TSTP Solution File: REL005+2 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL005+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 : n019.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 13:43:45 EDT 2023
% Result : Theorem 22.90s 3.30s
% Output : Proof 23.54s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL005+2 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n019.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Aug 25 19:14:43 EDT 2023
% 0.13/0.34 % CPUTime :
% 22.90/3.30 Command-line arguments: --ground-connectedness --complete-subsets
% 22.90/3.30
% 22.90/3.30 % SZS status Theorem
% 22.90/3.30
% 23.54/3.36 % SZS output start Proof
% 23.54/3.36 Take the following subset of the input axioms:
% 23.54/3.36 fof(composition_associativity, axiom, ![X0, X1, X2]: composition(X0, composition(X1, X2))=composition(composition(X0, X1), X2)).
% 23.54/3.36 fof(composition_identity, axiom, ![X0_2]: composition(X0_2, one)=X0_2).
% 23.54/3.36 fof(converse_additivity, axiom, ![X0_2, X1_2]: converse(join(X0_2, X1_2))=join(converse(X0_2), converse(X1_2))).
% 23.54/3.36 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)).
% 23.54/3.36 fof(converse_idempotence, axiom, ![X0_2]: converse(converse(X0_2))=X0_2).
% 23.54/3.36 fof(converse_multiplicativity, axiom, ![X0_2, X1_2]: converse(composition(X0_2, X1_2))=composition(converse(X1_2), converse(X0_2))).
% 23.54/3.36 fof(def_top, axiom, ![X0_2]: top=join(X0_2, complement(X0_2))).
% 23.54/3.36 fof(def_zero, axiom, ![X0_2]: zero=meet(X0_2, complement(X0_2))).
% 23.54/3.37 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)))).
% 23.54/3.37 fof(maddux1_join_commutativity, axiom, ![X0_2, X1_2]: join(X0_2, X1_2)=join(X1_2, X0_2)).
% 23.54/3.37 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)).
% 23.54/3.37 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)))).
% 23.54/3.37 fof(maddux4_definiton_of_meet, axiom, ![X0_2, X1_2]: meet(X0_2, X1_2)=complement(join(complement(X0_2), complement(X1_2)))).
% 23.54/3.37
% 23.54/3.37 Now clausify the problem and encode Horn clauses using encoding 3 of
% 23.54/3.37 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 23.54/3.37 We repeatedly replace C & s=t => u=v by the two clauses:
% 23.54/3.37 fresh(y, y, x1...xn) = u
% 23.54/3.37 C => fresh(s, t, x1...xn) = v
% 23.54/3.37 where fresh is a fresh function symbol and x1..xn are the free
% 23.54/3.37 variables of u and v.
% 23.54/3.37 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 23.54/3.37 input problem has no model of domain size 1).
% 23.54/3.37
% 23.54/3.37 The encoding turns the above axioms into the following unit equations and goals:
% 23.54/3.37
% 23.54/3.37 Axiom 1 (composition_identity): composition(X, one) = X.
% 23.54/3.37 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 23.54/3.37 Axiom 3 (converse_idempotence): converse(converse(X)) = X.
% 23.54/3.37 Axiom 4 (def_top): top = join(X, complement(X)).
% 23.54/3.37 Axiom 5 (def_zero): zero = meet(X, complement(X)).
% 23.54/3.37 Axiom 6 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 23.54/3.37 Axiom 7 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 23.54/3.37 Axiom 8 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 23.54/3.37 Axiom 9 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 23.54/3.37 Axiom 10 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 23.54/3.37 Axiom 11 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 23.54/3.37 Axiom 12 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 23.54/3.37
% 23.54/3.37 Lemma 13: complement(top) = zero.
% 23.54/3.37 Proof:
% 23.54/3.37 complement(top)
% 23.54/3.37 = { by axiom 4 (def_top) }
% 23.54/3.37 complement(join(complement(X), complement(complement(X))))
% 23.54/3.37 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 23.54/3.37 meet(X, complement(X))
% 23.54/3.37 = { by axiom 5 (def_zero) R->L }
% 23.54/3.37 zero
% 23.54/3.37
% 23.54/3.37 Lemma 14: join(X, join(Y, complement(X))) = join(Y, top).
% 23.54/3.37 Proof:
% 23.54/3.37 join(X, join(Y, complement(X)))
% 23.54/3.37 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 23.54/3.37 join(X, join(complement(X), Y))
% 23.54/3.37 = { by axiom 9 (maddux2_join_associativity) }
% 23.54/3.37 join(join(X, complement(X)), Y)
% 23.54/3.37 = { by axiom 4 (def_top) R->L }
% 23.54/3.37 join(top, Y)
% 23.54/3.37 = { by axiom 2 (maddux1_join_commutativity) }
% 23.54/3.37 join(Y, top)
% 23.54/3.37
% 23.54/3.37 Lemma 15: composition(converse(one), X) = X.
% 23.54/3.37 Proof:
% 23.54/3.37 composition(converse(one), X)
% 23.54/3.37 = { by axiom 3 (converse_idempotence) R->L }
% 23.54/3.37 composition(converse(one), converse(converse(X)))
% 23.54/3.37 = { by axiom 6 (converse_multiplicativity) R->L }
% 23.54/3.37 converse(composition(converse(X), one))
% 23.54/3.37 = { by axiom 1 (composition_identity) }
% 23.54/3.37 converse(converse(X))
% 23.54/3.37 = { by axiom 3 (converse_idempotence) }
% 23.54/3.37 X
% 23.54/3.37
% 23.54/3.37 Lemma 16: join(complement(X), complement(X)) = complement(X).
% 23.54/3.37 Proof:
% 23.54/3.37 join(complement(X), complement(X))
% 23.54/3.37 = { by lemma 15 R->L }
% 23.54/3.37 join(complement(X), composition(converse(one), complement(X)))
% 23.54/3.37 = { by lemma 15 R->L }
% 23.54/3.37 join(complement(X), composition(converse(one), complement(composition(converse(one), X))))
% 23.54/3.37 = { by axiom 1 (composition_identity) R->L }
% 23.54/3.37 join(complement(X), composition(converse(one), complement(composition(composition(converse(one), one), X))))
% 23.54/3.37 = { by axiom 7 (composition_associativity) R->L }
% 23.54/3.37 join(complement(X), composition(converse(one), complement(composition(converse(one), composition(one, X)))))
% 23.54/3.37 = { by lemma 15 }
% 23.54/3.37 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 23.54/3.37 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 23.54/3.37 join(composition(converse(one), complement(composition(one, X))), complement(X))
% 23.54/3.37 = { by axiom 11 (converse_cancellativity) }
% 23.54/3.37 complement(X)
% 23.54/3.37
% 23.54/3.37 Lemma 17: join(top, complement(X)) = top.
% 23.54/3.37 Proof:
% 23.54/3.37 join(top, complement(X))
% 23.54/3.37 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 23.54/3.37 join(complement(X), top)
% 23.54/3.37 = { by lemma 14 R->L }
% 23.54/3.37 join(X, join(complement(X), complement(X)))
% 23.54/3.37 = { by lemma 16 }
% 23.54/3.37 join(X, complement(X))
% 23.54/3.37 = { by axiom 4 (def_top) R->L }
% 23.54/3.37 top
% 23.54/3.37
% 23.54/3.37 Lemma 18: join(Y, top) = join(X, top).
% 23.54/3.37 Proof:
% 23.54/3.37 join(Y, top)
% 23.54/3.37 = { by lemma 17 R->L }
% 23.54/3.37 join(Y, join(top, complement(Y)))
% 23.54/3.37 = { by lemma 14 }
% 23.54/3.37 join(top, top)
% 23.54/3.37 = { by lemma 14 R->L }
% 23.54/3.37 join(X, join(top, complement(X)))
% 23.54/3.37 = { by lemma 17 }
% 23.54/3.37 join(X, top)
% 23.54/3.37
% 23.54/3.37 Lemma 19: join(X, top) = top.
% 23.54/3.37 Proof:
% 23.54/3.37 join(X, top)
% 23.54/3.37 = { by lemma 18 }
% 23.54/3.37 join(join(zero, zero), top)
% 23.54/3.37 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 23.54/3.37 join(top, join(zero, zero))
% 23.54/3.37 = { by lemma 13 R->L }
% 23.54/3.37 join(top, join(zero, complement(top)))
% 23.54/3.37 = { by lemma 13 R->L }
% 23.54/3.37 join(top, join(complement(top), complement(top)))
% 23.54/3.37 = { by lemma 16 }
% 23.54/3.37 join(top, complement(top))
% 23.54/3.37 = { by axiom 4 (def_top) R->L }
% 23.54/3.37 top
% 23.54/3.37
% 23.54/3.37 Lemma 20: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 23.54/3.37 Proof:
% 23.54/3.37 join(meet(X, Y), complement(join(complement(X), Y)))
% 23.54/3.37 = { by axiom 10 (maddux4_definiton_of_meet) }
% 23.54/3.37 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 23.54/3.37 = { by axiom 12 (maddux3_a_kind_of_de_Morgan) R->L }
% 23.54/3.37 X
% 23.54/3.37
% 23.54/3.37 Lemma 21: join(zero, meet(X, X)) = X.
% 23.54/3.37 Proof:
% 23.54/3.37 join(zero, meet(X, X))
% 23.54/3.37 = { by axiom 10 (maddux4_definiton_of_meet) }
% 23.54/3.37 join(zero, complement(join(complement(X), complement(X))))
% 23.54/3.37 = { by axiom 5 (def_zero) }
% 23.54/3.37 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 23.54/3.37 = { by lemma 20 }
% 23.54/3.37 X
% 23.54/3.37
% 23.54/3.37 Lemma 22: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 23.54/3.37 Proof:
% 23.54/3.37 join(zero, join(X, complement(complement(Y))))
% 23.54/3.37 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 23.54/3.37 join(zero, join(complement(complement(Y)), X))
% 23.54/3.37 = { by lemma 16 R->L }
% 23.54/3.37 join(zero, join(complement(join(complement(Y), complement(Y))), X))
% 23.54/3.37 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 23.54/3.37 join(zero, join(meet(Y, Y), X))
% 23.54/3.37 = { by axiom 9 (maddux2_join_associativity) }
% 23.54/3.37 join(join(zero, meet(Y, Y)), X)
% 23.54/3.37 = { by lemma 21 }
% 23.54/3.37 join(Y, X)
% 23.54/3.37 = { by axiom 2 (maddux1_join_commutativity) }
% 23.54/3.37 join(X, Y)
% 23.54/3.37
% 23.54/3.37 Lemma 23: join(zero, complement(complement(X))) = X.
% 23.54/3.37 Proof:
% 23.54/3.37 join(zero, complement(complement(X)))
% 23.54/3.37 = { by axiom 5 (def_zero) }
% 23.54/3.37 join(meet(X, complement(X)), complement(complement(X)))
% 23.54/3.37 = { by lemma 16 R->L }
% 23.54/3.37 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 23.54/3.37 = { by lemma 20 }
% 23.54/3.37 X
% 23.54/3.37
% 23.54/3.37 Lemma 24: join(zero, complement(X)) = complement(X).
% 23.54/3.37 Proof:
% 23.54/3.37 join(zero, complement(X))
% 23.54/3.37 = { by lemma 23 R->L }
% 23.54/3.37 join(zero, join(zero, complement(complement(complement(X)))))
% 23.54/3.37 = { by lemma 16 R->L }
% 23.54/3.37 join(zero, join(zero, join(complement(complement(complement(X))), complement(complement(complement(X))))))
% 23.54/3.37 = { by lemma 22 }
% 23.54/3.37 join(zero, join(complement(complement(complement(X))), complement(X)))
% 23.54/3.37 = { by axiom 2 (maddux1_join_commutativity) }
% 23.54/3.37 join(zero, join(complement(X), complement(complement(complement(X)))))
% 23.54/3.37 = { by lemma 22 }
% 23.54/3.37 join(complement(X), complement(X))
% 23.54/3.37 = { by lemma 16 }
% 23.54/3.37 complement(X)
% 23.54/3.37
% 23.54/3.37 Lemma 25: join(X, zero) = X.
% 23.54/3.37 Proof:
% 23.54/3.37 join(X, zero)
% 23.54/3.37 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 23.54/3.37 join(zero, X)
% 23.54/3.37 = { by lemma 22 R->L }
% 23.54/3.37 join(zero, join(zero, complement(complement(X))))
% 23.54/3.37 = { by lemma 24 }
% 23.54/3.37 join(zero, complement(complement(X)))
% 23.54/3.37 = { by lemma 23 }
% 23.54/3.37 X
% 23.54/3.37
% 23.54/3.37 Lemma 26: join(zero, X) = X.
% 23.54/3.37 Proof:
% 23.54/3.37 join(zero, X)
% 23.54/3.37 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 23.54/3.37 join(X, zero)
% 23.54/3.37 = { by lemma 25 }
% 23.54/3.37 X
% 23.54/3.37
% 23.54/3.37 Lemma 27: complement(complement(X)) = X.
% 23.54/3.37 Proof:
% 23.54/3.37 complement(complement(X))
% 23.54/3.37 = { by lemma 24 R->L }
% 23.54/3.37 join(zero, complement(complement(X)))
% 23.54/3.37 = { by lemma 23 }
% 23.54/3.37 X
% 23.54/3.37
% 23.54/3.37 Lemma 28: meet(Y, X) = meet(X, Y).
% 23.54/3.37 Proof:
% 23.54/3.37 meet(Y, X)
% 23.54/3.37 = { by axiom 10 (maddux4_definiton_of_meet) }
% 23.54/3.37 complement(join(complement(Y), complement(X)))
% 23.54/3.37 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 23.54/3.37 complement(join(complement(X), complement(Y)))
% 23.54/3.37 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 23.54/3.37 meet(X, Y)
% 23.54/3.37
% 23.54/3.37 Lemma 29: complement(join(zero, complement(X))) = meet(X, top).
% 23.54/3.37 Proof:
% 23.54/3.37 complement(join(zero, complement(X)))
% 23.54/3.37 = { by lemma 13 R->L }
% 23.54/3.37 complement(join(complement(top), complement(X)))
% 23.54/3.37 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 23.54/3.37 meet(top, X)
% 23.54/3.37 = { by lemma 28 R->L }
% 23.54/3.37 meet(X, top)
% 23.54/3.37
% 23.54/3.37 Lemma 30: join(X, join(complement(X), Y)) = top.
% 23.54/3.37 Proof:
% 23.54/3.37 join(X, join(complement(X), Y))
% 23.54/3.37 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 23.54/3.37 join(X, join(Y, complement(X)))
% 23.54/3.37 = { by lemma 14 }
% 23.54/3.37 join(Y, top)
% 23.54/3.37 = { by lemma 18 R->L }
% 23.54/3.37 join(Z, top)
% 23.54/3.37 = { by lemma 19 }
% 23.54/3.37 top
% 23.54/3.37
% 23.54/3.37 Lemma 31: join(X, complement(zero)) = top.
% 23.54/3.37 Proof:
% 23.54/3.37 join(X, complement(zero))
% 23.54/3.37 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 23.54/3.37 join(complement(zero), X)
% 23.54/3.37 = { by lemma 22 R->L }
% 23.54/3.37 join(zero, join(complement(zero), complement(complement(X))))
% 23.54/3.37 = { by lemma 30 }
% 23.54/3.37 top
% 23.54/3.37
% 23.54/3.37 Lemma 32: join(meet(X, Y), meet(X, complement(Y))) = X.
% 23.54/3.37 Proof:
% 23.54/3.37 join(meet(X, Y), meet(X, complement(Y)))
% 23.54/3.37 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 23.54/3.37 join(meet(X, complement(Y)), meet(X, Y))
% 23.54/3.37 = { by axiom 10 (maddux4_definiton_of_meet) }
% 23.54/3.37 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 23.54/3.37 = { by lemma 20 }
% 23.54/3.37 X
% 23.54/3.37
% 23.54/3.37 Lemma 33: meet(X, top) = X.
% 23.54/3.37 Proof:
% 23.54/3.37 meet(X, top)
% 23.54/3.37 = { by lemma 29 R->L }
% 23.54/3.37 complement(join(zero, complement(X)))
% 23.54/3.37 = { by lemma 24 R->L }
% 23.54/3.37 join(zero, complement(join(zero, complement(X))))
% 23.54/3.37 = { by lemma 29 }
% 23.54/3.37 join(zero, meet(X, top))
% 23.54/3.37 = { by lemma 31 R->L }
% 23.54/3.37 join(zero, meet(X, join(complement(zero), complement(zero))))
% 23.54/3.37 = { by lemma 16 }
% 23.54/3.37 join(zero, meet(X, complement(zero)))
% 23.54/3.37 = { by lemma 13 R->L }
% 23.54/3.37 join(complement(top), meet(X, complement(zero)))
% 23.54/3.37 = { by lemma 31 R->L }
% 23.54/3.37 join(complement(join(complement(X), complement(zero))), meet(X, complement(zero)))
% 23.54/3.37 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 23.54/3.37 join(meet(X, zero), meet(X, complement(zero)))
% 23.54/3.37 = { by lemma 32 }
% 23.54/3.37 X
% 23.54/3.37
% 23.54/3.37 Lemma 34: join(Y, join(Z, X)) = join(X, join(Y, Z)).
% 23.54/3.37 Proof:
% 23.54/3.37 join(Y, join(Z, X))
% 23.54/3.37 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 23.54/3.37 join(join(Z, X), Y)
% 23.54/3.37 = { by axiom 2 (maddux1_join_commutativity) }
% 23.54/3.37 join(join(X, Z), Y)
% 23.54/3.37 = { by axiom 9 (maddux2_join_associativity) R->L }
% 23.54/3.37 join(X, join(Z, Y))
% 23.54/3.37 = { by axiom 2 (maddux1_join_commutativity) }
% 23.54/3.37 join(X, join(Y, Z))
% 23.54/3.37
% 23.54/3.37 Lemma 35: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 23.54/3.37 Proof:
% 23.54/3.37 meet(X, join(complement(Y), complement(Z)))
% 23.54/3.37 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 23.54/3.37 meet(X, join(complement(Z), complement(Y)))
% 23.54/3.37 = { by lemma 28 }
% 23.54/3.37 meet(join(complement(Z), complement(Y)), X)
% 23.54/3.37 = { by axiom 10 (maddux4_definiton_of_meet) }
% 23.54/3.37 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 23.54/3.37 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 23.54/3.37 complement(join(meet(Z, Y), complement(X)))
% 23.54/3.37 = { by axiom 2 (maddux1_join_commutativity) }
% 23.54/3.37 complement(join(complement(X), meet(Z, Y)))
% 23.54/3.37 = { by lemma 28 R->L }
% 23.54/3.37 complement(join(complement(X), meet(Y, Z)))
% 23.54/3.37
% 23.54/3.37 Lemma 36: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 23.54/3.37 Proof:
% 23.54/3.37 complement(join(X, complement(Y)))
% 23.54/3.37 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 23.54/3.37 complement(join(complement(Y), X))
% 23.54/3.37 = { by lemma 33 R->L }
% 23.54/3.37 complement(join(complement(Y), meet(X, top)))
% 23.54/3.37 = { by lemma 28 R->L }
% 23.54/3.37 complement(join(complement(Y), meet(top, X)))
% 23.54/3.37 = { by lemma 35 R->L }
% 23.54/3.37 meet(Y, join(complement(top), complement(X)))
% 23.54/3.37 = { by lemma 13 }
% 23.54/3.37 meet(Y, join(zero, complement(X)))
% 23.54/3.37 = { by lemma 24 }
% 23.54/3.37 meet(Y, complement(X))
% 23.54/3.37
% 23.54/3.37 Lemma 37: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 23.54/3.37 Proof:
% 23.54/3.37 complement(meet(X, complement(Y)))
% 23.54/3.37 = { by lemma 26 R->L }
% 23.54/3.37 complement(join(zero, meet(X, complement(Y))))
% 23.54/3.37 = { by lemma 36 R->L }
% 23.54/3.37 complement(join(zero, complement(join(Y, complement(X)))))
% 23.54/3.37 = { by lemma 29 }
% 23.54/3.37 meet(join(Y, complement(X)), top)
% 23.54/3.37 = { by lemma 33 }
% 23.54/3.37 join(Y, complement(X))
% 23.54/3.37
% 23.54/3.37 Lemma 38: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 23.54/3.37 Proof:
% 23.54/3.37 complement(join(complement(X), Y))
% 23.54/3.37 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 23.54/3.37 complement(join(Y, complement(X)))
% 23.54/3.37 = { by lemma 36 }
% 23.54/3.37 meet(X, complement(Y))
% 23.54/3.37
% 23.54/3.37 Lemma 39: meet(X, join(X, complement(Y))) = X.
% 23.54/3.37 Proof:
% 23.54/3.37 meet(X, join(X, complement(Y)))
% 23.54/3.37 = { by lemma 37 R->L }
% 23.54/3.37 meet(X, complement(meet(Y, complement(X))))
% 23.54/3.37 = { by lemma 38 R->L }
% 23.54/3.37 complement(join(complement(X), meet(Y, complement(X))))
% 23.54/3.37 = { by lemma 24 R->L }
% 23.54/3.37 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 23.54/3.37 = { by lemma 13 R->L }
% 23.54/3.37 join(complement(top), complement(join(complement(X), meet(Y, complement(X)))))
% 23.54/3.37 = { by lemma 19 R->L }
% 23.54/3.37 join(complement(join(complement(Y), top)), complement(join(complement(X), meet(Y, complement(X)))))
% 23.54/3.38 = { by lemma 14 R->L }
% 23.54/3.38 join(complement(join(complement(X), join(complement(Y), complement(complement(X))))), complement(join(complement(X), meet(Y, complement(X)))))
% 23.54/3.38 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 23.54/3.38 join(complement(join(complement(X), join(complement(complement(X)), complement(Y)))), complement(join(complement(X), meet(Y, complement(X)))))
% 23.54/3.38 = { by lemma 21 R->L }
% 23.54/3.38 join(complement(join(complement(X), join(zero, meet(join(complement(complement(X)), complement(Y)), join(complement(complement(X)), complement(Y)))))), complement(join(complement(X), meet(Y, complement(X)))))
% 23.54/3.38 = { by lemma 35 }
% 23.54/3.38 join(complement(join(complement(X), join(zero, complement(join(complement(join(complement(complement(X)), complement(Y))), meet(complement(X), Y)))))), complement(join(complement(X), meet(Y, complement(X)))))
% 23.54/3.38 = { by lemma 24 }
% 23.54/3.38 join(complement(join(complement(X), complement(join(complement(join(complement(complement(X)), complement(Y))), meet(complement(X), Y))))), complement(join(complement(X), meet(Y, complement(X)))))
% 23.54/3.38 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 23.54/3.38 join(complement(join(complement(X), complement(join(meet(complement(X), Y), meet(complement(X), Y))))), complement(join(complement(X), meet(Y, complement(X)))))
% 23.54/3.38 = { by lemma 28 }
% 23.54/3.38 join(complement(join(complement(X), complement(join(meet(Y, complement(X)), meet(complement(X), Y))))), complement(join(complement(X), meet(Y, complement(X)))))
% 23.54/3.38 = { by lemma 28 }
% 23.54/3.38 join(complement(join(complement(X), complement(join(meet(Y, complement(X)), meet(Y, complement(X)))))), complement(join(complement(X), meet(Y, complement(X)))))
% 23.54/3.38 = { by axiom 10 (maddux4_definiton_of_meet) }
% 23.54/3.38 join(complement(join(complement(X), complement(join(meet(Y, complement(X)), complement(join(complement(Y), complement(complement(X)))))))), complement(join(complement(X), meet(Y, complement(X)))))
% 23.54/3.38 = { by axiom 10 (maddux4_definiton_of_meet) }
% 23.54/3.38 join(complement(join(complement(X), complement(join(complement(join(complement(Y), complement(complement(X)))), complement(join(complement(Y), complement(complement(X)))))))), complement(join(complement(X), meet(Y, complement(X)))))
% 23.54/3.38 = { by lemma 16 }
% 23.54/3.38 join(complement(join(complement(X), complement(complement(join(complement(Y), complement(complement(X))))))), complement(join(complement(X), meet(Y, complement(X)))))
% 23.54/3.38 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 23.54/3.38 join(complement(join(complement(X), complement(meet(Y, complement(X))))), complement(join(complement(X), meet(Y, complement(X)))))
% 23.54/3.38 = { by lemma 28 R->L }
% 23.54/3.38 join(complement(join(complement(X), complement(meet(complement(X), Y)))), complement(join(complement(X), meet(Y, complement(X)))))
% 23.54/3.38 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 23.54/3.38 join(meet(X, meet(complement(X), Y)), complement(join(complement(X), meet(Y, complement(X)))))
% 23.54/3.38 = { by lemma 28 R->L }
% 23.54/3.38 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 23.54/3.38 = { by lemma 20 }
% 23.54/3.38 X
% 23.54/3.38
% 23.54/3.38 Lemma 40: join(meet(X, Y), X) = X.
% 23.54/3.38 Proof:
% 23.54/3.38 join(meet(X, Y), X)
% 23.54/3.38 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 23.54/3.38 join(X, meet(X, Y))
% 23.54/3.38 = { by axiom 10 (maddux4_definiton_of_meet) }
% 23.54/3.38 join(X, complement(join(complement(X), complement(Y))))
% 23.54/3.38 = { by lemma 37 R->L }
% 23.54/3.38 complement(meet(join(complement(X), complement(Y)), complement(X)))
% 23.54/3.38 = { by lemma 28 R->L }
% 23.54/3.38 complement(meet(complement(X), join(complement(X), complement(Y))))
% 23.54/3.38 = { by lemma 39 }
% 23.54/3.38 complement(complement(X))
% 23.54/3.38 = { by lemma 27 }
% 23.54/3.38 X
% 23.54/3.38
% 23.54/3.38 Lemma 41: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 23.54/3.38 Proof:
% 23.54/3.38 meet(complement(X), complement(Y))
% 23.54/3.38 = { by lemma 28 }
% 23.54/3.38 meet(complement(Y), complement(X))
% 23.54/3.38 = { by lemma 24 R->L }
% 23.54/3.38 meet(join(zero, complement(Y)), complement(X))
% 23.54/3.38 = { by lemma 36 R->L }
% 23.54/3.38 complement(join(X, complement(join(zero, complement(Y)))))
% 23.54/3.38 = { by lemma 29 }
% 23.54/3.38 complement(join(X, meet(Y, top)))
% 23.54/3.38 = { by lemma 33 }
% 23.54/3.38 complement(join(X, Y))
% 23.54/3.38
% 23.54/3.38 Lemma 42: meet(Y, meet(Z, X)) = meet(X, meet(Y, Z)).
% 23.54/3.38 Proof:
% 23.54/3.38 meet(Y, meet(Z, X))
% 23.54/3.38 = { by lemma 33 R->L }
% 23.54/3.38 meet(meet(Y, top), meet(Z, X))
% 23.54/3.38 = { by lemma 29 R->L }
% 23.54/3.38 meet(complement(join(zero, complement(Y))), meet(Z, X))
% 23.54/3.38 = { by lemma 28 }
% 23.54/3.38 meet(complement(join(zero, complement(Y))), meet(X, Z))
% 23.54/3.38 = { by lemma 28 }
% 23.54/3.38 meet(meet(X, Z), complement(join(zero, complement(Y))))
% 23.54/3.38 = { by axiom 10 (maddux4_definiton_of_meet) }
% 23.54/3.38 meet(complement(join(complement(X), complement(Z))), complement(join(zero, complement(Y))))
% 23.54/3.38 = { by lemma 41 }
% 23.54/3.38 complement(join(join(complement(X), complement(Z)), join(zero, complement(Y))))
% 23.54/3.38 = { by axiom 9 (maddux2_join_associativity) R->L }
% 23.54/3.38 complement(join(complement(X), join(complement(Z), join(zero, complement(Y)))))
% 23.54/3.38 = { by lemma 38 }
% 23.54/3.38 meet(X, complement(join(complement(Z), join(zero, complement(Y)))))
% 23.54/3.38 = { by lemma 38 }
% 23.54/3.38 meet(X, meet(Z, complement(join(zero, complement(Y)))))
% 23.54/3.38 = { by lemma 29 }
% 23.54/3.38 meet(X, meet(Z, meet(Y, top)))
% 23.54/3.38 = { by lemma 33 }
% 23.54/3.38 meet(X, meet(Z, Y))
% 23.54/3.38 = { by lemma 28 R->L }
% 23.54/3.38 meet(X, meet(Y, Z))
% 23.54/3.38
% 23.54/3.38 Lemma 43: join(complement(converse(X)), converse(join(X, Y))) = top.
% 23.54/3.38 Proof:
% 23.54/3.38 join(complement(converse(X)), converse(join(X, Y)))
% 23.54/3.38 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 23.54/3.38 join(converse(join(X, Y)), complement(converse(X)))
% 23.54/3.38 = { by axiom 8 (converse_additivity) }
% 23.54/3.38 join(join(converse(X), converse(Y)), complement(converse(X)))
% 23.54/3.38 = { by axiom 9 (maddux2_join_associativity) R->L }
% 23.54/3.38 join(converse(X), join(converse(Y), complement(converse(X))))
% 23.54/3.38 = { by axiom 2 (maddux1_join_commutativity) }
% 23.54/3.38 join(converse(X), join(complement(converse(X)), converse(Y)))
% 23.54/3.38 = { by lemma 30 }
% 23.54/3.38 top
% 23.54/3.38
% 23.54/3.38 Lemma 44: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 23.54/3.38 Proof:
% 23.54/3.38 join(meet(X, Y), meet(Y, complement(X)))
% 23.54/3.38 = { by lemma 28 }
% 23.54/3.38 join(meet(Y, X), meet(Y, complement(X)))
% 23.54/3.38 = { by lemma 32 }
% 23.54/3.38 Y
% 23.54/3.38
% 23.54/3.38 Lemma 45: join(meet(X, Y), meet(complement(X), Y)) = Y.
% 23.54/3.38 Proof:
% 23.54/3.38 join(meet(X, Y), meet(complement(X), Y))
% 23.54/3.38 = { by lemma 28 }
% 23.54/3.38 join(meet(X, Y), meet(Y, complement(X)))
% 23.54/3.38 = { by lemma 44 }
% 23.54/3.38 Y
% 23.54/3.38
% 23.54/3.38 Lemma 46: meet(meet(converse(X), converse(Y)), complement(converse(meet(X, Y)))) = zero.
% 23.54/3.38 Proof:
% 23.54/3.38 meet(meet(converse(X), converse(Y)), complement(converse(meet(X, Y))))
% 23.54/3.38 = { by lemma 36 R->L }
% 23.54/3.38 complement(join(converse(meet(X, Y)), complement(meet(converse(X), converse(Y)))))
% 23.54/3.38 = { by axiom 3 (converse_idempotence) R->L }
% 23.54/3.38 complement(join(converse(meet(X, Y)), complement(converse(converse(meet(converse(X), converse(Y)))))))
% 23.54/3.38 = { by lemma 44 R->L }
% 23.54/3.38 complement(join(converse(meet(X, Y)), complement(converse(join(meet(complement(X), converse(meet(converse(X), converse(Y)))), meet(converse(meet(converse(X), converse(Y))), complement(complement(X))))))))
% 23.54/3.38 = { by lemma 28 }
% 23.54/3.38 complement(join(converse(meet(X, Y)), complement(converse(join(meet(converse(meet(converse(X), converse(Y))), complement(X)), meet(converse(meet(converse(X), converse(Y))), complement(complement(X))))))))
% 23.54/3.38 = { by lemma 36 R->L }
% 23.54/3.38 complement(join(converse(meet(X, Y)), complement(converse(join(complement(join(X, complement(converse(meet(converse(X), converse(Y)))))), meet(converse(meet(converse(X), converse(Y))), complement(complement(X))))))))
% 23.54/3.38 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 23.54/3.38 complement(join(converse(meet(X, Y)), complement(converse(join(complement(join(complement(converse(meet(converse(X), converse(Y)))), X)), meet(converse(meet(converse(X), converse(Y))), complement(complement(X))))))))
% 23.54/3.38 = { by axiom 3 (converse_idempotence) R->L }
% 23.54/3.38 complement(join(converse(meet(X, Y)), complement(converse(join(complement(join(complement(converse(meet(converse(X), converse(Y)))), converse(converse(X)))), meet(converse(meet(converse(X), converse(Y))), complement(complement(X))))))))
% 23.54/3.38 = { by lemma 32 R->L }
% 23.54/3.38 complement(join(converse(meet(X, Y)), complement(converse(join(complement(join(complement(converse(meet(converse(X), converse(Y)))), converse(join(meet(converse(X), converse(Y)), meet(converse(X), complement(converse(Y))))))), meet(converse(meet(converse(X), converse(Y))), complement(complement(X))))))))
% 23.54/3.38 = { by lemma 43 }
% 23.54/3.38 complement(join(converse(meet(X, Y)), complement(converse(join(complement(top), meet(converse(meet(converse(X), converse(Y))), complement(complement(X))))))))
% 23.54/3.38 = { by lemma 13 }
% 23.54/3.38 complement(join(converse(meet(X, Y)), complement(converse(join(zero, meet(converse(meet(converse(X), converse(Y))), complement(complement(X))))))))
% 23.54/3.38 = { by lemma 26 }
% 23.54/3.38 complement(join(converse(meet(X, Y)), complement(converse(meet(converse(meet(converse(X), converse(Y))), complement(complement(X)))))))
% 23.54/3.38 = { by lemma 27 }
% 23.54/3.38 complement(join(converse(meet(X, Y)), complement(converse(meet(converse(meet(converse(X), converse(Y))), X)))))
% 23.54/3.38 = { by lemma 28 R->L }
% 23.54/3.38 complement(join(converse(meet(X, Y)), complement(converse(meet(X, converse(meet(converse(X), converse(Y))))))))
% 23.54/3.38 = { by lemma 44 R->L }
% 23.54/3.38 complement(join(converse(meet(X, Y)), complement(converse(meet(X, join(meet(complement(Y), converse(meet(converse(X), converse(Y)))), meet(converse(meet(converse(X), converse(Y))), complement(complement(Y)))))))))
% 23.54/3.38 = { by lemma 28 }
% 23.54/3.38 complement(join(converse(meet(X, Y)), complement(converse(meet(X, join(meet(converse(meet(converse(X), converse(Y))), complement(Y)), meet(converse(meet(converse(X), converse(Y))), complement(complement(Y)))))))))
% 23.54/3.38 = { by lemma 36 R->L }
% 23.54/3.38 complement(join(converse(meet(X, Y)), complement(converse(meet(X, join(complement(join(Y, complement(converse(meet(converse(X), converse(Y)))))), meet(converse(meet(converse(X), converse(Y))), complement(complement(Y)))))))))
% 23.54/3.38 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 23.54/3.38 complement(join(converse(meet(X, Y)), complement(converse(meet(X, join(complement(join(complement(converse(meet(converse(X), converse(Y)))), Y)), meet(converse(meet(converse(X), converse(Y))), complement(complement(Y)))))))))
% 23.54/3.38 = { by axiom 3 (converse_idempotence) R->L }
% 23.54/3.38 complement(join(converse(meet(X, Y)), complement(converse(meet(X, join(complement(join(complement(converse(meet(converse(X), converse(Y)))), converse(converse(Y)))), meet(converse(meet(converse(X), converse(Y))), complement(complement(Y)))))))))
% 23.54/3.38 = { by lemma 44 R->L }
% 23.54/3.38 complement(join(converse(meet(X, Y)), complement(converse(meet(X, join(complement(join(complement(converse(meet(converse(X), converse(Y)))), converse(join(meet(converse(X), converse(Y)), meet(converse(Y), complement(converse(X))))))), meet(converse(meet(converse(X), converse(Y))), complement(complement(Y)))))))))
% 23.54/3.38 = { by lemma 43 }
% 23.54/3.38 complement(join(converse(meet(X, Y)), complement(converse(meet(X, join(complement(top), meet(converse(meet(converse(X), converse(Y))), complement(complement(Y)))))))))
% 23.54/3.38 = { by lemma 13 }
% 23.54/3.38 complement(join(converse(meet(X, Y)), complement(converse(meet(X, join(zero, meet(converse(meet(converse(X), converse(Y))), complement(complement(Y)))))))))
% 23.54/3.38 = { by lemma 26 }
% 23.54/3.38 complement(join(converse(meet(X, Y)), complement(converse(meet(X, meet(converse(meet(converse(X), converse(Y))), complement(complement(Y))))))))
% 23.54/3.38 = { by lemma 27 }
% 23.54/3.38 complement(join(converse(meet(X, Y)), complement(converse(meet(X, meet(converse(meet(converse(X), converse(Y))), Y))))))
% 23.54/3.38 = { by lemma 28 R->L }
% 23.54/3.38 complement(join(converse(meet(X, Y)), complement(converse(meet(X, meet(Y, converse(meet(converse(X), converse(Y)))))))))
% 23.54/3.38 = { by lemma 42 }
% 23.54/3.38 complement(join(converse(meet(X, Y)), complement(converse(meet(converse(meet(converse(X), converse(Y))), meet(X, Y))))))
% 23.54/3.38 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 23.54/3.38 complement(join(complement(converse(meet(converse(meet(converse(X), converse(Y))), meet(X, Y)))), converse(meet(X, Y))))
% 23.54/3.38 = { by lemma 44 R->L }
% 23.54/3.38 complement(join(complement(converse(meet(converse(meet(converse(X), converse(Y))), meet(X, Y)))), converse(join(meet(converse(meet(converse(X), converse(Y))), meet(X, Y)), meet(meet(X, Y), complement(converse(meet(converse(X), converse(Y)))))))))
% 23.54/3.38 = { by lemma 43 }
% 23.54/3.38 complement(top)
% 23.54/3.38 = { by lemma 13 }
% 23.54/3.38 zero
% 23.54/3.38
% 23.54/3.38 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))).
% 23.54/3.38 Proof:
% 23.54/3.38 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))))
% 23.54/3.39 = { by lemma 27 R->L }
% 23.54/3.39 tuple(complement(complement(join(meet(converse(x0), converse(x1)), converse(meet(x0, x1))))), join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))))
% 23.54/3.39 = { by lemma 41 R->L }
% 23.54/3.39 tuple(complement(meet(complement(meet(converse(x0), converse(x1))), complement(converse(meet(x0, x1))))), join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))))
% 23.54/3.39 = { by lemma 26 R->L }
% 23.54/3.39 tuple(complement(join(zero, meet(complement(meet(converse(x0), converse(x1))), complement(converse(meet(x0, x1)))))), join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))))
% 23.54/3.39 = { by lemma 46 R->L }
% 23.54/3.39 tuple(complement(join(meet(meet(converse(x0), converse(x1)), complement(converse(meet(x0, x1)))), meet(complement(meet(converse(x0), converse(x1))), complement(converse(meet(x0, x1)))))), join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))))
% 23.54/3.39 = { by lemma 45 }
% 23.54/3.39 tuple(complement(complement(converse(meet(x0, x1)))), join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))))
% 23.54/3.39 = { by lemma 27 }
% 23.54/3.39 tuple(converse(meet(x0, x1)), join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))))
% 23.54/3.39 = { by lemma 39 R->L }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), join(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), complement(join(zero, complement(converse(x0_2)))))))
% 23.54/3.39 = { by lemma 29 }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), join(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), meet(converse(x0_2), top))))
% 23.54/3.39 = { by lemma 33 }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), join(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), converse(x0_2))))
% 23.54/3.39 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), join(converse(x0_2), join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))))))
% 23.54/3.39 = { by lemma 34 }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), join(meet(converse(x0_2), converse(x1_2)), join(converse(x0_2), converse(meet(x0_2, x1_2))))))
% 23.54/3.39 = { by axiom 8 (converse_additivity) R->L }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), join(meet(converse(x0_2), converse(x1_2)), converse(join(x0_2, meet(x0_2, x1_2))))))
% 23.54/3.39 = { by axiom 2 (maddux1_join_commutativity) }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), join(meet(converse(x0_2), converse(x1_2)), converse(join(meet(x0_2, x1_2), x0_2)))))
% 23.54/3.39 = { by lemma 40 }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), join(meet(converse(x0_2), converse(x1_2)), converse(x0_2))))
% 23.54/3.39 = { by lemma 40 }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), converse(x0_2)))
% 23.54/3.39 = { by lemma 28 }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(converse(x0_2), join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2)))))
% 23.54/3.39 = { by lemma 44 R->L }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(converse(x0_2), join(meet(join(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), converse(meet(x1_2, complement(x0_2)))), join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2)))), meet(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), complement(join(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), converse(meet(x1_2, complement(x0_2)))))))))
% 23.54/3.39 = { by axiom 3 (converse_idempotence) R->L }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(converse(x0_2), join(meet(join(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), converse(meet(x1_2, complement(x0_2)))), join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2)))), meet(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), complement(join(converse(converse(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))))), converse(meet(x1_2, complement(x0_2)))))))))
% 23.54/3.39 = { by axiom 8 (converse_additivity) R->L }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(converse(x0_2), join(meet(join(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), converse(meet(x1_2, complement(x0_2)))), join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2)))), meet(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), complement(converse(join(converse(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2)))), meet(x1_2, complement(x0_2)))))))))
% 23.54/3.39 = { by axiom 3 (converse_idempotence) R->L }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(converse(x0_2), join(meet(join(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), converse(meet(x1_2, complement(x0_2)))), join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2)))), meet(converse(converse(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))))), complement(converse(join(converse(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2)))), meet(x1_2, complement(x0_2)))))))))
% 23.54/3.39 = { by lemma 38 R->L }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(converse(x0_2), join(meet(join(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), converse(meet(x1_2, complement(x0_2)))), join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2)))), complement(join(complement(converse(converse(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2)))))), converse(join(converse(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2)))), meet(x1_2, complement(x0_2)))))))))
% 23.54/3.39 = { by lemma 43 }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(converse(x0_2), join(meet(join(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), converse(meet(x1_2, complement(x0_2)))), join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2)))), complement(top))))
% 23.54/3.39 = { by lemma 13 }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(converse(x0_2), join(meet(join(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), converse(meet(x1_2, complement(x0_2)))), join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2)))), zero)))
% 23.54/3.39 = { by lemma 25 }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(converse(x0_2), meet(join(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), converse(meet(x1_2, complement(x0_2)))), join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))))))
% 23.54/3.39 = { by lemma 28 R->L }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(converse(x0_2), meet(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), join(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), converse(meet(x1_2, complement(x0_2)))))))
% 23.54/3.39 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(converse(x0_2), meet(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), join(converse(meet(x1_2, complement(x0_2))), join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2)))))))
% 23.54/3.39 = { by lemma 27 R->L }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(converse(x0_2), meet(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), join(converse(meet(x1_2, complement(x0_2))), complement(complement(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2)))))))))
% 23.54/3.39 = { by axiom 2 (maddux1_join_commutativity) }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(converse(x0_2), meet(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), join(converse(meet(x1_2, complement(x0_2))), complement(complement(join(meet(converse(x0_2), converse(x1_2)), converse(meet(x0_2, x1_2)))))))))
% 23.54/3.39 = { by lemma 41 R->L }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(converse(x0_2), meet(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), join(converse(meet(x1_2, complement(x0_2))), complement(meet(complement(meet(converse(x0_2), converse(x1_2))), complement(converse(meet(x0_2, x1_2)))))))))
% 23.54/3.39 = { by lemma 26 R->L }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(converse(x0_2), meet(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), join(converse(meet(x1_2, complement(x0_2))), complement(join(zero, meet(complement(meet(converse(x0_2), converse(x1_2))), complement(converse(meet(x0_2, x1_2))))))))))
% 23.54/3.39 = { by lemma 46 R->L }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(converse(x0_2), meet(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), join(converse(meet(x1_2, complement(x0_2))), complement(join(meet(meet(converse(x0_2), converse(x1_2)), complement(converse(meet(x0_2, x1_2)))), meet(complement(meet(converse(x0_2), converse(x1_2))), complement(converse(meet(x0_2, x1_2))))))))))
% 23.54/3.39 = { by lemma 45 }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(converse(x0_2), meet(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), join(converse(meet(x1_2, complement(x0_2))), complement(complement(converse(meet(x0_2, x1_2))))))))
% 23.54/3.39 = { by lemma 27 }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(converse(x0_2), meet(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), join(converse(meet(x1_2, complement(x0_2))), converse(meet(x0_2, x1_2))))))
% 23.54/3.39 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(converse(x0_2), meet(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), join(converse(meet(x0_2, x1_2)), converse(meet(x1_2, complement(x0_2)))))))
% 23.54/3.39 = { by axiom 8 (converse_additivity) R->L }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(converse(x0_2), meet(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), converse(join(meet(x0_2, x1_2), meet(x1_2, complement(x0_2)))))))
% 23.54/3.39 = { by lemma 44 }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(converse(x0_2), meet(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), converse(x1_2))))
% 23.54/3.39 = { by lemma 28 }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(converse(x0_2), meet(converse(x1_2), join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))))))
% 23.54/3.39 = { by lemma 42 }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), meet(converse(x0_2), converse(x1_2))))
% 23.54/3.39 = { by lemma 28 }
% 23.54/3.39 tuple(converse(meet(x0, x1)), meet(meet(converse(x0_2), converse(x1_2)), join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2)))))
% 23.54/3.39 = { by axiom 10 (maddux4_definiton_of_meet) }
% 23.54/3.39 tuple(converse(meet(x0, x1)), complement(join(complement(meet(converse(x0_2), converse(x1_2))), complement(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2)))))))
% 23.54/3.39 = { by lemma 24 R->L }
% 23.54/3.39 tuple(converse(meet(x0, x1)), join(zero, complement(join(complement(meet(converse(x0_2), converse(x1_2))), complement(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))))))))
% 23.54/3.39 = { by lemma 13 R->L }
% 23.54/3.39 tuple(converse(meet(x0, x1)), join(complement(top), complement(join(complement(meet(converse(x0_2), converse(x1_2))), complement(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))))))))
% 23.54/3.39 = { by lemma 30 R->L }
% 23.54/3.39 tuple(converse(meet(x0, x1)), join(complement(join(meet(converse(x0_2), converse(x1_2)), join(complement(meet(converse(x0_2), converse(x1_2))), converse(meet(x0_2, x1_2))))), complement(join(complement(meet(converse(x0_2), converse(x1_2))), complement(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))))))))
% 23.54/3.40 = { by lemma 34 R->L }
% 23.54/3.40 tuple(converse(meet(x0, x1)), join(complement(join(complement(meet(converse(x0_2), converse(x1_2))), join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))))), complement(join(complement(meet(converse(x0_2), converse(x1_2))), complement(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))))))))
% 23.54/3.40 = { by axiom 2 (maddux1_join_commutativity) }
% 23.54/3.40 tuple(converse(meet(x0, x1)), join(complement(join(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))), complement(meet(converse(x0_2), converse(x1_2))))), complement(join(complement(meet(converse(x0_2), converse(x1_2))), complement(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))))))))
% 23.54/3.40 = { by lemma 36 }
% 23.54/3.40 tuple(converse(meet(x0, x1)), join(meet(meet(converse(x0_2), converse(x1_2)), complement(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))))), complement(join(complement(meet(converse(x0_2), converse(x1_2))), complement(join(converse(meet(x0_2, x1_2)), meet(converse(x0_2), converse(x1_2))))))))
% 23.54/3.40 = { by lemma 20 }
% 23.54/3.40 tuple(converse(meet(x0, x1)), meet(converse(x0_2), converse(x1_2)))
% 23.54/3.40 % SZS output end Proof
% 23.54/3.40
% 23.54/3.40 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------