TSTP Solution File: REL030+2 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL030+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 : n004.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 13:44:14 EDT 2023
% Result : Theorem 41.12s 5.62s
% Output : Proof 42.81s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL030+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
% 0.12/0.34 % Computer : n004.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Fri Aug 25 20:49:08 EDT 2023
% 0.12/0.34 % CPUTime :
% 41.12/5.62 Command-line arguments: --no-flatten-goal
% 41.12/5.62
% 41.12/5.62 % SZS status Theorem
% 41.12/5.62
% 42.11/5.79 % SZS output start Proof
% 42.11/5.79 Take the following subset of the input axioms:
% 42.11/5.79 fof(composition_associativity, axiom, ![X0, X1, X2]: composition(X0, composition(X1, X2))=composition(composition(X0, X1), X2)).
% 42.11/5.79 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))).
% 42.11/5.79 fof(composition_identity, axiom, ![X0_2]: composition(X0_2, one)=X0_2).
% 42.11/5.79 fof(converse_additivity, axiom, ![X0_2, X1_2]: converse(join(X0_2, X1_2))=join(converse(X0_2), converse(X1_2))).
% 42.11/5.79 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)).
% 42.11/5.79 fof(converse_idempotence, axiom, ![X0_2]: converse(converse(X0_2))=X0_2).
% 42.11/5.79 fof(converse_multiplicativity, axiom, ![X0_2, X1_2]: converse(composition(X0_2, X1_2))=composition(converse(X1_2), converse(X0_2))).
% 42.11/5.79 fof(def_top, axiom, ![X0_2]: top=join(X0_2, complement(X0_2))).
% 42.11/5.79 fof(def_zero, axiom, ![X0_2]: zero=meet(X0_2, complement(X0_2))).
% 42.11/5.79 fof(goals, conjecture, ![X0_2, X1_2, X2_2]: (join(X0_2, one)=one => (join(meet(composition(X0_2, X1_2), complement(X2_2)), meet(composition(X0_2, X1_2), complement(composition(X0_2, X2_2))))=meet(composition(X0_2, X1_2), complement(composition(X0_2, X2_2))) & join(meet(composition(X0_2, X1_2), complement(composition(X0_2, X2_2))), meet(composition(X0_2, X1_2), complement(X2_2)))=meet(composition(X0_2, X1_2), complement(X2_2))))).
% 42.11/5.79 fof(maddux1_join_commutativity, axiom, ![X0_2, X1_2]: join(X0_2, X1_2)=join(X1_2, X0_2)).
% 42.11/5.79 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)).
% 42.11/5.79 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)))).
% 42.11/5.79 fof(maddux4_definiton_of_meet, axiom, ![X0_2, X1_2]: meet(X0_2, X1_2)=complement(join(complement(X0_2), complement(X1_2)))).
% 42.11/5.79
% 42.11/5.79 Now clausify the problem and encode Horn clauses using encoding 3 of
% 42.11/5.79 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 42.11/5.79 We repeatedly replace C & s=t => u=v by the two clauses:
% 42.11/5.79 fresh(y, y, x1...xn) = u
% 42.11/5.79 C => fresh(s, t, x1...xn) = v
% 42.11/5.79 where fresh is a fresh function symbol and x1..xn are the free
% 42.11/5.79 variables of u and v.
% 42.11/5.79 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 42.11/5.79 input problem has no model of domain size 1).
% 42.11/5.79
% 42.11/5.79 The encoding turns the above axioms into the following unit equations and goals:
% 42.11/5.79
% 42.11/5.79 Axiom 1 (converse_idempotence): converse(converse(X)) = X.
% 42.11/5.79 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 42.11/5.79 Axiom 3 (goals): join(x0, one) = one.
% 42.11/5.79 Axiom 4 (composition_identity): composition(X, one) = X.
% 42.11/5.79 Axiom 5 (def_zero): zero = meet(X, complement(X)).
% 42.11/5.79 Axiom 6 (def_top): top = join(X, complement(X)).
% 42.11/5.79 Axiom 7 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 42.11/5.79 Axiom 8 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 42.11/5.79 Axiom 9 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 42.11/5.79 Axiom 10 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 42.11/5.79 Axiom 11 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 42.11/5.79 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 42.11/5.79 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 42.11/5.79 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 42.11/5.79
% 42.11/5.80 Lemma 15: complement(top) = zero.
% 42.11/5.80 Proof:
% 42.11/5.80 complement(top)
% 42.11/5.80 = { by axiom 6 (def_top) }
% 42.11/5.80 complement(join(complement(X), complement(complement(X))))
% 42.11/5.80 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 42.11/5.80 meet(X, complement(X))
% 42.11/5.80 = { by axiom 5 (def_zero) R->L }
% 42.11/5.80 zero
% 42.11/5.80
% 42.11/5.80 Lemma 16: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 42.11/5.80 Proof:
% 42.11/5.80 converse(composition(converse(X), Y))
% 42.11/5.80 = { by axiom 9 (converse_multiplicativity) }
% 42.11/5.80 composition(converse(Y), converse(converse(X)))
% 42.11/5.80 = { by axiom 1 (converse_idempotence) }
% 42.11/5.80 composition(converse(Y), X)
% 42.11/5.80
% 42.11/5.80 Lemma 17: composition(converse(one), X) = X.
% 42.11/5.80 Proof:
% 42.11/5.80 composition(converse(one), X)
% 42.11/5.80 = { by lemma 16 R->L }
% 42.11/5.80 converse(composition(converse(X), one))
% 42.11/5.80 = { by axiom 4 (composition_identity) }
% 42.11/5.80 converse(converse(X))
% 42.11/5.80 = { by axiom 1 (converse_idempotence) }
% 42.11/5.80 X
% 42.11/5.80
% 42.11/5.80 Lemma 18: composition(one, X) = X.
% 42.11/5.80 Proof:
% 42.11/5.80 composition(one, X)
% 42.11/5.80 = { by lemma 17 R->L }
% 42.11/5.80 composition(converse(one), composition(one, X))
% 42.11/5.80 = { by axiom 10 (composition_associativity) }
% 42.11/5.80 composition(composition(converse(one), one), X)
% 42.11/5.80 = { by axiom 4 (composition_identity) }
% 42.11/5.80 composition(converse(one), X)
% 42.11/5.80 = { by lemma 17 }
% 42.11/5.80 X
% 42.11/5.80
% 42.11/5.80 Lemma 19: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 42.11/5.80 Proof:
% 42.11/5.80 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 42.11/5.80 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.11/5.80 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 42.11/5.80 = { by axiom 13 (converse_cancellativity) }
% 42.81/5.80 complement(X)
% 42.81/5.80
% 42.81/5.80 Lemma 20: join(complement(X), complement(X)) = complement(X).
% 42.81/5.80 Proof:
% 42.81/5.80 join(complement(X), complement(X))
% 42.81/5.80 = { by lemma 17 R->L }
% 42.81/5.80 join(complement(X), composition(converse(one), complement(X)))
% 42.81/5.80 = { by lemma 18 R->L }
% 42.81/5.80 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 42.81/5.80 = { by lemma 19 }
% 42.81/5.80 complement(X)
% 42.81/5.80
% 42.81/5.80 Lemma 21: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 42.81/5.80 Proof:
% 42.81/5.80 join(meet(X, Y), complement(join(complement(X), Y)))
% 42.81/5.80 = { by axiom 11 (maddux4_definiton_of_meet) }
% 42.81/5.80 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 42.81/5.80 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L }
% 42.81/5.80 X
% 42.81/5.80
% 42.81/5.80 Lemma 22: join(zero, meet(X, X)) = X.
% 42.81/5.80 Proof:
% 42.81/5.80 join(zero, meet(X, X))
% 42.81/5.80 = { by axiom 11 (maddux4_definiton_of_meet) }
% 42.81/5.80 join(zero, complement(join(complement(X), complement(X))))
% 42.81/5.80 = { by axiom 5 (def_zero) }
% 42.81/5.80 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 42.81/5.80 = { by lemma 21 }
% 42.81/5.80 X
% 42.81/5.80
% 42.81/5.80 Lemma 23: join(zero, join(X, meet(Y, Y))) = join(X, Y).
% 42.81/5.80 Proof:
% 42.81/5.80 join(zero, join(X, meet(Y, Y)))
% 42.81/5.80 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.80 join(zero, join(meet(Y, Y), X))
% 42.81/5.80 = { by axiom 8 (maddux2_join_associativity) }
% 42.81/5.80 join(join(zero, meet(Y, Y)), X)
% 42.81/5.80 = { by lemma 22 }
% 42.81/5.80 join(Y, X)
% 42.81/5.80 = { by axiom 2 (maddux1_join_commutativity) }
% 42.81/5.80 join(X, Y)
% 42.81/5.80
% 42.81/5.80 Lemma 24: join(X, join(Y, complement(X))) = join(Y, top).
% 42.81/5.80 Proof:
% 42.81/5.80 join(X, join(Y, complement(X)))
% 42.81/5.80 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.80 join(X, join(complement(X), Y))
% 42.81/5.80 = { by axiom 8 (maddux2_join_associativity) }
% 42.81/5.80 join(join(X, complement(X)), Y)
% 42.81/5.80 = { by axiom 6 (def_top) R->L }
% 42.81/5.80 join(top, Y)
% 42.81/5.80 = { by axiom 2 (maddux1_join_commutativity) }
% 42.81/5.80 join(Y, top)
% 42.81/5.80
% 42.81/5.80 Lemma 25: join(top, complement(X)) = top.
% 42.81/5.80 Proof:
% 42.81/5.80 join(top, complement(X))
% 42.81/5.80 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.80 join(complement(X), top)
% 42.81/5.80 = { by lemma 24 R->L }
% 42.81/5.80 join(X, join(complement(X), complement(X)))
% 42.81/5.80 = { by lemma 20 }
% 42.81/5.80 join(X, complement(X))
% 42.81/5.80 = { by axiom 6 (def_top) R->L }
% 42.81/5.80 top
% 42.81/5.80
% 42.81/5.80 Lemma 26: join(Y, top) = join(X, top).
% 42.81/5.80 Proof:
% 42.81/5.80 join(Y, top)
% 42.81/5.80 = { by lemma 25 R->L }
% 42.81/5.80 join(Y, join(top, complement(Y)))
% 42.81/5.80 = { by lemma 24 }
% 42.81/5.80 join(top, top)
% 42.81/5.80 = { by lemma 24 R->L }
% 42.81/5.80 join(X, join(top, complement(X)))
% 42.81/5.80 = { by lemma 25 }
% 42.81/5.80 join(X, top)
% 42.81/5.80
% 42.81/5.80 Lemma 27: composition(join(X, one), Y) = join(Y, composition(X, Y)).
% 42.81/5.80 Proof:
% 42.81/5.80 composition(join(X, one), Y)
% 42.81/5.80 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.80 composition(join(one, X), Y)
% 42.81/5.80 = { by axiom 12 (composition_distributivity) }
% 42.81/5.80 join(composition(one, Y), composition(X, Y))
% 42.81/5.80 = { by lemma 18 }
% 42.81/5.80 join(Y, composition(X, Y))
% 42.81/5.80
% 42.81/5.80 Lemma 28: converse(one) = one.
% 42.81/5.80 Proof:
% 42.81/5.80 converse(one)
% 42.81/5.80 = { by axiom 4 (composition_identity) R->L }
% 42.81/5.80 composition(converse(one), one)
% 42.81/5.80 = { by lemma 17 }
% 42.81/5.80 one
% 42.81/5.80
% 42.81/5.80 Lemma 29: join(X, zero) = join(X, X).
% 42.81/5.80 Proof:
% 42.81/5.80 join(X, zero)
% 42.81/5.80 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.80 join(zero, X)
% 42.81/5.80 = { by lemma 22 R->L }
% 42.81/5.80 join(zero, join(zero, meet(X, X)))
% 42.81/5.80 = { by axiom 11 (maddux4_definiton_of_meet) }
% 42.81/5.80 join(zero, join(zero, complement(join(complement(X), complement(X)))))
% 42.81/5.80 = { by lemma 20 R->L }
% 42.81/5.80 join(zero, join(zero, join(complement(join(complement(X), complement(X))), complement(join(complement(X), complement(X))))))
% 42.81/5.80 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 42.81/5.80 join(zero, join(zero, join(meet(X, X), complement(join(complement(X), complement(X))))))
% 42.81/5.80 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 42.81/5.80 join(zero, join(zero, join(meet(X, X), meet(X, X))))
% 42.81/5.80 = { by lemma 23 }
% 42.81/5.80 join(zero, join(meet(X, X), X))
% 42.81/5.80 = { by axiom 2 (maddux1_join_commutativity) }
% 42.81/5.80 join(zero, join(X, meet(X, X)))
% 42.81/5.80 = { by lemma 23 }
% 42.81/5.80 join(X, X)
% 42.81/5.80
% 42.81/5.80 Lemma 30: join(one, x0) = one.
% 42.81/5.80 Proof:
% 42.81/5.80 join(one, x0)
% 42.81/5.80 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.80 join(x0, one)
% 42.81/5.80 = { by axiom 3 (goals) }
% 42.81/5.80 one
% 42.81/5.80
% 42.81/5.80 Lemma 31: join(one, join(X, x0)) = join(X, one).
% 42.81/5.80 Proof:
% 42.81/5.80 join(one, join(X, x0))
% 42.81/5.80 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.80 join(one, join(x0, X))
% 42.81/5.80 = { by axiom 8 (maddux2_join_associativity) }
% 42.81/5.80 join(join(one, x0), X)
% 42.81/5.80 = { by lemma 30 }
% 42.81/5.80 join(one, X)
% 42.81/5.80 = { by axiom 2 (maddux1_join_commutativity) }
% 42.81/5.80 join(X, one)
% 42.81/5.80
% 42.81/5.80 Lemma 32: join(one, join(x0, X)) = join(X, one).
% 42.81/5.80 Proof:
% 42.81/5.80 join(one, join(x0, X))
% 42.81/5.80 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.80 join(one, join(X, x0))
% 42.81/5.80 = { by lemma 31 }
% 42.81/5.80 join(X, one)
% 42.81/5.80
% 42.81/5.80 Lemma 33: join(X, X) = X.
% 42.81/5.80 Proof:
% 42.81/5.80 join(X, X)
% 42.81/5.80 = { by lemma 17 R->L }
% 42.81/5.80 join(X, composition(converse(one), X))
% 42.81/5.80 = { by lemma 27 R->L }
% 42.81/5.80 composition(join(converse(one), one), X)
% 42.81/5.80 = { by lemma 28 }
% 42.81/5.80 composition(join(one, one), X)
% 42.81/5.80 = { by lemma 29 R->L }
% 42.81/5.80 composition(join(one, zero), X)
% 42.81/5.80 = { by axiom 2 (maddux1_join_commutativity) }
% 42.81/5.80 composition(join(zero, one), X)
% 42.81/5.80 = { by lemma 32 R->L }
% 42.81/5.80 composition(join(one, join(x0, zero)), X)
% 42.81/5.80 = { by lemma 29 }
% 42.81/5.80 composition(join(one, join(x0, x0)), X)
% 42.81/5.80 = { by lemma 31 }
% 42.81/5.80 composition(join(x0, one), X)
% 42.81/5.80 = { by axiom 2 (maddux1_join_commutativity) }
% 42.81/5.80 composition(join(one, x0), X)
% 42.81/5.80 = { by lemma 30 }
% 42.81/5.80 composition(one, X)
% 42.81/5.80 = { by lemma 18 }
% 42.81/5.80 X
% 42.81/5.80
% 42.81/5.80 Lemma 34: join(X, zero) = X.
% 42.81/5.80 Proof:
% 42.81/5.80 join(X, zero)
% 42.81/5.80 = { by lemma 29 }
% 42.81/5.80 join(X, X)
% 42.81/5.80 = { by lemma 33 }
% 42.81/5.80 X
% 42.81/5.80
% 42.81/5.80 Lemma 35: join(X, top) = top.
% 42.81/5.80 Proof:
% 42.81/5.80 join(X, top)
% 42.81/5.80 = { by lemma 26 }
% 42.81/5.80 join(join(zero, zero), top)
% 42.81/5.80 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.80 join(top, join(zero, zero))
% 42.81/5.80 = { by lemma 33 }
% 42.81/5.80 join(top, zero)
% 42.81/5.80 = { by lemma 34 }
% 42.81/5.80 top
% 42.81/5.80
% 42.81/5.80 Lemma 36: join(X, join(complement(X), Y)) = top.
% 42.81/5.80 Proof:
% 42.81/5.80 join(X, join(complement(X), Y))
% 42.81/5.80 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.80 join(X, join(Y, complement(X)))
% 42.81/5.80 = { by lemma 24 }
% 42.81/5.80 join(Y, top)
% 42.81/5.80 = { by lemma 26 R->L }
% 42.81/5.80 join(Z, top)
% 42.81/5.80 = { by lemma 35 }
% 42.81/5.80 top
% 42.81/5.80
% 42.81/5.80 Lemma 37: complement(zero) = top.
% 42.81/5.80 Proof:
% 42.81/5.80 complement(zero)
% 42.81/5.80 = { by lemma 20 R->L }
% 42.81/5.80 join(complement(zero), complement(zero))
% 42.81/5.80 = { by lemma 23 R->L }
% 42.81/5.80 join(zero, join(complement(zero), meet(complement(zero), complement(zero))))
% 42.81/5.80 = { by lemma 36 }
% 42.81/5.80 top
% 42.81/5.80
% 42.81/5.80 Lemma 38: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 42.81/5.80 Proof:
% 42.81/5.80 converse(join(X, converse(Y)))
% 42.81/5.80 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.80 converse(join(converse(Y), X))
% 42.81/5.80 = { by axiom 7 (converse_additivity) }
% 42.81/5.80 join(converse(converse(Y)), converse(X))
% 42.81/5.80 = { by axiom 1 (converse_idempotence) }
% 42.81/5.80 join(Y, converse(X))
% 42.81/5.80
% 42.81/5.80 Lemma 39: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 42.81/5.80 Proof:
% 42.81/5.80 converse(join(converse(X), Y))
% 42.81/5.80 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.80 converse(join(Y, converse(X)))
% 42.81/5.80 = { by lemma 38 }
% 42.81/5.80 join(X, converse(Y))
% 42.81/5.80
% 42.81/5.80 Lemma 40: join(X, converse(complement(converse(X)))) = converse(top).
% 42.81/5.80 Proof:
% 42.81/5.80 join(X, converse(complement(converse(X))))
% 42.81/5.80 = { by lemma 39 R->L }
% 42.81/5.80 converse(join(converse(X), complement(converse(X))))
% 42.81/5.80 = { by axiom 6 (def_top) R->L }
% 42.81/5.80 converse(top)
% 42.81/5.80
% 42.81/5.80 Lemma 41: join(X, converse(top)) = top.
% 42.81/5.80 Proof:
% 42.81/5.80 join(X, converse(top))
% 42.81/5.80 = { by lemma 40 R->L }
% 42.81/5.80 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 42.81/5.80 = { by lemma 36 }
% 42.81/5.80 top
% 42.81/5.80
% 42.81/5.80 Lemma 42: converse(top) = top.
% 42.81/5.80 Proof:
% 42.81/5.80 converse(top)
% 42.81/5.80 = { by lemma 35 R->L }
% 42.81/5.80 converse(join(X, top))
% 42.81/5.80 = { by axiom 7 (converse_additivity) }
% 42.81/5.80 join(converse(X), converse(top))
% 42.81/5.80 = { by lemma 41 }
% 42.81/5.80 top
% 42.81/5.80
% 42.81/5.80 Lemma 43: join(zero, X) = X.
% 42.81/5.80 Proof:
% 42.81/5.80 join(zero, X)
% 42.81/5.80 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.80 join(X, zero)
% 42.81/5.80 = { by lemma 34 }
% 42.81/5.80 X
% 42.81/5.80
% 42.81/5.80 Lemma 44: converse(zero) = zero.
% 42.81/5.80 Proof:
% 42.81/5.80 converse(zero)
% 42.81/5.80 = { by lemma 43 R->L }
% 42.81/5.80 join(zero, converse(zero))
% 42.81/5.80 = { by lemma 39 R->L }
% 42.81/5.80 converse(join(converse(zero), zero))
% 42.81/5.80 = { by lemma 29 }
% 42.81/5.80 converse(join(converse(zero), converse(zero)))
% 42.81/5.80 = { by lemma 38 }
% 42.81/5.80 join(zero, converse(converse(zero)))
% 42.81/5.80 = { by axiom 1 (converse_idempotence) }
% 42.81/5.80 join(zero, zero)
% 42.81/5.80 = { by lemma 33 }
% 42.81/5.80 zero
% 42.81/5.80
% 42.81/5.80 Lemma 45: meet(Y, X) = meet(X, Y).
% 42.81/5.80 Proof:
% 42.81/5.80 meet(Y, X)
% 42.81/5.80 = { by axiom 11 (maddux4_definiton_of_meet) }
% 42.81/5.80 complement(join(complement(Y), complement(X)))
% 42.81/5.80 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.80 complement(join(complement(X), complement(Y)))
% 42.81/5.80 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 42.81/5.80 meet(X, Y)
% 42.81/5.80
% 42.81/5.80 Lemma 46: complement(complement(X)) = meet(X, X).
% 42.81/5.80 Proof:
% 42.81/5.80 complement(complement(X))
% 42.81/5.80 = { by lemma 20 R->L }
% 42.81/5.80 complement(join(complement(X), complement(X)))
% 42.81/5.80 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 42.81/5.80 meet(X, X)
% 42.81/5.80
% 42.81/5.80 Lemma 47: complement(complement(X)) = X.
% 42.81/5.80 Proof:
% 42.81/5.80 complement(complement(X))
% 42.81/5.80 = { by lemma 43 R->L }
% 42.81/5.80 join(zero, complement(complement(X)))
% 42.81/5.80 = { by lemma 46 }
% 42.81/5.80 join(zero, meet(X, X))
% 42.81/5.80 = { by lemma 22 }
% 42.81/5.80 X
% 42.81/5.80
% 42.81/5.80 Lemma 48: complement(join(zero, complement(X))) = meet(X, top).
% 42.81/5.80 Proof:
% 42.81/5.80 complement(join(zero, complement(X)))
% 42.81/5.80 = { by lemma 15 R->L }
% 42.81/5.80 complement(join(complement(top), complement(X)))
% 42.81/5.80 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 42.81/5.80 meet(top, X)
% 42.81/5.80 = { by lemma 45 R->L }
% 42.81/5.80 meet(X, top)
% 42.81/5.80
% 42.81/5.80 Lemma 49: meet(X, top) = X.
% 42.81/5.80 Proof:
% 42.81/5.80 meet(X, top)
% 42.81/5.80 = { by lemma 45 }
% 42.81/5.80 meet(top, X)
% 42.81/5.80 = { by lemma 47 R->L }
% 42.81/5.80 meet(top, complement(complement(X)))
% 42.81/5.80 = { by lemma 45 }
% 42.81/5.80 meet(complement(complement(X)), top)
% 42.81/5.80 = { by lemma 48 R->L }
% 42.81/5.80 complement(join(zero, complement(complement(complement(X)))))
% 42.81/5.80 = { by lemma 46 }
% 42.81/5.80 complement(join(zero, meet(complement(X), complement(X))))
% 42.81/5.80 = { by lemma 22 }
% 42.81/5.80 complement(complement(X))
% 42.81/5.80 = { by lemma 47 }
% 42.81/5.80 X
% 42.81/5.80
% 42.81/5.80 Lemma 50: meet(top, X) = X.
% 42.81/5.80 Proof:
% 42.81/5.80 meet(top, X)
% 42.81/5.80 = { by lemma 45 }
% 42.81/5.80 meet(X, top)
% 42.81/5.80 = { by lemma 49 }
% 42.81/5.80 X
% 42.81/5.80
% 42.81/5.80 Lemma 51: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
% 42.81/5.80 Proof:
% 42.81/5.80 complement(join(complement(X), meet(Y, Z)))
% 42.81/5.80 = { by lemma 45 }
% 42.81/5.80 complement(join(complement(X), meet(Z, Y)))
% 42.81/5.80 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.80 complement(join(meet(Z, Y), complement(X)))
% 42.81/5.80 = { by axiom 11 (maddux4_definiton_of_meet) }
% 42.81/5.80 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 42.81/5.80 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 42.81/5.80 meet(join(complement(Z), complement(Y)), X)
% 42.81/5.80 = { by lemma 45 R->L }
% 42.81/5.80 meet(X, join(complement(Z), complement(Y)))
% 42.81/5.80 = { by axiom 2 (maddux1_join_commutativity) }
% 42.81/5.80 meet(X, join(complement(Y), complement(Z)))
% 42.81/5.80
% 42.81/5.80 Lemma 52: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 42.81/5.80 Proof:
% 42.81/5.80 join(complement(X), complement(Y))
% 42.81/5.80 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.80 join(complement(Y), complement(X))
% 42.81/5.80 = { by lemma 50 R->L }
% 42.81/5.80 meet(top, join(complement(Y), complement(X)))
% 42.81/5.80 = { by lemma 51 R->L }
% 42.81/5.80 complement(join(complement(top), meet(Y, X)))
% 42.81/5.80 = { by lemma 15 }
% 42.81/5.80 complement(join(zero, meet(Y, X)))
% 42.81/5.80 = { by lemma 43 }
% 42.81/5.80 complement(meet(Y, X))
% 42.81/5.80 = { by lemma 45 R->L }
% 42.81/5.80 complement(meet(X, Y))
% 42.81/5.80
% 42.81/5.80 Lemma 53: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 42.81/5.81 Proof:
% 42.81/5.81 complement(meet(X, complement(Y)))
% 42.81/5.81 = { by lemma 45 }
% 42.81/5.81 complement(meet(complement(Y), X))
% 42.81/5.81 = { by lemma 43 R->L }
% 42.81/5.81 complement(meet(join(zero, complement(Y)), X))
% 42.81/5.81 = { by lemma 52 R->L }
% 42.81/5.81 join(complement(join(zero, complement(Y))), complement(X))
% 42.81/5.81 = { by lemma 48 }
% 42.81/5.81 join(meet(Y, top), complement(X))
% 42.81/5.81 = { by lemma 49 }
% 42.81/5.81 join(Y, complement(X))
% 42.81/5.81
% 42.81/5.81 Lemma 54: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 42.81/5.81 Proof:
% 42.81/5.81 meet(Y, meet(X, Z))
% 42.81/5.81 = { by lemma 45 }
% 42.81/5.81 meet(Y, meet(Z, X))
% 42.81/5.81 = { by lemma 49 R->L }
% 42.81/5.81 meet(meet(Y, meet(Z, X)), top)
% 42.81/5.81 = { by lemma 48 R->L }
% 42.81/5.81 complement(join(zero, complement(meet(Y, meet(Z, X)))))
% 42.81/5.81 = { by lemma 45 }
% 42.81/5.81 complement(join(zero, complement(meet(Y, meet(X, Z)))))
% 42.81/5.81 = { by axiom 11 (maddux4_definiton_of_meet) }
% 42.81/5.81 complement(join(zero, complement(meet(Y, complement(join(complement(X), complement(Z)))))))
% 42.81/5.81 = { by lemma 53 }
% 42.81/5.81 complement(join(zero, join(join(complement(X), complement(Z)), complement(Y))))
% 42.81/5.81 = { by axiom 8 (maddux2_join_associativity) R->L }
% 42.81/5.81 complement(join(zero, join(complement(X), join(complement(Z), complement(Y)))))
% 42.81/5.81 = { by lemma 52 }
% 42.81/5.81 complement(join(zero, join(complement(X), complement(meet(Z, Y)))))
% 42.81/5.81 = { by lemma 52 }
% 42.81/5.81 complement(join(zero, complement(meet(X, meet(Z, Y)))))
% 42.81/5.81 = { by lemma 45 R->L }
% 42.81/5.81 complement(join(zero, complement(meet(X, meet(Y, Z)))))
% 42.81/5.81 = { by lemma 48 }
% 42.81/5.81 meet(meet(X, meet(Y, Z)), top)
% 42.81/5.81 = { by lemma 49 }
% 42.81/5.81 meet(X, meet(Y, Z))
% 42.81/5.81
% 42.81/5.81 Lemma 55: meet(meet(X, Y), Z) = meet(X, meet(Z, Y)).
% 42.81/5.81 Proof:
% 42.81/5.81 meet(meet(X, Y), Z)
% 42.81/5.81 = { by lemma 45 }
% 42.81/5.81 meet(Z, meet(X, Y))
% 42.81/5.81 = { by lemma 54 R->L }
% 42.81/5.81 meet(X, meet(Z, Y))
% 42.81/5.81
% 42.81/5.81 Lemma 56: join(meet(X, Y), meet(X, complement(Y))) = X.
% 42.81/5.81 Proof:
% 42.81/5.81 join(meet(X, Y), meet(X, complement(Y)))
% 42.81/5.81 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.81 join(meet(X, complement(Y)), meet(X, Y))
% 42.81/5.81 = { by axiom 11 (maddux4_definiton_of_meet) }
% 42.81/5.81 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 42.81/5.81 = { by lemma 21 }
% 42.81/5.81 X
% 42.81/5.81
% 42.81/5.81 Lemma 57: join(meet(X, Y), meet(complement(Y), X)) = X.
% 42.81/5.81 Proof:
% 42.81/5.81 join(meet(X, Y), meet(complement(Y), X))
% 42.81/5.81 = { by lemma 45 }
% 42.81/5.81 join(meet(X, Y), meet(X, complement(Y)))
% 42.81/5.81 = { by lemma 56 }
% 42.81/5.81 X
% 42.81/5.81
% 42.81/5.81 Lemma 58: meet(X, meet(X, X)) = X.
% 42.81/5.81 Proof:
% 42.81/5.81 meet(X, meet(X, X))
% 42.81/5.81 = { by lemma 46 R->L }
% 42.81/5.81 meet(X, complement(complement(X)))
% 42.81/5.81 = { by lemma 34 R->L }
% 42.81/5.81 join(meet(X, complement(complement(X))), zero)
% 42.81/5.81 = { by lemma 15 R->L }
% 42.81/5.81 join(meet(X, complement(complement(X))), complement(top))
% 42.81/5.81 = { by axiom 6 (def_top) }
% 42.81/5.81 join(meet(X, complement(complement(X))), complement(join(complement(X), complement(complement(X)))))
% 42.81/5.81 = { by lemma 21 }
% 42.81/5.81 X
% 42.81/5.81
% 42.81/5.81 Lemma 59: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 42.81/5.81 Proof:
% 42.81/5.81 complement(join(X, complement(Y)))
% 42.81/5.81 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.81 complement(join(complement(Y), X))
% 42.81/5.81 = { by lemma 58 R->L }
% 42.81/5.81 complement(join(complement(Y), meet(X, meet(X, X))))
% 42.81/5.81 = { by lemma 51 }
% 42.81/5.81 meet(Y, join(complement(X), complement(meet(X, X))))
% 42.81/5.81 = { by lemma 52 }
% 42.81/5.81 meet(Y, complement(meet(X, meet(X, X))))
% 42.81/5.81 = { by lemma 58 }
% 42.81/5.81 meet(Y, complement(X))
% 42.81/5.81
% 42.81/5.81 Lemma 60: meet(X, X) = X.
% 42.81/5.81 Proof:
% 42.81/5.81 meet(X, X)
% 42.81/5.81 = { by lemma 46 R->L }
% 42.81/5.81 complement(complement(X))
% 42.81/5.81 = { by lemma 47 }
% 42.81/5.81 X
% 42.81/5.81
% 42.81/5.81 Lemma 61: meet(X, join(X, complement(Y))) = X.
% 42.81/5.81 Proof:
% 42.81/5.81 meet(X, join(X, complement(Y)))
% 42.81/5.81 = { by lemma 53 R->L }
% 42.81/5.81 meet(X, complement(meet(Y, complement(X))))
% 42.81/5.81 = { by lemma 52 R->L }
% 42.81/5.81 meet(X, join(complement(Y), complement(complement(X))))
% 42.81/5.81 = { by lemma 51 R->L }
% 42.81/5.81 complement(join(complement(X), meet(Y, complement(X))))
% 42.81/5.81 = { by lemma 43 R->L }
% 42.81/5.81 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 42.81/5.81 = { by lemma 15 R->L }
% 42.81/5.81 join(complement(top), complement(join(complement(X), meet(Y, complement(X)))))
% 42.81/5.81 = { by lemma 35 R->L }
% 42.81/5.81 join(complement(join(complement(Y), top)), complement(join(complement(X), meet(Y, complement(X)))))
% 42.81/5.81 = { by lemma 24 R->L }
% 42.81/5.81 join(complement(join(complement(X), join(complement(Y), complement(complement(X))))), complement(join(complement(X), meet(Y, complement(X)))))
% 42.81/5.81 = { by lemma 52 }
% 42.81/5.81 join(complement(join(complement(X), complement(meet(Y, complement(X))))), complement(join(complement(X), meet(Y, complement(X)))))
% 42.81/5.81 = { by lemma 45 R->L }
% 42.81/5.81 join(complement(join(complement(X), complement(meet(complement(X), Y)))), complement(join(complement(X), meet(Y, complement(X)))))
% 42.81/5.81 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 42.81/5.81 join(meet(X, meet(complement(X), Y)), complement(join(complement(X), meet(Y, complement(X)))))
% 42.81/5.81 = { by lemma 45 R->L }
% 42.81/5.81 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 42.81/5.81 = { by lemma 21 }
% 42.81/5.81 X
% 42.81/5.81
% 42.81/5.81 Lemma 62: meet(X, join(Y, X)) = X.
% 42.81/5.81 Proof:
% 42.81/5.81 meet(X, join(Y, X))
% 42.81/5.81 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.81 meet(X, join(X, Y))
% 42.81/5.81 = { by lemma 60 R->L }
% 42.81/5.81 meet(X, join(X, meet(Y, Y)))
% 42.81/5.81 = { by lemma 46 R->L }
% 42.81/5.81 meet(X, join(X, complement(complement(Y))))
% 42.81/5.81 = { by lemma 61 }
% 42.81/5.81 X
% 42.81/5.81
% 42.81/5.81 Lemma 63: meet(X, join(Y, complement(X))) = meet(X, Y).
% 42.81/5.81 Proof:
% 42.81/5.81 meet(X, join(Y, complement(X)))
% 42.81/5.81 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.81 meet(X, join(complement(X), Y))
% 42.81/5.81 = { by lemma 43 R->L }
% 42.81/5.81 meet(X, join(zero, join(complement(X), Y)))
% 42.81/5.81 = { by axiom 8 (maddux2_join_associativity) }
% 42.81/5.81 meet(X, join(join(zero, complement(X)), Y))
% 42.81/5.81 = { by lemma 49 R->L }
% 42.81/5.81 meet(X, meet(join(join(zero, complement(X)), Y), top))
% 42.81/5.81 = { by lemma 55 R->L }
% 42.81/5.81 meet(meet(X, top), join(join(zero, complement(X)), Y))
% 42.81/5.81 = { by lemma 48 R->L }
% 42.81/5.81 meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), Y))
% 42.81/5.81 = { by lemma 47 R->L }
% 42.81/5.81 meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), complement(complement(Y))))
% 42.81/5.81 = { by lemma 57 R->L }
% 42.81/5.81 join(meet(meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), complement(complement(Y)))), Y), meet(complement(Y), meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), complement(complement(Y))))))
% 42.81/5.81 = { by lemma 45 }
% 42.81/5.81 join(meet(meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), complement(complement(Y)))), Y), meet(complement(Y), meet(join(join(zero, complement(X)), complement(complement(Y))), complement(join(zero, complement(X))))))
% 42.81/5.81 = { by lemma 54 }
% 42.81/5.81 join(meet(meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), complement(complement(Y)))), Y), meet(join(join(zero, complement(X)), complement(complement(Y))), meet(complement(Y), complement(join(zero, complement(X))))))
% 42.81/5.81 = { by lemma 59 R->L }
% 42.81/5.81 join(meet(meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), complement(complement(Y)))), Y), meet(join(join(zero, complement(X)), complement(complement(Y))), complement(join(join(zero, complement(X)), complement(complement(Y))))))
% 42.81/5.81 = { by axiom 5 (def_zero) R->L }
% 42.81/5.81 join(meet(meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), complement(complement(Y)))), Y), zero)
% 42.81/5.81 = { by lemma 34 }
% 42.81/5.81 meet(meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), complement(complement(Y)))), Y)
% 42.81/5.81 = { by lemma 55 }
% 42.81/5.81 meet(complement(join(zero, complement(X))), meet(Y, join(join(zero, complement(X)), complement(complement(Y)))))
% 42.81/5.81 = { by lemma 47 }
% 42.81/5.81 meet(complement(join(zero, complement(X))), meet(Y, join(join(zero, complement(X)), Y)))
% 42.81/5.81 = { by lemma 62 }
% 42.81/5.81 meet(complement(join(zero, complement(X))), Y)
% 42.81/5.81 = { by lemma 45 R->L }
% 42.81/5.81 meet(Y, complement(join(zero, complement(X))))
% 42.81/5.81 = { by lemma 48 }
% 42.81/5.81 meet(Y, meet(X, top))
% 42.81/5.81 = { by lemma 49 }
% 42.81/5.81 meet(Y, X)
% 42.81/5.81 = { by lemma 45 R->L }
% 42.81/5.81 meet(X, Y)
% 42.81/5.81
% 42.81/5.81 Lemma 64: meet(X, join(complement(X), Y)) = meet(X, Y).
% 42.81/5.81 Proof:
% 42.81/5.81 meet(X, join(complement(X), Y))
% 42.81/5.81 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.81 meet(X, join(Y, complement(X)))
% 42.81/5.81 = { by lemma 63 }
% 42.81/5.81 meet(X, Y)
% 42.81/5.81
% 42.81/5.81 Lemma 65: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 42.81/5.81 Proof:
% 42.81/5.81 converse(composition(X, converse(Y)))
% 42.81/5.81 = { by axiom 9 (converse_multiplicativity) }
% 42.81/5.81 composition(converse(converse(Y)), converse(X))
% 42.81/5.81 = { by axiom 1 (converse_idempotence) }
% 42.81/5.81 composition(Y, converse(X))
% 42.81/5.81
% 42.81/5.81 Lemma 66: join(converse(X), composition(Y, converse(Z))) = converse(join(X, composition(Z, converse(Y)))).
% 42.81/5.81 Proof:
% 42.81/5.81 join(converse(X), composition(Y, converse(Z)))
% 42.81/5.81 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.81 join(composition(Y, converse(Z)), converse(X))
% 42.81/5.81 = { by lemma 65 R->L }
% 42.81/5.81 join(converse(composition(Z, converse(Y))), converse(X))
% 42.81/5.81 = { by axiom 7 (converse_additivity) R->L }
% 42.81/5.81 converse(join(composition(Z, converse(Y)), X))
% 42.81/5.81 = { by axiom 2 (maddux1_join_commutativity) }
% 42.81/5.81 converse(join(X, composition(Z, converse(Y))))
% 42.81/5.81
% 42.81/5.81 Lemma 67: composition(join(X, converse(Y)), converse(Z)) = converse(composition(Z, join(Y, converse(X)))).
% 42.81/5.81 Proof:
% 42.81/5.81 composition(join(X, converse(Y)), converse(Z))
% 42.81/5.81 = { by lemma 38 R->L }
% 42.81/5.81 composition(converse(join(Y, converse(X))), converse(Z))
% 42.81/5.81 = { by axiom 9 (converse_multiplicativity) R->L }
% 42.81/5.81 converse(composition(Z, join(Y, converse(X))))
% 42.81/5.81
% 42.81/5.81 Lemma 68: join(composition(X, Y), composition(X, Z)) = composition(X, join(Y, Z)).
% 42.81/5.81 Proof:
% 42.81/5.81 join(composition(X, Y), composition(X, Z))
% 42.81/5.81 = { by axiom 1 (converse_idempotence) R->L }
% 42.81/5.81 join(composition(X, Y), composition(X, converse(converse(Z))))
% 42.81/5.81 = { by axiom 1 (converse_idempotence) R->L }
% 42.81/5.81 converse(converse(join(composition(X, Y), composition(X, converse(converse(Z))))))
% 42.81/5.81 = { by lemma 66 R->L }
% 42.81/5.81 converse(join(converse(composition(X, Y)), composition(converse(Z), converse(X))))
% 42.81/5.81 = { by axiom 9 (converse_multiplicativity) }
% 42.81/5.81 converse(join(composition(converse(Y), converse(X)), composition(converse(Z), converse(X))))
% 42.81/5.81 = { by axiom 12 (composition_distributivity) R->L }
% 42.81/5.81 converse(composition(join(converse(Y), converse(Z)), converse(X)))
% 42.81/5.81 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.81 converse(composition(join(converse(Z), converse(Y)), converse(X)))
% 42.81/5.81 = { by lemma 67 }
% 42.81/5.81 converse(converse(composition(X, join(Y, converse(converse(Z))))))
% 42.81/5.81 = { by axiom 1 (converse_idempotence) }
% 42.81/5.81 composition(X, join(Y, converse(converse(Z))))
% 42.81/5.81 = { by axiom 1 (converse_idempotence) }
% 42.81/5.81 composition(X, join(Y, Z))
% 42.81/5.81
% 42.81/5.81 Lemma 69: composition(join(one, Y), X) = join(X, composition(Y, X)).
% 42.81/5.81 Proof:
% 42.81/5.81 composition(join(one, Y), X)
% 42.81/5.81 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.81 composition(join(Y, one), X)
% 42.81/5.81 = { by lemma 27 }
% 42.81/5.81 join(X, composition(Y, X))
% 42.81/5.81
% 42.81/5.81 Lemma 70: join(X, composition(x0, X)) = X.
% 42.81/5.81 Proof:
% 42.81/5.81 join(X, composition(x0, X))
% 42.81/5.81 = { by lemma 69 R->L }
% 42.81/5.81 composition(join(one, x0), X)
% 42.81/5.81 = { by lemma 30 }
% 42.81/5.81 composition(one, X)
% 42.81/5.81 = { by lemma 18 }
% 42.81/5.81 X
% 42.81/5.81
% 42.81/5.81 Lemma 71: meet(X, composition(x0, top)) = composition(x0, X).
% 42.81/5.81 Proof:
% 42.81/5.81 meet(X, composition(x0, top))
% 42.81/5.81 = { by lemma 64 R->L }
% 42.81/5.81 meet(X, join(complement(X), composition(x0, top)))
% 42.81/5.81 = { by axiom 6 (def_top) }
% 42.81/5.81 meet(X, join(complement(X), composition(x0, join(complement(X), complement(complement(X))))))
% 42.81/5.81 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.81 meet(X, join(complement(X), composition(x0, join(complement(complement(X)), complement(X)))))
% 42.81/5.81 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.81 meet(X, join(composition(x0, join(complement(complement(X)), complement(X))), complement(X)))
% 42.81/5.81 = { by lemma 68 R->L }
% 42.81/5.81 meet(X, join(join(composition(x0, complement(complement(X))), composition(x0, complement(X))), complement(X)))
% 42.81/5.81 = { by axiom 8 (maddux2_join_associativity) R->L }
% 42.81/5.81 meet(X, join(composition(x0, complement(complement(X))), join(composition(x0, complement(X)), complement(X))))
% 42.81/5.81 = { by axiom 2 (maddux1_join_commutativity) }
% 42.81/5.81 meet(X, join(composition(x0, complement(complement(X))), join(complement(X), composition(x0, complement(X)))))
% 42.81/5.81 = { by lemma 70 }
% 42.81/5.81 meet(X, join(composition(x0, complement(complement(X))), complement(X)))
% 42.81/5.81 = { by axiom 2 (maddux1_join_commutativity) }
% 42.81/5.81 meet(X, join(complement(X), composition(x0, complement(complement(X)))))
% 42.81/5.81 = { by lemma 64 }
% 42.81/5.81 meet(X, composition(x0, complement(complement(X))))
% 42.81/5.81 = { by lemma 47 }
% 42.81/5.81 meet(X, composition(x0, X))
% 42.81/5.81 = { by lemma 45 }
% 42.81/5.81 meet(composition(x0, X), X)
% 42.81/5.81 = { by lemma 70 R->L }
% 42.81/5.81 meet(composition(x0, X), join(X, composition(x0, X)))
% 42.81/5.81 = { by lemma 62 }
% 42.81/5.81 composition(x0, X)
% 42.81/5.81
% 42.81/5.81 Lemma 72: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 42.81/5.81 Proof:
% 42.81/5.81 complement(meet(complement(X), Y))
% 42.81/5.81 = { by lemma 45 }
% 42.81/5.81 complement(meet(Y, complement(X)))
% 42.81/5.81 = { by lemma 53 }
% 42.81/5.81 join(X, complement(Y))
% 42.81/5.81
% 42.81/5.81 Lemma 73: complement(converse(X)) = converse(complement(X)).
% 42.81/5.81 Proof:
% 42.81/5.81 complement(converse(X))
% 42.81/5.81 = { by lemma 21 R->L }
% 42.81/5.81 complement(converse(join(meet(X, complement(converse(complement(converse(complement(X)))))), complement(join(complement(X), complement(converse(complement(converse(complement(X))))))))))
% 42.81/5.81 = { by lemma 59 R->L }
% 42.81/5.81 complement(converse(join(complement(join(converse(complement(converse(complement(X)))), complement(X))), complement(join(complement(X), complement(converse(complement(converse(complement(X))))))))))
% 42.81/5.81 = { by axiom 2 (maddux1_join_commutativity) }
% 42.81/5.81 complement(converse(join(complement(join(complement(X), converse(complement(converse(complement(X)))))), complement(join(complement(X), complement(converse(complement(converse(complement(X))))))))))
% 42.81/5.81 = { by lemma 40 }
% 42.81/5.82 complement(converse(join(complement(converse(top)), complement(join(complement(X), complement(converse(complement(converse(complement(X))))))))))
% 42.81/5.82 = { by lemma 42 }
% 42.81/5.82 complement(converse(join(complement(top), complement(join(complement(X), complement(converse(complement(converse(complement(X))))))))))
% 42.81/5.82 = { by lemma 15 }
% 42.81/5.82 complement(converse(join(zero, complement(join(complement(X), complement(converse(complement(converse(complement(X))))))))))
% 42.81/5.82 = { by lemma 43 }
% 42.81/5.82 complement(converse(complement(join(complement(X), complement(converse(complement(converse(complement(X)))))))))
% 42.81/5.82 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 42.81/5.82 complement(converse(meet(X, converse(complement(converse(complement(X)))))))
% 42.81/5.82 = { by lemma 34 R->L }
% 42.81/5.82 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), zero)))
% 42.81/5.82 = { by axiom 5 (def_zero) }
% 42.81/5.82 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), complement(complement(X))))))
% 42.81/5.82 = { by axiom 1 (converse_idempotence) R->L }
% 42.81/5.82 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(converse(complement(complement(X))))))))
% 42.81/5.82 = { by lemma 47 R->L }
% 42.81/5.82 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(complement(complement(converse(complement(complement(X))))))))))
% 42.81/5.82 = { by lemma 57 R->L }
% 42.81/5.82 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(complement(join(meet(complement(converse(complement(complement(X)))), converse(complement(converse(converse(complement(complement(X))))))), meet(complement(converse(complement(converse(converse(complement(complement(X))))))), complement(converse(complement(complement(X))))))))))))
% 42.81/5.82 = { by lemma 59 R->L }
% 42.81/5.82 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(complement(join(meet(complement(converse(complement(complement(X)))), converse(complement(converse(converse(complement(complement(X))))))), complement(join(converse(complement(complement(X))), complement(complement(converse(complement(converse(converse(complement(complement(X)))))))))))))))))
% 42.81/5.82 = { by lemma 59 }
% 42.81/5.82 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(meet(join(converse(complement(complement(X))), complement(complement(converse(complement(converse(converse(complement(complement(X))))))))), complement(meet(complement(converse(complement(complement(X)))), converse(complement(converse(converse(complement(complement(X))))))))))))))
% 42.81/5.82 = { by lemma 45 R->L }
% 42.81/5.82 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(meet(complement(meet(complement(converse(complement(complement(X)))), converse(complement(converse(converse(complement(complement(X)))))))), join(converse(complement(complement(X))), complement(complement(converse(complement(converse(converse(complement(complement(X)))))))))))))))
% 42.81/5.82 = { by lemma 72 }
% 42.81/5.82 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(meet(join(converse(complement(complement(X))), complement(converse(complement(converse(converse(complement(complement(X)))))))), join(converse(complement(complement(X))), complement(complement(converse(complement(converse(converse(complement(complement(X)))))))))))))))
% 42.81/5.82 = { by lemma 47 }
% 42.81/5.82 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(meet(join(converse(complement(complement(X))), complement(converse(complement(converse(converse(complement(complement(X)))))))), join(converse(complement(complement(X))), converse(complement(converse(converse(complement(complement(X)))))))))))))
% 42.81/5.82 = { by lemma 45 R->L }
% 42.81/5.82 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(meet(join(converse(complement(complement(X))), converse(complement(converse(converse(complement(complement(X))))))), join(converse(complement(complement(X))), complement(converse(complement(converse(converse(complement(complement(X))))))))))))))
% 42.81/5.82 = { by lemma 40 }
% 42.81/5.82 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(meet(converse(top), join(converse(complement(complement(X))), complement(converse(complement(converse(converse(complement(complement(X))))))))))))))
% 42.81/5.82 = { by lemma 42 }
% 42.81/5.82 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(meet(top, join(converse(complement(complement(X))), complement(converse(complement(converse(converse(complement(complement(X))))))))))))))
% 42.81/5.82 = { by lemma 50 }
% 42.81/5.82 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(join(converse(complement(complement(X))), complement(converse(complement(converse(converse(complement(complement(X)))))))))))))
% 42.81/5.82 = { by lemma 39 }
% 42.81/5.82 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), join(complement(complement(X)), converse(complement(converse(complement(converse(converse(complement(complement(X)))))))))))))
% 42.81/5.82 = { by axiom 1 (converse_idempotence) }
% 42.81/5.82 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), join(complement(complement(X)), converse(complement(converse(complement(complement(complement(X)))))))))))
% 42.81/5.82 = { by lemma 64 }
% 42.81/5.82 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(complement(converse(complement(complement(complement(X))))))))))
% 42.81/5.82 = { by lemma 47 }
% 42.81/5.82 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(complement(converse(complement(X))))))))
% 42.81/5.82 = { by lemma 45 }
% 42.81/5.82 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(converse(complement(converse(complement(X)))), complement(X)))))
% 42.81/5.82 = { by lemma 45 }
% 42.81/5.82 complement(converse(join(meet(converse(complement(converse(complement(X)))), X), meet(converse(complement(converse(complement(X)))), complement(X)))))
% 42.81/5.82 = { by lemma 56 }
% 42.81/5.82 complement(converse(converse(complement(converse(complement(X))))))
% 42.81/5.82 = { by axiom 1 (converse_idempotence) }
% 42.81/5.82 complement(complement(converse(complement(X))))
% 42.81/5.82 = { by lemma 46 }
% 42.81/5.82 meet(converse(complement(X)), converse(complement(X)))
% 42.81/5.82 = { by lemma 60 }
% 42.81/5.82 converse(complement(X))
% 42.81/5.82
% 42.81/5.82 Lemma 74: join(complement(one), composition(converse(X), complement(X))) = complement(one).
% 42.81/5.82 Proof:
% 42.81/5.82 join(complement(one), composition(converse(X), complement(X)))
% 42.81/5.82 = { by axiom 4 (composition_identity) R->L }
% 42.81/5.82 join(complement(one), composition(converse(X), complement(composition(X, one))))
% 42.81/5.82 = { by lemma 19 }
% 42.81/5.82 complement(one)
% 42.81/5.82
% 42.81/5.82 Lemma 75: join(complement(one), converse(complement(one))) = complement(one).
% 42.81/5.82 Proof:
% 42.81/5.82 join(complement(one), converse(complement(one)))
% 42.81/5.82 = { by axiom 4 (composition_identity) R->L }
% 42.81/5.82 join(complement(one), composition(converse(complement(one)), one))
% 42.81/5.82 = { by lemma 49 R->L }
% 42.81/5.82 join(complement(one), composition(converse(complement(one)), meet(one, top)))
% 42.81/5.82 = { by lemma 43 R->L }
% 42.81/5.82 join(complement(one), composition(converse(join(zero, complement(one))), meet(one, top)))
% 42.81/5.82 = { by lemma 48 R->L }
% 42.81/5.82 join(complement(one), composition(converse(join(zero, complement(one))), complement(join(zero, complement(one)))))
% 42.81/5.82 = { by lemma 74 }
% 42.81/5.82 complement(one)
% 42.81/5.82
% 42.81/5.82 Lemma 76: composition(converse(X), complement(composition(X, top))) = zero.
% 42.81/5.82 Proof:
% 42.81/5.82 composition(converse(X), complement(composition(X, top)))
% 42.81/5.82 = { by lemma 43 R->L }
% 42.81/5.82 join(zero, composition(converse(X), complement(composition(X, top))))
% 42.81/5.82 = { by lemma 15 R->L }
% 42.81/5.82 join(complement(top), composition(converse(X), complement(composition(X, top))))
% 42.81/5.82 = { by lemma 19 }
% 42.81/5.82 complement(top)
% 42.81/5.82 = { by lemma 15 }
% 42.81/5.82 zero
% 42.81/5.82
% 42.81/5.82 Lemma 77: join(top, X) = top.
% 42.81/5.82 Proof:
% 42.81/5.82 join(top, X)
% 42.81/5.82 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.82 join(X, top)
% 42.81/5.82 = { by lemma 26 R->L }
% 42.81/5.82 join(Y, top)
% 42.81/5.82 = { by lemma 35 }
% 42.81/5.82 top
% 42.81/5.82
% 42.81/5.82 Lemma 78: join(X, composition(top, X)) = composition(top, X).
% 42.81/5.82 Proof:
% 42.81/5.82 join(X, composition(top, X))
% 42.81/5.82 = { by lemma 42 R->L }
% 42.81/5.82 join(X, composition(converse(top), X))
% 42.81/5.82 = { by lemma 69 R->L }
% 42.81/5.82 composition(join(one, converse(top)), X)
% 42.81/5.82 = { by lemma 41 }
% 42.81/5.82 composition(top, X)
% 42.81/5.82
% 42.81/5.82 Lemma 79: composition(top, zero) = zero.
% 42.81/5.82 Proof:
% 42.81/5.82 composition(top, zero)
% 42.81/5.82 = { by lemma 42 R->L }
% 42.81/5.82 composition(converse(top), zero)
% 42.81/5.82 = { by lemma 43 R->L }
% 42.81/5.82 join(zero, composition(converse(top), zero))
% 42.81/5.82 = { by lemma 15 R->L }
% 42.81/5.82 join(complement(top), composition(converse(top), zero))
% 42.81/5.82 = { by lemma 15 R->L }
% 42.81/5.82 join(complement(top), composition(converse(top), complement(top)))
% 42.81/5.82 = { by lemma 77 R->L }
% 42.81/5.82 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 42.81/5.82 = { by lemma 78 }
% 42.81/5.82 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 42.81/5.82 = { by lemma 19 }
% 42.81/5.82 complement(top)
% 42.81/5.82 = { by lemma 15 }
% 42.81/5.82 zero
% 42.81/5.82
% 42.81/5.82 Lemma 80: join(one, converse(X)) = converse(join(X, one)).
% 42.81/5.82 Proof:
% 42.81/5.82 join(one, converse(X))
% 42.81/5.82 = { by lemma 28 R->L }
% 42.81/5.82 join(converse(one), converse(X))
% 42.81/5.82 = { by axiom 7 (converse_additivity) R->L }
% 42.81/5.82 converse(join(one, X))
% 42.81/5.82 = { by axiom 2 (maddux1_join_commutativity) }
% 42.81/5.82 converse(join(X, one))
% 42.81/5.82
% 42.81/5.82 Lemma 81: converse(join(one, complement(converse(x0)))) = top.
% 42.81/5.82 Proof:
% 42.81/5.82 converse(join(one, complement(converse(x0))))
% 42.81/5.82 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.82 converse(join(complement(converse(x0)), one))
% 42.81/5.82 = { by lemma 80 R->L }
% 42.81/5.82 join(one, converse(complement(converse(x0))))
% 42.81/5.82 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.82 join(converse(complement(converse(x0))), one)
% 42.81/5.82 = { by lemma 32 R->L }
% 42.81/5.82 join(one, join(x0, converse(complement(converse(x0)))))
% 42.81/5.82 = { by lemma 40 }
% 42.81/5.82 join(one, converse(top))
% 42.81/5.82 = { by lemma 80 }
% 42.81/5.82 converse(join(top, one))
% 42.81/5.82 = { by lemma 77 }
% 42.81/5.82 converse(top)
% 42.81/5.82 = { by lemma 42 }
% 42.81/5.82 top
% 42.81/5.82
% 42.81/5.82 Lemma 82: join(one, complement(converse(x0))) = top.
% 42.81/5.82 Proof:
% 42.81/5.82 join(one, complement(converse(x0)))
% 42.81/5.82 = { by axiom 1 (converse_idempotence) R->L }
% 42.81/5.82 converse(converse(join(one, complement(converse(x0)))))
% 42.81/5.82 = { by lemma 81 }
% 42.81/5.82 converse(top)
% 42.81/5.82 = { by lemma 42 }
% 42.81/5.82 top
% 42.81/5.82
% 42.81/5.82 Lemma 83: complement(meet(X, join(Y, complement(Z)))) = join(complement(X), meet(Z, complement(Y))).
% 42.81/5.82 Proof:
% 42.81/5.82 complement(meet(X, join(Y, complement(Z))))
% 42.81/5.82 = { by lemma 45 }
% 42.81/5.82 complement(meet(join(Y, complement(Z)), X))
% 42.81/5.82 = { by lemma 52 R->L }
% 42.81/5.82 join(complement(join(Y, complement(Z))), complement(X))
% 42.81/5.82 = { by lemma 59 }
% 42.81/5.82 join(meet(Z, complement(Y)), complement(X))
% 42.81/5.82 = { by axiom 2 (maddux1_join_commutativity) }
% 42.81/5.82 join(complement(X), meet(Z, complement(Y)))
% 42.81/5.82
% 42.81/5.82 Lemma 84: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 42.81/5.82 Proof:
% 42.81/5.82 meet(complement(X), complement(Y))
% 42.81/5.82 = { by lemma 45 }
% 42.81/5.82 meet(complement(Y), complement(X))
% 42.81/5.82 = { by lemma 43 R->L }
% 42.81/5.82 meet(join(zero, complement(Y)), complement(X))
% 42.81/5.82 = { by lemma 59 R->L }
% 42.81/5.82 complement(join(X, complement(join(zero, complement(Y)))))
% 42.81/5.82 = { by lemma 48 }
% 42.81/5.82 complement(join(X, meet(Y, top)))
% 42.81/5.82 = { by lemma 49 }
% 42.81/5.82 complement(join(X, Y))
% 42.81/5.83
% 42.81/5.83 Lemma 85: converse(x0) = x0.
% 42.81/5.83 Proof:
% 42.81/5.83 converse(x0)
% 42.81/5.83 = { by axiom 4 (composition_identity) R->L }
% 42.81/5.83 converse(composition(x0, one))
% 42.81/5.83 = { by lemma 71 R->L }
% 42.81/5.83 converse(meet(one, composition(x0, top)))
% 42.81/5.83 = { by lemma 60 R->L }
% 42.81/5.83 converse(meet(one, meet(composition(x0, top), composition(x0, top))))
% 42.81/5.83 = { by lemma 46 R->L }
% 42.81/5.83 converse(meet(one, complement(complement(composition(x0, top)))))
% 42.81/5.83 = { by lemma 59 R->L }
% 42.81/5.83 converse(complement(join(complement(composition(x0, top)), complement(one))))
% 42.81/5.83 = { by lemma 73 R->L }
% 42.81/5.83 complement(converse(join(complement(composition(x0, top)), complement(one))))
% 42.81/5.83 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.83 complement(converse(join(complement(one), complement(composition(x0, top)))))
% 42.81/5.83 = { by axiom 7 (converse_additivity) }
% 42.81/5.83 complement(join(converse(complement(one)), converse(complement(composition(x0, top)))))
% 42.81/5.83 = { by lemma 75 R->L }
% 42.81/5.83 complement(join(converse(join(complement(one), converse(complement(one)))), converse(complement(composition(x0, top)))))
% 42.81/5.83 = { by lemma 38 }
% 42.81/5.83 complement(join(join(complement(one), converse(complement(one))), converse(complement(composition(x0, top)))))
% 42.81/5.83 = { by lemma 75 }
% 42.81/5.83 complement(join(complement(one), converse(complement(composition(x0, top)))))
% 42.81/5.83 = { by axiom 2 (maddux1_join_commutativity) }
% 42.81/5.83 complement(join(converse(complement(composition(x0, top))), complement(one)))
% 42.81/5.83 = { by lemma 59 }
% 42.81/5.83 meet(one, complement(converse(complement(composition(x0, top)))))
% 42.81/5.83 = { by lemma 73 }
% 42.81/5.83 meet(one, converse(complement(complement(composition(x0, top)))))
% 42.81/5.83 = { by lemma 19 R->L }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(composition(x0, complement(composition(x0, top))))))))
% 42.81/5.83 = { by lemma 47 R->L }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(complement(composition(x0, complement(composition(x0, top))))))))))
% 42.81/5.83 = { by lemma 19 R->L }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), composition(converse(converse(complement(composition(x0, top)))), complement(composition(converse(complement(composition(x0, top))), composition(x0, complement(composition(x0, top)))))))))))))
% 42.81/5.83 = { by lemma 16 R->L }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), composition(converse(converse(complement(composition(x0, top)))), complement(converse(composition(converse(composition(x0, complement(composition(x0, top)))), complement(composition(x0, top)))))))))))))
% 42.81/5.83 = { by axiom 9 (converse_multiplicativity) }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), composition(converse(converse(complement(composition(x0, top)))), complement(converse(composition(composition(converse(complement(composition(x0, top))), converse(x0)), complement(composition(x0, top)))))))))))))
% 42.81/5.83 = { by axiom 10 (composition_associativity) R->L }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), composition(converse(converse(complement(composition(x0, top)))), complement(converse(composition(converse(complement(composition(x0, top))), composition(converse(x0), complement(composition(x0, top))))))))))))))
% 42.81/5.83 = { by lemma 76 }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), composition(converse(converse(complement(composition(x0, top)))), complement(converse(composition(converse(complement(composition(x0, top))), zero)))))))))))
% 42.81/5.83 = { by lemma 43 R->L }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), composition(converse(converse(complement(composition(x0, top)))), complement(converse(join(zero, composition(converse(complement(composition(x0, top))), zero))))))))))))
% 42.81/5.83 = { by lemma 79 R->L }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), composition(converse(converse(complement(composition(x0, top)))), complement(converse(join(composition(top, zero), composition(converse(complement(composition(x0, top))), zero))))))))))))
% 42.81/5.83 = { by axiom 12 (composition_distributivity) R->L }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), composition(converse(converse(complement(composition(x0, top)))), complement(converse(composition(join(top, converse(complement(composition(x0, top)))), zero)))))))))))
% 42.81/5.83 = { by lemma 77 }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), composition(converse(converse(complement(composition(x0, top)))), complement(converse(composition(top, zero)))))))))))
% 42.81/5.83 = { by lemma 79 }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), composition(converse(converse(complement(composition(x0, top)))), complement(converse(zero))))))))))
% 42.81/5.83 = { by lemma 44 }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), composition(converse(converse(complement(composition(x0, top)))), complement(zero)))))))))
% 42.81/5.83 = { by axiom 1 (converse_idempotence) }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), composition(complement(composition(x0, top)), complement(zero)))))))))
% 42.81/5.83 = { by lemma 37 }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), composition(complement(composition(x0, top)), top))))))))
% 42.81/5.83 = { by lemma 42 R->L }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), composition(complement(composition(x0, top)), converse(top)))))))))
% 42.81/5.83 = { by lemma 65 R->L }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), converse(composition(top, converse(complement(composition(x0, top))))))))))))
% 42.81/5.83 = { by lemma 78 R->L }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), converse(join(converse(complement(composition(x0, top))), composition(top, converse(complement(composition(x0, top)))))))))))))
% 42.81/5.83 = { by lemma 39 }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), join(complement(composition(x0, top)), converse(composition(top, converse(complement(composition(x0, top)))))))))))))
% 42.81/5.83 = { by lemma 65 }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), join(complement(composition(x0, top)), composition(complement(composition(x0, top)), converse(top))))))))))
% 42.81/5.83 = { by lemma 42 }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), join(complement(composition(x0, top)), composition(complement(composition(x0, top)), top)))))))))
% 42.81/5.83 = { by lemma 37 R->L }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), join(complement(composition(x0, top)), composition(complement(composition(x0, top)), complement(zero))))))))))
% 42.81/5.83 = { by axiom 1 (converse_idempotence) R->L }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), join(complement(composition(x0, top)), composition(converse(converse(complement(composition(x0, top)))), complement(zero))))))))))
% 42.81/5.83 = { by lemma 44 R->L }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), join(complement(composition(x0, top)), composition(converse(converse(complement(composition(x0, top)))), complement(converse(zero)))))))))))
% 42.81/5.83 = { by lemma 79 R->L }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), join(complement(composition(x0, top)), composition(converse(converse(complement(composition(x0, top)))), complement(converse(composition(top, zero))))))))))))
% 42.81/5.83 = { by lemma 76 R->L }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), join(complement(composition(x0, top)), composition(converse(converse(complement(composition(x0, top)))), complement(converse(composition(top, composition(converse(x0), complement(composition(x0, top)))))))))))))))
% 42.81/5.83 = { by lemma 82 R->L }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), join(complement(composition(x0, top)), composition(converse(converse(complement(composition(x0, top)))), complement(converse(composition(join(one, complement(converse(x0))), composition(converse(x0), complement(composition(x0, top)))))))))))))))
% 42.81/5.83 = { by axiom 10 (composition_associativity) }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), join(complement(composition(x0, top)), composition(converse(converse(complement(composition(x0, top)))), complement(converse(composition(composition(join(one, complement(converse(x0))), converse(x0)), complement(composition(x0, top))))))))))))))
% 42.81/5.83 = { by axiom 9 (converse_multiplicativity) }
% 42.81/5.83 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), join(complement(composition(x0, top)), composition(converse(converse(complement(composition(x0, top)))), complement(composition(converse(complement(composition(x0, top))), converse(composition(join(one, complement(converse(x0))), converse(x0))))))))))))))
% 42.81/5.83 = { by lemma 65 }
% 42.81/5.84 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), join(complement(composition(x0, top)), composition(converse(converse(complement(composition(x0, top)))), complement(composition(converse(complement(composition(x0, top))), composition(x0, converse(join(one, complement(converse(x0))))))))))))))))
% 42.81/5.84 = { by lemma 81 }
% 42.81/5.84 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), join(complement(composition(x0, top)), composition(converse(converse(complement(composition(x0, top)))), complement(composition(converse(complement(composition(x0, top))), composition(x0, top))))))))))))
% 42.81/5.84 = { by lemma 19 }
% 42.81/5.84 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, complement(composition(x0, top)))), complement(composition(x0, top)))))))))
% 42.81/5.84 = { by lemma 52 }
% 42.81/5.84 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(complement(meet(composition(x0, complement(composition(x0, top))), composition(x0, top)))))))))
% 42.81/5.84 = { by lemma 46 }
% 42.81/5.84 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(meet(meet(composition(x0, complement(composition(x0, top))), composition(x0, top)), meet(composition(x0, complement(composition(x0, top))), composition(x0, top))))))))
% 42.81/5.84 = { by lemma 55 }
% 42.81/5.84 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(meet(composition(x0, complement(composition(x0, top))), meet(meet(composition(x0, complement(composition(x0, top))), composition(x0, top)), composition(x0, top))))))))
% 42.81/5.84 = { by lemma 55 }
% 42.81/5.84 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(meet(composition(x0, complement(composition(x0, top))), meet(composition(x0, complement(composition(x0, top))), meet(composition(x0, top), composition(x0, top)))))))))
% 42.81/5.84 = { by lemma 45 }
% 42.81/5.84 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(meet(meet(composition(x0, complement(composition(x0, top))), meet(composition(x0, top), composition(x0, top))), composition(x0, complement(composition(x0, top)))))))))
% 42.81/5.84 = { by lemma 21 R->L }
% 42.81/5.84 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(meet(meet(composition(x0, complement(composition(x0, top))), meet(composition(x0, top), composition(x0, top))), join(meet(composition(x0, complement(composition(x0, top))), meet(composition(x0, top), composition(x0, top))), complement(join(complement(composition(x0, complement(composition(x0, top)))), meet(composition(x0, top), composition(x0, top)))))))))))
% 42.81/5.84 = { by lemma 61 }
% 42.81/5.84 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(meet(composition(x0, complement(composition(x0, top))), meet(composition(x0, top), composition(x0, top))))))))
% 42.81/5.84 = { by lemma 60 }
% 42.81/5.84 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(meet(composition(x0, complement(composition(x0, top))), composition(x0, top)))))))
% 42.81/5.84 = { by lemma 45 R->L }
% 42.81/5.84 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(meet(composition(x0, top), composition(x0, complement(composition(x0, top)))))))))
% 42.81/5.84 = { by lemma 64 R->L }
% 42.81/5.84 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(meet(composition(x0, top), join(complement(composition(x0, top)), composition(x0, complement(composition(x0, top))))))))))
% 42.81/5.84 = { by lemma 70 }
% 42.81/5.84 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(meet(composition(x0, top), complement(composition(x0, top))))))))
% 42.81/5.84 = { by axiom 5 (def_zero) R->L }
% 42.81/5.84 meet(one, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(zero)))))
% 42.81/5.84 = { by lemma 47 }
% 42.81/5.84 meet(one, converse(join(composition(x0, top), composition(converse(x0), complement(zero)))))
% 42.81/5.84 = { by lemma 37 }
% 42.81/5.84 meet(one, converse(join(composition(x0, top), composition(converse(x0), top))))
% 42.81/5.84 = { by axiom 12 (composition_distributivity) R->L }
% 42.81/5.84 meet(one, converse(composition(join(x0, converse(x0)), top)))
% 42.81/5.84 = { by lemma 81 R->L }
% 42.81/5.84 meet(one, converse(composition(join(x0, converse(x0)), converse(join(one, complement(converse(x0)))))))
% 42.81/5.84 = { by lemma 67 }
% 42.81/5.84 meet(one, converse(converse(composition(join(one, complement(converse(x0))), join(x0, converse(x0))))))
% 42.81/5.84 = { by lemma 82 }
% 42.81/5.84 meet(one, converse(converse(composition(top, join(x0, converse(x0))))))
% 42.81/5.84 = { by lemma 68 R->L }
% 42.81/5.84 meet(one, converse(converse(join(composition(top, x0), composition(top, converse(x0))))))
% 42.81/5.84 = { by lemma 37 R->L }
% 42.81/5.84 meet(one, converse(converse(join(composition(top, x0), composition(complement(zero), converse(x0))))))
% 42.81/5.84 = { by lemma 47 R->L }
% 42.81/5.84 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(zero), converse(x0))))))
% 42.81/5.84 = { by axiom 5 (def_zero) }
% 42.81/5.84 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(meet(composition(top, x0), complement(composition(top, x0)))), converse(x0))))))
% 42.81/5.84 = { by axiom 4 (composition_identity) R->L }
% 42.81/5.84 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(meet(composition(top, x0), composition(complement(composition(top, x0)), one))), converse(x0))))))
% 42.81/5.84 = { by lemma 30 R->L }
% 42.81/5.84 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(meet(composition(top, x0), composition(complement(composition(top, x0)), join(one, x0)))), converse(x0))))))
% 42.81/5.84 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.84 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(meet(composition(top, x0), composition(complement(composition(top, x0)), join(x0, one)))), converse(x0))))))
% 42.81/5.84 = { by lemma 28 R->L }
% 42.81/5.84 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(meet(composition(top, x0), composition(complement(composition(top, x0)), join(x0, converse(one))))), converse(x0))))))
% 42.81/5.84 = { by lemma 39 R->L }
% 42.81/5.84 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(meet(composition(top, x0), composition(complement(composition(top, x0)), converse(join(converse(x0), one))))), converse(x0))))))
% 42.81/5.84 = { by lemma 65 R->L }
% 42.81/5.84 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(meet(composition(top, x0), converse(composition(join(converse(x0), one), converse(complement(composition(top, x0))))))), converse(x0))))))
% 42.81/5.84 = { by lemma 27 }
% 42.81/5.84 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(meet(composition(top, x0), converse(join(converse(complement(composition(top, x0))), composition(converse(x0), converse(complement(composition(top, x0)))))))), converse(x0))))))
% 42.81/5.84 = { by lemma 39 }
% 42.81/5.84 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(meet(composition(top, x0), join(complement(composition(top, x0)), converse(composition(converse(x0), converse(complement(composition(top, x0)))))))), converse(x0))))))
% 42.81/5.84 = { by lemma 65 }
% 42.81/5.84 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(meet(composition(top, x0), join(complement(composition(top, x0)), composition(complement(composition(top, x0)), converse(converse(x0)))))), converse(x0))))))
% 42.81/5.84 = { by axiom 1 (converse_idempotence) }
% 42.81/5.84 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(meet(composition(top, x0), join(complement(composition(top, x0)), composition(complement(composition(top, x0)), x0)))), converse(x0))))))
% 42.81/5.84 = { by lemma 64 }
% 42.81/5.84 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(meet(composition(top, x0), composition(complement(composition(top, x0)), x0))), converse(x0))))))
% 42.81/5.84 = { by lemma 34 R->L }
% 42.81/5.84 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(join(meet(composition(top, x0), composition(complement(composition(top, x0)), x0)), zero)), converse(x0))))))
% 42.81/5.84 = { by lemma 15 R->L }
% 42.81/5.84 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(join(meet(composition(top, x0), composition(complement(composition(top, x0)), x0)), complement(top))), converse(x0))))))
% 42.81/5.84 = { by lemma 47 R->L }
% 42.81/5.84 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(join(meet(composition(top, x0), complement(complement(composition(complement(composition(top, x0)), x0)))), complement(top))), converse(x0))))))
% 42.81/5.84 = { by lemma 36 R->L }
% 42.81/5.84 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(join(meet(composition(top, x0), complement(complement(composition(complement(composition(top, x0)), x0)))), complement(join(composition(complement(composition(top, x0)), x0), join(complement(composition(complement(composition(top, x0)), x0)), composition(complement(complement(composition(top, x0))), x0)))))), converse(x0))))))
% 42.81/5.84 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.84 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(join(meet(composition(top, x0), complement(complement(composition(complement(composition(top, x0)), x0)))), complement(join(composition(complement(composition(top, x0)), x0), join(composition(complement(complement(composition(top, x0))), x0), complement(composition(complement(composition(top, x0)), x0))))))), converse(x0))))))
% 42.81/5.84 = { by axiom 8 (maddux2_join_associativity) }
% 42.81/5.84 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(join(meet(composition(top, x0), complement(complement(composition(complement(composition(top, x0)), x0)))), complement(join(join(composition(complement(composition(top, x0)), x0), composition(complement(complement(composition(top, x0))), x0)), complement(composition(complement(composition(top, x0)), x0)))))), converse(x0))))))
% 42.81/5.84 = { by axiom 12 (composition_distributivity) R->L }
% 42.81/5.84 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(join(meet(composition(top, x0), complement(complement(composition(complement(composition(top, x0)), x0)))), complement(join(composition(join(complement(composition(top, x0)), complement(complement(composition(top, x0)))), x0), complement(composition(complement(composition(top, x0)), x0)))))), converse(x0))))))
% 42.81/5.84 = { by axiom 2 (maddux1_join_commutativity) }
% 42.81/5.84 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(join(meet(composition(top, x0), complement(complement(composition(complement(composition(top, x0)), x0)))), complement(join(complement(composition(complement(composition(top, x0)), x0)), composition(join(complement(composition(top, x0)), complement(complement(composition(top, x0)))), x0))))), converse(x0))))))
% 42.81/5.84 = { by axiom 6 (def_top) R->L }
% 42.81/5.84 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(join(meet(composition(top, x0), complement(complement(composition(complement(composition(top, x0)), x0)))), complement(join(complement(composition(complement(composition(top, x0)), x0)), composition(top, x0))))), converse(x0))))))
% 42.81/5.84 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.84 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(join(complement(join(complement(composition(complement(composition(top, x0)), x0)), composition(top, x0))), meet(composition(top, x0), complement(complement(composition(complement(composition(top, x0)), x0)))))), converse(x0))))))
% 42.81/5.84 = { by lemma 83 R->L }
% 42.81/5.84 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(complement(meet(join(complement(composition(complement(composition(top, x0)), x0)), composition(top, x0)), join(complement(composition(complement(composition(top, x0)), x0)), complement(composition(top, x0)))))), converse(x0))))))
% 42.81/5.84 = { by lemma 52 R->L }
% 42.81/5.84 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(join(complement(join(complement(composition(complement(composition(top, x0)), x0)), composition(top, x0))), complement(join(complement(composition(complement(composition(top, x0)), x0)), complement(composition(top, x0)))))), converse(x0))))))
% 42.81/5.84 = { by lemma 84 R->L }
% 42.81/5.84 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(join(complement(join(complement(composition(complement(composition(top, x0)), x0)), composition(top, x0))), meet(complement(complement(composition(complement(composition(top, x0)), x0))), complement(complement(composition(top, x0)))))), converse(x0))))))
% 42.81/5.85 = { by lemma 83 R->L }
% 42.81/5.85 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(complement(meet(join(complement(composition(complement(composition(top, x0)), x0)), composition(top, x0)), join(complement(composition(top, x0)), complement(complement(complement(composition(complement(composition(top, x0)), x0)))))))), converse(x0))))))
% 42.81/5.85 = { by axiom 2 (maddux1_join_commutativity) }
% 42.81/5.85 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(complement(meet(join(complement(composition(complement(composition(top, x0)), x0)), composition(top, x0)), join(complement(complement(complement(composition(complement(composition(top, x0)), x0)))), complement(composition(top, x0)))))), converse(x0))))))
% 42.81/5.85 = { by lemma 52 R->L }
% 42.81/5.85 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(join(complement(join(complement(composition(complement(composition(top, x0)), x0)), composition(top, x0))), complement(join(complement(complement(complement(composition(complement(composition(top, x0)), x0)))), complement(composition(top, x0)))))), converse(x0))))))
% 42.81/5.85 = { by lemma 84 R->L }
% 42.81/5.85 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(join(meet(complement(complement(composition(complement(composition(top, x0)), x0))), complement(composition(top, x0))), complement(join(complement(complement(complement(composition(complement(composition(top, x0)), x0)))), complement(composition(top, x0)))))), converse(x0))))))
% 42.81/5.85 = { by lemma 21 }
% 42.81/5.85 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(complement(complement(composition(complement(composition(top, x0)), x0)))), converse(x0))))))
% 42.81/5.85 = { by lemma 47 }
% 42.81/5.85 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(complement(composition(complement(composition(top, x0)), x0)), converse(x0))))))
% 42.81/5.85 = { by lemma 66 R->L }
% 42.81/5.85 meet(one, converse(join(converse(complement(complement(composition(top, x0)))), composition(x0, converse(complement(composition(complement(composition(top, x0)), x0)))))))
% 42.81/5.85 = { by axiom 2 (maddux1_join_commutativity) }
% 42.81/5.85 meet(one, converse(join(composition(x0, converse(complement(composition(complement(composition(top, x0)), x0)))), converse(complement(complement(composition(top, x0)))))))
% 42.81/5.85 = { by lemma 73 R->L }
% 42.81/5.85 meet(one, converse(join(composition(x0, complement(converse(composition(complement(composition(top, x0)), x0)))), converse(complement(complement(composition(top, x0)))))))
% 42.81/5.85 = { by lemma 38 R->L }
% 42.81/5.85 meet(one, converse(converse(join(complement(complement(composition(top, x0))), converse(composition(x0, complement(converse(composition(complement(composition(top, x0)), x0)))))))))
% 42.81/5.85 = { by axiom 9 (converse_multiplicativity) }
% 42.81/5.85 meet(one, converse(converse(join(complement(complement(composition(top, x0))), composition(converse(complement(converse(composition(complement(composition(top, x0)), x0)))), converse(x0))))))
% 42.81/5.85 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.85 meet(one, converse(converse(join(composition(converse(complement(converse(composition(complement(composition(top, x0)), x0)))), converse(x0)), complement(complement(composition(top, x0)))))))
% 42.81/5.85 = { by axiom 7 (converse_additivity) }
% 42.81/5.85 meet(one, converse(join(converse(composition(converse(complement(converse(composition(complement(composition(top, x0)), x0)))), converse(x0))), converse(complement(complement(composition(top, x0)))))))
% 42.81/5.85 = { by lemma 16 }
% 42.81/5.85 meet(one, converse(join(composition(converse(converse(x0)), complement(converse(composition(complement(composition(top, x0)), x0)))), converse(complement(complement(composition(top, x0)))))))
% 42.81/5.85 = { by axiom 2 (maddux1_join_commutativity) }
% 42.81/5.85 meet(one, converse(join(converse(complement(complement(composition(top, x0)))), composition(converse(converse(x0)), complement(converse(composition(complement(composition(top, x0)), x0)))))))
% 42.81/5.85 = { by lemma 73 R->L }
% 42.81/5.85 meet(one, converse(join(complement(converse(complement(composition(top, x0)))), composition(converse(converse(x0)), complement(converse(composition(complement(composition(top, x0)), x0)))))))
% 42.81/5.85 = { by axiom 9 (converse_multiplicativity) }
% 42.81/5.85 meet(one, converse(join(complement(converse(complement(composition(top, x0)))), composition(converse(converse(x0)), complement(composition(converse(x0), converse(complement(composition(top, x0)))))))))
% 42.81/5.85 = { by lemma 19 }
% 42.81/5.85 meet(one, converse(complement(converse(complement(composition(top, x0))))))
% 42.81/5.85 = { by lemma 73 }
% 42.81/5.85 meet(one, converse(converse(complement(complement(composition(top, x0))))))
% 42.81/5.85 = { by lemma 47 }
% 42.81/5.85 meet(one, converse(converse(composition(top, x0))))
% 42.81/5.85 = { by axiom 1 (converse_idempotence) }
% 42.81/5.85 meet(one, composition(top, x0))
% 42.81/5.85 = { by lemma 64 R->L }
% 42.81/5.85 meet(one, join(complement(one), composition(top, x0)))
% 42.81/5.85 = { by lemma 47 R->L }
% 42.81/5.85 meet(one, join(complement(one), composition(top, complement(complement(x0)))))
% 42.81/5.85 = { by lemma 82 R->L }
% 42.81/5.85 meet(one, join(complement(one), composition(join(one, complement(converse(x0))), complement(complement(x0)))))
% 42.81/5.85 = { by lemma 69 }
% 42.81/5.85 meet(one, join(complement(one), join(complement(complement(x0)), composition(complement(converse(x0)), complement(complement(x0))))))
% 42.81/5.85 = { by lemma 73 }
% 42.81/5.85 meet(one, join(complement(one), join(complement(complement(x0)), composition(converse(complement(x0)), complement(complement(x0))))))
% 42.81/5.85 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.85 meet(one, join(complement(one), join(composition(converse(complement(x0)), complement(complement(x0))), complement(complement(x0)))))
% 42.81/5.85 = { by axiom 8 (maddux2_join_associativity) }
% 42.81/5.85 meet(one, join(join(complement(one), composition(converse(complement(x0)), complement(complement(x0)))), complement(complement(x0))))
% 42.81/5.85 = { by lemma 74 }
% 42.81/5.85 meet(one, join(complement(one), complement(complement(x0))))
% 42.81/5.85 = { by axiom 2 (maddux1_join_commutativity) }
% 42.81/5.85 meet(one, join(complement(complement(x0)), complement(one)))
% 42.81/5.85 = { by lemma 52 }
% 42.81/5.85 meet(one, complement(meet(complement(x0), one)))
% 42.81/5.85 = { by lemma 72 }
% 42.81/5.85 meet(one, join(x0, complement(one)))
% 42.81/5.85 = { by lemma 63 }
% 42.81/5.85 meet(one, x0)
% 42.81/5.85 = { by lemma 45 }
% 42.81/5.85 meet(x0, one)
% 42.81/5.85 = { by axiom 11 (maddux4_definiton_of_meet) }
% 42.81/5.85 complement(join(complement(x0), complement(one)))
% 42.81/5.85 = { by lemma 43 R->L }
% 42.81/5.85 join(zero, complement(join(complement(x0), complement(one))))
% 42.81/5.85 = { by lemma 15 R->L }
% 42.81/5.85 join(complement(top), complement(join(complement(x0), complement(one))))
% 42.81/5.85 = { by lemma 77 R->L }
% 42.81/5.85 join(complement(join(top, one)), complement(join(complement(x0), complement(one))))
% 42.81/5.85 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.85 join(complement(join(one, top)), complement(join(complement(x0), complement(one))))
% 42.81/5.85 = { by axiom 6 (def_top) }
% 42.81/5.85 join(complement(join(one, join(x0, complement(x0)))), complement(join(complement(x0), complement(one))))
% 42.81/5.85 = { by lemma 32 }
% 42.81/5.85 join(complement(join(complement(x0), one)), complement(join(complement(x0), complement(one))))
% 42.81/5.85 = { by axiom 2 (maddux1_join_commutativity) }
% 42.81/5.85 join(complement(join(one, complement(x0))), complement(join(complement(x0), complement(one))))
% 42.81/5.85 = { by lemma 59 }
% 42.81/5.85 join(meet(x0, complement(one)), complement(join(complement(x0), complement(one))))
% 42.81/5.85 = { by lemma 21 }
% 42.81/5.85 x0
% 42.81/5.85
% 42.81/5.85 Lemma 86: meet(composition(x0, X), Y) = meet(X, composition(x0, Y)).
% 42.81/5.85 Proof:
% 42.81/5.85 meet(composition(x0, X), Y)
% 42.81/5.85 = { by lemma 45 }
% 42.81/5.85 meet(Y, composition(x0, X))
% 42.81/5.85 = { by lemma 71 R->L }
% 42.81/5.85 meet(Y, meet(X, composition(x0, top)))
% 42.81/5.85 = { by lemma 54 }
% 42.81/5.85 meet(X, meet(Y, composition(x0, top)))
% 42.81/5.85 = { by lemma 71 }
% 42.81/5.85 meet(X, composition(x0, Y))
% 42.81/5.85
% 42.81/5.85 Lemma 87: composition(x0, complement(composition(x0, X))) = composition(x0, complement(X)).
% 42.81/5.85 Proof:
% 42.81/5.85 composition(x0, complement(composition(x0, X)))
% 42.81/5.85 = { by lemma 85 R->L }
% 42.81/5.85 composition(converse(x0), complement(composition(x0, X)))
% 42.81/5.85 = { by lemma 71 R->L }
% 42.81/5.85 composition(converse(x0), complement(meet(X, composition(x0, top))))
% 42.81/5.85 = { by lemma 52 R->L }
% 42.81/5.85 composition(converse(x0), join(complement(X), complement(composition(x0, top))))
% 42.81/5.85 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 42.81/5.85 composition(converse(x0), join(complement(composition(x0, top)), complement(X)))
% 42.81/5.85 = { by lemma 68 R->L }
% 42.81/5.85 join(composition(converse(x0), complement(composition(x0, top))), composition(converse(x0), complement(X)))
% 42.81/5.85 = { by lemma 76 }
% 42.81/5.85 join(zero, composition(converse(x0), complement(X)))
% 42.81/5.85 = { by lemma 43 }
% 42.81/5.85 composition(converse(x0), complement(X))
% 42.81/5.85 = { by lemma 85 }
% 42.81/5.85 composition(x0, complement(X))
% 42.81/5.85
% 42.81/5.85 Goal 1 (goals_1): tuple(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2))))) = tuple(meet(composition(x0, x1), complement(x2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))).
% 42.81/5.85 Proof:
% 42.81/5.85 tuple(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 42.81/5.85 = { by axiom 2 (maddux1_join_commutativity) }
% 42.81/5.85 tuple(join(meet(composition(x0, x1), complement(x2)), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 42.81/5.85 = { by lemma 86 }
% 42.81/5.85 tuple(join(meet(x1, composition(x0, complement(x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 42.81/5.85 = { by lemma 86 }
% 42.81/5.85 tuple(join(meet(x1, composition(x0, complement(x2))), meet(x1, composition(x0, complement(composition(x0, x2))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 42.81/5.85 = { by lemma 86 }
% 42.81/5.85 tuple(join(meet(x1, composition(x0, complement(x2))), meet(x1, composition(x0, complement(composition(x0, x2))))), join(meet(x1, composition(x0, complement(x2_2))), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 42.81/5.85 = { by lemma 86 }
% 42.81/5.85 tuple(join(meet(x1, composition(x0, complement(x2))), meet(x1, composition(x0, complement(composition(x0, x2))))), join(meet(x1, composition(x0, complement(x2_2))), meet(x1, composition(x0, complement(composition(x0, x2_2))))))
% 42.81/5.85 = { by lemma 87 }
% 42.81/5.85 tuple(join(meet(x1, composition(x0, complement(x2))), meet(x1, composition(x0, complement(x2)))), join(meet(x1, composition(x0, complement(x2_2))), meet(x1, composition(x0, complement(composition(x0, x2_2))))))
% 42.81/5.85 = { by lemma 33 }
% 42.81/5.85 tuple(meet(x1, composition(x0, complement(x2))), join(meet(x1, composition(x0, complement(x2_2))), meet(x1, composition(x0, complement(composition(x0, x2_2))))))
% 42.81/5.85 = { by lemma 87 }
% 42.81/5.85 tuple(meet(x1, composition(x0, complement(x2))), join(meet(x1, composition(x0, complement(x2_2))), meet(x1, composition(x0, complement(x2_2)))))
% 42.81/5.85 = { by lemma 33 }
% 42.81/5.85 tuple(meet(x1, composition(x0, complement(x2))), meet(x1, composition(x0, complement(x2_2))))
% 42.81/5.85 = { by lemma 87 R->L }
% 42.81/5.85 tuple(meet(x1, composition(x0, complement(x2))), meet(x1, composition(x0, complement(composition(x0, x2_2)))))
% 42.81/5.85 = { by lemma 86 R->L }
% 42.81/5.85 tuple(meet(x1, composition(x0, complement(x2))), meet(composition(x0, x1), complement(composition(x0, x2_2))))
% 42.81/5.85 = { by lemma 86 R->L }
% 42.81/5.85 tuple(meet(composition(x0, x1), complement(x2)), meet(composition(x0, x1), complement(composition(x0, x2_2))))
% 42.81/5.85 % SZS output end Proof
% 42.81/5.85
% 42.81/5.85 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------