TSTP Solution File: REL033+2 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL033+2 : TPTP v8.1.2. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n021.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 13:44:18 EDT 2023
% Result : Theorem 83.05s 11.11s
% Output : Proof 84.22s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL033+2 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.34 % Computer : n021.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Fri Aug 25 21:48:41 EDT 2023
% 0.12/0.34 % CPUTime :
% 83.05/11.11 Command-line arguments: --no-flatten-goal
% 83.05/11.11
% 83.05/11.11 % SZS status Theorem
% 83.05/11.11
% 83.81/11.20 % SZS output start Proof
% 83.81/11.20 Take the following subset of the input axioms:
% 83.81/11.20 fof(composition_associativity, axiom, ![X0, X1, X2]: composition(X0, composition(X1, X2))=composition(composition(X0, X1), X2)).
% 83.81/11.20 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))).
% 83.81/11.20 fof(composition_identity, axiom, ![X0_2]: composition(X0_2, one)=X0_2).
% 83.81/11.20 fof(converse_additivity, axiom, ![X0_2, X1_2]: converse(join(X0_2, X1_2))=join(converse(X0_2), converse(X1_2))).
% 83.81/11.20 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)).
% 83.81/11.20 fof(converse_idempotence, axiom, ![X0_2]: converse(converse(X0_2))=X0_2).
% 83.81/11.20 fof(converse_multiplicativity, axiom, ![X0_2, X1_2]: converse(composition(X0_2, X1_2))=composition(converse(X1_2), converse(X0_2))).
% 83.81/11.20 fof(def_top, axiom, ![X0_2]: top=join(X0_2, complement(X0_2))).
% 83.81/11.20 fof(def_zero, axiom, ![X0_2]: zero=meet(X0_2, complement(X0_2))).
% 83.81/11.20 fof(goals, conjecture, ![X0_2, X1_2, X2_2]: (composition(X0_2, top)=X0_2 => (join(composition(meet(X0_2, X1_2), X2_2), meet(X0_2, composition(X1_2, X2_2)))=meet(X0_2, composition(X1_2, X2_2)) & join(meet(X0_2, composition(X1_2, X2_2)), composition(meet(X0_2, X1_2), X2_2))=composition(meet(X0_2, X1_2), X2_2)))).
% 83.81/11.20 fof(maddux1_join_commutativity, axiom, ![X0_2, X1_2]: join(X0_2, X1_2)=join(X1_2, X0_2)).
% 83.81/11.20 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)).
% 83.81/11.20 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)))).
% 83.81/11.20 fof(maddux4_definiton_of_meet, axiom, ![X0_2, X1_2]: meet(X0_2, X1_2)=complement(join(complement(X0_2), complement(X1_2)))).
% 83.81/11.20
% 83.81/11.20 Now clausify the problem and encode Horn clauses using encoding 3 of
% 83.81/11.20 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 83.81/11.20 We repeatedly replace C & s=t => u=v by the two clauses:
% 83.81/11.20 fresh(y, y, x1...xn) = u
% 83.81/11.20 C => fresh(s, t, x1...xn) = v
% 83.81/11.20 where fresh is a fresh function symbol and x1..xn are the free
% 83.81/11.20 variables of u and v.
% 83.81/11.20 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 83.81/11.20 input problem has no model of domain size 1).
% 83.81/11.20
% 83.81/11.20 The encoding turns the above axioms into the following unit equations and goals:
% 83.81/11.20
% 83.81/11.20 Axiom 1 (converse_idempotence): converse(converse(X)) = X.
% 83.81/11.20 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 83.81/11.20 Axiom 3 (composition_identity): composition(X, one) = X.
% 83.81/11.20 Axiom 4 (goals): composition(x0, top) = x0.
% 83.81/11.20 Axiom 5 (def_zero): zero = meet(X, complement(X)).
% 83.81/11.20 Axiom 6 (def_top): top = join(X, complement(X)).
% 83.81/11.20 Axiom 7 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 83.81/11.20 Axiom 8 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 83.81/11.20 Axiom 9 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 83.81/11.20 Axiom 10 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 83.81/11.20 Axiom 11 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 83.81/11.20 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 83.81/11.20 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 83.81/11.20 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 83.81/11.20
% 83.81/11.20 Lemma 15: complement(top) = zero.
% 83.81/11.20 Proof:
% 83.81/11.20 complement(top)
% 83.81/11.20 = { by axiom 6 (def_top) }
% 83.81/11.20 complement(join(complement(X), complement(complement(X))))
% 83.81/11.20 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 83.81/11.20 meet(X, complement(X))
% 83.81/11.20 = { by axiom 5 (def_zero) R->L }
% 83.81/11.20 zero
% 83.81/11.20
% 83.81/11.20 Lemma 16: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 83.81/11.20 Proof:
% 83.81/11.20 converse(composition(converse(X), Y))
% 83.81/11.20 = { by axiom 9 (converse_multiplicativity) }
% 83.81/11.20 composition(converse(Y), converse(converse(X)))
% 83.81/11.20 = { by axiom 1 (converse_idempotence) }
% 83.81/11.20 composition(converse(Y), X)
% 83.81/11.20
% 83.81/11.20 Lemma 17: composition(converse(one), X) = X.
% 83.81/11.20 Proof:
% 83.81/11.20 composition(converse(one), X)
% 83.81/11.20 = { by lemma 16 R->L }
% 83.81/11.20 converse(composition(converse(X), one))
% 83.81/11.20 = { by axiom 3 (composition_identity) }
% 83.81/11.20 converse(converse(X))
% 83.81/11.20 = { by axiom 1 (converse_idempotence) }
% 83.81/11.20 X
% 83.81/11.20
% 83.81/11.20 Lemma 18: composition(one, X) = X.
% 83.81/11.20 Proof:
% 83.81/11.20 composition(one, X)
% 83.81/11.20 = { by lemma 17 R->L }
% 83.81/11.20 composition(converse(one), composition(one, X))
% 83.81/11.20 = { by axiom 10 (composition_associativity) }
% 83.81/11.20 composition(composition(converse(one), one), X)
% 83.81/11.20 = { by axiom 3 (composition_identity) }
% 83.81/11.20 composition(converse(one), X)
% 83.81/11.20 = { by lemma 17 }
% 83.81/11.20 X
% 83.81/11.20
% 83.81/11.20 Lemma 19: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 83.81/11.20 Proof:
% 83.81/11.20 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 83.81/11.20 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 83.81/11.20 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 83.81/11.20 = { by axiom 13 (converse_cancellativity) }
% 83.81/11.20 complement(X)
% 83.81/11.20
% 83.81/11.20 Lemma 20: join(complement(X), complement(X)) = complement(X).
% 83.81/11.20 Proof:
% 83.81/11.20 join(complement(X), complement(X))
% 83.81/11.20 = { by lemma 17 R->L }
% 83.81/11.20 join(complement(X), composition(converse(one), complement(X)))
% 83.81/11.20 = { by lemma 18 R->L }
% 83.81/11.20 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 83.81/11.20 = { by lemma 19 }
% 83.81/11.20 complement(X)
% 83.81/11.20
% 83.81/11.20 Lemma 21: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 83.81/11.20 Proof:
% 83.81/11.20 join(meet(X, Y), complement(join(complement(X), Y)))
% 83.81/11.20 = { by axiom 11 (maddux4_definiton_of_meet) }
% 83.81/11.20 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 83.81/11.20 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L }
% 83.81/11.20 X
% 83.81/11.20
% 83.81/11.20 Lemma 22: join(zero, meet(X, X)) = X.
% 83.81/11.20 Proof:
% 83.81/11.20 join(zero, meet(X, X))
% 83.81/11.20 = { by axiom 11 (maddux4_definiton_of_meet) }
% 83.81/11.20 join(zero, complement(join(complement(X), complement(X))))
% 83.81/11.20 = { by axiom 5 (def_zero) }
% 83.81/11.20 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 83.81/11.20 = { by lemma 21 }
% 83.81/11.20 X
% 83.81/11.20
% 83.81/11.20 Lemma 23: join(zero, join(X, meet(Y, Y))) = join(X, Y).
% 83.81/11.20 Proof:
% 83.81/11.20 join(zero, join(X, meet(Y, Y)))
% 83.81/11.20 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 83.81/11.20 join(zero, join(meet(Y, Y), X))
% 83.81/11.20 = { by axiom 8 (maddux2_join_associativity) }
% 83.81/11.20 join(join(zero, meet(Y, Y)), X)
% 83.81/11.20 = { by lemma 22 }
% 83.81/11.20 join(Y, X)
% 83.81/11.20 = { by axiom 2 (maddux1_join_commutativity) }
% 83.81/11.20 join(X, Y)
% 83.81/11.20
% 83.81/11.20 Lemma 24: join(X, join(Y, complement(X))) = join(Y, top).
% 83.81/11.20 Proof:
% 83.81/11.20 join(X, join(Y, complement(X)))
% 83.81/11.20 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 83.81/11.20 join(X, join(complement(X), Y))
% 83.81/11.20 = { by axiom 8 (maddux2_join_associativity) }
% 83.81/11.20 join(join(X, complement(X)), Y)
% 83.81/11.20 = { by axiom 6 (def_top) R->L }
% 83.81/11.20 join(top, Y)
% 83.81/11.20 = { by axiom 2 (maddux1_join_commutativity) }
% 83.81/11.20 join(Y, top)
% 83.81/11.20
% 83.81/11.20 Lemma 25: join(top, complement(X)) = top.
% 83.81/11.20 Proof:
% 83.81/11.20 join(top, complement(X))
% 83.81/11.20 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 83.81/11.20 join(complement(X), top)
% 83.81/11.20 = { by lemma 24 R->L }
% 83.81/11.20 join(X, join(complement(X), complement(X)))
% 83.81/11.20 = { by lemma 20 }
% 83.81/11.20 join(X, complement(X))
% 83.81/11.20 = { by axiom 6 (def_top) R->L }
% 83.81/11.20 top
% 83.81/11.20
% 83.81/11.20 Lemma 26: join(Y, top) = join(X, top).
% 83.81/11.20 Proof:
% 83.81/11.20 join(Y, top)
% 83.81/11.20 = { by lemma 25 R->L }
% 83.81/11.20 join(Y, join(top, complement(Y)))
% 83.81/11.20 = { by lemma 24 }
% 83.81/11.20 join(top, top)
% 83.81/11.20 = { by lemma 24 R->L }
% 83.81/11.20 join(X, join(top, complement(X)))
% 83.81/11.20 = { by lemma 25 }
% 83.81/11.20 join(X, top)
% 83.81/11.20
% 83.81/11.20 Lemma 27: join(X, top) = top.
% 83.81/11.20 Proof:
% 83.81/11.20 join(X, top)
% 83.81/11.20 = { by lemma 26 }
% 83.81/11.20 join(zero, top)
% 83.81/11.20 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 83.81/11.20 join(top, zero)
% 83.81/11.20 = { by lemma 15 R->L }
% 83.81/11.20 join(top, complement(top))
% 83.81/11.20 = { by axiom 6 (def_top) R->L }
% 83.81/11.20 top
% 83.81/11.20
% 83.81/11.20 Lemma 28: join(X, join(complement(X), Y)) = top.
% 83.81/11.20 Proof:
% 83.81/11.20 join(X, join(complement(X), Y))
% 83.81/11.20 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 83.81/11.20 join(X, join(Y, complement(X)))
% 83.81/11.20 = { by lemma 24 }
% 83.81/11.20 join(Y, top)
% 83.81/11.20 = { by lemma 26 R->L }
% 83.81/11.20 join(Z, top)
% 83.81/11.20 = { by lemma 27 }
% 83.81/11.20 top
% 83.81/11.20
% 83.81/11.20 Lemma 29: join(X, complement(zero)) = top.
% 83.81/11.20 Proof:
% 83.81/11.21 join(X, complement(zero))
% 83.81/11.21 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 83.81/11.21 join(complement(zero), X)
% 83.81/11.21 = { by lemma 23 R->L }
% 83.81/11.21 join(zero, join(complement(zero), meet(X, X)))
% 83.81/11.21 = { by lemma 28 }
% 83.81/11.21 top
% 83.81/11.21
% 83.81/11.21 Lemma 30: complement(zero) = top.
% 83.81/11.21 Proof:
% 83.81/11.21 complement(zero)
% 83.81/11.21 = { by lemma 20 R->L }
% 83.81/11.21 join(complement(zero), complement(zero))
% 83.81/11.21 = { by lemma 29 }
% 83.81/11.21 top
% 83.81/11.21
% 83.81/11.21 Lemma 31: converse(one) = one.
% 83.81/11.21 Proof:
% 83.81/11.21 converse(one)
% 83.81/11.21 = { by axiom 3 (composition_identity) R->L }
% 83.81/11.21 composition(converse(one), one)
% 83.81/11.21 = { by lemma 17 }
% 83.81/11.21 one
% 83.81/11.21
% 83.81/11.21 Lemma 32: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 83.81/11.21 Proof:
% 83.81/11.21 converse(join(X, converse(Y)))
% 83.81/11.21 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 83.81/11.21 converse(join(converse(Y), X))
% 83.81/11.21 = { by axiom 7 (converse_additivity) }
% 83.81/11.21 join(converse(converse(Y)), converse(X))
% 83.81/11.21 = { by axiom 1 (converse_idempotence) }
% 83.81/11.21 join(Y, converse(X))
% 83.81/11.21
% 83.81/11.21 Lemma 33: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 83.81/11.21 Proof:
% 83.81/11.21 converse(join(converse(X), Y))
% 83.81/11.21 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 83.81/11.21 converse(join(Y, converse(X)))
% 83.81/11.21 = { by lemma 32 }
% 83.81/11.21 join(X, converse(Y))
% 83.81/11.21
% 83.81/11.21 Lemma 34: join(X, converse(top)) = top.
% 83.81/11.21 Proof:
% 83.81/11.21 join(X, converse(top))
% 83.81/11.21 = { by axiom 6 (def_top) }
% 83.81/11.21 join(X, converse(join(converse(complement(X)), complement(converse(complement(X))))))
% 83.81/11.21 = { by lemma 33 }
% 83.81/11.21 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 83.81/11.21 = { by lemma 28 }
% 83.81/11.21 top
% 83.81/11.21
% 83.81/11.21 Lemma 35: converse(top) = top.
% 83.81/11.21 Proof:
% 83.81/11.21 converse(top)
% 83.81/11.21 = { by lemma 27 R->L }
% 83.81/11.21 converse(join(X, top))
% 83.81/11.21 = { by axiom 7 (converse_additivity) }
% 83.81/11.21 join(converse(X), converse(top))
% 83.81/11.21 = { by lemma 34 }
% 83.81/11.21 top
% 83.81/11.21
% 83.81/11.21 Lemma 36: complement(complement(X)) = meet(X, X).
% 83.81/11.21 Proof:
% 83.81/11.21 complement(complement(X))
% 83.81/11.21 = { by lemma 20 R->L }
% 83.81/11.21 complement(join(complement(X), complement(X)))
% 83.81/11.21 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 83.81/11.21 meet(X, X)
% 83.81/11.21
% 83.81/11.21 Lemma 37: meet(Y, X) = meet(X, Y).
% 83.81/11.21 Proof:
% 83.81/11.21 meet(Y, X)
% 83.81/11.21 = { by axiom 11 (maddux4_definiton_of_meet) }
% 83.81/11.21 complement(join(complement(Y), complement(X)))
% 83.81/11.21 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 83.81/11.21 complement(join(complement(X), complement(Y)))
% 83.81/11.21 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 83.81/11.21 meet(X, Y)
% 83.81/11.21
% 83.81/11.21 Lemma 38: complement(join(zero, complement(X))) = meet(X, top).
% 83.81/11.21 Proof:
% 83.81/11.21 complement(join(zero, complement(X)))
% 83.81/11.21 = { by lemma 15 R->L }
% 83.81/11.21 complement(join(complement(top), complement(X)))
% 83.81/11.21 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 83.81/11.21 meet(top, X)
% 83.81/11.21 = { by lemma 37 R->L }
% 83.81/11.21 meet(X, top)
% 83.81/11.21
% 83.81/11.21 Lemma 39: join(meet(X, Y), meet(X, complement(Y))) = X.
% 83.81/11.21 Proof:
% 83.81/11.21 join(meet(X, Y), meet(X, complement(Y)))
% 83.81/11.21 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 83.81/11.21 join(meet(X, complement(Y)), meet(X, Y))
% 83.81/11.21 = { by axiom 11 (maddux4_definiton_of_meet) }
% 83.81/11.21 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 83.81/11.21 = { by lemma 21 }
% 83.81/11.21 X
% 83.81/11.21
% 83.81/11.21 Lemma 40: join(zero, meet(X, top)) = X.
% 83.81/11.21 Proof:
% 83.81/11.21 join(zero, meet(X, top))
% 83.81/11.21 = { by lemma 30 R->L }
% 83.81/11.21 join(zero, meet(X, complement(zero)))
% 83.81/11.21 = { by lemma 15 R->L }
% 83.81/11.21 join(complement(top), meet(X, complement(zero)))
% 83.81/11.21 = { by lemma 29 R->L }
% 83.81/11.21 join(complement(join(complement(X), complement(zero))), meet(X, complement(zero)))
% 83.81/11.21 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 83.81/11.21 join(meet(X, zero), meet(X, complement(zero)))
% 83.81/11.21 = { by lemma 39 }
% 83.81/11.21 X
% 83.81/11.21
% 83.81/11.21 Lemma 41: join(zero, complement(X)) = complement(X).
% 83.81/11.21 Proof:
% 83.81/11.21 join(zero, complement(X))
% 83.81/11.21 = { by lemma 22 R->L }
% 83.81/11.21 join(zero, complement(join(zero, meet(X, X))))
% 83.81/11.21 = { by lemma 36 R->L }
% 83.81/11.21 join(zero, complement(join(zero, complement(complement(X)))))
% 83.81/11.21 = { by lemma 38 }
% 83.81/11.21 join(zero, meet(complement(X), top))
% 83.81/11.21 = { by lemma 40 }
% 83.81/11.21 complement(X)
% 83.81/11.21
% 83.81/11.21 Lemma 42: complement(complement(X)) = X.
% 83.81/11.21 Proof:
% 83.81/11.21 complement(complement(X))
% 83.81/11.21 = { by lemma 41 R->L }
% 83.81/11.21 join(zero, complement(complement(X)))
% 83.81/11.21 = { by lemma 36 }
% 83.81/11.21 join(zero, meet(X, X))
% 83.81/11.21 = { by lemma 22 }
% 83.81/11.21 X
% 83.81/11.21
% 83.81/11.21 Lemma 43: join(X, X) = X.
% 83.81/11.21 Proof:
% 83.81/11.21 join(X, X)
% 83.81/11.21 = { by lemma 42 R->L }
% 83.81/11.21 join(X, complement(complement(X)))
% 83.81/11.21 = { by lemma 42 R->L }
% 83.81/11.21 join(complement(complement(X)), complement(complement(X)))
% 83.81/11.21 = { by lemma 20 }
% 83.81/11.21 complement(complement(X))
% 83.81/11.21 = { by lemma 42 }
% 83.81/11.21 X
% 83.81/11.21
% 83.81/11.21 Lemma 44: join(X, zero) = X.
% 83.81/11.21 Proof:
% 83.81/11.21 join(X, zero)
% 83.81/11.21 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 83.81/11.21 join(zero, X)
% 83.81/11.21 = { by lemma 42 R->L }
% 83.81/11.21 join(zero, complement(complement(X)))
% 83.81/11.21 = { by lemma 36 }
% 83.81/11.21 join(zero, meet(X, X))
% 83.81/11.21 = { by lemma 22 }
% 83.81/11.21 X
% 83.81/11.21
% 83.81/11.21 Lemma 45: join(top, X) = top.
% 83.81/11.21 Proof:
% 83.81/11.21 join(top, X)
% 83.81/11.21 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 83.81/11.21 join(X, top)
% 83.81/11.21 = { by lemma 26 R->L }
% 83.81/11.21 join(Y, top)
% 83.81/11.21 = { by lemma 27 }
% 83.81/11.21 top
% 83.81/11.21
% 83.81/11.21 Lemma 46: join(zero, X) = X.
% 83.81/11.21 Proof:
% 83.81/11.21 join(zero, X)
% 83.81/11.21 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 83.81/11.21 join(X, zero)
% 83.81/11.21 = { by lemma 44 }
% 83.81/11.21 X
% 83.81/11.21
% 83.81/11.21 Lemma 47: meet(X, X) = X.
% 83.81/11.21 Proof:
% 83.81/11.21 meet(X, X)
% 83.81/11.21 = { by lemma 36 R->L }
% 83.81/11.21 complement(complement(X))
% 83.81/11.21 = { by lemma 42 }
% 83.81/11.21 X
% 83.81/11.21
% 83.81/11.21 Lemma 48: meet(X, top) = X.
% 83.81/11.21 Proof:
% 83.81/11.21 meet(X, top)
% 83.81/11.21 = { by lemma 38 R->L }
% 83.81/11.21 complement(join(zero, complement(X)))
% 83.81/11.21 = { by lemma 41 R->L }
% 83.81/11.21 join(zero, complement(join(zero, complement(X))))
% 83.81/11.21 = { by lemma 38 }
% 83.81/11.21 join(zero, meet(X, top))
% 83.81/11.21 = { by lemma 40 }
% 83.81/11.21 X
% 83.81/11.21
% 83.81/11.21 Lemma 49: meet(top, X) = X.
% 83.81/11.21 Proof:
% 83.81/11.21 meet(top, X)
% 83.81/11.21 = { by lemma 37 }
% 83.81/11.21 meet(X, top)
% 83.81/11.21 = { by lemma 48 }
% 83.81/11.21 X
% 83.81/11.21
% 83.81/11.21 Lemma 50: composition(join(one, Y), X) = join(X, composition(Y, X)).
% 83.81/11.21 Proof:
% 83.81/11.21 composition(join(one, Y), X)
% 83.81/11.21 = { by axiom 12 (composition_distributivity) }
% 83.81/11.21 join(composition(one, X), composition(Y, X))
% 83.81/11.21 = { by lemma 18 }
% 83.81/11.21 join(X, composition(Y, X))
% 83.81/11.21
% 83.81/11.21 Lemma 51: join(X, composition(top, X)) = composition(top, X).
% 83.81/11.21 Proof:
% 83.81/11.21 join(X, composition(top, X))
% 83.81/11.21 = { by lemma 35 R->L }
% 83.81/11.21 join(X, composition(converse(top), X))
% 83.81/11.21 = { by lemma 50 R->L }
% 83.81/11.21 composition(join(one, converse(top)), X)
% 83.81/11.21 = { by lemma 34 }
% 83.81/11.21 composition(top, X)
% 83.81/11.21
% 83.81/11.21 Lemma 52: composition(top, zero) = zero.
% 83.81/11.21 Proof:
% 83.81/11.21 composition(top, zero)
% 83.81/11.21 = { by lemma 35 R->L }
% 83.81/11.21 composition(converse(top), zero)
% 83.81/11.21 = { by lemma 46 R->L }
% 83.81/11.21 join(zero, composition(converse(top), zero))
% 83.81/11.21 = { by lemma 15 R->L }
% 83.81/11.21 join(complement(top), composition(converse(top), zero))
% 83.81/11.21 = { by lemma 15 R->L }
% 83.81/11.21 join(complement(top), composition(converse(top), complement(top)))
% 83.81/11.21 = { by lemma 45 R->L }
% 83.81/11.21 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 83.81/11.21 = { by lemma 51 }
% 83.81/11.21 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 83.81/11.21 = { by lemma 19 }
% 83.81/11.21 complement(top)
% 83.81/11.21 = { by lemma 15 }
% 83.81/11.21 zero
% 83.81/11.21
% 83.81/11.21 Lemma 53: composition(converse(x0), complement(x0)) = zero.
% 83.81/11.21 Proof:
% 83.81/11.21 composition(converse(x0), complement(x0))
% 83.81/11.21 = { by lemma 46 R->L }
% 83.81/11.21 join(zero, composition(converse(x0), complement(x0)))
% 83.81/11.21 = { by lemma 15 R->L }
% 83.81/11.21 join(complement(top), composition(converse(x0), complement(x0)))
% 83.81/11.21 = { by axiom 4 (goals) R->L }
% 83.81/11.21 join(complement(top), composition(converse(x0), complement(composition(x0, top))))
% 83.81/11.21 = { by lemma 19 }
% 83.81/11.21 complement(top)
% 83.81/11.21 = { by lemma 15 }
% 83.81/11.21 zero
% 83.81/11.21
% 83.81/11.21 Lemma 54: composition(converse(complement(x0)), composition(x0, X)) = converse(zero).
% 83.81/11.21 Proof:
% 83.81/11.21 composition(converse(complement(x0)), composition(x0, X))
% 83.81/11.21 = { by lemma 16 R->L }
% 83.81/11.21 converse(composition(converse(composition(x0, X)), complement(x0)))
% 83.81/11.21 = { by axiom 9 (converse_multiplicativity) }
% 83.81/11.21 converse(composition(composition(converse(X), converse(x0)), complement(x0)))
% 83.81/11.21 = { by axiom 10 (composition_associativity) R->L }
% 83.81/11.21 converse(composition(converse(X), composition(converse(x0), complement(x0))))
% 83.81/11.21 = { by lemma 53 }
% 83.81/11.21 converse(composition(converse(X), zero))
% 83.81/11.21 = { by lemma 46 R->L }
% 83.81/11.21 converse(join(zero, composition(converse(X), zero)))
% 83.81/11.21 = { by lemma 52 R->L }
% 83.81/11.21 converse(join(composition(top, zero), composition(converse(X), zero)))
% 83.81/11.21 = { by axiom 12 (composition_distributivity) R->L }
% 83.81/11.21 converse(composition(join(top, converse(X)), zero))
% 83.81/11.21 = { by lemma 45 }
% 83.81/11.21 converse(composition(top, zero))
% 83.81/11.21 = { by lemma 52 }
% 83.81/11.21 converse(zero)
% 83.81/11.21
% 83.81/11.21 Lemma 55: join(converse(zero), X) = X.
% 83.81/11.21 Proof:
% 83.81/11.21 join(converse(zero), X)
% 83.81/11.21 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 83.81/11.21 join(X, converse(zero))
% 83.81/11.21 = { by lemma 54 R->L }
% 83.81/11.21 join(X, composition(converse(complement(x0)), composition(x0, composition(Y, X))))
% 83.81/11.21 = { by axiom 10 (composition_associativity) }
% 83.81/11.21 join(X, composition(converse(complement(x0)), composition(composition(x0, Y), X)))
% 83.81/11.21 = { by axiom 10 (composition_associativity) }
% 83.81/11.21 join(X, composition(composition(converse(complement(x0)), composition(x0, Y)), X))
% 83.81/11.21 = { by lemma 54 }
% 83.81/11.21 join(X, composition(converse(zero), X))
% 83.81/11.21 = { by lemma 50 R->L }
% 83.81/11.21 composition(join(one, converse(zero)), X)
% 83.81/11.21 = { by lemma 31 R->L }
% 83.81/11.21 composition(join(converse(one), converse(zero)), X)
% 83.81/11.21 = { by axiom 7 (converse_additivity) R->L }
% 83.81/11.21 composition(converse(join(one, zero)), X)
% 83.81/11.21 = { by axiom 2 (maddux1_join_commutativity) }
% 83.81/11.21 composition(converse(join(zero, one)), X)
% 83.81/11.21 = { by lemma 46 }
% 83.81/11.21 composition(converse(one), X)
% 83.81/11.21 = { by lemma 17 }
% 83.81/11.21 X
% 83.81/11.21
% 83.81/11.21 Lemma 56: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 83.81/11.21 Proof:
% 83.81/11.21 join(meet(X, Y), meet(Y, complement(X)))
% 83.81/11.21 = { by lemma 37 }
% 83.81/11.21 join(meet(Y, X), meet(Y, complement(X)))
% 83.81/11.21 = { by lemma 39 }
% 83.81/11.21 Y
% 83.81/11.21
% 83.81/11.21 Lemma 57: join(meet(X, Y), meet(complement(X), Y)) = Y.
% 83.81/11.21 Proof:
% 83.81/11.21 join(meet(X, Y), meet(complement(X), Y))
% 83.81/11.21 = { by lemma 37 }
% 83.81/11.21 join(meet(X, Y), meet(Y, complement(X)))
% 83.81/11.21 = { by lemma 56 }
% 83.81/11.21 Y
% 83.81/11.21
% 83.81/11.21 Lemma 58: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
% 83.81/11.21 Proof:
% 83.81/11.21 complement(join(complement(X), meet(Y, Z)))
% 83.81/11.21 = { by lemma 37 }
% 83.81/11.21 complement(join(complement(X), meet(Z, Y)))
% 83.81/11.21 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 83.81/11.21 complement(join(meet(Z, Y), complement(X)))
% 83.81/11.21 = { by axiom 11 (maddux4_definiton_of_meet) }
% 83.81/11.21 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 83.81/11.21 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 83.81/11.21 meet(join(complement(Z), complement(Y)), X)
% 83.81/11.21 = { by lemma 37 R->L }
% 83.81/11.21 meet(X, join(complement(Z), complement(Y)))
% 83.81/11.21 = { by axiom 2 (maddux1_join_commutativity) }
% 83.81/11.21 meet(X, join(complement(Y), complement(Z)))
% 83.81/11.21
% 83.81/11.21 Lemma 59: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 83.81/11.21 Proof:
% 83.81/11.22 join(complement(X), complement(Y))
% 83.81/11.22 = { by lemma 49 R->L }
% 83.81/11.22 meet(top, join(complement(X), complement(Y)))
% 83.81/11.22 = { by lemma 58 R->L }
% 83.81/11.22 complement(join(complement(top), meet(X, Y)))
% 83.81/11.22 = { by lemma 15 }
% 83.81/11.22 complement(join(zero, meet(X, Y)))
% 83.81/11.22 = { by lemma 37 R->L }
% 83.81/11.22 complement(join(zero, meet(Y, X)))
% 83.81/11.22 = { by axiom 2 (maddux1_join_commutativity) }
% 83.81/11.22 complement(join(meet(Y, X), zero))
% 83.81/11.22 = { by lemma 44 }
% 83.81/11.22 complement(meet(Y, X))
% 83.81/11.22 = { by lemma 37 R->L }
% 83.81/11.22 complement(meet(X, Y))
% 83.81/11.22
% 83.81/11.22 Lemma 60: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 83.81/11.22 Proof:
% 83.81/11.22 complement(meet(X, complement(Y)))
% 83.81/11.22 = { by lemma 37 }
% 83.81/11.22 complement(meet(complement(Y), X))
% 83.81/11.22 = { by lemma 41 R->L }
% 83.81/11.22 complement(meet(join(zero, complement(Y)), X))
% 83.81/11.22 = { by lemma 59 R->L }
% 83.81/11.22 join(complement(join(zero, complement(Y))), complement(X))
% 83.81/11.22 = { by lemma 38 }
% 83.81/11.22 join(meet(Y, top), complement(X))
% 83.81/11.22 = { by lemma 48 }
% 83.81/11.22 join(Y, complement(X))
% 83.81/11.22
% 83.81/11.22 Lemma 61: meet(meet(X, Y), Z) = meet(X, meet(Z, Y)).
% 83.81/11.22 Proof:
% 83.81/11.22 meet(meet(X, Y), Z)
% 83.81/11.22 = { by lemma 37 }
% 83.81/11.22 meet(Z, meet(X, Y))
% 83.81/11.22 = { by lemma 47 R->L }
% 83.81/11.22 meet(meet(Z, meet(X, Y)), meet(Z, meet(X, Y)))
% 83.81/11.22 = { by lemma 36 R->L }
% 83.81/11.22 complement(complement(meet(Z, meet(X, Y))))
% 83.81/11.22 = { by lemma 37 }
% 83.81/11.22 complement(complement(meet(Z, meet(Y, X))))
% 83.81/11.22 = { by lemma 59 R->L }
% 83.81/11.22 complement(join(complement(Z), complement(meet(Y, X))))
% 83.81/11.22 = { by lemma 59 R->L }
% 83.81/11.22 complement(join(complement(Z), join(complement(Y), complement(X))))
% 83.81/11.22 = { by axiom 8 (maddux2_join_associativity) }
% 83.81/11.22 complement(join(join(complement(Z), complement(Y)), complement(X)))
% 83.81/11.22 = { by lemma 60 R->L }
% 83.81/11.22 complement(complement(meet(X, complement(join(complement(Z), complement(Y))))))
% 83.81/11.22 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 83.81/11.22 complement(complement(meet(X, meet(Z, Y))))
% 83.81/11.22 = { by lemma 37 R->L }
% 83.81/11.22 complement(complement(meet(X, meet(Y, Z))))
% 83.81/11.22 = { by lemma 42 }
% 83.81/11.22 meet(X, meet(Y, Z))
% 83.81/11.22 = { by lemma 37 R->L }
% 83.81/11.22 meet(X, meet(Z, Y))
% 83.81/11.22
% 83.81/11.22 Lemma 62: join(X, converse(join(Y, complement(converse(X))))) = top.
% 83.81/11.22 Proof:
% 83.81/11.22 join(X, converse(join(Y, complement(converse(X)))))
% 83.81/11.22 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 83.81/11.22 join(X, converse(join(complement(converse(X)), Y)))
% 83.81/11.22 = { by lemma 33 R->L }
% 83.81/11.22 converse(join(converse(X), join(complement(converse(X)), Y)))
% 83.81/11.22 = { by lemma 28 }
% 83.81/11.22 converse(top)
% 83.81/11.22 = { by lemma 35 }
% 83.81/11.22 top
% 83.81/11.22
% 83.81/11.22 Lemma 63: meet(X, converse(complement(converse(complement(X))))) = X.
% 83.81/11.22 Proof:
% 83.81/11.22 meet(X, converse(complement(converse(complement(X)))))
% 83.81/11.22 = { by lemma 20 R->L }
% 83.81/11.22 meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X))))))
% 83.81/11.22 = { by lemma 44 R->L }
% 83.81/11.22 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), zero)
% 83.81/11.22 = { by lemma 15 R->L }
% 83.81/11.22 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), complement(top))
% 83.81/11.22 = { by lemma 62 R->L }
% 83.81/11.22 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), complement(join(complement(X), converse(join(complement(converse(complement(X))), complement(converse(complement(X))))))))
% 83.81/11.22 = { by lemma 21 }
% 83.81/11.22 X
% 83.81/11.22
% 83.81/11.22 Lemma 64: complement(converse(complement(X))) = converse(X).
% 83.81/11.22 Proof:
% 83.81/11.22 complement(converse(complement(X)))
% 83.81/11.22 = { by axiom 1 (converse_idempotence) R->L }
% 83.81/11.22 converse(converse(complement(converse(complement(X)))))
% 83.81/11.22 = { by lemma 57 R->L }
% 83.81/11.22 converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(complement(converse(complement(X)))))))
% 83.81/11.22 = { by axiom 1 (converse_idempotence) R->L }
% 83.81/11.22 converse(join(meet(X, converse(complement(converse(complement(X))))), meet(converse(converse(complement(X))), converse(complement(converse(complement(X)))))))
% 83.81/11.22 = { by lemma 49 R->L }
% 83.81/11.22 converse(join(meet(X, converse(complement(converse(complement(X))))), meet(converse(converse(complement(X))), meet(top, converse(complement(converse(complement(X))))))))
% 84.22/11.22 = { by lemma 61 R->L }
% 84.22/11.22 converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), top)))
% 84.22/11.22 = { by lemma 28 R->L }
% 84.22/11.22 converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X)))), join(complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))))), converse(complement(join(complement(converse(complement(converse(complement(converse(complement(X))))))), converse(complement(X))))))))))
% 84.22/11.22 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 84.22/11.22 converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X)))), join(converse(complement(join(complement(converse(complement(converse(complement(converse(complement(X))))))), converse(complement(X))))), complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))))))))))
% 84.22/11.22 = { by axiom 8 (maddux2_join_associativity) }
% 84.22/11.22 converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(join(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X)))), converse(complement(join(complement(converse(complement(converse(complement(converse(complement(X))))))), converse(complement(X)))))), complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X)))))))))
% 84.22/11.22 = { by axiom 7 (converse_additivity) R->L }
% 84.22/11.22 converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(converse(join(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))), complement(join(complement(converse(complement(converse(complement(converse(complement(X))))))), converse(complement(X)))))), complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X)))))))))
% 84.22/11.22 = { by axiom 2 (maddux1_join_commutativity) }
% 84.22/11.22 converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))))), converse(join(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))), complement(join(complement(converse(complement(converse(complement(converse(complement(X))))))), converse(complement(X))))))))))
% 84.22/11.22 = { by lemma 21 }
% 84.22/11.22 converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))))), converse(converse(complement(converse(complement(converse(complement(X)))))))))))
% 84.22/11.22 = { by axiom 2 (maddux1_join_commutativity) }
% 84.22/11.22 converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(converse(converse(complement(converse(complement(converse(complement(X))))))), complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X)))))))))
% 84.22/11.22 = { by axiom 1 (converse_idempotence) }
% 84.22/11.22 converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(complement(converse(complement(converse(complement(X))))), complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X)))))))))
% 84.22/11.22 = { by lemma 37 R->L }
% 84.22/11.22 converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(complement(converse(complement(converse(complement(X))))), complement(converse(meet(converse(complement(X)), converse(complement(converse(complement(converse(complement(X)))))))))))))
% 84.22/11.22 = { by lemma 63 }
% 84.22/11.22 converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(complement(converse(complement(converse(complement(X))))), complement(converse(converse(complement(X))))))))
% 84.22/11.22 = { by lemma 59 }
% 84.22/11.22 converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), complement(meet(converse(complement(converse(complement(X)))), converse(converse(complement(X))))))))
% 84.22/11.22 = { by lemma 37 R->L }
% 84.22/11.22 converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), complement(meet(converse(converse(complement(X))), converse(complement(converse(complement(X)))))))))
% 84.22/11.22 = { by axiom 5 (def_zero) R->L }
% 84.22/11.22 converse(join(meet(X, converse(complement(converse(complement(X))))), zero))
% 84.22/11.22 = { by lemma 44 }
% 84.22/11.22 converse(meet(X, converse(complement(converse(complement(X))))))
% 84.22/11.22 = { by lemma 63 }
% 84.22/11.22 converse(X)
% 84.22/11.22
% 84.22/11.22 Lemma 65: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 84.22/11.22 Proof:
% 84.22/11.22 converse(composition(X, converse(Y)))
% 84.22/11.22 = { by axiom 9 (converse_multiplicativity) }
% 84.22/11.22 composition(converse(converse(Y)), converse(X))
% 84.22/11.22 = { by axiom 1 (converse_idempotence) }
% 84.22/11.22 composition(Y, converse(X))
% 84.22/11.22
% 84.22/11.22 Lemma 66: composition(complement(x0), top) = complement(x0).
% 84.22/11.22 Proof:
% 84.22/11.22 composition(complement(x0), top)
% 84.22/11.22 = { by lemma 35 R->L }
% 84.22/11.22 composition(complement(x0), converse(top))
% 84.22/11.22 = { by lemma 65 R->L }
% 84.22/11.22 converse(composition(top, converse(complement(x0))))
% 84.22/11.22 = { by lemma 51 R->L }
% 84.22/11.22 converse(join(converse(complement(x0)), composition(top, converse(complement(x0)))))
% 84.22/11.22 = { by lemma 33 }
% 84.22/11.22 join(complement(x0), converse(composition(top, converse(complement(x0)))))
% 84.22/11.22 = { by lemma 65 }
% 84.22/11.22 join(complement(x0), composition(complement(x0), converse(top)))
% 84.22/11.22 = { by lemma 35 }
% 84.22/11.22 join(complement(x0), composition(complement(x0), top))
% 84.22/11.22 = { by lemma 30 R->L }
% 84.22/11.22 join(complement(x0), composition(complement(x0), complement(zero)))
% 84.22/11.22 = { by lemma 55 R->L }
% 84.22/11.22 join(complement(x0), composition(complement(x0), complement(join(converse(zero), zero))))
% 84.22/11.22 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 84.22/11.22 join(complement(x0), composition(complement(x0), complement(join(zero, converse(zero)))))
% 84.22/11.22 = { by lemma 22 R->L }
% 84.22/11.22 join(complement(x0), composition(complement(x0), complement(join(zero, join(zero, meet(converse(zero), converse(zero)))))))
% 84.22/11.22 = { by axiom 11 (maddux4_definiton_of_meet) }
% 84.22/11.22 join(complement(x0), composition(complement(x0), complement(join(zero, join(zero, complement(join(complement(converse(zero)), complement(converse(zero)))))))))
% 84.22/11.22 = { by lemma 20 R->L }
% 84.22/11.22 join(complement(x0), composition(complement(x0), complement(join(zero, join(zero, join(complement(join(complement(converse(zero)), complement(converse(zero)))), complement(join(complement(converse(zero)), complement(converse(zero))))))))))
% 84.22/11.22 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 84.22/11.22 join(complement(x0), composition(complement(x0), complement(join(zero, join(zero, join(meet(converse(zero), converse(zero)), complement(join(complement(converse(zero)), complement(converse(zero))))))))))
% 84.22/11.22 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 84.22/11.22 join(complement(x0), composition(complement(x0), complement(join(zero, join(zero, join(meet(converse(zero), converse(zero)), meet(converse(zero), converse(zero))))))))
% 84.22/11.22 = { by lemma 23 }
% 84.22/11.23 join(complement(x0), composition(complement(x0), complement(join(zero, join(meet(converse(zero), converse(zero)), converse(zero))))))
% 84.22/11.23 = { by axiom 2 (maddux1_join_commutativity) }
% 84.22/11.23 join(complement(x0), composition(complement(x0), complement(join(zero, join(converse(zero), meet(converse(zero), converse(zero)))))))
% 84.22/11.23 = { by lemma 23 }
% 84.22/11.23 join(complement(x0), composition(complement(x0), complement(join(converse(zero), converse(zero)))))
% 84.22/11.23 = { by lemma 55 }
% 84.22/11.23 join(complement(x0), composition(complement(x0), complement(converse(zero))))
% 84.22/11.23 = { by axiom 1 (converse_idempotence) R->L }
% 84.22/11.23 join(complement(x0), composition(converse(converse(complement(x0))), complement(converse(zero))))
% 84.22/11.23 = { by lemma 53 R->L }
% 84.22/11.23 join(complement(x0), composition(converse(converse(complement(x0))), complement(converse(composition(converse(x0), complement(x0))))))
% 84.22/11.23 = { by lemma 16 }
% 84.22/11.23 join(complement(x0), composition(converse(converse(complement(x0))), complement(composition(converse(complement(x0)), x0))))
% 84.22/11.23 = { by lemma 19 }
% 84.22/11.23 complement(x0)
% 84.22/11.23
% 84.22/11.23 Lemma 67: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 84.22/11.23 Proof:
% 84.22/11.23 complement(meet(complement(X), Y))
% 84.22/11.23 = { by lemma 37 }
% 84.22/11.23 complement(meet(Y, complement(X)))
% 84.22/11.23 = { by lemma 60 }
% 84.22/11.23 join(X, complement(Y))
% 84.22/11.23
% 84.22/11.23 Lemma 68: join(composition(X, Y), join(Z, composition(W, Y))) = join(Z, composition(join(X, W), Y)).
% 84.22/11.23 Proof:
% 84.22/11.23 join(composition(X, Y), join(Z, composition(W, Y)))
% 84.22/11.23 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 84.22/11.23 join(composition(X, Y), join(composition(W, Y), Z))
% 84.22/11.23 = { by axiom 8 (maddux2_join_associativity) }
% 84.22/11.23 join(join(composition(X, Y), composition(W, Y)), Z)
% 84.22/11.23 = { by axiom 12 (composition_distributivity) R->L }
% 84.22/11.23 join(composition(join(X, W), Y), Z)
% 84.22/11.23 = { by axiom 2 (maddux1_join_commutativity) }
% 84.22/11.23 join(Z, composition(join(X, W), Y))
% 84.22/11.23
% 84.22/11.23 Lemma 69: join(complement(composition(X, Y)), composition(join(X, Z), Y)) = top.
% 84.22/11.23 Proof:
% 84.22/11.23 join(complement(composition(X, Y)), composition(join(X, Z), Y))
% 84.22/11.23 = { by lemma 68 R->L }
% 84.22/11.23 join(composition(X, Y), join(complement(composition(X, Y)), composition(Z, Y)))
% 84.22/11.23 = { by lemma 28 }
% 84.22/11.23 top
% 84.22/11.23
% 84.22/11.23 Lemma 70: meet(X, join(Y, complement(X))) = meet(X, Y).
% 84.22/11.23 Proof:
% 84.22/11.23 meet(X, join(Y, complement(X)))
% 84.22/11.23 = { by lemma 67 R->L }
% 84.22/11.23 meet(X, complement(meet(complement(Y), X)))
% 84.22/11.23 = { by lemma 39 R->L }
% 84.22/11.23 join(meet(meet(X, complement(meet(complement(Y), X))), Y), meet(meet(X, complement(meet(complement(Y), X))), complement(Y)))
% 84.22/11.23 = { by lemma 37 R->L }
% 84.22/11.23 join(meet(meet(X, complement(meet(complement(Y), X))), Y), meet(complement(Y), meet(X, complement(meet(complement(Y), X)))))
% 84.22/11.23 = { by lemma 59 R->L }
% 84.22/11.23 join(meet(meet(X, complement(meet(complement(Y), X))), Y), meet(complement(Y), meet(X, join(complement(complement(Y)), complement(X)))))
% 84.22/11.23 = { by axiom 11 (maddux4_definiton_of_meet) }
% 84.22/11.23 join(meet(meet(X, complement(meet(complement(Y), X))), Y), complement(join(complement(complement(Y)), complement(meet(X, join(complement(complement(Y)), complement(X)))))))
% 84.22/11.23 = { by lemma 59 R->L }
% 84.22/11.23 join(meet(meet(X, complement(meet(complement(Y), X))), Y), complement(join(complement(complement(Y)), join(complement(X), complement(join(complement(complement(Y)), complement(X)))))))
% 84.22/11.23 = { by axiom 8 (maddux2_join_associativity) }
% 84.22/11.23 join(meet(meet(X, complement(meet(complement(Y), X))), Y), complement(join(join(complement(complement(Y)), complement(X)), complement(join(complement(complement(Y)), complement(X))))))
% 84.22/11.23 = { by axiom 6 (def_top) R->L }
% 84.22/11.23 join(meet(meet(X, complement(meet(complement(Y), X))), Y), complement(top))
% 84.22/11.23 = { by lemma 15 }
% 84.22/11.23 join(meet(meet(X, complement(meet(complement(Y), X))), Y), zero)
% 84.22/11.23 = { by lemma 44 }
% 84.22/11.23 meet(meet(X, complement(meet(complement(Y), X))), Y)
% 84.22/11.23 = { by lemma 61 }
% 84.22/11.23 meet(X, meet(Y, complement(meet(complement(Y), X))))
% 84.22/11.23 = { by lemma 67 }
% 84.22/11.23 meet(X, meet(Y, join(Y, complement(X))))
% 84.22/11.23 = { by lemma 60 R->L }
% 84.22/11.23 meet(X, meet(Y, complement(meet(X, complement(Y)))))
% 84.22/11.23 = { by lemma 59 R->L }
% 84.22/11.23 meet(X, meet(Y, join(complement(X), complement(complement(Y)))))
% 84.22/11.23 = { by lemma 58 R->L }
% 84.22/11.23 meet(X, complement(join(complement(Y), meet(X, complement(Y)))))
% 84.22/11.23 = { by lemma 41 R->L }
% 84.22/11.23 meet(X, join(zero, complement(join(complement(Y), meet(X, complement(Y))))))
% 84.22/11.23 = { by lemma 15 R->L }
% 84.22/11.23 meet(X, join(complement(top), complement(join(complement(Y), meet(X, complement(Y))))))
% 84.22/11.23 = { by lemma 27 R->L }
% 84.22/11.23 meet(X, join(complement(join(complement(X), top)), complement(join(complement(Y), meet(X, complement(Y))))))
% 84.22/11.23 = { by lemma 24 R->L }
% 84.22/11.23 meet(X, join(complement(join(complement(Y), join(complement(X), complement(complement(Y))))), complement(join(complement(Y), meet(X, complement(Y))))))
% 84.22/11.23 = { by lemma 59 }
% 84.22/11.23 meet(X, join(complement(join(complement(Y), complement(meet(X, complement(Y))))), complement(join(complement(Y), meet(X, complement(Y))))))
% 84.22/11.23 = { by lemma 37 R->L }
% 84.22/11.23 meet(X, join(complement(join(complement(Y), complement(meet(complement(Y), X)))), complement(join(complement(Y), meet(X, complement(Y))))))
% 84.22/11.23 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 84.22/11.23 meet(X, join(meet(Y, meet(complement(Y), X)), complement(join(complement(Y), meet(X, complement(Y))))))
% 84.22/11.23 = { by lemma 37 R->L }
% 84.22/11.23 meet(X, join(meet(Y, meet(X, complement(Y))), complement(join(complement(Y), meet(X, complement(Y))))))
% 84.22/11.23 = { by lemma 21 }
% 84.22/11.23 meet(X, Y)
% 84.22/11.23
% 84.22/11.23 Lemma 71: meet(X, join(complement(X), Y)) = meet(X, Y).
% 84.22/11.23 Proof:
% 84.22/11.23 meet(X, join(complement(X), Y))
% 84.22/11.23 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 84.22/11.23 meet(X, join(Y, complement(X)))
% 84.22/11.23 = { by lemma 70 }
% 84.22/11.23 meet(X, Y)
% 84.22/11.23
% 84.22/11.23 Lemma 72: composition(join(X, complement(x0)), top) = join(complement(x0), composition(X, top)).
% 84.22/11.23 Proof:
% 84.22/11.23 composition(join(X, complement(x0)), top)
% 84.22/11.23 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 84.22/11.23 composition(join(complement(x0), X), top)
% 84.22/11.23 = { by axiom 12 (composition_distributivity) }
% 84.22/11.23 join(composition(complement(x0), top), composition(X, top))
% 84.22/11.23 = { by lemma 66 }
% 84.22/11.23 join(complement(x0), composition(X, top))
% 84.22/11.23
% 84.22/11.23 Lemma 73: join(complement(X), meet(X, one)) = join(one, complement(X)).
% 84.22/11.23 Proof:
% 84.22/11.23 join(complement(X), meet(X, one))
% 84.22/11.23 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 84.22/11.23 join(meet(X, one), complement(X))
% 84.22/11.23 = { by lemma 71 R->L }
% 84.22/11.23 join(meet(X, join(complement(X), one)), complement(X))
% 84.22/11.23 = { by lemma 21 R->L }
% 84.22/11.23 join(meet(X, join(complement(X), one)), join(meet(complement(X), converse(join(one, complement(converse(complement(complement(X))))))), complement(join(complement(complement(X)), converse(join(one, complement(converse(complement(complement(X))))))))))
% 84.22/11.23 = { by lemma 62 }
% 84.22/11.23 join(meet(X, join(complement(X), one)), join(meet(complement(X), converse(join(one, complement(converse(complement(complement(X))))))), complement(top)))
% 84.22/11.23 = { by lemma 15 }
% 84.22/11.23 join(meet(X, join(complement(X), one)), join(meet(complement(X), converse(join(one, complement(converse(complement(complement(X))))))), zero))
% 84.22/11.23 = { by lemma 44 }
% 84.22/11.23 join(meet(X, join(complement(X), one)), meet(complement(X), converse(join(one, complement(converse(complement(complement(X))))))))
% 84.22/11.23 = { by lemma 64 }
% 84.22/11.23 join(meet(X, join(complement(X), one)), meet(complement(X), converse(join(one, converse(complement(X))))))
% 84.22/11.23 = { by lemma 32 }
% 84.22/11.23 join(meet(X, join(complement(X), one)), meet(complement(X), join(complement(X), converse(one))))
% 84.22/11.23 = { by lemma 31 }
% 84.22/11.23 join(meet(X, join(complement(X), one)), meet(complement(X), join(complement(X), one)))
% 84.22/11.23 = { by lemma 57 }
% 84.22/11.23 join(complement(X), one)
% 84.22/11.23 = { by axiom 2 (maddux1_join_commutativity) }
% 84.22/11.23 join(one, complement(X))
% 84.22/11.23
% 84.22/11.23 Lemma 74: join(complement(x0), composition(join(X, complement(x0)), Y)) = join(complement(x0), composition(X, Y)).
% 84.22/11.23 Proof:
% 84.22/11.23 join(complement(x0), composition(join(X, complement(x0)), Y))
% 84.22/11.23 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 84.22/11.23 join(complement(x0), composition(join(complement(x0), X), Y))
% 84.22/11.23 = { by lemma 66 R->L }
% 84.22/11.23 join(composition(complement(x0), top), composition(join(complement(x0), X), Y))
% 84.22/11.23 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 84.22/11.23 join(composition(complement(x0), top), composition(join(X, complement(x0)), Y))
% 84.22/11.23 = { by lemma 68 R->L }
% 84.22/11.23 join(composition(X, Y), join(composition(complement(x0), top), composition(complement(x0), Y)))
% 84.22/11.23 = { by axiom 1 (converse_idempotence) R->L }
% 84.22/11.23 join(composition(X, Y), join(composition(complement(x0), top), composition(complement(x0), converse(converse(Y)))))
% 84.22/11.23 = { by axiom 1 (converse_idempotence) R->L }
% 84.22/11.23 join(composition(X, Y), converse(converse(join(composition(complement(x0), top), composition(complement(x0), converse(converse(Y)))))))
% 84.22/11.23 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 84.22/11.23 join(composition(X, Y), converse(converse(join(composition(complement(x0), converse(converse(Y))), composition(complement(x0), top)))))
% 84.22/11.23 = { by axiom 7 (converse_additivity) }
% 84.22/11.23 join(composition(X, Y), converse(join(converse(composition(complement(x0), converse(converse(Y)))), converse(composition(complement(x0), top)))))
% 84.22/11.23 = { by lemma 65 }
% 84.22/11.23 join(composition(X, Y), converse(join(composition(converse(Y), converse(complement(x0))), converse(composition(complement(x0), top)))))
% 84.22/11.23 = { by axiom 2 (maddux1_join_commutativity) }
% 84.22/11.23 join(composition(X, Y), converse(join(converse(composition(complement(x0), top)), composition(converse(Y), converse(complement(x0))))))
% 84.22/11.23 = { by axiom 9 (converse_multiplicativity) }
% 84.22/11.23 join(composition(X, Y), converse(join(composition(converse(top), converse(complement(x0))), composition(converse(Y), converse(complement(x0))))))
% 84.22/11.23 = { by axiom 12 (composition_distributivity) R->L }
% 84.22/11.23 join(composition(X, Y), converse(composition(join(converse(top), converse(Y)), converse(complement(x0)))))
% 84.22/11.23 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 84.22/11.23 join(composition(X, Y), converse(composition(join(converse(Y), converse(top)), converse(complement(x0)))))
% 84.22/11.23 = { by lemma 32 R->L }
% 84.22/11.23 join(composition(X, Y), converse(composition(converse(join(top, converse(converse(Y)))), converse(complement(x0)))))
% 84.22/11.23 = { by axiom 9 (converse_multiplicativity) R->L }
% 84.22/11.23 join(composition(X, Y), converse(converse(composition(complement(x0), join(top, converse(converse(Y)))))))
% 84.22/11.23 = { by axiom 1 (converse_idempotence) }
% 84.22/11.23 join(composition(X, Y), composition(complement(x0), join(top, converse(converse(Y)))))
% 84.22/11.23 = { by axiom 1 (converse_idempotence) }
% 84.22/11.23 join(composition(X, Y), composition(complement(x0), join(top, Y)))
% 84.22/11.23 = { by lemma 45 }
% 84.22/11.23 join(composition(X, Y), composition(complement(x0), top))
% 84.22/11.23 = { by lemma 66 }
% 84.22/11.23 join(composition(X, Y), complement(x0))
% 84.22/11.23 = { by axiom 2 (maddux1_join_commutativity) }
% 84.22/11.23 join(complement(x0), composition(X, Y))
% 84.22/11.23
% 84.22/11.23 Lemma 75: composition(meet(one, x0), X) = meet(X, x0).
% 84.22/11.23 Proof:
% 84.22/11.23 composition(meet(one, x0), X)
% 84.22/11.23 = { by lemma 48 R->L }
% 84.22/11.23 meet(composition(meet(one, x0), X), top)
% 84.22/11.23 = { by lemma 35 R->L }
% 84.22/11.23 meet(composition(meet(one, x0), X), converse(top))
% 84.22/11.23 = { by lemma 69 R->L }
% 84.22/11.23 meet(composition(meet(one, x0), X), converse(join(complement(composition(converse(X), converse(meet(one, x0)))), composition(join(converse(X), complement(converse(X))), converse(meet(one, x0))))))
% 84.22/11.23 = { by axiom 6 (def_top) R->L }
% 84.22/11.23 meet(composition(meet(one, x0), X), converse(join(complement(composition(converse(X), converse(meet(one, x0)))), composition(top, converse(meet(one, x0))))))
% 84.22/11.23 = { by axiom 2 (maddux1_join_commutativity) }
% 84.22/11.23 meet(composition(meet(one, x0), X), converse(join(composition(top, converse(meet(one, x0))), complement(composition(converse(X), converse(meet(one, x0)))))))
% 84.22/11.23 = { by axiom 9 (converse_multiplicativity) R->L }
% 84.22/11.23 meet(composition(meet(one, x0), X), converse(join(composition(top, converse(meet(one, x0))), complement(converse(composition(meet(one, x0), X))))))
% 84.22/11.24 = { by lemma 35 R->L }
% 84.22/11.24 meet(composition(meet(one, x0), X), converse(join(composition(converse(top), converse(meet(one, x0))), complement(converse(composition(meet(one, x0), X))))))
% 84.22/11.24 = { by axiom 9 (converse_multiplicativity) R->L }
% 84.22/11.24 meet(composition(meet(one, x0), X), converse(join(converse(composition(meet(one, x0), top)), complement(converse(composition(meet(one, x0), X))))))
% 84.22/11.24 = { by lemma 64 R->L }
% 84.22/11.24 meet(composition(meet(one, x0), X), converse(join(converse(composition(meet(one, x0), top)), complement(complement(converse(complement(composition(meet(one, x0), X))))))))
% 84.22/11.24 = { by lemma 36 }
% 84.22/11.24 meet(composition(meet(one, x0), X), converse(join(converse(composition(meet(one, x0), top)), meet(converse(complement(composition(meet(one, x0), X))), converse(complement(composition(meet(one, x0), X)))))))
% 84.22/11.24 = { by lemma 47 }
% 84.22/11.24 meet(composition(meet(one, x0), X), converse(join(converse(composition(meet(one, x0), top)), converse(complement(composition(meet(one, x0), X))))))
% 84.22/11.24 = { by axiom 7 (converse_additivity) R->L }
% 84.22/11.24 meet(composition(meet(one, x0), X), converse(converse(join(composition(meet(one, x0), top), complement(composition(meet(one, x0), X))))))
% 84.22/11.24 = { by axiom 1 (converse_idempotence) }
% 84.22/11.24 meet(composition(meet(one, x0), X), join(composition(meet(one, x0), top), complement(composition(meet(one, x0), X))))
% 84.22/11.24 = { by lemma 70 }
% 84.22/11.24 meet(composition(meet(one, x0), X), composition(meet(one, x0), top))
% 84.22/11.24 = { by lemma 48 R->L }
% 84.22/11.24 meet(composition(meet(one, x0), X), meet(composition(meet(one, x0), top), top))
% 84.22/11.24 = { by lemma 69 R->L }
% 84.22/11.24 meet(composition(meet(one, x0), X), meet(composition(meet(one, x0), top), join(complement(composition(meet(one, x0), top)), composition(join(meet(one, x0), meet(x0, complement(one))), top))))
% 84.22/11.24 = { by lemma 56 }
% 84.22/11.24 meet(composition(meet(one, x0), X), meet(composition(meet(one, x0), top), join(complement(composition(meet(one, x0), top)), composition(x0, top))))
% 84.22/11.24 = { by axiom 2 (maddux1_join_commutativity) }
% 84.22/11.24 meet(composition(meet(one, x0), X), meet(composition(meet(one, x0), top), join(composition(x0, top), complement(composition(meet(one, x0), top)))))
% 84.22/11.24 = { by lemma 37 R->L }
% 84.22/11.24 meet(composition(meet(one, x0), X), meet(composition(meet(one, x0), top), join(composition(x0, top), complement(composition(meet(x0, one), top)))))
% 84.22/11.24 = { by axiom 4 (goals) }
% 84.22/11.24 meet(composition(meet(one, x0), X), meet(composition(meet(one, x0), top), join(x0, complement(composition(meet(x0, one), top)))))
% 84.22/11.24 = { by lemma 37 R->L }
% 84.22/11.24 meet(composition(meet(one, x0), X), meet(composition(meet(one, x0), top), join(x0, complement(composition(meet(one, x0), top)))))
% 84.22/11.24 = { by lemma 70 }
% 84.22/11.24 meet(composition(meet(one, x0), X), meet(composition(meet(one, x0), top), x0))
% 84.22/11.24 = { by lemma 37 R->L }
% 84.22/11.24 meet(composition(meet(one, x0), X), meet(x0, composition(meet(one, x0), top)))
% 84.22/11.24 = { by lemma 71 R->L }
% 84.22/11.24 meet(composition(meet(one, x0), X), meet(x0, join(complement(x0), composition(meet(one, x0), top))))
% 84.22/11.24 = { by lemma 37 }
% 84.22/11.24 meet(composition(meet(one, x0), X), meet(x0, join(complement(x0), composition(meet(x0, one), top))))
% 84.22/11.24 = { by lemma 72 R->L }
% 84.22/11.24 meet(composition(meet(one, x0), X), meet(x0, composition(join(meet(x0, one), complement(x0)), top)))
% 84.22/11.24 = { by axiom 2 (maddux1_join_commutativity) }
% 84.22/11.24 meet(composition(meet(one, x0), X), meet(x0, composition(join(complement(x0), meet(x0, one)), top)))
% 84.22/11.24 = { by lemma 73 }
% 84.22/11.24 meet(composition(meet(one, x0), X), meet(x0, composition(join(one, complement(x0)), top)))
% 84.22/11.24 = { by lemma 72 }
% 84.22/11.24 meet(composition(meet(one, x0), X), meet(x0, join(complement(x0), composition(one, top))))
% 84.22/11.24 = { by lemma 18 }
% 84.22/11.24 meet(composition(meet(one, x0), X), meet(x0, join(complement(x0), top)))
% 84.22/11.24 = { by lemma 27 }
% 84.22/11.24 meet(composition(meet(one, x0), X), meet(x0, top))
% 84.22/11.24 = { by lemma 48 }
% 84.22/11.24 meet(composition(meet(one, x0), X), x0)
% 84.22/11.24 = { by lemma 37 R->L }
% 84.22/11.24 meet(x0, composition(meet(one, x0), X))
% 84.22/11.24 = { by lemma 71 R->L }
% 84.22/11.24 meet(x0, join(complement(x0), composition(meet(one, x0), X)))
% 84.22/11.24 = { by lemma 37 }
% 84.22/11.24 meet(x0, join(complement(x0), composition(meet(x0, one), X)))
% 84.22/11.24 = { by lemma 74 R->L }
% 84.22/11.24 meet(x0, join(complement(x0), composition(join(meet(x0, one), complement(x0)), X)))
% 84.22/11.24 = { by axiom 2 (maddux1_join_commutativity) }
% 84.22/11.24 meet(x0, join(complement(x0), composition(join(complement(x0), meet(x0, one)), X)))
% 84.22/11.24 = { by lemma 73 }
% 84.22/11.24 meet(x0, join(complement(x0), composition(join(one, complement(x0)), X)))
% 84.22/11.24 = { by lemma 74 }
% 84.22/11.24 meet(x0, join(complement(x0), composition(one, X)))
% 84.22/11.24 = { by lemma 18 }
% 84.22/11.24 meet(x0, join(complement(x0), X))
% 84.22/11.24 = { by axiom 2 (maddux1_join_commutativity) }
% 84.22/11.24 meet(x0, join(X, complement(x0)))
% 84.22/11.24 = { by lemma 70 }
% 84.22/11.24 meet(x0, X)
% 84.22/11.24 = { by lemma 37 R->L }
% 84.22/11.24 meet(X, x0)
% 84.22/11.24
% 84.22/11.24 Lemma 76: composition(meet(X, x0), Y) = meet(x0, composition(X, Y)).
% 84.22/11.24 Proof:
% 84.22/11.24 composition(meet(X, x0), Y)
% 84.22/11.24 = { by lemma 75 R->L }
% 84.22/11.24 composition(composition(meet(one, x0), X), Y)
% 84.22/11.24 = { by axiom 10 (composition_associativity) R->L }
% 84.22/11.24 composition(meet(one, x0), composition(X, Y))
% 84.22/11.24 = { by lemma 75 }
% 84.22/11.24 meet(composition(X, Y), x0)
% 84.22/11.24 = { by lemma 37 R->L }
% 84.22/11.24 meet(x0, composition(X, Y))
% 84.22/11.24
% 84.22/11.24 Goal 1 (goals_1): tuple(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2)))) = tuple(composition(meet(x0, x1), x2), meet(x0, composition(x1, x2_2))).
% 84.22/11.24 Proof:
% 84.22/11.24 tuple(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 84.22/11.24 = { by axiom 2 (maddux1_join_commutativity) }
% 84.22/11.24 tuple(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)), join(meet(x0, composition(x1, x2_2)), composition(meet(x0, x1), x2_2)))
% 84.22/11.24 = { by lemma 37 R->L }
% 84.22/11.24 tuple(join(meet(x0, composition(x1, x2)), composition(meet(x1, x0), x2)), join(meet(x0, composition(x1, x2_2)), composition(meet(x0, x1), x2_2)))
% 84.22/11.24 = { by lemma 37 R->L }
% 84.22/11.24 tuple(join(meet(x0, composition(x1, x2)), composition(meet(x1, x0), x2)), join(meet(x0, composition(x1, x2_2)), composition(meet(x1, x0), x2_2)))
% 84.22/11.24 = { by lemma 76 }
% 84.22/11.24 tuple(join(meet(x0, composition(x1, x2)), meet(x0, composition(x1, x2))), join(meet(x0, composition(x1, x2_2)), composition(meet(x1, x0), x2_2)))
% 84.22/11.24 = { by lemma 43 }
% 84.22/11.24 tuple(meet(x0, composition(x1, x2)), join(meet(x0, composition(x1, x2_2)), composition(meet(x1, x0), x2_2)))
% 84.22/11.24 = { by lemma 76 }
% 84.22/11.24 tuple(meet(x0, composition(x1, x2)), join(meet(x0, composition(x1, x2_2)), meet(x0, composition(x1, x2_2))))
% 84.22/11.24 = { by lemma 43 }
% 84.22/11.24 tuple(meet(x0, composition(x1, x2)), meet(x0, composition(x1, x2_2)))
% 84.22/11.24 = { by lemma 76 R->L }
% 84.22/11.24 tuple(composition(meet(x1, x0), x2), meet(x0, composition(x1, x2_2)))
% 84.22/11.24 = { by lemma 37 }
% 84.22/11.24 tuple(composition(meet(x0, x1), x2), meet(x0, composition(x1, x2_2)))
% 84.22/11.24 % SZS output end Proof
% 84.22/11.24
% 84.22/11.24 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------