TSTP Solution File: REL029+4 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL029+4 : 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 : n018.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 13:44:13 EDT 2023
% Result : Theorem 59.26s 8.07s
% Output : Proof 60.56s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL029+4 : 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
% 0.18/0.34 % Computer : n018.cluster.edu
% 0.18/0.34 % Model : x86_64 x86_64
% 0.18/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.34 % Memory : 8042.1875MB
% 0.18/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.18/0.34 % CPULimit : 300
% 0.18/0.34 % WCLimit : 300
% 0.18/0.34 % DateTime : Fri Aug 25 22:59:29 EDT 2023
% 0.18/0.34 % CPUTime :
% 59.26/8.07 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 59.26/8.07
% 59.26/8.07 % SZS status Theorem
% 59.26/8.07
% 60.56/8.23 % SZS output start Proof
% 60.56/8.23 Take the following subset of the input axioms:
% 60.56/8.23 fof(composition_associativity, axiom, ![X0, X1, X2]: composition(X0, composition(X1, X2))=composition(composition(X0, X1), X2)).
% 60.56/8.23 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))).
% 60.56/8.23 fof(composition_identity, axiom, ![X0_2]: composition(X0_2, one)=X0_2).
% 60.56/8.23 fof(converse_additivity, axiom, ![X0_2, X1_2]: converse(join(X0_2, X1_2))=join(converse(X0_2), converse(X1_2))).
% 60.56/8.23 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)).
% 60.56/8.23 fof(converse_idempotence, axiom, ![X0_2]: converse(converse(X0_2))=X0_2).
% 60.56/8.23 fof(converse_multiplicativity, axiom, ![X0_2, X1_2]: converse(composition(X0_2, X1_2))=composition(converse(X1_2), converse(X0_2))).
% 60.56/8.23 fof(dedekind_law, axiom, ![X0_2, X1_2, X2_2]: join(meet(composition(X0_2, X1_2), X2_2), composition(meet(X0_2, composition(X2_2, converse(X1_2))), meet(X1_2, composition(converse(X0_2), X2_2))))=composition(meet(X0_2, composition(X2_2, converse(X1_2))), meet(X1_2, composition(converse(X0_2), X2_2)))).
% 60.56/8.23 fof(def_top, axiom, ![X0_2]: top=join(X0_2, complement(X0_2))).
% 60.56/8.23 fof(def_zero, axiom, ![X0_2]: zero=meet(X0_2, complement(X0_2))).
% 60.56/8.23 fof(goals, conjecture, ![X0_2, X1_2, X2_2]: ((join(X0_2, one)=one & join(X1_2, one)=one) => (join(meet(composition(X0_2, X2_2), composition(X1_2, X2_2)), composition(meet(X0_2, X1_2), X2_2))=composition(meet(X0_2, X1_2), X2_2) & join(composition(meet(X0_2, X1_2), X2_2), meet(composition(X0_2, X2_2), composition(X1_2, X2_2)))=meet(composition(X0_2, X2_2), composition(X1_2, X2_2))))).
% 60.56/8.23 fof(maddux1_join_commutativity, axiom, ![X0_2, X1_2]: join(X0_2, X1_2)=join(X1_2, X0_2)).
% 60.56/8.23 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)).
% 60.56/8.23 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)))).
% 60.56/8.23 fof(maddux4_definiton_of_meet, axiom, ![X0_2, X1_2]: meet(X0_2, X1_2)=complement(join(complement(X0_2), complement(X1_2)))).
% 60.56/8.23
% 60.56/8.23 Now clausify the problem and encode Horn clauses using encoding 3 of
% 60.56/8.23 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 60.56/8.23 We repeatedly replace C & s=t => u=v by the two clauses:
% 60.56/8.23 fresh(y, y, x1...xn) = u
% 60.56/8.23 C => fresh(s, t, x1...xn) = v
% 60.56/8.23 where fresh is a fresh function symbol and x1..xn are the free
% 60.56/8.23 variables of u and v.
% 60.56/8.23 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 60.56/8.23 input problem has no model of domain size 1).
% 60.56/8.23
% 60.56/8.23 The encoding turns the above axioms into the following unit equations and goals:
% 60.56/8.23
% 60.56/8.23 Axiom 1 (composition_identity): composition(X, one) = X.
% 60.56/8.23 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 60.56/8.23 Axiom 3 (goals): join(x1, one) = one.
% 60.56/8.23 Axiom 4 (goals_1): join(x0, one) = one.
% 60.56/8.23 Axiom 5 (converse_idempotence): converse(converse(X)) = X.
% 60.56/8.23 Axiom 6 (def_zero): zero = meet(X, complement(X)).
% 60.56/8.23 Axiom 7 (def_top): top = join(X, complement(X)).
% 60.56/8.23 Axiom 8 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 60.56/8.23 Axiom 9 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 60.56/8.23 Axiom 10 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 60.56/8.23 Axiom 11 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 60.56/8.23 Axiom 12 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 60.56/8.23 Axiom 13 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 60.56/8.23 Axiom 14 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 60.56/8.23 Axiom 15 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 60.56/8.23 Axiom 16 (dedekind_law): join(meet(composition(X, Y), Z), composition(meet(X, composition(Z, converse(Y))), meet(Y, composition(converse(X), Z)))) = composition(meet(X, composition(Z, converse(Y))), meet(Y, composition(converse(X), Z))).
% 60.56/8.23
% 60.56/8.23 Lemma 17: complement(top) = zero.
% 60.56/8.23 Proof:
% 60.56/8.23 complement(top)
% 60.56/8.23 = { by axiom 7 (def_top) }
% 60.56/8.23 complement(join(complement(X), complement(complement(X))))
% 60.56/8.23 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 60.56/8.23 meet(X, complement(X))
% 60.56/8.23 = { by axiom 6 (def_zero) R->L }
% 60.56/8.23 zero
% 60.56/8.23
% 60.56/8.23 Lemma 18: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 60.56/8.23 Proof:
% 60.56/8.23 join(meet(X, Y), complement(join(complement(X), Y)))
% 60.56/8.23 = { by axiom 12 (maddux4_definiton_of_meet) }
% 60.56/8.23 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 60.56/8.23 = { by axiom 15 (maddux3_a_kind_of_de_Morgan) R->L }
% 60.56/8.23 X
% 60.56/8.23
% 60.56/8.23 Lemma 19: join(zero, meet(X, X)) = X.
% 60.56/8.23 Proof:
% 60.56/8.23 join(zero, meet(X, X))
% 60.56/8.23 = { by axiom 12 (maddux4_definiton_of_meet) }
% 60.56/8.23 join(zero, complement(join(complement(X), complement(X))))
% 60.56/8.23 = { by axiom 6 (def_zero) }
% 60.56/8.23 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 60.56/8.23 = { by lemma 18 }
% 60.56/8.23 X
% 60.56/8.23
% 60.56/8.23 Lemma 20: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 60.56/8.23 Proof:
% 60.56/8.23 converse(composition(converse(X), Y))
% 60.56/8.23 = { by axiom 8 (converse_multiplicativity) }
% 60.56/8.23 composition(converse(Y), converse(converse(X)))
% 60.56/8.24 = { by axiom 5 (converse_idempotence) }
% 60.56/8.24 composition(converse(Y), X)
% 60.56/8.24
% 60.56/8.24 Lemma 21: composition(converse(one), X) = X.
% 60.56/8.24 Proof:
% 60.56/8.24 composition(converse(one), X)
% 60.56/8.24 = { by lemma 20 R->L }
% 60.56/8.24 converse(composition(converse(X), one))
% 60.56/8.24 = { by axiom 1 (composition_identity) }
% 60.56/8.24 converse(converse(X))
% 60.56/8.24 = { by axiom 5 (converse_idempotence) }
% 60.56/8.24 X
% 60.56/8.24
% 60.56/8.24 Lemma 22: composition(one, X) = X.
% 60.56/8.24 Proof:
% 60.56/8.24 composition(one, X)
% 60.56/8.24 = { by lemma 21 R->L }
% 60.56/8.24 composition(converse(one), composition(one, X))
% 60.56/8.24 = { by axiom 9 (composition_associativity) }
% 60.56/8.24 composition(composition(converse(one), one), X)
% 60.56/8.24 = { by axiom 1 (composition_identity) }
% 60.56/8.24 composition(converse(one), X)
% 60.56/8.24 = { by lemma 21 }
% 60.56/8.24 X
% 60.56/8.24
% 60.56/8.24 Lemma 23: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 60.56/8.24 Proof:
% 60.56/8.24 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 60.56/8.24 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 60.56/8.24 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 60.56/8.24 = { by axiom 14 (converse_cancellativity) }
% 60.56/8.24 complement(X)
% 60.56/8.24
% 60.56/8.24 Lemma 24: join(complement(X), complement(X)) = complement(X).
% 60.56/8.24 Proof:
% 60.56/8.24 join(complement(X), complement(X))
% 60.56/8.24 = { by lemma 21 R->L }
% 60.56/8.24 join(complement(X), composition(converse(one), complement(X)))
% 60.56/8.24 = { by lemma 22 R->L }
% 60.56/8.24 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 60.56/8.24 = { by lemma 23 }
% 60.56/8.24 complement(X)
% 60.56/8.24
% 60.56/8.24 Lemma 25: join(zero, zero) = zero.
% 60.56/8.24 Proof:
% 60.56/8.24 join(zero, zero)
% 60.56/8.24 = { by lemma 17 R->L }
% 60.56/8.24 join(zero, complement(top))
% 60.56/8.24 = { by lemma 17 R->L }
% 60.56/8.24 join(complement(top), complement(top))
% 60.56/8.24 = { by lemma 24 }
% 60.56/8.24 complement(top)
% 60.56/8.24 = { by lemma 17 }
% 60.56/8.24 zero
% 60.56/8.24
% 60.56/8.24 Lemma 26: join(zero, join(zero, X)) = join(X, zero).
% 60.56/8.24 Proof:
% 60.56/8.24 join(zero, join(zero, X))
% 60.56/8.24 = { by axiom 11 (maddux2_join_associativity) }
% 60.56/8.24 join(join(zero, zero), X)
% 60.56/8.24 = { by lemma 25 }
% 60.56/8.24 join(zero, X)
% 60.56/8.24 = { by axiom 2 (maddux1_join_commutativity) }
% 60.56/8.24 join(X, zero)
% 60.56/8.24
% 60.56/8.24 Lemma 27: join(X, zero) = X.
% 60.56/8.24 Proof:
% 60.56/8.24 join(X, zero)
% 60.56/8.24 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 60.56/8.24 join(zero, X)
% 60.56/8.24 = { by lemma 19 R->L }
% 60.56/8.24 join(zero, join(zero, meet(X, X)))
% 60.56/8.24 = { by lemma 26 }
% 60.56/8.24 join(meet(X, X), zero)
% 60.56/8.24 = { by axiom 2 (maddux1_join_commutativity) }
% 60.56/8.24 join(zero, meet(X, X))
% 60.56/8.24 = { by lemma 19 }
% 60.56/8.24 X
% 60.56/8.24
% 60.56/8.24 Lemma 28: join(zero, X) = X.
% 60.56/8.24 Proof:
% 60.56/8.24 join(zero, X)
% 60.56/8.24 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 60.56/8.24 join(X, zero)
% 60.56/8.24 = { by lemma 27 }
% 60.56/8.24 X
% 60.56/8.24
% 60.56/8.24 Lemma 29: complement(zero) = top.
% 60.56/8.24 Proof:
% 60.56/8.24 complement(zero)
% 60.56/8.24 = { by lemma 28 R->L }
% 60.56/8.24 join(zero, complement(zero))
% 60.56/8.24 = { by axiom 7 (def_top) R->L }
% 60.56/8.24 top
% 60.56/8.24
% 60.56/8.24 Lemma 30: converse(one) = one.
% 60.56/8.24 Proof:
% 60.56/8.24 converse(one)
% 60.56/8.24 = { by axiom 1 (composition_identity) R->L }
% 60.56/8.24 composition(converse(one), one)
% 60.56/8.24 = { by lemma 21 }
% 60.56/8.24 one
% 60.56/8.24
% 60.56/8.24 Lemma 31: join(X, join(Y, complement(X))) = join(Y, top).
% 60.56/8.24 Proof:
% 60.56/8.24 join(X, join(Y, complement(X)))
% 60.56/8.24 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 60.56/8.24 join(X, join(complement(X), Y))
% 60.56/8.24 = { by axiom 11 (maddux2_join_associativity) }
% 60.56/8.24 join(join(X, complement(X)), Y)
% 60.56/8.24 = { by axiom 7 (def_top) R->L }
% 60.56/8.24 join(top, Y)
% 60.56/8.24 = { by axiom 2 (maddux1_join_commutativity) }
% 60.56/8.24 join(Y, top)
% 60.56/8.24
% 60.56/8.24 Lemma 32: join(top, complement(X)) = top.
% 60.56/8.24 Proof:
% 60.56/8.24 join(top, complement(X))
% 60.56/8.24 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 60.56/8.24 join(complement(X), top)
% 60.56/8.24 = { by lemma 31 R->L }
% 60.56/8.24 join(X, join(complement(X), complement(X)))
% 60.56/8.24 = { by lemma 24 }
% 60.56/8.24 join(X, complement(X))
% 60.56/8.24 = { by axiom 7 (def_top) R->L }
% 60.56/8.24 top
% 60.56/8.24
% 60.56/8.24 Lemma 33: join(Y, top) = join(X, top).
% 60.56/8.24 Proof:
% 60.56/8.24 join(Y, top)
% 60.56/8.24 = { by lemma 32 R->L }
% 60.56/8.24 join(Y, join(top, complement(Y)))
% 60.56/8.24 = { by lemma 31 }
% 60.56/8.24 join(top, top)
% 60.56/8.24 = { by lemma 31 R->L }
% 60.56/8.24 join(X, join(top, complement(X)))
% 60.56/8.24 = { by lemma 32 }
% 60.56/8.24 join(X, top)
% 60.56/8.24
% 60.56/8.24 Lemma 34: join(x1, join(one, X)) = join(X, one).
% 60.56/8.24 Proof:
% 60.56/8.24 join(x1, join(one, X))
% 60.56/8.24 = { by axiom 11 (maddux2_join_associativity) }
% 60.56/8.24 join(join(x1, one), X)
% 60.56/8.24 = { by axiom 3 (goals) }
% 60.56/8.24 join(one, X)
% 60.56/8.24 = { by axiom 2 (maddux1_join_commutativity) }
% 60.56/8.24 join(X, one)
% 60.56/8.24
% 60.56/8.24 Lemma 35: join(X, top) = top.
% 60.56/8.24 Proof:
% 60.56/8.24 join(X, top)
% 60.56/8.24 = { by lemma 33 }
% 60.56/8.24 join(x1, top)
% 60.56/8.24 = { by axiom 7 (def_top) }
% 60.56/8.24 join(x1, join(one, complement(one)))
% 60.56/8.24 = { by lemma 34 }
% 60.56/8.24 join(complement(one), one)
% 60.56/8.24 = { by axiom 2 (maddux1_join_commutativity) }
% 60.56/8.24 join(one, complement(one))
% 60.56/8.24 = { by axiom 7 (def_top) R->L }
% 60.56/8.24 top
% 60.56/8.24
% 60.56/8.24 Lemma 36: join(top, X) = top.
% 60.56/8.24 Proof:
% 60.56/8.24 join(top, X)
% 60.56/8.24 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 60.56/8.24 join(X, top)
% 60.56/8.24 = { by lemma 33 R->L }
% 60.56/8.24 join(Y, top)
% 60.56/8.24 = { by lemma 35 }
% 60.56/8.24 top
% 60.56/8.24
% 60.56/8.24 Lemma 37: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 60.56/8.24 Proof:
% 60.56/8.24 converse(join(X, converse(Y)))
% 60.56/8.24 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 60.56/8.24 converse(join(converse(Y), X))
% 60.56/8.24 = { by axiom 10 (converse_additivity) }
% 60.56/8.24 join(converse(converse(Y)), converse(X))
% 60.56/8.24 = { by axiom 5 (converse_idempotence) }
% 60.56/8.24 join(Y, converse(X))
% 60.56/8.24
% 60.56/8.24 Lemma 38: converse(top) = top.
% 60.56/8.24 Proof:
% 60.56/8.24 converse(top)
% 60.56/8.24 = { by lemma 36 R->L }
% 60.56/8.24 converse(join(top, converse(top)))
% 60.56/8.24 = { by lemma 37 }
% 60.56/8.24 join(top, converse(top))
% 60.56/8.24 = { by lemma 36 }
% 60.56/8.24 top
% 60.56/8.24
% 60.56/8.24 Lemma 39: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 60.56/8.24 Proof:
% 60.56/8.24 converse(join(converse(X), Y))
% 60.56/8.24 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 60.56/8.24 converse(join(Y, converse(X)))
% 60.56/8.24 = { by lemma 37 }
% 60.56/8.24 join(X, converse(Y))
% 60.56/8.24
% 60.56/8.24 Lemma 40: converse(zero) = zero.
% 60.56/8.24 Proof:
% 60.56/8.24 converse(zero)
% 60.56/8.24 = { by lemma 27 R->L }
% 60.56/8.24 join(converse(zero), zero)
% 60.56/8.24 = { by lemma 26 R->L }
% 60.56/8.24 join(zero, join(zero, converse(zero)))
% 60.56/8.24 = { by lemma 39 R->L }
% 60.56/8.24 join(zero, converse(join(converse(zero), zero)))
% 60.56/8.24 = { by lemma 27 }
% 60.56/8.24 join(zero, converse(converse(zero)))
% 60.56/8.24 = { by axiom 5 (converse_idempotence) }
% 60.56/8.24 join(zero, zero)
% 60.56/8.24 = { by lemma 25 }
% 60.56/8.24 zero
% 60.56/8.24
% 60.56/8.24 Lemma 41: complement(complement(X)) = meet(X, X).
% 60.56/8.24 Proof:
% 60.56/8.24 complement(complement(X))
% 60.56/8.24 = { by lemma 24 R->L }
% 60.56/8.24 complement(join(complement(X), complement(X)))
% 60.56/8.24 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 60.56/8.24 meet(X, X)
% 60.56/8.24
% 60.56/8.24 Lemma 42: complement(complement(X)) = X.
% 60.56/8.24 Proof:
% 60.56/8.24 complement(complement(X))
% 60.56/8.24 = { by lemma 28 R->L }
% 60.56/8.24 join(zero, complement(complement(X)))
% 60.56/8.24 = { by lemma 41 }
% 60.56/8.24 join(zero, meet(X, X))
% 60.56/8.24 = { by lemma 19 }
% 60.56/8.24 X
% 60.56/8.24
% 60.56/8.24 Lemma 43: meet(Y, X) = meet(X, Y).
% 60.56/8.24 Proof:
% 60.56/8.24 meet(Y, X)
% 60.56/8.24 = { by axiom 12 (maddux4_definiton_of_meet) }
% 60.56/8.24 complement(join(complement(Y), complement(X)))
% 60.56/8.24 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 60.56/8.24 complement(join(complement(X), complement(Y)))
% 60.56/8.24 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 60.56/8.24 meet(X, Y)
% 60.56/8.24
% 60.56/8.24 Lemma 44: join(meet(X, Y), meet(X, complement(Y))) = X.
% 60.56/8.24 Proof:
% 60.56/8.24 join(meet(X, Y), meet(X, complement(Y)))
% 60.56/8.24 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 60.56/8.24 join(meet(X, complement(Y)), meet(X, Y))
% 60.56/8.24 = { by axiom 12 (maddux4_definiton_of_meet) }
% 60.56/8.24 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 60.56/8.24 = { by lemma 18 }
% 60.56/8.24 X
% 60.56/8.24
% 60.56/8.24 Lemma 45: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 60.56/8.24 Proof:
% 60.56/8.24 join(meet(X, Y), meet(Y, complement(X)))
% 60.56/8.24 = { by lemma 43 }
% 60.56/8.24 join(meet(Y, X), meet(Y, complement(X)))
% 60.56/8.24 = { by lemma 44 }
% 60.56/8.24 Y
% 60.56/8.24
% 60.56/8.24 Lemma 46: join(meet(X, Y), meet(complement(X), Y)) = Y.
% 60.56/8.24 Proof:
% 60.56/8.24 join(meet(X, Y), meet(complement(X), Y))
% 60.56/8.24 = { by lemma 43 }
% 60.56/8.24 join(meet(X, Y), meet(Y, complement(X)))
% 60.56/8.24 = { by lemma 45 }
% 60.56/8.24 Y
% 60.56/8.24
% 60.56/8.24 Lemma 47: complement(join(zero, complement(X))) = meet(X, top).
% 60.56/8.24 Proof:
% 60.56/8.24 complement(join(zero, complement(X)))
% 60.56/8.24 = { by lemma 17 R->L }
% 60.56/8.24 complement(join(complement(top), complement(X)))
% 60.56/8.24 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 60.56/8.24 meet(top, X)
% 60.56/8.24 = { by lemma 43 R->L }
% 60.56/8.24 meet(X, top)
% 60.56/8.24
% 60.56/8.24 Lemma 48: meet(X, top) = X.
% 60.56/8.24 Proof:
% 60.56/8.24 meet(X, top)
% 60.56/8.24 = { by lemma 47 R->L }
% 60.56/8.24 complement(join(zero, complement(X)))
% 60.56/8.24 = { by lemma 28 }
% 60.56/8.24 complement(complement(X))
% 60.56/8.24 = { by lemma 42 }
% 60.56/8.24 X
% 60.56/8.24
% 60.56/8.24 Lemma 49: meet(top, X) = X.
% 60.56/8.24 Proof:
% 60.56/8.24 meet(top, X)
% 60.56/8.24 = { by lemma 43 }
% 60.56/8.24 meet(X, top)
% 60.56/8.24 = { by lemma 48 }
% 60.56/8.24 X
% 60.56/8.24
% 60.56/8.24 Lemma 50: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
% 60.56/8.24 Proof:
% 60.56/8.24 complement(join(complement(X), meet(Y, Z)))
% 60.56/8.24 = { by lemma 43 }
% 60.56/8.24 complement(join(complement(X), meet(Z, Y)))
% 60.56/8.24 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 60.56/8.24 complement(join(meet(Z, Y), complement(X)))
% 60.56/8.24 = { by axiom 12 (maddux4_definiton_of_meet) }
% 60.56/8.24 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 60.56/8.24 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 60.56/8.24 meet(join(complement(Z), complement(Y)), X)
% 60.56/8.24 = { by lemma 43 R->L }
% 60.56/8.24 meet(X, join(complement(Z), complement(Y)))
% 60.56/8.24 = { by axiom 2 (maddux1_join_commutativity) }
% 60.56/8.24 meet(X, join(complement(Y), complement(Z)))
% 60.56/8.24
% 60.56/8.24 Lemma 51: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 60.56/8.24 Proof:
% 60.56/8.24 join(complement(X), complement(Y))
% 60.56/8.24 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 60.56/8.24 join(complement(Y), complement(X))
% 60.56/8.24 = { by lemma 49 R->L }
% 60.56/8.24 meet(top, join(complement(Y), complement(X)))
% 60.56/8.24 = { by lemma 50 R->L }
% 60.56/8.24 complement(join(complement(top), meet(Y, X)))
% 60.56/8.24 = { by lemma 17 }
% 60.56/8.24 complement(join(zero, meet(Y, X)))
% 60.56/8.24 = { by lemma 28 }
% 60.56/8.24 complement(meet(Y, X))
% 60.56/8.24 = { by lemma 43 R->L }
% 60.56/8.24 complement(meet(X, Y))
% 60.56/8.24
% 60.56/8.24 Lemma 52: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 60.56/8.24 Proof:
% 60.56/8.24 complement(meet(X, complement(Y)))
% 60.56/8.24 = { by lemma 43 }
% 60.56/8.24 complement(meet(complement(Y), X))
% 60.56/8.24 = { by lemma 28 R->L }
% 60.56/8.24 complement(meet(join(zero, complement(Y)), X))
% 60.56/8.24 = { by lemma 51 R->L }
% 60.56/8.24 join(complement(join(zero, complement(Y))), complement(X))
% 60.56/8.24 = { by lemma 47 }
% 60.56/8.24 join(meet(Y, top), complement(X))
% 60.56/8.24 = { by lemma 48 }
% 60.56/8.24 join(Y, complement(X))
% 60.56/8.24
% 60.56/8.24 Lemma 53: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 60.56/8.24 Proof:
% 60.56/8.24 complement(join(X, complement(Y)))
% 60.56/8.24 = { by lemma 28 R->L }
% 60.56/8.24 complement(join(zero, join(X, complement(Y))))
% 60.56/8.24 = { by lemma 52 R->L }
% 60.56/8.24 complement(join(zero, complement(meet(Y, complement(X)))))
% 60.56/8.24 = { by lemma 47 }
% 60.56/8.24 meet(meet(Y, complement(X)), top)
% 60.56/8.24 = { by lemma 48 }
% 60.56/8.24 meet(Y, complement(X))
% 60.56/8.24
% 60.56/8.24 Lemma 54: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 60.56/8.24 Proof:
% 60.56/8.24 complement(join(complement(X), Y))
% 60.56/8.24 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 60.56/8.24 complement(join(Y, complement(X)))
% 60.56/8.24 = { by lemma 53 }
% 60.56/8.24 meet(X, complement(Y))
% 60.56/8.24
% 60.56/8.24 Lemma 55: join(X, converse(complement(converse(X)))) = top.
% 60.56/8.24 Proof:
% 60.56/8.24 join(X, converse(complement(converse(X))))
% 60.56/8.24 = { by lemma 39 R->L }
% 60.56/8.24 converse(join(converse(X), complement(converse(X))))
% 60.56/8.24 = { by axiom 7 (def_top) R->L }
% 60.56/8.24 converse(top)
% 60.56/8.24 = { by lemma 38 }
% 60.56/8.24 top
% 60.56/8.24
% 60.56/8.24 Lemma 56: join(X, composition(Y, X)) = composition(join(Y, one), X).
% 60.56/8.24 Proof:
% 60.56/8.24 join(X, composition(Y, X))
% 60.56/8.24 = { by lemma 22 R->L }
% 60.56/8.24 join(composition(one, X), composition(Y, X))
% 60.56/8.24 = { by axiom 13 (composition_distributivity) R->L }
% 60.56/8.24 composition(join(one, Y), X)
% 60.56/8.24 = { by axiom 2 (maddux1_join_commutativity) }
% 60.56/8.24 composition(join(Y, one), X)
% 60.56/8.24
% 60.56/8.24 Lemma 57: composition(top, zero) = zero.
% 60.56/8.24 Proof:
% 60.56/8.24 composition(top, zero)
% 60.56/8.24 = { by lemma 17 R->L }
% 60.56/8.24 composition(top, complement(top))
% 60.56/8.24 = { by lemma 36 R->L }
% 60.56/8.24 composition(join(top, one), complement(top))
% 60.56/8.24 = { by lemma 38 R->L }
% 60.56/8.24 composition(join(converse(top), one), complement(top))
% 60.56/8.24 = { by lemma 56 R->L }
% 60.56/8.24 join(complement(top), composition(converse(top), complement(top)))
% 60.56/8.24 = { by lemma 36 R->L }
% 60.56/8.24 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 60.56/8.24 = { by lemma 56 }
% 60.56/8.24 join(complement(top), composition(converse(top), complement(composition(join(top, one), top))))
% 60.56/8.24 = { by lemma 36 }
% 60.56/8.24 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 60.56/8.24 = { by lemma 23 }
% 60.56/8.24 complement(top)
% 60.56/8.24 = { by lemma 17 }
% 60.56/8.25 zero
% 60.56/8.25
% 60.56/8.25 Lemma 58: composition(X, zero) = zero.
% 60.56/8.25 Proof:
% 60.56/8.25 composition(X, zero)
% 60.56/8.25 = { by lemma 28 R->L }
% 60.56/8.25 join(zero, composition(X, zero))
% 60.56/8.25 = { by lemma 57 R->L }
% 60.56/8.25 join(composition(top, zero), composition(X, zero))
% 60.56/8.25 = { by axiom 13 (composition_distributivity) R->L }
% 60.56/8.25 composition(join(top, X), zero)
% 60.56/8.25 = { by lemma 36 }
% 60.56/8.25 composition(top, zero)
% 60.56/8.25 = { by lemma 57 }
% 60.56/8.25 zero
% 60.56/8.25
% 60.56/8.25 Lemma 59: composition(zero, X) = zero.
% 60.56/8.25 Proof:
% 60.56/8.25 composition(zero, X)
% 60.56/8.25 = { by lemma 40 R->L }
% 60.56/8.25 composition(converse(zero), X)
% 60.56/8.25 = { by lemma 20 R->L }
% 60.56/8.25 converse(composition(converse(X), zero))
% 60.56/8.25 = { by lemma 58 }
% 60.56/8.25 converse(zero)
% 60.56/8.25 = { by lemma 40 }
% 60.56/8.25 zero
% 60.56/8.25
% 60.56/8.25 Lemma 60: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 60.56/8.25 Proof:
% 60.56/8.25 meet(complement(X), complement(Y))
% 60.56/8.25 = { by lemma 43 }
% 60.56/8.25 meet(complement(Y), complement(X))
% 60.56/8.25 = { by lemma 28 R->L }
% 60.56/8.25 meet(join(zero, complement(Y)), complement(X))
% 60.56/8.25 = { by lemma 53 R->L }
% 60.56/8.25 complement(join(X, complement(join(zero, complement(Y)))))
% 60.56/8.25 = { by lemma 47 }
% 60.56/8.25 complement(join(X, meet(Y, top)))
% 60.56/8.25 = { by lemma 48 }
% 60.56/8.25 complement(join(X, Y))
% 60.56/8.25
% 60.56/8.25 Lemma 61: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 60.56/8.25 Proof:
% 60.56/8.25 complement(meet(complement(X), Y))
% 60.56/8.25 = { by lemma 43 }
% 60.56/8.25 complement(meet(Y, complement(X)))
% 60.56/8.25 = { by lemma 52 }
% 60.56/8.25 join(X, complement(Y))
% 60.56/8.25
% 60.56/8.25 Lemma 62: join(X, complement(meet(X, Y))) = top.
% 60.56/8.25 Proof:
% 60.56/8.25 join(X, complement(meet(X, Y)))
% 60.56/8.25 = { by lemma 43 }
% 60.56/8.25 join(X, complement(meet(Y, X)))
% 60.56/8.25 = { by lemma 51 R->L }
% 60.56/8.25 join(X, join(complement(Y), complement(X)))
% 60.56/8.25 = { by lemma 31 }
% 60.56/8.25 join(complement(Y), top)
% 60.56/8.25 = { by lemma 35 }
% 60.56/8.25 top
% 60.56/8.25
% 60.56/8.25 Lemma 63: meet(X, join(X, complement(Y))) = X.
% 60.56/8.25 Proof:
% 60.56/8.25 meet(X, join(X, complement(Y)))
% 60.56/8.25 = { by lemma 27 R->L }
% 60.56/8.25 join(meet(X, join(X, complement(Y))), zero)
% 60.56/8.25 = { by lemma 17 R->L }
% 60.56/8.25 join(meet(X, join(X, complement(Y))), complement(top))
% 60.56/8.25 = { by lemma 61 R->L }
% 60.56/8.25 join(meet(X, complement(meet(complement(X), Y))), complement(top))
% 60.56/8.25 = { by lemma 62 R->L }
% 60.56/8.25 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(meet(complement(X), Y)))))
% 60.56/8.25 = { by lemma 18 }
% 60.56/8.25 X
% 60.56/8.25
% 60.56/8.25 Lemma 64: join(X, meet(X, Y)) = X.
% 60.56/8.25 Proof:
% 60.56/8.25 join(X, meet(X, Y))
% 60.56/8.25 = { by axiom 12 (maddux4_definiton_of_meet) }
% 60.56/8.25 join(X, complement(join(complement(X), complement(Y))))
% 60.56/8.25 = { by lemma 61 R->L }
% 60.56/8.25 complement(meet(complement(X), join(complement(X), complement(Y))))
% 60.56/8.25 = { by lemma 63 }
% 60.56/8.25 complement(complement(X))
% 60.56/8.25 = { by lemma 42 }
% 60.56/8.25 X
% 60.56/8.25
% 60.56/8.25 Lemma 65: join(complement(X), meet(X, Y)) = join(Y, complement(X)).
% 60.56/8.25 Proof:
% 60.56/8.25 join(complement(X), meet(X, Y))
% 60.56/8.25 = { by lemma 43 }
% 60.56/8.25 join(complement(X), meet(Y, X))
% 60.56/8.25 = { by lemma 64 R->L }
% 60.56/8.25 join(join(complement(X), meet(complement(X), Y)), meet(Y, X))
% 60.56/8.25 = { by axiom 11 (maddux2_join_associativity) R->L }
% 60.56/8.25 join(complement(X), join(meet(complement(X), Y), meet(Y, X)))
% 60.56/8.25 = { by axiom 2 (maddux1_join_commutativity) }
% 60.56/8.25 join(complement(X), join(meet(Y, X), meet(complement(X), Y)))
% 60.56/8.25 = { by lemma 43 }
% 60.56/8.26 join(complement(X), join(meet(Y, X), meet(Y, complement(X))))
% 60.56/8.26 = { by lemma 44 }
% 60.56/8.26 join(complement(X), Y)
% 60.56/8.26 = { by axiom 2 (maddux1_join_commutativity) }
% 60.56/8.26 join(Y, complement(X))
% 60.56/8.26
% 60.56/8.26 Lemma 66: meet(X, complement(meet(X, Y))) = meet(X, complement(Y)).
% 60.56/8.26 Proof:
% 60.56/8.26 meet(X, complement(meet(X, Y)))
% 60.56/8.26 = { by lemma 51 R->L }
% 60.56/8.26 meet(X, join(complement(X), complement(Y)))
% 60.56/8.26 = { by lemma 50 R->L }
% 60.56/8.26 complement(join(complement(X), meet(X, Y)))
% 60.56/8.26 = { by lemma 65 }
% 60.56/8.26 complement(join(Y, complement(X)))
% 60.56/8.26 = { by lemma 53 }
% 60.56/8.26 meet(X, complement(Y))
% 60.56/8.26
% 60.56/8.26 Lemma 67: join(complement(one), converse(complement(one))) = complement(one).
% 60.56/8.26 Proof:
% 60.56/8.26 join(complement(one), converse(complement(one)))
% 60.56/8.26 = { by lemma 48 R->L }
% 60.56/8.26 join(complement(one), converse(meet(complement(one), top)))
% 60.56/8.26 = { by lemma 55 R->L }
% 60.56/8.26 join(complement(one), converse(meet(complement(one), join(one, converse(complement(converse(one)))))))
% 60.56/8.26 = { by lemma 30 }
% 60.56/8.26 join(complement(one), converse(meet(complement(one), join(one, converse(complement(one))))))
% 60.56/8.26 = { by lemma 43 }
% 60.56/8.26 join(complement(one), converse(meet(join(one, converse(complement(one))), complement(one))))
% 60.56/8.26 = { by lemma 53 R->L }
% 60.56/8.26 join(complement(one), converse(complement(join(one, complement(join(one, converse(complement(one))))))))
% 60.56/8.26 = { by lemma 60 R->L }
% 60.56/8.26 join(complement(one), converse(meet(complement(one), complement(complement(join(one, converse(complement(one))))))))
% 60.56/8.26 = { by lemma 60 R->L }
% 60.56/8.26 join(complement(one), converse(meet(complement(one), complement(meet(complement(one), complement(converse(complement(one))))))))
% 60.56/8.26 = { by lemma 66 }
% 60.56/8.26 join(complement(one), converse(meet(complement(one), complement(complement(converse(complement(one)))))))
% 60.56/8.26 = { by lemma 60 }
% 60.56/8.26 join(complement(one), converse(complement(join(one, complement(converse(complement(one)))))))
% 60.56/8.26 = { by lemma 53 }
% 60.56/8.26 join(complement(one), converse(meet(converse(complement(one)), complement(one))))
% 60.56/8.26 = { by lemma 39 R->L }
% 60.56/8.26 converse(join(converse(complement(one)), meet(converse(complement(one)), complement(one))))
% 60.56/8.26 = { by lemma 64 }
% 60.56/8.26 converse(converse(complement(one)))
% 60.56/8.26 = { by axiom 5 (converse_idempotence) }
% 60.56/8.26 complement(one)
% 60.56/8.26
% 60.56/8.26 Lemma 68: converse(complement(one)) = complement(one).
% 60.56/8.26 Proof:
% 60.56/8.26 converse(complement(one))
% 60.56/8.26 = { by lemma 67 R->L }
% 60.56/8.26 converse(join(complement(one), converse(complement(one))))
% 60.56/8.26 = { by lemma 37 }
% 60.56/8.26 join(complement(one), converse(complement(one)))
% 60.56/8.26 = { by lemma 67 }
% 60.56/8.26 complement(one)
% 60.56/8.26
% 60.56/8.26 Lemma 69: join(complement(one), composition(converse(X), complement(X))) = complement(one).
% 60.56/8.26 Proof:
% 60.56/8.26 join(complement(one), composition(converse(X), complement(X)))
% 60.56/8.26 = { by axiom 1 (composition_identity) R->L }
% 60.56/8.26 join(complement(one), composition(converse(X), complement(composition(X, one))))
% 60.56/8.26 = { by lemma 23 }
% 60.56/8.26 complement(one)
% 60.56/8.26
% 60.56/8.26 Lemma 70: converse(join(X, composition(converse(Y), Z))) = join(converse(X), composition(converse(Z), Y)).
% 60.56/8.26 Proof:
% 60.56/8.26 converse(join(X, composition(converse(Y), Z)))
% 60.56/8.26 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 60.56/8.26 converse(join(composition(converse(Y), Z), X))
% 60.56/8.26 = { by axiom 10 (converse_additivity) }
% 60.56/8.26 join(converse(composition(converse(Y), Z)), converse(X))
% 60.56/8.26 = { by lemma 20 }
% 60.56/8.26 join(composition(converse(Z), Y), converse(X))
% 60.56/8.26 = { by axiom 2 (maddux1_join_commutativity) }
% 60.56/8.26 join(converse(X), composition(converse(Z), Y))
% 60.56/8.26
% 60.56/8.26 Lemma 71: meet(X, complement(join(Y, X))) = zero.
% 60.56/8.26 Proof:
% 60.56/8.26 meet(X, complement(join(Y, X)))
% 60.56/8.26 = { by lemma 60 R->L }
% 60.56/8.26 meet(X, meet(complement(Y), complement(X)))
% 60.56/8.26 = { by lemma 43 }
% 60.56/8.26 meet(X, meet(complement(X), complement(Y)))
% 60.56/8.26 = { by axiom 12 (maddux4_definiton_of_meet) }
% 60.56/8.26 complement(join(complement(X), complement(meet(complement(X), complement(Y)))))
% 60.56/8.26 = { by lemma 62 }
% 60.56/8.26 complement(top)
% 60.56/8.26 = { by lemma 17 }
% 60.56/8.26 zero
% 60.56/8.26
% 60.56/8.26 Lemma 72: meet(one, composition(converse(complement(X)), X)) = zero.
% 60.56/8.26 Proof:
% 60.56/8.26 meet(one, composition(converse(complement(X)), X))
% 60.56/8.26 = { by lemma 43 }
% 60.56/8.26 meet(composition(converse(complement(X)), X), one)
% 60.56/8.26 = { by lemma 42 R->L }
% 60.56/8.26 meet(composition(converse(complement(X)), X), complement(complement(one)))
% 60.56/8.26 = { by lemma 68 R->L }
% 60.56/8.26 meet(composition(converse(complement(X)), X), complement(converse(complement(one))))
% 60.56/8.26 = { by lemma 69 R->L }
% 60.56/8.26 meet(composition(converse(complement(X)), X), complement(converse(join(complement(one), composition(converse(converse(converse(X))), complement(converse(converse(X))))))))
% 60.56/8.26 = { by axiom 5 (converse_idempotence) }
% 60.56/8.26 meet(composition(converse(complement(X)), X), complement(converse(join(complement(one), composition(converse(X), complement(converse(converse(X))))))))
% 60.56/8.26 = { by lemma 70 }
% 60.56/8.27 meet(composition(converse(complement(X)), X), complement(join(converse(complement(one)), composition(converse(complement(converse(converse(X)))), X))))
% 60.56/8.27 = { by lemma 68 }
% 60.56/8.27 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(complement(converse(converse(X)))), X))))
% 60.56/8.27 = { by axiom 5 (converse_idempotence) }
% 60.56/8.27 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(complement(X)), X))))
% 60.56/8.27 = { by lemma 71 }
% 60.56/8.27 zero
% 60.56/8.27
% 60.56/8.27 Lemma 73: converse(complement(converse(X))) = complement(X).
% 60.56/8.27 Proof:
% 60.56/8.27 converse(complement(converse(X)))
% 60.56/8.27 = { by lemma 42 R->L }
% 60.56/8.27 converse(complement(converse(complement(complement(X)))))
% 60.56/8.27 = { by lemma 46 R->L }
% 60.56/8.27 join(meet(X, converse(complement(converse(complement(complement(X)))))), meet(complement(X), converse(complement(converse(complement(complement(X)))))))
% 60.56/8.27 = { by lemma 42 R->L }
% 60.56/8.27 join(meet(X, converse(complement(converse(complement(complement(X)))))), meet(complement(X), complement(complement(converse(complement(converse(complement(complement(X)))))))))
% 60.56/8.27 = { by lemma 54 R->L }
% 60.56/8.27 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(join(complement(complement(X)), complement(converse(complement(converse(complement(complement(X)))))))))
% 60.56/8.27 = { by lemma 49 R->L }
% 60.56/8.27 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(meet(top, join(complement(complement(X)), complement(converse(complement(converse(complement(complement(X))))))))))
% 60.56/8.27 = { by lemma 55 R->L }
% 60.56/8.27 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(meet(join(complement(complement(X)), converse(complement(converse(complement(complement(X)))))), join(complement(complement(X)), complement(converse(complement(converse(complement(complement(X))))))))))
% 60.56/8.27 = { by lemma 43 }
% 60.56/8.27 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(meet(join(complement(complement(X)), complement(converse(complement(converse(complement(complement(X))))))), join(complement(complement(X)), converse(complement(converse(complement(complement(X)))))))))
% 60.56/8.27 = { by lemma 42 R->L }
% 60.56/8.27 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(meet(join(complement(complement(X)), complement(converse(complement(converse(complement(complement(X))))))), join(complement(complement(X)), complement(complement(converse(complement(converse(complement(complement(X)))))))))))
% 60.56/8.27 = { by lemma 52 R->L }
% 60.56/8.27 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(meet(complement(meet(converse(complement(converse(complement(complement(X))))), complement(complement(complement(X))))), join(complement(complement(X)), complement(complement(converse(complement(converse(complement(complement(X)))))))))))
% 60.56/8.27 = { by lemma 43 }
% 60.56/8.27 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(meet(join(complement(complement(X)), complement(complement(converse(complement(converse(complement(complement(X)))))))), complement(meet(converse(complement(converse(complement(complement(X))))), complement(complement(complement(X))))))))
% 60.56/8.27 = { by lemma 53 R->L }
% 60.56/8.27 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(join(meet(converse(complement(converse(complement(complement(X))))), complement(complement(complement(X)))), complement(join(complement(complement(X)), complement(complement(converse(complement(converse(complement(complement(X)))))))))))))
% 60.56/8.27 = { by lemma 53 }
% 60.56/8.27 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(join(meet(converse(complement(converse(complement(complement(X))))), complement(complement(complement(X)))), meet(complement(converse(complement(converse(complement(complement(X)))))), complement(complement(complement(X))))))))
% 60.56/8.27 = { by lemma 46 }
% 60.56/8.27 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(complement(complement(complement(X))))))
% 60.56/8.27 = { by lemma 42 }
% 60.56/8.27 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(complement(X))))
% 60.56/8.27 = { by lemma 42 }
% 60.56/8.27 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(X))
% 60.56/8.27 = { by lemma 42 }
% 60.56/8.27 join(meet(X, converse(complement(converse(X)))), complement(X))
% 60.56/8.27 = { by lemma 27 R->L }
% 60.56/8.27 join(join(meet(X, converse(complement(converse(X)))), zero), complement(X))
% 60.56/8.27 = { by lemma 59 R->L }
% 60.56/8.27 join(join(meet(X, converse(complement(converse(X)))), composition(zero, meet(X, composition(converse(one), converse(complement(converse(X))))))), complement(X))
% 60.56/8.27 = { by lemma 22 R->L }
% 60.56/8.27 join(join(meet(composition(one, X), converse(complement(converse(X)))), composition(zero, meet(X, composition(converse(one), converse(complement(converse(X))))))), complement(X))
% 60.56/8.27 = { by lemma 72 R->L }
% 60.56/8.27 join(join(meet(composition(one, X), converse(complement(converse(X)))), composition(meet(one, composition(converse(complement(converse(X))), converse(X))), meet(X, composition(converse(one), converse(complement(converse(X))))))), complement(X))
% 60.56/8.27 = { by axiom 16 (dedekind_law) }
% 60.56/8.27 join(composition(meet(one, composition(converse(complement(converse(X))), converse(X))), meet(X, composition(converse(one), converse(complement(converse(X)))))), complement(X))
% 60.56/8.27 = { by lemma 72 }
% 60.56/8.27 join(composition(zero, meet(X, composition(converse(one), converse(complement(converse(X)))))), complement(X))
% 60.56/8.27 = { by lemma 59 }
% 60.56/8.27 join(zero, complement(X))
% 60.56/8.27 = { by lemma 28 }
% 60.56/8.27 complement(X)
% 60.56/8.27
% 60.56/8.27 Lemma 74: complement(converse(X)) = converse(complement(X)).
% 60.56/8.27 Proof:
% 60.56/8.27 complement(converse(X))
% 60.56/8.27 = { by axiom 5 (converse_idempotence) R->L }
% 60.56/8.27 converse(converse(complement(converse(X))))
% 60.56/8.27 = { by lemma 73 }
% 60.56/8.27 converse(complement(X))
% 60.56/8.27
% 60.56/8.27 Lemma 75: join(complement(X), meet(Y, X)) = join(Y, complement(X)).
% 60.56/8.27 Proof:
% 60.56/8.27 join(complement(X), meet(Y, X))
% 60.56/8.27 = { by lemma 43 }
% 60.56/8.27 join(complement(X), meet(X, Y))
% 60.56/8.27 = { by lemma 65 }
% 60.56/8.27 join(Y, complement(X))
% 60.56/8.27
% 60.56/8.27 Lemma 76: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 60.56/8.27 Proof:
% 60.56/8.27 meet(Y, meet(X, Z))
% 60.56/8.27 = { by lemma 43 }
% 60.56/8.27 meet(Y, meet(Z, X))
% 60.56/8.27 = { by lemma 48 R->L }
% 60.56/8.27 meet(meet(Y, meet(Z, X)), top)
% 60.56/8.27 = { by lemma 47 R->L }
% 60.56/8.27 complement(join(zero, complement(meet(Y, meet(Z, X)))))
% 60.56/8.27 = { by lemma 43 }
% 60.56/8.27 complement(join(zero, complement(meet(Y, meet(X, Z)))))
% 60.56/8.27 = { by axiom 12 (maddux4_definiton_of_meet) }
% 60.56/8.27 complement(join(zero, complement(meet(Y, complement(join(complement(X), complement(Z)))))))
% 60.56/8.27 = { by lemma 52 }
% 60.56/8.27 complement(join(zero, join(join(complement(X), complement(Z)), complement(Y))))
% 60.56/8.27 = { by axiom 11 (maddux2_join_associativity) R->L }
% 60.56/8.27 complement(join(zero, join(complement(X), join(complement(Z), complement(Y)))))
% 60.56/8.27 = { by lemma 51 }
% 60.56/8.27 complement(join(zero, join(complement(X), complement(meet(Z, Y)))))
% 60.56/8.27 = { by lemma 51 }
% 60.56/8.27 complement(join(zero, complement(meet(X, meet(Z, Y)))))
% 60.56/8.27 = { by lemma 43 R->L }
% 60.56/8.27 complement(join(zero, complement(meet(X, meet(Y, Z)))))
% 60.56/8.27 = { by lemma 47 }
% 60.56/8.27 meet(meet(X, meet(Y, Z)), top)
% 60.56/8.27 = { by lemma 48 }
% 60.56/8.27 meet(X, meet(Y, Z))
% 60.56/8.27
% 60.56/8.27 Lemma 77: meet(X, join(X, Y)) = X.
% 60.56/8.27 Proof:
% 60.56/8.27 meet(X, join(X, Y))
% 60.56/8.27 = { by lemma 48 R->L }
% 60.56/8.27 meet(X, join(X, meet(Y, top)))
% 60.56/8.27 = { by lemma 47 R->L }
% 60.56/8.27 meet(X, join(X, complement(join(zero, complement(Y)))))
% 60.56/8.27 = { by lemma 63 }
% 60.56/8.27 X
% 60.56/8.27
% 60.56/8.27 Lemma 78: meet(X, meet(Y, composition(top, X))) = meet(X, Y).
% 60.56/8.27 Proof:
% 60.56/8.27 meet(X, meet(Y, composition(top, X)))
% 60.56/8.27 = { by lemma 76 }
% 60.56/8.27 meet(Y, meet(X, composition(top, X)))
% 60.56/8.27 = { by lemma 36 R->L }
% 60.56/8.27 meet(Y, meet(X, composition(join(top, one), X)))
% 60.56/8.27 = { by lemma 56 R->L }
% 60.56/8.27 meet(Y, meet(X, join(X, composition(top, X))))
% 60.56/8.27 = { by lemma 77 }
% 60.56/8.27 meet(Y, X)
% 60.56/8.27 = { by lemma 43 R->L }
% 60.56/8.27 meet(X, Y)
% 60.56/8.27
% 60.56/8.27 Lemma 79: meet(X, join(complement(X), Y)) = meet(X, Y).
% 60.56/8.27 Proof:
% 60.56/8.27 meet(X, join(complement(X), Y))
% 60.56/8.27 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 60.56/8.27 meet(X, join(Y, complement(X)))
% 60.56/8.27 = { by lemma 52 R->L }
% 60.56/8.27 meet(X, complement(meet(X, complement(Y))))
% 60.56/8.27 = { by lemma 66 }
% 60.56/8.27 meet(X, complement(complement(Y)))
% 60.56/8.27 = { by lemma 42 }
% 60.56/8.27 meet(X, Y)
% 60.56/8.27
% 60.56/8.27 Lemma 80: converse(join(X, one)) = join(one, converse(X)).
% 60.56/8.27 Proof:
% 60.56/8.27 converse(join(X, one))
% 60.56/8.27 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 60.56/8.27 converse(join(one, X))
% 60.56/8.27 = { by axiom 10 (converse_additivity) }
% 60.56/8.27 join(converse(one), converse(X))
% 60.56/8.27 = { by lemma 30 }
% 60.56/8.27 join(one, converse(X))
% 60.56/8.27
% 60.56/8.27 Lemma 81: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 60.56/8.27 Proof:
% 60.56/8.27 converse(composition(X, converse(Y)))
% 60.56/8.27 = { by axiom 8 (converse_multiplicativity) }
% 60.56/8.27 composition(converse(converse(Y)), converse(X))
% 60.56/8.27 = { by axiom 5 (converse_idempotence) }
% 60.56/8.27 composition(Y, converse(X))
% 60.56/8.27
% 60.56/8.27 Lemma 82: join(composition(X, join(Z, one)), Y) = join(X, join(Y, composition(X, Z))).
% 60.56/8.27 Proof:
% 60.56/8.27 join(composition(X, join(Z, one)), Y)
% 60.56/8.27 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 60.56/8.27 join(composition(X, join(one, Z)), Y)
% 60.56/8.27 = { by axiom 5 (converse_idempotence) R->L }
% 60.56/8.27 join(composition(X, join(one, converse(converse(Z)))), Y)
% 60.56/8.27 = { by lemma 80 R->L }
% 60.56/8.27 join(composition(X, converse(join(converse(Z), one))), Y)
% 60.56/8.27 = { by lemma 81 R->L }
% 60.56/8.27 join(converse(composition(join(converse(Z), one), converse(X))), Y)
% 60.56/8.27 = { by lemma 56 R->L }
% 60.56/8.27 join(converse(join(converse(X), composition(converse(Z), converse(X)))), Y)
% 60.56/8.27 = { by lemma 39 }
% 60.56/8.27 join(join(X, converse(composition(converse(Z), converse(X)))), Y)
% 60.56/8.27 = { by lemma 81 }
% 60.56/8.27 join(join(X, composition(X, converse(converse(Z)))), Y)
% 60.56/8.27 = { by axiom 5 (converse_idempotence) }
% 60.56/8.27 join(join(X, composition(X, Z)), Y)
% 60.56/8.27 = { by axiom 11 (maddux2_join_associativity) R->L }
% 60.56/8.27 join(X, join(composition(X, Z), Y))
% 60.56/8.27 = { by axiom 2 (maddux1_join_commutativity) }
% 60.56/8.27 join(X, join(Y, composition(X, Z)))
% 60.56/8.27
% 60.56/8.27 Lemma 83: meet(x1, meet(one, X)) = meet(X, x1).
% 60.56/8.27 Proof:
% 60.56/8.27 meet(x1, meet(one, X))
% 60.56/8.27 = { by lemma 43 }
% 60.56/8.27 meet(x1, meet(X, one))
% 60.56/8.27 = { by lemma 76 }
% 60.56/8.27 meet(X, meet(x1, one))
% 60.56/8.27 = { by axiom 12 (maddux4_definiton_of_meet) }
% 60.56/8.27 meet(X, complement(join(complement(x1), complement(one))))
% 60.56/8.27 = { by lemma 28 R->L }
% 60.56/8.27 meet(X, join(zero, complement(join(complement(x1), complement(one)))))
% 60.56/8.27 = { by lemma 17 R->L }
% 60.56/8.27 meet(X, join(complement(top), complement(join(complement(x1), complement(one)))))
% 60.56/8.27 = { by lemma 35 R->L }
% 60.56/8.27 meet(X, join(complement(join(one, top)), complement(join(complement(x1), complement(one)))))
% 60.56/8.27 = { by lemma 31 R->L }
% 60.56/8.27 meet(X, join(complement(join(x1, join(one, complement(x1)))), complement(join(complement(x1), complement(one)))))
% 60.56/8.27 = { by lemma 34 }
% 60.56/8.27 meet(X, join(complement(join(complement(x1), one)), complement(join(complement(x1), complement(one)))))
% 60.56/8.27 = { by axiom 2 (maddux1_join_commutativity) }
% 60.56/8.27 meet(X, join(complement(join(one, complement(x1))), complement(join(complement(x1), complement(one)))))
% 60.56/8.27 = { by lemma 53 }
% 60.56/8.27 meet(X, join(meet(x1, complement(one)), complement(join(complement(x1), complement(one)))))
% 60.56/8.27 = { by lemma 18 }
% 60.56/8.27 meet(X, x1)
% 60.56/8.27
% 60.56/8.27 Lemma 84: meet(one, composition(converse(X), complement(X))) = zero.
% 60.56/8.27 Proof:
% 60.56/8.27 meet(one, composition(converse(X), complement(X)))
% 60.56/8.27 = { by lemma 43 }
% 60.56/8.27 meet(composition(converse(X), complement(X)), one)
% 60.56/8.27 = { by lemma 42 R->L }
% 60.56/8.27 meet(composition(converse(X), complement(X)), complement(complement(one)))
% 60.56/8.27 = { by lemma 69 R->L }
% 60.56/8.27 meet(composition(converse(X), complement(X)), complement(join(complement(one), composition(converse(X), complement(X)))))
% 60.56/8.27 = { by lemma 71 }
% 60.56/8.27 zero
% 60.56/8.27
% 60.56/8.27 Lemma 85: meet(X, zero) = zero.
% 60.56/8.27 Proof:
% 60.56/8.27 meet(X, zero)
% 60.56/8.27 = { by lemma 43 }
% 60.56/8.27 meet(zero, X)
% 60.56/8.27 = { by axiom 12 (maddux4_definiton_of_meet) }
% 60.56/8.27 complement(join(complement(zero), complement(X)))
% 60.56/8.27 = { by lemma 29 }
% 60.56/8.27 complement(join(top, complement(X)))
% 60.56/8.27 = { by lemma 32 }
% 60.56/8.27 complement(top)
% 60.56/8.27 = { by lemma 17 }
% 60.56/8.27 zero
% 60.56/8.27
% 60.56/8.27 Lemma 86: meet(x1, composition(converse(X), complement(X))) = zero.
% 60.56/8.27 Proof:
% 60.56/8.27 meet(x1, composition(converse(X), complement(X)))
% 60.56/8.27 = { by lemma 43 }
% 60.56/8.27 meet(composition(converse(X), complement(X)), x1)
% 60.56/8.27 = { by lemma 83 R->L }
% 60.56/8.27 meet(x1, meet(one, composition(converse(X), complement(X))))
% 60.56/8.27 = { by lemma 84 }
% 60.56/8.27 meet(x1, zero)
% 60.56/8.27 = { by lemma 85 }
% 60.56/8.27 zero
% 60.56/8.27
% 60.56/8.27 Lemma 87: meet(X, composition(top, x1)) = composition(X, x1).
% 60.56/8.27 Proof:
% 60.56/8.27 meet(X, composition(top, x1))
% 60.56/8.27 = { by lemma 79 R->L }
% 60.56/8.27 meet(X, join(complement(X), composition(top, x1)))
% 60.56/8.28 = { by axiom 7 (def_top) }
% 60.56/8.28 meet(X, join(complement(X), composition(join(complement(X), complement(complement(X))), x1)))
% 60.56/8.28 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 60.56/8.28 meet(X, join(complement(X), composition(join(complement(complement(X)), complement(X)), x1)))
% 60.56/8.28 = { by axiom 13 (composition_distributivity) }
% 60.56/8.28 meet(X, join(complement(X), join(composition(complement(complement(X)), x1), composition(complement(X), x1))))
% 60.56/8.28 = { by lemma 82 R->L }
% 60.56/8.28 meet(X, join(composition(complement(X), join(x1, one)), composition(complement(complement(X)), x1)))
% 60.56/8.28 = { by axiom 3 (goals) }
% 60.56/8.28 meet(X, join(composition(complement(X), one), composition(complement(complement(X)), x1)))
% 60.56/8.28 = { by axiom 1 (composition_identity) }
% 60.56/8.28 meet(X, join(complement(X), composition(complement(complement(X)), x1)))
% 60.56/8.28 = { by lemma 79 }
% 60.56/8.28 meet(X, composition(complement(complement(X)), x1))
% 60.56/8.28 = { by lemma 42 }
% 60.56/8.28 meet(X, composition(X, x1))
% 60.56/8.28 = { by lemma 43 }
% 60.56/8.28 meet(composition(X, x1), X)
% 60.56/8.28 = { by lemma 42 R->L }
% 60.56/8.28 meet(composition(X, x1), complement(complement(X)))
% 60.56/8.28 = { by lemma 28 R->L }
% 60.56/8.28 join(zero, meet(composition(X, x1), complement(complement(X))))
% 60.56/8.28 = { by lemma 58 R->L }
% 60.56/8.28 join(composition(meet(X, composition(complement(X), converse(x1))), zero), meet(composition(X, x1), complement(complement(X))))
% 60.56/8.28 = { by lemma 86 R->L }
% 60.56/8.28 join(composition(meet(X, composition(complement(X), converse(x1))), meet(x1, composition(converse(X), complement(X)))), meet(composition(X, x1), complement(complement(X))))
% 60.56/8.28 = { by axiom 16 (dedekind_law) R->L }
% 60.56/8.28 join(join(meet(composition(X, x1), complement(X)), composition(meet(X, composition(complement(X), converse(x1))), meet(x1, composition(converse(X), complement(X))))), meet(composition(X, x1), complement(complement(X))))
% 60.56/8.28 = { by lemma 86 }
% 60.56/8.28 join(join(meet(composition(X, x1), complement(X)), composition(meet(X, composition(complement(X), converse(x1))), zero)), meet(composition(X, x1), complement(complement(X))))
% 60.56/8.28 = { by lemma 58 }
% 60.56/8.28 join(join(meet(composition(X, x1), complement(X)), zero), meet(composition(X, x1), complement(complement(X))))
% 60.56/8.28 = { by lemma 27 }
% 60.56/8.28 join(meet(composition(X, x1), complement(X)), meet(composition(X, x1), complement(complement(X))))
% 60.56/8.28 = { by lemma 43 R->L }
% 60.56/8.28 join(meet(complement(X), composition(X, x1)), meet(composition(X, x1), complement(complement(X))))
% 60.56/8.28 = { by lemma 45 }
% 60.56/8.28 composition(X, x1)
% 60.56/8.28
% 60.56/8.28 Lemma 88: join(x1, converse(x1)) = converse(x1).
% 60.56/8.28 Proof:
% 60.56/8.28 join(x1, converse(x1))
% 60.56/8.28 = { by lemma 42 R->L }
% 60.56/8.28 join(x1, converse(complement(complement(x1))))
% 60.56/8.28 = { by lemma 74 R->L }
% 60.56/8.28 join(x1, complement(converse(complement(x1))))
% 60.56/8.28 = { by lemma 75 R->L }
% 60.56/8.28 join(complement(converse(complement(x1))), meet(x1, converse(complement(x1))))
% 60.56/8.28 = { by lemma 78 R->L }
% 60.56/8.28 join(complement(converse(complement(x1))), meet(x1, meet(converse(complement(x1)), composition(top, x1))))
% 60.56/8.28 = { by lemma 87 }
% 60.56/8.28 join(complement(converse(complement(x1))), meet(x1, composition(converse(complement(x1)), x1)))
% 60.56/8.28 = { by lemma 43 }
% 60.56/8.28 join(complement(converse(complement(x1))), meet(composition(converse(complement(x1)), x1), x1))
% 60.56/8.28 = { by lemma 83 R->L }
% 60.56/8.28 join(complement(converse(complement(x1))), meet(x1, meet(one, composition(converse(complement(x1)), x1))))
% 60.56/8.28 = { by lemma 72 }
% 60.56/8.28 join(complement(converse(complement(x1))), meet(x1, zero))
% 60.56/8.28 = { by lemma 85 }
% 60.56/8.28 join(complement(converse(complement(x1))), zero)
% 60.56/8.28 = { by lemma 27 }
% 60.56/8.28 complement(converse(complement(x1)))
% 60.56/8.28 = { by lemma 74 }
% 60.56/8.28 converse(complement(complement(x1)))
% 60.56/8.28 = { by lemma 42 }
% 60.56/8.28 converse(x1)
% 60.56/8.28
% 60.56/8.28 Lemma 89: converse(x1) = x1.
% 60.56/8.28 Proof:
% 60.56/8.28 converse(x1)
% 60.56/8.28 = { by lemma 88 R->L }
% 60.56/8.28 join(x1, converse(x1))
% 60.56/8.28 = { by lemma 37 R->L }
% 60.56/8.28 converse(join(x1, converse(x1)))
% 60.56/8.28 = { by lemma 88 }
% 60.56/8.28 converse(converse(x1))
% 60.56/8.28 = { by axiom 5 (converse_idempotence) }
% 60.56/8.28 x1
% 60.56/8.28
% 60.56/8.28 Lemma 90: join(one, converse(x1)) = one.
% 60.56/8.28 Proof:
% 60.56/8.28 join(one, converse(x1))
% 60.56/8.28 = { by lemma 80 R->L }
% 60.56/8.28 converse(join(x1, one))
% 60.56/8.28 = { by axiom 3 (goals) }
% 60.56/8.28 converse(one)
% 60.56/8.28 = { by lemma 30 }
% 60.56/8.28 one
% 60.56/8.28
% 60.56/8.28 Lemma 91: join(one, converse(x0)) = one.
% 60.56/8.28 Proof:
% 60.56/8.28 join(one, converse(x0))
% 60.56/8.28 = { by lemma 80 R->L }
% 60.56/8.28 converse(join(x0, one))
% 60.56/8.28 = { by axiom 4 (goals_1) }
% 60.56/8.28 converse(one)
% 60.56/8.28 = { by lemma 30 }
% 60.56/8.28 one
% 60.56/8.28
% 60.56/8.28 Lemma 92: meet(X, join(Y, X)) = X.
% 60.56/8.28 Proof:
% 60.56/8.28 meet(X, join(Y, X))
% 60.56/8.28 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 60.56/8.28 meet(X, join(X, Y))
% 60.56/8.28 = { by lemma 77 }
% 60.56/8.28 X
% 60.56/8.28
% 60.56/8.28 Lemma 93: join(x0, join(one, X)) = join(X, one).
% 60.56/8.28 Proof:
% 60.56/8.28 join(x0, join(one, X))
% 60.56/8.28 = { by axiom 11 (maddux2_join_associativity) }
% 60.56/8.28 join(join(x0, one), X)
% 60.56/8.28 = { by axiom 4 (goals_1) }
% 60.56/8.28 join(one, X)
% 60.56/8.28 = { by axiom 2 (maddux1_join_commutativity) }
% 60.56/8.28 join(X, one)
% 60.56/8.28
% 60.56/8.28 Lemma 94: converse(meet(X, converse(Y))) = meet(Y, converse(X)).
% 60.56/8.28 Proof:
% 60.56/8.28 converse(meet(X, converse(Y)))
% 60.56/8.28 = { by lemma 42 R->L }
% 60.56/8.28 converse(complement(complement(meet(X, converse(Y)))))
% 60.56/8.28 = { by lemma 51 R->L }
% 60.56/8.28 converse(complement(join(complement(X), complement(converse(Y)))))
% 60.56/8.28 = { by lemma 74 }
% 60.56/8.28 converse(complement(join(complement(X), converse(complement(Y)))))
% 60.56/8.28 = { by lemma 37 R->L }
% 60.56/8.28 converse(complement(converse(join(complement(Y), converse(complement(X))))))
% 60.56/8.28 = { by lemma 73 }
% 60.56/8.28 complement(join(complement(Y), converse(complement(X))))
% 60.56/8.28 = { by lemma 54 }
% 60.56/8.28 meet(Y, complement(converse(complement(X))))
% 60.56/8.28 = { by lemma 74 }
% 60.56/8.28 meet(Y, converse(complement(complement(X))))
% 60.56/8.28 = { by lemma 42 }
% 60.56/8.28 meet(Y, converse(X))
% 60.56/8.28
% 60.56/8.28 Lemma 95: meet(Y, composition(X, x1)) = meet(X, composition(Y, x1)).
% 60.56/8.28 Proof:
% 60.56/8.28 meet(Y, composition(X, x1))
% 60.56/8.28 = { by lemma 87 R->L }
% 60.56/8.28 meet(Y, meet(X, composition(top, x1)))
% 60.56/8.28 = { by lemma 76 R->L }
% 60.56/8.28 meet(X, meet(Y, composition(top, x1)))
% 60.56/8.28 = { by lemma 87 }
% 60.56/8.28 meet(X, composition(Y, x1))
% 60.56/8.28
% 60.56/8.28 Lemma 96: meet(composition(top, x1), X) = composition(X, x1).
% 60.56/8.28 Proof:
% 60.56/8.28 meet(composition(top, x1), X)
% 60.56/8.28 = { by lemma 43 }
% 60.56/8.28 meet(X, composition(top, x1))
% 60.56/8.28 = { by lemma 87 }
% 60.56/8.28 composition(X, x1)
% 60.56/8.28
% 60.56/8.28 Lemma 97: converse(meet(x0, x1)) = composition(converse(x0), x1).
% 60.56/8.28 Proof:
% 60.56/8.28 converse(meet(x0, x1))
% 60.56/8.28 = { by lemma 89 R->L }
% 60.56/8.28 converse(meet(x0, converse(x1)))
% 60.56/8.28 = { by lemma 94 }
% 60.56/8.28 meet(x1, converse(x0))
% 60.56/8.28 = { by lemma 43 }
% 60.56/8.28 meet(converse(x0), x1)
% 60.56/8.28 = { by lemma 22 R->L }
% 60.56/8.28 meet(converse(x0), composition(one, x1))
% 60.56/8.28 = { by lemma 95 R->L }
% 60.56/8.28 meet(one, composition(converse(x0), x1))
% 60.56/8.28 = { by lemma 96 R->L }
% 60.56/8.28 meet(one, meet(composition(top, x1), converse(x0)))
% 60.56/8.28 = { by lemma 76 }
% 60.56/8.28 meet(composition(top, x1), meet(one, converse(x0)))
% 60.56/8.28 = { by lemma 96 }
% 60.56/8.28 composition(meet(one, converse(x0)), x1)
% 60.56/8.28 = { by lemma 43 R->L }
% 60.56/8.28 composition(meet(converse(x0), one), x1)
% 60.56/8.28 = { by lemma 91 R->L }
% 60.56/8.28 composition(meet(converse(x0), join(one, converse(x0))), x1)
% 60.56/8.28 = { by lemma 92 }
% 60.56/8.28 composition(converse(x0), x1)
% 60.56/8.28
% 60.56/8.28 Lemma 98: join(X, meet(Y, X)) = X.
% 60.56/8.28 Proof:
% 60.56/8.28 join(X, meet(Y, X))
% 60.56/8.28 = { by lemma 43 R->L }
% 60.56/8.28 join(X, meet(X, Y))
% 60.56/8.28 = { by lemma 64 }
% 60.56/8.28 X
% 60.56/8.28
% 60.56/8.28 Lemma 99: join(composition(join(Z, one), X), Y) = join(X, join(Y, composition(Z, X))).
% 60.56/8.28 Proof:
% 60.56/8.28 join(composition(join(Z, one), X), Y)
% 60.56/8.28 = { by lemma 56 R->L }
% 60.56/8.28 join(join(X, composition(Z, X)), Y)
% 60.56/8.28 = { by axiom 11 (maddux2_join_associativity) R->L }
% 60.56/8.28 join(X, join(composition(Z, X), Y))
% 60.56/8.28 = { by axiom 2 (maddux1_join_commutativity) }
% 60.56/8.28 join(X, join(Y, composition(Z, X)))
% 60.56/8.28
% 60.56/8.28 Lemma 100: composition(meet(X, x1), x1) = meet(X, x1).
% 60.56/8.28 Proof:
% 60.56/8.28 composition(meet(X, x1), x1)
% 60.56/8.28 = { by lemma 87 R->L }
% 60.56/8.28 meet(meet(X, x1), composition(top, x1))
% 60.56/8.28 = { by lemma 36 R->L }
% 60.56/8.28 meet(meet(X, x1), composition(join(top, one), x1))
% 60.56/8.28 = { by lemma 56 R->L }
% 60.56/8.28 meet(meet(X, x1), join(x1, composition(top, x1)))
% 60.56/8.28 = { by lemma 98 R->L }
% 60.56/8.28 meet(meet(X, x1), join(join(x1, meet(X, x1)), composition(top, x1)))
% 60.56/8.28 = { by axiom 11 (maddux2_join_associativity) R->L }
% 60.56/8.28 meet(meet(X, x1), join(x1, join(meet(X, x1), composition(top, x1))))
% 60.56/8.28 = { by lemma 99 R->L }
% 60.56/8.28 meet(meet(X, x1), join(composition(join(top, one), x1), meet(X, x1)))
% 60.56/8.28 = { by lemma 92 }
% 60.56/8.28 meet(X, x1)
% 60.56/8.28
% 60.56/8.28 Lemma 101: meet(meet(X, x1), converse(meet(x0, x1))) = meet(meet(X, x1), converse(x0)).
% 60.56/8.28 Proof:
% 60.56/8.28 meet(meet(X, x1), converse(meet(x0, x1)))
% 60.56/8.28 = { by lemma 97 }
% 60.56/8.28 meet(meet(X, x1), composition(converse(x0), x1))
% 60.56/8.28 = { by lemma 95 }
% 60.56/8.28 meet(converse(x0), composition(meet(X, x1), x1))
% 60.56/8.28 = { by lemma 100 }
% 60.56/8.28 meet(converse(x0), meet(X, x1))
% 60.56/8.28 = { by lemma 43 R->L }
% 60.56/8.28 meet(meet(X, x1), converse(x0))
% 60.56/8.28
% 60.56/8.28 Lemma 102: converse(meet(converse(X), Y)) = meet(X, converse(Y)).
% 60.56/8.28 Proof:
% 60.56/8.28 converse(meet(converse(X), Y))
% 60.56/8.28 = { by lemma 43 }
% 60.56/8.28 converse(meet(Y, converse(X)))
% 60.56/8.28 = { by lemma 94 }
% 60.56/8.28 meet(X, converse(Y))
% 60.56/8.28
% 60.56/8.28 Lemma 103: meet(X, meet(Y, Z)) = meet(Y, meet(Z, X)).
% 60.56/8.28 Proof:
% 60.56/8.28 meet(X, meet(Y, Z))
% 60.56/8.28 = { by lemma 76 }
% 60.56/8.28 meet(Y, meet(X, Z))
% 60.56/8.28 = { by lemma 43 }
% 60.56/8.28 meet(Y, meet(Z, X))
% 60.56/8.28
% 60.56/8.28 Lemma 104: join(converse(X), join(Y, converse(Z))) = join(Y, converse(join(X, Z))).
% 60.56/8.28 Proof:
% 60.56/8.28 join(converse(X), join(Y, converse(Z)))
% 60.56/8.28 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 60.56/8.28 join(converse(X), join(converse(Z), Y))
% 60.56/8.28 = { by axiom 11 (maddux2_join_associativity) }
% 60.56/8.28 join(join(converse(X), converse(Z)), Y)
% 60.56/8.28 = { by axiom 10 (converse_additivity) R->L }
% 60.56/8.28 join(converse(join(X, Z)), Y)
% 60.56/8.28 = { by axiom 2 (maddux1_join_commutativity) }
% 60.56/8.28 join(Y, converse(join(X, Z)))
% 60.56/8.28
% 60.56/8.28 Lemma 105: meet(one, converse(meet(meet(X, x1), Y))) = converse(meet(Y, meet(X, x1))).
% 60.56/8.28 Proof:
% 60.56/8.28 meet(one, converse(meet(meet(X, x1), Y)))
% 60.56/8.28 = { by lemma 43 }
% 60.56/8.28 meet(one, converse(meet(Y, meet(X, x1))))
% 60.56/8.28 = { by lemma 43 }
% 60.56/8.28 meet(converse(meet(Y, meet(X, x1))), one)
% 60.56/8.28 = { by lemma 90 R->L }
% 60.56/8.28 meet(converse(meet(Y, meet(X, x1))), join(one, converse(x1)))
% 60.56/8.28 = { by lemma 64 R->L }
% 60.56/8.28 meet(converse(meet(Y, meet(X, x1))), join(one, converse(join(x1, meet(x1, meet(Y, X))))))
% 60.56/8.28 = { by lemma 103 }
% 60.56/8.28 meet(converse(meet(Y, meet(X, x1))), join(one, converse(join(x1, meet(Y, meet(X, x1))))))
% 60.56/8.28 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 60.56/8.28 meet(converse(meet(Y, meet(X, x1))), join(one, converse(join(meet(Y, meet(X, x1)), x1))))
% 60.56/8.28 = { by lemma 104 R->L }
% 60.56/8.28 meet(converse(meet(Y, meet(X, x1))), join(converse(meet(Y, meet(X, x1))), join(one, converse(x1))))
% 60.56/8.28 = { by lemma 90 }
% 60.56/8.28 meet(converse(meet(Y, meet(X, x1))), join(converse(meet(Y, meet(X, x1))), one))
% 60.56/8.28 = { by axiom 2 (maddux1_join_commutativity) }
% 60.56/8.28 meet(converse(meet(Y, meet(X, x1))), join(one, converse(meet(Y, meet(X, x1)))))
% 60.56/8.28 = { by lemma 92 }
% 60.56/8.28 converse(meet(Y, meet(X, x1)))
% 60.56/8.28
% 60.56/8.28 Lemma 106: meet(meet(x0, x1), converse(x0)) = meet(x0, converse(meet(x0, x1))).
% 60.56/8.28 Proof:
% 60.56/8.28 meet(meet(x0, x1), converse(x0))
% 60.56/8.28 = { by lemma 101 R->L }
% 60.56/8.28 meet(meet(x0, x1), converse(meet(x0, x1)))
% 60.56/8.28 = { by lemma 102 R->L }
% 60.56/8.28 converse(meet(converse(meet(x0, x1)), meet(x0, x1)))
% 60.56/8.28 = { by lemma 105 R->L }
% 60.56/8.28 meet(one, converse(meet(meet(x0, x1), converse(meet(x0, x1)))))
% 60.56/8.28 = { by lemma 101 }
% 60.56/8.28 meet(one, converse(meet(meet(x0, x1), converse(x0))))
% 60.56/8.28 = { by lemma 105 }
% 60.56/8.28 converse(meet(converse(x0), meet(x0, x1)))
% 60.56/8.28 = { by lemma 102 }
% 60.56/8.28 meet(x0, converse(meet(x0, x1)))
% 60.56/8.28
% 60.56/8.28 Lemma 107: meet(Z, meet(Y, X)) = meet(X, meet(Y, Z)).
% 60.56/8.28 Proof:
% 60.56/8.28 meet(Z, meet(Y, X))
% 60.56/8.28 = { by lemma 103 R->L }
% 60.56/8.28 meet(X, meet(Z, Y))
% 60.56/8.28 = { by lemma 43 R->L }
% 60.56/8.28 meet(X, meet(Y, Z))
% 60.56/8.28
% 60.56/8.28 Lemma 108: meet(X, complement(meet(Y, X))) = meet(X, complement(Y)).
% 60.56/8.28 Proof:
% 60.56/8.29 meet(X, complement(meet(Y, X)))
% 60.56/8.29 = { by lemma 43 }
% 60.56/8.29 meet(X, complement(meet(X, Y)))
% 60.56/8.29 = { by lemma 66 }
% 60.56/8.29 meet(X, complement(Y))
% 60.56/8.29
% 60.56/8.29 Lemma 109: meet(x0, meet(one, X)) = meet(X, x0).
% 60.56/8.29 Proof:
% 60.56/8.29 meet(x0, meet(one, X))
% 60.56/8.29 = { by lemma 43 }
% 60.56/8.29 meet(x0, meet(X, one))
% 60.56/8.29 = { by lemma 76 }
% 60.56/8.29 meet(X, meet(x0, one))
% 60.56/8.29 = { by axiom 12 (maddux4_definiton_of_meet) }
% 60.56/8.29 meet(X, complement(join(complement(x0), complement(one))))
% 60.56/8.29 = { by lemma 28 R->L }
% 60.56/8.29 meet(X, join(zero, complement(join(complement(x0), complement(one)))))
% 60.56/8.29 = { by lemma 17 R->L }
% 60.56/8.29 meet(X, join(complement(top), complement(join(complement(x0), complement(one)))))
% 60.56/8.29 = { by lemma 35 R->L }
% 60.56/8.29 meet(X, join(complement(join(one, top)), complement(join(complement(x0), complement(one)))))
% 60.56/8.29 = { by lemma 31 R->L }
% 60.56/8.29 meet(X, join(complement(join(x0, join(one, complement(x0)))), complement(join(complement(x0), complement(one)))))
% 60.56/8.29 = { by lemma 93 }
% 60.56/8.29 meet(X, join(complement(join(complement(x0), one)), complement(join(complement(x0), complement(one)))))
% 60.56/8.29 = { by axiom 2 (maddux1_join_commutativity) }
% 60.56/8.29 meet(X, join(complement(join(one, complement(x0))), complement(join(complement(x0), complement(one)))))
% 60.56/8.29 = { by lemma 53 }
% 60.56/8.29 meet(X, join(meet(x0, complement(one)), complement(join(complement(x0), complement(one)))))
% 60.56/8.29 = { by lemma 18 }
% 60.56/8.29 meet(X, x0)
% 60.56/8.29
% 60.56/8.29 Lemma 110: meet(x0, composition(converse(X), complement(X))) = zero.
% 60.56/8.29 Proof:
% 60.56/8.29 meet(x0, composition(converse(X), complement(X)))
% 60.56/8.29 = { by lemma 43 }
% 60.56/8.29 meet(composition(converse(X), complement(X)), x0)
% 60.56/8.29 = { by lemma 109 R->L }
% 60.56/8.29 meet(x0, meet(one, composition(converse(X), complement(X))))
% 60.56/8.29 = { by lemma 84 }
% 60.56/8.29 meet(x0, zero)
% 60.56/8.29 = { by lemma 85 }
% 60.56/8.29 zero
% 60.56/8.29
% 60.56/8.29 Lemma 111: meet(X, composition(X, x0)) = composition(X, x0).
% 60.56/8.29 Proof:
% 60.56/8.29 meet(X, composition(X, x0))
% 60.56/8.29 = { by lemma 43 }
% 60.56/8.29 meet(composition(X, x0), X)
% 60.56/8.29 = { by lemma 42 R->L }
% 60.56/8.29 meet(composition(X, x0), complement(complement(X)))
% 60.56/8.29 = { by lemma 28 R->L }
% 60.56/8.29 join(zero, meet(composition(X, x0), complement(complement(X))))
% 60.56/8.29 = { by lemma 58 R->L }
% 60.56/8.29 join(composition(meet(X, composition(complement(X), converse(x0))), zero), meet(composition(X, x0), complement(complement(X))))
% 60.56/8.29 = { by lemma 110 R->L }
% 60.56/8.29 join(composition(meet(X, composition(complement(X), converse(x0))), meet(x0, composition(converse(X), complement(X)))), meet(composition(X, x0), complement(complement(X))))
% 60.56/8.29 = { by axiom 16 (dedekind_law) R->L }
% 60.56/8.29 join(join(meet(composition(X, x0), complement(X)), composition(meet(X, composition(complement(X), converse(x0))), meet(x0, composition(converse(X), complement(X))))), meet(composition(X, x0), complement(complement(X))))
% 60.56/8.29 = { by lemma 110 }
% 60.56/8.29 join(join(meet(composition(X, x0), complement(X)), composition(meet(X, composition(complement(X), converse(x0))), zero)), meet(composition(X, x0), complement(complement(X))))
% 60.56/8.29 = { by lemma 58 }
% 60.56/8.29 join(join(meet(composition(X, x0), complement(X)), zero), meet(composition(X, x0), complement(complement(X))))
% 60.56/8.29 = { by lemma 27 }
% 60.56/8.29 join(meet(composition(X, x0), complement(X)), meet(composition(X, x0), complement(complement(X))))
% 60.56/8.29 = { by lemma 43 R->L }
% 60.56/8.29 join(meet(complement(X), composition(X, x0)), meet(composition(X, x0), complement(complement(X))))
% 60.56/8.29 = { by lemma 45 }
% 60.56/8.29 composition(X, x0)
% 60.56/8.29
% 60.56/8.29 Lemma 112: meet(X, composition(top, x0)) = composition(X, x0).
% 60.56/8.29 Proof:
% 60.56/8.29 meet(X, composition(top, x0))
% 60.56/8.29 = { by lemma 79 R->L }
% 60.56/8.29 meet(X, join(complement(X), composition(top, x0)))
% 60.56/8.29 = { by axiom 7 (def_top) }
% 60.56/8.29 meet(X, join(complement(X), composition(join(complement(X), complement(complement(X))), x0)))
% 60.56/8.29 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 60.56/8.29 meet(X, join(complement(X), composition(join(complement(complement(X)), complement(X)), x0)))
% 60.56/8.29 = { by axiom 13 (composition_distributivity) }
% 60.56/8.29 meet(X, join(complement(X), join(composition(complement(complement(X)), x0), composition(complement(X), x0))))
% 60.56/8.29 = { by lemma 82 R->L }
% 60.56/8.29 meet(X, join(composition(complement(X), join(x0, one)), composition(complement(complement(X)), x0)))
% 60.56/8.29 = { by axiom 4 (goals_1) }
% 60.56/8.29 meet(X, join(composition(complement(X), one), composition(complement(complement(X)), x0)))
% 60.56/8.29 = { by axiom 1 (composition_identity) }
% 60.56/8.29 meet(X, join(complement(X), composition(complement(complement(X)), x0)))
% 60.56/8.29 = { by lemma 79 }
% 60.56/8.29 meet(X, composition(complement(complement(X)), x0))
% 60.56/8.29 = { by lemma 42 }
% 60.56/8.29 meet(X, composition(X, x0))
% 60.56/8.29 = { by lemma 111 }
% 60.56/8.29 composition(X, x0)
% 60.56/8.29
% 60.56/8.29 Lemma 113: meet(x0, converse(meet(x0, x1))) = meet(x0, x1).
% 60.56/8.29 Proof:
% 60.56/8.29 meet(x0, converse(meet(x0, x1)))
% 60.56/8.29 = { by lemma 106 R->L }
% 60.56/8.29 meet(meet(x0, x1), converse(x0))
% 60.56/8.29 = { by lemma 42 R->L }
% 60.56/8.29 meet(meet(x0, x1), converse(complement(complement(x0))))
% 60.56/8.29 = { by lemma 74 R->L }
% 60.56/8.29 meet(meet(x0, x1), complement(converse(complement(x0))))
% 60.56/8.29 = { by lemma 43 }
% 60.56/8.29 meet(complement(converse(complement(x0))), meet(x0, x1))
% 60.56/8.29 = { by lemma 107 }
% 60.56/8.29 meet(x1, meet(x0, complement(converse(complement(x0)))))
% 60.56/8.29 = { by lemma 108 R->L }
% 60.56/8.29 meet(x1, meet(x0, complement(meet(converse(complement(x0)), x0))))
% 60.56/8.29 = { by lemma 107 R->L }
% 60.56/8.29 meet(complement(meet(converse(complement(x0)), x0)), meet(x0, x1))
% 60.56/8.29 = { by lemma 43 R->L }
% 60.56/8.29 meet(meet(x0, x1), complement(meet(converse(complement(x0)), x0)))
% 60.56/8.29 = { by lemma 43 }
% 60.56/8.29 meet(meet(x0, x1), complement(meet(x0, converse(complement(x0)))))
% 60.56/8.29 = { by lemma 78 R->L }
% 60.56/8.29 meet(meet(x0, x1), complement(meet(x0, meet(converse(complement(x0)), composition(top, x0)))))
% 60.56/8.29 = { by lemma 112 }
% 60.56/8.29 meet(meet(x0, x1), complement(meet(x0, composition(converse(complement(x0)), x0))))
% 60.56/8.29 = { by lemma 43 }
% 60.56/8.29 meet(meet(x0, x1), complement(meet(composition(converse(complement(x0)), x0), x0)))
% 60.56/8.29 = { by lemma 109 R->L }
% 60.56/8.29 meet(meet(x0, x1), complement(meet(x0, meet(one, composition(converse(complement(x0)), x0)))))
% 60.56/8.29 = { by lemma 72 }
% 60.56/8.29 meet(meet(x0, x1), complement(meet(x0, zero)))
% 60.56/8.29 = { by lemma 85 }
% 60.56/8.29 meet(meet(x0, x1), complement(zero))
% 60.56/8.29 = { by lemma 29 }
% 60.56/8.29 meet(meet(x0, x1), top)
% 60.56/8.29 = { by lemma 48 }
% 60.56/8.29 meet(x0, x1)
% 60.56/8.29
% 60.56/8.29 Lemma 114: converse(meet(x0, x1)) = meet(x0, x1).
% 60.56/8.29 Proof:
% 60.56/8.29 converse(meet(x0, x1))
% 60.56/8.29 = { by lemma 92 R->L }
% 60.56/8.29 meet(converse(meet(x0, x1)), join(one, converse(meet(x0, x1))))
% 60.56/8.29 = { by lemma 80 R->L }
% 60.56/8.29 meet(converse(meet(x0, x1)), converse(join(meet(x0, x1), one)))
% 60.56/8.29 = { by lemma 93 R->L }
% 60.56/8.29 meet(converse(meet(x0, x1)), converse(join(x0, join(one, meet(x0, x1)))))
% 60.56/8.29 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 60.56/8.29 meet(converse(meet(x0, x1)), converse(join(x0, join(meet(x0, x1), one))))
% 60.56/8.29 = { by axiom 11 (maddux2_join_associativity) }
% 60.56/8.29 meet(converse(meet(x0, x1)), converse(join(join(x0, meet(x0, x1)), one)))
% 60.56/8.29 = { by lemma 64 }
% 60.56/8.29 meet(converse(meet(x0, x1)), converse(join(x0, one)))
% 60.56/8.29 = { by axiom 4 (goals_1) }
% 60.56/8.29 meet(converse(meet(x0, x1)), converse(one))
% 60.56/8.29 = { by lemma 30 }
% 60.56/8.29 meet(converse(meet(x0, x1)), one)
% 60.56/8.29 = { by lemma 43 R->L }
% 60.56/8.29 meet(one, converse(meet(x0, x1)))
% 60.56/8.29 = { by lemma 113 R->L }
% 60.56/8.29 meet(one, converse(meet(x0, converse(meet(x0, x1)))))
% 60.56/8.29 = { by lemma 43 }
% 60.56/8.29 meet(one, converse(meet(converse(meet(x0, x1)), x0)))
% 60.56/8.29 = { by lemma 43 }
% 60.56/8.29 meet(converse(meet(converse(meet(x0, x1)), x0)), one)
% 60.56/8.29 = { by lemma 91 R->L }
% 60.56/8.29 meet(converse(meet(converse(meet(x0, x1)), x0)), join(one, converse(x0)))
% 60.56/8.29 = { by lemma 64 R->L }
% 60.56/8.29 meet(converse(meet(converse(meet(x0, x1)), x0)), join(one, converse(join(x0, meet(x0, converse(meet(x0, x1)))))))
% 60.56/8.29 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 60.56/8.29 meet(converse(meet(converse(meet(x0, x1)), x0)), join(one, converse(join(meet(x0, converse(meet(x0, x1))), x0))))
% 60.56/8.29 = { by lemma 104 R->L }
% 60.56/8.29 meet(converse(meet(converse(meet(x0, x1)), x0)), join(converse(meet(x0, converse(meet(x0, x1)))), join(one, converse(x0))))
% 60.56/8.29 = { by lemma 91 }
% 60.56/8.29 meet(converse(meet(converse(meet(x0, x1)), x0)), join(converse(meet(x0, converse(meet(x0, x1)))), one))
% 60.56/8.29 = { by axiom 2 (maddux1_join_commutativity) }
% 60.56/8.29 meet(converse(meet(converse(meet(x0, x1)), x0)), join(one, converse(meet(x0, converse(meet(x0, x1))))))
% 60.56/8.29 = { by lemma 43 R->L }
% 60.56/8.29 meet(converse(meet(converse(meet(x0, x1)), x0)), join(one, converse(meet(converse(meet(x0, x1)), x0))))
% 60.56/8.29 = { by lemma 92 }
% 60.56/8.29 converse(meet(converse(meet(x0, x1)), x0))
% 60.56/8.29 = { by lemma 102 }
% 60.56/8.29 meet(meet(x0, x1), converse(x0))
% 60.56/8.29 = { by lemma 106 }
% 60.56/8.29 meet(x0, converse(meet(x0, x1)))
% 60.56/8.29 = { by lemma 113 }
% 60.56/8.29 meet(x0, x1)
% 60.56/8.29
% 60.56/8.29 Lemma 115: join(X, join(Y, Z)) = join(Y, join(X, Z)).
% 60.56/8.29 Proof:
% 60.56/8.29 join(X, join(Y, Z))
% 60.56/8.29 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 60.56/8.29 join(join(Y, Z), X)
% 60.56/8.29 = { by axiom 11 (maddux2_join_associativity) R->L }
% 60.56/8.29 join(Y, join(Z, X))
% 60.56/8.29 = { by axiom 2 (maddux1_join_commutativity) }
% 60.56/8.29 join(Y, join(X, Z))
% 60.56/8.29
% 60.56/8.29 Lemma 116: join(X, join(Y, composition(x1, X))) = join(X, Y).
% 60.56/8.29 Proof:
% 60.56/8.29 join(X, join(Y, composition(x1, X)))
% 60.56/8.29 = { by lemma 99 R->L }
% 60.56/8.29 join(composition(join(x1, one), X), Y)
% 60.56/8.29 = { by axiom 3 (goals) }
% 60.56/8.29 join(composition(one, X), Y)
% 60.56/8.29 = { by lemma 22 }
% 60.56/8.29 join(X, Y)
% 60.56/8.29
% 60.56/8.29 Lemma 117: meet(X, join(Y, join(Z, X))) = X.
% 60.56/8.29 Proof:
% 60.56/8.29 meet(X, join(Y, join(Z, X)))
% 60.56/8.29 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 60.56/8.29 meet(X, join(Y, join(X, Z)))
% 60.56/8.29 = { by lemma 115 R->L }
% 60.56/8.29 meet(X, join(X, join(Y, Z)))
% 60.56/8.29 = { by lemma 77 }
% 60.56/8.29 X
% 60.56/8.29
% 60.56/8.29 Lemma 118: meet(X, composition(x1, X)) = composition(x1, X).
% 60.56/8.29 Proof:
% 60.56/8.29 meet(X, composition(x1, X))
% 60.56/8.29 = { by lemma 43 }
% 60.56/8.29 meet(composition(x1, X), X)
% 60.56/8.29 = { by lemma 27 R->L }
% 60.56/8.29 meet(composition(x1, X), join(X, zero))
% 60.56/8.29 = { by lemma 116 R->L }
% 60.56/8.29 meet(composition(x1, X), join(X, join(zero, composition(x1, X))))
% 60.56/8.29 = { by lemma 117 }
% 60.56/8.29 composition(x1, X)
% 60.56/8.29
% 60.56/8.29 Lemma 119: join(composition(X, Y), composition(Z, Y)) = composition(join(Z, X), Y).
% 60.56/8.29 Proof:
% 60.56/8.29 join(composition(X, Y), composition(Z, Y))
% 60.56/8.29 = { by axiom 13 (composition_distributivity) R->L }
% 60.56/8.29 composition(join(X, Z), Y)
% 60.56/8.29 = { by axiom 2 (maddux1_join_commutativity) }
% 60.56/8.29 composition(join(Z, X), Y)
% 60.56/8.29
% 60.56/8.29 Lemma 120: composition(converse(X), complement(composition(X, top))) = zero.
% 60.56/8.29 Proof:
% 60.56/8.29 composition(converse(X), complement(composition(X, top)))
% 60.56/8.29 = { by lemma 28 R->L }
% 60.56/8.29 join(zero, composition(converse(X), complement(composition(X, top))))
% 60.56/8.29 = { by lemma 17 R->L }
% 60.56/8.29 join(complement(top), composition(converse(X), complement(composition(X, top))))
% 60.56/8.29 = { by lemma 23 }
% 60.56/8.29 complement(top)
% 60.56/8.29 = { by lemma 17 }
% 60.56/8.29 zero
% 60.56/8.29
% 60.56/8.29 Lemma 121: meet(composition(X, Y), complement(composition(X, top))) = zero.
% 60.56/8.29 Proof:
% 60.56/8.29 meet(composition(X, Y), complement(composition(X, top)))
% 60.56/8.29 = { by lemma 27 R->L }
% 60.56/8.29 join(meet(composition(X, Y), complement(composition(X, top))), zero)
% 60.56/8.29 = { by lemma 58 R->L }
% 60.56/8.29 join(meet(composition(X, Y), complement(composition(X, top))), composition(meet(X, composition(complement(composition(X, top)), converse(Y))), zero))
% 60.56/8.29 = { by lemma 85 R->L }
% 60.56/8.29 join(meet(composition(X, Y), complement(composition(X, top))), composition(meet(X, composition(complement(composition(X, top)), converse(Y))), meet(Y, zero)))
% 60.56/8.29 = { by lemma 120 R->L }
% 60.56/8.29 join(meet(composition(X, Y), complement(composition(X, top))), composition(meet(X, composition(complement(composition(X, top)), converse(Y))), meet(Y, composition(converse(X), complement(composition(X, top))))))
% 60.56/8.29 = { by axiom 16 (dedekind_law) }
% 60.56/8.29 composition(meet(X, composition(complement(composition(X, top)), converse(Y))), meet(Y, composition(converse(X), complement(composition(X, top)))))
% 60.56/8.29 = { by lemma 120 }
% 60.56/8.29 composition(meet(X, composition(complement(composition(X, top)), converse(Y))), meet(Y, zero))
% 60.56/8.29 = { by lemma 85 }
% 60.56/8.29 composition(meet(X, composition(complement(composition(X, top)), converse(Y))), zero)
% 60.56/8.29 = { by lemma 58 }
% 60.56/8.29 zero
% 60.56/8.29
% 60.56/8.29 Lemma 122: meet(composition(meet(X, Y), Z), composition(X, Z)) = composition(meet(X, Y), Z).
% 60.56/8.29 Proof:
% 60.56/8.29 meet(composition(meet(X, Y), Z), composition(X, Z))
% 60.56/8.29 = { by lemma 64 R->L }
% 60.56/8.29 meet(composition(meet(X, Y), Z), composition(join(X, meet(X, Y)), Z))
% 60.56/8.30 = { by lemma 119 R->L }
% 60.56/8.30 meet(composition(meet(X, Y), Z), join(composition(meet(X, Y), Z), composition(X, Z)))
% 60.56/8.30 = { by lemma 77 }
% 60.56/8.30 composition(meet(X, Y), Z)
% 60.56/8.30
% 60.56/8.30 Lemma 123: meet(composition(meet(x0, x1), X), composition(meet(x0, x1), Y)) = meet(Y, composition(meet(x0, x1), X)).
% 60.56/8.30 Proof:
% 60.56/8.30 meet(composition(meet(x0, x1), X), composition(meet(x0, x1), Y))
% 60.56/8.30 = { by lemma 114 R->L }
% 60.56/8.30 meet(composition(meet(x0, x1), X), composition(converse(meet(x0, x1)), Y))
% 60.56/8.30 = { by lemma 20 R->L }
% 60.56/8.30 meet(composition(meet(x0, x1), X), converse(composition(converse(Y), meet(x0, x1))))
% 60.56/8.30 = { by lemma 100 R->L }
% 60.56/8.30 meet(composition(meet(x0, x1), X), converse(composition(converse(Y), composition(meet(x0, x1), x1))))
% 60.56/8.30 = { by lemma 96 R->L }
% 60.56/8.30 meet(composition(meet(x0, x1), X), converse(composition(converse(Y), meet(composition(top, x1), meet(x0, x1)))))
% 60.56/8.30 = { by lemma 103 R->L }
% 60.56/8.30 meet(composition(meet(x0, x1), X), converse(composition(converse(Y), meet(x1, meet(composition(top, x1), x0)))))
% 60.56/8.30 = { by lemma 96 }
% 60.56/8.30 meet(composition(meet(x0, x1), X), converse(composition(converse(Y), meet(x1, composition(x0, x1)))))
% 60.56/8.30 = { by lemma 43 }
% 60.56/8.30 meet(composition(meet(x0, x1), X), converse(composition(converse(Y), meet(composition(x0, x1), x1))))
% 60.56/8.30 = { by lemma 27 R->L }
% 60.56/8.30 meet(composition(meet(x0, x1), X), converse(composition(converse(Y), meet(composition(x0, x1), join(x1, zero)))))
% 60.56/8.30 = { by lemma 22 R->L }
% 60.56/8.30 meet(composition(meet(x0, x1), X), converse(composition(converse(Y), meet(composition(x0, x1), join(composition(one, x1), zero)))))
% 60.56/8.30 = { by axiom 4 (goals_1) R->L }
% 60.56/8.30 meet(composition(meet(x0, x1), X), converse(composition(converse(Y), meet(composition(x0, x1), join(composition(join(x0, one), x1), zero)))))
% 60.56/8.30 = { by lemma 99 }
% 60.56/8.30 meet(composition(meet(x0, x1), X), converse(composition(converse(Y), meet(composition(x0, x1), join(x1, join(zero, composition(x0, x1)))))))
% 60.56/8.30 = { by lemma 117 }
% 60.56/8.30 meet(composition(meet(x0, x1), X), converse(composition(converse(Y), composition(x0, x1))))
% 60.56/8.30 = { by axiom 9 (composition_associativity) }
% 60.56/8.30 meet(composition(meet(x0, x1), X), converse(composition(composition(converse(Y), x0), x1)))
% 60.56/8.30 = { by lemma 96 R->L }
% 60.56/8.30 meet(composition(meet(x0, x1), X), converse(meet(composition(top, x1), composition(converse(Y), x0))))
% 60.56/8.30 = { by lemma 112 R->L }
% 60.56/8.30 meet(composition(meet(x0, x1), X), converse(meet(composition(top, x1), meet(converse(Y), composition(top, x0)))))
% 60.56/8.30 = { by lemma 76 }
% 60.56/8.30 meet(composition(meet(x0, x1), X), converse(meet(converse(Y), meet(composition(top, x1), composition(top, x0)))))
% 60.56/8.30 = { by lemma 112 }
% 60.56/8.30 meet(composition(meet(x0, x1), X), converse(meet(converse(Y), composition(composition(top, x1), x0))))
% 60.56/8.30 = { by axiom 9 (composition_associativity) R->L }
% 60.56/8.30 meet(composition(meet(x0, x1), X), converse(meet(converse(Y), composition(top, composition(x1, x0)))))
% 60.56/8.30 = { by lemma 89 R->L }
% 60.56/8.30 meet(composition(meet(x0, x1), X), converse(meet(converse(Y), composition(top, composition(converse(x1), x0)))))
% 60.56/8.30 = { by lemma 20 R->L }
% 60.56/8.30 meet(composition(meet(x0, x1), X), converse(meet(converse(Y), composition(top, converse(composition(converse(x0), x1))))))
% 60.56/8.30 = { by lemma 97 R->L }
% 60.56/8.30 meet(composition(meet(x0, x1), X), converse(meet(converse(Y), composition(top, converse(converse(meet(x0, x1)))))))
% 60.56/8.30 = { by axiom 5 (converse_idempotence) }
% 60.56/8.30 meet(composition(meet(x0, x1), X), converse(meet(converse(Y), composition(top, meet(x0, x1)))))
% 60.56/8.30 = { by lemma 102 }
% 60.56/8.30 meet(composition(meet(x0, x1), X), meet(Y, converse(composition(top, meet(x0, x1)))))
% 60.56/8.30 = { by axiom 8 (converse_multiplicativity) }
% 60.56/8.30 meet(composition(meet(x0, x1), X), meet(Y, composition(converse(meet(x0, x1)), converse(top))))
% 60.56/8.30 = { by lemma 38 }
% 60.56/8.30 meet(composition(meet(x0, x1), X), meet(Y, composition(converse(meet(x0, x1)), top)))
% 60.56/8.30 = { by lemma 114 }
% 60.56/8.30 meet(composition(meet(x0, x1), X), meet(Y, composition(meet(x0, x1), top)))
% 60.56/8.30 = { by lemma 76 }
% 60.56/8.30 meet(Y, meet(composition(meet(x0, x1), X), composition(meet(x0, x1), top)))
% 60.56/8.30 = { by lemma 43 }
% 60.56/8.30 meet(Y, meet(composition(meet(x0, x1), top), composition(meet(x0, x1), X)))
% 60.56/8.30 = { by lemma 27 R->L }
% 60.56/8.30 meet(Y, join(meet(composition(meet(x0, x1), top), composition(meet(x0, x1), X)), zero))
% 60.56/8.30 = { by lemma 121 R->L }
% 60.56/8.30 meet(Y, join(meet(composition(meet(x0, x1), top), composition(meet(x0, x1), X)), meet(composition(meet(x0, x1), X), complement(composition(meet(x0, x1), top)))))
% 60.56/8.30 = { by lemma 45 }
% 60.56/8.30 meet(Y, composition(meet(x0, x1), X))
% 60.56/8.30
% 60.56/8.30 Lemma 124: join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y)) = meet(composition(X, Y), composition(Z, Y)).
% 60.56/8.30 Proof:
% 60.56/8.30 join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y))
% 60.56/8.30 = { by lemma 122 R->L }
% 60.56/8.30 join(meet(composition(X, Y), composition(Z, Y)), meet(composition(meet(X, Z), Y), composition(X, Y)))
% 60.56/8.30 = { by lemma 43 }
% 60.56/8.30 join(meet(composition(X, Y), composition(Z, Y)), meet(composition(X, Y), composition(meet(X, Z), Y)))
% 60.56/8.30 = { by lemma 77 R->L }
% 60.56/8.30 join(meet(composition(X, Y), composition(Z, Y)), meet(composition(X, Y), meet(composition(meet(X, Z), Y), join(composition(meet(X, Z), Y), composition(Z, Y)))))
% 60.56/8.30 = { by lemma 119 }
% 60.56/8.30 join(meet(composition(X, Y), composition(Z, Y)), meet(composition(X, Y), meet(composition(meet(X, Z), Y), composition(join(Z, meet(X, Z)), Y))))
% 60.56/8.30 = { by lemma 98 }
% 60.56/8.30 join(meet(composition(X, Y), composition(Z, Y)), meet(composition(X, Y), meet(composition(meet(X, Z), Y), composition(Z, Y))))
% 60.56/8.30 = { by lemma 76 }
% 60.56/8.30 join(meet(composition(X, Y), composition(Z, Y)), meet(composition(meet(X, Z), Y), meet(composition(X, Y), composition(Z, Y))))
% 60.56/8.30 = { by lemma 43 R->L }
% 60.56/8.30 join(meet(composition(X, Y), composition(Z, Y)), meet(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y)))
% 60.56/8.30 = { by lemma 64 }
% 60.56/8.30 meet(composition(X, Y), composition(Z, Y))
% 60.56/8.30
% 60.56/8.30 Lemma 125: meet(W, join(meet(composition(Z, Y), composition(X, Y)), composition(meet(Z, X), Y))) = meet(composition(X, Y), meet(composition(Z, Y), W)).
% 60.56/8.30 Proof:
% 60.56/8.30 meet(W, join(meet(composition(Z, Y), composition(X, Y)), composition(meet(Z, X), Y)))
% 60.56/8.30 = { by lemma 124 }
% 60.56/8.30 meet(W, meet(composition(Z, Y), composition(X, Y)))
% 60.56/8.30 = { by lemma 43 R->L }
% 60.56/8.30 meet(W, meet(composition(X, Y), composition(Z, Y)))
% 60.56/8.30 = { by lemma 76 R->L }
% 60.56/8.30 meet(composition(X, Y), meet(W, composition(Z, Y)))
% 60.56/8.30 = { by lemma 43 R->L }
% 60.56/8.30 meet(composition(X, Y), meet(composition(Z, Y), W))
% 60.56/8.30
% 60.56/8.30 Goal 1 (goals_2): tuple(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), join(composition(meet(x0, x1), x2), meet(composition(x0, x2), composition(x1, x2)))) = tuple(composition(meet(x0, x1), x2), meet(composition(x0, x2), composition(x1, x2))).
% 60.56/8.30 Proof:
% 60.56/8.30 tuple(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), join(composition(meet(x0, x1), x2), meet(composition(x0, x2), composition(x1, x2))))
% 60.56/8.30 = { by axiom 2 (maddux1_join_commutativity) }
% 60.56/8.30 tuple(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.30 = { by lemma 43 }
% 60.56/8.30 tuple(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x1, x0), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.30 = { by axiom 12 (maddux4_definiton_of_meet) }
% 60.56/8.30 tuple(join(meet(composition(x0, x2), composition(x1, x2)), composition(complement(join(complement(x1), complement(x0))), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.30 = { by lemma 24 R->L }
% 60.56/8.30 tuple(join(meet(composition(x0, x2), composition(x1, x2)), composition(join(complement(join(complement(x1), complement(x0))), complement(join(complement(x1), complement(x0)))), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.30 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 60.56/8.30 tuple(join(meet(composition(x0, x2), composition(x1, x2)), composition(join(meet(x1, x0), complement(join(complement(x1), complement(x0)))), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.30 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 60.56/8.30 tuple(join(meet(composition(x0, x2), composition(x1, x2)), composition(join(meet(x1, x0), meet(x1, x0)), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.30 = { by lemma 43 R->L }
% 60.56/8.30 tuple(join(meet(composition(x0, x2), composition(x1, x2)), composition(join(meet(x1, x0), meet(x0, x1)), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.30 = { by lemma 43 R->L }
% 60.56/8.30 tuple(join(meet(composition(x0, x2), composition(x1, x2)), composition(join(meet(x0, x1), meet(x0, x1)), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.30 = { by lemma 119 R->L }
% 60.56/8.30 tuple(join(meet(composition(x0, x2), composition(x1, x2)), join(composition(meet(x0, x1), x2), composition(meet(x0, x1), x2))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.30 = { by lemma 115 }
% 60.56/8.30 tuple(join(composition(meet(x0, x1), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.30 = { by axiom 2 (maddux1_join_commutativity) }
% 60.56/8.30 tuple(join(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), composition(meet(x0, x1), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.30 = { by lemma 42 R->L }
% 60.56/8.30 tuple(join(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(complement(composition(meet(x0, x1), x2)))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.30 = { by lemma 75 R->L }
% 60.56/8.30 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(composition(meet(x0, x1), x2)))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.30 = { by lemma 122 R->L }
% 60.56/8.30 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(meet(x0, x1), x2), composition(x0, x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.30 = { by lemma 43 }
% 60.56/8.30 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x0, x2), composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.30 = { by lemma 123 R->L }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(meet(x0, x1), x2), composition(meet(x0, x1), composition(x0, x2)))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by lemma 100 R->L }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(meet(x0, x1), x2), composition(composition(meet(x0, x1), x1), composition(x0, x2)))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by axiom 9 (composition_associativity) R->L }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(meet(x0, x1), x2), composition(meet(x0, x1), composition(x1, composition(x0, x2))))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by lemma 123 }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x1, composition(x0, x2)), composition(meet(x0, x1), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by lemma 27 R->L }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x1, composition(x0, x2)), composition(join(meet(x0, x1), zero), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by axiom 6 (def_zero) }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x1, composition(x0, x2)), composition(join(meet(x0, x1), meet(composition(x1, x0), complement(composition(x1, x0)))), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by lemma 111 R->L }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x1, composition(x0, x2)), composition(join(meet(x0, x1), meet(composition(x1, x0), complement(meet(x1, composition(x1, x0))))), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by lemma 118 R->L }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x1, composition(x0, x2)), composition(join(meet(x0, x1), meet(composition(x1, x0), complement(meet(x1, meet(x0, composition(x1, x0)))))), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by lemma 107 R->L }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x1, composition(x0, x2)), composition(join(meet(x0, x1), meet(composition(x1, x0), complement(meet(composition(x1, x0), meet(x0, x1))))), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by lemma 43 R->L }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x1, composition(x0, x2)), composition(join(meet(x0, x1), meet(composition(x1, x0), complement(meet(meet(x0, x1), composition(x1, x0))))), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by lemma 108 }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x1, composition(x0, x2)), composition(join(meet(x0, x1), meet(composition(x1, x0), complement(meet(x0, x1)))), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by lemma 43 }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x1, composition(x0, x2)), composition(join(meet(x0, x1), meet(complement(meet(x0, x1)), composition(x1, x0))), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by lemma 19 R->L }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x1, composition(x0, x2)), composition(join(join(zero, meet(meet(x0, x1), meet(x0, x1))), meet(complement(meet(x0, x1)), composition(x1, x0))), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by lemma 28 }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x1, composition(x0, x2)), composition(join(meet(meet(x0, x1), meet(x0, x1)), meet(complement(meet(x0, x1)), composition(x1, x0))), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by lemma 41 R->L }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x1, composition(x0, x2)), composition(join(complement(complement(meet(x0, x1))), meet(complement(meet(x0, x1)), composition(x1, x0))), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by lemma 65 }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x1, composition(x0, x2)), composition(join(composition(x1, x0), complement(complement(meet(x0, x1)))), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by lemma 42 }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x1, composition(x0, x2)), composition(join(composition(x1, x0), meet(x0, x1)), x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by lemma 119 R->L }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x1, composition(x0, x2)), join(composition(meet(x0, x1), x2), composition(composition(x1, x0), x2)))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by axiom 9 (composition_associativity) R->L }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x1, composition(x0, x2)), join(composition(meet(x0, x1), x2), composition(x1, composition(x0, x2))))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by lemma 92 }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(composition(x1, composition(x0, x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by lemma 118 R->L }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x0, x2), composition(x1, composition(x0, x2)))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by lemma 42 R->L }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x0, x2), composition(x1, complement(complement(composition(x0, x2)))))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by lemma 79 R->L }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x0, x2), join(complement(composition(x0, x2)), composition(x1, complement(complement(composition(x0, x2))))))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by lemma 116 R->L }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x0, x2), join(complement(composition(x0, x2)), join(composition(x1, complement(complement(composition(x0, x2)))), composition(x1, complement(composition(x0, x2))))))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by axiom 5 (converse_idempotence) R->L }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x0, x2), join(complement(composition(x0, x2)), join(composition(x1, complement(complement(composition(x0, x2)))), composition(converse(converse(x1)), complement(composition(x0, x2))))))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by axiom 5 (converse_idempotence) R->L }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x0, x2), join(complement(composition(x0, x2)), join(converse(converse(composition(x1, complement(complement(composition(x0, x2)))))), composition(converse(converse(x1)), complement(composition(x0, x2))))))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by lemma 70 R->L }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x0, x2), join(complement(composition(x0, x2)), converse(join(converse(composition(x1, complement(complement(composition(x0, x2))))), composition(converse(complement(composition(x0, x2))), converse(x1))))))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by axiom 8 (converse_multiplicativity) }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x0, x2), join(complement(composition(x0, x2)), converse(join(composition(converse(complement(complement(composition(x0, x2)))), converse(x1)), composition(converse(complement(composition(x0, x2))), converse(x1))))))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by axiom 13 (composition_distributivity) R->L }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x0, x2), join(complement(composition(x0, x2)), converse(composition(join(converse(complement(complement(composition(x0, x2)))), converse(complement(composition(x0, x2)))), converse(x1)))))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by axiom 8 (converse_multiplicativity) }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x0, x2), join(complement(composition(x0, x2)), composition(converse(converse(x1)), converse(join(converse(complement(complement(composition(x0, x2)))), converse(complement(composition(x0, x2))))))))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by lemma 37 }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x0, x2), join(complement(composition(x0, x2)), composition(converse(converse(x1)), join(complement(composition(x0, x2)), converse(converse(complement(complement(composition(x0, x2)))))))))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by axiom 5 (converse_idempotence) }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x0, x2), join(complement(composition(x0, x2)), composition(x1, join(complement(composition(x0, x2)), converse(converse(complement(complement(composition(x0, x2)))))))))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by axiom 5 (converse_idempotence) }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x0, x2), join(complement(composition(x0, x2)), composition(x1, join(complement(composition(x0, x2)), complement(complement(composition(x0, x2)))))))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by axiom 7 (def_top) R->L }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x0, x2), join(complement(composition(x0, x2)), composition(x1, top)))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by lemma 79 }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x0, x2), composition(x1, top))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by lemma 43 }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(meet(composition(x1, top), composition(x0, x2))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by lemma 43 }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(complement(meet(composition(x1, top), composition(x0, x2))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.31 = { by lemma 125 }
% 60.56/8.31 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(composition(x1, x2), meet(composition(x0, x2), complement(meet(composition(x1, top), composition(x0, x2)))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.32 = { by lemma 108 }
% 60.56/8.32 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(composition(x1, x2), meet(composition(x0, x2), complement(composition(x1, top))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.32 = { by lemma 125 R->L }
% 60.56/8.32 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(complement(composition(x1, top)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.32 = { by lemma 43 }
% 60.56/8.32 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), complement(composition(x1, top)))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.32 = { by lemma 124 }
% 60.56/8.32 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(meet(composition(x0, x2), composition(x1, x2)), complement(composition(x1, top)))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.32 = { by lemma 18 R->L }
% 60.56/8.32 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(meet(composition(x0, x2), composition(x1, x2)), complement(composition(x1, x2))), complement(join(complement(meet(composition(x0, x2), composition(x1, x2))), complement(composition(x1, x2))))), complement(composition(x1, top)))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.32 = { by lemma 53 R->L }
% 60.56/8.32 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(complement(join(composition(x1, x2), complement(meet(composition(x0, x2), composition(x1, x2))))), complement(join(complement(meet(composition(x0, x2), composition(x1, x2))), complement(composition(x1, x2))))), complement(composition(x1, top)))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.32 = { by lemma 43 R->L }
% 60.56/8.32 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(complement(join(composition(x1, x2), complement(meet(composition(x1, x2), composition(x0, x2))))), complement(join(complement(meet(composition(x0, x2), composition(x1, x2))), complement(composition(x1, x2))))), complement(composition(x1, top)))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.32 = { by lemma 62 }
% 60.56/8.32 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(complement(top), complement(join(complement(meet(composition(x0, x2), composition(x1, x2))), complement(composition(x1, x2))))), complement(composition(x1, top)))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.32 = { by lemma 17 }
% 60.56/8.32 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(zero, complement(join(complement(meet(composition(x0, x2), composition(x1, x2))), complement(composition(x1, x2))))), complement(composition(x1, top)))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.32 = { by lemma 28 }
% 60.56/8.32 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(complement(join(complement(meet(composition(x0, x2), composition(x1, x2))), complement(composition(x1, x2)))), complement(composition(x1, top)))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.32 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 60.56/8.32 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(meet(meet(composition(x0, x2), composition(x1, x2)), composition(x1, x2)), complement(composition(x1, top)))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.32 = { by lemma 43 }
% 60.56/8.32 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(complement(composition(x1, top)), meet(meet(composition(x0, x2), composition(x1, x2)), composition(x1, x2)))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.32 = { by lemma 76 R->L }
% 60.56/8.32 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(meet(composition(x0, x2), composition(x1, x2)), meet(complement(composition(x1, top)), composition(x1, x2)))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.32 = { by lemma 124 R->L }
% 60.56/8.32 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), meet(complement(composition(x1, top)), composition(x1, x2)))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.32 = { by lemma 43 R->L }
% 60.56/8.32 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), meet(composition(x1, x2), complement(composition(x1, top))))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.32 = { by lemma 121 }
% 60.56/8.32 tuple(join(complement(complement(composition(meet(x0, x1), x2))), meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), zero)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.32 = { by lemma 85 }
% 60.56/8.32 tuple(join(complement(complement(composition(meet(x0, x1), x2))), zero), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.32 = { by lemma 27 }
% 60.56/8.32 tuple(complement(complement(composition(meet(x0, x1), x2))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.32 = { by lemma 42 }
% 60.56/8.32 tuple(composition(meet(x0, x1), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 60.56/8.32 = { by lemma 124 }
% 60.56/8.32 tuple(composition(meet(x0, x1), x2), meet(composition(x0, x2), composition(x1, x2)))
% 60.56/8.32 % SZS output end Proof
% 60.56/8.32
% 60.56/8.32 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------