. 62.68/8.24 Proof: 62.68/8.24 complement(top) 62.68/8.24 = { by axiom 5 (def_top) R->L } 62.68/8.24 complement(join(complement(X), complement(complement(X)))) 62.68/8.24 = { by axiom 10 (maddux4_definiton_of_meet) R->L } 62.68/8.24 meet(X, complement(X)) 62.68/8.24 = { by axiom 4 (def_zero) R->L } 62.68/8.24 zero 62.68/8.24 62.68/8.24 Lemma 27: complement(join(zero, complement(X))) = meet(X, top). 62.68/8.24 Proof: 62.68/8.24 complement(join(zero, complement(X))) 62.68/8.24 = { by lemma 26 R->L } 62.68/8.24 complement(join(complement(top), complement(X))) 62.68/8.24 = { by axiom 10 (maddux4_definiton_of_meet) R->L } 62.68/8.24 meet(top, X) 62.68/8.24 = { by lemma 21 R->L } 62.68/8.24 meet(X, top) 62.68/8.24 62.68/8.24 Lemma 28: join(zero, join(X, meet(Y, Y))) = join(X, Y). 62.68/8.24 Proof: 62.68/8.24 join(zero, join(X, meet(Y, Y))) 62.68/8.24 = { by axiom 2 (maddux1_join_commutativity) R->L } 62.68/8.24 join(zero, join(meet(Y, Y), X)) 62.68/8.24 = { by axiom 7 (maddux2_join_associativity) R->L } 62.68/8.24 join(join(zero, meet(Y, Y)), X) 62.68/8.24 = { by lemma 25 } 62.68/8.24 join(Y, X) 62.68/8.24 = { by axiom 2 (maddux1_join_commutativity) } 62.68/8.24 join(X, Y) 62.68/8.24 62.68/8.24 Lemma 29: join(X, join(Y, complement(X))) = join(Y, top). 62.68/8.25 Proof: 62.68/8.25 join(X, join(Y, complement(X))) 62.68/8.25 = { by axiom 2 (maddux1_join_commutativity) R->L } 62.68/8.25 join(X, join(complement(X), Y)) 62.68/8.25 = { by axiom 7 (maddux2_join_associativity) R->L } 62.68/8.25 join(join(X, complement(X)), Y) 62.68/8.25 = { by axiom 5 (def_top) } 62.68/8.25 join(top, Y) 62.68/8.25 = { by axiom 2 (maddux1_join_commutativity) } 62.68/8.25 join(Y, top) 62.68/8.25 62.68/8.25 Lemma 30: join(top, complement(X)) = top. 62.68/8.25 Proof: 62.68/8.25 join(top, complement(X)) 62.68/8.25 = { by axiom 2 (maddux1_join_commutativity) R->L } 62.68/8.25 join(complement(X), top) 62.68/8.25 = { by lemma 29 R->L } 62.68/8.25 join(X, join(complement(X), complement(X))) 62.68/8.25 = { by lemma 20 } 62.68/8.25 join(X, complement(X)) 62.68/8.25 = { by axiom 5 (def_top) } 62.68/8.25 top 62.68/8.25 62.68/8.25 Lemma 31: join(Y, top) = join(X, top). 62.68/8.25 Proof: 62.68/8.25 join(Y, top) 62.68/8.25 = { by lemma 30 R->L } 62.68/8.25 join(Y, join(top, complement(Y))) 62.68/8.25 = { by lemma 29 } 62.68/8.25 join(top, top) 62.68/8.25 = { by lemma 29 R->L } 62.68/8.25 join(X, join(top, complement(X))) 62.68/8.25 = { by lemma 30 } 62.68/8.25 join(X, top) 62.68/8.25 62.68/8.25 Lemma 32: join(X, top) = top. 62.68/8.25 Proof: 62.68/8.25 join(X, top) 62.68/8.25 = { by lemma 31 } 62.68/8.25 join(complement(Y), top) 62.68/8.25 = { by axiom 2 (maddux1_join_commutativity) R->L } 62.68/8.25 join(top, complement(Y)) 62.68/8.25 = { by lemma 30 } 62.68/8.25 top 62.68/8.25 62.68/8.25 Lemma 33: join(X, join(complement(X), Y)) = top. 62.68/8.25 Proof: 62.68/8.25 join(X, join(complement(X), Y)) 62.68/8.25 = { by axiom 2 (maddux1_join_commutativity) R->L } 62.68/8.25 join(X, join(Y, complement(X))) 62.68/8.25 = { by lemma 29 } 62.68/8.25 join(Y, top) 62.68/8.25 = { by lemma 31 R->L } 62.68/8.25 join(Z, top) 62.68/8.25 = { by lemma 32 } 62.68/8.25 top 62.68/8.25 62.68/8.25 Lemma 34: join(X, complement(zero)) = top. 62.68/8.25 Proof: 62.68/8.25 join(X, complement(zero)) 62.68/8.25 = { by axiom 2 (maddux1_join_commutativity) R->L } 62.68/8.25 join(complement(zero), X) 62.68/8.25 = { by lemma 28 R->L } 62.68/8.25 join(zero, join(complement(zero), meet(X, X))) 62.68/8.25 = { by lemma 33 } 62.68/8.25 top 62.68/8.25 62.68/8.25 Lemma 35: join(meet(X, Y), meet(X, complement(Y))) = X. 62.68/8.25 Proof: 62.68/8.25 join(meet(X, Y), meet(X, complement(Y))) 62.68/8.25 = { by axiom 2 (maddux1_join_commutativity) R->L } 62.68/8.25 join(meet(X, complement(Y)), meet(X, Y)) 62.68/8.25 = { by axiom 10 (maddux4_definiton_of_meet) } 62.68/8.25 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y)))) 62.68/8.25 = { by lemma 24 } 62.68/8.25 X 62.68/8.25 62.68/8.25 Lemma 36: join(zero, meet(X, top)) = X. 62.68/8.25 Proof: 62.68/8.25 join(zero, meet(X, top)) 62.68/8.25 = { by lemma 34 R->L } 62.68/8.25 join(zero, meet(X, join(complement(zero), complement(zero)))) 62.68/8.25 = { by lemma 20 } 62.68/8.25 join(zero, meet(X, complement(zero))) 62.68/8.25 = { by lemma 26 R->L } 62.68/8.25 join(complement(top), meet(X, complement(zero))) 62.68/8.25 = { by lemma 34 R->L } 62.68/8.25 join(complement(join(complement(X), complement(zero))), meet(X, complement(zero))) 62.68/8.25 = { by axiom 10 (maddux4_definiton_of_meet) R->L } 62.68/8.25 join(meet(X, zero), meet(X, complement(zero))) 62.68/8.25 = { by lemma 35 } 62.68/8.25 X 62.68/8.25 62.68/8.25 Lemma 37: join(zero, complement(X)) = complement(X). 62.68/8.25 Proof: 62.68/8.25 join(zero, complement(X)) 62.68/8.25 = { by lemma 25 R->L } 62.68/8.25 join(zero, complement(join(zero, meet(X, X)))) 62.68/8.25 = { by lemma 23 R->L } 62.68/8.25 join(zero, complement(join(zero, complement(complement(X))))) 62.68/8.25 = { by lemma 27 } 62.68/8.25 join(zero, meet(complement(X), top)) 62.68/8.25 = { by lemma 36 } 62.68/8.25 complement(X) 62.68/8.25 62.68/8.25 Lemma 38: complement(complement(X)) = X. 62.68/8.25 Proof: 62.68/8.25 complement(complement(X)) 62.68/8.25 = { by lemma 23 } 62.68/8.25 meet(X, X) 62.68/8.25 = { by lemma 22 R->L } 62.68/8.25 complement(join(complement(X), complement(X))) 62.68/8.25 = { by lemma 37 R->L } 62.68/8.25 join(zero, complement(join(complement(X), complement(X)))) 62.68/8.25 = { by lemma 22 } 62.68/8.25 join(zero, meet(X, X)) 62.68/8.25 = { by lemma 25 } 62.68/8.25 X 62.68/8.25 62.68/8.25 Lemma 39: join(X, zero) = X. 62.68/8.25 Proof: 62.68/8.25 join(X, zero) 62.68/8.25 = { by axiom 2 (maddux1_join_commutativity) R->L } 62.68/8.25 join(zero, X) 62.68/8.25 = { by lemma 38 R->L } 62.68/8.25 join(zero, complement(complement(X))) 62.68/8.25 = { by lemma 23 } 62.68/8.25 join(zero, meet(X, X)) 62.68/8.25 = { by lemma 25 } 62.68/8.25 X 62.68/8.25 62.68/8.25 Lemma 40: join(zero, X) = X. 62.68/8.25 Proof: 62.68/8.25 join(zero, X) 62.68/8.25 = { by axiom 2 (maddux1_join_commutativity) R->L } 62.68/8.25 join(X, zero) 62.68/8.25 = { by lemma 39 } 62.68/8.25 X 62.68/8.25 62.68/8.25 Lemma 41: converse(join(X, converse(Y))) = join(Y, converse(X)). 62.68/8.25 Proof: 62.68/8.25 converse(join(X, converse(Y))) 62.68/8.25 = { by axiom 2 (maddux1_join_commutativity) R->L } 62.68/8.25 converse(join(converse(Y), X)) 62.68/8.25 = { by axiom 6 (converse_additivity) } 62.68/8.25 join(converse(converse(Y)), converse(X)) 62.68/8.25 = { by axiom 1 (converse_idempotence) } 62.68/8.25 join(Y, converse(X)) 62.68/8.25 62.68/8.25 Lemma 42: converse(join(converse(X), Y)) = join(X, converse(Y)). 62.68/8.25 Proof: 62.68/8.25 converse(join(converse(X), Y)) 62.68/8.25 = { by axiom 2 (maddux1_join_commutativity) R->L } 62.68/8.25 converse(join(Y, converse(X))) 62.68/8.25 = { by lemma 41 } 62.68/8.25 join(X, converse(Y)) 62.68/8.25 62.68/8.25 Lemma 43: converse(zero) = zero. 62.68/8.25 Proof: 62.68/8.25 converse(zero) 62.68/8.25 = { by lemma 40 R->L } 62.68/8.25 join(zero, converse(zero)) 62.68/8.25 = { by lemma 42 R->L } 62.68/8.25 converse(join(converse(zero), zero)) 62.68/8.25 = { by axiom 2 (maddux1_join_commutativity) R->L } 62.68/8.25 converse(join(zero, converse(zero))) 62.68/8.25 = { by lemma 25 R->L } 62.68/8.25 converse(join(zero, join(zero, meet(converse(zero), converse(zero))))) 62.68/8.25 = { by axiom 10 (maddux4_definiton_of_meet) } 62.68/8.25 converse(join(zero, join(zero, complement(join(complement(converse(zero)), complement(converse(zero))))))) 62.68/8.25 = { by lemma 20 R->L } 62.68/8.25 converse(join(zero, join(zero, join(complement(join(complement(converse(zero)), complement(converse(zero)))), complement(join(complement(converse(zero)), complement(converse(zero)))))))) 62.68/8.25 = { by axiom 10 (maddux4_definiton_of_meet) R->L } 62.68/8.25 converse(join(zero, join(zero, join(meet(converse(zero), converse(zero)), complement(join(complement(converse(zero)), complement(converse(zero)))))))) 62.68/8.25 = { by axiom 10 (maddux4_definiton_of_meet) R->L } 62.68/8.25 converse(join(zero, join(zero, join(meet(converse(zero), converse(zero)), meet(converse(zero), converse(zero)))))) 62.68/8.25 = { by lemma 28 } 62.68/8.25 converse(join(zero, join(meet(converse(zero), converse(zero)), converse(zero)))) 62.68/8.25 = { by axiom 2 (maddux1_join_commutativity) } 62.68/8.25 converse(join(zero, join(converse(zero), meet(converse(zero), converse(zero))))) 62.68/8.25 = { by lemma 28 } 62.68/8.25 converse(join(converse(zero), converse(zero))) 62.68/8.25 = { by lemma 41 } 62.68/8.25 join(zero, converse(converse(zero))) 62.68/8.25 = { by axiom 1 (converse_idempotence) } 62.68/8.25 join(zero, zero) 62.68/8.25 = { by lemma 38 R->L } 62.68/8.25 join(zero, complement(complement(zero))) 62.68/8.25 = { by lemma 38 R->L } 62.68/8.25 join(complement(complement(zero)), complement(complement(zero))) 62.68/8.25 = { by lemma 20 } 62.68/8.25 complement(complement(zero)) 62.68/8.25 = { by lemma 38 } 62.68/8.25 zero 62.68/8.25 62.68/8.25 Lemma 44: meet(X, X) = X. 62.68/8.25 Proof: 62.68/8.25 meet(X, X) 62.68/8.25 = { by lemma 23 R->L } 62.68/8.25 complement(complement(X)) 62.68/8.25 = { by lemma 38 } 62.68/8.25 X 62.68/8.25 62.68/8.25 Lemma 45: meet(X, top) = X. 62.68/8.25 Proof: 62.68/8.25 meet(X, top) 62.68/8.25 = { by lemma 27 R->L } 62.68/8.25 complement(join(zero, complement(X))) 62.68/8.25 = { by lemma 37 R->L } 62.68/8.25 join(zero, complement(join(zero, complement(X)))) 62.68/8.25 = { by lemma 27 } 62.68/8.25 join(zero, meet(X, top)) 62.68/8.25 = { by lemma 36 } 62.68/8.25 X 62.68/8.25 62.68/8.25 Lemma 46: meet(top, X) = X. 62.68/8.25 Proof: 62.68/8.25 meet(top, X) 62.68/8.25 = { by lemma 21 } 62.68/8.25 meet(X, top) 62.68/8.25 = { by lemma 45 } 62.68/8.25 X 62.68/8.25 62.68/8.25 Lemma 47: join(complement(X), complement(Y)) = complement(meet(X, Y)). 62.68/8.25 Proof: 62.68/8.25 join(complement(X), complement(Y)) 62.68/8.25 = { by lemma 46 R->L } 62.68/8.25 meet(top, join(complement(X), complement(Y))) 62.68/8.25 = { by axiom 2 (maddux1_join_commutativity) R->L } 62.68/8.25 meet(top, join(complement(Y), complement(X))) 62.68/8.25 = { by lemma 21 } 62.68/8.25 meet(join(complement(Y), complement(X)), top) 62.68/8.25 = { by axiom 10 (maddux4_definiton_of_meet) } 62.68/8.25 complement(join(complement(join(complement(Y), complement(X))), complement(top))) 62.68/8.25 = { by axiom 10 (maddux4_definiton_of_meet) R->L } 62.68/8.25 complement(join(meet(Y, X), complement(top))) 62.68/8.25 = { by axiom 2 (maddux1_join_commutativity) } 62.68/8.25 complement(join(complement(top), meet(Y, X))) 62.68/8.25 = { by lemma 21 R->L } 62.68/8.25 complement(join(complement(top), meet(X, Y))) 62.68/8.25 = { by lemma 26 } 62.68/8.25 complement(join(zero, meet(X, Y))) 62.68/8.25 = { by lemma 21 R->L } 62.68/8.25 complement(join(zero, meet(Y, X))) 62.68/8.25 = { by axiom 2 (maddux1_join_commutativity) } 62.68/8.25 complement(join(meet(Y, X), zero)) 62.68/8.25 = { by lemma 39 } 62.68/8.25 complement(meet(Y, X)) 62.68/8.25 = { by lemma 21 R->L } 62.68/8.25 complement(meet(X, Y)) 62.68/8.25 62.68/8.25 Lemma 48: complement(meet(X, complement(Y))) = join(Y, complement(X)). 62.68/8.25 Proof: 62.68/8.25 complement(meet(X, complement(Y))) 62.68/8.25 = { by lemma 21 } 62.68/8.25 complement(meet(complement(Y), X)) 62.68/8.25 = { by lemma 37 R->L } 62.68/8.25 complement(meet(join(zero, complement(Y)), X)) 62.68/8.25 = { by lemma 47 R->L } 62.68/8.25 join(complement(join(zero, complement(Y))), complement(X)) 62.68/8.25 = { by lemma 27 } 62.68/8.25 join(meet(Y, top), complement(X)) 62.68/8.25 = { by lemma 45 } 62.68/8.25 join(Y, complement(X)) 62.68/8.25 62.68/8.25 Lemma 49: complement(join(X, complement(Y))) = meet(Y, complement(X)). 62.68/8.25 Proof: 62.68/8.25 complement(join(X, complement(Y))) 62.68/8.25 = { by lemma 48 R->L } 62.68/8.25 complement(complement(meet(Y, complement(X)))) 62.68/8.25 = { by lemma 23 } 62.68/8.25 meet(meet(Y, complement(X)), meet(Y, complement(X))) 62.68/8.25 = { by lemma 44 } 62.68/8.25 meet(Y, complement(X)) 62.68/8.25 62.68/8.25 Lemma 50: converse(composition(X, converse(Y))) = composition(Y, converse(X)). 62.68/8.25 Proof: 62.68/8.25 converse(composition(X, converse(Y))) 62.68/8.25 = { by axiom 8 (converse_multiplicativity) } 62.68/8.25 composition(converse(converse(Y)), converse(X)) 62.68/8.25 = { by axiom 1 (converse_idempotence) } 62.68/8.25 composition(Y, converse(X)) 62.68/8.25 62.68/8.25 Lemma 51: composition(join(X, one), Y) = join(Y, composition(X, Y)). 62.68/8.25 Proof: 62.68/8.25 composition(join(X, one), Y) 62.68/8.25 = { by axiom 2 (maddux1_join_commutativity) R->L } 62.68/8.25 composition(join(one, X), Y) 62.68/8.25 = { by axiom 11 (composition_distributivity) } 62.68/8.25 join(composition(one, Y), composition(X, Y)) 62.68/8.25 = { by lemma 18 } 62.68/8.25 join(Y, composition(X, Y)) 62.68/8.25 62.68/8.25 Lemma 52: composition(zero, X) = zero. 62.68/8.25 Proof: 62.68/8.25 composition(zero, X) 62.68/8.25 = { by lemma 40 R->L } 62.68/8.25 join(zero, composition(zero, X)) 62.68/8.26 = { by axiom 1 (converse_idempotence) R->L } 62.68/8.26 join(zero, composition(zero, converse(converse(X)))) 62.68/8.26 = { by lemma 50 R->L } 62.68/8.26 join(zero, converse(composition(converse(X), converse(zero)))) 62.68/8.26 = { by lemma 42 R->L } 62.68/8.26 converse(join(converse(zero), composition(converse(X), converse(zero)))) 62.68/8.26 = { by lemma 51 R->L } 62.68/8.26 converse(composition(join(converse(X), one), converse(zero))) 62.68/8.26 = { by lemma 50 } 62.68/8.26 composition(zero, converse(join(converse(X), one))) 62.68/8.26 = { by lemma 42 } 62.68/8.26 composition(zero, join(X, converse(one))) 62.68/8.26 = { by axiom 3 (composition_identity) R->L } 62.68/8.26 composition(zero, join(X, composition(converse(one), one))) 62.68/8.26 = { by lemma 17 } 62.68/8.26 composition(zero, join(X, one)) 62.68/8.26 = { by lemma 43 R->L } 62.68/8.26 composition(converse(zero), join(X, one)) 62.68/8.26 = { by lemma 26 R->L } 62.68/8.26 composition(converse(complement(top)), join(X, one)) 62.68/8.26 = { by lemma 32 R->L } 62.68/8.26 composition(converse(complement(join(composition(X, top), top))), join(X, one)) 62.68/8.26 = { by axiom 2 (maddux1_join_commutativity) } 62.68/8.26 composition(converse(complement(join(top, composition(X, top)))), join(X, one)) 62.68/8.26 = { by lemma 51 R->L } 62.68/8.26 composition(converse(complement(composition(join(X, one), top))), join(X, one)) 62.68/8.26 = { by lemma 16 R->L } 62.68/8.26 converse(composition(converse(join(X, one)), complement(composition(join(X, one), top)))) 62.68/8.26 = { by lemma 40 R->L } 62.68/8.26 converse(join(zero, composition(converse(join(X, one)), complement(composition(join(X, one), top))))) 62.68/8.26 = { by lemma 26 R->L } 62.68/8.26 converse(join(complement(top), composition(converse(join(X, one)), complement(composition(join(X, one), top))))) 62.68/8.26 = { by lemma 19 } 62.68/8.26 converse(complement(top)) 62.68/8.26 = { by lemma 26 } 62.68/8.26 converse(zero) 62.68/8.26 = { by lemma 43 } 62.68/8.26 zero 62.68/8.26 62.68/8.26 Lemma 53: join(complement(one), composition(converse(X), complement(X))) = complement(one). 62.68/8.26 Proof: 62.68/8.26 join(complement(one), composition(converse(X), complement(X))) 62.68/8.26 = { by axiom 3 (composition_identity) R->L } 62.68/8.26 join(complement(one), composition(converse(X), complement(composition(X, one)))) 62.68/8.26 = { by lemma 19 } 62.68/8.26 complement(one) 62.68/8.26 62.68/8.26 Lemma 54: join(complement(one), converse(complement(one))) = complement(one). 62.68/8.26 Proof: 62.68/8.26 join(complement(one), converse(complement(one))) 62.68/8.26 = { by axiom 3 (composition_identity) R->L } 62.68/8.26 join(complement(one), composition(converse(complement(one)), one)) 62.68/8.26 = { by lemma 45 R->L } 62.68/8.26 join(complement(one), composition(converse(complement(one)), meet(one, top))) 62.68/8.26 = { by lemma 37 R->L } 62.68/8.26 join(complement(one), composition(converse(join(zero, complement(one))), meet(one, top))) 62.68/8.26 = { by lemma 27 R->L } 62.68/8.26 join(complement(one), composition(converse(join(zero, complement(one))), complement(join(zero, complement(one))))) 62.68/8.26 = { by lemma 53 } 62.68/8.26 complement(one) 62.68/8.26 62.68/8.26 Lemma 55: meet(one, composition(converse(complement(X)), X)) = zero. 62.68/8.26 Proof: 62.68/8.26 meet(one, composition(converse(complement(X)), X)) 62.68/8.26 = { by lemma 46 R->L } 62.68/8.26 meet(one, meet(top, composition(converse(complement(X)), X))) 62.68/8.26 = { by lemma 21 } 62.68/8.26 meet(one, meet(composition(converse(complement(X)), X), top)) 62.68/8.26 = { by lemma 38 R->L } 62.68/8.26 complement(complement(meet(one, meet(composition(converse(complement(X)), X), top)))) 62.68/8.26 = { by lemma 21 } 62.68/8.26 complement(complement(meet(one, meet(top, composition(converse(complement(X)), X))))) 62.68/8.26 = { by axiom 10 (maddux4_definiton_of_meet) } 62.68/8.26 complement(complement(meet(one, complement(join(complement(top), complement(composition(converse(complement(X)), X))))))) 62.68/8.26 = { by lemma 48 } 62.68/8.26 complement(join(join(complement(top), complement(composition(converse(complement(X)), X))), complement(one))) 62.68/8.26 = { by axiom 7 (maddux2_join_associativity) } 62.68/8.26 complement(join(complement(top), join(complement(composition(converse(complement(X)), X)), complement(one)))) 62.68/8.26 = { by lemma 47 } 62.68/8.26 complement(join(complement(top), complement(meet(composition(converse(complement(X)), X), one)))) 62.68/8.26 = { by lemma 47 } 62.68/8.26 complement(complement(meet(top, meet(composition(converse(complement(X)), X), one)))) 62.68/8.26 = { by lemma 21 R->L } 62.68/8.26 complement(complement(meet(top, meet(one, composition(converse(complement(X)), X))))) 62.68/8.26 = { by lemma 23 } 62.68/8.26 meet(meet(top, meet(one, composition(converse(complement(X)), X))), meet(top, meet(one, composition(converse(complement(X)), X)))) 62.68/8.26 = { by lemma 44 } 62.68/8.26 meet(top, meet(one, composition(converse(complement(X)), X))) 62.68/8.26 = { by lemma 21 R->L } 62.68/8.26 meet(meet(one, composition(converse(complement(X)), X)), top) 62.68/8.26 = { by lemma 33 R->L } 62.68/8.26 meet(meet(one, composition(converse(complement(X)), X)), join(converse(composition(converse(X), complement(X))), join(complement(converse(composition(converse(X), complement(X)))), converse(complement(one))))) 62.68/8.26 = { by axiom 2 (maddux1_join_commutativity) R->L } 62.68/8.26 meet(meet(one, composition(converse(complement(X)), X)), join(converse(composition(converse(X), complement(X))), join(converse(complement(one)), complement(converse(composition(converse(X), complement(X))))))) 62.68/8.26 = { by axiom 2 (maddux1_join_commutativity) R->L } 62.68/8.26 meet(meet(one, composition(converse(complement(X)), X)), join(join(converse(complement(one)), complement(converse(composition(converse(X), complement(X))))), converse(composition(converse(X), complement(X))))) 62.68/8.26 = { by axiom 7 (maddux2_join_associativity) } 62.68/8.26 meet(meet(one, composition(converse(complement(X)), X)), join(converse(complement(one)), join(complement(converse(composition(converse(X), complement(X)))), converse(composition(converse(X), complement(X)))))) 62.68/8.26 = { by axiom 2 (maddux1_join_commutativity) } 62.68/8.26 meet(meet(one, composition(converse(complement(X)), X)), join(converse(complement(one)), join(converse(composition(converse(X), complement(X))), complement(converse(composition(converse(X), complement(X))))))) 62.68/8.26 = { by axiom 7 (maddux2_join_associativity) R->L } 62.68/8.26 meet(meet(one, composition(converse(complement(X)), X)), join(join(converse(complement(one)), converse(composition(converse(X), complement(X)))), complement(converse(composition(converse(X), complement(X)))))) 62.68/8.26 = { by axiom 6 (converse_additivity) R->L } 62.68/8.26 meet(meet(one, composition(converse(complement(X)), X)), join(converse(join(complement(one), composition(converse(X), complement(X)))), complement(converse(composition(converse(X), complement(X)))))) 62.68/8.26 = { by axiom 2 (maddux1_join_commutativity) } 62.68/8.26 meet(meet(one, composition(converse(complement(X)), X)), join(complement(converse(composition(converse(X), complement(X)))), converse(join(complement(one), composition(converse(X), complement(X)))))) 62.68/8.26 = { by lemma 53 } 62.68/8.26 meet(meet(one, composition(converse(complement(X)), X)), join(complement(converse(composition(converse(X), complement(X)))), converse(complement(one)))) 62.68/8.26 = { by lemma 16 } 62.68/8.26 meet(meet(one, composition(converse(complement(X)), X)), join(complement(composition(converse(complement(X)), X)), converse(complement(one)))) 62.68/8.26 = { by lemma 54 R->L } 62.68/8.26 meet(meet(one, composition(converse(complement(X)), X)), join(complement(composition(converse(complement(X)), X)), converse(join(complement(one), converse(complement(one)))))) 62.68/8.26 = { by lemma 41 } 62.68/8.26 meet(meet(one, composition(converse(complement(X)), X)), join(complement(composition(converse(complement(X)), X)), join(complement(one), converse(complement(one))))) 62.68/8.26 = { by lemma 54 } 62.68/8.26 meet(meet(one, composition(converse(complement(X)), X)), join(complement(composition(converse(complement(X)), X)), complement(one))) 62.68/8.26 = { by lemma 47 } 62.68/8.26 meet(meet(one, composition(converse(complement(X)), X)), complement(meet(composition(converse(complement(X)), X), one))) 62.68/8.26 = { by lemma 21 R->L } 62.68/8.26 meet(meet(one, composition(converse(complement(X)), X)), complement(meet(one, composition(converse(complement(X)), X)))) 62.68/8.26 = { by axiom 4 (def_zero) R->L } 62.68/8.26 zero 62.68/8.26 62.68/8.26 Lemma 56: complement(converse(X)) = converse(complement(X)). 62.68/8.26 Proof: 62.68/8.26 complement(converse(X)) 62.68/8.26 = { by axiom 1 (converse_idempotence) R->L } 62.68/8.26 converse(converse(complement(converse(X)))) 62.68/8.26 = { by lemma 38 R->L } 62.68/8.26 converse(converse(complement(converse(complement(complement(X)))))) 62.68/8.26 = { by lemma 35 R->L } 62.68/8.26 converse(join(meet(converse(complement(converse(complement(complement(X))))), X), meet(converse(complement(converse(complement(complement(X))))), complement(X)))) 62.68/8.26 = { by lemma 21 R->L } 62.68/8.26 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), meet(converse(complement(converse(complement(complement(X))))), complement(X)))) 62.68/8.26 = { by lemma 21 R->L } 62.68/8.26 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), meet(complement(X), converse(complement(converse(complement(complement(X)))))))) 62.68/8.26 = { by lemma 38 R->L } 62.68/8.26 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), meet(complement(X), complement(complement(converse(complement(converse(complement(complement(X)))))))))) 62.68/8.27 = { by lemma 49 R->L } 62.68/8.27 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(join(complement(converse(complement(converse(complement(complement(X)))))), complement(complement(X)))))) 62.68/8.27 = { by axiom 2 (maddux1_join_commutativity) } 62.68/8.27 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(join(complement(complement(X)), complement(converse(complement(converse(complement(complement(X)))))))))) 62.68/8.27 = { by lemma 45 R->L } 62.68/8.27 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(join(complement(complement(X)), meet(complement(converse(complement(converse(complement(complement(X)))))), top))))) 62.68/8.27 = { by lemma 33 R->L } 62.68/8.27 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(join(complement(complement(X)), meet(complement(converse(complement(converse(complement(complement(X)))))), join(converse(Y), join(complement(converse(Y)), converse(complement(converse(complement(converse(Y)))))))))))) 62.68/8.27 = { by lemma 42 R->L } 62.68/8.27 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(join(complement(complement(X)), meet(complement(converse(complement(converse(complement(complement(X)))))), join(converse(Y), converse(join(converse(complement(converse(Y))), complement(converse(complement(converse(Y)))))))))))) 62.68/8.27 = { by axiom 5 (def_top) } 62.68/8.27 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(join(complement(complement(X)), meet(complement(converse(complement(converse(complement(complement(X)))))), join(converse(Y), converse(top))))))) 62.68/8.27 = { by axiom 6 (converse_additivity) R->L } 62.68/8.27 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(join(complement(complement(X)), meet(complement(converse(complement(converse(complement(complement(X)))))), converse(join(Y, top))))))) 62.68/8.27 = { by lemma 32 } 62.68/8.27 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(join(complement(complement(X)), meet(complement(converse(complement(converse(complement(complement(X)))))), converse(top)))))) 62.68/8.27 = { by axiom 5 (def_top) R->L } 62.68/8.27 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(join(complement(complement(X)), meet(complement(converse(complement(converse(complement(complement(X)))))), converse(join(converse(complement(complement(X))), complement(converse(complement(complement(X))))))))))) 62.68/8.27 = { by lemma 42 } 62.68/8.27 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(join(complement(complement(X)), meet(complement(converse(complement(converse(complement(complement(X)))))), join(complement(complement(X)), converse(complement(converse(complement(complement(X))))))))))) 62.68/8.27 = { by axiom 2 (maddux1_join_commutativity) R->L } 62.68/8.27 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(join(complement(complement(X)), meet(complement(converse(complement(converse(complement(complement(X)))))), join(converse(complement(converse(complement(complement(X))))), complement(complement(X)))))))) 62.68/8.27 = { by lemma 21 } 62.68/8.27 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(join(complement(complement(X)), meet(join(converse(complement(converse(complement(complement(X))))), complement(complement(X))), complement(converse(complement(converse(complement(complement(X))))))))))) 62.68/8.27 = { by lemma 49 R->L } 62.68/8.27 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(join(complement(complement(X)), complement(join(converse(complement(converse(complement(complement(X))))), complement(join(converse(complement(converse(complement(complement(X))))), complement(complement(X)))))))))) 62.68/8.27 = { by lemma 44 R->L } 62.68/8.27 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(join(complement(complement(X)), complement(join(converse(complement(converse(complement(complement(X))))), complement(join(converse(complement(converse(complement(complement(X))))), meet(complement(complement(X)), complement(complement(X))))))))))) 62.68/8.27 = { by lemma 23 R->L } 62.68/8.27 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(join(complement(complement(X)), complement(join(converse(complement(converse(complement(complement(X))))), complement(join(converse(complement(converse(complement(complement(X))))), complement(complement(complement(complement(X)))))))))))) 62.68/8.27 = { by lemma 49 } 62.68/8.27 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(join(complement(complement(X)), complement(join(converse(complement(converse(complement(complement(X))))), meet(complement(complement(complement(X))), complement(converse(complement(converse(complement(complement(X))))))))))))) 62.68/8.27 = { by lemma 21 R->L } 62.68/8.27 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(join(complement(complement(X)), complement(join(converse(complement(converse(complement(complement(X))))), meet(complement(converse(complement(converse(complement(complement(X)))))), complement(complement(complement(X)))))))))) 62.68/8.27 = { by lemma 48 R->L } 62.68/8.27 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(meet(join(converse(complement(converse(complement(complement(X))))), meet(complement(converse(complement(converse(complement(complement(X)))))), complement(complement(complement(X))))), complement(complement(complement(X)))))))) 62.68/8.27 = { by lemma 21 R->L } 62.68/8.27 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(meet(complement(complement(complement(X))), join(converse(complement(converse(complement(complement(X))))), meet(complement(converse(complement(converse(complement(complement(X)))))), complement(complement(complement(X)))))))))) 62.68/8.27 = { by lemma 21 } 62.68/8.27 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(meet(complement(complement(complement(X))), join(converse(complement(converse(complement(complement(X))))), meet(complement(complement(complement(X))), complement(converse(complement(converse(complement(complement(X))))))))))))) 62.68/8.27 = { by lemma 37 R->L } 62.68/8.27 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(meet(complement(complement(complement(X))), join(converse(complement(converse(complement(complement(X))))), meet(complement(complement(complement(X))), join(zero, complement(converse(complement(converse(complement(complement(X)))))))))))))) 62.68/8.27 = { by lemma 45 R->L } 62.68/8.27 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(meet(complement(complement(complement(X))), join(meet(converse(complement(converse(complement(complement(X))))), top), meet(complement(complement(complement(X))), join(zero, complement(converse(complement(converse(complement(complement(X)))))))))))))) 62.68/8.27 = { by lemma 27 R->L } 62.68/8.27 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(meet(complement(complement(complement(X))), join(complement(join(zero, complement(converse(complement(converse(complement(complement(X)))))))), meet(complement(complement(complement(X))), join(zero, complement(converse(complement(converse(complement(complement(X)))))))))))))) 62.68/8.28 = { by lemma 39 R->L } 62.68/8.28 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(join(meet(complement(complement(complement(X))), join(complement(join(zero, complement(converse(complement(converse(complement(complement(X)))))))), meet(complement(complement(complement(X))), join(zero, complement(converse(complement(converse(complement(complement(X)))))))))), zero))))) 62.68/8.28 = { by lemma 26 R->L } 62.68/8.28 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(join(meet(complement(complement(complement(X))), join(complement(join(zero, complement(converse(complement(converse(complement(complement(X)))))))), meet(complement(complement(complement(X))), join(zero, complement(converse(complement(converse(complement(complement(X)))))))))), complement(top)))))) 62.68/8.28 = { by axiom 5 (def_top) R->L } 62.68/8.28 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(join(meet(complement(complement(complement(X))), join(complement(join(zero, complement(converse(complement(converse(complement(complement(X)))))))), meet(complement(complement(complement(X))), join(zero, complement(converse(complement(converse(complement(complement(X)))))))))), complement(join(join(complement(complement(complement(complement(X)))), complement(join(zero, complement(converse(complement(converse(complement(complement(X))))))))), complement(join(complement(complement(complement(complement(X)))), complement(join(zero, complement(converse(complement(converse(complement(complement(X))))))))))))))))) 62.68/8.28 = { by axiom 10 (maddux4_definiton_of_meet) R->L } 62.68/8.28 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(join(meet(complement(complement(complement(X))), join(complement(join(zero, complement(converse(complement(converse(complement(complement(X)))))))), meet(complement(complement(complement(X))), join(zero, complement(converse(complement(converse(complement(complement(X)))))))))), complement(join(join(complement(complement(complement(complement(X)))), complement(join(zero, complement(converse(complement(converse(complement(complement(X))))))))), meet(complement(complement(complement(X))), join(zero, complement(converse(complement(converse(complement(complement(X))))))))))))))) 62.68/8.28 = { by axiom 7 (maddux2_join_associativity) } 62.68/8.28 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(join(meet(complement(complement(complement(X))), join(complement(join(zero, complement(converse(complement(converse(complement(complement(X)))))))), meet(complement(complement(complement(X))), join(zero, complement(converse(complement(converse(complement(complement(X)))))))))), complement(join(complement(complement(complement(complement(X)))), join(complement(join(zero, complement(converse(complement(converse(complement(complement(X)))))))), meet(complement(complement(complement(X))), join(zero, complement(converse(complement(converse(complement(complement(X)))))))))))))))) 62.68/8.28 = { by lemma 24 } 62.68/8.28 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(complement(complement(complement(X))))))) 62.68/8.28 = { by lemma 38 } 62.68/8.28 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(complement(X))))) 62.68/8.28 = { by lemma 38 } 62.68/8.28 converse(join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(X))) 62.68/8.28 = { by lemma 38 } 62.68/8.28 converse(join(meet(X, converse(complement(converse(X)))), complement(X))) 62.68/8.28 = { by lemma 39 R->L } 62.68/8.28 converse(join(join(meet(X, converse(complement(converse(X)))), zero), complement(X))) 62.68/8.28 = { by lemma 52 R->L } 62.68/8.28 converse(join(join(meet(X, converse(complement(converse(X)))), composition(zero, meet(X, composition(converse(one), converse(complement(converse(X))))))), complement(X))) 62.68/8.28 = { by lemma 18 R->L } 62.68/8.28 converse(join(join(meet(composition(one, X), converse(complement(converse(X)))), composition(zero, meet(X, composition(converse(one), converse(complement(converse(X))))))), complement(X))) 62.68/8.28 = { by lemma 55 R->L } 62.68/8.28 converse(join(join(meet(composition(one, X), converse(complement(converse(X)))), composition(meet(one, composition(converse(complement(converse(X))), converse(X))), meet(X, composition(converse(one), converse(complement(converse(X))))))), complement(X))) 62.68/8.28 = { by axiom 15 (dedekind_law) R->L } 62.68/8.28 converse(join(composition(meet(one, composition(converse(complement(converse(X))), converse(X))), meet(X, composition(converse(one), converse(complement(converse(X)))))), complement(X))) 62.68/8.28 = { by lemma 55 } 62.68/8.28 converse(join(composition(zero, meet(X, composition(converse(one), converse(complement(converse(X)))))), complement(X))) 62.68/8.28 = { by lemma 52 } 62.68/8.28 converse(join(zero, complement(X))) 62.68/8.28 = { by lemma 40 } 62.68/8.28 converse(complement(X)) 62.68/8.28 62.68/8.28 Lemma 57: join(X, composition(complement(composition(complement(X), Y)), converse(Y))) = X. 62.68/8.28 Proof: 62.68/8.28 join(X, composition(complement(composition(complement(X), Y)), converse(Y))) 62.68/8.28 = { by axiom 1 (converse_idempotence) R->L } 62.68/8.28 converse(converse(join(X, composition(complement(composition(complement(X), Y)), converse(Y))))) 62.68/8.28 = { by lemma 38 R->L } 62.68/8.28 converse(converse(join(complement(complement(X)), composition(complement(composition(complement(X), Y)), converse(Y))))) 62.68/8.28 = { by axiom 2 (maddux1_join_commutativity) R->L } 62.68/8.28 converse(converse(join(composition(complement(composition(complement(X), Y)), converse(Y)), complement(complement(X))))) 62.68/8.28 = { by axiom 6 (converse_additivity) } 62.68/8.28 converse(join(converse(composition(complement(composition(complement(X), Y)), converse(Y))), converse(complement(complement(X))))) 62.68/8.28 = { by lemma 50 } 62.68/8.28 converse(join(composition(Y, converse(complement(composition(complement(X), Y)))), converse(complement(complement(X))))) 62.68/8.28 = { by lemma 56 R->L } 62.68/8.28 converse(join(composition(Y, complement(converse(composition(complement(X), Y)))), converse(complement(complement(X))))) 62.68/8.28 = { by lemma 41 R->L } 62.68/8.28 converse(converse(join(complement(complement(X)), converse(composition(Y, complement(converse(composition(complement(X), Y)))))))) 62.68/8.28 = { by axiom 8 (converse_multiplicativity) } 62.68/8.28 converse(converse(join(complement(complement(X)), composition(converse(complement(converse(composition(complement(X), Y)))), converse(Y))))) 62.68/8.28 = { by axiom 2 (maddux1_join_commutativity) R->L } 62.68/8.28 converse(converse(join(composition(converse(complement(converse(composition(complement(X), Y)))), converse(Y)), complement(complement(X))))) 62.68/8.28 = { by axiom 6 (converse_additivity) } 62.68/8.28 converse(join(converse(composition(converse(complement(converse(composition(complement(X), Y)))), converse(Y))), converse(complement(complement(X))))) 62.68/8.28 = { by lemma 16 } 62.68/8.28 converse(join(composition(converse(converse(Y)), complement(converse(composition(complement(X), Y)))), converse(complement(complement(X))))) 62.68/8.28 = { by axiom 2 (maddux1_join_commutativity) } 62.68/8.28 converse(join(converse(complement(complement(X))), composition(converse(converse(Y)), complement(converse(composition(complement(X), Y)))))) 62.68/8.28 = { by lemma 56 R->L } 62.68/8.28 converse(join(complement(converse(complement(X))), composition(converse(converse(Y)), complement(converse(composition(complement(X), Y)))))) 62.68/8.28 = { by axiom 8 (converse_multiplicativity) } 62.68/8.28 converse(join(complement(converse(complement(X))), composition(converse(converse(Y)), complement(composition(converse(Y), converse(complement(X))))))) 62.68/8.28 = { by lemma 19 } 62.68/8.28 converse(complement(converse(complement(X)))) 62.68/8.28 = { by lemma 56 } 62.68/8.28 converse(converse(complement(complement(X)))) 62.68/8.28 = { by lemma 38 } 62.68/8.28 converse(converse(X)) 62.68/8.28 = { by axiom 1 (converse_idempotence) } 62.68/8.28 X 62.68/8.28 62.68/8.28 Goal 1 (goals_1): x0 = join(composition(x2, converse(x1)), x0). 62.68/8.28 Proof: 62.68/8.28 x0 62.68/8.28 = { by lemma 57 R->L } 62.68/8.28 join(x0, composition(complement(composition(complement(x0), x1)), converse(x1))) 62.68/8.28 = { by lemma 35 R->L } 62.68/8.28 join(x0, composition(complement(join(meet(composition(complement(x0), x1), x2), meet(composition(complement(x0), x1), complement(x2)))), converse(x1))) 62.68/8.28 = { by axiom 2 (maddux1_join_commutativity) } 62.68/8.28 join(x0, composition(complement(join(meet(composition(complement(x0), x1), complement(x2)), meet(composition(complement(x0), x1), x2))), converse(x1))) 62.68/8.28 = { by lemma 21 R->L } 62.68/8.28 join(x0, composition(complement(join(meet(complement(x2), composition(complement(x0), x1)), meet(composition(complement(x0), x1), x2))), converse(x1))) 62.68/8.28 = { by lemma 22 R->L } 62.68/8.28 join(x0, composition(complement(join(meet(complement(x2), composition(complement(x0), x1)), complement(join(complement(x2), complement(composition(complement(x0), x1)))))), converse(x1))) 62.68/8.28 = { by axiom 12 (goals) R->L } 62.68/8.28 join(x0, composition(complement(join(meet(complement(x2), composition(complement(x0), x1)), complement(join(join(composition(complement(x0), x1), complement(x2)), complement(composition(complement(x0), x1)))))), converse(x1))) 62.68/8.28 = { by axiom 2 (maddux1_join_commutativity) } 62.68/8.29 join(x0, composition(complement(join(meet(complement(x2), composition(complement(x0), x1)), complement(join(join(complement(x2), composition(complement(x0), x1)), complement(composition(complement(x0), x1)))))), converse(x1))) 62.68/8.29 = { by axiom 7 (maddux2_join_associativity) } 62.68/8.29 join(x0, composition(complement(join(meet(complement(x2), composition(complement(x0), x1)), complement(join(complement(x2), join(composition(complement(x0), x1), complement(composition(complement(x0), x1))))))), converse(x1))) 62.68/8.29 = { by axiom 5 (def_top) } 62.68/8.29 join(x0, composition(complement(join(meet(complement(x2), composition(complement(x0), x1)), complement(join(complement(x2), top)))), converse(x1))) 62.68/8.29 = { by axiom 2 (maddux1_join_commutativity) } 62.68/8.29 join(x0, composition(complement(join(meet(complement(x2), composition(complement(x0), x1)), complement(join(top, complement(x2))))), converse(x1))) 62.68/8.29 = { by lemma 30 } 62.68/8.29 join(x0, composition(complement(join(meet(complement(x2), composition(complement(x0), x1)), complement(top))), converse(x1))) 62.68/8.29 = { by lemma 26 } 62.68/8.29 join(x0, composition(complement(join(meet(complement(x2), composition(complement(x0), x1)), zero)), converse(x1))) 62.68/8.29 = { by lemma 39 } 62.68/8.29 join(x0, composition(complement(meet(complement(x2), composition(complement(x0), x1))), converse(x1))) 62.68/8.29 = { by lemma 21 } 62.68/8.29 join(x0, composition(complement(meet(composition(complement(x0), x1), complement(x2))), converse(x1))) 62.68/8.29 = { by lemma 48 } 62.68/8.29 join(x0, composition(join(x2, complement(composition(complement(x0), x1))), converse(x1))) 62.68/8.29 = { by axiom 11 (composition_distributivity) } 62.68/8.29 join(x0, join(composition(x2, converse(x1)), composition(complement(composition(complement(x0), x1)), converse(x1)))) 62.68/8.29 = { by axiom 2 (maddux1_join_commutativity) R->L } 62.68/8.29 join(x0, join(composition(complement(composition(complement(x0), x1)), converse(x1)), composition(x2, converse(x1)))) 62.68/8.29 = { by axiom 7 (maddux2_join_associativity) R->L } 62.68/8.29 join(join(x0, composition(complement(composition(complement(x0), x1)), converse(x1))), composition(x2, converse(x1))) 62.68/8.29 = { by lemma 57 } 62.68/8.29 join(x0, composition(x2, converse(x1))) 62.68/8.29 = { by axiom 2 (maddux1_join_commutativity) } 62.68/8.29 join(composition(x2, converse(x1)), x0) 62.68/8.29 % SZS output end Proof 62.68/8.29 62.68/8.29 RESULT: Theorem (the conjecture is true). 62.68/8.30 EOF