TSTP Solution File: REL026+2 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL026+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 : n025.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 13:44:07 EDT 2023
% Result : Theorem 11.87s 1.94s
% Output : Proof 13.01s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL026+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
% 0.13/0.35 % Computer : n025.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Fri Aug 25 22:06:37 EDT 2023
% 0.20/0.35 % CPUTime :
% 11.87/1.94 Command-line arguments: --flatten
% 11.87/1.94
% 11.87/1.94 % SZS status Theorem
% 11.87/1.94
% 13.01/2.02 % SZS output start Proof
% 13.01/2.02 Take the following subset of the input axioms:
% 13.01/2.02 fof(composition_associativity, axiom, ![X0, X1, X2]: composition(X0, composition(X1, X2))=composition(composition(X0, X1), X2)).
% 13.01/2.02 fof(composition_distributivity, axiom, ![X0_2, X1_2, X2_2]: composition(join(X0_2, X1_2), X2_2)=join(composition(X0_2, X2_2), composition(X1_2, X2_2))).
% 13.01/2.02 fof(composition_identity, axiom, ![X0_2]: composition(X0_2, one)=X0_2).
% 13.01/2.02 fof(converse_additivity, axiom, ![X0_2, X1_2]: converse(join(X0_2, X1_2))=join(converse(X0_2), converse(X1_2))).
% 13.01/2.02 fof(converse_cancellativity, axiom, ![X0_2, X1_2]: join(composition(converse(X0_2), complement(composition(X0_2, X1_2))), complement(X1_2))=complement(X1_2)).
% 13.01/2.02 fof(converse_idempotence, axiom, ![X0_2]: converse(converse(X0_2))=X0_2).
% 13.01/2.02 fof(converse_multiplicativity, axiom, ![X0_2, X1_2]: converse(composition(X0_2, X1_2))=composition(converse(X1_2), converse(X0_2))).
% 13.01/2.02 fof(def_top, axiom, ![X0_2]: top=join(X0_2, complement(X0_2))).
% 13.01/2.02 fof(def_zero, axiom, ![X0_2]: zero=meet(X0_2, complement(X0_2))).
% 13.01/2.02 fof(goals, conjecture, ![X0_2, X1_2]: (join(X0_2, one)=one => (join(meet(composition(X0_2, top), X1_2), composition(X0_2, X1_2))=composition(X0_2, X1_2) & join(composition(X0_2, X1_2), meet(composition(X0_2, top), X1_2))=meet(composition(X0_2, top), X1_2)))).
% 13.01/2.02 fof(maddux1_join_commutativity, axiom, ![X0_2, X1_2]: join(X0_2, X1_2)=join(X1_2, X0_2)).
% 13.01/2.02 fof(maddux2_join_associativity, axiom, ![X0_2, X1_2, X2_2]: join(X0_2, join(X1_2, X2_2))=join(join(X0_2, X1_2), X2_2)).
% 13.01/2.02 fof(maddux3_a_kind_of_de_Morgan, axiom, ![X0_2, X1_2]: X0_2=join(complement(join(complement(X0_2), complement(X1_2))), complement(join(complement(X0_2), X1_2)))).
% 13.01/2.02 fof(maddux4_definiton_of_meet, axiom, ![X0_2, X1_2]: meet(X0_2, X1_2)=complement(join(complement(X0_2), complement(X1_2)))).
% 13.01/2.02
% 13.01/2.02 Now clausify the problem and encode Horn clauses using encoding 3 of
% 13.01/2.02 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 13.01/2.02 We repeatedly replace C & s=t => u=v by the two clauses:
% 13.01/2.02 fresh(y, y, x1...xn) = u
% 13.01/2.02 C => fresh(s, t, x1...xn) = v
% 13.01/2.02 where fresh is a fresh function symbol and x1..xn are the free
% 13.01/2.02 variables of u and v.
% 13.01/2.02 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 13.01/2.02 input problem has no model of domain size 1).
% 13.01/2.02
% 13.01/2.02 The encoding turns the above axioms into the following unit equations and goals:
% 13.01/2.02
% 13.01/2.02 Axiom 1 (converse_idempotence): converse(converse(X)) = X.
% 13.01/2.02 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 13.01/2.02 Axiom 3 (goals): join(x0, one) = one.
% 13.01/2.02 Axiom 4 (composition_identity): composition(X, one) = X.
% 13.01/2.02 Axiom 5 (def_zero): zero = meet(X, complement(X)).
% 13.01/2.02 Axiom 6 (def_top): top = join(X, complement(X)).
% 13.01/2.02 Axiom 7 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 13.01/2.02 Axiom 8 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 13.01/2.02 Axiom 9 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 13.01/2.02 Axiom 10 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 13.01/2.02 Axiom 11 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 13.01/2.02 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 13.01/2.02 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 13.01/2.02 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 13.01/2.02
% 13.01/2.02 Lemma 15: complement(top) = zero.
% 13.01/2.02 Proof:
% 13.01/2.02 complement(top)
% 13.01/2.02 = { by axiom 6 (def_top) }
% 13.01/2.02 complement(join(complement(X), complement(complement(X))))
% 13.01/2.02 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 13.01/2.02 meet(X, complement(X))
% 13.01/2.02 = { by axiom 5 (def_zero) R->L }
% 13.01/2.02 zero
% 13.01/2.02
% 13.01/2.02 Lemma 16: join(X, join(Y, complement(X))) = join(Y, top).
% 13.01/2.02 Proof:
% 13.01/2.02 join(X, join(Y, complement(X)))
% 13.01/2.02 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.02 join(X, join(complement(X), Y))
% 13.01/2.02 = { by axiom 8 (maddux2_join_associativity) }
% 13.01/2.02 join(join(X, complement(X)), Y)
% 13.01/2.02 = { by axiom 6 (def_top) R->L }
% 13.01/2.02 join(top, Y)
% 13.01/2.02 = { by axiom 2 (maddux1_join_commutativity) }
% 13.01/2.02 join(Y, top)
% 13.01/2.02
% 13.01/2.02 Lemma 17: composition(X, join(x0, one)) = X.
% 13.01/2.02 Proof:
% 13.01/2.02 composition(X, join(x0, one))
% 13.01/2.02 = { by axiom 3 (goals) }
% 13.01/2.02 composition(X, one)
% 13.01/2.02 = { by axiom 4 (composition_identity) }
% 13.01/2.02 X
% 13.01/2.02
% 13.01/2.02 Lemma 18: composition(converse(join(x0, one)), X) = X.
% 13.01/2.02 Proof:
% 13.01/2.02 composition(converse(join(x0, one)), X)
% 13.01/2.02 = { by axiom 1 (converse_idempotence) R->L }
% 13.01/2.02 composition(converse(join(x0, one)), converse(converse(X)))
% 13.01/2.02 = { by axiom 9 (converse_multiplicativity) R->L }
% 13.01/2.02 converse(composition(converse(X), join(x0, one)))
% 13.01/2.02 = { by lemma 17 }
% 13.01/2.02 converse(converse(X))
% 13.01/2.02 = { by axiom 1 (converse_idempotence) }
% 13.01/2.02 X
% 13.01/2.02
% 13.01/2.02 Lemma 19: composition(join(x0, one), X) = X.
% 13.01/2.02 Proof:
% 13.01/2.02 composition(join(x0, one), X)
% 13.01/2.03 = { by lemma 18 R->L }
% 13.01/2.03 composition(converse(join(x0, one)), composition(join(x0, one), X))
% 13.01/2.03 = { by axiom 10 (composition_associativity) }
% 13.01/2.03 composition(composition(converse(join(x0, one)), join(x0, one)), X)
% 13.01/2.03 = { by lemma 17 }
% 13.01/2.03 composition(converse(join(x0, one)), X)
% 13.01/2.03 = { by lemma 18 }
% 13.01/2.03 X
% 13.01/2.03
% 13.01/2.03 Lemma 20: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 13.01/2.03 Proof:
% 13.01/2.03 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 13.01/2.03 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.03 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 13.01/2.03 = { by axiom 13 (converse_cancellativity) }
% 13.01/2.03 complement(X)
% 13.01/2.03
% 13.01/2.03 Lemma 21: join(complement(X), complement(X)) = complement(X).
% 13.01/2.03 Proof:
% 13.01/2.03 join(complement(X), complement(X))
% 13.01/2.03 = { by lemma 18 R->L }
% 13.01/2.03 join(complement(X), composition(converse(join(x0, one)), complement(X)))
% 13.01/2.03 = { by lemma 19 R->L }
% 13.01/2.03 join(complement(X), composition(converse(join(x0, one)), complement(composition(join(x0, one), X))))
% 13.01/2.03 = { by lemma 20 }
% 13.01/2.03 complement(X)
% 13.01/2.03
% 13.01/2.03 Lemma 22: join(top, complement(X)) = top.
% 13.01/2.03 Proof:
% 13.01/2.03 join(top, complement(X))
% 13.01/2.03 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.03 join(complement(X), top)
% 13.01/2.03 = { by lemma 16 R->L }
% 13.01/2.03 join(X, join(complement(X), complement(X)))
% 13.01/2.03 = { by lemma 21 }
% 13.01/2.03 join(X, complement(X))
% 13.01/2.03 = { by axiom 6 (def_top) R->L }
% 13.01/2.03 top
% 13.01/2.03
% 13.01/2.03 Lemma 23: join(Y, top) = join(X, top).
% 13.01/2.03 Proof:
% 13.01/2.03 join(Y, top)
% 13.01/2.03 = { by lemma 22 R->L }
% 13.01/2.03 join(Y, join(top, complement(Y)))
% 13.01/2.03 = { by lemma 16 }
% 13.01/2.03 join(top, top)
% 13.01/2.03 = { by lemma 16 R->L }
% 13.01/2.03 join(X, join(top, complement(X)))
% 13.01/2.03 = { by lemma 22 }
% 13.01/2.03 join(X, top)
% 13.01/2.03
% 13.01/2.03 Lemma 24: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 13.01/2.03 Proof:
% 13.01/2.03 join(meet(X, Y), complement(join(complement(X), Y)))
% 13.01/2.03 = { by axiom 11 (maddux4_definiton_of_meet) }
% 13.01/2.03 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 13.01/2.03 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L }
% 13.01/2.03 X
% 13.01/2.03
% 13.01/2.03 Lemma 25: join(zero, meet(X, X)) = X.
% 13.01/2.03 Proof:
% 13.01/2.03 join(zero, meet(X, X))
% 13.01/2.03 = { by axiom 11 (maddux4_definiton_of_meet) }
% 13.01/2.03 join(zero, complement(join(complement(X), complement(X))))
% 13.01/2.03 = { by axiom 5 (def_zero) }
% 13.01/2.03 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 13.01/2.03 = { by lemma 24 }
% 13.01/2.03 X
% 13.01/2.03
% 13.01/2.03 Lemma 26: join(zero, join(X, meet(Y, Y))) = join(X, Y).
% 13.01/2.03 Proof:
% 13.01/2.03 join(zero, join(X, meet(Y, Y)))
% 13.01/2.03 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.03 join(zero, join(meet(Y, Y), X))
% 13.01/2.03 = { by axiom 8 (maddux2_join_associativity) }
% 13.01/2.03 join(join(zero, meet(Y, Y)), X)
% 13.01/2.03 = { by lemma 25 }
% 13.01/2.03 join(Y, X)
% 13.01/2.03 = { by axiom 2 (maddux1_join_commutativity) }
% 13.01/2.03 join(X, Y)
% 13.01/2.03
% 13.01/2.03 Lemma 27: join(X, zero) = join(X, X).
% 13.01/2.03 Proof:
% 13.01/2.03 join(X, zero)
% 13.01/2.03 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.03 join(zero, X)
% 13.01/2.03 = { by lemma 25 R->L }
% 13.01/2.03 join(zero, join(zero, meet(X, X)))
% 13.01/2.03 = { by axiom 11 (maddux4_definiton_of_meet) }
% 13.01/2.03 join(zero, join(zero, complement(join(complement(X), complement(X)))))
% 13.01/2.03 = { by lemma 21 R->L }
% 13.01/2.03 join(zero, join(zero, join(complement(join(complement(X), complement(X))), complement(join(complement(X), complement(X))))))
% 13.01/2.03 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 13.01/2.03 join(zero, join(zero, join(meet(X, X), complement(join(complement(X), complement(X))))))
% 13.01/2.03 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 13.01/2.03 join(zero, join(zero, join(meet(X, X), meet(X, X))))
% 13.01/2.03 = { by lemma 26 }
% 13.01/2.03 join(zero, join(meet(X, X), X))
% 13.01/2.03 = { by axiom 2 (maddux1_join_commutativity) }
% 13.01/2.03 join(zero, join(X, meet(X, X)))
% 13.01/2.03 = { by lemma 26 }
% 13.01/2.03 join(X, X)
% 13.01/2.03
% 13.01/2.03 Lemma 28: composition(join(X, join(x0, one)), Y) = join(Y, composition(X, Y)).
% 13.01/2.03 Proof:
% 13.01/2.03 composition(join(X, join(x0, one)), Y)
% 13.01/2.03 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.03 composition(join(join(x0, one), X), Y)
% 13.01/2.03 = { by axiom 12 (composition_distributivity) }
% 13.01/2.03 join(composition(join(x0, one), Y), composition(X, Y))
% 13.01/2.03 = { by lemma 19 }
% 13.01/2.03 join(Y, composition(X, Y))
% 13.01/2.03
% 13.01/2.03 Lemma 29: converse(join(x0, one)) = join(x0, one).
% 13.01/2.03 Proof:
% 13.01/2.03 converse(join(x0, one))
% 13.01/2.03 = { by lemma 17 R->L }
% 13.01/2.03 composition(converse(join(x0, one)), join(x0, one))
% 13.01/2.03 = { by lemma 18 }
% 13.01/2.03 join(x0, one)
% 13.01/2.03
% 13.01/2.03 Lemma 30: join(join(x0, one), join(X, Y)) = join(X, join(Y, one)).
% 13.01/2.03 Proof:
% 13.01/2.03 join(join(x0, one), join(X, Y))
% 13.01/2.03 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.03 join(join(x0, one), join(Y, X))
% 13.01/2.03 = { by axiom 3 (goals) }
% 13.01/2.03 join(one, join(Y, X))
% 13.01/2.03 = { by axiom 8 (maddux2_join_associativity) }
% 13.01/2.03 join(join(one, Y), X)
% 13.01/2.03 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.03 join(join(Y, one), X)
% 13.01/2.03 = { by axiom 2 (maddux1_join_commutativity) }
% 13.01/2.03 join(X, join(Y, one))
% 13.01/2.03
% 13.01/2.03 Lemma 31: join(join(x0, one), X) = join(X, one).
% 13.01/2.03 Proof:
% 13.01/2.03 join(join(x0, one), X)
% 13.01/2.03 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.03 join(X, join(x0, one))
% 13.01/2.03 = { by axiom 3 (goals) }
% 13.01/2.03 join(X, one)
% 13.01/2.03
% 13.01/2.03 Lemma 32: join(X, X) = X.
% 13.01/2.03 Proof:
% 13.01/2.03 join(X, X)
% 13.01/2.03 = { by lemma 18 R->L }
% 13.01/2.03 join(X, composition(converse(join(x0, one)), X))
% 13.01/2.03 = { by lemma 28 R->L }
% 13.01/2.03 composition(join(converse(join(x0, one)), join(x0, one)), X)
% 13.01/2.03 = { by lemma 29 }
% 13.01/2.03 composition(join(join(x0, one), join(x0, one)), X)
% 13.01/2.03 = { by lemma 27 R->L }
% 13.01/2.03 composition(join(join(x0, one), zero), X)
% 13.01/2.03 = { by axiom 2 (maddux1_join_commutativity) }
% 13.01/2.03 composition(join(zero, join(x0, one)), X)
% 13.01/2.03 = { by lemma 30 R->L }
% 13.01/2.03 composition(join(join(x0, one), join(zero, x0)), X)
% 13.01/2.03 = { by axiom 2 (maddux1_join_commutativity) }
% 13.01/2.03 composition(join(join(x0, one), join(x0, zero)), X)
% 13.01/2.03 = { by lemma 27 }
% 13.01/2.03 composition(join(join(x0, one), join(x0, x0)), X)
% 13.01/2.03 = { by lemma 30 }
% 13.01/2.03 composition(join(x0, join(x0, one)), X)
% 13.01/2.03 = { by axiom 2 (maddux1_join_commutativity) }
% 13.01/2.03 composition(join(join(x0, one), x0), X)
% 13.01/2.03 = { by lemma 31 }
% 13.01/2.03 composition(join(x0, one), X)
% 13.01/2.03 = { by lemma 19 }
% 13.01/2.03 X
% 13.01/2.03
% 13.01/2.03 Lemma 33: join(X, zero) = X.
% 13.01/2.03 Proof:
% 13.01/2.03 join(X, zero)
% 13.01/2.03 = { by lemma 27 }
% 13.01/2.03 join(X, X)
% 13.01/2.03 = { by lemma 32 }
% 13.01/2.03 X
% 13.01/2.03
% 13.01/2.03 Lemma 34: join(zero, X) = X.
% 13.01/2.03 Proof:
% 13.01/2.03 join(zero, X)
% 13.01/2.03 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.03 join(X, zero)
% 13.01/2.03 = { by lemma 33 }
% 13.01/2.03 X
% 13.01/2.03
% 13.01/2.03 Lemma 35: join(X, top) = top.
% 13.01/2.03 Proof:
% 13.01/2.03 join(X, top)
% 13.01/2.03 = { by lemma 23 }
% 13.01/2.03 join(zero, top)
% 13.01/2.03 = { by lemma 34 }
% 13.01/2.03 top
% 13.01/2.03
% 13.01/2.03 Lemma 36: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 13.01/2.03 Proof:
% 13.01/2.03 converse(join(X, converse(Y)))
% 13.01/2.03 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.03 converse(join(converse(Y), X))
% 13.01/2.03 = { by axiom 7 (converse_additivity) }
% 13.01/2.03 join(converse(converse(Y)), converse(X))
% 13.01/2.03 = { by axiom 1 (converse_idempotence) }
% 13.01/2.03 join(Y, converse(X))
% 13.01/2.03
% 13.01/2.03 Lemma 37: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 13.01/2.03 Proof:
% 13.01/2.03 converse(join(converse(X), Y))
% 13.01/2.03 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.03 converse(join(Y, converse(X)))
% 13.01/2.03 = { by lemma 36 }
% 13.01/2.03 join(X, converse(Y))
% 13.01/2.03
% 13.01/2.03 Lemma 38: join(X, converse(complement(converse(X)))) = converse(top).
% 13.01/2.03 Proof:
% 13.01/2.03 join(X, converse(complement(converse(X))))
% 13.01/2.03 = { by lemma 37 R->L }
% 13.01/2.03 converse(join(converse(X), complement(converse(X))))
% 13.01/2.03 = { by axiom 6 (def_top) R->L }
% 13.01/2.03 converse(top)
% 13.01/2.03
% 13.01/2.03 Lemma 39: join(X, join(complement(X), Y)) = top.
% 13.01/2.03 Proof:
% 13.01/2.03 join(X, join(complement(X), Y))
% 13.01/2.03 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.03 join(X, join(Y, complement(X)))
% 13.01/2.03 = { by lemma 16 }
% 13.01/2.03 join(Y, top)
% 13.01/2.03 = { by lemma 23 R->L }
% 13.01/2.03 join(Z, top)
% 13.01/2.03 = { by lemma 35 }
% 13.01/2.03 top
% 13.01/2.03
% 13.01/2.03 Lemma 40: join(X, converse(top)) = top.
% 13.01/2.03 Proof:
% 13.01/2.03 join(X, converse(top))
% 13.01/2.03 = { by lemma 38 R->L }
% 13.01/2.03 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 13.01/2.03 = { by lemma 39 }
% 13.01/2.03 top
% 13.01/2.03
% 13.01/2.03 Lemma 41: converse(top) = top.
% 13.01/2.03 Proof:
% 13.01/2.03 converse(top)
% 13.01/2.03 = { by lemma 35 R->L }
% 13.01/2.03 converse(join(X, top))
% 13.01/2.03 = { by axiom 7 (converse_additivity) }
% 13.01/2.03 join(converse(X), converse(top))
% 13.01/2.03 = { by lemma 40 }
% 13.01/2.03 top
% 13.01/2.03
% 13.01/2.03 Lemma 42: join(top, X) = top.
% 13.01/2.03 Proof:
% 13.01/2.03 join(top, X)
% 13.01/2.03 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.03 join(X, top)
% 13.01/2.03 = { by lemma 23 R->L }
% 13.01/2.03 join(Y, top)
% 13.01/2.03 = { by lemma 35 }
% 13.01/2.03 top
% 13.01/2.03
% 13.01/2.03 Lemma 43: complement(complement(X)) = meet(X, X).
% 13.01/2.03 Proof:
% 13.01/2.03 complement(complement(X))
% 13.01/2.03 = { by lemma 21 R->L }
% 13.01/2.03 complement(join(complement(X), complement(X)))
% 13.01/2.03 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 13.01/2.03 meet(X, X)
% 13.01/2.03
% 13.01/2.03 Lemma 44: complement(complement(X)) = X.
% 13.01/2.03 Proof:
% 13.01/2.03 complement(complement(X))
% 13.01/2.03 = { by lemma 34 R->L }
% 13.01/2.03 join(zero, complement(complement(X)))
% 13.01/2.03 = { by lemma 43 }
% 13.01/2.03 join(zero, meet(X, X))
% 13.01/2.03 = { by lemma 25 }
% 13.01/2.03 X
% 13.01/2.03
% 13.01/2.03 Lemma 45: meet(X, X) = X.
% 13.01/2.03 Proof:
% 13.01/2.03 meet(X, X)
% 13.01/2.03 = { by lemma 43 R->L }
% 13.01/2.03 complement(complement(X))
% 13.01/2.03 = { by lemma 44 }
% 13.01/2.03 X
% 13.01/2.03
% 13.01/2.03 Lemma 46: meet(Y, X) = meet(X, Y).
% 13.01/2.03 Proof:
% 13.01/2.03 meet(Y, X)
% 13.01/2.03 = { by axiom 11 (maddux4_definiton_of_meet) }
% 13.01/2.03 complement(join(complement(Y), complement(X)))
% 13.01/2.03 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.03 complement(join(complement(X), complement(Y)))
% 13.01/2.03 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 13.01/2.03 meet(X, Y)
% 13.01/2.03
% 13.01/2.03 Lemma 47: complement(join(zero, complement(X))) = meet(X, top).
% 13.01/2.03 Proof:
% 13.01/2.03 complement(join(zero, complement(X)))
% 13.01/2.03 = { by lemma 15 R->L }
% 13.01/2.03 complement(join(complement(top), complement(X)))
% 13.01/2.03 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 13.01/2.03 meet(top, X)
% 13.01/2.03 = { by lemma 46 R->L }
% 13.01/2.03 meet(X, top)
% 13.01/2.03
% 13.01/2.03 Lemma 48: meet(X, top) = X.
% 13.01/2.03 Proof:
% 13.01/2.03 meet(X, top)
% 13.01/2.03 = { by lemma 46 }
% 13.01/2.03 meet(top, X)
% 13.01/2.03 = { by lemma 44 R->L }
% 13.01/2.03 meet(top, complement(complement(X)))
% 13.01/2.03 = { by lemma 46 }
% 13.01/2.03 meet(complement(complement(X)), top)
% 13.01/2.03 = { by lemma 47 R->L }
% 13.01/2.03 complement(join(zero, complement(complement(complement(X)))))
% 13.01/2.03 = { by lemma 43 }
% 13.01/2.03 complement(join(zero, meet(complement(X), complement(X))))
% 13.01/2.03 = { by lemma 25 }
% 13.01/2.03 complement(complement(X))
% 13.01/2.03 = { by lemma 44 }
% 13.01/2.03 X
% 13.01/2.03
% 13.01/2.03 Lemma 49: meet(top, X) = X.
% 13.01/2.03 Proof:
% 13.01/2.03 meet(top, X)
% 13.01/2.03 = { by lemma 46 }
% 13.01/2.03 meet(X, top)
% 13.01/2.03 = { by lemma 48 }
% 13.01/2.03 X
% 13.01/2.03
% 13.01/2.03 Lemma 50: composition(join(join(x0, one), Y), X) = join(X, composition(Y, X)).
% 13.01/2.03 Proof:
% 13.01/2.03 composition(join(join(x0, one), Y), X)
% 13.01/2.03 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.03 composition(join(Y, join(x0, one)), X)
% 13.01/2.03 = { by lemma 28 }
% 13.01/2.03 join(X, composition(Y, X))
% 13.01/2.03
% 13.01/2.03 Lemma 51: composition(top, zero) = zero.
% 13.01/2.03 Proof:
% 13.01/2.03 composition(top, zero)
% 13.01/2.03 = { by lemma 41 R->L }
% 13.01/2.03 composition(converse(top), zero)
% 13.01/2.03 = { by lemma 34 R->L }
% 13.01/2.03 join(zero, composition(converse(top), zero))
% 13.01/2.03 = { by lemma 15 R->L }
% 13.01/2.03 join(complement(top), composition(converse(top), zero))
% 13.01/2.03 = { by lemma 15 R->L }
% 13.01/2.03 join(complement(top), composition(converse(top), complement(top)))
% 13.01/2.03 = { by lemma 42 R->L }
% 13.01/2.03 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 13.01/2.03 = { by lemma 41 R->L }
% 13.01/2.03 join(complement(top), composition(converse(top), complement(join(top, composition(converse(top), top)))))
% 13.01/2.03 = { by lemma 50 R->L }
% 13.01/2.03 join(complement(top), composition(converse(top), complement(composition(join(join(x0, one), converse(top)), top))))
% 13.01/2.03 = { by lemma 40 }
% 13.01/2.03 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 13.01/2.03 = { by lemma 20 }
% 13.01/2.03 complement(top)
% 13.01/2.03 = { by lemma 15 }
% 13.01/2.03 zero
% 13.01/2.03
% 13.01/2.03 Lemma 52: join(X, join(X, Y)) = join(X, Y).
% 13.01/2.03 Proof:
% 13.01/2.03 join(X, join(X, Y))
% 13.01/2.03 = { by axiom 8 (maddux2_join_associativity) }
% 13.01/2.03 join(join(X, X), Y)
% 13.01/2.03 = { by lemma 32 }
% 13.01/2.03 join(X, Y)
% 13.01/2.03
% 13.01/2.03 Lemma 53: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 13.01/2.03 Proof:
% 13.01/2.03 join(Y, join(X, Z))
% 13.01/2.03 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.03 join(join(X, Z), Y)
% 13.01/2.03 = { by axiom 8 (maddux2_join_associativity) R->L }
% 13.01/2.03 join(X, join(Z, Y))
% 13.01/2.03 = { by axiom 2 (maddux1_join_commutativity) }
% 13.01/2.03 join(X, join(Y, Z))
% 13.01/2.03
% 13.01/2.03 Lemma 54: join(Z, join(X, Y)) = join(X, join(Y, Z)).
% 13.01/2.03 Proof:
% 13.01/2.03 join(Z, join(X, Y))
% 13.01/2.03 = { by lemma 53 }
% 13.01/2.03 join(X, join(Z, Y))
% 13.01/2.03 = { by axiom 2 (maddux1_join_commutativity) }
% 13.01/2.03 join(X, join(Y, Z))
% 13.01/2.03
% 13.01/2.03 Lemma 55: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
% 13.01/2.03 Proof:
% 13.01/2.03 complement(join(complement(X), meet(Y, Z)))
% 13.01/2.03 = { by lemma 46 }
% 13.01/2.03 complement(join(complement(X), meet(Z, Y)))
% 13.01/2.03 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.03 complement(join(meet(Z, Y), complement(X)))
% 13.01/2.03 = { by axiom 11 (maddux4_definiton_of_meet) }
% 13.01/2.03 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 13.01/2.03 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 13.01/2.03 meet(join(complement(Z), complement(Y)), X)
% 13.01/2.03 = { by lemma 46 R->L }
% 13.01/2.03 meet(X, join(complement(Z), complement(Y)))
% 13.01/2.03 = { by axiom 2 (maddux1_join_commutativity) }
% 13.01/2.03 meet(X, join(complement(Y), complement(Z)))
% 13.01/2.03
% 13.01/2.04 Lemma 56: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 13.01/2.04 Proof:
% 13.01/2.04 join(complement(X), complement(Y))
% 13.01/2.04 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.04 join(complement(Y), complement(X))
% 13.01/2.04 = { by lemma 49 R->L }
% 13.01/2.04 meet(top, join(complement(Y), complement(X)))
% 13.01/2.04 = { by lemma 55 R->L }
% 13.01/2.04 complement(join(complement(top), meet(Y, X)))
% 13.01/2.04 = { by lemma 15 }
% 13.01/2.04 complement(join(zero, meet(Y, X)))
% 13.01/2.04 = { by lemma 34 }
% 13.01/2.04 complement(meet(Y, X))
% 13.01/2.04 = { by lemma 46 R->L }
% 13.01/2.04 complement(meet(X, Y))
% 13.01/2.04
% 13.01/2.04 Lemma 57: meet(X, meet(X, X)) = X.
% 13.01/2.04 Proof:
% 13.01/2.04 meet(X, meet(X, X))
% 13.01/2.04 = { by lemma 43 R->L }
% 13.01/2.04 meet(X, complement(complement(X)))
% 13.01/2.04 = { by lemma 33 R->L }
% 13.01/2.04 join(meet(X, complement(complement(X))), zero)
% 13.01/2.04 = { by lemma 15 R->L }
% 13.01/2.04 join(meet(X, complement(complement(X))), complement(top))
% 13.01/2.04 = { by axiom 6 (def_top) }
% 13.01/2.04 join(meet(X, complement(complement(X))), complement(join(complement(X), complement(complement(X)))))
% 13.01/2.04 = { by lemma 24 }
% 13.01/2.04 X
% 13.01/2.04
% 13.01/2.04 Lemma 58: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 13.01/2.04 Proof:
% 13.01/2.04 complement(join(X, complement(Y)))
% 13.01/2.04 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.04 complement(join(complement(Y), X))
% 13.01/2.04 = { by lemma 57 R->L }
% 13.01/2.04 complement(join(complement(Y), meet(X, meet(X, X))))
% 13.01/2.04 = { by lemma 55 }
% 13.01/2.04 meet(Y, join(complement(X), complement(meet(X, X))))
% 13.01/2.04 = { by lemma 56 }
% 13.01/2.04 meet(Y, complement(meet(X, meet(X, X))))
% 13.01/2.04 = { by lemma 57 }
% 13.01/2.04 meet(Y, complement(X))
% 13.01/2.04
% 13.01/2.04 Lemma 59: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 13.01/2.04 Proof:
% 13.01/2.04 complement(meet(X, complement(Y)))
% 13.01/2.04 = { by lemma 46 }
% 13.01/2.04 complement(meet(complement(Y), X))
% 13.01/2.04 = { by lemma 34 R->L }
% 13.01/2.04 complement(meet(join(zero, complement(Y)), X))
% 13.01/2.04 = { by lemma 56 R->L }
% 13.01/2.04 join(complement(join(zero, complement(Y))), complement(X))
% 13.01/2.04 = { by lemma 47 }
% 13.01/2.04 join(meet(Y, top), complement(X))
% 13.01/2.04 = { by lemma 48 }
% 13.01/2.04 join(Y, complement(X))
% 13.01/2.04
% 13.01/2.04 Lemma 60: meet(X, join(X, Y)) = X.
% 13.01/2.04 Proof:
% 13.01/2.04 meet(X, join(X, Y))
% 13.01/2.04 = { by lemma 45 R->L }
% 13.01/2.04 meet(X, join(X, meet(Y, Y)))
% 13.01/2.04 = { by lemma 43 R->L }
% 13.01/2.04 meet(X, join(X, complement(complement(Y))))
% 13.01/2.04 = { by lemma 59 R->L }
% 13.01/2.04 meet(X, complement(meet(complement(Y), complement(X))))
% 13.01/2.04 = { by lemma 56 R->L }
% 13.01/2.04 meet(X, join(complement(complement(Y)), complement(complement(X))))
% 13.01/2.04 = { by lemma 55 R->L }
% 13.01/2.04 complement(join(complement(X), meet(complement(Y), complement(X))))
% 13.01/2.04 = { by lemma 34 R->L }
% 13.01/2.04 join(zero, complement(join(complement(X), meet(complement(Y), complement(X)))))
% 13.01/2.04 = { by lemma 15 R->L }
% 13.01/2.04 join(complement(top), complement(join(complement(X), meet(complement(Y), complement(X)))))
% 13.01/2.04 = { by lemma 35 R->L }
% 13.01/2.04 join(complement(join(complement(complement(Y)), top)), complement(join(complement(X), meet(complement(Y), complement(X)))))
% 13.01/2.04 = { by lemma 16 R->L }
% 13.01/2.04 join(complement(join(complement(X), join(complement(complement(Y)), complement(complement(X))))), complement(join(complement(X), meet(complement(Y), complement(X)))))
% 13.01/2.04 = { by lemma 56 }
% 13.01/2.04 join(complement(join(complement(X), complement(meet(complement(Y), complement(X))))), complement(join(complement(X), meet(complement(Y), complement(X)))))
% 13.01/2.04 = { by lemma 46 R->L }
% 13.01/2.04 join(complement(join(complement(X), complement(meet(complement(X), complement(Y))))), complement(join(complement(X), meet(complement(Y), complement(X)))))
% 13.01/2.04 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 13.01/2.04 join(meet(X, meet(complement(X), complement(Y))), complement(join(complement(X), meet(complement(Y), complement(X)))))
% 13.01/2.04 = { by lemma 46 R->L }
% 13.01/2.04 join(meet(X, meet(complement(Y), complement(X))), complement(join(complement(X), meet(complement(Y), complement(X)))))
% 13.01/2.04 = { by lemma 24 }
% 13.01/2.04 X
% 13.01/2.04
% 13.01/2.04 Lemma 61: meet(X, join(Y, X)) = X.
% 13.01/2.04 Proof:
% 13.01/2.04 meet(X, join(Y, X))
% 13.01/2.04 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.04 meet(X, join(X, Y))
% 13.01/2.04 = { by lemma 60 }
% 13.01/2.04 X
% 13.01/2.04
% 13.01/2.04 Lemma 62: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 13.01/2.04 Proof:
% 13.01/2.04 meet(Y, meet(X, Z))
% 13.01/2.04 = { by lemma 46 }
% 13.01/2.04 meet(Y, meet(Z, X))
% 13.01/2.04 = { by lemma 48 R->L }
% 13.01/2.04 meet(meet(Y, meet(Z, X)), top)
% 13.01/2.04 = { by lemma 47 R->L }
% 13.01/2.04 complement(join(zero, complement(meet(Y, meet(Z, X)))))
% 13.01/2.04 = { by lemma 46 }
% 13.01/2.04 complement(join(zero, complement(meet(Y, meet(X, Z)))))
% 13.01/2.04 = { by axiom 11 (maddux4_definiton_of_meet) }
% 13.01/2.04 complement(join(zero, complement(meet(Y, complement(join(complement(X), complement(Z)))))))
% 13.01/2.04 = { by lemma 59 }
% 13.01/2.04 complement(join(zero, join(join(complement(X), complement(Z)), complement(Y))))
% 13.01/2.04 = { by axiom 8 (maddux2_join_associativity) R->L }
% 13.01/2.04 complement(join(zero, join(complement(X), join(complement(Z), complement(Y)))))
% 13.01/2.04 = { by lemma 56 }
% 13.01/2.04 complement(join(zero, join(complement(X), complement(meet(Z, Y)))))
% 13.01/2.04 = { by lemma 56 }
% 13.01/2.04 complement(join(zero, complement(meet(X, meet(Z, Y)))))
% 13.01/2.04 = { by lemma 46 R->L }
% 13.01/2.04 complement(join(zero, complement(meet(X, meet(Y, Z)))))
% 13.01/2.04 = { by lemma 47 }
% 13.01/2.04 meet(meet(X, meet(Y, Z)), top)
% 13.01/2.04 = { by lemma 48 }
% 13.01/2.04 meet(X, meet(Y, Z))
% 13.01/2.04
% 13.01/2.04 Lemma 63: meet(meet(X, Y), Z) = meet(X, meet(Z, Y)).
% 13.01/2.04 Proof:
% 13.01/2.04 meet(meet(X, Y), Z)
% 13.01/2.04 = { by lemma 46 }
% 13.01/2.04 meet(Z, meet(X, Y))
% 13.01/2.04 = { by lemma 62 R->L }
% 13.01/2.04 meet(X, meet(Z, Y))
% 13.01/2.04
% 13.01/2.04 Lemma 64: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 13.01/2.04 Proof:
% 13.01/2.04 converse(composition(X, converse(Y)))
% 13.01/2.04 = { by axiom 9 (converse_multiplicativity) }
% 13.01/2.04 composition(converse(converse(Y)), converse(X))
% 13.01/2.04 = { by axiom 1 (converse_idempotence) }
% 13.01/2.04 composition(Y, converse(X))
% 13.01/2.04
% 13.01/2.04 Lemma 65: join(meet(X, Y), meet(X, complement(Y))) = X.
% 13.01/2.04 Proof:
% 13.01/2.04 join(meet(X, Y), meet(X, complement(Y)))
% 13.01/2.04 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.04 join(meet(X, complement(Y)), meet(X, Y))
% 13.01/2.04 = { by axiom 11 (maddux4_definiton_of_meet) }
% 13.01/2.04 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 13.01/2.04 = { by lemma 24 }
% 13.01/2.04 X
% 13.01/2.04
% 13.01/2.04 Lemma 66: join(meet(X, Y), meet(complement(Y), X)) = X.
% 13.01/2.04 Proof:
% 13.01/2.04 join(meet(X, Y), meet(complement(Y), X))
% 13.01/2.04 = { by lemma 46 }
% 13.01/2.04 join(meet(X, Y), meet(X, complement(Y)))
% 13.01/2.04 = { by lemma 65 }
% 13.01/2.04 X
% 13.01/2.04
% 13.01/2.04 Lemma 67: meet(complement(X), join(X, Y)) = meet(Y, complement(X)).
% 13.01/2.04 Proof:
% 13.01/2.04 meet(complement(X), join(X, Y))
% 13.01/2.04 = { by lemma 44 R->L }
% 13.01/2.04 meet(complement(X), join(X, complement(complement(Y))))
% 13.01/2.04 = { by lemma 66 R->L }
% 13.01/2.04 join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(Y), meet(complement(X), join(X, complement(complement(Y))))))
% 13.01/2.04 = { by lemma 46 }
% 13.01/2.04 join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(Y), meet(join(X, complement(complement(Y))), complement(X))))
% 13.01/2.04 = { by lemma 62 }
% 13.01/2.04 join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(join(X, complement(complement(Y))), meet(complement(Y), complement(X))))
% 13.01/2.04 = { by lemma 58 R->L }
% 13.01/2.04 join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(join(X, complement(complement(Y))), complement(join(X, complement(complement(Y))))))
% 13.01/2.04 = { by axiom 5 (def_zero) R->L }
% 13.01/2.04 join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), zero)
% 13.01/2.04 = { by lemma 33 }
% 13.01/2.04 meet(meet(complement(X), join(X, complement(complement(Y)))), Y)
% 13.01/2.04 = { by lemma 63 }
% 13.01/2.04 meet(complement(X), meet(Y, join(X, complement(complement(Y)))))
% 13.01/2.04 = { by lemma 44 }
% 13.01/2.04 meet(complement(X), meet(Y, join(X, Y)))
% 13.01/2.04 = { by lemma 61 }
% 13.01/2.04 meet(complement(X), Y)
% 13.01/2.04 = { by lemma 46 R->L }
% 13.01/2.04 meet(Y, complement(X))
% 13.01/2.04
% 13.01/2.04 Lemma 68: meet(X, join(complement(X), Y)) = meet(X, Y).
% 13.01/2.04 Proof:
% 13.01/2.04 meet(X, join(complement(X), Y))
% 13.01/2.04 = { by lemma 34 R->L }
% 13.01/2.04 meet(X, join(zero, join(complement(X), Y)))
% 13.01/2.04 = { by axiom 8 (maddux2_join_associativity) }
% 13.01/2.04 meet(X, join(join(zero, complement(X)), Y))
% 13.01/2.04 = { by lemma 48 R->L }
% 13.01/2.04 meet(X, meet(join(join(zero, complement(X)), Y), top))
% 13.01/2.04 = { by lemma 63 R->L }
% 13.01/2.04 meet(meet(X, top), join(join(zero, complement(X)), Y))
% 13.01/2.04 = { by lemma 47 R->L }
% 13.01/2.04 meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), Y))
% 13.01/2.04 = { by lemma 67 }
% 13.01/2.04 meet(Y, complement(join(zero, complement(X))))
% 13.01/2.04 = { by lemma 47 }
% 13.01/2.04 meet(Y, meet(X, top))
% 13.01/2.04 = { by lemma 48 }
% 13.01/2.04 meet(Y, X)
% 13.01/2.04 = { by lemma 46 R->L }
% 13.01/2.04 meet(X, Y)
% 13.01/2.04
% 13.01/2.04 Lemma 69: join(join(X, composition(x0, Y)), Y) = join(X, Y).
% 13.01/2.04 Proof:
% 13.01/2.04 join(join(X, composition(x0, Y)), Y)
% 13.01/2.04 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.04 join(Y, join(X, composition(x0, Y)))
% 13.01/2.04 = { by lemma 54 }
% 13.01/2.04 join(X, join(composition(x0, Y), Y))
% 13.01/2.04 = { by lemma 19 R->L }
% 13.01/2.04 join(X, join(composition(x0, Y), composition(join(x0, one), Y)))
% 13.01/2.04 = { by axiom 3 (goals) }
% 13.01/2.04 join(X, join(composition(x0, Y), composition(one, Y)))
% 13.01/2.04 = { by axiom 12 (composition_distributivity) R->L }
% 13.01/2.04 join(X, composition(join(x0, one), Y))
% 13.01/2.04 = { by lemma 19 }
% 13.01/2.04 join(X, Y)
% 13.01/2.04
% 13.01/2.04 Lemma 70: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 13.01/2.04 Proof:
% 13.01/2.04 join(meet(X, Y), meet(Y, complement(X)))
% 13.01/2.04 = { by lemma 46 }
% 13.01/2.04 join(meet(Y, X), meet(Y, complement(X)))
% 13.01/2.04 = { by lemma 65 }
% 13.01/2.04 Y
% 13.01/2.04
% 13.01/2.04 Lemma 71: meet(join(X, Y), join(X, complement(Y))) = X.
% 13.01/2.04 Proof:
% 13.01/2.04 meet(join(X, Y), join(X, complement(Y)))
% 13.01/2.04 = { by lemma 46 }
% 13.01/2.04 meet(join(X, complement(Y)), join(X, Y))
% 13.01/2.04 = { by lemma 44 R->L }
% 13.01/2.04 meet(join(X, complement(Y)), join(X, complement(complement(Y))))
% 13.01/2.04 = { by lemma 59 R->L }
% 13.01/2.04 meet(complement(meet(Y, complement(X))), join(X, complement(complement(Y))))
% 13.01/2.04 = { by lemma 46 R->L }
% 13.01/2.04 meet(complement(meet(complement(X), Y)), join(X, complement(complement(Y))))
% 13.01/2.04 = { by lemma 46 }
% 13.01/2.04 meet(join(X, complement(complement(Y))), complement(meet(complement(X), Y)))
% 13.01/2.04 = { by lemma 58 R->L }
% 13.01/2.04 complement(join(meet(complement(X), Y), complement(join(X, complement(complement(Y))))))
% 13.01/2.04 = { by lemma 58 }
% 13.01/2.04 complement(join(meet(complement(X), Y), meet(complement(Y), complement(X))))
% 13.01/2.04 = { by lemma 66 }
% 13.01/2.04 complement(complement(X))
% 13.01/2.04 = { by lemma 44 }
% 13.01/2.04 X
% 13.01/2.04
% 13.01/2.04 Lemma 72: join(meet(X, Y), complement(join(Y, complement(X)))) = X.
% 13.01/2.04 Proof:
% 13.01/2.04 join(meet(X, Y), complement(join(Y, complement(X))))
% 13.01/2.04 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.04 join(meet(X, Y), complement(join(complement(X), Y)))
% 13.01/2.04 = { by lemma 24 }
% 13.01/2.04 X
% 13.01/2.04
% 13.01/2.04 Lemma 73: join(composition(X, top), composition(composition(X, Y), top)) = composition(X, top).
% 13.01/2.04 Proof:
% 13.01/2.04 join(composition(X, top), composition(composition(X, Y), top))
% 13.01/2.04 = { by lemma 39 R->L }
% 13.01/2.04 join(composition(X, top), composition(composition(X, Y), join(zero, join(complement(zero), meet(complement(zero), complement(zero))))))
% 13.01/2.04 = { by lemma 26 }
% 13.01/2.04 join(composition(X, top), composition(composition(X, Y), join(complement(zero), complement(zero))))
% 13.01/2.04 = { by lemma 21 }
% 13.01/2.04 join(composition(X, top), composition(composition(X, Y), complement(zero)))
% 13.01/2.04 = { by axiom 1 (converse_idempotence) R->L }
% 13.01/2.04 join(composition(X, top), composition(converse(converse(composition(X, Y))), complement(zero)))
% 13.01/2.04 = { by lemma 44 R->L }
% 13.01/2.04 join(complement(complement(composition(X, top))), composition(converse(converse(composition(X, Y))), complement(zero)))
% 13.01/2.04 = { by lemma 51 R->L }
% 13.01/2.04 join(complement(complement(composition(X, top))), composition(converse(converse(composition(X, Y))), complement(composition(top, zero))))
% 13.01/2.04 = { by lemma 42 R->L }
% 13.01/2.04 join(complement(complement(composition(X, top))), composition(converse(converse(composition(X, Y))), complement(composition(join(top, converse(Y)), zero))))
% 13.01/2.04 = { by axiom 12 (composition_distributivity) }
% 13.01/2.04 join(complement(complement(composition(X, top))), composition(converse(converse(composition(X, Y))), complement(join(composition(top, zero), composition(converse(Y), zero)))))
% 13.01/2.04 = { by lemma 51 }
% 13.01/2.04 join(complement(complement(composition(X, top))), composition(converse(converse(composition(X, Y))), complement(join(zero, composition(converse(Y), zero)))))
% 13.01/2.04 = { by lemma 34 }
% 13.01/2.04 join(complement(complement(composition(X, top))), composition(converse(converse(composition(X, Y))), complement(composition(converse(Y), zero))))
% 13.01/2.04 = { by lemma 15 R->L }
% 13.01/2.04 join(complement(complement(composition(X, top))), composition(converse(converse(composition(X, Y))), complement(composition(converse(Y), complement(top)))))
% 13.01/2.04 = { by lemma 20 R->L }
% 13.01/2.04 join(complement(complement(composition(X, top))), composition(converse(converse(composition(X, Y))), complement(composition(converse(Y), join(complement(top), composition(converse(X), complement(composition(X, top))))))))
% 13.01/2.04 = { by lemma 15 }
% 13.01/2.04 join(complement(complement(composition(X, top))), composition(converse(converse(composition(X, Y))), complement(composition(converse(Y), join(zero, composition(converse(X), complement(composition(X, top))))))))
% 13.01/2.04 = { by lemma 34 }
% 13.01/2.04 join(complement(complement(composition(X, top))), composition(converse(converse(composition(X, Y))), complement(composition(converse(Y), composition(converse(X), complement(composition(X, top)))))))
% 13.01/2.04 = { by axiom 10 (composition_associativity) }
% 13.01/2.04 join(complement(complement(composition(X, top))), composition(converse(converse(composition(X, Y))), complement(composition(composition(converse(Y), converse(X)), complement(composition(X, top))))))
% 13.01/2.04 = { by axiom 9 (converse_multiplicativity) R->L }
% 13.01/2.04 join(complement(complement(composition(X, top))), composition(converse(converse(composition(X, Y))), complement(composition(converse(composition(X, Y)), complement(composition(X, top))))))
% 13.01/2.04 = { by lemma 20 }
% 13.01/2.04 complement(complement(composition(X, top)))
% 13.01/2.04 = { by lemma 44 }
% 13.01/2.04 composition(X, top)
% 13.01/2.04
% 13.01/2.04 Lemma 74: meet(join(meet(composition(X, top), Y), composition(X, Z)), composition(X, top)) = join(meet(composition(X, top), Y), composition(X, Z)).
% 13.01/2.04 Proof:
% 13.01/2.04 meet(join(meet(composition(X, top), Y), composition(X, Z)), composition(X, top))
% 13.01/2.04 = { by lemma 73 R->L }
% 13.01/2.05 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(composition(X, top), composition(composition(X, Z), top)))
% 13.01/2.05 = { by lemma 42 R->L }
% 13.01/2.05 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(composition(X, top), composition(composition(X, Z), join(top, join(x0, one)))))
% 13.01/2.05 = { by lemma 29 R->L }
% 13.01/2.05 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(composition(X, top), composition(composition(X, Z), join(top, converse(join(x0, one))))))
% 13.01/2.05 = { by lemma 37 R->L }
% 13.01/2.05 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(composition(X, top), composition(composition(X, Z), converse(join(converse(top), join(x0, one))))))
% 13.01/2.05 = { by lemma 64 R->L }
% 13.01/2.05 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(composition(X, top), converse(composition(join(converse(top), join(x0, one)), converse(composition(X, Z))))))
% 13.01/2.05 = { by lemma 28 }
% 13.01/2.05 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(composition(X, top), converse(join(converse(composition(X, Z)), composition(converse(top), converse(composition(X, Z)))))))
% 13.01/2.05 = { by lemma 37 }
% 13.01/2.05 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(composition(X, top), join(composition(X, Z), converse(composition(converse(top), converse(composition(X, Z)))))))
% 13.01/2.05 = { by lemma 64 }
% 13.01/2.05 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(composition(X, top), join(composition(X, Z), composition(composition(X, Z), converse(converse(top))))))
% 13.01/2.05 = { by axiom 1 (converse_idempotence) }
% 13.01/2.05 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(composition(X, top), join(composition(X, Z), composition(composition(X, Z), top))))
% 13.01/2.05 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.05 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(composition(X, top), join(composition(composition(X, Z), top), composition(X, Z))))
% 13.01/2.05 = { by axiom 8 (maddux2_join_associativity) }
% 13.01/2.05 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(join(composition(X, top), composition(composition(X, Z), top)), composition(X, Z)))
% 13.01/2.05 = { by lemma 73 }
% 13.01/2.05 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(composition(X, top), composition(X, Z)))
% 13.01/2.05 = { by lemma 72 R->L }
% 13.01/2.05 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(join(meet(composition(X, top), Y), complement(join(Y, complement(composition(X, top))))), composition(X, Z)))
% 13.01/2.05 = { by lemma 52 R->L }
% 13.01/2.05 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(join(meet(composition(X, top), Y), join(meet(composition(X, top), Y), complement(join(Y, complement(composition(X, top)))))), composition(X, Z)))
% 13.01/2.05 = { by lemma 72 }
% 13.01/2.05 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(join(meet(composition(X, top), Y), composition(X, top)), composition(X, Z)))
% 13.01/2.05 = { by axiom 8 (maddux2_join_associativity) R->L }
% 13.01/2.05 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(meet(composition(X, top), Y), join(composition(X, top), composition(X, Z))))
% 13.01/2.05 = { by axiom 2 (maddux1_join_commutativity) }
% 13.01/2.05 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(meet(composition(X, top), Y), join(composition(X, Z), composition(X, top))))
% 13.01/2.05 = { by lemma 54 R->L }
% 13.01/2.05 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(composition(X, top), join(meet(composition(X, top), Y), composition(X, Z))))
% 13.01/2.05 = { by axiom 2 (maddux1_join_commutativity) }
% 13.01/2.05 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(join(meet(composition(X, top), Y), composition(X, Z)), composition(X, top)))
% 13.01/2.05 = { by lemma 60 }
% 13.01/2.05 join(meet(composition(X, top), Y), composition(X, Z))
% 13.01/2.05
% 13.01/2.05 Lemma 75: join(join(meet(X, Y), composition(x0, Y)), Y) = Y.
% 13.01/2.05 Proof:
% 13.01/2.05 join(join(meet(X, Y), composition(x0, Y)), Y)
% 13.01/2.05 = { by lemma 69 }
% 13.01/2.05 join(meet(X, Y), Y)
% 13.01/2.05 = { by lemma 70 R->L }
% 13.01/2.05 join(meet(X, Y), join(meet(X, Y), meet(Y, complement(X))))
% 13.01/2.05 = { by lemma 52 }
% 13.01/2.05 join(meet(X, Y), meet(Y, complement(X)))
% 13.01/2.05 = { by lemma 70 }
% 13.01/2.05 Y
% 13.01/2.05
% 13.01/2.05 Lemma 76: join(meet(composition(x0, top), X), composition(x0, X)) = meet(composition(x0, top), X).
% 13.01/2.05 Proof:
% 13.01/2.05 join(meet(composition(x0, top), X), composition(x0, X))
% 13.01/2.05 = { by lemma 74 R->L }
% 13.01/2.05 meet(join(meet(composition(x0, top), X), composition(x0, X)), composition(x0, top))
% 13.01/2.05 = { by lemma 46 }
% 13.01/2.05 meet(composition(x0, top), join(meet(composition(x0, top), X), composition(x0, X)))
% 13.01/2.05 = { by lemma 60 R->L }
% 13.01/2.05 meet(composition(x0, top), meet(join(meet(composition(x0, top), X), composition(x0, X)), join(join(meet(composition(x0, top), X), composition(x0, X)), X)))
% 13.01/2.05 = { by lemma 75 }
% 13.01/2.05 meet(composition(x0, top), meet(join(meet(composition(x0, top), X), composition(x0, X)), X))
% 13.01/2.05 = { by lemma 62 }
% 13.01/2.05 meet(join(meet(composition(x0, top), X), composition(x0, X)), meet(composition(x0, top), X))
% 13.01/2.05 = { by lemma 46 }
% 13.01/2.05 meet(meet(composition(x0, top), X), join(meet(composition(x0, top), X), composition(x0, X)))
% 13.01/2.05 = { by axiom 11 (maddux4_definiton_of_meet) }
% 13.01/2.05 complement(join(complement(meet(composition(x0, top), X)), complement(join(meet(composition(x0, top), X), composition(x0, X)))))
% 13.01/2.05 = { by lemma 34 R->L }
% 13.01/2.05 join(zero, complement(join(complement(meet(composition(x0, top), X)), complement(join(meet(composition(x0, top), X), composition(x0, X))))))
% 13.01/2.05 = { by lemma 15 R->L }
% 13.01/2.05 join(complement(top), complement(join(complement(meet(composition(x0, top), X)), complement(join(meet(composition(x0, top), X), composition(x0, X))))))
% 13.01/2.05 = { by lemma 35 R->L }
% 13.01/2.05 join(complement(join(composition(x0, X), top)), complement(join(complement(meet(composition(x0, top), X)), complement(join(meet(composition(x0, top), X), composition(x0, X))))))
% 13.01/2.05 = { by lemma 16 R->L }
% 13.01/2.05 join(complement(join(meet(composition(x0, top), X), join(composition(x0, X), complement(meet(composition(x0, top), X))))), complement(join(complement(meet(composition(x0, top), X)), complement(join(meet(composition(x0, top), X), composition(x0, X))))))
% 13.01/2.05 = { by lemma 54 R->L }
% 13.01/2.05 join(complement(join(complement(meet(composition(x0, top), X)), join(meet(composition(x0, top), X), composition(x0, X)))), complement(join(complement(meet(composition(x0, top), X)), complement(join(meet(composition(x0, top), X), composition(x0, X))))))
% 13.01/2.05 = { by axiom 2 (maddux1_join_commutativity) }
% 13.01/2.05 join(complement(join(join(meet(composition(x0, top), X), composition(x0, X)), complement(meet(composition(x0, top), X)))), complement(join(complement(meet(composition(x0, top), X)), complement(join(meet(composition(x0, top), X), composition(x0, X))))))
% 13.01/2.05 = { by lemma 58 }
% 13.01/2.05 join(meet(meet(composition(x0, top), X), complement(join(meet(composition(x0, top), X), composition(x0, X)))), complement(join(complement(meet(composition(x0, top), X)), complement(join(meet(composition(x0, top), X), composition(x0, X))))))
% 13.01/2.05 = { by lemma 24 }
% 13.01/2.05 meet(composition(x0, top), X)
% 13.01/2.05
% 13.01/2.05 Lemma 77: join(composition(X, Y), composition(X, complement(join(meet(composition(x0, top), Y), composition(x0, Y))))) = composition(X, top).
% 13.01/2.05 Proof:
% 13.01/2.05 join(composition(X, Y), composition(X, complement(join(meet(composition(x0, top), Y), composition(x0, Y)))))
% 13.01/2.05 = { by lemma 76 }
% 13.01/2.05 join(composition(X, Y), composition(X, complement(meet(composition(x0, top), Y))))
% 13.01/2.05 = { by axiom 1 (converse_idempotence) R->L }
% 13.01/2.05 join(composition(X, Y), composition(X, converse(converse(complement(meet(composition(x0, top), Y))))))
% 13.01/2.05 = { by lemma 45 R->L }
% 13.01/2.05 join(composition(X, Y), composition(X, converse(meet(converse(complement(meet(composition(x0, top), Y))), converse(complement(meet(composition(x0, top), Y)))))))
% 13.01/2.05 = { by lemma 43 R->L }
% 13.01/2.05 join(composition(X, Y), composition(X, converse(complement(complement(converse(complement(meet(composition(x0, top), Y))))))))
% 13.01/2.05 = { by axiom 1 (converse_idempotence) R->L }
% 13.01/2.05 join(composition(X, Y), composition(X, converse(complement(converse(converse(complement(converse(complement(meet(composition(x0, top), Y))))))))))
% 13.01/2.05 = { by lemma 70 R->L }
% 13.01/2.05 join(composition(X, Y), composition(X, converse(complement(converse(join(meet(meet(composition(x0, top), Y), converse(complement(converse(complement(meet(composition(x0, top), Y)))))), meet(converse(complement(converse(complement(meet(composition(x0, top), Y))))), complement(meet(composition(x0, top), Y)))))))))
% 13.01/2.05 = { by lemma 46 R->L }
% 13.01/2.05 join(composition(X, Y), composition(X, converse(complement(converse(join(meet(meet(composition(x0, top), Y), converse(complement(converse(complement(meet(composition(x0, top), Y)))))), meet(complement(meet(composition(x0, top), Y)), converse(complement(converse(complement(meet(composition(x0, top), Y))))))))))))
% 13.01/2.05 = { by lemma 44 R->L }
% 13.01/2.05 join(composition(X, Y), composition(X, converse(complement(converse(join(meet(meet(composition(x0, top), Y), converse(complement(converse(complement(meet(composition(x0, top), Y)))))), meet(complement(meet(composition(x0, top), Y)), converse(complement(converse(complement(complement(complement(meet(composition(x0, top), Y))))))))))))))
% 13.01/2.05 = { by lemma 68 R->L }
% 13.01/2.05 join(composition(X, Y), composition(X, converse(complement(converse(join(meet(meet(composition(x0, top), Y), converse(complement(converse(complement(meet(composition(x0, top), Y)))))), meet(complement(meet(composition(x0, top), Y)), join(complement(complement(meet(composition(x0, top), Y))), converse(complement(converse(complement(complement(complement(meet(composition(x0, top), Y)))))))))))))))
% 13.01/2.05 = { by axiom 1 (converse_idempotence) R->L }
% 13.01/2.05 join(composition(X, Y), composition(X, converse(complement(converse(join(meet(meet(composition(x0, top), Y), converse(complement(converse(complement(meet(composition(x0, top), Y)))))), meet(complement(meet(composition(x0, top), Y)), join(complement(complement(meet(composition(x0, top), Y))), converse(complement(converse(complement(converse(converse(complement(complement(meet(composition(x0, top), Y)))))))))))))))))
% 13.01/2.05 = { by lemma 37 R->L }
% 13.01/2.05 join(composition(X, Y), composition(X, converse(complement(converse(join(meet(meet(composition(x0, top), Y), converse(complement(converse(complement(meet(composition(x0, top), Y)))))), meet(complement(meet(composition(x0, top), Y)), converse(join(converse(complement(complement(meet(composition(x0, top), Y)))), complement(converse(complement(converse(converse(complement(complement(meet(composition(x0, top), Y)))))))))))))))))
% 13.01/2.05 = { by lemma 49 R->L }
% 13.01/2.05 join(composition(X, Y), composition(X, converse(complement(converse(join(meet(meet(composition(x0, top), Y), converse(complement(converse(complement(meet(composition(x0, top), Y)))))), meet(complement(meet(composition(x0, top), Y)), converse(meet(top, join(converse(complement(complement(meet(composition(x0, top), Y)))), complement(converse(complement(converse(converse(complement(complement(meet(composition(x0, top), Y))))))))))))))))))
% 13.01/2.05 = { by lemma 41 R->L }
% 13.01/2.05 join(composition(X, Y), composition(X, converse(complement(converse(join(meet(meet(composition(x0, top), Y), converse(complement(converse(complement(meet(composition(x0, top), Y)))))), meet(complement(meet(composition(x0, top), Y)), converse(meet(converse(top), join(converse(complement(complement(meet(composition(x0, top), Y)))), complement(converse(complement(converse(converse(complement(complement(meet(composition(x0, top), Y))))))))))))))))))
% 13.01/2.05 = { by lemma 38 R->L }
% 13.01/2.05 join(composition(X, Y), composition(X, converse(complement(converse(join(meet(meet(composition(x0, top), Y), converse(complement(converse(complement(meet(composition(x0, top), Y)))))), meet(complement(meet(composition(x0, top), Y)), converse(meet(join(converse(complement(complement(meet(composition(x0, top), Y)))), converse(complement(converse(converse(complement(complement(meet(composition(x0, top), Y)))))))), join(converse(complement(complement(meet(composition(x0, top), Y)))), complement(converse(complement(converse(converse(complement(complement(meet(composition(x0, top), Y))))))))))))))))))
% 13.01/2.05 = { by lemma 71 }
% 13.01/2.05 join(composition(X, Y), composition(X, converse(complement(converse(join(meet(meet(composition(x0, top), Y), converse(complement(converse(complement(meet(composition(x0, top), Y)))))), meet(complement(meet(composition(x0, top), Y)), converse(converse(complement(complement(meet(composition(x0, top), Y))))))))))))
% 13.01/2.05 = { by axiom 1 (converse_idempotence) }
% 13.01/2.05 join(composition(X, Y), composition(X, converse(complement(converse(join(meet(meet(composition(x0, top), Y), converse(complement(converse(complement(meet(composition(x0, top), Y)))))), meet(complement(meet(composition(x0, top), Y)), complement(complement(meet(composition(x0, top), Y))))))))))
% 13.01/2.05 = { by axiom 5 (def_zero) R->L }
% 13.01/2.05 join(composition(X, Y), composition(X, converse(complement(converse(join(meet(meet(composition(x0, top), Y), converse(complement(converse(complement(meet(composition(x0, top), Y)))))), zero))))))
% 13.01/2.05 = { by lemma 33 }
% 13.01/2.05 join(composition(X, Y), composition(X, converse(complement(converse(meet(meet(composition(x0, top), Y), converse(complement(converse(complement(meet(composition(x0, top), Y)))))))))))
% 13.01/2.05 = { by axiom 11 (maddux4_definiton_of_meet) }
% 13.01/2.05 join(composition(X, Y), composition(X, converse(complement(converse(complement(join(complement(meet(composition(x0, top), Y)), complement(converse(complement(converse(complement(meet(composition(x0, top), Y)))))))))))))
% 13.01/2.05 = { by lemma 34 R->L }
% 13.01/2.05 join(composition(X, Y), composition(X, converse(complement(converse(join(zero, complement(join(complement(meet(composition(x0, top), Y)), complement(converse(complement(converse(complement(meet(composition(x0, top), Y))))))))))))))
% 13.01/2.06 = { by lemma 15 R->L }
% 13.01/2.06 join(composition(X, Y), composition(X, converse(complement(converse(join(complement(top), complement(join(complement(meet(composition(x0, top), Y)), complement(converse(complement(converse(complement(meet(composition(x0, top), Y))))))))))))))
% 13.01/2.06 = { by lemma 41 R->L }
% 13.01/2.06 join(composition(X, Y), composition(X, converse(complement(converse(join(complement(converse(top)), complement(join(complement(meet(composition(x0, top), Y)), complement(converse(complement(converse(complement(meet(composition(x0, top), Y))))))))))))))
% 13.01/2.06 = { by lemma 38 R->L }
% 13.01/2.06 join(composition(X, Y), composition(X, converse(complement(converse(join(complement(join(complement(meet(composition(x0, top), Y)), converse(complement(converse(complement(meet(composition(x0, top), Y))))))), complement(join(complement(meet(composition(x0, top), Y)), complement(converse(complement(converse(complement(meet(composition(x0, top), Y))))))))))))))
% 13.01/2.06 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.06 join(composition(X, Y), composition(X, converse(complement(converse(join(complement(join(converse(complement(converse(complement(meet(composition(x0, top), Y))))), complement(meet(composition(x0, top), Y)))), complement(join(complement(meet(composition(x0, top), Y)), complement(converse(complement(converse(complement(meet(composition(x0, top), Y))))))))))))))
% 13.01/2.06 = { by lemma 58 }
% 13.01/2.06 join(composition(X, Y), composition(X, converse(complement(converse(join(meet(meet(composition(x0, top), Y), complement(converse(complement(converse(complement(meet(composition(x0, top), Y))))))), complement(join(complement(meet(composition(x0, top), Y)), complement(converse(complement(converse(complement(meet(composition(x0, top), Y))))))))))))))
% 13.01/2.06 = { by lemma 24 }
% 13.01/2.06 join(composition(X, Y), composition(X, converse(complement(converse(meet(composition(x0, top), Y))))))
% 13.01/2.06 = { by axiom 1 (converse_idempotence) R->L }
% 13.01/2.06 converse(converse(join(composition(X, Y), composition(X, converse(complement(converse(meet(composition(x0, top), Y))))))))
% 13.01/2.06 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.06 converse(converse(join(composition(X, converse(complement(converse(meet(composition(x0, top), Y))))), composition(X, Y))))
% 13.01/2.06 = { by axiom 7 (converse_additivity) }
% 13.01/2.06 converse(join(converse(composition(X, converse(complement(converse(meet(composition(x0, top), Y)))))), converse(composition(X, Y))))
% 13.01/2.06 = { by lemma 64 }
% 13.01/2.06 converse(join(composition(complement(converse(meet(composition(x0, top), Y))), converse(X)), converse(composition(X, Y))))
% 13.01/2.06 = { by axiom 2 (maddux1_join_commutativity) }
% 13.01/2.06 converse(join(converse(composition(X, Y)), composition(complement(converse(meet(composition(x0, top), Y))), converse(X))))
% 13.01/2.06 = { by axiom 9 (converse_multiplicativity) }
% 13.01/2.06 converse(join(composition(converse(Y), converse(X)), composition(complement(converse(meet(composition(x0, top), Y))), converse(X))))
% 13.01/2.06 = { by axiom 12 (composition_distributivity) R->L }
% 13.01/2.06 converse(composition(join(converse(Y), complement(converse(meet(composition(x0, top), Y)))), converse(X)))
% 13.01/2.06 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.06 converse(composition(join(complement(converse(meet(composition(x0, top), Y))), converse(Y)), converse(X)))
% 13.01/2.06 = { by lemma 36 R->L }
% 13.01/2.06 converse(composition(converse(join(Y, converse(complement(converse(meet(composition(x0, top), Y)))))), converse(X)))
% 13.01/2.06 = { by axiom 9 (converse_multiplicativity) R->L }
% 13.01/2.06 converse(converse(composition(X, join(Y, converse(complement(converse(meet(composition(x0, top), Y))))))))
% 13.01/2.06 = { by axiom 1 (converse_idempotence) }
% 13.01/2.06 composition(X, join(Y, converse(complement(converse(meet(composition(x0, top), Y))))))
% 13.01/2.06 = { by lemma 37 R->L }
% 13.01/2.06 composition(X, converse(join(converse(Y), complement(converse(meet(composition(x0, top), Y))))))
% 13.01/2.06 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.06 composition(X, converse(join(complement(converse(meet(composition(x0, top), Y))), converse(Y))))
% 13.01/2.06 = { by lemma 75 R->L }
% 13.01/2.06 composition(X, converse(join(complement(converse(meet(composition(x0, top), Y))), converse(join(join(meet(composition(x0, top), Y), composition(x0, Y)), Y)))))
% 13.01/2.06 = { by lemma 69 }
% 13.01/2.06 composition(X, converse(join(complement(converse(meet(composition(x0, top), Y))), converse(join(meet(composition(x0, top), Y), Y)))))
% 13.01/2.06 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.06 composition(X, converse(join(converse(join(meet(composition(x0, top), Y), Y)), complement(converse(meet(composition(x0, top), Y))))))
% 13.01/2.06 = { by axiom 7 (converse_additivity) }
% 13.01/2.06 composition(X, converse(join(join(converse(meet(composition(x0, top), Y)), converse(Y)), complement(converse(meet(composition(x0, top), Y))))))
% 13.01/2.06 = { by axiom 8 (maddux2_join_associativity) R->L }
% 13.01/2.06 composition(X, converse(join(converse(meet(composition(x0, top), Y)), join(converse(Y), complement(converse(meet(composition(x0, top), Y)))))))
% 13.01/2.06 = { by axiom 2 (maddux1_join_commutativity) }
% 13.01/2.06 composition(X, converse(join(converse(meet(composition(x0, top), Y)), join(complement(converse(meet(composition(x0, top), Y))), converse(Y)))))
% 13.01/2.06 = { by lemma 39 }
% 13.01/2.06 composition(X, converse(top))
% 13.01/2.06 = { by lemma 41 }
% 13.01/2.06 composition(X, top)
% 13.01/2.06
% 13.01/2.06 Goal 1 (goals_1): tuple(join(meet(composition(x0, top), x1), composition(x0, x1)), join(composition(x0, x1), meet(composition(x0, top), x1))) = tuple(composition(x0, x1), meet(composition(x0, top), x1)).
% 13.01/2.06 Proof:
% 13.01/2.06 tuple(join(meet(composition(x0, top), x1), composition(x0, x1)), join(composition(x0, x1), meet(composition(x0, top), x1)))
% 13.01/2.06 = { by axiom 2 (maddux1_join_commutativity) }
% 13.01/2.06 tuple(join(meet(composition(x0, top), x1), composition(x0, x1)), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 13.01/2.06 = { by lemma 74 R->L }
% 13.01/2.06 tuple(meet(join(meet(composition(x0, top), x1), composition(x0, x1)), composition(x0, top)), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 13.01/2.06 = { by lemma 46 }
% 13.01/2.06 tuple(meet(composition(x0, top), join(meet(composition(x0, top), x1), composition(x0, x1))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 13.01/2.06 = { by lemma 68 R->L }
% 13.01/2.06 tuple(meet(composition(x0, top), join(complement(composition(x0, top)), join(meet(composition(x0, top), x1), composition(x0, x1)))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 13.01/2.06 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.06 tuple(meet(composition(x0, top), join(join(meet(composition(x0, top), x1), composition(x0, x1)), complement(composition(x0, top)))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 13.01/2.06 = { by axiom 8 (maddux2_join_associativity) R->L }
% 13.01/2.06 tuple(meet(composition(x0, top), join(meet(composition(x0, top), x1), join(composition(x0, x1), complement(composition(x0, top))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 13.01/2.06 = { by axiom 2 (maddux1_join_commutativity) }
% 13.01/2.06 tuple(meet(composition(x0, top), join(meet(composition(x0, top), x1), join(complement(composition(x0, top)), composition(x0, x1)))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 13.01/2.06 = { by lemma 76 R->L }
% 13.01/2.06 tuple(meet(composition(x0, top), join(join(meet(composition(x0, top), x1), composition(x0, x1)), join(complement(composition(x0, top)), composition(x0, x1)))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 13.01/2.06 = { by axiom 8 (maddux2_join_associativity) }
% 13.01/2.06 tuple(meet(composition(x0, top), join(join(join(meet(composition(x0, top), x1), composition(x0, x1)), complement(composition(x0, top))), composition(x0, x1))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 13.01/2.06 = { by lemma 59 R->L }
% 13.01/2.06 tuple(meet(composition(x0, top), join(complement(meet(composition(x0, top), complement(join(meet(composition(x0, top), x1), composition(x0, x1))))), composition(x0, x1))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 13.01/2.06 = { by lemma 77 R->L }
% 13.01/2.06 tuple(meet(join(composition(x0, x1), composition(x0, complement(join(meet(composition(x0, top), x1), composition(x0, x1))))), join(complement(meet(composition(x0, top), complement(join(meet(composition(x0, top), x1), composition(x0, x1))))), composition(x0, x1))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 13.01/2.06 = { by lemma 61 R->L }
% 13.01/2.06 tuple(meet(join(composition(x0, x1), meet(composition(x0, complement(join(meet(composition(x0, top), x1), composition(x0, x1)))), join(complement(join(meet(composition(x0, top), x1), composition(x0, x1))), composition(x0, complement(join(meet(composition(x0, top), x1), composition(x0, x1))))))), join(complement(meet(composition(x0, top), complement(join(meet(composition(x0, top), x1), composition(x0, x1))))), composition(x0, x1))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 13.01/2.06 = { by lemma 50 R->L }
% 13.01/2.06 tuple(meet(join(composition(x0, x1), meet(composition(x0, complement(join(meet(composition(x0, top), x1), composition(x0, x1)))), composition(join(join(x0, one), x0), complement(join(meet(composition(x0, top), x1), composition(x0, x1)))))), join(complement(meet(composition(x0, top), complement(join(meet(composition(x0, top), x1), composition(x0, x1))))), composition(x0, x1))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 13.01/2.06 = { by lemma 31 }
% 13.01/2.06 tuple(meet(join(composition(x0, x1), meet(composition(x0, complement(join(meet(composition(x0, top), x1), composition(x0, x1)))), composition(join(x0, one), complement(join(meet(composition(x0, top), x1), composition(x0, x1)))))), join(complement(meet(composition(x0, top), complement(join(meet(composition(x0, top), x1), composition(x0, x1))))), composition(x0, x1))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 13.01/2.06 = { by lemma 19 }
% 13.01/2.06 tuple(meet(join(composition(x0, x1), meet(composition(x0, complement(join(meet(composition(x0, top), x1), composition(x0, x1)))), complement(join(meet(composition(x0, top), x1), composition(x0, x1))))), join(complement(meet(composition(x0, top), complement(join(meet(composition(x0, top), x1), composition(x0, x1))))), composition(x0, x1))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 13.01/2.06 = { by lemma 67 R->L }
% 13.01/2.06 tuple(meet(join(composition(x0, x1), meet(complement(join(meet(composition(x0, top), x1), composition(x0, x1))), join(join(meet(composition(x0, top), x1), composition(x0, x1)), composition(x0, complement(join(meet(composition(x0, top), x1), composition(x0, x1))))))), join(complement(meet(composition(x0, top), complement(join(meet(composition(x0, top), x1), composition(x0, x1))))), composition(x0, x1))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 13.01/2.06 = { by axiom 2 (maddux1_join_commutativity) }
% 13.01/2.06 tuple(meet(join(composition(x0, x1), meet(complement(join(meet(composition(x0, top), x1), composition(x0, x1))), join(composition(x0, complement(join(meet(composition(x0, top), x1), composition(x0, x1)))), join(meet(composition(x0, top), x1), composition(x0, x1))))), join(complement(meet(composition(x0, top), complement(join(meet(composition(x0, top), x1), composition(x0, x1))))), composition(x0, x1))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 13.01/2.06 = { by lemma 76 }
% 13.01/2.06 tuple(meet(join(composition(x0, x1), meet(complement(meet(composition(x0, top), x1)), join(composition(x0, complement(join(meet(composition(x0, top), x1), composition(x0, x1)))), join(meet(composition(x0, top), x1), composition(x0, x1))))), join(complement(meet(composition(x0, top), complement(join(meet(composition(x0, top), x1), composition(x0, x1))))), composition(x0, x1))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 13.01/2.06 = { by lemma 53 }
% 13.01/2.06 tuple(meet(join(composition(x0, x1), meet(complement(meet(composition(x0, top), x1)), join(meet(composition(x0, top), x1), join(composition(x0, complement(join(meet(composition(x0, top), x1), composition(x0, x1)))), composition(x0, x1))))), join(complement(meet(composition(x0, top), complement(join(meet(composition(x0, top), x1), composition(x0, x1))))), composition(x0, x1))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 13.01/2.06 = { by lemma 67 }
% 13.01/2.06 tuple(meet(join(composition(x0, x1), meet(join(composition(x0, complement(join(meet(composition(x0, top), x1), composition(x0, x1)))), composition(x0, x1)), complement(meet(composition(x0, top), x1)))), join(complement(meet(composition(x0, top), complement(join(meet(composition(x0, top), x1), composition(x0, x1))))), composition(x0, x1))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 13.01/2.06 = { by lemma 46 R->L }
% 13.01/2.06 tuple(meet(join(composition(x0, x1), meet(complement(meet(composition(x0, top), x1)), join(composition(x0, complement(join(meet(composition(x0, top), x1), composition(x0, x1)))), composition(x0, x1)))), join(complement(meet(composition(x0, top), complement(join(meet(composition(x0, top), x1), composition(x0, x1))))), composition(x0, x1))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 13.01/2.06 = { by lemma 76 R->L }
% 13.01/2.06 tuple(meet(join(composition(x0, x1), meet(complement(join(meet(composition(x0, top), x1), composition(x0, x1))), join(composition(x0, complement(join(meet(composition(x0, top), x1), composition(x0, x1)))), composition(x0, x1)))), join(complement(meet(composition(x0, top), complement(join(meet(composition(x0, top), x1), composition(x0, x1))))), composition(x0, x1))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 13.01/2.06 = { by axiom 2 (maddux1_join_commutativity) }
% 13.01/2.06 tuple(meet(join(composition(x0, x1), meet(complement(join(meet(composition(x0, top), x1), composition(x0, x1))), join(composition(x0, x1), composition(x0, complement(join(meet(composition(x0, top), x1), composition(x0, x1))))))), join(complement(meet(composition(x0, top), complement(join(meet(composition(x0, top), x1), composition(x0, x1))))), composition(x0, x1))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 13.01/2.06 = { by lemma 77 }
% 13.01/2.06 tuple(meet(join(composition(x0, x1), meet(complement(join(meet(composition(x0, top), x1), composition(x0, x1))), composition(x0, top))), join(complement(meet(composition(x0, top), complement(join(meet(composition(x0, top), x1), composition(x0, x1))))), composition(x0, x1))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 13.01/2.06 = { by lemma 46 R->L }
% 13.01/2.06 tuple(meet(join(composition(x0, x1), meet(composition(x0, top), complement(join(meet(composition(x0, top), x1), composition(x0, x1))))), join(complement(meet(composition(x0, top), complement(join(meet(composition(x0, top), x1), composition(x0, x1))))), composition(x0, x1))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 13.01/2.06 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 13.01/2.06 tuple(meet(join(composition(x0, x1), meet(composition(x0, top), complement(join(meet(composition(x0, top), x1), composition(x0, x1))))), join(composition(x0, x1), complement(meet(composition(x0, top), complement(join(meet(composition(x0, top), x1), composition(x0, x1))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 13.01/2.06 = { by lemma 71 }
% 13.01/2.06 tuple(composition(x0, x1), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 13.01/2.06 = { by lemma 76 }
% 13.01/2.06 tuple(composition(x0, x1), meet(composition(x0, top), x1))
% 13.01/2.06 % SZS output end Proof
% 13.01/2.06
% 13.01/2.06 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------