TSTP Solution File: REL027+2 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL027+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 : n002.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 13:44:09 EDT 2023
% Result : Theorem 14.22s 2.18s
% Output : Proof 15.25s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : REL027+2 : TPTP v8.1.2. Released v4.0.0.
% 0.07/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.33 % Computer : n002.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Aug 25 20:24:47 EDT 2023
% 0.12/0.34 % CPUTime :
% 14.22/2.18 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 14.22/2.18
% 14.22/2.18 % SZS status Theorem
% 14.22/2.18
% 14.64/2.25 % SZS output start Proof
% 14.64/2.25 Take the following subset of the input axioms:
% 14.64/2.26 fof(composition_associativity, axiom, ![X0, X1, X2]: composition(X0, composition(X1, X2))=composition(composition(X0, X1), X2)).
% 14.64/2.26 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))).
% 14.64/2.26 fof(composition_identity, axiom, ![X0_2]: composition(X0_2, one)=X0_2).
% 14.64/2.26 fof(converse_additivity, axiom, ![X0_2, X1_2]: converse(join(X0_2, X1_2))=join(converse(X0_2), converse(X1_2))).
% 14.64/2.26 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)).
% 14.64/2.26 fof(converse_idempotence, axiom, ![X0_2]: converse(converse(X0_2))=X0_2).
% 14.64/2.26 fof(converse_multiplicativity, axiom, ![X0_2, X1_2]: converse(composition(X0_2, X1_2))=composition(converse(X1_2), converse(X0_2))).
% 14.64/2.26 fof(def_top, axiom, ![X0_2]: top=join(X0_2, complement(X0_2))).
% 14.64/2.26 fof(def_zero, axiom, ![X0_2]: zero=meet(X0_2, complement(X0_2))).
% 14.64/2.26 fof(goals, conjecture, ![X0_2]: (join(X0_2, one)=one => (join(meet(complement(composition(X0_2, top)), one), meet(complement(X0_2), one))=meet(complement(X0_2), one) & join(meet(complement(X0_2), one), meet(complement(composition(X0_2, top)), one))=meet(complement(composition(X0_2, top)), one)))).
% 14.64/2.26 fof(maddux1_join_commutativity, axiom, ![X0_2, X1_2]: join(X0_2, X1_2)=join(X1_2, X0_2)).
% 14.64/2.26 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)).
% 14.64/2.26 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)))).
% 14.64/2.26 fof(maddux4_definiton_of_meet, axiom, ![X0_2, X1_2]: meet(X0_2, X1_2)=complement(join(complement(X0_2), complement(X1_2)))).
% 14.64/2.26
% 14.64/2.26 Now clausify the problem and encode Horn clauses using encoding 3 of
% 14.64/2.26 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 14.64/2.26 We repeatedly replace C & s=t => u=v by the two clauses:
% 14.64/2.26 fresh(y, y, x1...xn) = u
% 14.64/2.26 C => fresh(s, t, x1...xn) = v
% 14.64/2.26 where fresh is a fresh function symbol and x1..xn are the free
% 14.64/2.26 variables of u and v.
% 14.64/2.26 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 14.64/2.26 input problem has no model of domain size 1).
% 14.64/2.26
% 14.64/2.26 The encoding turns the above axioms into the following unit equations and goals:
% 14.64/2.26
% 14.64/2.26 Axiom 1 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 14.64/2.26 Axiom 2 (goals): join(x0, one) = one.
% 14.64/2.26 Axiom 3 (composition_identity): composition(X, one) = X.
% 14.64/2.26 Axiom 4 (converse_idempotence): converse(converse(X)) = X.
% 14.64/2.26 Axiom 5 (def_top): top = join(X, complement(X)).
% 14.64/2.26 Axiom 6 (def_zero): zero = meet(X, complement(X)).
% 14.64/2.26 Axiom 7 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 14.64/2.26 Axiom 8 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 14.64/2.26 Axiom 9 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 14.64/2.26 Axiom 10 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 14.64/2.26 Axiom 11 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 14.64/2.26 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 14.64/2.26 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 14.64/2.26 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 14.64/2.26
% 14.64/2.26 Lemma 15: complement(top) = zero.
% 14.64/2.26 Proof:
% 14.64/2.26 complement(top)
% 14.64/2.26 = { by axiom 5 (def_top) }
% 14.64/2.26 complement(join(complement(X), complement(complement(X))))
% 14.64/2.26 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 14.64/2.26 meet(X, complement(X))
% 14.64/2.26 = { by axiom 6 (def_zero) R->L }
% 14.64/2.26 zero
% 14.64/2.26
% 14.64/2.26 Lemma 16: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 14.64/2.26 Proof:
% 14.64/2.26 join(meet(X, Y), complement(join(complement(X), Y)))
% 14.64/2.26 = { by axiom 11 (maddux4_definiton_of_meet) }
% 14.64/2.26 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 14.64/2.26 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L }
% 14.64/2.26 X
% 14.64/2.26
% 14.64/2.26 Lemma 17: join(zero, meet(X, X)) = X.
% 14.64/2.26 Proof:
% 14.64/2.26 join(zero, meet(X, X))
% 14.64/2.26 = { by axiom 11 (maddux4_definiton_of_meet) }
% 14.64/2.26 join(zero, complement(join(complement(X), complement(X))))
% 14.64/2.26 = { by axiom 6 (def_zero) }
% 14.64/2.26 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 14.64/2.26 = { by lemma 16 }
% 14.64/2.26 X
% 14.64/2.26
% 14.64/2.26 Lemma 18: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 14.64/2.26 Proof:
% 14.64/2.26 converse(composition(converse(X), Y))
% 14.64/2.26 = { by axiom 9 (converse_multiplicativity) }
% 14.64/2.26 composition(converse(Y), converse(converse(X)))
% 14.64/2.26 = { by axiom 4 (converse_idempotence) }
% 14.64/2.26 composition(converse(Y), X)
% 14.64/2.26
% 14.64/2.26 Lemma 19: composition(converse(one), X) = X.
% 14.64/2.26 Proof:
% 14.64/2.26 composition(converse(one), X)
% 14.64/2.26 = { by lemma 18 R->L }
% 14.64/2.26 converse(composition(converse(X), one))
% 14.64/2.26 = { by axiom 3 (composition_identity) }
% 14.64/2.26 converse(converse(X))
% 14.64/2.26 = { by axiom 4 (converse_idempotence) }
% 14.64/2.26 X
% 14.64/2.26
% 14.64/2.26 Lemma 20: composition(one, X) = X.
% 14.64/2.26 Proof:
% 14.64/2.26 composition(one, X)
% 14.64/2.26 = { by lemma 19 R->L }
% 14.64/2.26 composition(converse(one), composition(one, X))
% 14.64/2.26 = { by axiom 10 (composition_associativity) }
% 14.64/2.26 composition(composition(converse(one), one), X)
% 14.64/2.26 = { by axiom 3 (composition_identity) }
% 14.64/2.26 composition(converse(one), X)
% 14.64/2.26 = { by lemma 19 }
% 14.64/2.26 X
% 14.64/2.26
% 14.64/2.26 Lemma 21: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 14.64/2.26 Proof:
% 14.64/2.26 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 14.64/2.26 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.64/2.26 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 14.64/2.26 = { by axiom 13 (converse_cancellativity) }
% 14.64/2.26 complement(X)
% 14.64/2.26
% 14.64/2.26 Lemma 22: join(complement(X), complement(X)) = complement(X).
% 14.64/2.26 Proof:
% 14.64/2.26 join(complement(X), complement(X))
% 14.64/2.26 = { by lemma 19 R->L }
% 14.64/2.26 join(complement(X), composition(converse(one), complement(X)))
% 14.64/2.26 = { by lemma 20 R->L }
% 14.64/2.26 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 14.64/2.26 = { by lemma 21 }
% 14.64/2.26 complement(X)
% 14.64/2.26
% 14.64/2.26 Lemma 23: join(zero, zero) = zero.
% 14.64/2.26 Proof:
% 14.64/2.26 join(zero, zero)
% 14.64/2.26 = { by lemma 15 R->L }
% 14.64/2.26 join(zero, complement(top))
% 14.64/2.26 = { by lemma 15 R->L }
% 14.64/2.26 join(complement(top), complement(top))
% 14.64/2.26 = { by lemma 22 }
% 14.64/2.26 complement(top)
% 14.64/2.26 = { by lemma 15 }
% 14.64/2.26 zero
% 14.64/2.26
% 14.64/2.26 Lemma 24: join(zero, join(zero, X)) = join(X, zero).
% 14.64/2.26 Proof:
% 14.64/2.26 join(zero, join(zero, X))
% 14.64/2.26 = { by axiom 8 (maddux2_join_associativity) }
% 14.64/2.26 join(join(zero, zero), X)
% 14.64/2.26 = { by lemma 23 }
% 14.64/2.26 join(zero, X)
% 14.64/2.26 = { by axiom 1 (maddux1_join_commutativity) }
% 14.64/2.26 join(X, zero)
% 14.64/2.26
% 14.64/2.26 Lemma 25: join(zero, complement(complement(X))) = X.
% 14.64/2.26 Proof:
% 14.64/2.26 join(zero, complement(complement(X)))
% 14.64/2.26 = { by axiom 6 (def_zero) }
% 14.64/2.26 join(meet(X, complement(X)), complement(complement(X)))
% 14.64/2.26 = { by lemma 22 R->L }
% 14.64/2.26 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 14.64/2.26 = { by lemma 16 }
% 14.64/2.26 X
% 14.64/2.26
% 14.64/2.26 Lemma 26: join(X, zero) = X.
% 14.64/2.26 Proof:
% 14.64/2.26 join(X, zero)
% 14.64/2.26 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.64/2.26 join(zero, X)
% 14.64/2.26 = { by lemma 17 R->L }
% 14.64/2.26 join(zero, join(zero, meet(X, X)))
% 14.64/2.26 = { by lemma 24 }
% 14.64/2.26 join(meet(X, X), zero)
% 14.64/2.26 = { by axiom 11 (maddux4_definiton_of_meet) }
% 14.64/2.26 join(complement(join(complement(X), complement(X))), zero)
% 14.64/2.26 = { by lemma 22 }
% 14.64/2.26 join(complement(complement(X)), zero)
% 14.64/2.26 = { by axiom 1 (maddux1_join_commutativity) }
% 14.64/2.26 join(zero, complement(complement(X)))
% 14.64/2.26 = { by lemma 25 }
% 14.64/2.26 X
% 14.64/2.26
% 14.64/2.26 Lemma 27: join(zero, X) = X.
% 14.64/2.26 Proof:
% 14.64/2.26 join(zero, X)
% 14.64/2.26 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.64/2.26 join(X, zero)
% 14.64/2.26 = { by lemma 26 }
% 14.64/2.26 X
% 14.64/2.26
% 14.64/2.26 Lemma 28: complement(zero) = top.
% 14.64/2.26 Proof:
% 14.64/2.26 complement(zero)
% 14.64/2.26 = { by lemma 27 R->L }
% 14.64/2.26 join(zero, complement(zero))
% 14.64/2.26 = { by axiom 5 (def_top) R->L }
% 14.64/2.26 top
% 14.64/2.26
% 14.64/2.26 Lemma 29: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 14.64/2.26 Proof:
% 14.64/2.26 converse(join(X, converse(Y)))
% 14.64/2.26 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.64/2.26 converse(join(converse(Y), X))
% 14.64/2.26 = { by axiom 7 (converse_additivity) }
% 14.64/2.26 join(converse(converse(Y)), converse(X))
% 14.64/2.26 = { by axiom 4 (converse_idempotence) }
% 14.64/2.26 join(Y, converse(X))
% 14.64/2.26
% 14.64/2.26 Lemma 30: join(X, join(Y, complement(X))) = join(Y, top).
% 14.64/2.26 Proof:
% 14.64/2.26 join(X, join(Y, complement(X)))
% 14.64/2.26 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.64/2.26 join(X, join(complement(X), Y))
% 14.64/2.26 = { by axiom 8 (maddux2_join_associativity) }
% 14.64/2.26 join(join(X, complement(X)), Y)
% 14.64/2.26 = { by axiom 5 (def_top) R->L }
% 14.64/2.26 join(top, Y)
% 14.64/2.26 = { by axiom 1 (maddux1_join_commutativity) }
% 14.64/2.26 join(Y, top)
% 14.64/2.26
% 14.64/2.26 Lemma 31: join(top, complement(X)) = top.
% 14.64/2.26 Proof:
% 14.64/2.26 join(top, complement(X))
% 14.64/2.26 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.64/2.26 join(complement(X), top)
% 14.64/2.26 = { by lemma 30 R->L }
% 14.64/2.26 join(X, join(complement(X), complement(X)))
% 14.64/2.26 = { by lemma 22 }
% 14.64/2.26 join(X, complement(X))
% 14.64/2.26 = { by axiom 5 (def_top) R->L }
% 14.64/2.26 top
% 14.64/2.26
% 14.64/2.26 Lemma 32: join(Y, top) = join(X, top).
% 14.64/2.26 Proof:
% 14.64/2.26 join(Y, top)
% 14.64/2.26 = { by lemma 31 R->L }
% 14.64/2.26 join(Y, join(top, complement(Y)))
% 14.64/2.26 = { by lemma 30 }
% 14.64/2.26 join(top, top)
% 14.64/2.26 = { by lemma 30 R->L }
% 14.64/2.26 join(X, join(top, complement(X)))
% 14.64/2.26 = { by lemma 31 }
% 14.64/2.26 join(X, top)
% 14.64/2.26
% 14.64/2.26 Lemma 33: join(x0, join(one, X)) = join(X, one).
% 14.64/2.26 Proof:
% 14.64/2.26 join(x0, join(one, X))
% 14.64/2.26 = { by axiom 8 (maddux2_join_associativity) }
% 14.64/2.26 join(join(x0, one), X)
% 14.64/2.26 = { by axiom 2 (goals) }
% 14.64/2.26 join(one, X)
% 14.64/2.26 = { by axiom 1 (maddux1_join_commutativity) }
% 14.64/2.26 join(X, one)
% 14.64/2.26
% 14.64/2.26 Lemma 34: join(X, top) = top.
% 14.64/2.26 Proof:
% 14.64/2.26 join(X, top)
% 14.64/2.26 = { by lemma 32 }
% 14.64/2.26 join(x0, top)
% 14.64/2.26 = { by axiom 5 (def_top) }
% 14.64/2.26 join(x0, join(one, complement(one)))
% 14.64/2.26 = { by lemma 33 }
% 14.64/2.26 join(complement(one), one)
% 14.64/2.26 = { by axiom 1 (maddux1_join_commutativity) }
% 14.64/2.26 join(one, complement(one))
% 14.64/2.26 = { by axiom 5 (def_top) R->L }
% 14.64/2.26 top
% 14.64/2.26
% 14.64/2.26 Lemma 35: join(top, X) = top.
% 14.64/2.26 Proof:
% 14.64/2.26 join(top, X)
% 14.64/2.26 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.64/2.26 join(X, top)
% 14.64/2.26 = { by lemma 32 R->L }
% 14.64/2.26 join(Y, top)
% 14.64/2.26 = { by lemma 34 }
% 14.64/2.26 top
% 14.64/2.26
% 14.64/2.26 Lemma 36: join(X, converse(top)) = converse(top).
% 14.64/2.27 Proof:
% 14.64/2.27 join(X, converse(top))
% 14.64/2.27 = { by lemma 29 R->L }
% 14.64/2.27 converse(join(top, converse(X)))
% 14.64/2.27 = { by lemma 35 }
% 14.64/2.27 converse(top)
% 14.64/2.27
% 14.64/2.27 Lemma 37: converse(top) = top.
% 14.64/2.27 Proof:
% 14.64/2.27 converse(top)
% 14.64/2.27 = { by lemma 36 R->L }
% 14.64/2.27 join(X, converse(top))
% 14.64/2.27 = { by lemma 36 R->L }
% 14.64/2.27 join(X, join(complement(X), converse(top)))
% 14.64/2.27 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.64/2.27 join(X, join(converse(top), complement(X)))
% 14.64/2.27 = { by lemma 30 }
% 14.64/2.27 join(converse(top), top)
% 14.64/2.27 = { by lemma 34 }
% 14.64/2.27 top
% 14.64/2.27
% 14.64/2.27 Lemma 38: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 14.64/2.27 Proof:
% 14.64/2.27 converse(join(converse(X), Y))
% 14.64/2.27 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.64/2.27 converse(join(Y, converse(X)))
% 14.64/2.27 = { by lemma 29 }
% 14.64/2.27 join(X, converse(Y))
% 14.64/2.27
% 14.64/2.27 Lemma 39: converse(zero) = zero.
% 14.64/2.27 Proof:
% 14.64/2.27 converse(zero)
% 14.64/2.27 = { by lemma 26 R->L }
% 14.64/2.27 join(converse(zero), zero)
% 14.64/2.27 = { by lemma 24 R->L }
% 14.64/2.27 join(zero, join(zero, converse(zero)))
% 14.64/2.27 = { by lemma 38 R->L }
% 14.64/2.27 join(zero, converse(join(converse(zero), zero)))
% 14.64/2.27 = { by lemma 26 }
% 14.64/2.27 join(zero, converse(converse(zero)))
% 14.64/2.27 = { by axiom 4 (converse_idempotence) }
% 14.64/2.27 join(zero, zero)
% 14.64/2.27 = { by lemma 23 }
% 14.64/2.27 zero
% 14.64/2.27
% 14.64/2.27 Lemma 40: complement(complement(X)) = X.
% 14.64/2.27 Proof:
% 14.64/2.27 complement(complement(X))
% 14.64/2.27 = { by lemma 27 R->L }
% 14.64/2.27 join(zero, complement(complement(X)))
% 14.64/2.27 = { by lemma 25 }
% 14.64/2.27 X
% 14.64/2.27
% 14.64/2.27 Lemma 41: meet(Y, X) = meet(X, Y).
% 14.64/2.27 Proof:
% 14.64/2.27 meet(Y, X)
% 14.64/2.27 = { by axiom 11 (maddux4_definiton_of_meet) }
% 14.64/2.27 complement(join(complement(Y), complement(X)))
% 14.64/2.27 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.64/2.27 complement(join(complement(X), complement(Y)))
% 14.64/2.27 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 14.64/2.27 meet(X, Y)
% 14.64/2.27
% 14.64/2.27 Lemma 42: complement(join(zero, complement(X))) = meet(X, top).
% 14.64/2.27 Proof:
% 14.64/2.27 complement(join(zero, complement(X)))
% 14.64/2.27 = { by lemma 15 R->L }
% 14.64/2.27 complement(join(complement(top), complement(X)))
% 14.64/2.27 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 14.64/2.27 meet(top, X)
% 14.64/2.27 = { by lemma 41 R->L }
% 14.64/2.27 meet(X, top)
% 14.64/2.27
% 14.64/2.27 Lemma 43: meet(X, top) = X.
% 14.64/2.27 Proof:
% 14.64/2.27 meet(X, top)
% 14.64/2.27 = { by lemma 42 R->L }
% 14.64/2.27 complement(join(zero, complement(X)))
% 14.64/2.27 = { by lemma 27 }
% 14.64/2.27 complement(complement(X))
% 14.64/2.27 = { by lemma 40 }
% 14.64/2.27 X
% 14.64/2.27
% 14.64/2.27 Lemma 44: composition(join(X, one), Y) = join(Y, composition(X, Y)).
% 14.64/2.27 Proof:
% 14.64/2.27 composition(join(X, one), Y)
% 14.64/2.27 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.64/2.27 composition(join(one, X), Y)
% 14.64/2.27 = { by axiom 12 (composition_distributivity) }
% 14.64/2.27 join(composition(one, Y), composition(X, Y))
% 14.64/2.27 = { by lemma 20 }
% 14.64/2.27 join(Y, composition(X, Y))
% 14.64/2.27
% 14.64/2.27 Lemma 45: composition(join(one, Y), X) = join(X, composition(Y, X)).
% 14.64/2.27 Proof:
% 14.64/2.27 composition(join(one, Y), X)
% 14.64/2.27 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.64/2.27 composition(join(Y, one), X)
% 14.64/2.27 = { by lemma 44 }
% 14.64/2.27 join(X, composition(Y, X))
% 14.64/2.27
% 14.64/2.27 Lemma 46: composition(top, zero) = zero.
% 14.64/2.27 Proof:
% 14.64/2.27 composition(top, zero)
% 14.64/2.27 = { by lemma 37 R->L }
% 14.64/2.27 composition(converse(top), zero)
% 14.64/2.27 = { by lemma 27 R->L }
% 14.64/2.27 join(zero, composition(converse(top), zero))
% 14.64/2.27 = { by lemma 15 R->L }
% 14.64/2.27 join(complement(top), composition(converse(top), zero))
% 14.64/2.27 = { by lemma 15 R->L }
% 14.64/2.27 join(complement(top), composition(converse(top), complement(top)))
% 14.64/2.27 = { by lemma 35 R->L }
% 14.64/2.27 join(complement(top), composition(converse(top), complement(join(top, composition(complement(x0), top)))))
% 14.64/2.27 = { by lemma 45 R->L }
% 14.64/2.27 join(complement(top), composition(converse(top), complement(composition(join(one, complement(x0)), top))))
% 14.64/2.27 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.64/2.27 join(complement(top), composition(converse(top), complement(composition(join(complement(x0), one), top))))
% 14.64/2.27 = { by lemma 33 R->L }
% 14.64/2.27 join(complement(top), composition(converse(top), complement(composition(join(x0, join(one, complement(x0))), top))))
% 14.64/2.27 = { by lemma 30 }
% 14.64/2.27 join(complement(top), composition(converse(top), complement(composition(join(one, top), top))))
% 14.64/2.27 = { by lemma 34 }
% 14.64/2.27 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 14.64/2.27 = { by lemma 21 }
% 14.64/2.27 complement(top)
% 14.64/2.27 = { by lemma 15 }
% 14.64/2.27 zero
% 14.64/2.27
% 14.64/2.27 Lemma 47: converse(composition(top, X)) = composition(converse(X), top).
% 14.64/2.27 Proof:
% 14.64/2.27 converse(composition(top, X))
% 14.64/2.27 = { by axiom 9 (converse_multiplicativity) }
% 14.64/2.27 composition(converse(X), converse(top))
% 14.64/2.27 = { by lemma 37 }
% 14.64/2.27 composition(converse(X), top)
% 14.64/2.27
% 14.64/2.27 Lemma 48: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 14.64/2.27 Proof:
% 14.64/2.27 meet(X, join(complement(Y), complement(Z)))
% 14.64/2.27 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.64/2.27 meet(X, join(complement(Z), complement(Y)))
% 14.64/2.27 = { by lemma 41 }
% 14.64/2.27 meet(join(complement(Z), complement(Y)), X)
% 14.64/2.27 = { by axiom 11 (maddux4_definiton_of_meet) }
% 14.64/2.27 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 14.64/2.27 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 14.64/2.27 complement(join(meet(Z, Y), complement(X)))
% 14.64/2.27 = { by axiom 1 (maddux1_join_commutativity) }
% 14.64/2.27 complement(join(complement(X), meet(Z, Y)))
% 14.64/2.27 = { by lemma 41 R->L }
% 14.64/2.27 complement(join(complement(X), meet(Y, Z)))
% 14.64/2.27
% 14.64/2.27 Lemma 49: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 14.64/2.27 Proof:
% 14.64/2.27 complement(join(X, complement(Y)))
% 14.64/2.27 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.64/2.27 complement(join(complement(Y), X))
% 14.64/2.27 = { by lemma 43 R->L }
% 14.64/2.27 complement(join(complement(Y), meet(X, top)))
% 14.64/2.27 = { by lemma 41 R->L }
% 14.64/2.27 complement(join(complement(Y), meet(top, X)))
% 14.64/2.27 = { by lemma 48 R->L }
% 14.64/2.27 meet(Y, join(complement(top), complement(X)))
% 14.64/2.27 = { by lemma 15 }
% 14.64/2.27 meet(Y, join(zero, complement(X)))
% 14.64/2.27 = { by lemma 27 }
% 14.64/2.27 meet(Y, complement(X))
% 14.64/2.27
% 14.64/2.27 Lemma 50: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 14.64/2.27 Proof:
% 14.64/2.27 complement(meet(X, complement(Y)))
% 14.64/2.27 = { by lemma 27 R->L }
% 14.64/2.27 complement(join(zero, meet(X, complement(Y))))
% 14.64/2.27 = { by lemma 49 R->L }
% 14.64/2.27 complement(join(zero, complement(join(Y, complement(X)))))
% 14.64/2.27 = { by lemma 42 }
% 14.64/2.27 meet(join(Y, complement(X)), top)
% 14.64/2.27 = { by lemma 43 }
% 14.64/2.27 join(Y, complement(X))
% 14.64/2.27
% 14.64/2.27 Lemma 51: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 14.64/2.27 Proof:
% 14.64/2.27 complement(meet(complement(X), Y))
% 14.64/2.27 = { by lemma 41 }
% 14.64/2.27 complement(meet(Y, complement(X)))
% 14.64/2.27 = { by lemma 50 }
% 14.64/2.27 join(X, complement(Y))
% 14.64/2.27
% 14.64/2.27 Lemma 52: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 14.64/2.27 Proof:
% 14.64/2.27 converse(composition(X, converse(Y)))
% 14.64/2.27 = { by axiom 9 (converse_multiplicativity) }
% 14.64/2.27 composition(converse(converse(Y)), converse(X))
% 14.64/2.27 = { by axiom 4 (converse_idempotence) }
% 14.64/2.27 composition(Y, converse(X))
% 14.64/2.27
% 14.64/2.27 Lemma 53: composition(converse(X), complement(composition(X, top))) = zero.
% 14.64/2.27 Proof:
% 14.64/2.27 composition(converse(X), complement(composition(X, top)))
% 14.64/2.27 = { by lemma 27 R->L }
% 14.64/2.28 join(zero, composition(converse(X), complement(composition(X, top))))
% 14.64/2.28 = { by lemma 15 R->L }
% 14.64/2.28 join(complement(top), composition(converse(X), complement(composition(X, top))))
% 14.64/2.28 = { by lemma 21 }
% 14.64/2.28 complement(top)
% 14.64/2.28 = { by lemma 15 }
% 14.64/2.28 zero
% 14.64/2.28
% 14.64/2.28 Lemma 54: join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)) = meet(complement(x0), one).
% 14.64/2.28 Proof:
% 14.64/2.28 join(meet(complement(composition(x0, top)), one), meet(complement(x0), one))
% 14.64/2.28 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.64/2.28 join(meet(complement(x0), one), meet(complement(composition(x0, top)), one))
% 14.64/2.28 = { by lemma 16 R->L }
% 14.64/2.28 join(meet(complement(x0), one), join(meet(meet(complement(composition(x0, top)), one), complement(meet(complement(x0), one))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.28 = { by lemma 49 R->L }
% 14.64/2.28 join(meet(complement(x0), one), join(complement(join(meet(complement(x0), one), complement(meet(complement(composition(x0, top)), one)))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.28 = { by lemma 41 R->L }
% 14.64/2.28 join(meet(complement(x0), one), join(complement(join(meet(complement(x0), one), complement(meet(one, complement(composition(x0, top)))))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.28 = { by lemma 50 }
% 14.64/2.28 join(meet(complement(x0), one), join(complement(join(meet(complement(x0), one), join(composition(x0, top), complement(one)))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.28 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.64/2.28 join(meet(complement(x0), one), join(complement(join(meet(complement(x0), one), join(complement(one), composition(x0, top)))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.28 = { by axiom 8 (maddux2_join_associativity) }
% 14.64/2.28 join(meet(complement(x0), one), join(complement(join(join(meet(complement(x0), one), complement(one)), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.28 = { by axiom 2 (goals) R->L }
% 14.64/2.28 join(meet(complement(x0), one), join(complement(join(join(meet(complement(x0), one), complement(join(x0, one))), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.28 = { by lemma 40 R->L }
% 14.64/2.28 join(meet(complement(x0), one), join(complement(join(join(meet(complement(x0), one), complement(join(complement(complement(x0)), one))), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.28 = { by lemma 16 }
% 14.64/2.28 join(meet(complement(x0), one), join(complement(join(complement(x0), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.28 = { by lemma 21 R->L }
% 14.64/2.28 join(meet(complement(x0), one), join(complement(join(join(complement(x0), composition(converse(converse(complement(composition(x0, top)))), complement(composition(converse(complement(composition(x0, top))), x0)))), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.28 = { by lemma 18 R->L }
% 14.64/2.28 join(meet(complement(x0), one), join(complement(join(join(complement(x0), composition(converse(converse(complement(composition(x0, top)))), complement(converse(composition(converse(x0), complement(composition(x0, top))))))), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.28 = { by lemma 53 }
% 14.64/2.28 join(meet(complement(x0), one), join(complement(join(join(complement(x0), composition(converse(converse(complement(composition(x0, top)))), complement(converse(zero)))), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.28 = { by lemma 39 }
% 14.64/2.28 join(meet(complement(x0), one), join(complement(join(join(complement(x0), composition(converse(converse(complement(composition(x0, top)))), complement(zero))), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.28 = { by axiom 4 (converse_idempotence) }
% 14.64/2.28 join(meet(complement(x0), one), join(complement(join(join(complement(x0), composition(complement(composition(x0, top)), complement(zero))), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.28 = { by lemma 28 }
% 14.64/2.28 join(meet(complement(x0), one), join(complement(join(join(complement(x0), composition(complement(composition(x0, top)), top)), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.28 = { by axiom 4 (converse_idempotence) R->L }
% 14.64/2.28 join(meet(complement(x0), one), join(complement(join(join(complement(x0), composition(converse(converse(complement(composition(x0, top)))), top)), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.28 = { by lemma 47 R->L }
% 14.64/2.28 join(meet(complement(x0), one), join(complement(join(join(complement(x0), converse(composition(top, converse(complement(composition(x0, top)))))), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.28 = { by lemma 34 R->L }
% 14.64/2.28 join(meet(complement(x0), one), join(complement(join(join(complement(x0), converse(composition(join(one, top), converse(complement(composition(x0, top)))))), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.28 = { by lemma 45 }
% 14.64/2.29 join(meet(complement(x0), one), join(complement(join(join(complement(x0), converse(join(converse(complement(composition(x0, top))), composition(top, converse(complement(composition(x0, top))))))), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.29 = { by lemma 38 }
% 14.64/2.29 join(meet(complement(x0), one), join(complement(join(join(complement(x0), join(complement(composition(x0, top)), converse(composition(top, converse(complement(composition(x0, top))))))), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.29 = { by lemma 47 }
% 14.64/2.29 join(meet(complement(x0), one), join(complement(join(join(complement(x0), join(complement(composition(x0, top)), composition(converse(converse(complement(composition(x0, top)))), top))), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.29 = { by axiom 4 (converse_idempotence) }
% 14.64/2.29 join(meet(complement(x0), one), join(complement(join(join(complement(x0), join(complement(composition(x0, top)), composition(complement(composition(x0, top)), top))), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.29 = { by lemma 28 R->L }
% 14.64/2.29 join(meet(complement(x0), one), join(complement(join(join(complement(x0), join(complement(composition(x0, top)), composition(complement(composition(x0, top)), complement(zero)))), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.29 = { by axiom 4 (converse_idempotence) R->L }
% 14.64/2.29 join(meet(complement(x0), one), join(complement(join(join(complement(x0), join(complement(composition(x0, top)), composition(converse(converse(complement(composition(x0, top)))), complement(zero)))), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.29 = { by lemma 39 R->L }
% 14.64/2.29 join(meet(complement(x0), one), join(complement(join(join(complement(x0), join(complement(composition(x0, top)), composition(converse(converse(complement(composition(x0, top)))), complement(converse(zero))))), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.29 = { by lemma 46 R->L }
% 14.64/2.29 join(meet(complement(x0), one), join(complement(join(join(complement(x0), join(complement(composition(x0, top)), composition(converse(converse(complement(composition(x0, top)))), complement(converse(composition(top, zero)))))), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.29 = { by lemma 35 R->L }
% 14.64/2.29 join(meet(complement(x0), one), join(complement(join(join(complement(x0), join(complement(composition(x0, top)), composition(converse(converse(complement(composition(x0, top)))), complement(converse(composition(join(top, converse(top)), zero)))))), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.29 = { by axiom 12 (composition_distributivity) }
% 14.64/2.29 join(meet(complement(x0), one), join(complement(join(join(complement(x0), join(complement(composition(x0, top)), composition(converse(converse(complement(composition(x0, top)))), complement(converse(join(composition(top, zero), composition(converse(top), zero))))))), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.29 = { by lemma 46 }
% 14.64/2.29 join(meet(complement(x0), one), join(complement(join(join(complement(x0), join(complement(composition(x0, top)), composition(converse(converse(complement(composition(x0, top)))), complement(converse(join(zero, composition(converse(top), zero))))))), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.29 = { by lemma 27 }
% 14.64/2.29 join(meet(complement(x0), one), join(complement(join(join(complement(x0), join(complement(composition(x0, top)), composition(converse(converse(complement(composition(x0, top)))), complement(converse(composition(converse(top), zero)))))), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.29 = { by lemma 53 R->L }
% 14.64/2.29 join(meet(complement(x0), one), join(complement(join(join(complement(x0), join(complement(composition(x0, top)), composition(converse(converse(complement(composition(x0, top)))), complement(converse(composition(converse(top), composition(converse(x0), complement(composition(x0, top))))))))), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.29 = { by axiom 10 (composition_associativity) }
% 14.64/2.29 join(meet(complement(x0), one), join(complement(join(join(complement(x0), join(complement(composition(x0, top)), composition(converse(converse(complement(composition(x0, top)))), complement(converse(composition(composition(converse(top), converse(x0)), complement(composition(x0, top)))))))), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.29 = { by axiom 9 (converse_multiplicativity) R->L }
% 14.64/2.29 join(meet(complement(x0), one), join(complement(join(join(complement(x0), join(complement(composition(x0, top)), composition(converse(converse(complement(composition(x0, top)))), complement(converse(composition(converse(composition(x0, top)), complement(composition(x0, top)))))))), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.29 = { by lemma 18 }
% 14.64/2.29 join(meet(complement(x0), one), join(complement(join(join(complement(x0), join(complement(composition(x0, top)), composition(converse(converse(complement(composition(x0, top)))), complement(composition(converse(complement(composition(x0, top))), composition(x0, top)))))), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.29 = { by lemma 21 }
% 14.64/2.29 join(meet(complement(x0), one), join(complement(join(join(complement(x0), complement(composition(x0, top))), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.29 = { by axiom 1 (maddux1_join_commutativity) }
% 14.64/2.29 join(meet(complement(x0), one), join(complement(join(join(complement(composition(x0, top)), complement(x0)), composition(x0, top))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.29 = { by axiom 8 (maddux2_join_associativity) R->L }
% 14.64/2.29 join(meet(complement(x0), one), join(complement(join(complement(composition(x0, top)), join(complement(x0), composition(x0, top)))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.29 = { by axiom 1 (maddux1_join_commutativity) }
% 14.64/2.29 join(meet(complement(x0), one), join(complement(join(complement(composition(x0, top)), join(composition(x0, top), complement(x0)))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.29 = { by axiom 8 (maddux2_join_associativity) }
% 14.64/2.29 join(meet(complement(x0), one), join(complement(join(join(complement(composition(x0, top)), composition(x0, top)), complement(x0))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.29 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.64/2.29 join(meet(complement(x0), one), join(complement(join(join(composition(x0, top), complement(composition(x0, top))), complement(x0))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.29 = { by axiom 5 (def_top) R->L }
% 14.64/2.29 join(meet(complement(x0), one), join(complement(join(top, complement(x0))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.29 = { by axiom 1 (maddux1_join_commutativity) }
% 14.64/2.29 join(meet(complement(x0), one), join(complement(join(complement(x0), top)), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.29 = { by lemma 34 }
% 14.64/2.29 join(meet(complement(x0), one), join(complement(top), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.29 = { by lemma 15 }
% 14.64/2.29 join(meet(complement(x0), one), join(zero, complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one))))))
% 14.64/2.29 = { by lemma 27 }
% 14.64/2.29 join(meet(complement(x0), one), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(meet(complement(x0), one)))))
% 14.64/2.29 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 14.64/2.29 join(meet(complement(x0), one), meet(meet(complement(composition(x0, top)), one), meet(complement(x0), one)))
% 14.64/2.29 = { by lemma 41 }
% 14.64/2.29 join(meet(complement(x0), one), meet(meet(complement(x0), one), meet(complement(composition(x0, top)), one)))
% 14.64/2.29 = { by axiom 11 (maddux4_definiton_of_meet) }
% 14.64/2.29 join(meet(complement(x0), one), complement(join(complement(meet(complement(x0), one)), complement(meet(complement(composition(x0, top)), one)))))
% 14.64/2.29 = { by lemma 51 R->L }
% 14.64/2.29 complement(meet(complement(meet(complement(x0), one)), join(complement(meet(complement(x0), one)), complement(meet(complement(composition(x0, top)), one)))))
% 14.64/2.29 = { by lemma 26 R->L }
% 14.64/2.29 complement(join(meet(complement(meet(complement(x0), one)), join(complement(meet(complement(x0), one)), complement(meet(complement(composition(x0, top)), one)))), zero))
% 14.64/2.29 = { by lemma 15 R->L }
% 14.64/2.29 complement(join(meet(complement(meet(complement(x0), one)), join(complement(meet(complement(x0), one)), complement(meet(complement(composition(x0, top)), one)))), complement(top)))
% 14.64/2.29 = { by lemma 51 R->L }
% 14.64/2.29 complement(join(meet(complement(meet(complement(x0), one)), complement(meet(complement(complement(meet(complement(x0), one))), meet(complement(composition(x0, top)), one)))), complement(top)))
% 14.64/2.29 = { by lemma 34 R->L }
% 14.64/2.29 complement(join(meet(complement(meet(complement(x0), one)), complement(meet(complement(complement(meet(complement(x0), one))), meet(complement(composition(x0, top)), one)))), complement(join(complement(meet(complement(composition(x0, top)), one)), top))))
% 14.64/2.29 = { by lemma 30 R->L }
% 14.64/2.30 complement(join(meet(complement(meet(complement(x0), one)), complement(meet(complement(complement(meet(complement(x0), one))), meet(complement(composition(x0, top)), one)))), complement(join(complement(complement(meet(complement(x0), one))), join(complement(meet(complement(composition(x0, top)), one)), complement(complement(complement(meet(complement(x0), one)))))))))
% 14.64/2.30 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.64/2.30 complement(join(meet(complement(meet(complement(x0), one)), complement(meet(complement(complement(meet(complement(x0), one))), meet(complement(composition(x0, top)), one)))), complement(join(complement(complement(meet(complement(x0), one))), join(complement(complement(complement(meet(complement(x0), one)))), complement(meet(complement(composition(x0, top)), one)))))))
% 14.64/2.30 = { by lemma 17 R->L }
% 14.64/2.30 complement(join(meet(complement(meet(complement(x0), one)), complement(meet(complement(complement(meet(complement(x0), one))), meet(complement(composition(x0, top)), one)))), complement(join(complement(complement(meet(complement(x0), one))), join(zero, meet(join(complement(complement(complement(meet(complement(x0), one)))), complement(meet(complement(composition(x0, top)), one))), join(complement(complement(complement(meet(complement(x0), one)))), complement(meet(complement(composition(x0, top)), one)))))))))
% 14.64/2.30 = { by lemma 48 }
% 14.64/2.30 complement(join(meet(complement(meet(complement(x0), one)), complement(meet(complement(complement(meet(complement(x0), one))), meet(complement(composition(x0, top)), one)))), complement(join(complement(complement(meet(complement(x0), one))), join(zero, complement(join(complement(join(complement(complement(complement(meet(complement(x0), one)))), complement(meet(complement(composition(x0, top)), one)))), meet(complement(complement(meet(complement(x0), one))), meet(complement(composition(x0, top)), one)))))))))
% 14.64/2.30 = { by lemma 27 }
% 14.64/2.30 complement(join(meet(complement(meet(complement(x0), one)), complement(meet(complement(complement(meet(complement(x0), one))), meet(complement(composition(x0, top)), one)))), complement(join(complement(complement(meet(complement(x0), one))), complement(join(complement(join(complement(complement(complement(meet(complement(x0), one)))), complement(meet(complement(composition(x0, top)), one)))), meet(complement(complement(meet(complement(x0), one))), meet(complement(composition(x0, top)), one))))))))
% 14.64/2.30 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 14.64/2.30 complement(join(meet(complement(meet(complement(x0), one)), complement(meet(complement(complement(meet(complement(x0), one))), meet(complement(composition(x0, top)), one)))), complement(join(complement(complement(meet(complement(x0), one))), complement(join(meet(complement(complement(meet(complement(x0), one))), meet(complement(composition(x0, top)), one)), meet(complement(complement(meet(complement(x0), one))), meet(complement(composition(x0, top)), one))))))))
% 14.64/2.30 = { by lemma 41 }
% 14.64/2.30 complement(join(meet(complement(meet(complement(x0), one)), complement(meet(complement(complement(meet(complement(x0), one))), meet(complement(composition(x0, top)), one)))), complement(join(complement(complement(meet(complement(x0), one))), complement(join(meet(meet(complement(composition(x0, top)), one), complement(complement(meet(complement(x0), one)))), meet(complement(complement(meet(complement(x0), one))), meet(complement(composition(x0, top)), one))))))))
% 14.64/2.30 = { by lemma 41 }
% 14.64/2.30 complement(join(meet(complement(meet(complement(x0), one)), complement(meet(complement(complement(meet(complement(x0), one))), meet(complement(composition(x0, top)), one)))), complement(join(complement(complement(meet(complement(x0), one))), complement(join(meet(meet(complement(composition(x0, top)), one), complement(complement(meet(complement(x0), one)))), meet(meet(complement(composition(x0, top)), one), complement(complement(meet(complement(x0), one))))))))))
% 14.64/2.30 = { by axiom 11 (maddux4_definiton_of_meet) }
% 14.64/2.30 complement(join(meet(complement(meet(complement(x0), one)), complement(meet(complement(complement(meet(complement(x0), one))), meet(complement(composition(x0, top)), one)))), complement(join(complement(complement(meet(complement(x0), one))), complement(join(meet(meet(complement(composition(x0, top)), one), complement(complement(meet(complement(x0), one)))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(complement(complement(meet(complement(x0), one))))))))))))
% 14.64/2.30 = { by axiom 11 (maddux4_definiton_of_meet) }
% 14.64/2.30 complement(join(meet(complement(meet(complement(x0), one)), complement(meet(complement(complement(meet(complement(x0), one))), meet(complement(composition(x0, top)), one)))), complement(join(complement(complement(meet(complement(x0), one))), complement(join(complement(join(complement(meet(complement(composition(x0, top)), one)), complement(complement(complement(meet(complement(x0), one)))))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(complement(complement(meet(complement(x0), one))))))))))))
% 14.64/2.30 = { by lemma 22 }
% 14.64/2.30 complement(join(meet(complement(meet(complement(x0), one)), complement(meet(complement(complement(meet(complement(x0), one))), meet(complement(composition(x0, top)), one)))), complement(join(complement(complement(meet(complement(x0), one))), complement(complement(join(complement(meet(complement(composition(x0, top)), one)), complement(complement(complement(meet(complement(x0), one)))))))))))
% 14.64/2.30 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 14.64/2.30 complement(join(meet(complement(meet(complement(x0), one)), complement(meet(complement(complement(meet(complement(x0), one))), meet(complement(composition(x0, top)), one)))), complement(join(complement(complement(meet(complement(x0), one))), complement(meet(meet(complement(composition(x0, top)), one), complement(complement(meet(complement(x0), one)))))))))
% 14.64/2.30 = { by lemma 41 R->L }
% 14.64/2.30 complement(join(meet(complement(meet(complement(x0), one)), complement(meet(complement(complement(meet(complement(x0), one))), meet(complement(composition(x0, top)), one)))), complement(join(complement(complement(meet(complement(x0), one))), complement(meet(complement(complement(meet(complement(x0), one))), meet(complement(composition(x0, top)), one)))))))
% 14.64/2.30 = { by lemma 16 }
% 14.64/2.30 complement(complement(meet(complement(x0), one)))
% 14.64/2.30 = { by lemma 40 }
% 14.64/2.30 meet(complement(x0), one)
% 14.64/2.30
% 14.64/2.30 Goal 1 (goals_1): tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(x0), one), meet(complement(composition(x0, top)), one))) = tuple(meet(complement(x0), one), meet(complement(composition(x0, top)), one)).
% 14.64/2.30 Proof:
% 14.64/2.30 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(x0), one), meet(complement(composition(x0, top)), one)))
% 14.64/2.30 = { by axiom 1 (maddux1_join_commutativity) }
% 14.64/2.30 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)))
% 15.25/2.30 = { by lemma 16 R->L }
% 15.25/2.30 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), complement(meet(complement(composition(x0, top)), one))), complement(join(complement(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one))), complement(meet(complement(composition(x0, top)), one))))))
% 15.25/2.30 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 15.25/2.30 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), complement(meet(complement(composition(x0, top)), one))), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), meet(complement(composition(x0, top)), one))))
% 15.25/2.30 = { by axiom 1 (maddux1_join_commutativity) }
% 15.25/2.30 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), meet(complement(composition(x0, top)), one)), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), complement(meet(complement(composition(x0, top)), one)))))
% 15.25/2.30 = { by lemma 41 }
% 15.25/2.30 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(meet(complement(composition(x0, top)), one), join(meet(complement(composition(x0, top)), one), meet(complement(x0), one))), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), complement(meet(complement(composition(x0, top)), one)))))
% 15.25/2.30 = { by axiom 11 (maddux4_definiton_of_meet) }
% 15.25/2.30 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(complement(join(complement(meet(complement(composition(x0, top)), one)), complement(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one))))), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), complement(meet(complement(composition(x0, top)), one)))))
% 15.25/2.30 = { by lemma 27 R->L }
% 15.25/2.30 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(join(zero, complement(join(complement(meet(complement(composition(x0, top)), one)), complement(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)))))), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), complement(meet(complement(composition(x0, top)), one)))))
% 15.25/2.30 = { by lemma 15 R->L }
% 15.25/2.30 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(join(complement(top), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)))))), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), complement(meet(complement(composition(x0, top)), one)))))
% 15.25/2.30 = { by lemma 35 R->L }
% 15.25/2.30 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(join(complement(join(top, meet(complement(x0), one))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)))))), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), complement(meet(complement(composition(x0, top)), one)))))
% 15.25/2.30 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 15.25/2.30 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(join(complement(join(meet(complement(x0), one), top)), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)))))), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), complement(meet(complement(composition(x0, top)), one)))))
% 15.25/2.30 = { by lemma 30 R->L }
% 15.25/2.30 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(join(complement(join(meet(complement(composition(x0, top)), one), join(meet(complement(x0), one), complement(meet(complement(composition(x0, top)), one))))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)))))), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), complement(meet(complement(composition(x0, top)), one)))))
% 15.25/2.30 = { by axiom 8 (maddux2_join_associativity) }
% 15.25/2.30 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(join(complement(join(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), complement(meet(complement(composition(x0, top)), one)))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)))))), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), complement(meet(complement(composition(x0, top)), one)))))
% 15.25/2.30 = { by lemma 49 }
% 15.25/2.30 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(join(meet(meet(complement(composition(x0, top)), one), complement(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)))), complement(join(complement(meet(complement(composition(x0, top)), one)), complement(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)))))), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), complement(meet(complement(composition(x0, top)), one)))))
% 15.25/2.30 = { by lemma 16 }
% 15.25/2.30 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(composition(x0, top)), one), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), complement(meet(complement(composition(x0, top)), one)))))
% 15.25/2.30 = { by lemma 41 R->L }
% 15.25/2.30 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(composition(x0, top)), one), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), complement(meet(one, complement(composition(x0, top)))))))
% 15.25/2.30 = { by lemma 50 }
% 15.25/2.30 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(composition(x0, top)), one), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(composition(x0, top), complement(one)))))
% 15.25/2.30 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 15.25/2.30 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(composition(x0, top)), one), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(complement(one), composition(x0, top)))))
% 15.25/2.31 = { by axiom 5 (def_top) }
% 15.25/2.31 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(composition(x0, top)), one), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(complement(one), composition(x0, join(one, complement(one)))))))
% 15.25/2.31 = { by axiom 4 (converse_idempotence) R->L }
% 15.25/2.31 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(composition(x0, top)), one), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(complement(one), composition(x0, join(one, converse(converse(complement(one)))))))))
% 15.25/2.31 = { by lemma 19 R->L }
% 15.25/2.31 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(composition(x0, top)), one), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(complement(one), composition(x0, join(composition(converse(one), one), converse(converse(complement(one)))))))))
% 15.25/2.31 = { by axiom 3 (composition_identity) }
% 15.25/2.31 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(composition(x0, top)), one), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(complement(one), composition(x0, join(converse(one), converse(converse(complement(one)))))))))
% 15.25/2.31 = { by axiom 7 (converse_additivity) R->L }
% 15.25/2.31 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(composition(x0, top)), one), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(complement(one), composition(x0, converse(join(one, converse(complement(one)))))))))
% 15.25/2.31 = { by axiom 1 (maddux1_join_commutativity) }
% 15.25/2.31 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(composition(x0, top)), one), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(complement(one), composition(x0, converse(join(converse(complement(one)), one)))))))
% 15.25/2.31 = { by lemma 52 R->L }
% 15.25/2.31 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(composition(x0, top)), one), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(complement(one), converse(composition(join(converse(complement(one)), one), converse(x0)))))))
% 15.25/2.31 = { by lemma 44 }
% 15.25/2.31 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(composition(x0, top)), one), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(complement(one), converse(join(converse(x0), composition(converse(complement(one)), converse(x0))))))))
% 15.25/2.31 = { by lemma 38 }
% 15.25/2.31 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(composition(x0, top)), one), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(complement(one), join(x0, converse(composition(converse(complement(one)), converse(x0))))))))
% 15.25/2.31 = { by lemma 52 }
% 15.25/2.31 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(composition(x0, top)), one), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(complement(one), join(x0, composition(x0, converse(converse(complement(one)))))))))
% 15.25/2.31 = { by axiom 4 (converse_idempotence) }
% 15.25/2.31 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(composition(x0, top)), one), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(complement(one), join(x0, composition(x0, complement(one)))))))
% 15.25/2.31 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 15.25/2.31 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(composition(x0, top)), one), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(complement(one), join(composition(x0, complement(one)), x0)))))
% 15.25/2.31 = { by axiom 8 (maddux2_join_associativity) }
% 15.25/2.31 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(composition(x0, top)), one), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(join(complement(one), composition(x0, complement(one))), x0))))
% 15.25/2.31 = { by lemma 44 R->L }
% 15.25/2.31 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(composition(x0, top)), one), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(composition(join(x0, one), complement(one)), x0))))
% 15.25/2.31 = { by axiom 2 (goals) }
% 15.25/2.31 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(composition(x0, top)), one), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(composition(one, complement(one)), x0))))
% 15.25/2.31 = { by lemma 20 }
% 15.25/2.31 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(composition(x0, top)), one), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(complement(one), x0))))
% 15.25/2.31 = { by axiom 1 (maddux1_join_commutativity) }
% 15.25/2.31 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(composition(x0, top)), one), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(x0, complement(one)))))
% 15.25/2.31 = { by lemma 50 R->L }
% 15.25/2.31 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(composition(x0, top)), one), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), complement(meet(one, complement(x0))))))
% 15.25/2.31 = { by lemma 41 }
% 15.25/2.31 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(composition(x0, top)), one), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), complement(meet(complement(x0), one)))))
% 15.25/2.31 = { by lemma 54 R->L }
% 15.25/2.31 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(composition(x0, top)), one), meet(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), complement(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one))))))
% 15.25/2.31 = { by axiom 6 (def_zero) R->L }
% 15.25/2.31 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), join(meet(complement(composition(x0, top)), one), zero))
% 15.25/2.31 = { by lemma 26 }
% 15.25/2.31 tuple(join(meet(complement(composition(x0, top)), one), meet(complement(x0), one)), meet(complement(composition(x0, top)), one))
% 15.25/2.31 = { by lemma 54 }
% 15.25/2.31 tuple(meet(complement(x0), one), meet(complement(composition(x0, top)), one))
% 15.25/2.31 % SZS output end Proof
% 15.25/2.31
% 15.25/2.31 RESULT: Theorem (the conjecture is true).
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