TSTP Solution File: REL026-2 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL026-2 : 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 : n027.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 13:44:06 EDT 2023
% Result : Unsatisfiable 12.60s 1.98s
% Output : Proof 12.60s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : REL026-2 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.35 % Computer : n027.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Fri Aug 25 22:28:34 EDT 2023
% 0.14/0.35 % CPUTime :
% 12.60/1.98 Command-line arguments: --flatten
% 12.60/1.98
% 12.60/1.98 % SZS status Unsatisfiable
% 12.60/1.98
% 12.60/2.06 % SZS output start Proof
% 12.60/2.06 Take the following subset of the input axioms:
% 12.60/2.06 fof(composition_associativity_5, axiom, ![A, B, C]: composition(A, composition(B, C))=composition(composition(A, B), C)).
% 12.60/2.06 fof(composition_distributivity_7, axiom, ![A2, B2, C2]: composition(join(A2, B2), C2)=join(composition(A2, C2), composition(B2, C2))).
% 12.60/2.06 fof(composition_identity_6, axiom, ![A2]: composition(A2, one)=A2).
% 12.60/2.06 fof(converse_additivity_9, axiom, ![A2, B2]: converse(join(A2, B2))=join(converse(A2), converse(B2))).
% 12.60/2.06 fof(converse_cancellativity_11, axiom, ![A2, B2]: join(composition(converse(A2), complement(composition(A2, B2))), complement(B2))=complement(B2)).
% 12.60/2.06 fof(converse_idempotence_8, axiom, ![A2]: converse(converse(A2))=A2).
% 12.60/2.06 fof(converse_multiplicativity_10, axiom, ![A2, B2]: converse(composition(A2, B2))=composition(converse(B2), converse(A2))).
% 12.60/2.06 fof(def_top_12, axiom, ![A2]: top=join(A2, complement(A2))).
% 12.60/2.06 fof(def_zero_13, axiom, ![A2]: zero=meet(A2, complement(A2))).
% 12.60/2.06 fof(goals_14, negated_conjecture, join(sk1, one)=one).
% 12.60/2.06 fof(goals_15, negated_conjecture, join(composition(sk1, sk2), meet(composition(sk1, top), sk2))!=meet(composition(sk1, top), sk2) | join(meet(composition(sk1, top), sk2), composition(sk1, sk2))!=composition(sk1, sk2)).
% 12.60/2.06 fof(maddux1_join_commutativity_1, axiom, ![A2, B2]: join(A2, B2)=join(B2, A2)).
% 12.60/2.06 fof(maddux2_join_associativity_2, axiom, ![A2, B2, C2]: join(A2, join(B2, C2))=join(join(A2, B2), C2)).
% 12.60/2.06 fof(maddux3_a_kind_of_de_Morgan_3, axiom, ![A2, B2]: A2=join(complement(join(complement(A2), complement(B2))), complement(join(complement(A2), B2)))).
% 12.60/2.06 fof(maddux4_definiton_of_meet_4, axiom, ![A2, B2]: meet(A2, B2)=complement(join(complement(A2), complement(B2)))).
% 12.60/2.06
% 12.60/2.06 Now clausify the problem and encode Horn clauses using encoding 3 of
% 12.60/2.06 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 12.60/2.06 We repeatedly replace C & s=t => u=v by the two clauses:
% 12.60/2.06 fresh(y, y, x1...xn) = u
% 12.60/2.06 C => fresh(s, t, x1...xn) = v
% 12.60/2.06 where fresh is a fresh function symbol and x1..xn are the free
% 12.60/2.06 variables of u and v.
% 12.60/2.06 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 12.60/2.06 input problem has no model of domain size 1).
% 12.60/2.06
% 12.60/2.06 The encoding turns the above axioms into the following unit equations and goals:
% 12.60/2.06
% 12.60/2.06 Axiom 1 (converse_idempotence_8): converse(converse(X)) = X.
% 12.60/2.06 Axiom 2 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 12.60/2.06 Axiom 3 (goals_14): join(sk1, one) = one.
% 12.60/2.06 Axiom 4 (composition_identity_6): composition(X, one) = X.
% 12.60/2.06 Axiom 5 (def_zero_13): zero = meet(X, complement(X)).
% 12.60/2.06 Axiom 6 (def_top_12): top = join(X, complement(X)).
% 12.60/2.06 Axiom 7 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 12.60/2.06 Axiom 8 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 12.60/2.06 Axiom 9 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 12.60/2.06 Axiom 10 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 12.60/2.06 Axiom 11 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 12.60/2.06 Axiom 12 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 12.60/2.06 Axiom 13 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 12.60/2.06 Axiom 14 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 12.60/2.06
% 12.60/2.06 Lemma 15: complement(top) = zero.
% 12.60/2.06 Proof:
% 12.60/2.06 complement(top)
% 12.60/2.06 = { by axiom 6 (def_top_12) }
% 12.60/2.06 complement(join(complement(X), complement(complement(X))))
% 12.60/2.06 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 12.60/2.06 meet(X, complement(X))
% 12.60/2.06 = { by axiom 5 (def_zero_13) R->L }
% 12.60/2.06 zero
% 12.60/2.06
% 12.60/2.06 Lemma 16: join(X, join(Y, complement(X))) = join(Y, top).
% 12.60/2.06 Proof:
% 12.60/2.06 join(X, join(Y, complement(X)))
% 12.60/2.06 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.06 join(X, join(complement(X), Y))
% 12.60/2.06 = { by axiom 8 (maddux2_join_associativity_2) }
% 12.60/2.06 join(join(X, complement(X)), Y)
% 12.60/2.06 = { by axiom 6 (def_top_12) R->L }
% 12.60/2.06 join(top, Y)
% 12.60/2.06 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.60/2.06 join(Y, top)
% 12.60/2.06
% 12.60/2.06 Lemma 17: composition(X, join(sk1, one)) = X.
% 12.60/2.06 Proof:
% 12.60/2.06 composition(X, join(sk1, one))
% 12.60/2.06 = { by axiom 3 (goals_14) }
% 12.60/2.06 composition(X, one)
% 12.60/2.06 = { by axiom 4 (composition_identity_6) }
% 12.60/2.06 X
% 12.60/2.06
% 12.60/2.06 Lemma 18: composition(converse(join(sk1, one)), X) = X.
% 12.60/2.06 Proof:
% 12.60/2.06 composition(converse(join(sk1, one)), X)
% 12.60/2.06 = { by axiom 1 (converse_idempotence_8) R->L }
% 12.60/2.06 composition(converse(join(sk1, one)), converse(converse(X)))
% 12.60/2.06 = { by axiom 9 (converse_multiplicativity_10) R->L }
% 12.60/2.06 converse(composition(converse(X), join(sk1, one)))
% 12.60/2.06 = { by lemma 17 }
% 12.60/2.06 converse(converse(X))
% 12.60/2.06 = { by axiom 1 (converse_idempotence_8) }
% 12.60/2.06 X
% 12.60/2.06
% 12.60/2.06 Lemma 19: composition(join(sk1, one), X) = X.
% 12.60/2.06 Proof:
% 12.60/2.06 composition(join(sk1, one), X)
% 12.60/2.06 = { by lemma 18 R->L }
% 12.60/2.06 composition(converse(join(sk1, one)), composition(join(sk1, one), X))
% 12.60/2.06 = { by axiom 10 (composition_associativity_5) }
% 12.60/2.06 composition(composition(converse(join(sk1, one)), join(sk1, one)), X)
% 12.60/2.06 = { by lemma 17 }
% 12.60/2.07 composition(converse(join(sk1, one)), X)
% 12.60/2.07 = { by lemma 18 }
% 12.60/2.07 X
% 12.60/2.07
% 12.60/2.07 Lemma 20: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 12.60/2.07 Proof:
% 12.60/2.07 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 12.60/2.07 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.07 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 12.60/2.07 = { by axiom 13 (converse_cancellativity_11) }
% 12.60/2.07 complement(X)
% 12.60/2.07
% 12.60/2.07 Lemma 21: join(complement(X), complement(X)) = complement(X).
% 12.60/2.07 Proof:
% 12.60/2.07 join(complement(X), complement(X))
% 12.60/2.07 = { by lemma 18 R->L }
% 12.60/2.07 join(complement(X), composition(converse(join(sk1, one)), complement(X)))
% 12.60/2.07 = { by lemma 19 R->L }
% 12.60/2.07 join(complement(X), composition(converse(join(sk1, one)), complement(composition(join(sk1, one), X))))
% 12.60/2.07 = { by lemma 20 }
% 12.60/2.07 complement(X)
% 12.60/2.07
% 12.60/2.07 Lemma 22: join(top, complement(X)) = top.
% 12.60/2.07 Proof:
% 12.60/2.07 join(top, complement(X))
% 12.60/2.07 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.07 join(complement(X), top)
% 12.60/2.07 = { by lemma 16 R->L }
% 12.60/2.07 join(X, join(complement(X), complement(X)))
% 12.60/2.07 = { by lemma 21 }
% 12.60/2.07 join(X, complement(X))
% 12.60/2.07 = { by axiom 6 (def_top_12) R->L }
% 12.60/2.07 top
% 12.60/2.07
% 12.60/2.07 Lemma 23: join(Y, top) = join(X, top).
% 12.60/2.07 Proof:
% 12.60/2.07 join(Y, top)
% 12.60/2.07 = { by lemma 22 R->L }
% 12.60/2.07 join(Y, join(top, complement(Y)))
% 12.60/2.07 = { by lemma 16 }
% 12.60/2.07 join(top, top)
% 12.60/2.07 = { by lemma 16 R->L }
% 12.60/2.07 join(X, join(top, complement(X)))
% 12.60/2.07 = { by lemma 22 }
% 12.60/2.07 join(X, top)
% 12.60/2.07
% 12.60/2.07 Lemma 24: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 12.60/2.07 Proof:
% 12.60/2.07 join(meet(X, Y), complement(join(complement(X), Y)))
% 12.60/2.07 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 12.60/2.07 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 12.60/2.07 = { by axiom 14 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 12.60/2.07 X
% 12.60/2.07
% 12.60/2.07 Lemma 25: join(zero, meet(X, X)) = X.
% 12.60/2.07 Proof:
% 12.60/2.07 join(zero, meet(X, X))
% 12.60/2.07 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 12.60/2.07 join(zero, complement(join(complement(X), complement(X))))
% 12.60/2.07 = { by axiom 5 (def_zero_13) }
% 12.60/2.07 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 12.60/2.07 = { by lemma 24 }
% 12.60/2.07 X
% 12.60/2.07
% 12.60/2.07 Lemma 26: join(zero, join(X, meet(Y, Y))) = join(X, Y).
% 12.60/2.07 Proof:
% 12.60/2.07 join(zero, join(X, meet(Y, Y)))
% 12.60/2.07 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.07 join(zero, join(meet(Y, Y), X))
% 12.60/2.07 = { by axiom 8 (maddux2_join_associativity_2) }
% 12.60/2.07 join(join(zero, meet(Y, Y)), X)
% 12.60/2.07 = { by lemma 25 }
% 12.60/2.07 join(Y, X)
% 12.60/2.07 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.60/2.07 join(X, Y)
% 12.60/2.07
% 12.60/2.07 Lemma 27: join(X, zero) = join(X, X).
% 12.60/2.07 Proof:
% 12.60/2.07 join(X, zero)
% 12.60/2.07 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.07 join(zero, X)
% 12.60/2.07 = { by lemma 25 R->L }
% 12.60/2.07 join(zero, join(zero, meet(X, X)))
% 12.60/2.07 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 12.60/2.07 join(zero, join(zero, complement(join(complement(X), complement(X)))))
% 12.60/2.07 = { by lemma 21 R->L }
% 12.60/2.07 join(zero, join(zero, join(complement(join(complement(X), complement(X))), complement(join(complement(X), complement(X))))))
% 12.60/2.07 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 12.60/2.07 join(zero, join(zero, join(meet(X, X), complement(join(complement(X), complement(X))))))
% 12.60/2.07 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 12.60/2.07 join(zero, join(zero, join(meet(X, X), meet(X, X))))
% 12.60/2.07 = { by lemma 26 }
% 12.60/2.07 join(zero, join(meet(X, X), X))
% 12.60/2.07 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.60/2.07 join(zero, join(X, meet(X, X)))
% 12.60/2.07 = { by lemma 26 }
% 12.60/2.07 join(X, X)
% 12.60/2.07
% 12.60/2.07 Lemma 28: composition(join(X, join(sk1, one)), Y) = join(Y, composition(X, Y)).
% 12.60/2.07 Proof:
% 12.60/2.07 composition(join(X, join(sk1, one)), Y)
% 12.60/2.07 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.07 composition(join(join(sk1, one), X), Y)
% 12.60/2.07 = { by axiom 12 (composition_distributivity_7) }
% 12.60/2.07 join(composition(join(sk1, one), Y), composition(X, Y))
% 12.60/2.07 = { by lemma 19 }
% 12.60/2.07 join(Y, composition(X, Y))
% 12.60/2.07
% 12.60/2.07 Lemma 29: converse(join(sk1, one)) = join(sk1, one).
% 12.60/2.07 Proof:
% 12.60/2.07 converse(join(sk1, one))
% 12.60/2.07 = { by lemma 17 R->L }
% 12.60/2.07 composition(converse(join(sk1, one)), join(sk1, one))
% 12.60/2.07 = { by lemma 18 }
% 12.60/2.07 join(sk1, one)
% 12.60/2.07
% 12.60/2.07 Lemma 30: join(join(sk1, one), join(X, Y)) = join(X, join(Y, one)).
% 12.60/2.07 Proof:
% 12.60/2.07 join(join(sk1, one), join(X, Y))
% 12.60/2.07 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.07 join(join(sk1, one), join(Y, X))
% 12.60/2.07 = { by axiom 3 (goals_14) }
% 12.60/2.07 join(one, join(Y, X))
% 12.60/2.07 = { by axiom 8 (maddux2_join_associativity_2) }
% 12.60/2.07 join(join(one, Y), X)
% 12.60/2.07 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.07 join(join(Y, one), X)
% 12.60/2.07 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.60/2.07 join(X, join(Y, one))
% 12.60/2.07
% 12.60/2.07 Lemma 31: join(join(sk1, one), X) = join(X, one).
% 12.60/2.07 Proof:
% 12.60/2.07 join(join(sk1, one), X)
% 12.60/2.07 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.07 join(X, join(sk1, one))
% 12.60/2.07 = { by axiom 3 (goals_14) }
% 12.60/2.07 join(X, one)
% 12.60/2.07
% 12.60/2.07 Lemma 32: join(X, X) = X.
% 12.60/2.07 Proof:
% 12.60/2.07 join(X, X)
% 12.60/2.07 = { by lemma 18 R->L }
% 12.60/2.07 join(X, composition(converse(join(sk1, one)), X))
% 12.60/2.07 = { by lemma 28 R->L }
% 12.60/2.07 composition(join(converse(join(sk1, one)), join(sk1, one)), X)
% 12.60/2.07 = { by lemma 29 }
% 12.60/2.07 composition(join(join(sk1, one), join(sk1, one)), X)
% 12.60/2.07 = { by lemma 27 R->L }
% 12.60/2.07 composition(join(join(sk1, one), zero), X)
% 12.60/2.07 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.60/2.07 composition(join(zero, join(sk1, one)), X)
% 12.60/2.07 = { by lemma 30 R->L }
% 12.60/2.07 composition(join(join(sk1, one), join(zero, sk1)), X)
% 12.60/2.07 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.60/2.07 composition(join(join(sk1, one), join(sk1, zero)), X)
% 12.60/2.07 = { by lemma 27 }
% 12.60/2.07 composition(join(join(sk1, one), join(sk1, sk1)), X)
% 12.60/2.07 = { by lemma 30 }
% 12.60/2.07 composition(join(sk1, join(sk1, one)), X)
% 12.60/2.07 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.60/2.07 composition(join(join(sk1, one), sk1), X)
% 12.60/2.07 = { by lemma 31 }
% 12.60/2.07 composition(join(sk1, one), X)
% 12.60/2.07 = { by lemma 19 }
% 12.60/2.07 X
% 12.60/2.07
% 12.60/2.07 Lemma 33: join(X, zero) = X.
% 12.60/2.07 Proof:
% 12.60/2.07 join(X, zero)
% 12.60/2.07 = { by lemma 27 }
% 12.60/2.07 join(X, X)
% 12.60/2.07 = { by lemma 32 }
% 12.60/2.07 X
% 12.60/2.07
% 12.60/2.07 Lemma 34: join(zero, X) = X.
% 12.60/2.07 Proof:
% 12.60/2.07 join(zero, X)
% 12.60/2.07 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.07 join(X, zero)
% 12.60/2.07 = { by lemma 33 }
% 12.60/2.07 X
% 12.60/2.07
% 12.60/2.07 Lemma 35: join(X, top) = top.
% 12.60/2.07 Proof:
% 12.60/2.07 join(X, top)
% 12.60/2.07 = { by lemma 23 }
% 12.60/2.07 join(zero, top)
% 12.60/2.07 = { by lemma 34 }
% 12.60/2.07 top
% 12.60/2.07
% 12.60/2.07 Lemma 36: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 12.60/2.07 Proof:
% 12.60/2.07 converse(join(X, converse(Y)))
% 12.60/2.07 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.07 converse(join(converse(Y), X))
% 12.60/2.07 = { by axiom 7 (converse_additivity_9) }
% 12.60/2.07 join(converse(converse(Y)), converse(X))
% 12.60/2.07 = { by axiom 1 (converse_idempotence_8) }
% 12.60/2.07 join(Y, converse(X))
% 12.60/2.07
% 12.60/2.07 Lemma 37: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 12.60/2.07 Proof:
% 12.60/2.07 converse(join(converse(X), Y))
% 12.60/2.07 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.07 converse(join(Y, converse(X)))
% 12.60/2.07 = { by lemma 36 }
% 12.60/2.07 join(X, converse(Y))
% 12.60/2.07
% 12.60/2.07 Lemma 38: join(X, converse(complement(converse(X)))) = converse(top).
% 12.60/2.07 Proof:
% 12.60/2.07 join(X, converse(complement(converse(X))))
% 12.60/2.07 = { by lemma 37 R->L }
% 12.60/2.07 converse(join(converse(X), complement(converse(X))))
% 12.60/2.07 = { by axiom 6 (def_top_12) R->L }
% 12.60/2.07 converse(top)
% 12.60/2.07
% 12.60/2.07 Lemma 39: join(X, join(complement(X), Y)) = top.
% 12.60/2.07 Proof:
% 12.60/2.07 join(X, join(complement(X), Y))
% 12.60/2.07 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.07 join(X, join(Y, complement(X)))
% 12.60/2.07 = { by lemma 16 }
% 12.60/2.07 join(Y, top)
% 12.60/2.07 = { by lemma 23 R->L }
% 12.60/2.07 join(Z, top)
% 12.60/2.07 = { by lemma 35 }
% 12.60/2.07 top
% 12.60/2.07
% 12.60/2.07 Lemma 40: join(X, converse(top)) = top.
% 12.60/2.07 Proof:
% 12.60/2.07 join(X, converse(top))
% 12.60/2.07 = { by lemma 38 R->L }
% 12.60/2.07 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 12.60/2.07 = { by lemma 39 }
% 12.60/2.07 top
% 12.60/2.07
% 12.60/2.07 Lemma 41: converse(top) = top.
% 12.60/2.07 Proof:
% 12.60/2.07 converse(top)
% 12.60/2.07 = { by lemma 35 R->L }
% 12.60/2.07 converse(join(X, top))
% 12.60/2.07 = { by axiom 7 (converse_additivity_9) }
% 12.60/2.07 join(converse(X), converse(top))
% 12.60/2.07 = { by lemma 40 }
% 12.60/2.07 top
% 12.60/2.07
% 12.60/2.07 Lemma 42: join(top, X) = top.
% 12.60/2.07 Proof:
% 12.60/2.07 join(top, X)
% 12.60/2.07 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.07 join(X, top)
% 12.60/2.07 = { by lemma 23 R->L }
% 12.60/2.07 join(Y, top)
% 12.60/2.07 = { by lemma 35 }
% 12.60/2.07 top
% 12.60/2.07
% 12.60/2.07 Lemma 43: complement(complement(X)) = meet(X, X).
% 12.60/2.07 Proof:
% 12.60/2.07 complement(complement(X))
% 12.60/2.07 = { by lemma 21 R->L }
% 12.60/2.07 complement(join(complement(X), complement(X)))
% 12.60/2.07 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 12.60/2.07 meet(X, X)
% 12.60/2.07
% 12.60/2.07 Lemma 44: complement(complement(X)) = X.
% 12.60/2.07 Proof:
% 12.60/2.07 complement(complement(X))
% 12.60/2.07 = { by lemma 34 R->L }
% 12.60/2.07 join(zero, complement(complement(X)))
% 12.60/2.07 = { by lemma 43 }
% 12.60/2.07 join(zero, meet(X, X))
% 12.60/2.07 = { by lemma 25 }
% 12.60/2.07 X
% 12.60/2.07
% 12.60/2.07 Lemma 45: meet(X, X) = X.
% 12.60/2.07 Proof:
% 12.60/2.07 meet(X, X)
% 12.60/2.07 = { by lemma 43 R->L }
% 12.60/2.07 complement(complement(X))
% 12.60/2.07 = { by lemma 44 }
% 12.60/2.07 X
% 12.60/2.07
% 12.60/2.07 Lemma 46: meet(Y, X) = meet(X, Y).
% 12.60/2.07 Proof:
% 12.60/2.07 meet(Y, X)
% 12.60/2.07 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 12.60/2.07 complement(join(complement(Y), complement(X)))
% 12.60/2.07 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.07 complement(join(complement(X), complement(Y)))
% 12.60/2.07 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 12.60/2.07 meet(X, Y)
% 12.60/2.07
% 12.60/2.07 Lemma 47: complement(join(zero, complement(X))) = meet(X, top).
% 12.60/2.07 Proof:
% 12.60/2.07 complement(join(zero, complement(X)))
% 12.60/2.07 = { by lemma 15 R->L }
% 12.60/2.07 complement(join(complement(top), complement(X)))
% 12.60/2.07 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 12.60/2.07 meet(top, X)
% 12.60/2.07 = { by lemma 46 R->L }
% 12.60/2.07 meet(X, top)
% 12.60/2.07
% 12.60/2.07 Lemma 48: meet(X, top) = X.
% 12.60/2.07 Proof:
% 12.60/2.07 meet(X, top)
% 12.60/2.07 = { by lemma 46 }
% 12.60/2.07 meet(top, X)
% 12.60/2.07 = { by lemma 44 R->L }
% 12.60/2.07 meet(top, complement(complement(X)))
% 12.60/2.07 = { by lemma 46 }
% 12.60/2.07 meet(complement(complement(X)), top)
% 12.60/2.07 = { by lemma 47 R->L }
% 12.60/2.07 complement(join(zero, complement(complement(complement(X)))))
% 12.60/2.07 = { by lemma 43 }
% 12.60/2.07 complement(join(zero, meet(complement(X), complement(X))))
% 12.60/2.07 = { by lemma 25 }
% 12.60/2.07 complement(complement(X))
% 12.60/2.07 = { by lemma 44 }
% 12.60/2.07 X
% 12.60/2.07
% 12.60/2.07 Lemma 49: meet(top, X) = X.
% 12.60/2.07 Proof:
% 12.60/2.07 meet(top, X)
% 12.60/2.07 = { by lemma 46 }
% 12.60/2.07 meet(X, top)
% 12.60/2.07 = { by lemma 48 }
% 12.60/2.07 X
% 12.60/2.07
% 12.60/2.08 Lemma 50: composition(join(join(sk1, one), Y), X) = join(X, composition(Y, X)).
% 12.60/2.08 Proof:
% 12.60/2.08 composition(join(join(sk1, one), Y), X)
% 12.60/2.08 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.08 composition(join(Y, join(sk1, one)), X)
% 12.60/2.08 = { by lemma 28 }
% 12.60/2.08 join(X, composition(Y, X))
% 12.60/2.08
% 12.60/2.08 Lemma 51: composition(top, zero) = zero.
% 12.60/2.08 Proof:
% 12.60/2.08 composition(top, zero)
% 12.60/2.08 = { by lemma 41 R->L }
% 12.60/2.08 composition(converse(top), zero)
% 12.60/2.08 = { by lemma 34 R->L }
% 12.60/2.08 join(zero, composition(converse(top), zero))
% 12.60/2.08 = { by lemma 15 R->L }
% 12.60/2.08 join(complement(top), composition(converse(top), zero))
% 12.60/2.08 = { by lemma 15 R->L }
% 12.60/2.08 join(complement(top), composition(converse(top), complement(top)))
% 12.60/2.08 = { by lemma 42 R->L }
% 12.60/2.08 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 12.60/2.08 = { by lemma 41 R->L }
% 12.60/2.08 join(complement(top), composition(converse(top), complement(join(top, composition(converse(top), top)))))
% 12.60/2.08 = { by lemma 50 R->L }
% 12.60/2.08 join(complement(top), composition(converse(top), complement(composition(join(join(sk1, one), converse(top)), top))))
% 12.60/2.08 = { by lemma 40 }
% 12.60/2.08 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 12.60/2.08 = { by lemma 20 }
% 12.60/2.08 complement(top)
% 12.60/2.08 = { by lemma 15 }
% 12.60/2.08 zero
% 12.60/2.08
% 12.60/2.08 Lemma 52: join(X, join(X, Y)) = join(X, Y).
% 12.60/2.08 Proof:
% 12.60/2.08 join(X, join(X, Y))
% 12.60/2.08 = { by axiom 8 (maddux2_join_associativity_2) }
% 12.60/2.08 join(join(X, X), Y)
% 12.60/2.08 = { by lemma 32 }
% 12.60/2.08 join(X, Y)
% 12.60/2.08
% 12.60/2.08 Lemma 53: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 12.60/2.08 Proof:
% 12.60/2.08 join(Y, join(X, Z))
% 12.60/2.08 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.08 join(join(X, Z), Y)
% 12.60/2.08 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 12.60/2.08 join(X, join(Z, Y))
% 12.60/2.08 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.60/2.08 join(X, join(Y, Z))
% 12.60/2.08
% 12.60/2.08 Lemma 54: join(Z, join(X, Y)) = join(X, join(Y, Z)).
% 12.60/2.08 Proof:
% 12.60/2.08 join(Z, join(X, Y))
% 12.60/2.08 = { by lemma 53 }
% 12.60/2.08 join(X, join(Z, Y))
% 12.60/2.08 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.60/2.08 join(X, join(Y, Z))
% 12.60/2.08
% 12.60/2.08 Lemma 55: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
% 12.60/2.08 Proof:
% 12.60/2.08 complement(join(complement(X), meet(Y, Z)))
% 12.60/2.08 = { by lemma 46 }
% 12.60/2.08 complement(join(complement(X), meet(Z, Y)))
% 12.60/2.08 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.08 complement(join(meet(Z, Y), complement(X)))
% 12.60/2.08 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 12.60/2.08 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 12.60/2.08 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 12.60/2.08 meet(join(complement(Z), complement(Y)), X)
% 12.60/2.08 = { by lemma 46 R->L }
% 12.60/2.08 meet(X, join(complement(Z), complement(Y)))
% 12.60/2.08 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.60/2.08 meet(X, join(complement(Y), complement(Z)))
% 12.60/2.08
% 12.60/2.08 Lemma 56: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 12.60/2.08 Proof:
% 12.60/2.08 join(complement(X), complement(Y))
% 12.60/2.08 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.08 join(complement(Y), complement(X))
% 12.60/2.08 = { by lemma 49 R->L }
% 12.60/2.08 meet(top, join(complement(Y), complement(X)))
% 12.60/2.08 = { by lemma 55 R->L }
% 12.60/2.08 complement(join(complement(top), meet(Y, X)))
% 12.60/2.08 = { by lemma 15 }
% 12.60/2.08 complement(join(zero, meet(Y, X)))
% 12.60/2.08 = { by lemma 34 }
% 12.60/2.08 complement(meet(Y, X))
% 12.60/2.08 = { by lemma 46 R->L }
% 12.60/2.08 complement(meet(X, Y))
% 12.60/2.08
% 12.60/2.08 Lemma 57: meet(X, meet(X, X)) = X.
% 12.60/2.08 Proof:
% 12.60/2.08 meet(X, meet(X, X))
% 12.60/2.08 = { by lemma 43 R->L }
% 12.60/2.08 meet(X, complement(complement(X)))
% 12.60/2.08 = { by lemma 33 R->L }
% 12.60/2.08 join(meet(X, complement(complement(X))), zero)
% 12.60/2.08 = { by lemma 15 R->L }
% 12.60/2.08 join(meet(X, complement(complement(X))), complement(top))
% 12.60/2.08 = { by axiom 6 (def_top_12) }
% 12.60/2.08 join(meet(X, complement(complement(X))), complement(join(complement(X), complement(complement(X)))))
% 12.60/2.08 = { by lemma 24 }
% 12.60/2.08 X
% 12.60/2.08
% 12.60/2.08 Lemma 58: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 12.60/2.08 Proof:
% 12.60/2.08 complement(join(X, complement(Y)))
% 12.60/2.08 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.08 complement(join(complement(Y), X))
% 12.60/2.08 = { by lemma 57 R->L }
% 12.60/2.08 complement(join(complement(Y), meet(X, meet(X, X))))
% 12.60/2.08 = { by lemma 55 }
% 12.60/2.08 meet(Y, join(complement(X), complement(meet(X, X))))
% 12.60/2.08 = { by lemma 56 }
% 12.60/2.08 meet(Y, complement(meet(X, meet(X, X))))
% 12.60/2.08 = { by lemma 57 }
% 12.60/2.08 meet(Y, complement(X))
% 12.60/2.08
% 12.60/2.08 Lemma 59: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 12.60/2.08 Proof:
% 12.60/2.08 complement(meet(X, complement(Y)))
% 12.60/2.08 = { by lemma 46 }
% 12.60/2.08 complement(meet(complement(Y), X))
% 12.60/2.08 = { by lemma 34 R->L }
% 12.60/2.08 complement(meet(join(zero, complement(Y)), X))
% 12.60/2.08 = { by lemma 56 R->L }
% 12.60/2.08 join(complement(join(zero, complement(Y))), complement(X))
% 12.60/2.08 = { by lemma 47 }
% 12.60/2.08 join(meet(Y, top), complement(X))
% 12.60/2.08 = { by lemma 48 }
% 12.60/2.08 join(Y, complement(X))
% 12.60/2.08
% 12.60/2.08 Lemma 60: meet(X, join(X, Y)) = X.
% 12.60/2.08 Proof:
% 12.60/2.08 meet(X, join(X, Y))
% 12.60/2.08 = { by lemma 45 R->L }
% 12.60/2.08 meet(X, join(X, meet(Y, Y)))
% 12.60/2.08 = { by lemma 43 R->L }
% 12.60/2.08 meet(X, join(X, complement(complement(Y))))
% 12.60/2.08 = { by lemma 59 R->L }
% 12.60/2.08 meet(X, complement(meet(complement(Y), complement(X))))
% 12.60/2.08 = { by lemma 56 R->L }
% 12.60/2.08 meet(X, join(complement(complement(Y)), complement(complement(X))))
% 12.60/2.08 = { by lemma 55 R->L }
% 12.60/2.08 complement(join(complement(X), meet(complement(Y), complement(X))))
% 12.60/2.08 = { by lemma 34 R->L }
% 12.60/2.08 join(zero, complement(join(complement(X), meet(complement(Y), complement(X)))))
% 12.60/2.08 = { by lemma 15 R->L }
% 12.60/2.08 join(complement(top), complement(join(complement(X), meet(complement(Y), complement(X)))))
% 12.60/2.08 = { by lemma 35 R->L }
% 12.60/2.08 join(complement(join(complement(complement(Y)), top)), complement(join(complement(X), meet(complement(Y), complement(X)))))
% 12.60/2.08 = { by lemma 16 R->L }
% 12.60/2.08 join(complement(join(complement(X), join(complement(complement(Y)), complement(complement(X))))), complement(join(complement(X), meet(complement(Y), complement(X)))))
% 12.60/2.08 = { by lemma 56 }
% 12.60/2.08 join(complement(join(complement(X), complement(meet(complement(Y), complement(X))))), complement(join(complement(X), meet(complement(Y), complement(X)))))
% 12.60/2.08 = { by lemma 46 R->L }
% 12.60/2.08 join(complement(join(complement(X), complement(meet(complement(X), complement(Y))))), complement(join(complement(X), meet(complement(Y), complement(X)))))
% 12.60/2.08 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 12.60/2.08 join(meet(X, meet(complement(X), complement(Y))), complement(join(complement(X), meet(complement(Y), complement(X)))))
% 12.60/2.08 = { by lemma 46 R->L }
% 12.60/2.08 join(meet(X, meet(complement(Y), complement(X))), complement(join(complement(X), meet(complement(Y), complement(X)))))
% 12.60/2.08 = { by lemma 24 }
% 12.60/2.08 X
% 12.60/2.08
% 12.60/2.08 Lemma 61: meet(X, join(Y, X)) = X.
% 12.60/2.08 Proof:
% 12.60/2.08 meet(X, join(Y, X))
% 12.60/2.08 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.08 meet(X, join(X, Y))
% 12.60/2.08 = { by lemma 60 }
% 12.60/2.08 X
% 12.60/2.08
% 12.60/2.08 Lemma 62: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 12.60/2.08 Proof:
% 12.60/2.08 meet(Y, meet(X, Z))
% 12.60/2.08 = { by lemma 46 }
% 12.60/2.08 meet(Y, meet(Z, X))
% 12.60/2.08 = { by lemma 48 R->L }
% 12.60/2.08 meet(meet(Y, meet(Z, X)), top)
% 12.60/2.08 = { by lemma 47 R->L }
% 12.60/2.08 complement(join(zero, complement(meet(Y, meet(Z, X)))))
% 12.60/2.08 = { by lemma 46 }
% 12.60/2.08 complement(join(zero, complement(meet(Y, meet(X, Z)))))
% 12.60/2.08 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 12.60/2.08 complement(join(zero, complement(meet(Y, complement(join(complement(X), complement(Z)))))))
% 12.60/2.08 = { by lemma 59 }
% 12.60/2.08 complement(join(zero, join(join(complement(X), complement(Z)), complement(Y))))
% 12.60/2.08 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 12.60/2.08 complement(join(zero, join(complement(X), join(complement(Z), complement(Y)))))
% 12.60/2.08 = { by lemma 56 }
% 12.60/2.08 complement(join(zero, join(complement(X), complement(meet(Z, Y)))))
% 12.60/2.08 = { by lemma 56 }
% 12.60/2.08 complement(join(zero, complement(meet(X, meet(Z, Y)))))
% 12.60/2.08 = { by lemma 46 R->L }
% 12.60/2.08 complement(join(zero, complement(meet(X, meet(Y, Z)))))
% 12.60/2.08 = { by lemma 47 }
% 12.60/2.08 meet(meet(X, meet(Y, Z)), top)
% 12.60/2.08 = { by lemma 48 }
% 12.60/2.08 meet(X, meet(Y, Z))
% 12.60/2.08
% 12.60/2.08 Lemma 63: meet(meet(X, Y), Z) = meet(X, meet(Z, Y)).
% 12.60/2.08 Proof:
% 12.60/2.08 meet(meet(X, Y), Z)
% 12.60/2.08 = { by lemma 46 }
% 12.60/2.08 meet(Z, meet(X, Y))
% 12.60/2.08 = { by lemma 62 R->L }
% 12.60/2.08 meet(X, meet(Z, Y))
% 12.60/2.08
% 12.60/2.08 Lemma 64: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 12.60/2.08 Proof:
% 12.60/2.08 converse(composition(X, converse(Y)))
% 12.60/2.08 = { by axiom 9 (converse_multiplicativity_10) }
% 12.60/2.08 composition(converse(converse(Y)), converse(X))
% 12.60/2.08 = { by axiom 1 (converse_idempotence_8) }
% 12.60/2.08 composition(Y, converse(X))
% 12.60/2.08
% 12.60/2.08 Lemma 65: join(meet(X, Y), meet(X, complement(Y))) = X.
% 12.60/2.08 Proof:
% 12.60/2.08 join(meet(X, Y), meet(X, complement(Y)))
% 12.60/2.08 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.08 join(meet(X, complement(Y)), meet(X, Y))
% 12.60/2.08 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 12.60/2.08 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 12.60/2.08 = { by lemma 24 }
% 12.60/2.08 X
% 12.60/2.08
% 12.60/2.08 Lemma 66: join(meet(X, Y), meet(complement(Y), X)) = X.
% 12.60/2.08 Proof:
% 12.60/2.08 join(meet(X, Y), meet(complement(Y), X))
% 12.60/2.08 = { by lemma 46 }
% 12.60/2.08 join(meet(X, Y), meet(X, complement(Y)))
% 12.60/2.08 = { by lemma 65 }
% 12.60/2.08 X
% 12.60/2.08
% 12.60/2.08 Lemma 67: meet(complement(X), join(X, Y)) = meet(Y, complement(X)).
% 12.60/2.08 Proof:
% 12.60/2.08 meet(complement(X), join(X, Y))
% 12.60/2.08 = { by lemma 44 R->L }
% 12.60/2.08 meet(complement(X), join(X, complement(complement(Y))))
% 12.60/2.08 = { by lemma 66 R->L }
% 12.60/2.08 join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(Y), meet(complement(X), join(X, complement(complement(Y))))))
% 12.60/2.08 = { by lemma 46 }
% 12.60/2.08 join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(Y), meet(join(X, complement(complement(Y))), complement(X))))
% 12.60/2.08 = { by lemma 62 }
% 12.60/2.08 join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(join(X, complement(complement(Y))), meet(complement(Y), complement(X))))
% 12.60/2.08 = { by lemma 58 R->L }
% 12.60/2.08 join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(join(X, complement(complement(Y))), complement(join(X, complement(complement(Y))))))
% 12.60/2.08 = { by axiom 5 (def_zero_13) R->L }
% 12.60/2.08 join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), zero)
% 12.60/2.08 = { by lemma 33 }
% 12.60/2.08 meet(meet(complement(X), join(X, complement(complement(Y)))), Y)
% 12.60/2.08 = { by lemma 63 }
% 12.60/2.08 meet(complement(X), meet(Y, join(X, complement(complement(Y)))))
% 12.60/2.08 = { by lemma 44 }
% 12.60/2.08 meet(complement(X), meet(Y, join(X, Y)))
% 12.60/2.08 = { by lemma 61 }
% 12.60/2.08 meet(complement(X), Y)
% 12.60/2.08 = { by lemma 46 R->L }
% 12.60/2.08 meet(Y, complement(X))
% 12.60/2.08
% 12.60/2.08 Lemma 68: meet(X, join(complement(X), Y)) = meet(X, Y).
% 12.60/2.08 Proof:
% 12.60/2.08 meet(X, join(complement(X), Y))
% 12.60/2.08 = { by lemma 34 R->L }
% 12.60/2.08 meet(X, join(zero, join(complement(X), Y)))
% 12.60/2.08 = { by axiom 8 (maddux2_join_associativity_2) }
% 12.60/2.08 meet(X, join(join(zero, complement(X)), Y))
% 12.60/2.08 = { by lemma 48 R->L }
% 12.60/2.08 meet(X, meet(join(join(zero, complement(X)), Y), top))
% 12.60/2.08 = { by lemma 63 R->L }
% 12.60/2.08 meet(meet(X, top), join(join(zero, complement(X)), Y))
% 12.60/2.08 = { by lemma 47 R->L }
% 12.60/2.08 meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), Y))
% 12.60/2.09 = { by lemma 67 }
% 12.60/2.09 meet(Y, complement(join(zero, complement(X))))
% 12.60/2.09 = { by lemma 47 }
% 12.60/2.09 meet(Y, meet(X, top))
% 12.60/2.09 = { by lemma 48 }
% 12.60/2.09 meet(Y, X)
% 12.60/2.09 = { by lemma 46 R->L }
% 12.60/2.09 meet(X, Y)
% 12.60/2.09
% 12.60/2.09 Lemma 69: join(join(X, composition(sk1, Y)), Y) = join(X, Y).
% 12.60/2.09 Proof:
% 12.60/2.09 join(join(X, composition(sk1, Y)), Y)
% 12.60/2.09 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.09 join(Y, join(X, composition(sk1, Y)))
% 12.60/2.09 = { by lemma 54 }
% 12.60/2.09 join(X, join(composition(sk1, Y), Y))
% 12.60/2.09 = { by lemma 19 R->L }
% 12.60/2.09 join(X, join(composition(sk1, Y), composition(join(sk1, one), Y)))
% 12.60/2.09 = { by axiom 3 (goals_14) }
% 12.60/2.09 join(X, join(composition(sk1, Y), composition(one, Y)))
% 12.60/2.09 = { by axiom 12 (composition_distributivity_7) R->L }
% 12.60/2.09 join(X, composition(join(sk1, one), Y))
% 12.60/2.09 = { by lemma 19 }
% 12.60/2.09 join(X, Y)
% 12.60/2.09
% 12.60/2.09 Lemma 70: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 12.60/2.09 Proof:
% 12.60/2.09 join(meet(X, Y), meet(Y, complement(X)))
% 12.60/2.09 = { by lemma 46 }
% 12.60/2.09 join(meet(Y, X), meet(Y, complement(X)))
% 12.60/2.09 = { by lemma 65 }
% 12.60/2.09 Y
% 12.60/2.09
% 12.60/2.09 Lemma 71: meet(join(X, Y), join(X, complement(Y))) = X.
% 12.60/2.09 Proof:
% 12.60/2.09 meet(join(X, Y), join(X, complement(Y)))
% 12.60/2.09 = { by lemma 46 }
% 12.60/2.09 meet(join(X, complement(Y)), join(X, Y))
% 12.60/2.09 = { by lemma 44 R->L }
% 12.60/2.09 meet(join(X, complement(Y)), join(X, complement(complement(Y))))
% 12.60/2.09 = { by lemma 59 R->L }
% 12.60/2.09 meet(complement(meet(Y, complement(X))), join(X, complement(complement(Y))))
% 12.60/2.09 = { by lemma 46 R->L }
% 12.60/2.09 meet(complement(meet(complement(X), Y)), join(X, complement(complement(Y))))
% 12.60/2.09 = { by lemma 46 }
% 12.60/2.09 meet(join(X, complement(complement(Y))), complement(meet(complement(X), Y)))
% 12.60/2.09 = { by lemma 58 R->L }
% 12.60/2.09 complement(join(meet(complement(X), Y), complement(join(X, complement(complement(Y))))))
% 12.60/2.09 = { by lemma 58 }
% 12.60/2.09 complement(join(meet(complement(X), Y), meet(complement(Y), complement(X))))
% 12.60/2.09 = { by lemma 66 }
% 12.60/2.09 complement(complement(X))
% 12.60/2.09 = { by lemma 44 }
% 12.60/2.09 X
% 12.60/2.09
% 12.60/2.09 Lemma 72: join(meet(X, Y), complement(join(Y, complement(X)))) = X.
% 12.60/2.09 Proof:
% 12.60/2.09 join(meet(X, Y), complement(join(Y, complement(X))))
% 12.60/2.09 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.09 join(meet(X, Y), complement(join(complement(X), Y)))
% 12.60/2.09 = { by lemma 24 }
% 12.60/2.09 X
% 12.60/2.09
% 12.60/2.09 Lemma 73: join(composition(X, top), composition(composition(X, Y), top)) = composition(X, top).
% 12.60/2.09 Proof:
% 12.60/2.09 join(composition(X, top), composition(composition(X, Y), top))
% 12.60/2.09 = { by lemma 39 R->L }
% 12.60/2.09 join(composition(X, top), composition(composition(X, Y), join(zero, join(complement(zero), meet(complement(zero), complement(zero))))))
% 12.60/2.09 = { by lemma 26 }
% 12.60/2.09 join(composition(X, top), composition(composition(X, Y), join(complement(zero), complement(zero))))
% 12.60/2.09 = { by lemma 21 }
% 12.60/2.09 join(composition(X, top), composition(composition(X, Y), complement(zero)))
% 12.60/2.09 = { by axiom 1 (converse_idempotence_8) R->L }
% 12.60/2.09 join(composition(X, top), composition(converse(converse(composition(X, Y))), complement(zero)))
% 12.60/2.09 = { by lemma 44 R->L }
% 12.60/2.09 join(complement(complement(composition(X, top))), composition(converse(converse(composition(X, Y))), complement(zero)))
% 12.60/2.09 = { by lemma 51 R->L }
% 12.60/2.09 join(complement(complement(composition(X, top))), composition(converse(converse(composition(X, Y))), complement(composition(top, zero))))
% 12.60/2.09 = { by lemma 42 R->L }
% 12.60/2.09 join(complement(complement(composition(X, top))), composition(converse(converse(composition(X, Y))), complement(composition(join(top, converse(Y)), zero))))
% 12.60/2.09 = { by axiom 12 (composition_distributivity_7) }
% 12.60/2.09 join(complement(complement(composition(X, top))), composition(converse(converse(composition(X, Y))), complement(join(composition(top, zero), composition(converse(Y), zero)))))
% 12.60/2.09 = { by lemma 51 }
% 12.60/2.09 join(complement(complement(composition(X, top))), composition(converse(converse(composition(X, Y))), complement(join(zero, composition(converse(Y), zero)))))
% 12.60/2.09 = { by lemma 34 }
% 12.60/2.09 join(complement(complement(composition(X, top))), composition(converse(converse(composition(X, Y))), complement(composition(converse(Y), zero))))
% 12.60/2.09 = { by lemma 15 R->L }
% 12.60/2.09 join(complement(complement(composition(X, top))), composition(converse(converse(composition(X, Y))), complement(composition(converse(Y), complement(top)))))
% 12.60/2.09 = { by lemma 20 R->L }
% 12.60/2.09 join(complement(complement(composition(X, top))), composition(converse(converse(composition(X, Y))), complement(composition(converse(Y), join(complement(top), composition(converse(X), complement(composition(X, top))))))))
% 12.60/2.09 = { by lemma 15 }
% 12.60/2.09 join(complement(complement(composition(X, top))), composition(converse(converse(composition(X, Y))), complement(composition(converse(Y), join(zero, composition(converse(X), complement(composition(X, top))))))))
% 12.60/2.09 = { by lemma 34 }
% 12.60/2.09 join(complement(complement(composition(X, top))), composition(converse(converse(composition(X, Y))), complement(composition(converse(Y), composition(converse(X), complement(composition(X, top)))))))
% 12.60/2.09 = { by axiom 10 (composition_associativity_5) }
% 12.60/2.09 join(complement(complement(composition(X, top))), composition(converse(converse(composition(X, Y))), complement(composition(composition(converse(Y), converse(X)), complement(composition(X, top))))))
% 12.60/2.09 = { by axiom 9 (converse_multiplicativity_10) R->L }
% 12.60/2.09 join(complement(complement(composition(X, top))), composition(converse(converse(composition(X, Y))), complement(composition(converse(composition(X, Y)), complement(composition(X, top))))))
% 12.60/2.09 = { by lemma 20 }
% 12.60/2.09 complement(complement(composition(X, top)))
% 12.60/2.09 = { by lemma 44 }
% 12.60/2.09 composition(X, top)
% 12.60/2.09
% 12.60/2.09 Lemma 74: meet(join(meet(composition(X, top), Y), composition(X, Z)), composition(X, top)) = join(meet(composition(X, top), Y), composition(X, Z)).
% 12.60/2.09 Proof:
% 12.60/2.09 meet(join(meet(composition(X, top), Y), composition(X, Z)), composition(X, top))
% 12.60/2.09 = { by lemma 73 R->L }
% 12.60/2.09 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(composition(X, top), composition(composition(X, Z), top)))
% 12.60/2.09 = { by lemma 42 R->L }
% 12.60/2.09 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(composition(X, top), composition(composition(X, Z), join(top, join(sk1, one)))))
% 12.60/2.09 = { by lemma 29 R->L }
% 12.60/2.09 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(composition(X, top), composition(composition(X, Z), join(top, converse(join(sk1, one))))))
% 12.60/2.09 = { by lemma 37 R->L }
% 12.60/2.09 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(composition(X, top), composition(composition(X, Z), converse(join(converse(top), join(sk1, one))))))
% 12.60/2.09 = { by lemma 64 R->L }
% 12.60/2.09 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(composition(X, top), converse(composition(join(converse(top), join(sk1, one)), converse(composition(X, Z))))))
% 12.60/2.09 = { by lemma 28 }
% 12.60/2.09 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(composition(X, top), converse(join(converse(composition(X, Z)), composition(converse(top), converse(composition(X, Z)))))))
% 12.60/2.09 = { by lemma 37 }
% 12.60/2.09 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(composition(X, top), join(composition(X, Z), converse(composition(converse(top), converse(composition(X, Z)))))))
% 12.60/2.09 = { by lemma 64 }
% 12.60/2.09 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(composition(X, top), join(composition(X, Z), composition(composition(X, Z), converse(converse(top))))))
% 12.60/2.09 = { by axiom 1 (converse_idempotence_8) }
% 12.60/2.09 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(composition(X, top), join(composition(X, Z), composition(composition(X, Z), top))))
% 12.60/2.09 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.09 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(composition(X, top), join(composition(composition(X, Z), top), composition(X, Z))))
% 12.60/2.09 = { by axiom 8 (maddux2_join_associativity_2) }
% 12.60/2.09 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(join(composition(X, top), composition(composition(X, Z), top)), composition(X, Z)))
% 12.60/2.09 = { by lemma 73 }
% 12.60/2.09 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(composition(X, top), composition(X, Z)))
% 12.60/2.09 = { by lemma 72 R->L }
% 12.60/2.09 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(join(meet(composition(X, top), Y), complement(join(Y, complement(composition(X, top))))), composition(X, Z)))
% 12.60/2.09 = { by lemma 52 R->L }
% 12.60/2.09 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(join(meet(composition(X, top), Y), join(meet(composition(X, top), Y), complement(join(Y, complement(composition(X, top)))))), composition(X, Z)))
% 12.60/2.09 = { by lemma 72 }
% 12.60/2.09 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(join(meet(composition(X, top), Y), composition(X, top)), composition(X, Z)))
% 12.60/2.09 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 12.60/2.09 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(meet(composition(X, top), Y), join(composition(X, top), composition(X, Z))))
% 12.60/2.09 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.60/2.09 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(meet(composition(X, top), Y), join(composition(X, Z), composition(X, top))))
% 12.60/2.09 = { by lemma 54 R->L }
% 12.60/2.09 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(composition(X, top), join(meet(composition(X, top), Y), composition(X, Z))))
% 12.60/2.09 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.60/2.09 meet(join(meet(composition(X, top), Y), composition(X, Z)), join(join(meet(composition(X, top), Y), composition(X, Z)), composition(X, top)))
% 12.60/2.09 = { by lemma 60 }
% 12.60/2.09 join(meet(composition(X, top), Y), composition(X, Z))
% 12.60/2.09
% 12.60/2.09 Lemma 75: join(join(meet(X, Y), composition(sk1, Y)), Y) = Y.
% 12.60/2.09 Proof:
% 12.60/2.09 join(join(meet(X, Y), composition(sk1, Y)), Y)
% 12.60/2.09 = { by lemma 69 }
% 12.60/2.09 join(meet(X, Y), Y)
% 12.60/2.09 = { by lemma 70 R->L }
% 12.60/2.09 join(meet(X, Y), join(meet(X, Y), meet(Y, complement(X))))
% 12.60/2.09 = { by lemma 52 }
% 12.60/2.09 join(meet(X, Y), meet(Y, complement(X)))
% 12.60/2.09 = { by lemma 70 }
% 12.60/2.09 Y
% 12.60/2.09
% 12.60/2.09 Lemma 76: join(meet(composition(sk1, top), X), composition(sk1, X)) = meet(composition(sk1, top), X).
% 12.60/2.09 Proof:
% 12.60/2.09 join(meet(composition(sk1, top), X), composition(sk1, X))
% 12.60/2.09 = { by lemma 74 R->L }
% 12.60/2.09 meet(join(meet(composition(sk1, top), X), composition(sk1, X)), composition(sk1, top))
% 12.60/2.09 = { by lemma 46 }
% 12.60/2.09 meet(composition(sk1, top), join(meet(composition(sk1, top), X), composition(sk1, X)))
% 12.60/2.09 = { by lemma 60 R->L }
% 12.60/2.09 meet(composition(sk1, top), meet(join(meet(composition(sk1, top), X), composition(sk1, X)), join(join(meet(composition(sk1, top), X), composition(sk1, X)), X)))
% 12.60/2.09 = { by lemma 75 }
% 12.60/2.09 meet(composition(sk1, top), meet(join(meet(composition(sk1, top), X), composition(sk1, X)), X))
% 12.60/2.09 = { by lemma 62 }
% 12.60/2.09 meet(join(meet(composition(sk1, top), X), composition(sk1, X)), meet(composition(sk1, top), X))
% 12.60/2.09 = { by lemma 46 }
% 12.60/2.09 meet(meet(composition(sk1, top), X), join(meet(composition(sk1, top), X), composition(sk1, X)))
% 12.60/2.09 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 12.60/2.09 complement(join(complement(meet(composition(sk1, top), X)), complement(join(meet(composition(sk1, top), X), composition(sk1, X)))))
% 12.60/2.09 = { by lemma 34 R->L }
% 12.60/2.09 join(zero, complement(join(complement(meet(composition(sk1, top), X)), complement(join(meet(composition(sk1, top), X), composition(sk1, X))))))
% 12.60/2.09 = { by lemma 15 R->L }
% 12.60/2.09 join(complement(top), complement(join(complement(meet(composition(sk1, top), X)), complement(join(meet(composition(sk1, top), X), composition(sk1, X))))))
% 12.60/2.09 = { by lemma 35 R->L }
% 12.60/2.09 join(complement(join(composition(sk1, X), top)), complement(join(complement(meet(composition(sk1, top), X)), complement(join(meet(composition(sk1, top), X), composition(sk1, X))))))
% 12.60/2.09 = { by lemma 16 R->L }
% 12.60/2.09 join(complement(join(meet(composition(sk1, top), X), join(composition(sk1, X), complement(meet(composition(sk1, top), X))))), complement(join(complement(meet(composition(sk1, top), X)), complement(join(meet(composition(sk1, top), X), composition(sk1, X))))))
% 12.60/2.09 = { by lemma 54 R->L }
% 12.60/2.09 join(complement(join(complement(meet(composition(sk1, top), X)), join(meet(composition(sk1, top), X), composition(sk1, X)))), complement(join(complement(meet(composition(sk1, top), X)), complement(join(meet(composition(sk1, top), X), composition(sk1, X))))))
% 12.60/2.09 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.60/2.09 join(complement(join(join(meet(composition(sk1, top), X), composition(sk1, X)), complement(meet(composition(sk1, top), X)))), complement(join(complement(meet(composition(sk1, top), X)), complement(join(meet(composition(sk1, top), X), composition(sk1, X))))))
% 12.60/2.09 = { by lemma 58 }
% 12.60/2.09 join(meet(meet(composition(sk1, top), X), complement(join(meet(composition(sk1, top), X), composition(sk1, X)))), complement(join(complement(meet(composition(sk1, top), X)), complement(join(meet(composition(sk1, top), X), composition(sk1, X))))))
% 12.60/2.09 = { by lemma 24 }
% 12.60/2.10 meet(composition(sk1, top), X)
% 12.60/2.10
% 12.60/2.10 Lemma 77: join(composition(X, Y), composition(X, complement(join(meet(composition(sk1, top), Y), composition(sk1, Y))))) = composition(X, top).
% 12.60/2.10 Proof:
% 12.60/2.10 join(composition(X, Y), composition(X, complement(join(meet(composition(sk1, top), Y), composition(sk1, Y)))))
% 12.60/2.10 = { by lemma 76 }
% 12.60/2.10 join(composition(X, Y), composition(X, complement(meet(composition(sk1, top), Y))))
% 12.60/2.10 = { by axiom 1 (converse_idempotence_8) R->L }
% 12.60/2.10 join(composition(X, Y), composition(X, converse(converse(complement(meet(composition(sk1, top), Y))))))
% 12.60/2.10 = { by lemma 45 R->L }
% 12.60/2.10 join(composition(X, Y), composition(X, converse(meet(converse(complement(meet(composition(sk1, top), Y))), converse(complement(meet(composition(sk1, top), Y)))))))
% 12.60/2.10 = { by lemma 43 R->L }
% 12.60/2.10 join(composition(X, Y), composition(X, converse(complement(complement(converse(complement(meet(composition(sk1, top), Y))))))))
% 12.60/2.10 = { by axiom 1 (converse_idempotence_8) R->L }
% 12.60/2.10 join(composition(X, Y), composition(X, converse(complement(converse(converse(complement(converse(complement(meet(composition(sk1, top), Y))))))))))
% 12.60/2.10 = { by lemma 70 R->L }
% 12.60/2.10 join(composition(X, Y), composition(X, converse(complement(converse(join(meet(meet(composition(sk1, top), Y), converse(complement(converse(complement(meet(composition(sk1, top), Y)))))), meet(converse(complement(converse(complement(meet(composition(sk1, top), Y))))), complement(meet(composition(sk1, top), Y)))))))))
% 12.60/2.10 = { by lemma 46 R->L }
% 12.60/2.10 join(composition(X, Y), composition(X, converse(complement(converse(join(meet(meet(composition(sk1, top), Y), converse(complement(converse(complement(meet(composition(sk1, top), Y)))))), meet(complement(meet(composition(sk1, top), Y)), converse(complement(converse(complement(meet(composition(sk1, top), Y))))))))))))
% 12.60/2.10 = { by lemma 44 R->L }
% 12.60/2.10 join(composition(X, Y), composition(X, converse(complement(converse(join(meet(meet(composition(sk1, top), Y), converse(complement(converse(complement(meet(composition(sk1, top), Y)))))), meet(complement(meet(composition(sk1, top), Y)), converse(complement(converse(complement(complement(complement(meet(composition(sk1, top), Y))))))))))))))
% 12.60/2.10 = { by lemma 68 R->L }
% 12.60/2.10 join(composition(X, Y), composition(X, converse(complement(converse(join(meet(meet(composition(sk1, top), Y), converse(complement(converse(complement(meet(composition(sk1, top), Y)))))), meet(complement(meet(composition(sk1, top), Y)), join(complement(complement(meet(composition(sk1, top), Y))), converse(complement(converse(complement(complement(complement(meet(composition(sk1, top), Y)))))))))))))))
% 12.60/2.10 = { by axiom 1 (converse_idempotence_8) R->L }
% 12.60/2.10 join(composition(X, Y), composition(X, converse(complement(converse(join(meet(meet(composition(sk1, top), Y), converse(complement(converse(complement(meet(composition(sk1, top), Y)))))), meet(complement(meet(composition(sk1, top), Y)), join(complement(complement(meet(composition(sk1, top), Y))), converse(complement(converse(complement(converse(converse(complement(complement(meet(composition(sk1, top), Y)))))))))))))))))
% 12.60/2.10 = { by lemma 37 R->L }
% 12.60/2.10 join(composition(X, Y), composition(X, converse(complement(converse(join(meet(meet(composition(sk1, top), Y), converse(complement(converse(complement(meet(composition(sk1, top), Y)))))), meet(complement(meet(composition(sk1, top), Y)), converse(join(converse(complement(complement(meet(composition(sk1, top), Y)))), complement(converse(complement(converse(converse(complement(complement(meet(composition(sk1, top), Y)))))))))))))))))
% 12.60/2.10 = { by lemma 49 R->L }
% 12.60/2.10 join(composition(X, Y), composition(X, converse(complement(converse(join(meet(meet(composition(sk1, top), Y), converse(complement(converse(complement(meet(composition(sk1, top), Y)))))), meet(complement(meet(composition(sk1, top), Y)), converse(meet(top, join(converse(complement(complement(meet(composition(sk1, top), Y)))), complement(converse(complement(converse(converse(complement(complement(meet(composition(sk1, top), Y))))))))))))))))))
% 12.60/2.10 = { by lemma 41 R->L }
% 12.60/2.10 join(composition(X, Y), composition(X, converse(complement(converse(join(meet(meet(composition(sk1, top), Y), converse(complement(converse(complement(meet(composition(sk1, top), Y)))))), meet(complement(meet(composition(sk1, top), Y)), converse(meet(converse(top), join(converse(complement(complement(meet(composition(sk1, top), Y)))), complement(converse(complement(converse(converse(complement(complement(meet(composition(sk1, top), Y))))))))))))))))))
% 12.60/2.10 = { by lemma 38 R->L }
% 12.60/2.10 join(composition(X, Y), composition(X, converse(complement(converse(join(meet(meet(composition(sk1, top), Y), converse(complement(converse(complement(meet(composition(sk1, top), Y)))))), meet(complement(meet(composition(sk1, top), Y)), converse(meet(join(converse(complement(complement(meet(composition(sk1, top), Y)))), converse(complement(converse(converse(complement(complement(meet(composition(sk1, top), Y)))))))), join(converse(complement(complement(meet(composition(sk1, top), Y)))), complement(converse(complement(converse(converse(complement(complement(meet(composition(sk1, top), Y))))))))))))))))))
% 12.60/2.10 = { by lemma 71 }
% 12.60/2.10 join(composition(X, Y), composition(X, converse(complement(converse(join(meet(meet(composition(sk1, top), Y), converse(complement(converse(complement(meet(composition(sk1, top), Y)))))), meet(complement(meet(composition(sk1, top), Y)), converse(converse(complement(complement(meet(composition(sk1, top), Y))))))))))))
% 12.60/2.10 = { by axiom 1 (converse_idempotence_8) }
% 12.60/2.10 join(composition(X, Y), composition(X, converse(complement(converse(join(meet(meet(composition(sk1, top), Y), converse(complement(converse(complement(meet(composition(sk1, top), Y)))))), meet(complement(meet(composition(sk1, top), Y)), complement(complement(meet(composition(sk1, top), Y))))))))))
% 12.60/2.10 = { by axiom 5 (def_zero_13) R->L }
% 12.60/2.10 join(composition(X, Y), composition(X, converse(complement(converse(join(meet(meet(composition(sk1, top), Y), converse(complement(converse(complement(meet(composition(sk1, top), Y)))))), zero))))))
% 12.60/2.10 = { by lemma 33 }
% 12.60/2.10 join(composition(X, Y), composition(X, converse(complement(converse(meet(meet(composition(sk1, top), Y), converse(complement(converse(complement(meet(composition(sk1, top), Y)))))))))))
% 12.60/2.10 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 12.60/2.10 join(composition(X, Y), composition(X, converse(complement(converse(complement(join(complement(meet(composition(sk1, top), Y)), complement(converse(complement(converse(complement(meet(composition(sk1, top), Y)))))))))))))
% 12.60/2.10 = { by lemma 34 R->L }
% 12.60/2.10 join(composition(X, Y), composition(X, converse(complement(converse(join(zero, complement(join(complement(meet(composition(sk1, top), Y)), complement(converse(complement(converse(complement(meet(composition(sk1, top), Y))))))))))))))
% 12.60/2.10 = { by lemma 15 R->L }
% 12.60/2.10 join(composition(X, Y), composition(X, converse(complement(converse(join(complement(top), complement(join(complement(meet(composition(sk1, top), Y)), complement(converse(complement(converse(complement(meet(composition(sk1, top), Y))))))))))))))
% 12.60/2.10 = { by lemma 41 R->L }
% 12.60/2.10 join(composition(X, Y), composition(X, converse(complement(converse(join(complement(converse(top)), complement(join(complement(meet(composition(sk1, top), Y)), complement(converse(complement(converse(complement(meet(composition(sk1, top), Y))))))))))))))
% 12.60/2.10 = { by lemma 38 R->L }
% 12.60/2.10 join(composition(X, Y), composition(X, converse(complement(converse(join(complement(join(complement(meet(composition(sk1, top), Y)), converse(complement(converse(complement(meet(composition(sk1, top), Y))))))), complement(join(complement(meet(composition(sk1, top), Y)), complement(converse(complement(converse(complement(meet(composition(sk1, top), Y))))))))))))))
% 12.60/2.10 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.10 join(composition(X, Y), composition(X, converse(complement(converse(join(complement(join(converse(complement(converse(complement(meet(composition(sk1, top), Y))))), complement(meet(composition(sk1, top), Y)))), complement(join(complement(meet(composition(sk1, top), Y)), complement(converse(complement(converse(complement(meet(composition(sk1, top), Y))))))))))))))
% 12.60/2.10 = { by lemma 58 }
% 12.60/2.10 join(composition(X, Y), composition(X, converse(complement(converse(join(meet(meet(composition(sk1, top), Y), complement(converse(complement(converse(complement(meet(composition(sk1, top), Y))))))), complement(join(complement(meet(composition(sk1, top), Y)), complement(converse(complement(converse(complement(meet(composition(sk1, top), Y))))))))))))))
% 12.60/2.10 = { by lemma 24 }
% 12.60/2.10 join(composition(X, Y), composition(X, converse(complement(converse(meet(composition(sk1, top), Y))))))
% 12.60/2.10 = { by axiom 1 (converse_idempotence_8) R->L }
% 12.60/2.10 converse(converse(join(composition(X, Y), composition(X, converse(complement(converse(meet(composition(sk1, top), Y))))))))
% 12.60/2.10 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.10 converse(converse(join(composition(X, converse(complement(converse(meet(composition(sk1, top), Y))))), composition(X, Y))))
% 12.60/2.10 = { by axiom 7 (converse_additivity_9) }
% 12.60/2.10 converse(join(converse(composition(X, converse(complement(converse(meet(composition(sk1, top), Y)))))), converse(composition(X, Y))))
% 12.60/2.10 = { by lemma 64 }
% 12.60/2.10 converse(join(composition(complement(converse(meet(composition(sk1, top), Y))), converse(X)), converse(composition(X, Y))))
% 12.60/2.10 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.60/2.10 converse(join(converse(composition(X, Y)), composition(complement(converse(meet(composition(sk1, top), Y))), converse(X))))
% 12.60/2.10 = { by axiom 9 (converse_multiplicativity_10) }
% 12.60/2.10 converse(join(composition(converse(Y), converse(X)), composition(complement(converse(meet(composition(sk1, top), Y))), converse(X))))
% 12.60/2.10 = { by axiom 12 (composition_distributivity_7) R->L }
% 12.60/2.10 converse(composition(join(converse(Y), complement(converse(meet(composition(sk1, top), Y)))), converse(X)))
% 12.60/2.10 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.10 converse(composition(join(complement(converse(meet(composition(sk1, top), Y))), converse(Y)), converse(X)))
% 12.60/2.10 = { by lemma 36 R->L }
% 12.60/2.10 converse(composition(converse(join(Y, converse(complement(converse(meet(composition(sk1, top), Y)))))), converse(X)))
% 12.60/2.10 = { by axiom 9 (converse_multiplicativity_10) R->L }
% 12.60/2.10 converse(converse(composition(X, join(Y, converse(complement(converse(meet(composition(sk1, top), Y))))))))
% 12.60/2.10 = { by axiom 1 (converse_idempotence_8) }
% 12.60/2.10 composition(X, join(Y, converse(complement(converse(meet(composition(sk1, top), Y))))))
% 12.60/2.10 = { by lemma 37 R->L }
% 12.60/2.10 composition(X, converse(join(converse(Y), complement(converse(meet(composition(sk1, top), Y))))))
% 12.60/2.10 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.10 composition(X, converse(join(complement(converse(meet(composition(sk1, top), Y))), converse(Y))))
% 12.60/2.10 = { by lemma 75 R->L }
% 12.60/2.10 composition(X, converse(join(complement(converse(meet(composition(sk1, top), Y))), converse(join(join(meet(composition(sk1, top), Y), composition(sk1, Y)), Y)))))
% 12.60/2.10 = { by lemma 69 }
% 12.60/2.10 composition(X, converse(join(complement(converse(meet(composition(sk1, top), Y))), converse(join(meet(composition(sk1, top), Y), Y)))))
% 12.60/2.10 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.10 composition(X, converse(join(converse(join(meet(composition(sk1, top), Y), Y)), complement(converse(meet(composition(sk1, top), Y))))))
% 12.60/2.10 = { by axiom 7 (converse_additivity_9) }
% 12.60/2.10 composition(X, converse(join(join(converse(meet(composition(sk1, top), Y)), converse(Y)), complement(converse(meet(composition(sk1, top), Y))))))
% 12.60/2.10 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 12.60/2.10 composition(X, converse(join(converse(meet(composition(sk1, top), Y)), join(converse(Y), complement(converse(meet(composition(sk1, top), Y)))))))
% 12.60/2.10 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.60/2.10 composition(X, converse(join(converse(meet(composition(sk1, top), Y)), join(complement(converse(meet(composition(sk1, top), Y))), converse(Y)))))
% 12.60/2.10 = { by lemma 39 }
% 12.60/2.10 composition(X, converse(top))
% 12.60/2.10 = { by lemma 41 }
% 12.60/2.11 composition(X, top)
% 12.60/2.11
% 12.60/2.11 Goal 1 (goals_15): tuple(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), join(composition(sk1, sk2), meet(composition(sk1, top), sk2))) = tuple(composition(sk1, sk2), meet(composition(sk1, top), sk2)).
% 12.60/2.11 Proof:
% 12.60/2.11 tuple(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), join(composition(sk1, sk2), meet(composition(sk1, top), sk2)))
% 12.60/2.11 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.60/2.11 tuple(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 12.60/2.11 = { by lemma 74 R->L }
% 12.60/2.11 tuple(meet(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), composition(sk1, top)), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 12.60/2.11 = { by lemma 46 }
% 12.60/2.11 tuple(meet(composition(sk1, top), join(meet(composition(sk1, top), sk2), composition(sk1, sk2))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 12.60/2.11 = { by lemma 68 R->L }
% 12.60/2.11 tuple(meet(composition(sk1, top), join(complement(composition(sk1, top)), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 12.60/2.11 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.11 tuple(meet(composition(sk1, top), join(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), complement(composition(sk1, top)))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 12.60/2.11 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 12.60/2.11 tuple(meet(composition(sk1, top), join(meet(composition(sk1, top), sk2), join(composition(sk1, sk2), complement(composition(sk1, top))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 12.60/2.11 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.60/2.11 tuple(meet(composition(sk1, top), join(meet(composition(sk1, top), sk2), join(complement(composition(sk1, top)), composition(sk1, sk2)))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 12.60/2.11 = { by lemma 76 R->L }
% 12.60/2.11 tuple(meet(composition(sk1, top), join(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), join(complement(composition(sk1, top)), composition(sk1, sk2)))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 12.60/2.11 = { by axiom 8 (maddux2_join_associativity_2) }
% 12.60/2.11 tuple(meet(composition(sk1, top), join(join(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), complement(composition(sk1, top))), composition(sk1, sk2))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 12.60/2.11 = { by lemma 59 R->L }
% 12.60/2.11 tuple(meet(composition(sk1, top), join(complement(meet(composition(sk1, top), complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), composition(sk1, sk2))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 12.60/2.11 = { by lemma 77 R->L }
% 12.60/2.11 tuple(meet(join(composition(sk1, sk2), composition(sk1, complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), join(complement(meet(composition(sk1, top), complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), composition(sk1, sk2))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 12.60/2.11 = { by lemma 61 R->L }
% 12.60/2.11 tuple(meet(join(composition(sk1, sk2), meet(composition(sk1, complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))), join(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))), composition(sk1, complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))))), join(complement(meet(composition(sk1, top), complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), composition(sk1, sk2))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 12.60/2.11 = { by lemma 50 R->L }
% 12.60/2.11 tuple(meet(join(composition(sk1, sk2), meet(composition(sk1, complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))), composition(join(join(sk1, one), sk1), complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))))), join(complement(meet(composition(sk1, top), complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), composition(sk1, sk2))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 12.60/2.11 = { by lemma 31 }
% 12.60/2.11 tuple(meet(join(composition(sk1, sk2), meet(composition(sk1, complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))), composition(join(sk1, one), complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))))), join(complement(meet(composition(sk1, top), complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), composition(sk1, sk2))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 12.60/2.11 = { by lemma 19 }
% 12.60/2.11 tuple(meet(join(composition(sk1, sk2), meet(composition(sk1, complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))), complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), join(complement(meet(composition(sk1, top), complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), composition(sk1, sk2))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 12.60/2.11 = { by lemma 67 R->L }
% 12.60/2.11 tuple(meet(join(composition(sk1, sk2), meet(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))), join(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), composition(sk1, complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))))), join(complement(meet(composition(sk1, top), complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), composition(sk1, sk2))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 12.60/2.11 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.60/2.11 tuple(meet(join(composition(sk1, sk2), meet(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))), join(composition(sk1, complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), join(complement(meet(composition(sk1, top), complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), composition(sk1, sk2))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 12.60/2.11 = { by lemma 76 }
% 12.60/2.11 tuple(meet(join(composition(sk1, sk2), meet(complement(meet(composition(sk1, top), sk2)), join(composition(sk1, complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), join(complement(meet(composition(sk1, top), complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), composition(sk1, sk2))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 12.60/2.11 = { by lemma 53 }
% 12.60/2.11 tuple(meet(join(composition(sk1, sk2), meet(complement(meet(composition(sk1, top), sk2)), join(meet(composition(sk1, top), sk2), join(composition(sk1, complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))), composition(sk1, sk2))))), join(complement(meet(composition(sk1, top), complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), composition(sk1, sk2))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 12.60/2.11 = { by lemma 67 }
% 12.60/2.11 tuple(meet(join(composition(sk1, sk2), meet(join(composition(sk1, complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))), composition(sk1, sk2)), complement(meet(composition(sk1, top), sk2)))), join(complement(meet(composition(sk1, top), complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), composition(sk1, sk2))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 12.60/2.11 = { by lemma 46 R->L }
% 12.60/2.11 tuple(meet(join(composition(sk1, sk2), meet(complement(meet(composition(sk1, top), sk2)), join(composition(sk1, complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))), composition(sk1, sk2)))), join(complement(meet(composition(sk1, top), complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), composition(sk1, sk2))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 12.60/2.11 = { by lemma 76 R->L }
% 12.60/2.11 tuple(meet(join(composition(sk1, sk2), meet(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))), join(composition(sk1, complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))), composition(sk1, sk2)))), join(complement(meet(composition(sk1, top), complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), composition(sk1, sk2))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 12.60/2.11 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.60/2.11 tuple(meet(join(composition(sk1, sk2), meet(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))), join(composition(sk1, sk2), composition(sk1, complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))))), join(complement(meet(composition(sk1, top), complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), composition(sk1, sk2))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 12.60/2.11 = { by lemma 77 }
% 12.60/2.11 tuple(meet(join(composition(sk1, sk2), meet(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))), composition(sk1, top))), join(complement(meet(composition(sk1, top), complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), composition(sk1, sk2))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 12.60/2.11 = { by lemma 46 R->L }
% 12.60/2.11 tuple(meet(join(composition(sk1, sk2), meet(composition(sk1, top), complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), join(complement(meet(composition(sk1, top), complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), composition(sk1, sk2))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 12.60/2.11 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.60/2.11 tuple(meet(join(composition(sk1, sk2), meet(composition(sk1, top), complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), join(composition(sk1, sk2), complement(meet(composition(sk1, top), complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 12.60/2.11 = { by lemma 71 }
% 12.60/2.11 tuple(composition(sk1, sk2), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 12.60/2.11 = { by lemma 76 }
% 12.60/2.11 tuple(composition(sk1, sk2), meet(composition(sk1, top), sk2))
% 12.60/2.11 % SZS output end Proof
% 12.60/2.11
% 12.60/2.11 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------