TSTP Solution File: REL029+2 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL029+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 : n010.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 13:44:12 EDT 2023
% Result : Theorem 52.02s 7.08s
% Output : Proof 53.75s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : REL029+2 : TPTP v8.1.2. Released v4.0.0.
% 0.11/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.33 % Computer : n010.cluster.edu
% 0.13/0.33 % Model : x86_64 x86_64
% 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.33 % Memory : 8042.1875MB
% 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.33 % CPULimit : 300
% 0.13/0.33 % WCLimit : 300
% 0.13/0.33 % DateTime : Fri Aug 25 19:26:35 EDT 2023
% 0.13/0.34 % CPUTime :
% 52.02/7.08 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 52.02/7.08
% 52.02/7.08 % SZS status Theorem
% 52.02/7.09
% 53.00/7.19 % SZS output start Proof
% 53.00/7.19 Take the following subset of the input axioms:
% 53.00/7.19 fof(composition_associativity, axiom, ![X0, X1, X2]: composition(X0, composition(X1, X2))=composition(composition(X0, X1), X2)).
% 53.00/7.19 fof(composition_distributivity, axiom, ![X0_2, X1_2, X2_2]: composition(join(X0_2, X1_2), X2_2)=join(composition(X0_2, X2_2), composition(X1_2, X2_2))).
% 53.00/7.19 fof(composition_identity, axiom, ![X0_2]: composition(X0_2, one)=X0_2).
% 53.00/7.19 fof(converse_additivity, axiom, ![X0_2, X1_2]: converse(join(X0_2, X1_2))=join(converse(X0_2), converse(X1_2))).
% 53.00/7.19 fof(converse_cancellativity, axiom, ![X0_2, X1_2]: join(composition(converse(X0_2), complement(composition(X0_2, X1_2))), complement(X1_2))=complement(X1_2)).
% 53.00/7.19 fof(converse_idempotence, axiom, ![X0_2]: converse(converse(X0_2))=X0_2).
% 53.00/7.19 fof(converse_multiplicativity, axiom, ![X0_2, X1_2]: converse(composition(X0_2, X1_2))=composition(converse(X1_2), converse(X0_2))).
% 53.00/7.19 fof(def_top, axiom, ![X0_2]: top=join(X0_2, complement(X0_2))).
% 53.00/7.19 fof(def_zero, axiom, ![X0_2]: zero=meet(X0_2, complement(X0_2))).
% 53.00/7.19 fof(goals, conjecture, ![X0_2, X1_2, X2_2]: ((join(X0_2, one)=one & join(X1_2, one)=one) => (join(meet(composition(X0_2, X2_2), composition(X1_2, X2_2)), composition(meet(X0_2, X1_2), X2_2))=composition(meet(X0_2, X1_2), X2_2) & join(composition(meet(X0_2, X1_2), X2_2), meet(composition(X0_2, X2_2), composition(X1_2, X2_2)))=meet(composition(X0_2, X2_2), composition(X1_2, X2_2))))).
% 53.00/7.19 fof(maddux1_join_commutativity, axiom, ![X0_2, X1_2]: join(X0_2, X1_2)=join(X1_2, X0_2)).
% 53.00/7.19 fof(maddux2_join_associativity, axiom, ![X0_2, X1_2, X2_2]: join(X0_2, join(X1_2, X2_2))=join(join(X0_2, X1_2), X2_2)).
% 53.00/7.19 fof(maddux3_a_kind_of_de_Morgan, axiom, ![X0_2, X1_2]: X0_2=join(complement(join(complement(X0_2), complement(X1_2))), complement(join(complement(X0_2), X1_2)))).
% 53.00/7.19 fof(maddux4_definiton_of_meet, axiom, ![X0_2, X1_2]: meet(X0_2, X1_2)=complement(join(complement(X0_2), complement(X1_2)))).
% 53.00/7.19
% 53.00/7.19 Now clausify the problem and encode Horn clauses using encoding 3 of
% 53.00/7.19 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 53.00/7.19 We repeatedly replace C & s=t => u=v by the two clauses:
% 53.00/7.19 fresh(y, y, x1...xn) = u
% 53.00/7.19 C => fresh(s, t, x1...xn) = v
% 53.00/7.19 where fresh is a fresh function symbol and x1..xn are the free
% 53.00/7.19 variables of u and v.
% 53.00/7.19 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 53.00/7.19 input problem has no model of domain size 1).
% 53.00/7.19
% 53.00/7.19 The encoding turns the above axioms into the following unit equations and goals:
% 53.00/7.19
% 53.00/7.19 Axiom 1 (composition_identity): composition(X, one) = X.
% 53.00/7.19 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 53.00/7.19 Axiom 3 (goals): join(x1, one) = one.
% 53.00/7.19 Axiom 4 (goals_1): join(x0, one) = one.
% 53.00/7.19 Axiom 5 (converse_idempotence): converse(converse(X)) = X.
% 53.00/7.19 Axiom 6 (def_top): top = join(X, complement(X)).
% 53.00/7.19 Axiom 7 (def_zero): zero = meet(X, complement(X)).
% 53.00/7.19 Axiom 8 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 53.00/7.19 Axiom 9 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 53.00/7.19 Axiom 10 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 53.00/7.19 Axiom 11 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 53.00/7.19 Axiom 12 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 53.00/7.19 Axiom 13 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 53.00/7.19 Axiom 14 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 53.00/7.19 Axiom 15 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 53.00/7.19
% 53.00/7.19 Lemma 16: complement(top) = zero.
% 53.00/7.19 Proof:
% 53.00/7.19 complement(top)
% 53.00/7.19 = { by axiom 6 (def_top) }
% 53.00/7.19 complement(join(complement(X), complement(complement(X))))
% 53.00/7.19 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 53.00/7.19 meet(X, complement(X))
% 53.00/7.19 = { by axiom 7 (def_zero) R->L }
% 53.00/7.19 zero
% 53.00/7.19
% 53.00/7.19 Lemma 17: join(X, join(Y, complement(X))) = join(Y, top).
% 53.00/7.19 Proof:
% 53.00/7.19 join(X, join(Y, complement(X)))
% 53.00/7.19 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 53.00/7.19 join(X, join(complement(X), Y))
% 53.00/7.19 = { by axiom 11 (maddux2_join_associativity) }
% 53.00/7.19 join(join(X, complement(X)), Y)
% 53.00/7.19 = { by axiom 6 (def_top) R->L }
% 53.00/7.19 join(top, Y)
% 53.00/7.19 = { by axiom 2 (maddux1_join_commutativity) }
% 53.00/7.19 join(Y, top)
% 53.00/7.19
% 53.00/7.19 Lemma 18: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 53.00/7.19 Proof:
% 53.00/7.19 converse(composition(converse(X), Y))
% 53.00/7.19 = { by axiom 8 (converse_multiplicativity) }
% 53.00/7.19 composition(converse(Y), converse(converse(X)))
% 53.00/7.19 = { by axiom 5 (converse_idempotence) }
% 53.00/7.19 composition(converse(Y), X)
% 53.00/7.19
% 53.00/7.19 Lemma 19: composition(converse(one), X) = X.
% 53.00/7.19 Proof:
% 53.00/7.19 composition(converse(one), X)
% 53.00/7.19 = { by lemma 18 R->L }
% 53.00/7.19 converse(composition(converse(X), one))
% 53.00/7.19 = { by axiom 1 (composition_identity) }
% 53.00/7.19 converse(converse(X))
% 53.00/7.19 = { by axiom 5 (converse_idempotence) }
% 53.00/7.19 X
% 53.00/7.19
% 53.00/7.19 Lemma 20: composition(X, composition(one, Y)) = composition(X, Y).
% 53.00/7.19 Proof:
% 53.00/7.19 composition(X, composition(one, Y))
% 53.00/7.19 = { by axiom 9 (composition_associativity) }
% 53.00/7.19 composition(composition(X, one), Y)
% 53.00/7.19 = { by axiom 1 (composition_identity) }
% 53.00/7.19 composition(X, Y)
% 53.00/7.19
% 53.00/7.19 Lemma 21: composition(one, X) = X.
% 53.00/7.19 Proof:
% 53.00/7.19 composition(one, X)
% 53.00/7.19 = { by lemma 19 R->L }
% 53.00/7.19 composition(converse(one), composition(one, X))
% 53.00/7.19 = { by lemma 20 }
% 53.00/7.19 composition(converse(one), X)
% 53.00/7.19 = { by lemma 19 }
% 53.00/7.19 X
% 53.00/7.19
% 53.00/7.19 Lemma 22: join(complement(X), complement(X)) = complement(X).
% 53.00/7.19 Proof:
% 53.00/7.19 join(complement(X), complement(X))
% 53.00/7.19 = { by lemma 19 R->L }
% 53.00/7.19 join(complement(X), composition(converse(one), complement(X)))
% 53.00/7.19 = { by lemma 21 R->L }
% 53.00/7.19 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 53.00/7.19 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 53.00/7.19 join(composition(converse(one), complement(composition(one, X))), complement(X))
% 53.00/7.19 = { by axiom 14 (converse_cancellativity) }
% 53.00/7.19 complement(X)
% 53.00/7.19
% 53.00/7.19 Lemma 23: join(top, complement(X)) = top.
% 53.00/7.19 Proof:
% 53.00/7.19 join(top, complement(X))
% 53.00/7.19 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 53.00/7.19 join(complement(X), top)
% 53.00/7.19 = { by lemma 17 R->L }
% 53.00/7.19 join(X, join(complement(X), complement(X)))
% 53.00/7.19 = { by lemma 22 }
% 53.00/7.19 join(X, complement(X))
% 53.00/7.19 = { by axiom 6 (def_top) R->L }
% 53.00/7.19 top
% 53.00/7.19
% 53.00/7.19 Lemma 24: join(X, top) = top.
% 53.00/7.19 Proof:
% 53.00/7.19 join(X, top)
% 53.00/7.19 = { by lemma 23 R->L }
% 53.00/7.19 join(X, join(top, complement(X)))
% 53.00/7.19 = { by lemma 17 }
% 53.00/7.19 join(top, top)
% 53.00/7.19 = { by lemma 17 R->L }
% 53.00/7.19 join(x1, join(top, complement(x1)))
% 53.00/7.19 = { by lemma 23 }
% 53.00/7.19 join(x1, top)
% 53.00/7.20 = { by axiom 6 (def_top) }
% 53.00/7.20 join(x1, join(one, complement(one)))
% 53.00/7.20 = { by axiom 11 (maddux2_join_associativity) }
% 53.00/7.20 join(join(x1, one), complement(one))
% 53.00/7.20 = { by axiom 3 (goals) }
% 53.00/7.20 join(one, complement(one))
% 53.00/7.20 = { by axiom 6 (def_top) R->L }
% 53.00/7.20 top
% 53.00/7.20
% 53.00/7.20 Lemma 25: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 53.00/7.20 Proof:
% 53.00/7.20 join(meet(X, Y), complement(join(complement(X), Y)))
% 53.00/7.20 = { by axiom 12 (maddux4_definiton_of_meet) }
% 53.00/7.20 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 53.00/7.20 = { by axiom 15 (maddux3_a_kind_of_de_Morgan) R->L }
% 53.00/7.20 X
% 53.00/7.20
% 53.00/7.20 Lemma 26: join(zero, meet(X, X)) = X.
% 53.00/7.20 Proof:
% 53.00/7.20 join(zero, meet(X, X))
% 53.00/7.20 = { by axiom 12 (maddux4_definiton_of_meet) }
% 53.00/7.20 join(zero, complement(join(complement(X), complement(X))))
% 53.00/7.20 = { by axiom 7 (def_zero) }
% 53.00/7.20 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 53.00/7.20 = { by lemma 25 }
% 53.00/7.20 X
% 53.00/7.20
% 53.00/7.20 Lemma 27: join(X, zero) = X.
% 53.00/7.20 Proof:
% 53.00/7.20 join(X, zero)
% 53.00/7.20 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 53.00/7.20 join(zero, X)
% 53.00/7.20 = { by lemma 26 R->L }
% 53.00/7.20 join(zero, join(zero, meet(X, X)))
% 53.00/7.20 = { by axiom 11 (maddux2_join_associativity) }
% 53.00/7.20 join(join(zero, zero), meet(X, X))
% 53.00/7.20 = { by lemma 16 R->L }
% 53.00/7.20 join(join(zero, complement(top)), meet(X, X))
% 53.00/7.20 = { by lemma 16 R->L }
% 53.00/7.20 join(join(complement(top), complement(top)), meet(X, X))
% 53.00/7.20 = { by lemma 22 }
% 53.00/7.20 join(complement(top), meet(X, X))
% 53.00/7.20 = { by lemma 16 }
% 53.00/7.20 join(zero, meet(X, X))
% 53.00/7.20 = { by lemma 26 }
% 53.00/7.20 X
% 53.00/7.20
% 53.00/7.20 Lemma 28: join(zero, X) = X.
% 53.00/7.20 Proof:
% 53.00/7.20 join(zero, X)
% 53.00/7.20 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 53.00/7.20 join(X, zero)
% 53.00/7.20 = { by lemma 27 }
% 53.00/7.20 X
% 53.00/7.20
% 53.00/7.20 Lemma 29: complement(complement(X)) = meet(X, X).
% 53.00/7.20 Proof:
% 53.00/7.20 complement(complement(X))
% 53.00/7.20 = { by lemma 22 R->L }
% 53.00/7.20 complement(join(complement(X), complement(X)))
% 53.00/7.20 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 53.00/7.20 meet(X, X)
% 53.00/7.20
% 53.00/7.20 Lemma 30: complement(complement(X)) = X.
% 53.00/7.20 Proof:
% 53.00/7.20 complement(complement(X))
% 53.00/7.20 = { by lemma 28 R->L }
% 53.00/7.20 join(zero, complement(complement(X)))
% 53.00/7.20 = { by lemma 29 }
% 53.00/7.20 join(zero, meet(X, X))
% 53.00/7.20 = { by lemma 26 }
% 53.00/7.20 X
% 53.00/7.20
% 53.00/7.20 Lemma 31: meet(Y, X) = meet(X, Y).
% 53.00/7.20 Proof:
% 53.00/7.20 meet(Y, X)
% 53.00/7.20 = { by axiom 12 (maddux4_definiton_of_meet) }
% 53.00/7.20 complement(join(complement(Y), complement(X)))
% 53.00/7.20 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 53.00/7.20 complement(join(complement(X), complement(Y)))
% 53.00/7.20 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 53.00/7.20 meet(X, Y)
% 53.00/7.20
% 53.00/7.20 Lemma 32: complement(join(zero, complement(X))) = meet(X, top).
% 53.00/7.20 Proof:
% 53.00/7.20 complement(join(zero, complement(X)))
% 53.00/7.20 = { by lemma 16 R->L }
% 53.00/7.20 complement(join(complement(top), complement(X)))
% 53.00/7.20 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 53.00/7.20 meet(top, X)
% 53.00/7.20 = { by lemma 31 R->L }
% 53.00/7.20 meet(X, top)
% 53.00/7.20
% 53.00/7.20 Lemma 33: meet(X, top) = X.
% 53.00/7.20 Proof:
% 53.00/7.20 meet(X, top)
% 53.00/7.20 = { by lemma 32 R->L }
% 53.00/7.20 complement(join(zero, complement(X)))
% 53.00/7.20 = { by lemma 28 }
% 53.00/7.20 complement(complement(X))
% 53.00/7.20 = { by lemma 30 }
% 53.00/7.20 X
% 53.00/7.20
% 53.00/7.20 Lemma 34: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
% 53.00/7.20 Proof:
% 53.00/7.20 complement(join(complement(X), meet(Y, Z)))
% 53.00/7.20 = { by lemma 31 }
% 53.00/7.20 complement(join(complement(X), meet(Z, Y)))
% 53.00/7.20 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 53.00/7.20 complement(join(meet(Z, Y), complement(X)))
% 53.00/7.20 = { by axiom 12 (maddux4_definiton_of_meet) }
% 53.00/7.20 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 53.00/7.20 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 53.00/7.20 meet(join(complement(Z), complement(Y)), X)
% 53.00/7.20 = { by lemma 31 R->L }
% 53.00/7.20 meet(X, join(complement(Z), complement(Y)))
% 53.00/7.20 = { by axiom 2 (maddux1_join_commutativity) }
% 53.00/7.20 meet(X, join(complement(Y), complement(Z)))
% 53.00/7.20
% 53.00/7.20 Lemma 35: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 53.00/7.20 Proof:
% 53.00/7.20 join(complement(X), complement(Y))
% 53.00/7.20 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 53.00/7.20 join(complement(Y), complement(X))
% 53.00/7.20 = { by lemma 33 R->L }
% 53.00/7.20 meet(join(complement(Y), complement(X)), top)
% 53.00/7.20 = { by lemma 31 R->L }
% 53.00/7.20 meet(top, join(complement(Y), complement(X)))
% 53.00/7.20 = { by lemma 34 R->L }
% 53.00/7.20 complement(join(complement(top), meet(Y, X)))
% 53.00/7.20 = { by lemma 16 }
% 53.00/7.20 complement(join(zero, meet(Y, X)))
% 53.00/7.20 = { by lemma 28 }
% 53.00/7.20 complement(meet(Y, X))
% 53.00/7.20 = { by lemma 31 R->L }
% 53.00/7.20 complement(meet(X, Y))
% 53.00/7.20
% 53.00/7.20 Lemma 36: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 53.00/7.20 Proof:
% 53.00/7.20 complement(meet(X, complement(Y)))
% 53.00/7.20 = { by lemma 31 }
% 53.00/7.20 complement(meet(complement(Y), X))
% 53.00/7.20 = { by lemma 28 R->L }
% 53.00/7.20 complement(meet(join(zero, complement(Y)), X))
% 53.00/7.20 = { by lemma 35 R->L }
% 53.00/7.20 join(complement(join(zero, complement(Y))), complement(X))
% 53.00/7.20 = { by lemma 32 }
% 53.00/7.20 join(meet(Y, top), complement(X))
% 53.00/7.20 = { by lemma 33 }
% 53.00/7.20 join(Y, complement(X))
% 53.00/7.20
% 53.00/7.20 Lemma 37: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 53.00/7.20 Proof:
% 53.00/7.20 complement(meet(complement(X), Y))
% 53.00/7.20 = { by lemma 31 }
% 53.00/7.20 complement(meet(Y, complement(X)))
% 53.00/7.20 = { by lemma 36 }
% 53.00/7.20 join(X, complement(Y))
% 53.00/7.20
% 53.00/7.20 Lemma 38: join(X, complement(meet(X, Y))) = top.
% 53.00/7.20 Proof:
% 53.00/7.20 join(X, complement(meet(X, Y)))
% 53.00/7.20 = { by lemma 31 }
% 53.00/7.20 join(X, complement(meet(Y, X)))
% 53.00/7.20 = { by lemma 35 R->L }
% 53.00/7.20 join(X, join(complement(Y), complement(X)))
% 53.00/7.20 = { by lemma 17 }
% 53.00/7.20 join(complement(Y), top)
% 53.00/7.20 = { by lemma 24 }
% 53.00/7.20 top
% 53.00/7.20
% 53.00/7.20 Lemma 39: meet(X, join(X, complement(Y))) = X.
% 53.00/7.20 Proof:
% 53.00/7.20 meet(X, join(X, complement(Y)))
% 53.00/7.20 = { by lemma 27 R->L }
% 53.00/7.20 join(meet(X, join(X, complement(Y))), zero)
% 53.00/7.20 = { by lemma 16 R->L }
% 53.00/7.20 join(meet(X, join(X, complement(Y))), complement(top))
% 53.00/7.20 = { by lemma 37 R->L }
% 53.00/7.20 join(meet(X, complement(meet(complement(X), Y))), complement(top))
% 53.00/7.20 = { by lemma 38 R->L }
% 53.00/7.20 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(meet(complement(X), Y)))))
% 53.00/7.20 = { by lemma 25 }
% 53.00/7.20 X
% 53.00/7.20
% 53.00/7.20 Lemma 40: join(X, meet(X, Y)) = X.
% 53.00/7.20 Proof:
% 53.00/7.20 join(X, meet(X, Y))
% 53.00/7.20 = { by axiom 12 (maddux4_definiton_of_meet) }
% 53.00/7.20 join(X, complement(join(complement(X), complement(Y))))
% 53.00/7.20 = { by lemma 37 R->L }
% 53.00/7.20 complement(meet(complement(X), join(complement(X), complement(Y))))
% 53.00/7.20 = { by lemma 39 }
% 53.00/7.20 complement(complement(X))
% 53.00/7.20 = { by lemma 30 }
% 53.00/7.20 X
% 53.00/7.20
% 53.00/7.20 Lemma 41: join(X, meet(Y, X)) = X.
% 53.00/7.20 Proof:
% 53.00/7.20 join(X, meet(Y, X))
% 53.00/7.20 = { by lemma 31 R->L }
% 53.00/7.20 join(X, meet(X, Y))
% 53.00/7.20 = { by lemma 40 }
% 53.00/7.20 X
% 53.00/7.20
% 53.00/7.20 Lemma 42: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 53.00/7.20 Proof:
% 53.00/7.20 converse(composition(X, converse(Y)))
% 53.00/7.20 = { by axiom 8 (converse_multiplicativity) }
% 53.00/7.20 composition(converse(converse(Y)), converse(X))
% 53.00/7.20 = { by axiom 5 (converse_idempotence) }
% 53.00/7.20 composition(Y, converse(X))
% 53.00/7.20
% 53.00/7.20 Lemma 43: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 53.00/7.20 Proof:
% 53.00/7.20 converse(join(X, converse(Y)))
% 53.00/7.20 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 53.00/7.20 converse(join(converse(Y), X))
% 53.00/7.20 = { by axiom 10 (converse_additivity) }
% 53.00/7.20 join(converse(converse(Y)), converse(X))
% 53.00/7.20 = { by axiom 5 (converse_idempotence) }
% 53.00/7.20 join(Y, converse(X))
% 53.00/7.20
% 53.00/7.20 Lemma 44: join(X, composition(Y, X)) = composition(join(Y, one), X).
% 53.00/7.20 Proof:
% 53.00/7.20 join(X, composition(Y, X))
% 53.00/7.20 = { by lemma 21 R->L }
% 53.00/7.20 join(composition(one, X), composition(Y, X))
% 53.00/7.20 = { by axiom 13 (composition_distributivity) R->L }
% 53.00/7.20 composition(join(one, Y), X)
% 53.00/7.20 = { by axiom 2 (maddux1_join_commutativity) }
% 53.00/7.20 composition(join(Y, one), X)
% 53.00/7.20
% 53.00/7.20 Lemma 45: join(X, composition(X, Y)) = composition(X, join(Y, one)).
% 53.00/7.20 Proof:
% 53.00/7.20 join(X, composition(X, Y))
% 53.00/7.20 = { by axiom 5 (converse_idempotence) R->L }
% 53.00/7.20 join(X, composition(X, converse(converse(Y))))
% 53.00/7.20 = { by lemma 42 R->L }
% 53.00/7.20 join(X, converse(composition(converse(Y), converse(X))))
% 53.00/7.20 = { by lemma 43 R->L }
% 53.00/7.20 converse(join(composition(converse(Y), converse(X)), converse(X)))
% 53.00/7.20 = { by axiom 2 (maddux1_join_commutativity) }
% 53.00/7.20 converse(join(converse(X), composition(converse(Y), converse(X))))
% 53.00/7.20 = { by lemma 44 }
% 53.00/7.20 converse(composition(join(converse(Y), one), converse(X)))
% 53.00/7.20 = { by lemma 42 }
% 53.00/7.20 composition(X, converse(join(converse(Y), one)))
% 53.00/7.20 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 53.00/7.20 composition(X, converse(join(one, converse(Y))))
% 53.00/7.20 = { by axiom 10 (converse_additivity) }
% 53.00/7.20 composition(X, join(converse(one), converse(converse(Y))))
% 53.00/7.20 = { by axiom 1 (composition_identity) R->L }
% 53.00/7.20 composition(X, join(composition(converse(one), one), converse(converse(Y))))
% 53.00/7.20 = { by lemma 19 }
% 53.00/7.20 composition(X, join(one, converse(converse(Y))))
% 53.00/7.20 = { by axiom 5 (converse_idempotence) }
% 53.00/7.20 composition(X, join(one, Y))
% 53.00/7.20 = { by axiom 2 (maddux1_join_commutativity) }
% 53.00/7.20 composition(X, join(Y, one))
% 53.00/7.20
% 53.00/7.20 Lemma 46: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 53.00/7.20 Proof:
% 53.00/7.20 join(Y, join(X, Z))
% 53.00/7.20 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 53.00/7.20 join(join(X, Z), Y)
% 53.00/7.20 = { by axiom 11 (maddux2_join_associativity) R->L }
% 53.00/7.20 join(X, join(Z, Y))
% 53.00/7.20 = { by axiom 2 (maddux1_join_commutativity) }
% 53.00/7.20 join(X, join(Y, Z))
% 53.00/7.20
% 53.00/7.20 Lemma 47: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 53.00/7.20 Proof:
% 53.00/7.20 complement(join(X, complement(Y)))
% 53.00/7.20 = { by lemma 28 R->L }
% 53.00/7.20 complement(join(zero, join(X, complement(Y))))
% 53.00/7.20 = { by lemma 36 R->L }
% 53.00/7.20 complement(join(zero, complement(meet(Y, complement(X)))))
% 53.00/7.20 = { by lemma 32 }
% 53.00/7.20 meet(meet(Y, complement(X)), top)
% 53.00/7.20 = { by lemma 33 }
% 53.00/7.20 meet(Y, complement(X))
% 53.00/7.20
% 53.00/7.20 Lemma 48: meet(X, join(X, Y)) = X.
% 53.00/7.20 Proof:
% 53.00/7.20 meet(X, join(X, Y))
% 53.00/7.20 = { by lemma 33 R->L }
% 53.00/7.20 meet(X, join(X, meet(Y, top)))
% 53.00/7.20 = { by lemma 32 R->L }
% 53.00/7.20 meet(X, join(X, complement(join(zero, complement(Y)))))
% 53.00/7.20 = { by lemma 39 }
% 53.00/7.20 X
% 53.00/7.20
% 53.00/7.20 Lemma 49: join(composition(X, join(Z, one)), Y) = join(X, join(Y, composition(X, Z))).
% 53.00/7.20 Proof:
% 53.00/7.20 join(composition(X, join(Z, one)), Y)
% 53.00/7.20 = { by lemma 45 R->L }
% 53.00/7.20 join(join(X, composition(X, Z)), Y)
% 53.00/7.20 = { by axiom 11 (maddux2_join_associativity) R->L }
% 53.00/7.20 join(X, join(composition(X, Z), Y))
% 53.00/7.20 = { by axiom 2 (maddux1_join_commutativity) }
% 53.00/7.20 join(X, join(Y, composition(X, Z)))
% 53.00/7.20
% 53.00/7.20 Lemma 50: join(X, join(Y, composition(X, x0))) = join(X, Y).
% 53.00/7.20 Proof:
% 53.00/7.20 join(X, join(Y, composition(X, x0)))
% 53.00/7.20 = { by lemma 49 R->L }
% 53.00/7.20 join(composition(X, join(x0, one)), Y)
% 53.00/7.20 = { by axiom 4 (goals_1) }
% 53.00/7.20 join(composition(X, one), Y)
% 53.00/7.20 = { by axiom 1 (composition_identity) }
% 53.00/7.20 join(X, Y)
% 53.00/7.20
% 53.00/7.20 Lemma 51: meet(X, join(Y, join(Z, X))) = X.
% 53.00/7.20 Proof:
% 53.00/7.20 meet(X, join(Y, join(Z, X)))
% 53.00/7.20 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 53.00/7.20 meet(X, join(Y, join(X, Z)))
% 53.00/7.20 = { by lemma 46 }
% 53.00/7.20 meet(X, join(X, join(Y, Z)))
% 53.00/7.20 = { by lemma 48 }
% 53.00/7.20 X
% 53.00/7.20
% 53.00/7.20 Lemma 52: meet(X, composition(X, x0)) = composition(X, x0).
% 53.00/7.20 Proof:
% 53.00/7.20 meet(X, composition(X, x0))
% 53.00/7.20 = { by lemma 31 }
% 53.00/7.20 meet(composition(X, x0), X)
% 53.00/7.20 = { by lemma 27 R->L }
% 53.00/7.20 meet(composition(X, x0), join(X, zero))
% 53.00/7.20 = { by lemma 50 R->L }
% 53.00/7.20 meet(composition(X, x0), join(X, join(zero, composition(X, x0))))
% 53.00/7.20 = { by lemma 51 }
% 53.00/7.20 composition(X, x0)
% 53.00/7.20
% 53.00/7.21 Lemma 53: meet(X, composition(top, X)) = X.
% 53.00/7.21 Proof:
% 53.00/7.21 meet(X, composition(top, X))
% 53.00/7.21 = { by lemma 24 R->L }
% 53.00/7.21 meet(X, composition(join(one, top), X))
% 53.00/7.21 = { by axiom 2 (maddux1_join_commutativity) }
% 53.00/7.21 meet(X, composition(join(top, one), X))
% 53.00/7.21 = { by lemma 44 R->L }
% 53.00/7.21 meet(X, join(X, composition(top, X)))
% 53.00/7.21 = { by lemma 48 }
% 53.00/7.21 X
% 53.00/7.21
% 53.00/7.21 Lemma 54: join(X, join(Y, composition(x1, X))) = join(X, Y).
% 53.00/7.21 Proof:
% 53.00/7.21 join(X, join(Y, composition(x1, X)))
% 53.00/7.21 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 53.00/7.21 join(X, join(composition(x1, X), Y))
% 53.00/7.21 = { by axiom 11 (maddux2_join_associativity) }
% 53.00/7.21 join(join(X, composition(x1, X)), Y)
% 53.00/7.21 = { by lemma 44 }
% 53.00/7.21 join(composition(join(x1, one), X), Y)
% 53.00/7.21 = { by axiom 3 (goals) }
% 53.00/7.21 join(composition(one, X), Y)
% 53.00/7.21 = { by lemma 21 }
% 53.00/7.21 join(X, Y)
% 53.00/7.21
% 53.00/7.21 Lemma 55: meet(X, composition(x1, X)) = composition(x1, X).
% 53.00/7.21 Proof:
% 53.00/7.21 meet(X, composition(x1, X))
% 53.00/7.21 = { by lemma 31 }
% 53.00/7.21 meet(composition(x1, X), X)
% 53.00/7.21 = { by lemma 27 R->L }
% 53.00/7.21 meet(composition(x1, X), join(X, zero))
% 53.00/7.21 = { by lemma 54 R->L }
% 53.00/7.21 meet(composition(x1, X), join(X, join(zero, composition(x1, X))))
% 53.00/7.21 = { by lemma 51 }
% 53.00/7.21 composition(x1, X)
% 53.00/7.21
% 53.00/7.21 Lemma 56: join(complement(X), meet(X, Y)) = join(Y, complement(X)).
% 53.00/7.21 Proof:
% 53.00/7.21 join(complement(X), meet(X, Y))
% 53.00/7.21 = { by lemma 31 }
% 53.00/7.21 join(complement(X), meet(Y, X))
% 53.00/7.21 = { by lemma 40 R->L }
% 53.00/7.21 join(join(complement(X), meet(complement(X), Y)), meet(Y, X))
% 53.00/7.21 = { by axiom 11 (maddux2_join_associativity) R->L }
% 53.00/7.21 join(complement(X), join(meet(complement(X), Y), meet(Y, X)))
% 53.00/7.21 = { by axiom 2 (maddux1_join_commutativity) }
% 53.00/7.21 join(complement(X), join(meet(Y, X), meet(complement(X), Y)))
% 53.00/7.21 = { by lemma 31 }
% 53.00/7.21 join(complement(X), join(meet(Y, X), meet(Y, complement(X))))
% 53.00/7.21 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 53.00/7.21 join(complement(X), join(meet(Y, complement(X)), meet(Y, X)))
% 53.00/7.21 = { by axiom 12 (maddux4_definiton_of_meet) }
% 53.00/7.21 join(complement(X), join(meet(Y, complement(X)), complement(join(complement(Y), complement(X)))))
% 53.00/7.21 = { by lemma 25 }
% 53.00/7.21 join(complement(X), Y)
% 53.00/7.21 = { by axiom 2 (maddux1_join_commutativity) }
% 53.00/7.21 join(Y, complement(X))
% 53.00/7.21
% 53.00/7.21 Lemma 57: meet(X, complement(meet(X, Y))) = meet(X, complement(Y)).
% 53.00/7.21 Proof:
% 53.00/7.21 meet(X, complement(meet(X, Y)))
% 53.00/7.21 = { by lemma 35 R->L }
% 53.00/7.21 meet(X, join(complement(X), complement(Y)))
% 53.00/7.21 = { by lemma 34 R->L }
% 53.00/7.21 complement(join(complement(X), meet(X, Y)))
% 53.00/7.21 = { by lemma 56 }
% 53.00/7.21 complement(join(Y, complement(X)))
% 53.00/7.21 = { by lemma 47 }
% 53.00/7.21 meet(X, complement(Y))
% 53.00/7.21
% 53.00/7.21 Lemma 58: meet(X, join(complement(X), Y)) = meet(X, Y).
% 53.00/7.21 Proof:
% 53.00/7.21 meet(X, join(complement(X), Y))
% 53.00/7.21 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 53.00/7.21 meet(X, join(Y, complement(X)))
% 53.00/7.21 = { by lemma 36 R->L }
% 53.00/7.21 meet(X, complement(meet(X, complement(Y))))
% 53.00/7.21 = { by lemma 57 }
% 53.00/7.21 meet(X, complement(complement(Y)))
% 53.00/7.21 = { by lemma 30 }
% 53.00/7.21 meet(X, Y)
% 53.00/7.21
% 53.00/7.21 Lemma 59: meet(X, composition(x1, top)) = composition(x1, X).
% 53.00/7.21 Proof:
% 53.00/7.21 meet(X, composition(x1, top))
% 53.00/7.21 = { by lemma 58 R->L }
% 53.00/7.21 meet(X, join(complement(X), composition(x1, top)))
% 53.00/7.21 = { by axiom 6 (def_top) }
% 53.00/7.21 meet(X, join(complement(X), composition(x1, join(complement(X), complement(complement(X))))))
% 53.00/7.21 = { by axiom 5 (converse_idempotence) R->L }
% 53.00/7.21 meet(X, join(complement(X), composition(x1, join(complement(X), converse(converse(complement(complement(X))))))))
% 53.00/7.21 = { by axiom 5 (converse_idempotence) R->L }
% 53.00/7.21 meet(X, join(complement(X), composition(converse(converse(x1)), join(complement(X), converse(converse(complement(complement(X))))))))
% 53.00/7.21 = { by lemma 43 R->L }
% 53.00/7.21 meet(X, join(complement(X), composition(converse(converse(x1)), converse(join(converse(complement(complement(X))), converse(complement(X)))))))
% 53.00/7.21 = { by axiom 8 (converse_multiplicativity) R->L }
% 53.00/7.21 meet(X, join(complement(X), converse(composition(join(converse(complement(complement(X))), converse(complement(X))), converse(x1)))))
% 53.00/7.21 = { by axiom 13 (composition_distributivity) }
% 53.00/7.21 meet(X, join(complement(X), converse(join(composition(converse(complement(complement(X))), converse(x1)), composition(converse(complement(X)), converse(x1))))))
% 53.00/7.21 = { by axiom 8 (converse_multiplicativity) R->L }
% 53.00/7.21 meet(X, join(complement(X), converse(join(converse(composition(x1, complement(complement(X)))), composition(converse(complement(X)), converse(x1))))))
% 53.00/7.21 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 53.00/7.21 meet(X, join(complement(X), converse(join(composition(converse(complement(X)), converse(x1)), converse(composition(x1, complement(complement(X))))))))
% 53.00/7.21 = { by axiom 10 (converse_additivity) }
% 53.00/7.21 meet(X, join(complement(X), join(converse(composition(converse(complement(X)), converse(x1))), converse(converse(composition(x1, complement(complement(X))))))))
% 53.00/7.21 = { by lemma 18 }
% 53.00/7.21 meet(X, join(complement(X), join(composition(converse(converse(x1)), complement(X)), converse(converse(composition(x1, complement(complement(X))))))))
% 53.00/7.21 = { by axiom 2 (maddux1_join_commutativity) }
% 53.00/7.21 meet(X, join(complement(X), join(converse(converse(composition(x1, complement(complement(X))))), composition(converse(converse(x1)), complement(X)))))
% 53.00/7.21 = { by axiom 5 (converse_idempotence) }
% 53.00/7.21 meet(X, join(complement(X), join(composition(x1, complement(complement(X))), composition(converse(converse(x1)), complement(X)))))
% 53.00/7.21 = { by axiom 5 (converse_idempotence) }
% 53.00/7.21 meet(X, join(complement(X), join(composition(x1, complement(complement(X))), composition(x1, complement(X)))))
% 53.00/7.21 = { by lemma 54 }
% 53.00/7.21 meet(X, join(complement(X), composition(x1, complement(complement(X)))))
% 53.00/7.21 = { by lemma 58 }
% 53.00/7.21 meet(X, composition(x1, complement(complement(X))))
% 53.00/7.21 = { by lemma 30 }
% 53.00/7.21 meet(X, composition(x1, X))
% 53.00/7.21 = { by lemma 55 }
% 53.00/7.21 composition(x1, X)
% 53.00/7.21
% 53.00/7.21 Lemma 60: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 53.00/7.21 Proof:
% 53.00/7.21 meet(Y, meet(X, Z))
% 53.00/7.21 = { by lemma 31 }
% 53.00/7.21 meet(Y, meet(Z, X))
% 53.00/7.21 = { by lemma 33 R->L }
% 53.00/7.21 meet(meet(Y, meet(Z, X)), top)
% 53.00/7.21 = { by lemma 32 R->L }
% 53.00/7.21 complement(join(zero, complement(meet(Y, meet(Z, X)))))
% 53.00/7.21 = { by lemma 31 }
% 53.00/7.21 complement(join(zero, complement(meet(Y, meet(X, Z)))))
% 53.00/7.21 = { by axiom 12 (maddux4_definiton_of_meet) }
% 53.00/7.21 complement(join(zero, complement(meet(Y, complement(join(complement(X), complement(Z)))))))
% 53.00/7.21 = { by lemma 36 }
% 53.00/7.21 complement(join(zero, join(join(complement(X), complement(Z)), complement(Y))))
% 53.00/7.21 = { by axiom 11 (maddux2_join_associativity) R->L }
% 53.00/7.21 complement(join(zero, join(complement(X), join(complement(Z), complement(Y)))))
% 53.00/7.21 = { by lemma 35 }
% 53.00/7.21 complement(join(zero, join(complement(X), complement(meet(Z, Y)))))
% 53.00/7.21 = { by lemma 35 }
% 53.00/7.21 complement(join(zero, complement(meet(X, meet(Z, Y)))))
% 53.00/7.21 = { by lemma 31 R->L }
% 53.00/7.21 complement(join(zero, complement(meet(X, meet(Y, Z)))))
% 53.00/7.21 = { by lemma 32 }
% 53.00/7.21 meet(meet(X, meet(Y, Z)), top)
% 53.00/7.21 = { by lemma 33 }
% 53.00/7.21 meet(X, meet(Y, Z))
% 53.00/7.21
% 53.00/7.21 Lemma 61: meet(Z, meet(Y, X)) = meet(X, meet(Y, Z)).
% 53.00/7.21 Proof:
% 53.00/7.21 meet(Z, meet(Y, X))
% 53.00/7.21 = { by lemma 31 R->L }
% 53.00/7.21 meet(Z, meet(X, Y))
% 53.00/7.21 = { by lemma 60 R->L }
% 53.00/7.21 meet(X, meet(Z, Y))
% 53.00/7.21 = { by lemma 31 R->L }
% 53.00/7.21 meet(X, meet(Y, Z))
% 53.00/7.21
% 53.00/7.21 Lemma 62: join(X, join(Y, composition(X, x1))) = join(X, Y).
% 53.00/7.21 Proof:
% 53.00/7.21 join(X, join(Y, composition(X, x1)))
% 53.00/7.21 = { by lemma 49 R->L }
% 53.00/7.21 join(composition(X, join(x1, one)), Y)
% 53.00/7.21 = { by axiom 3 (goals) }
% 53.00/7.21 join(composition(X, one), Y)
% 53.00/7.21 = { by axiom 1 (composition_identity) }
% 53.00/7.21 join(X, Y)
% 53.00/7.21
% 53.00/7.21 Lemma 63: join(composition(X, Y), composition(Z, Y)) = composition(join(Z, X), Y).
% 53.00/7.21 Proof:
% 53.00/7.21 join(composition(X, Y), composition(Z, Y))
% 53.00/7.21 = { by axiom 13 (composition_distributivity) R->L }
% 53.00/7.21 composition(join(X, Z), Y)
% 53.00/7.21 = { by axiom 2 (maddux1_join_commutativity) }
% 53.00/7.21 composition(join(Z, X), Y)
% 53.00/7.21
% 53.00/7.21 Lemma 64: join(composition(meet(X, Y), Z), composition(Y, Z)) = composition(Y, Z).
% 53.00/7.21 Proof:
% 53.00/7.21 join(composition(meet(X, Y), Z), composition(Y, Z))
% 53.00/7.21 = { by lemma 63 }
% 53.00/7.21 composition(join(Y, meet(X, Y)), Z)
% 53.00/7.21 = { by lemma 41 }
% 53.00/7.21 composition(Y, Z)
% 53.00/7.21
% 53.00/7.21 Lemma 65: join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y)) = meet(composition(X, Y), composition(Z, Y)).
% 53.00/7.21 Proof:
% 53.00/7.21 join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y))
% 53.00/7.21 = { by lemma 48 R->L }
% 53.00/7.21 join(meet(composition(X, Y), composition(Z, Y)), meet(composition(meet(X, Z), Y), join(composition(meet(X, Z), Y), composition(X, Y))))
% 53.00/7.21 = { by lemma 63 }
% 53.00/7.21 join(meet(composition(X, Y), composition(Z, Y)), meet(composition(meet(X, Z), Y), composition(join(X, meet(X, Z)), Y)))
% 53.00/7.21 = { by lemma 40 }
% 53.00/7.21 join(meet(composition(X, Y), composition(Z, Y)), meet(composition(meet(X, Z), Y), composition(X, Y)))
% 53.00/7.21 = { by lemma 31 }
% 53.00/7.21 join(meet(composition(X, Y), composition(Z, Y)), meet(composition(X, Y), composition(meet(X, Z), Y)))
% 53.00/7.21 = { by lemma 48 R->L }
% 53.00/7.21 join(meet(composition(X, Y), composition(Z, Y)), meet(composition(X, Y), meet(composition(meet(X, Z), Y), join(composition(meet(X, Z), Y), composition(Z, Y)))))
% 53.00/7.21 = { by lemma 64 }
% 53.00/7.21 join(meet(composition(X, Y), composition(Z, Y)), meet(composition(X, Y), meet(composition(meet(X, Z), Y), composition(Z, Y))))
% 53.00/7.21 = { by lemma 60 }
% 53.00/7.21 join(meet(composition(X, Y), composition(Z, Y)), meet(composition(meet(X, Z), Y), meet(composition(X, Y), composition(Z, Y))))
% 53.00/7.21 = { by lemma 31 R->L }
% 53.00/7.21 join(meet(composition(X, Y), composition(Z, Y)), meet(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y)))
% 53.00/7.21 = { by lemma 40 }
% 53.00/7.21 meet(composition(X, Y), composition(Z, Y))
% 53.00/7.21
% 53.00/7.21 Goal 1 (goals_2): tuple(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), join(composition(meet(x0, x1), x2), meet(composition(x0, x2), composition(x1, x2)))) = tuple(composition(meet(x0, x1), x2), meet(composition(x0, x2), composition(x1, x2))).
% 53.00/7.21 Proof:
% 53.00/7.21 tuple(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), join(composition(meet(x0, x1), x2), meet(composition(x0, x2), composition(x1, x2))))
% 53.00/7.21 = { by axiom 2 (maddux1_join_commutativity) }
% 53.00/7.21 tuple(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.00/7.21 = { by lemma 48 R->L }
% 53.00/7.21 tuple(meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), join(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), composition(x1, x2))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.00/7.21 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 53.00/7.21 tuple(meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), join(composition(x1, x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.00/7.21 = { by lemma 46 }
% 53.00/7.21 tuple(meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), join(meet(composition(x0, x2), composition(x1, x2)), join(composition(x1, x2), composition(meet(x0, x1), x2)))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.00/7.21 = { by axiom 2 (maddux1_join_commutativity) }
% 53.00/7.21 tuple(meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), join(meet(composition(x0, x2), composition(x1, x2)), join(composition(meet(x0, x1), x2), composition(x1, x2)))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.00/7.21 = { by lemma 64 }
% 53.00/7.21 tuple(meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(x1, x2))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.00/7.21 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 53.00/7.21 tuple(meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), join(composition(x1, x2), meet(composition(x0, x2), composition(x1, x2)))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.00/7.21 = { by lemma 31 R->L }
% 53.00/7.21 tuple(meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), join(composition(x1, x2), meet(composition(x1, x2), composition(x0, x2)))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.21 = { by lemma 40 }
% 53.75/7.21 tuple(meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), composition(x1, x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.21 = { by lemma 53 R->L }
% 53.75/7.22 tuple(meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), composition(meet(x1, composition(top, x1)), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 58 R->L }
% 53.75/7.22 tuple(meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), composition(meet(x1, join(complement(x1), composition(top, x1))), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by axiom 6 (def_top) }
% 53.75/7.22 tuple(meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), composition(meet(x1, join(complement(x1), composition(join(complement(x1), complement(complement(x1))), x1))), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 53.75/7.22 tuple(meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), composition(meet(x1, join(complement(x1), composition(join(complement(complement(x1)), complement(x1)), x1))), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by axiom 13 (composition_distributivity) }
% 53.75/7.22 tuple(meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), composition(meet(x1, join(complement(x1), join(composition(complement(complement(x1)), x1), composition(complement(x1), x1)))), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 62 }
% 53.75/7.22 tuple(meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), composition(meet(x1, join(complement(x1), composition(complement(complement(x1)), x1))), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 58 }
% 53.75/7.22 tuple(meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), composition(meet(x1, composition(complement(complement(x1)), x1)), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 30 }
% 53.75/7.22 tuple(meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), composition(meet(x1, composition(x1, x1)), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 31 }
% 53.75/7.22 tuple(meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), composition(meet(composition(x1, x1), x1), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 27 R->L }
% 53.75/7.22 tuple(meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), composition(meet(composition(x1, x1), join(x1, zero)), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 62 R->L }
% 53.75/7.22 tuple(meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), composition(meet(composition(x1, x1), join(x1, join(zero, composition(x1, x1)))), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 51 }
% 53.75/7.22 tuple(meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), composition(composition(x1, x1), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by axiom 9 (composition_associativity) R->L }
% 53.75/7.22 tuple(meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), composition(x1, composition(x1, x2))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 59 R->L }
% 53.75/7.22 tuple(meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), meet(composition(x1, x2), composition(x1, top))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 31 }
% 53.75/7.22 tuple(meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), meet(composition(x1, top), composition(x1, x2))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 65 }
% 53.75/7.22 tuple(meet(meet(composition(x0, x2), composition(x1, x2)), meet(composition(x1, top), composition(x1, x2))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 60 }
% 53.75/7.22 tuple(meet(composition(x1, top), meet(meet(composition(x0, x2), composition(x1, x2)), composition(x1, x2))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 31 R->L }
% 53.75/7.22 tuple(meet(meet(meet(composition(x0, x2), composition(x1, x2)), composition(x1, x2)), composition(x1, top)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by axiom 12 (maddux4_definiton_of_meet) }
% 53.75/7.22 tuple(meet(complement(join(complement(meet(composition(x0, x2), composition(x1, x2))), complement(composition(x1, x2)))), composition(x1, top)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 28 R->L }
% 53.75/7.22 tuple(meet(join(zero, complement(join(complement(meet(composition(x0, x2), composition(x1, x2))), complement(composition(x1, x2))))), composition(x1, top)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 16 R->L }
% 53.75/7.22 tuple(meet(join(complement(top), complement(join(complement(meet(composition(x0, x2), composition(x1, x2))), complement(composition(x1, x2))))), composition(x1, top)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 38 R->L }
% 53.75/7.22 tuple(meet(join(complement(join(composition(x1, x2), complement(meet(composition(x1, x2), composition(x0, x2))))), complement(join(complement(meet(composition(x0, x2), composition(x1, x2))), complement(composition(x1, x2))))), composition(x1, top)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 31 }
% 53.75/7.22 tuple(meet(join(complement(join(composition(x1, x2), complement(meet(composition(x0, x2), composition(x1, x2))))), complement(join(complement(meet(composition(x0, x2), composition(x1, x2))), complement(composition(x1, x2))))), composition(x1, top)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 47 }
% 53.75/7.22 tuple(meet(join(meet(meet(composition(x0, x2), composition(x1, x2)), complement(composition(x1, x2))), complement(join(complement(meet(composition(x0, x2), composition(x1, x2))), complement(composition(x1, x2))))), composition(x1, top)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 25 }
% 53.75/7.22 tuple(meet(meet(composition(x0, x2), composition(x1, x2)), composition(x1, top)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 65 R->L }
% 53.75/7.22 tuple(meet(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), composition(x1, top)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 31 }
% 53.75/7.22 tuple(meet(composition(x1, top), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 65 }
% 53.75/7.22 tuple(meet(composition(x1, top), meet(composition(x0, x2), composition(x1, x2))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 31 R->L }
% 53.75/7.22 tuple(meet(composition(x1, top), meet(composition(x1, x2), composition(x0, x2))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 60 R->L }
% 53.75/7.22 tuple(meet(composition(x1, x2), meet(composition(x1, top), composition(x0, x2))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 31 R->L }
% 53.75/7.22 tuple(meet(composition(x1, x2), meet(composition(x0, x2), composition(x1, top))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 59 }
% 53.75/7.22 tuple(meet(composition(x1, x2), composition(x1, composition(x0, x2))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 31 }
% 53.75/7.22 tuple(meet(composition(x1, composition(x0, x2)), composition(x1, x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 20 R->L }
% 53.75/7.22 tuple(meet(composition(x1, composition(x0, x2)), composition(x1, composition(one, x2))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by axiom 4 (goals_1) R->L }
% 53.75/7.22 tuple(meet(composition(x1, composition(x0, x2)), composition(x1, composition(join(x0, one), x2))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by axiom 9 (composition_associativity) }
% 53.75/7.22 tuple(meet(composition(x1, composition(x0, x2)), composition(composition(x1, join(x0, one)), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 45 R->L }
% 53.75/7.22 tuple(meet(composition(x1, composition(x0, x2)), composition(join(x1, composition(x1, x0)), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 53.75/7.22 tuple(meet(composition(x1, composition(x0, x2)), composition(join(composition(x1, x0), x1), x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 63 R->L }
% 53.75/7.22 tuple(meet(composition(x1, composition(x0, x2)), join(composition(x1, x2), composition(composition(x1, x0), x2))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by axiom 9 (composition_associativity) R->L }
% 53.75/7.22 tuple(meet(composition(x1, composition(x0, x2)), join(composition(x1, x2), composition(x1, composition(x0, x2)))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 53.75/7.22 tuple(meet(composition(x1, composition(x0, x2)), join(composition(x1, composition(x0, x2)), composition(x1, x2))), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 48 }
% 53.75/7.22 tuple(composition(x1, composition(x0, x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 41 R->L }
% 53.75/7.22 tuple(composition(join(x1, meet(x0, x1)), composition(x0, x2)), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by axiom 9 (composition_associativity) }
% 53.75/7.22 tuple(composition(composition(join(x1, meet(x0, x1)), x0), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 53.75/7.22 tuple(composition(composition(join(meet(x0, x1), x1), x0), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by axiom 13 (composition_distributivity) }
% 53.75/7.22 tuple(composition(join(composition(meet(x0, x1), x0), composition(x1, x0)), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 52 R->L }
% 53.75/7.22 tuple(composition(join(meet(meet(x0, x1), composition(meet(x0, x1), x0)), composition(x1, x0)), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 30 R->L }
% 53.75/7.22 tuple(composition(join(meet(meet(x0, x1), composition(complement(complement(meet(x0, x1))), x0)), composition(x1, x0)), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 58 R->L }
% 53.75/7.22 tuple(composition(join(meet(meet(x0, x1), join(complement(meet(x0, x1)), composition(complement(complement(meet(x0, x1))), x0))), composition(x1, x0)), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 50 R->L }
% 53.75/7.22 tuple(composition(join(meet(meet(x0, x1), join(complement(meet(x0, x1)), join(composition(complement(complement(meet(x0, x1))), x0), composition(complement(meet(x0, x1)), x0)))), composition(x1, x0)), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by axiom 13 (composition_distributivity) R->L }
% 53.75/7.22 tuple(composition(join(meet(meet(x0, x1), join(complement(meet(x0, x1)), composition(join(complement(complement(meet(x0, x1))), complement(meet(x0, x1))), x0))), composition(x1, x0)), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by axiom 2 (maddux1_join_commutativity) }
% 53.75/7.22 tuple(composition(join(meet(meet(x0, x1), join(complement(meet(x0, x1)), composition(join(complement(meet(x0, x1)), complement(complement(meet(x0, x1)))), x0))), composition(x1, x0)), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by axiom 6 (def_top) R->L }
% 53.75/7.22 tuple(composition(join(meet(meet(x0, x1), join(complement(meet(x0, x1)), composition(top, x0))), composition(x1, x0)), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 58 }
% 53.75/7.22 tuple(composition(join(meet(meet(x0, x1), composition(top, x0)), composition(x1, x0)), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 31 }
% 53.75/7.22 tuple(composition(join(meet(composition(top, x0), meet(x0, x1)), composition(x1, x0)), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 61 }
% 53.75/7.22 tuple(composition(join(meet(x1, meet(x0, composition(top, x0))), composition(x1, x0)), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 53 }
% 53.75/7.22 tuple(composition(join(meet(x1, x0), composition(x1, x0)), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by lemma 31 }
% 53.75/7.22 tuple(composition(join(meet(x0, x1), composition(x1, x0)), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.22 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 53.75/7.23 tuple(composition(join(composition(x1, x0), meet(x0, x1)), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.23 = { by lemma 30 R->L }
% 53.75/7.23 tuple(composition(join(composition(x1, x0), complement(complement(meet(x0, x1)))), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.23 = { by lemma 56 R->L }
% 53.75/7.23 tuple(composition(join(complement(complement(meet(x0, x1))), meet(complement(meet(x0, x1)), composition(x1, x0))), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.23 = { by lemma 29 }
% 53.75/7.23 tuple(composition(join(meet(meet(x0, x1), meet(x0, x1)), meet(complement(meet(x0, x1)), composition(x1, x0))), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.23 = { by lemma 28 R->L }
% 53.75/7.23 tuple(composition(join(join(zero, meet(meet(x0, x1), meet(x0, x1))), meet(complement(meet(x0, x1)), composition(x1, x0))), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.23 = { by lemma 26 }
% 53.75/7.23 tuple(composition(join(meet(x0, x1), meet(complement(meet(x0, x1)), composition(x1, x0))), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.23 = { by lemma 31 }
% 53.75/7.23 tuple(composition(join(meet(x0, x1), meet(composition(x1, x0), complement(meet(x0, x1)))), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.23 = { by lemma 57 R->L }
% 53.75/7.23 tuple(composition(join(meet(x0, x1), meet(composition(x1, x0), complement(meet(composition(x1, x0), meet(x0, x1))))), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.23 = { by lemma 61 }
% 53.75/7.23 tuple(composition(join(meet(x0, x1), meet(composition(x1, x0), complement(meet(x1, meet(x0, composition(x1, x0)))))), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.23 = { by lemma 55 }
% 53.75/7.23 tuple(composition(join(meet(x0, x1), meet(composition(x1, x0), complement(meet(x1, composition(x1, x0))))), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.23 = { by lemma 52 }
% 53.75/7.23 tuple(composition(join(meet(x0, x1), meet(composition(x1, x0), complement(composition(x1, x0)))), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.23 = { by axiom 7 (def_zero) R->L }
% 53.75/7.23 tuple(composition(join(meet(x0, x1), zero), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.23 = { by lemma 27 }
% 53.75/7.23 tuple(composition(meet(x0, x1), x2), join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)))
% 53.75/7.23 = { by lemma 65 }
% 53.75/7.23 tuple(composition(meet(x0, x1), x2), meet(composition(x0, x2), composition(x1, x2)))
% 53.75/7.23 % SZS output end Proof
% 53.75/7.23
% 53.75/7.23 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------