TSTP Solution File: REL025-2 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL025-2 : 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 : n016.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 24.02s 3.43s
% Output : Proof 25.51s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : REL025-2 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.12 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.10/0.32 % Computer : n016.cluster.edu
% 0.10/0.32 % Model : x86_64 x86_64
% 0.10/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.32 % Memory : 8042.1875MB
% 0.10/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.10/0.32 % CPULimit : 300
% 0.10/0.32 % WCLimit : 300
% 0.10/0.32 % DateTime : Fri Aug 25 19:36:25 EDT 2023
% 0.10/0.32 % CPUTime :
% 24.02/3.43 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 24.02/3.43
% 24.02/3.43 % SZS status Unsatisfiable
% 24.02/3.43
% 24.80/3.55 % SZS output start Proof
% 24.80/3.55 Take the following subset of the input axioms:
% 24.80/3.56 fof(composition_associativity_5, axiom, ![A, B, C]: composition(A, composition(B, C))=composition(composition(A, B), C)).
% 24.80/3.56 fof(composition_distributivity_7, axiom, ![A2, B2, C2]: composition(join(A2, B2), C2)=join(composition(A2, C2), composition(B2, C2))).
% 24.80/3.56 fof(composition_identity_6, axiom, ![A2]: composition(A2, one)=A2).
% 24.80/3.56 fof(converse_additivity_9, axiom, ![A2, B2]: converse(join(A2, B2))=join(converse(A2), converse(B2))).
% 24.80/3.56 fof(converse_cancellativity_11, axiom, ![A2, B2]: join(composition(converse(A2), complement(composition(A2, B2))), complement(B2))=complement(B2)).
% 24.80/3.56 fof(converse_idempotence_8, axiom, ![A2]: converse(converse(A2))=A2).
% 24.80/3.56 fof(converse_multiplicativity_10, axiom, ![A2, B2]: converse(composition(A2, B2))=composition(converse(B2), converse(A2))).
% 24.80/3.56 fof(def_top_12, axiom, ![A2]: top=join(A2, complement(A2))).
% 24.80/3.56 fof(def_zero_13, axiom, ![A2]: zero=meet(A2, complement(A2))).
% 24.80/3.56 fof(goals_14, negated_conjecture, join(sk1, one)=one).
% 24.80/3.56 fof(goals_17, negated_conjecture, join(sk1, converse(sk1))!=converse(sk1) | join(converse(sk1), sk1)!=sk1).
% 24.80/3.56 fof(maddux1_join_commutativity_1, axiom, ![A2, B2]: join(A2, B2)=join(B2, A2)).
% 24.80/3.56 fof(maddux2_join_associativity_2, axiom, ![A2, B2, C2]: join(A2, join(B2, C2))=join(join(A2, B2), C2)).
% 24.80/3.56 fof(maddux3_a_kind_of_de_Morgan_3, axiom, ![A2, B2]: A2=join(complement(join(complement(A2), complement(B2))), complement(join(complement(A2), B2)))).
% 24.80/3.56 fof(maddux4_definiton_of_meet_4, axiom, ![A2, B2]: meet(A2, B2)=complement(join(complement(A2), complement(B2)))).
% 24.80/3.56
% 24.80/3.56 Now clausify the problem and encode Horn clauses using encoding 3 of
% 24.80/3.56 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 24.80/3.56 We repeatedly replace C & s=t => u=v by the two clauses:
% 24.80/3.56 fresh(y, y, x1...xn) = u
% 24.80/3.56 C => fresh(s, t, x1...xn) = v
% 24.80/3.56 where fresh is a fresh function symbol and x1..xn are the free
% 24.80/3.56 variables of u and v.
% 24.80/3.56 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 24.80/3.56 input problem has no model of domain size 1).
% 24.80/3.56
% 24.80/3.56 The encoding turns the above axioms into the following unit equations and goals:
% 24.80/3.56
% 24.80/3.56 Axiom 1 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 24.80/3.56 Axiom 2 (goals_14): join(sk1, one) = one.
% 24.80/3.56 Axiom 3 (converse_idempotence_8): converse(converse(X)) = X.
% 24.80/3.56 Axiom 4 (composition_identity_6): composition(X, one) = X.
% 24.80/3.56 Axiom 5 (def_top_12): top = join(X, complement(X)).
% 24.80/3.56 Axiom 6 (def_zero_13): zero = meet(X, complement(X)).
% 24.80/3.56 Axiom 7 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 24.80/3.56 Axiom 8 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 24.80/3.56 Axiom 9 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 24.80/3.56 Axiom 10 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 24.80/3.56 Axiom 11 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 24.80/3.56 Axiom 12 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 24.80/3.56 Axiom 13 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 24.80/3.56 Axiom 14 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 24.80/3.56
% 24.80/3.56 Lemma 15: complement(top) = zero.
% 24.80/3.56 Proof:
% 24.80/3.56 complement(top)
% 24.80/3.56 = { by axiom 5 (def_top_12) }
% 24.80/3.56 complement(join(complement(X), complement(complement(X))))
% 24.80/3.56 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 24.80/3.56 meet(X, complement(X))
% 24.80/3.56 = { by axiom 6 (def_zero_13) R->L }
% 24.80/3.56 zero
% 24.80/3.56
% 24.80/3.56 Lemma 16: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 24.80/3.56 Proof:
% 24.80/3.56 converse(composition(converse(X), Y))
% 24.80/3.56 = { by axiom 9 (converse_multiplicativity_10) }
% 24.80/3.56 composition(converse(Y), converse(converse(X)))
% 24.80/3.56 = { by axiom 3 (converse_idempotence_8) }
% 24.80/3.56 composition(converse(Y), X)
% 24.80/3.56
% 24.80/3.56 Lemma 17: composition(converse(one), X) = X.
% 24.80/3.56 Proof:
% 24.80/3.56 composition(converse(one), X)
% 24.80/3.56 = { by lemma 16 R->L }
% 24.80/3.56 converse(composition(converse(X), one))
% 24.80/3.56 = { by axiom 4 (composition_identity_6) }
% 24.80/3.56 converse(converse(X))
% 24.80/3.56 = { by axiom 3 (converse_idempotence_8) }
% 24.80/3.56 X
% 24.80/3.56
% 24.80/3.56 Lemma 18: converse(one) = one.
% 24.80/3.56 Proof:
% 24.80/3.56 converse(one)
% 24.80/3.56 = { by axiom 4 (composition_identity_6) R->L }
% 24.80/3.56 composition(converse(one), one)
% 24.80/3.56 = { by lemma 17 }
% 24.80/3.56 one
% 24.80/3.56
% 24.80/3.56 Lemma 19: join(X, join(Y, complement(X))) = join(Y, top).
% 24.80/3.56 Proof:
% 24.80/3.56 join(X, join(Y, complement(X)))
% 24.80/3.56 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 24.80/3.56 join(X, join(complement(X), Y))
% 24.80/3.56 = { by axiom 8 (maddux2_join_associativity_2) }
% 24.80/3.56 join(join(X, complement(X)), Y)
% 24.80/3.56 = { by axiom 5 (def_top_12) R->L }
% 24.80/3.56 join(top, Y)
% 24.80/3.56 = { by axiom 1 (maddux1_join_commutativity_1) }
% 24.80/3.56 join(Y, top)
% 24.80/3.56
% 24.80/3.56 Lemma 20: composition(one, X) = X.
% 24.80/3.56 Proof:
% 24.80/3.56 composition(one, X)
% 24.80/3.56 = { by lemma 17 R->L }
% 24.80/3.56 composition(converse(one), composition(one, X))
% 24.80/3.56 = { by axiom 10 (composition_associativity_5) }
% 24.80/3.56 composition(composition(converse(one), one), X)
% 24.80/3.56 = { by axiom 4 (composition_identity_6) }
% 24.80/3.56 composition(converse(one), X)
% 24.80/3.56 = { by lemma 17 }
% 24.80/3.56 X
% 24.80/3.56
% 24.80/3.56 Lemma 21: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 24.80/3.56 Proof:
% 24.80/3.56 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 24.80/3.56 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 24.80/3.56 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 24.80/3.56 = { by axiom 13 (converse_cancellativity_11) }
% 24.80/3.56 complement(X)
% 24.80/3.56
% 24.80/3.56 Lemma 22: join(complement(X), complement(X)) = complement(X).
% 24.80/3.56 Proof:
% 24.80/3.56 join(complement(X), complement(X))
% 24.80/3.56 = { by lemma 17 R->L }
% 24.80/3.56 join(complement(X), composition(converse(one), complement(X)))
% 24.80/3.56 = { by lemma 20 R->L }
% 24.80/3.56 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 24.80/3.56 = { by lemma 21 }
% 24.80/3.56 complement(X)
% 24.80/3.56
% 24.80/3.56 Lemma 23: join(top, complement(X)) = top.
% 24.80/3.56 Proof:
% 24.80/3.56 join(top, complement(X))
% 24.80/3.56 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 24.80/3.56 join(complement(X), top)
% 24.80/3.56 = { by lemma 19 R->L }
% 24.80/3.56 join(X, join(complement(X), complement(X)))
% 24.80/3.56 = { by lemma 22 }
% 24.80/3.56 join(X, complement(X))
% 24.80/3.56 = { by axiom 5 (def_top_12) R->L }
% 24.80/3.56 top
% 24.80/3.56
% 24.80/3.56 Lemma 24: join(Y, top) = join(X, top).
% 24.80/3.56 Proof:
% 24.80/3.56 join(Y, top)
% 24.80/3.56 = { by lemma 23 R->L }
% 24.80/3.56 join(Y, join(top, complement(Y)))
% 24.80/3.56 = { by lemma 19 }
% 24.80/3.56 join(top, top)
% 24.80/3.56 = { by lemma 19 R->L }
% 24.80/3.56 join(X, join(top, complement(X)))
% 24.80/3.56 = { by lemma 23 }
% 24.80/3.56 join(X, top)
% 24.80/3.56
% 24.80/3.56 Lemma 25: join(sk1, join(one, X)) = join(X, one).
% 24.80/3.56 Proof:
% 24.80/3.56 join(sk1, join(one, X))
% 24.80/3.56 = { by axiom 8 (maddux2_join_associativity_2) }
% 24.80/3.56 join(join(sk1, one), X)
% 24.80/3.56 = { by axiom 2 (goals_14) }
% 24.80/3.56 join(one, X)
% 24.80/3.56 = { by axiom 1 (maddux1_join_commutativity_1) }
% 24.80/3.56 join(X, one)
% 24.80/3.56
% 24.80/3.56 Lemma 26: join(X, top) = top.
% 24.80/3.56 Proof:
% 24.80/3.56 join(X, top)
% 24.80/3.56 = { by lemma 24 }
% 24.80/3.56 join(sk1, top)
% 24.80/3.56 = { by axiom 5 (def_top_12) }
% 24.80/3.56 join(sk1, join(one, complement(one)))
% 24.80/3.56 = { by lemma 25 }
% 24.80/3.56 join(complement(one), one)
% 24.80/3.56 = { by axiom 1 (maddux1_join_commutativity_1) }
% 24.80/3.56 join(one, complement(one))
% 24.80/3.56 = { by axiom 5 (def_top_12) R->L }
% 24.80/3.56 top
% 24.80/3.56
% 24.80/3.56 Lemma 27: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 24.80/3.56 Proof:
% 24.80/3.56 join(meet(X, Y), complement(join(complement(X), Y)))
% 24.80/3.56 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 24.80/3.56 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 24.80/3.56 = { by axiom 14 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 24.80/3.56 X
% 24.80/3.56
% 24.80/3.56 Lemma 28: join(zero, meet(X, X)) = X.
% 24.80/3.56 Proof:
% 24.80/3.56 join(zero, meet(X, X))
% 24.80/3.56 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 24.80/3.56 join(zero, complement(join(complement(X), complement(X))))
% 24.80/3.56 = { by axiom 6 (def_zero_13) }
% 24.80/3.56 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 24.80/3.56 = { by lemma 27 }
% 24.80/3.56 X
% 24.80/3.56
% 24.80/3.56 Lemma 29: join(zero, complement(complement(X))) = X.
% 24.80/3.56 Proof:
% 24.80/3.56 join(zero, complement(complement(X)))
% 24.80/3.56 = { by axiom 6 (def_zero_13) }
% 24.80/3.56 join(meet(X, complement(X)), complement(complement(X)))
% 24.80/3.56 = { by lemma 22 R->L }
% 24.80/3.56 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 24.80/3.56 = { by lemma 27 }
% 24.80/3.56 X
% 24.80/3.56
% 24.80/3.56 Lemma 30: join(X, zero) = X.
% 24.80/3.56 Proof:
% 24.80/3.56 join(X, zero)
% 24.80/3.56 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 24.80/3.56 join(zero, X)
% 24.80/3.56 = { by lemma 28 R->L }
% 24.80/3.56 join(zero, join(zero, meet(X, X)))
% 24.80/3.56 = { by axiom 8 (maddux2_join_associativity_2) }
% 24.80/3.56 join(join(zero, zero), meet(X, X))
% 24.80/3.56 = { by lemma 15 R->L }
% 24.80/3.56 join(join(zero, complement(top)), meet(X, X))
% 24.80/3.56 = { by lemma 15 R->L }
% 24.80/3.56 join(join(complement(top), complement(top)), meet(X, X))
% 24.80/3.56 = { by lemma 22 }
% 24.80/3.56 join(complement(top), meet(X, X))
% 24.80/3.56 = { by lemma 15 }
% 24.80/3.56 join(zero, meet(X, X))
% 24.80/3.56 = { by axiom 1 (maddux1_join_commutativity_1) }
% 24.80/3.56 join(meet(X, X), zero)
% 24.80/3.56 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 24.80/3.56 join(complement(join(complement(X), complement(X))), zero)
% 24.80/3.56 = { by lemma 22 }
% 24.80/3.56 join(complement(complement(X)), zero)
% 24.80/3.56 = { by axiom 1 (maddux1_join_commutativity_1) }
% 24.80/3.56 join(zero, complement(complement(X)))
% 24.80/3.56 = { by lemma 29 }
% 24.80/3.56 X
% 24.80/3.56
% 24.80/3.56 Lemma 31: join(top, X) = top.
% 24.80/3.56 Proof:
% 24.80/3.56 join(top, X)
% 24.80/3.56 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 24.80/3.57 join(X, top)
% 24.80/3.57 = { by lemma 24 R->L }
% 24.80/3.57 join(Y, top)
% 24.80/3.57 = { by lemma 26 }
% 24.80/3.57 top
% 24.80/3.57
% 24.80/3.57 Lemma 32: join(zero, X) = X.
% 24.80/3.57 Proof:
% 24.80/3.57 join(zero, X)
% 24.80/3.57 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 24.80/3.57 join(X, zero)
% 24.80/3.57 = { by lemma 30 }
% 24.80/3.57 X
% 24.80/3.57
% 24.80/3.57 Lemma 33: complement(complement(X)) = X.
% 24.80/3.57 Proof:
% 24.80/3.57 complement(complement(X))
% 24.80/3.57 = { by lemma 32 R->L }
% 24.80/3.57 join(zero, complement(complement(X)))
% 24.80/3.57 = { by lemma 29 }
% 24.80/3.57 X
% 24.80/3.57
% 24.80/3.57 Lemma 34: meet(X, X) = X.
% 24.80/3.57 Proof:
% 24.80/3.57 meet(X, X)
% 24.80/3.57 = { by lemma 32 R->L }
% 24.80/3.57 join(zero, meet(X, X))
% 24.80/3.57 = { by lemma 28 }
% 24.80/3.57 X
% 24.80/3.57
% 24.80/3.57 Lemma 35: meet(Y, X) = meet(X, Y).
% 24.80/3.57 Proof:
% 24.80/3.57 meet(Y, X)
% 24.80/3.57 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 24.80/3.57 complement(join(complement(Y), complement(X)))
% 24.80/3.57 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 24.80/3.57 complement(join(complement(X), complement(Y)))
% 24.80/3.57 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 24.80/3.57 meet(X, Y)
% 24.80/3.57
% 24.80/3.57 Lemma 36: complement(join(zero, complement(X))) = meet(X, top).
% 24.80/3.57 Proof:
% 24.80/3.57 complement(join(zero, complement(X)))
% 24.80/3.57 = { by lemma 15 R->L }
% 24.80/3.57 complement(join(complement(top), complement(X)))
% 24.80/3.57 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 24.80/3.57 meet(top, X)
% 24.80/3.57 = { by lemma 35 R->L }
% 24.80/3.57 meet(X, top)
% 24.80/3.57
% 24.80/3.57 Lemma 37: meet(X, top) = X.
% 24.80/3.57 Proof:
% 24.80/3.57 meet(X, top)
% 24.80/3.57 = { by lemma 36 R->L }
% 24.80/3.57 complement(join(zero, complement(X)))
% 24.80/3.57 = { by lemma 32 }
% 24.80/3.57 complement(complement(X))
% 24.80/3.57 = { by lemma 33 }
% 24.80/3.57 X
% 24.80/3.57
% 24.80/3.57 Lemma 38: composition(join(X, one), Y) = join(Y, composition(X, Y)).
% 24.80/3.57 Proof:
% 24.80/3.57 composition(join(X, one), Y)
% 24.80/3.57 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 24.80/3.57 composition(join(one, X), Y)
% 24.80/3.57 = { by axiom 12 (composition_distributivity_7) }
% 24.80/3.57 join(composition(one, Y), composition(X, Y))
% 24.80/3.57 = { by lemma 20 }
% 24.80/3.57 join(Y, composition(X, Y))
% 24.80/3.57
% 24.80/3.57 Lemma 39: composition(join(one, Y), X) = join(X, composition(Y, X)).
% 24.80/3.57 Proof:
% 24.80/3.57 composition(join(one, Y), X)
% 24.80/3.57 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 24.80/3.57 composition(join(Y, one), X)
% 24.80/3.57 = { by lemma 38 }
% 24.80/3.57 join(X, composition(Y, X))
% 24.80/3.57
% 24.80/3.57 Lemma 40: join(X, composition(top, X)) = composition(top, X).
% 24.80/3.57 Proof:
% 24.80/3.57 join(X, composition(top, X))
% 24.80/3.57 = { by lemma 39 R->L }
% 24.80/3.57 composition(join(one, top), X)
% 24.80/3.57 = { by lemma 26 }
% 24.80/3.57 composition(top, X)
% 24.80/3.57
% 24.80/3.57 Lemma 41: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 24.80/3.57 Proof:
% 24.80/3.57 converse(join(X, converse(Y)))
% 24.80/3.57 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 24.80/3.57 converse(join(converse(Y), X))
% 24.80/3.57 = { by axiom 7 (converse_additivity_9) }
% 24.80/3.57 join(converse(converse(Y)), converse(X))
% 24.80/3.57 = { by axiom 3 (converse_idempotence_8) }
% 24.80/3.57 join(Y, converse(X))
% 24.80/3.57
% 24.80/3.57 Lemma 42: composition(top, zero) = zero.
% 24.80/3.57 Proof:
% 24.80/3.57 composition(top, zero)
% 24.80/3.57 = { by lemma 40 R->L }
% 24.80/3.57 join(zero, composition(top, zero))
% 24.80/3.57 = { by lemma 31 R->L }
% 24.80/3.57 join(zero, composition(join(top, converse(top)), zero))
% 24.80/3.57 = { by lemma 41 R->L }
% 24.80/3.57 join(zero, composition(converse(join(top, converse(top))), zero))
% 24.80/3.57 = { by lemma 31 }
% 24.80/3.57 join(zero, composition(converse(top), zero))
% 24.80/3.57 = { by lemma 15 R->L }
% 24.80/3.57 join(complement(top), composition(converse(top), zero))
% 24.80/3.57 = { by lemma 15 R->L }
% 24.80/3.57 join(complement(top), composition(converse(top), complement(top)))
% 24.80/3.57 = { by lemma 31 R->L }
% 24.80/3.57 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 24.80/3.57 = { by lemma 40 }
% 24.80/3.57 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 24.80/3.57 = { by lemma 21 }
% 24.80/3.57 complement(top)
% 24.80/3.57 = { by lemma 15 }
% 24.80/3.57 zero
% 24.80/3.57
% 24.80/3.57 Lemma 43: join(meet(X, Y), meet(X, Y)) = meet(X, Y).
% 24.80/3.57 Proof:
% 24.80/3.57 join(meet(X, Y), meet(X, Y))
% 24.80/3.57 = { by lemma 35 }
% 24.80/3.57 join(meet(Y, X), meet(X, Y))
% 24.80/3.57 = { by lemma 35 }
% 24.80/3.57 join(meet(Y, X), meet(Y, X))
% 24.80/3.57 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 24.80/3.57 join(meet(Y, X), complement(join(complement(Y), complement(X))))
% 24.80/3.57 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 24.80/3.57 join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X))))
% 24.80/3.57 = { by lemma 22 }
% 24.80/3.57 complement(join(complement(Y), complement(X)))
% 24.80/3.57 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 24.80/3.57 meet(Y, X)
% 24.80/3.57 = { by lemma 35 R->L }
% 24.80/3.57 meet(X, Y)
% 24.80/3.57
% 24.80/3.57 Lemma 44: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 24.80/3.57 Proof:
% 24.80/3.57 meet(X, join(complement(Y), complement(Z)))
% 24.80/3.57 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 24.80/3.57 meet(X, join(complement(Z), complement(Y)))
% 24.80/3.57 = { by lemma 35 }
% 24.80/3.57 meet(join(complement(Z), complement(Y)), X)
% 24.80/3.57 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 24.80/3.57 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 24.80/3.57 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 24.80/3.57 complement(join(meet(Z, Y), complement(X)))
% 24.80/3.57 = { by axiom 1 (maddux1_join_commutativity_1) }
% 24.80/3.57 complement(join(complement(X), meet(Z, Y)))
% 24.80/3.57 = { by lemma 35 R->L }
% 24.80/3.57 complement(join(complement(X), meet(Y, Z)))
% 24.80/3.57
% 24.80/3.57 Lemma 45: join(X, complement(meet(X, Y))) = top.
% 24.80/3.57 Proof:
% 24.80/3.57 join(X, complement(meet(X, Y)))
% 24.80/3.57 = { by lemma 43 R->L }
% 24.80/3.57 join(X, complement(join(meet(X, Y), meet(X, Y))))
% 24.80/3.57 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 24.80/3.57 join(X, complement(join(complement(join(complement(X), complement(Y))), meet(X, Y))))
% 24.80/3.57 = { by lemma 32 R->L }
% 24.80/3.57 join(X, join(zero, complement(join(complement(join(complement(X), complement(Y))), meet(X, Y)))))
% 24.80/3.57 = { by lemma 44 R->L }
% 24.80/3.57 join(X, join(zero, meet(join(complement(X), complement(Y)), join(complement(X), complement(Y)))))
% 24.80/3.57 = { by lemma 28 }
% 24.80/3.57 join(X, join(complement(X), complement(Y)))
% 24.80/3.57 = { by axiom 1 (maddux1_join_commutativity_1) }
% 24.80/3.57 join(X, join(complement(Y), complement(X)))
% 24.80/3.57 = { by lemma 19 }
% 24.80/3.57 join(complement(Y), top)
% 24.80/3.57 = { by lemma 26 }
% 24.80/3.57 top
% 24.80/3.57
% 24.80/3.57 Lemma 46: meet(X, meet(Y, complement(X))) = zero.
% 24.80/3.57 Proof:
% 24.80/3.57 meet(X, meet(Y, complement(X)))
% 24.80/3.57 = { by lemma 35 }
% 24.80/3.57 meet(X, meet(complement(X), Y))
% 24.80/3.57 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 24.80/3.57 complement(join(complement(X), complement(meet(complement(X), Y))))
% 24.80/3.57 = { by lemma 45 }
% 24.80/3.57 complement(top)
% 24.80/3.57 = { by lemma 15 }
% 24.80/3.57 zero
% 24.80/3.57
% 24.80/3.57 Lemma 47: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 24.80/3.57 Proof:
% 24.80/3.57 complement(join(X, complement(Y)))
% 24.80/3.57 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 24.80/3.57 complement(join(complement(Y), X))
% 24.80/3.57 = { by lemma 37 R->L }
% 24.80/3.57 complement(join(complement(Y), meet(X, top)))
% 24.80/3.57 = { by lemma 35 R->L }
% 24.80/3.57 complement(join(complement(Y), meet(top, X)))
% 24.80/3.57 = { by lemma 44 R->L }
% 24.80/3.57 meet(Y, join(complement(top), complement(X)))
% 24.80/3.57 = { by lemma 15 }
% 24.80/3.57 meet(Y, join(zero, complement(X)))
% 24.80/3.57 = { by lemma 32 }
% 24.80/3.57 meet(Y, complement(X))
% 24.80/3.57
% 24.80/3.57 Lemma 48: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 24.80/3.57 Proof:
% 24.80/3.57 complement(join(complement(X), Y))
% 24.80/3.57 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 24.80/3.57 complement(join(Y, complement(X)))
% 24.80/3.57 = { by lemma 47 }
% 24.80/3.57 meet(X, complement(Y))
% 24.80/3.57
% 24.80/3.57 Lemma 49: converse(composition(Y, converse(X))) = composition(X, converse(Y)).
% 24.80/3.57 Proof:
% 24.80/3.57 converse(composition(Y, converse(X)))
% 24.80/3.57 = { by axiom 9 (converse_multiplicativity_10) }
% 24.80/3.57 composition(converse(converse(X)), converse(Y))
% 24.80/3.57 = { by axiom 3 (converse_idempotence_8) }
% 24.80/3.57 composition(X, converse(Y))
% 24.80/3.57
% 24.80/3.57 Lemma 50: meet(X, complement(composition(sk1, top))) = meet(X, complement(composition(sk1, X))).
% 24.80/3.57 Proof:
% 24.80/3.57 meet(X, complement(composition(sk1, top)))
% 24.80/3.57 = { by lemma 48 R->L }
% 24.80/3.57 complement(join(complement(X), composition(sk1, top)))
% 24.80/3.57 = { by axiom 5 (def_top_12) }
% 24.80/3.57 complement(join(complement(X), composition(sk1, join(complement(X), complement(complement(X))))))
% 24.80/3.57 = { by axiom 3 (converse_idempotence_8) R->L }
% 24.80/3.57 complement(join(complement(X), composition(sk1, join(complement(X), converse(converse(complement(complement(X))))))))
% 24.80/3.57 = { by lemma 41 R->L }
% 24.80/3.57 complement(join(complement(X), composition(sk1, converse(join(converse(complement(complement(X))), converse(complement(X)))))))
% 24.80/3.57 = { by lemma 49 R->L }
% 24.80/3.57 complement(join(complement(X), converse(composition(join(converse(complement(complement(X))), converse(complement(X))), converse(sk1)))))
% 24.80/3.57 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 24.80/3.57 complement(join(complement(X), converse(composition(join(converse(complement(X)), converse(complement(complement(X)))), converse(sk1)))))
% 24.80/3.57 = { by axiom 12 (composition_distributivity_7) }
% 24.80/3.57 complement(join(complement(X), converse(join(composition(converse(complement(X)), converse(sk1)), composition(converse(complement(complement(X))), converse(sk1))))))
% 24.80/3.57 = { by axiom 9 (converse_multiplicativity_10) R->L }
% 24.80/3.58 complement(join(complement(X), converse(join(converse(composition(sk1, complement(X))), composition(converse(complement(complement(X))), converse(sk1))))))
% 24.80/3.58 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 24.80/3.58 complement(join(complement(X), converse(join(composition(converse(complement(complement(X))), converse(sk1)), converse(composition(sk1, complement(X)))))))
% 24.80/3.58 = { by axiom 7 (converse_additivity_9) }
% 24.80/3.58 complement(join(complement(X), join(converse(composition(converse(complement(complement(X))), converse(sk1))), converse(converse(composition(sk1, complement(X)))))))
% 24.80/3.58 = { by lemma 16 }
% 24.80/3.58 complement(join(complement(X), join(composition(converse(converse(sk1)), complement(complement(X))), converse(converse(composition(sk1, complement(X)))))))
% 24.80/3.58 = { by axiom 1 (maddux1_join_commutativity_1) }
% 24.80/3.58 complement(join(complement(X), join(converse(converse(composition(sk1, complement(X)))), composition(converse(converse(sk1)), complement(complement(X))))))
% 24.80/3.58 = { by axiom 3 (converse_idempotence_8) }
% 24.80/3.58 complement(join(complement(X), join(composition(sk1, complement(X)), composition(converse(converse(sk1)), complement(complement(X))))))
% 24.80/3.58 = { by axiom 3 (converse_idempotence_8) }
% 24.80/3.58 complement(join(complement(X), join(composition(sk1, complement(X)), composition(sk1, complement(complement(X))))))
% 24.80/3.58 = { by axiom 8 (maddux2_join_associativity_2) }
% 24.80/3.58 complement(join(join(complement(X), composition(sk1, complement(X))), composition(sk1, complement(complement(X)))))
% 24.80/3.58 = { by lemma 38 R->L }
% 24.80/3.58 complement(join(composition(join(sk1, one), complement(X)), composition(sk1, complement(complement(X)))))
% 24.80/3.58 = { by axiom 2 (goals_14) }
% 24.80/3.58 complement(join(composition(one, complement(X)), composition(sk1, complement(complement(X)))))
% 24.80/3.58 = { by lemma 20 }
% 24.80/3.58 complement(join(complement(X), composition(sk1, complement(complement(X)))))
% 24.80/3.58 = { by lemma 48 }
% 24.80/3.58 meet(X, complement(composition(sk1, complement(complement(X)))))
% 24.80/3.58 = { by lemma 33 }
% 24.80/3.58 meet(X, complement(composition(sk1, X)))
% 24.80/3.58
% 24.80/3.58 Lemma 51: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 24.80/3.58 Proof:
% 24.80/3.58 meet(complement(X), complement(Y))
% 24.80/3.58 = { by lemma 35 }
% 24.80/3.58 meet(complement(Y), complement(X))
% 24.80/3.58 = { by lemma 32 R->L }
% 24.80/3.58 meet(join(zero, complement(Y)), complement(X))
% 24.80/3.58 = { by lemma 47 R->L }
% 24.80/3.58 complement(join(X, complement(join(zero, complement(Y)))))
% 24.80/3.58 = { by lemma 36 }
% 24.80/3.58 complement(join(X, meet(Y, top)))
% 24.80/3.58 = { by lemma 37 }
% 24.80/3.58 complement(join(X, Y))
% 24.80/3.58
% 24.80/3.58 Lemma 52: meet(Y, meet(Z, X)) = meet(X, meet(Y, Z)).
% 24.80/3.58 Proof:
% 24.80/3.58 meet(Y, meet(Z, X))
% 24.80/3.58 = { by lemma 37 R->L }
% 24.80/3.58 meet(meet(Y, top), meet(Z, X))
% 24.80/3.58 = { by lemma 36 R->L }
% 24.80/3.58 meet(complement(join(zero, complement(Y))), meet(Z, X))
% 24.80/3.58 = { by lemma 35 }
% 24.80/3.58 meet(complement(join(zero, complement(Y))), meet(X, Z))
% 24.80/3.58 = { by lemma 35 }
% 24.80/3.58 meet(meet(X, Z), complement(join(zero, complement(Y))))
% 24.80/3.58 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 24.80/3.58 meet(complement(join(complement(X), complement(Z))), complement(join(zero, complement(Y))))
% 24.80/3.58 = { by lemma 51 }
% 24.80/3.58 complement(join(join(complement(X), complement(Z)), join(zero, complement(Y))))
% 24.80/3.58 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 24.80/3.58 complement(join(complement(X), join(complement(Z), join(zero, complement(Y)))))
% 24.80/3.58 = { by lemma 48 }
% 24.80/3.58 meet(X, complement(join(complement(Z), join(zero, complement(Y)))))
% 24.80/3.58 = { by lemma 48 }
% 24.80/3.58 meet(X, meet(Z, complement(join(zero, complement(Y)))))
% 24.80/3.58 = { by lemma 36 }
% 24.80/3.58 meet(X, meet(Z, meet(Y, top)))
% 24.80/3.58 = { by lemma 37 }
% 24.80/3.58 meet(X, meet(Z, Y))
% 24.80/3.58 = { by lemma 35 R->L }
% 24.80/3.58 meet(X, meet(Y, Z))
% 24.80/3.58
% 24.80/3.58 Lemma 53: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 24.80/3.58 Proof:
% 24.80/3.58 complement(meet(X, complement(Y)))
% 24.80/3.58 = { by lemma 32 R->L }
% 24.80/3.58 complement(join(zero, meet(X, complement(Y))))
% 24.80/3.58 = { by lemma 47 R->L }
% 24.80/3.58 complement(join(zero, complement(join(Y, complement(X)))))
% 24.80/3.58 = { by lemma 36 }
% 24.80/3.58 meet(join(Y, complement(X)), top)
% 24.80/3.58 = { by lemma 37 }
% 24.80/3.58 join(Y, complement(X))
% 24.80/3.58
% 24.80/3.58 Lemma 54: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 24.80/3.58 Proof:
% 24.80/3.58 complement(meet(complement(X), Y))
% 24.80/3.58 = { by lemma 35 }
% 24.80/3.58 complement(meet(Y, complement(X)))
% 24.80/3.58 = { by lemma 53 }
% 24.80/3.58 join(X, complement(Y))
% 24.80/3.58
% 24.80/3.58 Lemma 55: meet(X, join(X, complement(Y))) = X.
% 24.80/3.58 Proof:
% 24.80/3.58 meet(X, join(X, complement(Y)))
% 24.80/3.58 = { by lemma 30 R->L }
% 24.80/3.58 join(meet(X, join(X, complement(Y))), zero)
% 24.80/3.58 = { by lemma 15 R->L }
% 24.80/3.58 join(meet(X, join(X, complement(Y))), complement(top))
% 24.80/3.58 = { by lemma 54 R->L }
% 24.80/3.58 join(meet(X, complement(meet(complement(X), Y))), complement(top))
% 24.80/3.58 = { by lemma 45 R->L }
% 24.80/3.58 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(meet(complement(X), Y)))))
% 24.80/3.58 = { by lemma 27 }
% 24.80/3.58 X
% 24.80/3.58
% 24.80/3.58 Lemma 56: join(complement(X), meet(X, Y)) = join(Y, complement(X)).
% 24.80/3.58 Proof:
% 24.80/3.58 join(complement(X), meet(X, Y))
% 24.80/3.58 = { by lemma 35 }
% 24.80/3.58 join(complement(X), meet(Y, X))
% 24.80/3.58 = { by lemma 33 R->L }
% 24.80/3.58 join(complement(complement(complement(X))), meet(Y, X))
% 24.80/3.58 = { by lemma 55 R->L }
% 24.80/3.58 join(complement(meet(complement(complement(X)), join(complement(complement(X)), complement(Y)))), meet(Y, X))
% 24.80/3.58 = { by lemma 54 }
% 24.80/3.58 join(join(complement(X), complement(join(complement(complement(X)), complement(Y)))), meet(Y, X))
% 24.80/3.58 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 24.80/3.58 join(join(complement(X), meet(complement(X), Y)), meet(Y, X))
% 24.80/3.58 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 24.80/3.58 join(complement(X), join(meet(complement(X), Y), meet(Y, X)))
% 24.80/3.58 = { by axiom 1 (maddux1_join_commutativity_1) }
% 24.80/3.58 join(complement(X), join(meet(Y, X), meet(complement(X), Y)))
% 24.80/3.58 = { by lemma 35 }
% 24.80/3.58 join(complement(X), join(meet(Y, X), meet(Y, complement(X))))
% 24.80/3.58 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 24.80/3.58 join(complement(X), join(meet(Y, complement(X)), meet(Y, X)))
% 24.80/3.58 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 24.80/3.58 join(complement(X), join(meet(Y, complement(X)), complement(join(complement(Y), complement(X)))))
% 24.80/3.58 = { by lemma 27 }
% 24.80/3.58 join(complement(X), Y)
% 24.80/3.58 = { by axiom 1 (maddux1_join_commutativity_1) }
% 24.80/3.58 join(Y, complement(X))
% 24.80/3.58
% 24.80/3.58 Lemma 57: join(complement(X), meet(Y, X)) = join(Y, complement(X)).
% 24.80/3.58 Proof:
% 24.80/3.58 join(complement(X), meet(Y, X))
% 24.80/3.58 = { by lemma 35 }
% 24.80/3.58 join(complement(X), meet(X, Y))
% 24.80/3.58 = { by lemma 56 }
% 24.80/3.58 join(Y, complement(X))
% 24.80/3.58
% 24.80/3.58 Lemma 58: join(X, composition(sk1, X)) = X.
% 24.80/3.58 Proof:
% 24.80/3.58 join(X, composition(sk1, X))
% 24.80/3.58 = { by lemma 38 R->L }
% 24.80/3.58 composition(join(sk1, one), X)
% 24.80/3.58 = { by axiom 2 (goals_14) }
% 24.80/3.58 composition(one, X)
% 24.80/3.58 = { by lemma 20 }
% 24.80/3.58 X
% 24.80/3.58
% 24.80/3.58 Lemma 59: composition(sk1, sk1) = sk1.
% 24.80/3.58 Proof:
% 24.80/3.58 composition(sk1, sk1)
% 24.80/3.58 = { by lemma 33 R->L }
% 24.80/3.58 complement(complement(composition(sk1, sk1)))
% 24.80/3.58 = { by lemma 30 R->L }
% 24.80/3.58 join(complement(complement(composition(sk1, sk1))), zero)
% 24.80/3.58 = { by lemma 46 R->L }
% 24.80/3.58 join(complement(complement(composition(sk1, sk1))), meet(composition(sk1, one), meet(one, complement(composition(sk1, one)))))
% 24.80/3.58 = { by lemma 50 R->L }
% 24.80/3.58 join(complement(complement(composition(sk1, sk1))), meet(composition(sk1, one), meet(one, complement(composition(sk1, top)))))
% 24.80/3.58 = { by axiom 4 (composition_identity_6) }
% 24.80/3.58 join(complement(complement(composition(sk1, sk1))), meet(sk1, meet(one, complement(composition(sk1, top)))))
% 24.80/3.58 = { by lemma 52 }
% 24.80/3.58 join(complement(complement(composition(sk1, sk1))), meet(complement(composition(sk1, top)), meet(sk1, one)))
% 24.80/3.58 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 24.80/3.58 join(complement(complement(composition(sk1, sk1))), meet(complement(composition(sk1, top)), complement(join(complement(sk1), complement(one)))))
% 24.80/3.58 = { by lemma 32 R->L }
% 24.80/3.58 join(complement(complement(composition(sk1, sk1))), meet(complement(composition(sk1, top)), join(zero, complement(join(complement(sk1), complement(one))))))
% 24.80/3.58 = { by lemma 15 R->L }
% 24.80/3.58 join(complement(complement(composition(sk1, sk1))), meet(complement(composition(sk1, top)), join(complement(top), complement(join(complement(sk1), complement(one))))))
% 24.80/3.58 = { by lemma 26 R->L }
% 24.80/3.58 join(complement(complement(composition(sk1, sk1))), meet(complement(composition(sk1, top)), join(complement(join(one, top)), complement(join(complement(sk1), complement(one))))))
% 24.80/3.58 = { by lemma 19 R->L }
% 24.80/3.58 join(complement(complement(composition(sk1, sk1))), meet(complement(composition(sk1, top)), join(complement(join(sk1, join(one, complement(sk1)))), complement(join(complement(sk1), complement(one))))))
% 24.80/3.58 = { by lemma 25 }
% 24.80/3.58 join(complement(complement(composition(sk1, sk1))), meet(complement(composition(sk1, top)), join(complement(join(complement(sk1), one)), complement(join(complement(sk1), complement(one))))))
% 24.80/3.58 = { by axiom 1 (maddux1_join_commutativity_1) }
% 24.80/3.58 join(complement(complement(composition(sk1, sk1))), meet(complement(composition(sk1, top)), join(complement(join(one, complement(sk1))), complement(join(complement(sk1), complement(one))))))
% 24.80/3.58 = { by lemma 47 }
% 24.80/3.58 join(complement(complement(composition(sk1, sk1))), meet(complement(composition(sk1, top)), join(meet(sk1, complement(one)), complement(join(complement(sk1), complement(one))))))
% 24.80/3.58 = { by lemma 27 }
% 24.80/3.58 join(complement(complement(composition(sk1, sk1))), meet(complement(composition(sk1, top)), sk1))
% 24.80/3.58 = { by lemma 35 R->L }
% 24.80/3.58 join(complement(complement(composition(sk1, sk1))), meet(sk1, complement(composition(sk1, top))))
% 24.80/3.58 = { by lemma 50 }
% 24.80/3.58 join(complement(complement(composition(sk1, sk1))), meet(sk1, complement(composition(sk1, sk1))))
% 24.80/3.58 = { by lemma 57 }
% 24.80/3.58 join(sk1, complement(complement(composition(sk1, sk1))))
% 24.80/3.58 = { by lemma 33 }
% 24.80/3.58 join(sk1, composition(sk1, sk1))
% 24.80/3.58 = { by lemma 58 }
% 24.80/3.58 sk1
% 24.80/3.59
% 24.80/3.59 Lemma 60: meet(complement(X), join(X, Y)) = meet(Y, complement(X)).
% 24.80/3.59 Proof:
% 24.80/3.59 meet(complement(X), join(X, Y))
% 24.80/3.59 = { by lemma 35 }
% 24.80/3.59 meet(join(X, Y), complement(X))
% 24.80/3.59 = { by lemma 47 R->L }
% 24.80/3.59 complement(join(X, complement(join(X, Y))))
% 24.80/3.59 = { by lemma 51 R->L }
% 24.80/3.59 meet(complement(X), complement(complement(join(X, Y))))
% 24.80/3.59 = { by lemma 51 R->L }
% 24.80/3.59 meet(complement(X), complement(meet(complement(X), complement(Y))))
% 24.80/3.59 = { by lemma 48 R->L }
% 24.80/3.59 complement(join(complement(complement(X)), meet(complement(X), complement(Y))))
% 24.80/3.59 = { by lemma 56 }
% 24.80/3.59 complement(join(complement(Y), complement(complement(X))))
% 24.80/3.59 = { by lemma 47 }
% 24.80/3.59 meet(complement(X), complement(complement(Y)))
% 24.80/3.59 = { by lemma 51 }
% 24.80/3.59 complement(join(X, complement(Y)))
% 24.80/3.59 = { by lemma 47 }
% 24.80/3.59 meet(Y, complement(X))
% 24.80/3.59
% 24.80/3.59 Lemma 61: meet(X, composition(sk1, top)) = composition(sk1, X).
% 24.80/3.59 Proof:
% 24.80/3.59 meet(X, composition(sk1, top))
% 24.80/3.59 = { by lemma 35 }
% 24.80/3.59 meet(composition(sk1, top), X)
% 24.80/3.59 = { by lemma 33 R->L }
% 24.80/3.59 meet(composition(sk1, top), complement(complement(X)))
% 24.80/3.59 = { by lemma 60 R->L }
% 24.80/3.59 meet(complement(complement(X)), join(complement(X), composition(sk1, top)))
% 24.80/3.59 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 24.80/3.59 meet(complement(complement(X)), join(composition(sk1, top), complement(X)))
% 24.80/3.59 = { by lemma 53 R->L }
% 24.80/3.59 meet(complement(complement(X)), complement(meet(X, complement(composition(sk1, top)))))
% 24.80/3.59 = { by lemma 50 }
% 24.80/3.59 meet(complement(complement(X)), complement(meet(X, complement(composition(sk1, X)))))
% 24.80/3.59 = { by lemma 53 }
% 24.80/3.59 meet(complement(complement(X)), join(composition(sk1, X), complement(X)))
% 24.80/3.59 = { by axiom 1 (maddux1_join_commutativity_1) }
% 24.80/3.59 meet(complement(complement(X)), join(complement(X), composition(sk1, X)))
% 24.80/3.59 = { by lemma 60 }
% 24.80/3.59 meet(composition(sk1, X), complement(complement(X)))
% 24.80/3.59 = { by lemma 33 }
% 24.80/3.59 meet(composition(sk1, X), X)
% 24.80/3.59 = { by lemma 58 R->L }
% 24.80/3.59 meet(composition(sk1, X), join(X, composition(sk1, X)))
% 24.80/3.59 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 24.80/3.59 meet(composition(sk1, X), join(composition(sk1, X), X))
% 24.80/3.59 = { by lemma 37 R->L }
% 24.80/3.59 meet(composition(sk1, X), join(composition(sk1, X), meet(X, top)))
% 24.80/3.59 = { by lemma 36 R->L }
% 24.80/3.59 meet(composition(sk1, X), join(composition(sk1, X), complement(join(zero, complement(X)))))
% 24.80/3.59 = { by lemma 55 }
% 24.80/3.59 composition(sk1, X)
% 24.80/3.59
% 24.80/3.59 Lemma 62: meet(X, composition(sk1, Y)) = composition(sk1, meet(X, Y)).
% 24.80/3.59 Proof:
% 24.80/3.59 meet(X, composition(sk1, Y))
% 24.80/3.59 = { by lemma 61 R->L }
% 24.80/3.59 meet(X, meet(Y, composition(sk1, top)))
% 24.80/3.59 = { by lemma 52 }
% 24.80/3.59 meet(composition(sk1, top), meet(X, Y))
% 24.80/3.59 = { by lemma 35 R->L }
% 24.80/3.59 meet(meet(X, Y), composition(sk1, top))
% 24.80/3.59 = { by lemma 61 }
% 24.80/3.59 composition(sk1, meet(X, Y))
% 24.80/3.59
% 24.80/3.59 Lemma 63: join(converse(sk1), sk1) = converse(sk1).
% 24.80/3.59 Proof:
% 24.80/3.59 join(converse(sk1), sk1)
% 24.80/3.59 = { by axiom 1 (maddux1_join_commutativity_1) }
% 24.80/3.59 join(sk1, converse(sk1))
% 24.80/3.59 = { by lemma 33 R->L }
% 24.80/3.59 join(sk1, complement(complement(converse(sk1))))
% 24.80/3.59 = { by lemma 57 R->L }
% 24.80/3.59 join(complement(complement(converse(sk1))), meet(sk1, complement(converse(sk1))))
% 24.80/3.59 = { by lemma 35 }
% 24.80/3.59 join(complement(complement(converse(sk1))), meet(complement(converse(sk1)), sk1))
% 24.80/3.59 = { by lemma 59 R->L }
% 24.80/3.59 join(complement(complement(converse(sk1))), meet(complement(converse(sk1)), composition(sk1, sk1)))
% 24.80/3.59 = { by lemma 62 }
% 24.80/3.59 join(complement(complement(converse(sk1))), composition(sk1, meet(complement(converse(sk1)), sk1)))
% 24.80/3.59 = { by lemma 35 }
% 24.80/3.59 join(complement(complement(converse(sk1))), composition(sk1, meet(sk1, complement(converse(sk1)))))
% 24.80/3.59 = { by lemma 62 R->L }
% 24.80/3.59 join(complement(complement(converse(sk1))), meet(sk1, composition(sk1, complement(converse(sk1)))))
% 24.80/3.59 = { by lemma 35 R->L }
% 24.80/3.59 join(complement(complement(converse(sk1))), meet(composition(sk1, complement(converse(sk1))), sk1))
% 24.80/3.59 = { by axiom 4 (composition_identity_6) R->L }
% 24.80/3.59 join(complement(complement(converse(sk1))), meet(composition(sk1, complement(converse(sk1))), composition(sk1, one)))
% 24.80/3.59 = { by lemma 62 }
% 24.80/3.59 join(complement(complement(converse(sk1))), composition(sk1, meet(composition(sk1, complement(converse(sk1))), one)))
% 24.80/3.59 = { by lemma 33 R->L }
% 24.80/3.59 join(complement(complement(converse(sk1))), composition(sk1, meet(composition(sk1, complement(converse(sk1))), complement(complement(one)))))
% 24.80/3.59 = { by lemma 21 R->L }
% 24.80/3.59 join(complement(complement(converse(sk1))), composition(sk1, meet(composition(sk1, complement(converse(sk1))), complement(join(complement(one), composition(converse(converse(sk1)), complement(composition(converse(sk1), one))))))))
% 24.80/3.59 = { by axiom 4 (composition_identity_6) }
% 24.80/3.59 join(complement(complement(converse(sk1))), composition(sk1, meet(composition(sk1, complement(converse(sk1))), complement(join(complement(one), composition(converse(converse(sk1)), complement(converse(sk1))))))))
% 24.80/3.59 = { by axiom 3 (converse_idempotence_8) }
% 24.80/3.59 join(complement(complement(converse(sk1))), composition(sk1, meet(composition(sk1, complement(converse(sk1))), complement(join(complement(one), composition(sk1, complement(converse(sk1))))))))
% 24.80/3.59 = { by lemma 51 R->L }
% 24.80/3.59 join(complement(complement(converse(sk1))), composition(sk1, meet(composition(sk1, complement(converse(sk1))), meet(complement(complement(one)), complement(composition(sk1, complement(converse(sk1))))))))
% 24.80/3.59 = { by lemma 46 }
% 24.80/3.59 join(complement(complement(converse(sk1))), composition(sk1, zero))
% 24.80/3.59 = { by lemma 32 R->L }
% 24.80/3.59 join(complement(complement(converse(sk1))), join(zero, composition(sk1, zero)))
% 24.80/3.59 = { by lemma 42 R->L }
% 24.80/3.59 join(complement(complement(converse(sk1))), join(composition(top, zero), composition(sk1, zero)))
% 24.80/3.59 = { by axiom 12 (composition_distributivity_7) R->L }
% 24.80/3.59 join(complement(complement(converse(sk1))), composition(join(top, sk1), zero))
% 24.80/3.59 = { by lemma 31 }
% 24.80/3.59 join(complement(complement(converse(sk1))), composition(top, zero))
% 24.80/3.59 = { by lemma 42 }
% 24.80/3.59 join(complement(complement(converse(sk1))), zero)
% 24.80/3.59 = { by lemma 30 }
% 24.80/3.59 complement(complement(converse(sk1)))
% 24.80/3.59 = { by lemma 33 }
% 24.80/3.59 converse(sk1)
% 24.80/3.59
% 24.80/3.59 Lemma 64: join(Y, join(Z, X)) = join(X, join(Y, Z)).
% 24.80/3.59 Proof:
% 24.80/3.59 join(Y, join(Z, X))
% 24.80/3.59 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 24.80/3.59 join(join(Z, X), Y)
% 24.80/3.59 = { by axiom 1 (maddux1_join_commutativity_1) }
% 24.80/3.59 join(join(X, Z), Y)
% 24.80/3.59 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 24.80/3.59 join(X, join(Z, Y))
% 24.80/3.59 = { by axiom 1 (maddux1_join_commutativity_1) }
% 24.80/3.59 join(X, join(Y, Z))
% 24.80/3.59
% 24.80/3.59 Lemma 65: converse(join(converse(X), Y)) = join(converse(Y), X).
% 24.80/3.59 Proof:
% 24.80/3.59 converse(join(converse(X), Y))
% 24.80/3.59 = { by axiom 1 (maddux1_join_commutativity_1) }
% 24.80/3.59 converse(join(Y, converse(X)))
% 24.80/3.59 = { by lemma 41 }
% 24.80/3.59 join(X, converse(Y))
% 24.80/3.59 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 24.80/3.59 join(converse(Y), X)
% 24.80/3.59
% 24.80/3.59 Lemma 66: composition(join(converse(sk1), sk1), sk1) = sk1.
% 24.80/3.59 Proof:
% 24.80/3.59 composition(join(converse(sk1), sk1), sk1)
% 24.80/3.59 = { by lemma 65 R->L }
% 24.80/3.59 composition(converse(join(converse(sk1), sk1)), sk1)
% 24.80/3.59 = { by lemma 16 R->L }
% 24.80/3.59 converse(composition(converse(sk1), join(converse(sk1), sk1)))
% 24.80/3.59 = { by axiom 1 (maddux1_join_commutativity_1) }
% 24.80/3.59 converse(composition(converse(sk1), join(sk1, converse(sk1))))
% 24.80/3.59 = { by lemma 34 R->L }
% 24.80/3.59 converse(composition(converse(sk1), join(sk1, meet(converse(sk1), converse(sk1)))))
% 24.80/3.59 = { by lemma 43 R->L }
% 24.80/3.59 converse(composition(converse(sk1), join(sk1, join(meet(converse(sk1), converse(sk1)), meet(converse(sk1), converse(sk1))))))
% 24.80/3.59 = { by lemma 34 }
% 24.80/3.59 converse(composition(converse(sk1), join(sk1, join(converse(sk1), meet(converse(sk1), converse(sk1))))))
% 24.80/3.59 = { by lemma 34 }
% 24.80/3.59 converse(composition(converse(sk1), join(sk1, join(converse(sk1), converse(sk1)))))
% 24.80/3.59 = { by lemma 64 R->L }
% 24.80/3.59 converse(composition(converse(sk1), join(converse(sk1), join(converse(sk1), sk1))))
% 24.80/3.59 = { by axiom 1 (maddux1_join_commutativity_1) }
% 24.80/3.59 converse(composition(converse(sk1), join(join(converse(sk1), sk1), converse(sk1))))
% 24.80/3.59 = { by lemma 41 R->L }
% 24.80/3.59 converse(composition(converse(sk1), converse(join(sk1, converse(join(converse(sk1), sk1))))))
% 24.80/3.59 = { by axiom 9 (converse_multiplicativity_10) R->L }
% 24.80/3.59 converse(converse(composition(join(sk1, converse(join(converse(sk1), sk1))), sk1)))
% 24.80/3.59 = { by axiom 12 (composition_distributivity_7) }
% 24.80/3.59 converse(converse(join(composition(sk1, sk1), composition(converse(join(converse(sk1), sk1)), sk1))))
% 24.80/3.59 = { by lemma 59 }
% 24.80/3.59 converse(converse(join(sk1, composition(converse(join(converse(sk1), sk1)), sk1))))
% 24.80/3.59 = { by axiom 7 (converse_additivity_9) }
% 24.80/3.59 converse(join(converse(sk1), converse(composition(converse(join(converse(sk1), sk1)), sk1))))
% 24.80/3.59 = { by lemma 16 }
% 24.80/3.59 converse(join(converse(sk1), composition(converse(sk1), join(converse(sk1), sk1))))
% 24.80/3.59 = { by lemma 65 R->L }
% 24.80/3.59 converse(join(converse(sk1), composition(converse(sk1), converse(join(converse(sk1), sk1)))))
% 24.80/3.59 = { by lemma 49 R->L }
% 24.80/3.59 converse(join(converse(sk1), converse(composition(join(converse(sk1), sk1), converse(converse(sk1))))))
% 24.80/3.59 = { by lemma 41 R->L }
% 24.80/3.59 converse(converse(join(composition(join(converse(sk1), sk1), converse(converse(sk1))), converse(converse(sk1)))))
% 24.80/3.59 = { by axiom 1 (maddux1_join_commutativity_1) }
% 24.80/3.59 converse(converse(join(converse(converse(sk1)), composition(join(converse(sk1), sk1), converse(converse(sk1))))))
% 24.80/3.59 = { by lemma 39 R->L }
% 24.80/3.59 converse(converse(composition(join(one, join(converse(sk1), sk1)), converse(converse(sk1)))))
% 24.80/3.59 = { by lemma 64 }
% 24.80/3.59 converse(converse(composition(join(sk1, join(one, converse(sk1))), converse(converse(sk1)))))
% 24.80/3.59 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 24.80/3.59 converse(converse(composition(join(sk1, join(converse(sk1), one)), converse(converse(sk1)))))
% 24.80/3.59 = { by lemma 18 R->L }
% 24.80/3.59 converse(converse(composition(join(sk1, join(converse(sk1), converse(one))), converse(converse(sk1)))))
% 24.80/3.60 = { by axiom 7 (converse_additivity_9) R->L }
% 24.80/3.60 converse(converse(composition(join(sk1, converse(join(sk1, one))), converse(converse(sk1)))))
% 24.80/3.60 = { by axiom 2 (goals_14) }
% 24.80/3.60 converse(converse(composition(join(sk1, converse(one)), converse(converse(sk1)))))
% 24.80/3.60 = { by lemma 18 }
% 25.51/3.60 converse(converse(composition(join(sk1, one), converse(converse(sk1)))))
% 25.51/3.60 = { by axiom 2 (goals_14) }
% 25.51/3.60 converse(converse(composition(one, converse(converse(sk1)))))
% 25.51/3.60 = { by lemma 20 }
% 25.51/3.60 converse(converse(converse(converse(sk1))))
% 25.51/3.60 = { by axiom 3 (converse_idempotence_8) }
% 25.51/3.60 converse(converse(sk1))
% 25.51/3.60 = { by axiom 3 (converse_idempotence_8) }
% 25.51/3.60 sk1
% 25.51/3.60
% 25.51/3.60 Goal 1 (goals_17): tuple(join(converse(sk1), sk1), join(sk1, converse(sk1))) = tuple(sk1, converse(sk1)).
% 25.51/3.60 Proof:
% 25.51/3.60 tuple(join(converse(sk1), sk1), join(sk1, converse(sk1)))
% 25.51/3.60 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 25.51/3.60 tuple(join(converse(sk1), sk1), join(converse(sk1), sk1))
% 25.51/3.60 = { by lemma 63 }
% 25.51/3.60 tuple(converse(sk1), join(converse(sk1), sk1))
% 25.51/3.60 = { by lemma 66 R->L }
% 25.51/3.60 tuple(converse(composition(join(converse(sk1), sk1), sk1)), join(converse(sk1), sk1))
% 25.51/3.60 = { by lemma 63 }
% 25.51/3.60 tuple(converse(composition(converse(sk1), sk1)), join(converse(sk1), sk1))
% 25.51/3.60 = { by lemma 16 }
% 25.51/3.60 tuple(composition(converse(sk1), sk1), join(converse(sk1), sk1))
% 25.51/3.60 = { by lemma 63 R->L }
% 25.51/3.60 tuple(composition(join(converse(sk1), sk1), sk1), join(converse(sk1), sk1))
% 25.51/3.60 = { by lemma 66 }
% 25.51/3.60 tuple(sk1, join(converse(sk1), sk1))
% 25.51/3.60 = { by lemma 63 }
% 25.51/3.60 tuple(sk1, converse(sk1))
% 25.51/3.60 % SZS output end Proof
% 25.51/3.60
% 25.51/3.60 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------