TSTP Solution File: REL005+4 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL005+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 : n025.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 13:43:46 EDT 2023
% Result : Theorem 10.37s 1.69s
% Output : Proof 10.58s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL005+4 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.12 % 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 : n025.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 : Sat Aug 26 02:59:53 EDT 2023
% 0.12/0.34 % CPUTime :
% 10.37/1.69 Command-line arguments: --no-flatten-goal
% 10.37/1.69
% 10.37/1.69 % SZS status Theorem
% 10.37/1.69
% 10.58/1.75 % SZS output start Proof
% 10.58/1.75 Take the following subset of the input axioms:
% 10.58/1.75 fof(composition_associativity, axiom, ![X0, X1, X2]: composition(X0, composition(X1, X2))=composition(composition(X0, X1), X2)).
% 10.58/1.75 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))).
% 10.58/1.75 fof(composition_identity, axiom, ![X0_2]: composition(X0_2, one)=X0_2).
% 10.58/1.75 fof(converse_additivity, axiom, ![X0_2, X1_2]: converse(join(X0_2, X1_2))=join(converse(X0_2), converse(X1_2))).
% 10.58/1.75 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)).
% 10.58/1.75 fof(converse_idempotence, axiom, ![X0_2]: converse(converse(X0_2))=X0_2).
% 10.58/1.75 fof(converse_multiplicativity, axiom, ![X0_2, X1_2]: converse(composition(X0_2, X1_2))=composition(converse(X1_2), converse(X0_2))).
% 10.58/1.75 fof(def_top, axiom, ![X0_2]: top=join(X0_2, complement(X0_2))).
% 10.58/1.75 fof(def_zero, axiom, ![X0_2]: zero=meet(X0_2, complement(X0_2))).
% 10.58/1.75 fof(goals, conjecture, ![X0_2, X1_2]: (join(converse(meet(X0_2, X1_2)), meet(converse(X0_2), converse(X1_2)))=meet(converse(X0_2), converse(X1_2)) & join(meet(converse(X0_2), converse(X1_2)), converse(meet(X0_2, X1_2)))=converse(meet(X0_2, X1_2)))).
% 10.58/1.75 fof(maddux1_join_commutativity, axiom, ![X0_2, X1_2]: join(X0_2, X1_2)=join(X1_2, X0_2)).
% 10.58/1.75 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)).
% 10.58/1.75 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)))).
% 10.58/1.75 fof(maddux4_definiton_of_meet, axiom, ![X0_2, X1_2]: meet(X0_2, X1_2)=complement(join(complement(X0_2), complement(X1_2)))).
% 10.58/1.75 fof(modular_law_2, axiom, ![X0_2, X1_2, X2_2]: join(meet(composition(X0_2, X1_2), X2_2), meet(composition(meet(X0_2, composition(X2_2, converse(X1_2))), X1_2), X2_2))=meet(composition(meet(X0_2, composition(X2_2, converse(X1_2))), X1_2), X2_2)).
% 10.58/1.75
% 10.58/1.75 Now clausify the problem and encode Horn clauses using encoding 3 of
% 10.58/1.75 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 10.58/1.75 We repeatedly replace C & s=t => u=v by the two clauses:
% 10.58/1.75 fresh(y, y, x1...xn) = u
% 10.58/1.75 C => fresh(s, t, x1...xn) = v
% 10.58/1.75 where fresh is a fresh function symbol and x1..xn are the free
% 10.58/1.75 variables of u and v.
% 10.58/1.75 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 10.58/1.75 input problem has no model of domain size 1).
% 10.58/1.75
% 10.58/1.75 The encoding turns the above axioms into the following unit equations and goals:
% 10.58/1.75
% 10.58/1.75 Axiom 1 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 10.58/1.75 Axiom 2 (converse_idempotence): converse(converse(X)) = X.
% 10.58/1.75 Axiom 3 (composition_identity): composition(X, one) = X.
% 10.58/1.75 Axiom 4 (def_top): top = join(X, complement(X)).
% 10.58/1.75 Axiom 5 (def_zero): zero = meet(X, complement(X)).
% 10.58/1.75 Axiom 6 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 10.58/1.75 Axiom 7 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 10.58/1.75 Axiom 8 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 10.58/1.75 Axiom 9 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 10.58/1.75 Axiom 10 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 10.58/1.75 Axiom 11 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 10.58/1.75 Axiom 12 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 10.58/1.75 Axiom 13 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 10.58/1.75 Axiom 14 (modular_law_2): join(meet(composition(X, Y), Z), meet(composition(meet(X, composition(Z, converse(Y))), Y), Z)) = meet(composition(meet(X, composition(Z, converse(Y))), Y), Z).
% 10.58/1.75
% 10.58/1.75 Lemma 15: complement(top) = zero.
% 10.58/1.75 Proof:
% 10.58/1.75 complement(top)
% 10.58/1.75 = { by axiom 4 (def_top) }
% 10.58/1.75 complement(join(complement(X), complement(complement(X))))
% 10.58/1.75 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 10.58/1.75 meet(X, complement(X))
% 10.58/1.75 = { by axiom 5 (def_zero) R->L }
% 10.58/1.75 zero
% 10.58/1.75
% 10.58/1.75 Lemma 16: join(X, join(Y, complement(X))) = join(Y, top).
% 10.58/1.75 Proof:
% 10.58/1.75 join(X, join(Y, complement(X)))
% 10.58/1.75 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 10.58/1.75 join(X, join(complement(X), Y))
% 10.58/1.75 = { by axiom 7 (maddux2_join_associativity) }
% 10.58/1.75 join(join(X, complement(X)), Y)
% 10.58/1.75 = { by axiom 4 (def_top) R->L }
% 10.58/1.75 join(top, Y)
% 10.58/1.75 = { by axiom 1 (maddux1_join_commutativity) }
% 10.58/1.75 join(Y, top)
% 10.58/1.75
% 10.58/1.75 Lemma 17: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 10.58/1.75 Proof:
% 10.58/1.75 converse(composition(converse(X), Y))
% 10.58/1.75 = { by axiom 8 (converse_multiplicativity) }
% 10.58/1.75 composition(converse(Y), converse(converse(X)))
% 10.58/1.75 = { by axiom 2 (converse_idempotence) }
% 10.58/1.75 composition(converse(Y), X)
% 10.58/1.75
% 10.58/1.75 Lemma 18: composition(converse(one), X) = X.
% 10.58/1.75 Proof:
% 10.58/1.75 composition(converse(one), X)
% 10.58/1.75 = { by lemma 17 R->L }
% 10.58/1.75 converse(composition(converse(X), one))
% 10.58/1.75 = { by axiom 3 (composition_identity) }
% 10.58/1.75 converse(converse(X))
% 10.58/1.75 = { by axiom 2 (converse_idempotence) }
% 10.58/1.75 X
% 10.58/1.75
% 10.58/1.75 Lemma 19: composition(one, X) = X.
% 10.58/1.75 Proof:
% 10.58/1.75 composition(one, X)
% 10.58/1.75 = { by lemma 18 R->L }
% 10.58/1.75 composition(converse(one), composition(one, X))
% 10.58/1.75 = { by axiom 9 (composition_associativity) }
% 10.58/1.75 composition(composition(converse(one), one), X)
% 10.58/1.75 = { by axiom 3 (composition_identity) }
% 10.58/1.75 composition(converse(one), X)
% 10.58/1.75 = { by lemma 18 }
% 10.58/1.75 X
% 10.58/1.75
% 10.58/1.75 Lemma 20: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 10.58/1.75 Proof:
% 10.58/1.75 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 10.58/1.75 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 10.58/1.75 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 10.58/1.75 = { by axiom 12 (converse_cancellativity) }
% 10.58/1.76 complement(X)
% 10.58/1.76
% 10.58/1.76 Lemma 21: join(complement(X), complement(X)) = complement(X).
% 10.58/1.76 Proof:
% 10.58/1.76 join(complement(X), complement(X))
% 10.58/1.76 = { by lemma 18 R->L }
% 10.58/1.76 join(complement(X), composition(converse(one), complement(X)))
% 10.58/1.76 = { by lemma 19 R->L }
% 10.58/1.76 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 10.58/1.76 = { by lemma 20 }
% 10.58/1.76 complement(X)
% 10.58/1.76
% 10.58/1.76 Lemma 22: join(top, complement(X)) = top.
% 10.58/1.76 Proof:
% 10.58/1.76 join(top, complement(X))
% 10.58/1.76 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 10.58/1.76 join(complement(X), top)
% 10.58/1.76 = { by lemma 16 R->L }
% 10.58/1.76 join(X, join(complement(X), complement(X)))
% 10.58/1.76 = { by lemma 21 }
% 10.58/1.76 join(X, complement(X))
% 10.58/1.76 = { by axiom 4 (def_top) R->L }
% 10.58/1.76 top
% 10.58/1.76
% 10.58/1.76 Lemma 23: join(Y, top) = join(X, top).
% 10.58/1.76 Proof:
% 10.58/1.76 join(Y, top)
% 10.58/1.76 = { by lemma 22 R->L }
% 10.58/1.76 join(Y, join(top, complement(Y)))
% 10.58/1.76 = { by lemma 16 }
% 10.58/1.76 join(top, top)
% 10.58/1.76 = { by lemma 16 R->L }
% 10.58/1.76 join(X, join(top, complement(X)))
% 10.58/1.76 = { by lemma 22 }
% 10.58/1.76 join(X, top)
% 10.58/1.76
% 10.58/1.76 Lemma 24: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 10.58/1.76 Proof:
% 10.58/1.76 join(meet(X, Y), complement(join(complement(X), Y)))
% 10.58/1.76 = { by axiom 10 (maddux4_definiton_of_meet) }
% 10.58/1.76 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 10.58/1.76 = { by axiom 13 (maddux3_a_kind_of_de_Morgan) R->L }
% 10.58/1.76 X
% 10.58/1.76
% 10.58/1.76 Lemma 25: join(zero, meet(X, X)) = X.
% 10.58/1.76 Proof:
% 10.58/1.76 join(zero, meet(X, X))
% 10.58/1.76 = { by axiom 10 (maddux4_definiton_of_meet) }
% 10.58/1.76 join(zero, complement(join(complement(X), complement(X))))
% 10.58/1.76 = { by axiom 5 (def_zero) }
% 10.58/1.76 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 10.58/1.76 = { by lemma 24 }
% 10.58/1.76 X
% 10.58/1.76
% 10.58/1.76 Lemma 26: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 10.58/1.76 Proof:
% 10.58/1.76 join(zero, join(X, complement(complement(Y))))
% 10.58/1.76 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 10.58/1.76 join(zero, join(complement(complement(Y)), X))
% 10.58/1.76 = { by lemma 21 R->L }
% 10.58/1.76 join(zero, join(complement(join(complement(Y), complement(Y))), X))
% 10.58/1.76 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 10.58/1.76 join(zero, join(meet(Y, Y), X))
% 10.58/1.76 = { by axiom 7 (maddux2_join_associativity) }
% 10.58/1.76 join(join(zero, meet(Y, Y)), X)
% 10.58/1.76 = { by lemma 25 }
% 10.58/1.76 join(Y, X)
% 10.58/1.76 = { by axiom 1 (maddux1_join_commutativity) }
% 10.58/1.76 join(X, Y)
% 10.58/1.76
% 10.58/1.76 Lemma 27: join(zero, complement(complement(X))) = X.
% 10.58/1.76 Proof:
% 10.58/1.76 join(zero, complement(complement(X)))
% 10.58/1.76 = { by axiom 5 (def_zero) }
% 10.58/1.76 join(meet(X, complement(X)), complement(complement(X)))
% 10.58/1.76 = { by lemma 21 R->L }
% 10.58/1.76 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 10.58/1.76 = { by lemma 24 }
% 10.58/1.76 X
% 10.58/1.76
% 10.58/1.76 Lemma 28: join(X, zero) = join(X, X).
% 10.58/1.76 Proof:
% 10.58/1.76 join(X, zero)
% 10.58/1.76 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 10.58/1.76 join(zero, X)
% 10.58/1.76 = { by lemma 27 R->L }
% 10.58/1.76 join(zero, join(zero, complement(complement(X))))
% 10.58/1.76 = { by lemma 21 R->L }
% 10.58/1.76 join(zero, join(zero, join(complement(complement(X)), complement(complement(X)))))
% 10.58/1.76 = { by lemma 26 }
% 10.58/1.76 join(zero, join(complement(complement(X)), X))
% 10.58/1.76 = { by axiom 1 (maddux1_join_commutativity) }
% 10.58/1.76 join(zero, join(X, complement(complement(X))))
% 10.58/1.76 = { by lemma 26 }
% 10.58/1.76 join(X, X)
% 10.58/1.76
% 10.58/1.76 Lemma 29: join(zero, complement(X)) = complement(X).
% 10.58/1.76 Proof:
% 10.58/1.76 join(zero, complement(X))
% 10.58/1.76 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 10.58/1.76 join(complement(X), zero)
% 10.58/1.76 = { by lemma 28 }
% 10.58/1.76 join(complement(X), complement(X))
% 10.58/1.76 = { by lemma 21 }
% 10.58/1.76 complement(X)
% 10.58/1.76
% 10.58/1.76 Lemma 30: join(X, zero) = X.
% 10.58/1.76 Proof:
% 10.58/1.76 join(X, zero)
% 10.58/1.76 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 10.58/1.76 join(zero, X)
% 10.58/1.76 = { by lemma 26 R->L }
% 10.58/1.76 join(zero, join(zero, complement(complement(X))))
% 10.58/1.76 = { by lemma 29 }
% 10.58/1.76 join(zero, complement(complement(X)))
% 10.58/1.76 = { by lemma 27 }
% 10.58/1.76 X
% 10.58/1.76
% 10.58/1.76 Lemma 31: join(X, top) = top.
% 10.58/1.76 Proof:
% 10.58/1.76 join(X, top)
% 10.58/1.76 = { by lemma 23 }
% 10.58/1.76 join(join(zero, zero), top)
% 10.58/1.76 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 10.58/1.76 join(top, join(zero, zero))
% 10.58/1.76 = { by lemma 30 }
% 10.58/1.76 join(top, zero)
% 10.58/1.76 = { by lemma 30 }
% 10.58/1.76 top
% 10.58/1.76
% 10.58/1.76 Lemma 32: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 10.58/1.76 Proof:
% 10.58/1.76 converse(join(X, converse(Y)))
% 10.58/1.76 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 10.58/1.76 converse(join(converse(Y), X))
% 10.58/1.76 = { by axiom 6 (converse_additivity) }
% 10.58/1.76 join(converse(converse(Y)), converse(X))
% 10.58/1.76 = { by axiom 2 (converse_idempotence) }
% 10.58/1.76 join(Y, converse(X))
% 10.58/1.76
% 10.58/1.76 Lemma 33: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 10.58/1.76 Proof:
% 10.58/1.76 converse(join(converse(X), Y))
% 10.58/1.76 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 10.58/1.76 converse(join(Y, converse(X)))
% 10.58/1.76 = { by lemma 32 }
% 10.58/1.76 join(X, converse(Y))
% 10.58/1.76
% 10.58/1.76 Lemma 34: join(X, converse(complement(converse(X)))) = converse(top).
% 10.58/1.76 Proof:
% 10.58/1.76 join(X, converse(complement(converse(X))))
% 10.58/1.76 = { by lemma 33 R->L }
% 10.58/1.76 converse(join(converse(X), complement(converse(X))))
% 10.58/1.76 = { by axiom 4 (def_top) R->L }
% 10.58/1.76 converse(top)
% 10.58/1.76
% 10.58/1.76 Lemma 35: join(X, join(complement(X), Y)) = top.
% 10.58/1.76 Proof:
% 10.58/1.76 join(X, join(complement(X), Y))
% 10.58/1.76 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 10.58/1.76 join(X, join(Y, complement(X)))
% 10.58/1.76 = { by lemma 16 }
% 10.58/1.76 join(Y, top)
% 10.58/1.76 = { by lemma 23 R->L }
% 10.58/1.76 join(Z, top)
% 10.58/1.76 = { by lemma 31 }
% 10.58/1.76 top
% 10.58/1.76
% 10.58/1.76 Lemma 36: join(X, converse(top)) = top.
% 10.58/1.76 Proof:
% 10.58/1.76 join(X, converse(top))
% 10.58/1.76 = { by lemma 34 R->L }
% 10.58/1.76 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 10.58/1.76 = { by lemma 35 }
% 10.58/1.76 top
% 10.58/1.76
% 10.58/1.76 Lemma 37: converse(top) = top.
% 10.58/1.76 Proof:
% 10.58/1.76 converse(top)
% 10.58/1.76 = { by lemma 31 R->L }
% 10.58/1.76 converse(join(X, top))
% 10.58/1.76 = { by axiom 6 (converse_additivity) }
% 10.58/1.76 join(converse(X), converse(top))
% 10.58/1.76 = { by lemma 36 }
% 10.58/1.76 top
% 10.58/1.76
% 10.58/1.76 Lemma 38: join(zero, X) = X.
% 10.58/1.76 Proof:
% 10.58/1.76 join(zero, X)
% 10.58/1.76 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 10.58/1.76 join(X, zero)
% 10.58/1.76 = { by lemma 30 }
% 10.58/1.76 X
% 10.58/1.76
% 10.58/1.76 Lemma 39: meet(Y, X) = meet(X, Y).
% 10.58/1.76 Proof:
% 10.58/1.76 meet(Y, X)
% 10.58/1.76 = { by axiom 10 (maddux4_definiton_of_meet) }
% 10.58/1.76 complement(join(complement(Y), complement(X)))
% 10.58/1.76 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 10.58/1.76 complement(join(complement(X), complement(Y)))
% 10.58/1.76 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 10.58/1.76 meet(X, Y)
% 10.58/1.76
% 10.58/1.76 Lemma 40: complement(join(zero, complement(X))) = meet(X, top).
% 10.58/1.76 Proof:
% 10.58/1.76 complement(join(zero, complement(X)))
% 10.58/1.76 = { by lemma 15 R->L }
% 10.58/1.76 complement(join(complement(top), complement(X)))
% 10.58/1.76 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 10.58/1.76 meet(top, X)
% 10.58/1.76 = { by lemma 39 R->L }
% 10.58/1.76 meet(X, top)
% 10.58/1.76
% 10.58/1.76 Lemma 41: join(X, complement(zero)) = top.
% 10.58/1.76 Proof:
% 10.58/1.76 join(X, complement(zero))
% 10.58/1.76 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 10.58/1.76 join(complement(zero), X)
% 10.58/1.76 = { by lemma 26 R->L }
% 10.58/1.76 join(zero, join(complement(zero), complement(complement(X))))
% 10.58/1.76 = { by lemma 35 }
% 10.58/1.76 top
% 10.58/1.76
% 10.58/1.76 Lemma 42: meet(X, zero) = zero.
% 10.58/1.76 Proof:
% 10.58/1.76 meet(X, zero)
% 10.58/1.76 = { by axiom 10 (maddux4_definiton_of_meet) }
% 10.58/1.76 complement(join(complement(X), complement(zero)))
% 10.58/1.76 = { by lemma 41 }
% 10.58/1.76 complement(top)
% 10.58/1.76 = { by lemma 15 }
% 10.58/1.76 zero
% 10.58/1.76
% 10.58/1.76 Lemma 43: join(meet(X, Y), meet(X, complement(Y))) = X.
% 10.58/1.76 Proof:
% 10.58/1.76 join(meet(X, Y), meet(X, complement(Y)))
% 10.58/1.76 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 10.58/1.76 join(meet(X, complement(Y)), meet(X, Y))
% 10.58/1.76 = { by axiom 10 (maddux4_definiton_of_meet) }
% 10.58/1.76 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 10.58/1.76 = { by lemma 24 }
% 10.58/1.76 X
% 10.58/1.76
% 10.58/1.76 Lemma 44: meet(X, top) = X.
% 10.58/1.76 Proof:
% 10.58/1.76 meet(X, top)
% 10.58/1.76 = { by lemma 40 R->L }
% 10.58/1.76 complement(join(zero, complement(X)))
% 10.58/1.76 = { by lemma 29 R->L }
% 10.58/1.76 join(zero, complement(join(zero, complement(X))))
% 10.58/1.76 = { by lemma 40 }
% 10.58/1.76 join(zero, meet(X, top))
% 10.58/1.76 = { by lemma 41 R->L }
% 10.58/1.76 join(zero, meet(X, join(complement(zero), complement(zero))))
% 10.58/1.76 = { by lemma 21 }
% 10.58/1.76 join(zero, meet(X, complement(zero)))
% 10.58/1.76 = { by lemma 42 R->L }
% 10.58/1.76 join(meet(X, zero), meet(X, complement(zero)))
% 10.58/1.76 = { by lemma 43 }
% 10.58/1.76 X
% 10.58/1.76
% 10.58/1.76 Lemma 45: join(meet(X, Y), meet(X, Y)) = meet(X, Y).
% 10.58/1.76 Proof:
% 10.58/1.76 join(meet(X, Y), meet(X, Y))
% 10.58/1.76 = { by lemma 39 }
% 10.58/1.76 join(meet(Y, X), meet(X, Y))
% 10.58/1.76 = { by lemma 39 }
% 10.58/1.76 join(meet(Y, X), meet(Y, X))
% 10.58/1.76 = { by axiom 10 (maddux4_definiton_of_meet) }
% 10.58/1.76 join(meet(Y, X), complement(join(complement(Y), complement(X))))
% 10.58/1.76 = { by axiom 10 (maddux4_definiton_of_meet) }
% 10.58/1.76 join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X))))
% 10.58/1.76 = { by lemma 21 }
% 10.58/1.76 complement(join(complement(Y), complement(X)))
% 10.58/1.76 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 10.58/1.76 meet(Y, X)
% 10.58/1.76 = { by lemma 39 R->L }
% 10.58/1.76 meet(X, Y)
% 10.58/1.76
% 10.58/1.76 Lemma 46: join(X, X) = X.
% 10.58/1.76 Proof:
% 10.58/1.76 join(X, X)
% 10.58/1.76 = { by lemma 44 R->L }
% 10.58/1.76 join(X, meet(X, top))
% 10.58/1.76 = { by lemma 44 R->L }
% 10.58/1.76 join(meet(X, top), meet(X, top))
% 10.58/1.76 = { by lemma 45 }
% 10.58/1.76 meet(X, top)
% 10.58/1.76 = { by lemma 44 }
% 10.58/1.76 X
% 10.58/1.76
% 10.58/1.76 Lemma 47: converse(zero) = zero.
% 10.58/1.76 Proof:
% 10.58/1.76 converse(zero)
% 10.58/1.76 = { by lemma 38 R->L }
% 10.58/1.76 join(zero, converse(zero))
% 10.58/1.76 = { by lemma 33 R->L }
% 10.58/1.76 converse(join(converse(zero), zero))
% 10.58/1.76 = { by lemma 28 }
% 10.58/1.76 converse(join(converse(zero), converse(zero)))
% 10.58/1.76 = { by lemma 32 }
% 10.58/1.76 join(zero, converse(converse(zero)))
% 10.58/1.76 = { by axiom 2 (converse_idempotence) }
% 10.58/1.76 join(zero, zero)
% 10.58/1.76 = { by lemma 46 }
% 10.58/1.76 zero
% 10.58/1.76
% 10.58/1.76 Lemma 48: join(top, X) = top.
% 10.58/1.76 Proof:
% 10.58/1.76 join(top, X)
% 10.58/1.76 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 10.58/1.76 join(X, top)
% 10.58/1.76 = { by lemma 23 R->L }
% 10.58/1.76 join(Y, top)
% 10.58/1.76 = { by lemma 31 }
% 10.58/1.76 top
% 10.58/1.76
% 10.58/1.76 Lemma 49: complement(complement(X)) = X.
% 10.58/1.76 Proof:
% 10.58/1.76 complement(complement(X))
% 10.58/1.76 = { by lemma 29 R->L }
% 10.58/1.76 join(zero, complement(complement(X)))
% 10.58/1.76 = { by lemma 27 }
% 10.58/1.76 X
% 10.58/1.76
% 10.58/1.76 Lemma 50: meet(zero, X) = zero.
% 10.58/1.76 Proof:
% 10.58/1.76 meet(zero, X)
% 10.58/1.76 = { by lemma 39 }
% 10.58/1.76 meet(X, zero)
% 10.58/1.76 = { by lemma 42 }
% 10.58/1.76 zero
% 10.58/1.76
% 10.58/1.76 Lemma 51: composition(top, zero) = zero.
% 10.58/1.76 Proof:
% 10.58/1.76 composition(top, zero)
% 10.58/1.76 = { by lemma 37 R->L }
% 10.58/1.76 composition(converse(top), zero)
% 10.58/1.76 = { by lemma 38 R->L }
% 10.58/1.76 join(zero, composition(converse(top), zero))
% 10.58/1.76 = { by lemma 15 R->L }
% 10.58/1.76 join(complement(top), composition(converse(top), zero))
% 10.58/1.76 = { by lemma 15 R->L }
% 10.58/1.76 join(complement(top), composition(converse(top), complement(top)))
% 10.58/1.76 = { by lemma 48 R->L }
% 10.58/1.76 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 10.58/1.76 = { by lemma 37 R->L }
% 10.58/1.76 join(complement(top), composition(converse(top), complement(join(top, composition(converse(top), top)))))
% 10.58/1.76 = { by lemma 19 R->L }
% 10.58/1.76 join(complement(top), composition(converse(top), complement(join(composition(one, top), composition(converse(top), top)))))
% 10.58/1.76 = { by axiom 11 (composition_distributivity) R->L }
% 10.58/1.76 join(complement(top), composition(converse(top), complement(composition(join(one, converse(top)), top))))
% 10.58/1.76 = { by lemma 36 }
% 10.58/1.76 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 10.58/1.76 = { by lemma 20 }
% 10.58/1.76 complement(top)
% 10.58/1.76 = { by lemma 15 }
% 10.58/1.76 zero
% 10.58/1.76
% 10.58/1.76 Lemma 52: composition(zero, X) = zero.
% 10.58/1.76 Proof:
% 10.58/1.76 composition(zero, X)
% 10.58/1.76 = { by lemma 47 R->L }
% 10.58/1.76 composition(converse(zero), X)
% 10.58/1.76 = { by lemma 17 R->L }
% 10.58/1.76 converse(composition(converse(X), zero))
% 10.58/1.76 = { by lemma 38 R->L }
% 10.58/1.76 converse(join(zero, composition(converse(X), zero)))
% 10.58/1.76 = { by lemma 51 R->L }
% 10.58/1.76 converse(join(composition(top, zero), composition(converse(X), zero)))
% 10.58/1.76 = { by axiom 11 (composition_distributivity) R->L }
% 10.58/1.76 converse(composition(join(top, converse(X)), zero))
% 10.58/1.76 = { by lemma 48 }
% 10.58/1.76 converse(composition(top, zero))
% 10.58/1.76 = { by lemma 51 }
% 10.58/1.76 converse(zero)
% 10.58/1.76 = { by lemma 47 }
% 10.58/1.76 zero
% 10.58/1.76
% 10.58/1.76 Lemma 53: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 10.58/1.76 Proof:
% 10.58/1.76 meet(X, join(complement(Y), complement(Z)))
% 10.58/1.76 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 10.58/1.76 meet(X, join(complement(Z), complement(Y)))
% 10.58/1.76 = { by lemma 39 }
% 10.58/1.76 meet(join(complement(Z), complement(Y)), X)
% 10.58/1.76 = { by axiom 10 (maddux4_definiton_of_meet) }
% 10.58/1.76 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 10.58/1.76 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 10.58/1.76 complement(join(meet(Z, Y), complement(X)))
% 10.58/1.76 = { by axiom 1 (maddux1_join_commutativity) }
% 10.58/1.76 complement(join(complement(X), meet(Z, Y)))
% 10.58/1.76 = { by lemma 39 R->L }
% 10.58/1.76 complement(join(complement(X), meet(Y, Z)))
% 10.58/1.76
% 10.58/1.76 Lemma 54: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 10.58/1.76 Proof:
% 10.58/1.76 join(complement(X), complement(Y))
% 10.58/1.76 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 10.58/1.76 join(complement(Y), complement(X))
% 10.58/1.76 = { by lemma 25 R->L }
% 10.58/1.76 join(zero, meet(join(complement(Y), complement(X)), join(complement(Y), complement(X))))
% 10.58/1.76 = { by lemma 53 }
% 10.58/1.76 join(zero, complement(join(complement(join(complement(Y), complement(X))), meet(Y, X))))
% 10.58/1.76 = { by lemma 29 }
% 10.58/1.76 complement(join(complement(join(complement(Y), complement(X))), meet(Y, X)))
% 10.58/1.76 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 10.58/1.76 complement(join(meet(Y, X), meet(Y, X)))
% 10.58/1.76 = { by lemma 45 }
% 10.58/1.76 complement(meet(Y, X))
% 10.58/1.76 = { by lemma 39 R->L }
% 10.58/1.76 complement(meet(X, Y))
% 10.58/1.76
% 10.58/1.77 Lemma 55: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 10.58/1.77 Proof:
% 10.58/1.77 complement(join(X, complement(Y)))
% 10.58/1.77 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 10.58/1.77 complement(join(complement(Y), X))
% 10.58/1.77 = { by lemma 44 R->L }
% 10.58/1.77 complement(join(complement(Y), meet(X, top)))
% 10.58/1.77 = { by lemma 39 R->L }
% 10.58/1.77 complement(join(complement(Y), meet(top, X)))
% 10.58/1.77 = { by lemma 53 R->L }
% 10.58/1.77 meet(Y, join(complement(top), complement(X)))
% 10.58/1.77 = { by lemma 15 }
% 10.58/1.77 meet(Y, join(zero, complement(X)))
% 10.58/1.77 = { by lemma 29 }
% 10.58/1.77 meet(Y, complement(X))
% 10.58/1.77
% 10.58/1.77 Lemma 56: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 10.58/1.77 Proof:
% 10.58/1.77 complement(join(complement(X), Y))
% 10.58/1.77 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 10.58/1.77 complement(join(Y, complement(X)))
% 10.58/1.77 = { by lemma 55 }
% 10.58/1.77 meet(X, complement(Y))
% 10.58/1.77
% 10.58/1.77 Lemma 57: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 10.58/1.77 Proof:
% 10.58/1.77 complement(meet(X, complement(Y)))
% 10.58/1.77 = { by lemma 38 R->L }
% 10.58/1.77 complement(join(zero, meet(X, complement(Y))))
% 10.58/1.77 = { by lemma 55 R->L }
% 10.58/1.77 complement(join(zero, complement(join(Y, complement(X)))))
% 10.58/1.77 = { by lemma 40 }
% 10.58/1.77 meet(join(Y, complement(X)), top)
% 10.58/1.77 = { by lemma 44 }
% 10.58/1.77 join(Y, complement(X))
% 10.58/1.77
% 10.58/1.77 Lemma 58: meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 10.58/1.77 Proof:
% 10.58/1.77 meet(meet(X, Y), Z)
% 10.58/1.77 = { by lemma 39 }
% 10.58/1.77 meet(Z, meet(X, Y))
% 10.58/1.77 = { by lemma 39 }
% 10.58/1.77 meet(Z, meet(Y, X))
% 10.58/1.77 = { by lemma 44 R->L }
% 10.58/1.77 meet(Z, meet(Y, meet(X, top)))
% 10.58/1.77 = { by lemma 40 R->L }
% 10.58/1.77 meet(Z, meet(Y, complement(join(zero, complement(X)))))
% 10.58/1.77 = { by lemma 56 R->L }
% 10.58/1.77 meet(Z, complement(join(complement(Y), join(zero, complement(X)))))
% 10.58/1.77 = { by lemma 56 R->L }
% 10.58/1.77 complement(join(complement(Z), join(complement(Y), join(zero, complement(X)))))
% 10.58/1.77 = { by axiom 7 (maddux2_join_associativity) }
% 10.58/1.77 complement(join(join(complement(Z), complement(Y)), join(zero, complement(X))))
% 10.58/1.77 = { by lemma 44 R->L }
% 10.58/1.77 complement(join(join(complement(Z), complement(Y)), meet(join(zero, complement(X)), top)))
% 10.58/1.77 = { by lemma 40 R->L }
% 10.58/1.77 complement(join(join(complement(Z), complement(Y)), complement(join(zero, complement(join(zero, complement(X)))))))
% 10.58/1.77 = { by lemma 55 }
% 10.58/1.77 meet(join(zero, complement(join(zero, complement(X)))), complement(join(complement(Z), complement(Y))))
% 10.58/1.77 = { by lemma 29 }
% 10.58/1.77 meet(complement(join(zero, complement(X))), complement(join(complement(Z), complement(Y))))
% 10.58/1.77 = { by lemma 39 R->L }
% 10.58/1.77 meet(complement(join(complement(Z), complement(Y))), complement(join(zero, complement(X))))
% 10.58/1.77 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 10.58/1.77 meet(meet(Z, Y), complement(join(zero, complement(X))))
% 10.58/1.77 = { by lemma 39 R->L }
% 10.58/1.77 meet(complement(join(zero, complement(X))), meet(Z, Y))
% 10.58/1.77 = { by lemma 39 R->L }
% 10.58/1.77 meet(complement(join(zero, complement(X))), meet(Y, Z))
% 10.58/1.77 = { by lemma 40 }
% 10.58/1.77 meet(meet(X, top), meet(Y, Z))
% 10.58/1.77 = { by lemma 44 }
% 10.58/1.77 meet(X, meet(Y, Z))
% 10.58/1.77
% 10.58/1.77 Lemma 59: join(complement(X), meet(complement(Y), Z)) = complement(meet(X, join(Y, complement(Z)))).
% 10.58/1.77 Proof:
% 10.58/1.77 join(complement(X), meet(complement(Y), Z))
% 10.58/1.77 = { by lemma 39 }
% 10.58/1.77 join(complement(X), meet(Z, complement(Y)))
% 10.58/1.77 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 10.58/1.77 join(meet(Z, complement(Y)), complement(X))
% 10.58/1.77 = { by lemma 55 R->L }
% 10.58/1.77 join(complement(join(Y, complement(Z))), complement(X))
% 10.58/1.77 = { by lemma 54 }
% 10.58/1.77 complement(meet(join(Y, complement(Z)), X))
% 10.58/1.77 = { by lemma 39 R->L }
% 10.58/1.77 complement(meet(X, join(Y, complement(Z))))
% 10.58/1.77
% 10.58/1.77 Lemma 60: complement(meet(Y, join(X, complement(Y)))) = complement(meet(X, join(Y, complement(X)))).
% 10.58/1.77 Proof:
% 10.58/1.77 complement(meet(Y, join(X, complement(Y))))
% 10.58/1.77 = { by lemma 59 R->L }
% 10.58/1.77 join(complement(Y), meet(complement(X), Y))
% 10.58/1.77 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 10.58/1.77 join(meet(complement(X), Y), complement(Y))
% 10.58/1.77 = { by lemma 43 R->L }
% 10.58/1.77 join(meet(complement(X), Y), join(meet(complement(Y), X), meet(complement(Y), complement(X))))
% 10.58/1.77 = { by lemma 39 R->L }
% 10.58/1.77 join(meet(complement(X), Y), join(meet(complement(Y), X), meet(complement(X), complement(Y))))
% 10.58/1.77 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 10.58/1.77 join(meet(complement(X), Y), join(meet(complement(X), complement(Y)), meet(complement(Y), X)))
% 10.58/1.77 = { by axiom 7 (maddux2_join_associativity) }
% 10.58/1.77 join(join(meet(complement(X), Y), meet(complement(X), complement(Y))), meet(complement(Y), X))
% 10.58/1.77 = { by lemma 43 }
% 10.58/1.77 join(complement(X), meet(complement(Y), X))
% 10.58/1.77 = { by lemma 59 }
% 10.58/1.77 complement(meet(X, join(Y, complement(X))))
% 10.58/1.77
% 10.58/1.77 Lemma 61: meet(Y, join(X, complement(Y))) = meet(X, join(Y, complement(X))).
% 10.58/1.77 Proof:
% 10.58/1.77 meet(Y, join(X, complement(Y)))
% 10.58/1.77 = { by lemma 44 R->L }
% 10.58/1.77 meet(Y, meet(join(X, complement(Y)), top))
% 10.58/1.77 = { by lemma 58 R->L }
% 10.58/1.77 meet(meet(Y, join(X, complement(Y))), top)
% 10.58/1.77 = { by lemma 40 R->L }
% 10.58/1.77 complement(join(zero, complement(meet(Y, join(X, complement(Y))))))
% 10.58/1.77 = { by lemma 60 }
% 10.58/1.77 complement(join(zero, complement(meet(X, join(Y, complement(X))))))
% 10.58/1.77 = { by lemma 40 }
% 10.58/1.77 meet(meet(X, join(Y, complement(X))), top)
% 10.58/1.77 = { by lemma 58 }
% 10.58/1.77 meet(X, meet(join(Y, complement(X)), top))
% 10.58/1.77 = { by lemma 44 }
% 10.58/1.77 meet(X, join(Y, complement(X)))
% 10.58/1.77
% 10.58/1.77 Lemma 62: meet(X, meet(Y, complement(X))) = zero.
% 10.58/1.77 Proof:
% 10.58/1.77 meet(X, meet(Y, complement(X)))
% 10.58/1.77 = { by lemma 39 }
% 10.58/1.77 meet(X, meet(complement(X), Y))
% 10.58/1.77 = { by axiom 10 (maddux4_definiton_of_meet) }
% 10.58/1.77 complement(join(complement(X), complement(meet(complement(X), Y))))
% 10.58/1.77 = { by lemma 39 }
% 10.58/1.77 complement(join(complement(X), complement(meet(Y, complement(X)))))
% 10.58/1.77 = { by lemma 54 R->L }
% 10.58/1.77 complement(join(complement(X), join(complement(Y), complement(complement(X)))))
% 10.58/1.77 = { by lemma 16 }
% 10.58/1.77 complement(join(complement(Y), top))
% 10.58/1.77 = { by lemma 31 }
% 10.58/1.77 complement(top)
% 10.58/1.77 = { by lemma 15 }
% 10.58/1.77 zero
% 10.58/1.77
% 10.58/1.77 Lemma 63: meet(X, join(Y, complement(X))) = meet(X, Y).
% 10.58/1.77 Proof:
% 10.58/1.77 meet(X, join(Y, complement(X)))
% 10.58/1.77 = { by lemma 61 }
% 10.58/1.77 meet(Y, join(X, complement(Y)))
% 10.58/1.77 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 10.58/1.77 meet(Y, join(complement(Y), X))
% 10.58/1.77 = { by lemma 39 }
% 10.58/1.77 meet(join(complement(Y), X), Y)
% 10.58/1.77 = { by lemma 24 R->L }
% 10.58/1.77 meet(join(complement(Y), X), join(meet(Y, X), complement(join(complement(Y), X))))
% 10.58/1.77 = { by lemma 61 R->L }
% 10.58/1.77 meet(meet(Y, X), join(join(complement(Y), X), complement(meet(Y, X))))
% 10.58/1.77 = { by lemma 58 }
% 10.58/1.77 meet(Y, meet(X, join(join(complement(Y), X), complement(meet(Y, X)))))
% 10.58/1.77 = { by axiom 7 (maddux2_join_associativity) R->L }
% 10.58/1.77 meet(Y, meet(X, join(complement(Y), join(X, complement(meet(Y, X))))))
% 10.58/1.77 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 10.58/1.77 meet(Y, meet(X, join(join(X, complement(meet(Y, X))), complement(Y))))
% 10.58/1.77 = { by axiom 7 (maddux2_join_associativity) R->L }
% 10.58/1.77 meet(Y, meet(X, join(X, join(complement(meet(Y, X)), complement(Y)))))
% 10.58/1.77 = { by axiom 1 (maddux1_join_commutativity) }
% 10.58/1.77 meet(Y, meet(X, join(X, join(complement(Y), complement(meet(Y, X))))))
% 10.58/1.77 = { by lemma 44 R->L }
% 10.58/1.77 meet(Y, meet(X, join(X, meet(join(complement(Y), complement(meet(Y, X))), top))))
% 10.58/1.77 = { by lemma 40 R->L }
% 10.58/1.77 meet(Y, meet(X, join(X, complement(join(zero, complement(join(complement(Y), complement(meet(Y, X)))))))))
% 10.58/1.77 = { by lemma 57 R->L }
% 10.58/1.77 meet(Y, meet(X, complement(meet(join(zero, complement(join(complement(Y), complement(meet(Y, X))))), complement(X)))))
% 10.58/1.77 = { by lemma 56 R->L }
% 10.58/1.77 meet(Y, complement(join(complement(X), meet(join(zero, complement(join(complement(Y), complement(meet(Y, X))))), complement(X)))))
% 10.58/1.77 = { by lemma 29 R->L }
% 10.58/1.77 meet(Y, join(zero, complement(join(complement(X), meet(join(zero, complement(join(complement(Y), complement(meet(Y, X))))), complement(X))))))
% 10.58/1.77 = { by lemma 62 R->L }
% 10.58/1.77 meet(Y, join(meet(X, meet(join(zero, complement(join(complement(Y), complement(meet(Y, X))))), complement(X))), complement(join(complement(X), meet(join(zero, complement(join(complement(Y), complement(meet(Y, X))))), complement(X))))))
% 10.58/1.77 = { by lemma 24 }
% 10.58/1.77 meet(Y, X)
% 10.58/1.77 = { by lemma 39 R->L }
% 10.58/1.77 meet(X, Y)
% 10.58/1.77
% 10.58/1.77 Lemma 64: meet(X, join(complement(X), Y)) = meet(X, Y).
% 10.58/1.77 Proof:
% 10.58/1.77 meet(X, join(complement(X), Y))
% 10.58/1.77 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 10.58/1.77 meet(X, join(Y, complement(X)))
% 10.58/1.77 = { by lemma 63 }
% 10.58/1.77 meet(X, Y)
% 10.58/1.77
% 10.58/1.77 Lemma 65: meet(one, composition(converse(complement(X)), X)) = zero.
% 10.58/1.77 Proof:
% 10.58/1.77 meet(one, composition(converse(complement(X)), X))
% 10.58/1.77 = { by lemma 39 }
% 10.58/1.77 meet(composition(converse(complement(X)), X), one)
% 10.58/1.77 = { by lemma 49 R->L }
% 10.58/1.77 meet(composition(converse(complement(X)), X), complement(complement(one)))
% 10.58/1.77 = { by lemma 20 R->L }
% 10.58/1.77 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(join(zero, complement(X))), complement(composition(join(zero, complement(X)), one))))))
% 10.58/1.77 = { by axiom 3 (composition_identity) }
% 10.58/1.77 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(join(zero, complement(X))), complement(join(zero, complement(X)))))))
% 10.58/1.77 = { by lemma 40 }
% 10.58/1.77 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(join(zero, complement(X))), meet(X, top)))))
% 10.58/1.77 = { by lemma 29 }
% 10.58/1.77 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(complement(X)), meet(X, top)))))
% 10.58/1.77 = { by lemma 44 }
% 10.58/1.77 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(complement(X)), X))))
% 10.58/1.77 = { by lemma 56 }
% 10.58/1.77 meet(composition(converse(complement(X)), X), meet(one, complement(composition(converse(complement(X)), X))))
% 10.58/1.77 = { by lemma 62 }
% 10.58/1.77 zero
% 10.58/1.77
% 10.58/1.77 Lemma 66: meet(converse(X), converse(Y)) = converse(meet(X, Y)).
% 10.58/1.77 Proof:
% 10.58/1.77 meet(converse(X), converse(Y))
% 10.58/1.77 = { by lemma 39 }
% 10.58/1.77 meet(converse(Y), converse(X))
% 10.58/1.77 = { by lemma 64 R->L }
% 10.58/1.77 meet(converse(Y), join(complement(converse(Y)), converse(X)))
% 10.58/1.77 = { by lemma 33 R->L }
% 10.58/1.77 meet(converse(Y), converse(join(converse(complement(converse(Y))), X)))
% 10.58/1.77 = { by lemma 49 R->L }
% 10.58/1.77 meet(converse(Y), converse(join(converse(complement(converse(Y))), complement(complement(X)))))
% 10.58/1.77 = { by lemma 57 R->L }
% 10.58/1.77 meet(converse(Y), converse(complement(meet(complement(X), complement(converse(complement(converse(Y))))))))
% 10.58/1.77 = { by lemma 29 R->L }
% 10.58/1.77 meet(converse(Y), converse(complement(meet(complement(X), join(zero, complement(converse(complement(converse(Y)))))))))
% 10.58/1.77 = { by lemma 63 R->L }
% 10.58/1.77 meet(converse(Y), converse(complement(meet(complement(X), join(join(zero, complement(converse(complement(converse(Y))))), complement(complement(X)))))))
% 10.58/1.77 = { by lemma 60 }
% 10.58/1.77 meet(converse(Y), converse(complement(meet(join(zero, complement(converse(complement(converse(Y))))), join(complement(X), complement(join(zero, complement(converse(complement(converse(Y)))))))))))
% 10.58/1.77 = { by lemma 40 }
% 10.58/1.77 meet(converse(Y), converse(complement(meet(join(zero, complement(converse(complement(converse(Y))))), join(complement(X), meet(converse(complement(converse(Y))), top))))))
% 10.58/1.77 = { by lemma 29 }
% 10.58/1.77 meet(converse(Y), converse(complement(meet(complement(converse(complement(converse(Y)))), join(complement(X), meet(converse(complement(converse(Y))), top))))))
% 10.58/1.77 = { by lemma 39 }
% 10.58/1.77 meet(converse(Y), converse(complement(meet(join(complement(X), meet(converse(complement(converse(Y))), top)), complement(converse(complement(converse(Y))))))))
% 10.58/1.77 = { by lemma 57 }
% 10.58/1.77 meet(converse(Y), converse(join(converse(complement(converse(Y))), complement(join(complement(X), meet(converse(complement(converse(Y))), top))))))
% 10.58/1.77 = { by lemma 44 }
% 10.58/1.77 meet(converse(Y), converse(join(converse(complement(converse(Y))), complement(join(complement(X), converse(complement(converse(Y))))))))
% 10.58/1.77 = { by axiom 1 (maddux1_join_commutativity) }
% 10.58/1.77 meet(converse(Y), converse(join(converse(complement(converse(Y))), complement(join(converse(complement(converse(Y))), complement(X))))))
% 10.58/1.77 = { by lemma 55 }
% 10.58/1.77 meet(converse(Y), converse(join(converse(complement(converse(Y))), meet(X, complement(converse(complement(converse(Y))))))))
% 10.58/1.77 = { by lemma 33 }
% 10.58/1.77 meet(converse(Y), join(complement(converse(Y)), converse(meet(X, complement(converse(complement(converse(Y))))))))
% 10.58/1.77 = { by lemma 64 }
% 10.58/1.77 meet(converse(Y), converse(meet(X, complement(converse(complement(converse(Y)))))))
% 10.58/1.77 = { by lemma 29 R->L }
% 10.58/1.77 meet(converse(Y), converse(meet(X, complement(converse(join(zero, complement(converse(Y))))))))
% 10.58/1.77 = { by lemma 24 R->L }
% 10.58/1.77 meet(converse(Y), converse(meet(X, complement(converse(join(meet(join(zero, complement(converse(Y))), complement(converse(complement(converse(complement(join(zero, complement(converse(Y))))))))), complement(join(complement(join(zero, complement(converse(Y)))), complement(converse(complement(converse(complement(join(zero, complement(converse(Y))))))))))))))))
% 10.58/1.78 = { by lemma 56 R->L }
% 10.58/1.78 meet(converse(Y), converse(meet(X, complement(converse(join(complement(join(complement(join(zero, complement(converse(Y)))), converse(complement(converse(complement(join(zero, complement(converse(Y))))))))), complement(join(complement(join(zero, complement(converse(Y)))), complement(converse(complement(converse(complement(join(zero, complement(converse(Y))))))))))))))))
% 10.58/1.78 = { by lemma 34 }
% 10.58/1.78 meet(converse(Y), converse(meet(X, complement(converse(join(complement(converse(top)), complement(join(complement(join(zero, complement(converse(Y)))), complement(converse(complement(converse(complement(join(zero, complement(converse(Y))))))))))))))))
% 10.58/1.78 = { by lemma 37 }
% 10.58/1.78 meet(converse(Y), converse(meet(X, complement(converse(join(complement(top), complement(join(complement(join(zero, complement(converse(Y)))), complement(converse(complement(converse(complement(join(zero, complement(converse(Y))))))))))))))))
% 10.58/1.78 = { by lemma 15 }
% 10.58/1.78 meet(converse(Y), converse(meet(X, complement(converse(join(zero, complement(join(complement(join(zero, complement(converse(Y)))), complement(converse(complement(converse(complement(join(zero, complement(converse(Y))))))))))))))))
% 10.58/1.78 = { by lemma 29 }
% 10.58/1.78 meet(converse(Y), converse(meet(X, complement(converse(complement(join(complement(join(zero, complement(converse(Y)))), complement(converse(complement(converse(complement(join(zero, complement(converse(Y)))))))))))))))
% 10.58/1.78 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 10.58/1.78 meet(converse(Y), converse(meet(X, complement(converse(meet(join(zero, complement(converse(Y))), converse(complement(converse(complement(join(zero, complement(converse(Y)))))))))))))
% 10.58/1.78 = { by lemma 30 R->L }
% 10.58/1.78 meet(converse(Y), converse(meet(X, complement(converse(join(meet(join(zero, complement(converse(Y))), converse(complement(converse(complement(join(zero, complement(converse(Y)))))))), zero))))))
% 10.58/1.78 = { by lemma 50 R->L }
% 10.58/1.78 meet(converse(Y), converse(meet(X, complement(converse(join(meet(join(zero, complement(converse(Y))), converse(complement(converse(complement(join(zero, complement(converse(Y)))))))), meet(zero, converse(complement(converse(complement(join(zero, complement(converse(Y))))))))))))))
% 10.58/1.78 = { by lemma 52 R->L }
% 10.58/1.78 meet(converse(Y), converse(meet(X, complement(converse(join(meet(join(zero, complement(converse(Y))), converse(complement(converse(complement(join(zero, complement(converse(Y)))))))), meet(composition(zero, complement(join(zero, complement(converse(Y))))), converse(complement(converse(complement(join(zero, complement(converse(Y))))))))))))))
% 10.58/1.78 = { by lemma 65 R->L }
% 10.58/1.78 meet(converse(Y), converse(meet(X, complement(converse(join(meet(join(zero, complement(converse(Y))), converse(complement(converse(complement(join(zero, complement(converse(Y)))))))), meet(composition(meet(one, composition(converse(complement(converse(complement(join(zero, complement(converse(Y))))))), converse(complement(join(zero, complement(converse(Y))))))), complement(join(zero, complement(converse(Y))))), converse(complement(converse(complement(join(zero, complement(converse(Y))))))))))))))
% 10.58/1.78 = { by axiom 14 (modular_law_2) R->L }
% 10.58/1.78 meet(converse(Y), converse(meet(X, complement(converse(join(meet(join(zero, complement(converse(Y))), converse(complement(converse(complement(join(zero, complement(converse(Y)))))))), join(meet(composition(one, complement(join(zero, complement(converse(Y))))), converse(complement(converse(complement(join(zero, complement(converse(Y)))))))), meet(composition(meet(one, composition(converse(complement(converse(complement(join(zero, complement(converse(Y))))))), converse(complement(join(zero, complement(converse(Y))))))), complement(join(zero, complement(converse(Y))))), converse(complement(converse(complement(join(zero, complement(converse(Y)))))))))))))))
% 10.58/1.78 = { by lemma 65 }
% 10.58/1.78 meet(converse(Y), converse(meet(X, complement(converse(join(meet(join(zero, complement(converse(Y))), converse(complement(converse(complement(join(zero, complement(converse(Y)))))))), join(meet(composition(one, complement(join(zero, complement(converse(Y))))), converse(complement(converse(complement(join(zero, complement(converse(Y)))))))), meet(composition(zero, complement(join(zero, complement(converse(Y))))), converse(complement(converse(complement(join(zero, complement(converse(Y)))))))))))))))
% 10.58/1.78 = { by lemma 19 }
% 10.58/1.78 meet(converse(Y), converse(meet(X, complement(converse(join(meet(join(zero, complement(converse(Y))), converse(complement(converse(complement(join(zero, complement(converse(Y)))))))), join(meet(complement(join(zero, complement(converse(Y)))), converse(complement(converse(complement(join(zero, complement(converse(Y)))))))), meet(composition(zero, complement(join(zero, complement(converse(Y))))), converse(complement(converse(complement(join(zero, complement(converse(Y)))))))))))))))
% 10.58/1.78 = { by lemma 52 }
% 10.58/1.78 meet(converse(Y), converse(meet(X, complement(converse(join(meet(join(zero, complement(converse(Y))), converse(complement(converse(complement(join(zero, complement(converse(Y)))))))), join(meet(complement(join(zero, complement(converse(Y)))), converse(complement(converse(complement(join(zero, complement(converse(Y)))))))), meet(zero, converse(complement(converse(complement(join(zero, complement(converse(Y)))))))))))))))
% 10.58/1.78 = { by lemma 50 }
% 10.58/1.78 meet(converse(Y), converse(meet(X, complement(converse(join(meet(join(zero, complement(converse(Y))), converse(complement(converse(complement(join(zero, complement(converse(Y)))))))), join(meet(complement(join(zero, complement(converse(Y)))), converse(complement(converse(complement(join(zero, complement(converse(Y)))))))), zero)))))))
% 10.58/1.78 = { by lemma 30 }
% 10.58/1.78 meet(converse(Y), converse(meet(X, complement(converse(join(meet(join(zero, complement(converse(Y))), converse(complement(converse(complement(join(zero, complement(converse(Y)))))))), meet(complement(join(zero, complement(converse(Y)))), converse(complement(converse(complement(join(zero, complement(converse(Y))))))))))))))
% 10.58/1.78 = { by lemma 39 }
% 10.58/1.78 meet(converse(Y), converse(meet(X, complement(converse(join(meet(join(zero, complement(converse(Y))), converse(complement(converse(complement(join(zero, complement(converse(Y)))))))), meet(converse(complement(converse(complement(join(zero, complement(converse(Y))))))), complement(join(zero, complement(converse(Y)))))))))))
% 10.58/1.78 = { by lemma 39 }
% 10.58/1.78 meet(converse(Y), converse(meet(X, complement(converse(join(meet(converse(complement(converse(complement(join(zero, complement(converse(Y))))))), join(zero, complement(converse(Y)))), meet(converse(complement(converse(complement(join(zero, complement(converse(Y))))))), complement(join(zero, complement(converse(Y)))))))))))
% 10.58/1.78 = { by lemma 43 }
% 10.58/1.78 meet(converse(Y), converse(meet(X, complement(converse(converse(complement(converse(complement(join(zero, complement(converse(Y))))))))))))
% 10.58/1.78 = { by axiom 2 (converse_idempotence) }
% 10.58/1.78 meet(converse(Y), converse(meet(X, complement(complement(converse(complement(join(zero, complement(converse(Y))))))))))
% 10.58/1.78 = { by lemma 40 }
% 10.58/1.78 meet(converse(Y), converse(meet(X, complement(complement(converse(meet(converse(Y), top)))))))
% 10.58/1.78 = { by lemma 44 }
% 10.58/1.78 meet(converse(Y), converse(meet(X, complement(complement(converse(converse(Y)))))))
% 10.58/1.78 = { by lemma 49 }
% 10.58/1.78 meet(converse(Y), converse(meet(X, converse(converse(Y)))))
% 10.58/1.78 = { by lemma 39 }
% 10.58/1.78 meet(converse(Y), converse(meet(converse(converse(Y)), X)))
% 10.58/1.78 = { by axiom 2 (converse_idempotence) R->L }
% 10.58/1.78 meet(converse(converse(converse(Y))), converse(meet(converse(converse(Y)), X)))
% 10.58/1.78 = { by lemma 39 }
% 10.58/1.78 meet(converse(meet(converse(converse(Y)), X)), converse(converse(converse(Y))))
% 10.58/1.78 = { by lemma 24 R->L }
% 10.58/1.78 meet(converse(meet(converse(converse(Y)), X)), converse(join(meet(converse(converse(Y)), X), complement(join(complement(converse(converse(Y))), X)))))
% 10.58/1.78 = { by lemma 30 R->L }
% 10.58/1.78 join(meet(converse(meet(converse(converse(Y)), X)), converse(join(meet(converse(converse(Y)), X), complement(join(complement(converse(converse(Y))), X))))), zero)
% 10.58/1.78 = { by lemma 15 R->L }
% 10.58/1.78 join(meet(converse(meet(converse(converse(Y)), X)), converse(join(meet(converse(converse(Y)), X), complement(join(complement(converse(converse(Y))), X))))), complement(top))
% 10.58/1.78 = { by lemma 35 R->L }
% 10.58/1.78 join(meet(converse(meet(converse(converse(Y)), X)), converse(join(meet(converse(converse(Y)), X), complement(join(complement(converse(converse(Y))), X))))), complement(join(converse(meet(converse(converse(Y)), X)), join(complement(converse(meet(converse(converse(Y)), X))), converse(complement(join(complement(converse(converse(Y))), X)))))))
% 10.58/1.78 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 10.58/1.78 join(meet(converse(meet(converse(converse(Y)), X)), converse(join(meet(converse(converse(Y)), X), complement(join(complement(converse(converse(Y))), X))))), complement(join(converse(meet(converse(converse(Y)), X)), join(converse(complement(join(complement(converse(converse(Y))), X))), complement(converse(meet(converse(converse(Y)), X)))))))
% 10.58/1.78 = { by axiom 7 (maddux2_join_associativity) }
% 10.58/1.78 join(meet(converse(meet(converse(converse(Y)), X)), converse(join(meet(converse(converse(Y)), X), complement(join(complement(converse(converse(Y))), X))))), complement(join(join(converse(meet(converse(converse(Y)), X)), converse(complement(join(complement(converse(converse(Y))), X)))), complement(converse(meet(converse(converse(Y)), X))))))
% 10.58/1.78 = { by axiom 6 (converse_additivity) R->L }
% 10.58/1.78 join(meet(converse(meet(converse(converse(Y)), X)), converse(join(meet(converse(converse(Y)), X), complement(join(complement(converse(converse(Y))), X))))), complement(join(converse(join(meet(converse(converse(Y)), X), complement(join(complement(converse(converse(Y))), X)))), complement(converse(meet(converse(converse(Y)), X))))))
% 10.58/1.78 = { by axiom 1 (maddux1_join_commutativity) }
% 10.58/1.78 join(meet(converse(meet(converse(converse(Y)), X)), converse(join(meet(converse(converse(Y)), X), complement(join(complement(converse(converse(Y))), X))))), complement(join(complement(converse(meet(converse(converse(Y)), X))), converse(join(meet(converse(converse(Y)), X), complement(join(complement(converse(converse(Y))), X)))))))
% 10.58/1.78 = { by lemma 24 }
% 10.58/1.78 converse(meet(converse(converse(Y)), X))
% 10.58/1.78 = { by lemma 39 R->L }
% 10.58/1.78 converse(meet(X, converse(converse(Y))))
% 10.58/1.78 = { by axiom 2 (converse_idempotence) }
% 10.58/1.78 converse(meet(X, Y))
% 10.58/1.78
% 10.58/1.78 Goal 1 (goals): tuple(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1))), join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2)))) = tuple(converse(meet(x0, x1)), meet(converse(x0_2), converse(x1_2))).
% 10.58/1.78 Proof:
% 10.58/1.78 tuple(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1))), join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))))
% 10.58/1.78 = { by axiom 1 (maddux1_join_commutativity) }
% 10.58/1.78 tuple(join(converse(meet(x0, x1)), meet(converse(x0), converse(x1))), join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))))
% 10.58/1.78 = { by lemma 66 }
% 10.58/1.78 tuple(join(converse(meet(x0, x1)), converse(meet(x0, x1))), join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))))
% 10.58/1.78 = { by lemma 46 }
% 10.58/1.78 tuple(converse(meet(x0, x1)), join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))))
% 10.58/1.78 = { by lemma 66 }
% 10.58/1.78 tuple(converse(meet(x0, x1)), join(converse(meet(x0_2, x1_2)), converse(meet(x0_2, x1_2))))
% 10.58/1.78 = { by lemma 46 }
% 10.58/1.78 tuple(converse(meet(x0, x1)), converse(meet(x0_2, x1_2)))
% 10.58/1.78 = { by lemma 66 R->L }
% 10.58/1.78 tuple(converse(meet(x0, x1)), meet(converse(x0_2), converse(x1_2)))
% 10.58/1.78 % SZS output end Proof
% 10.58/1.78
% 10.58/1.78 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------