TSTP Solution File: REL046+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL046+1 : 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 : n007.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 13:44:31 EDT 2023
% Result : Theorem 0.20s 0.49s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : REL046+1 : TPTP v8.1.2. Released v4.0.0.
% 0.07/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.35 % Computer : n007.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Fri Aug 25 19:43:26 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.20/0.49 Command-line arguments: --no-flatten-goal
% 0.20/0.49
% 0.20/0.49 % SZS status Theorem
% 0.20/0.49
% 0.20/0.51 % SZS output start Proof
% 0.20/0.51 Take the following subset of the input axioms:
% 0.20/0.52 fof(composition_associativity, axiom, ![X0, X1, X2]: composition(X0, composition(X1, X2))=composition(composition(X0, X1), X2)).
% 0.20/0.52 fof(composition_identity, axiom, ![X0_2]: composition(X0_2, one)=X0_2).
% 0.20/0.52 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)).
% 0.20/0.52 fof(converse_idempotence, axiom, ![X0_2]: converse(converse(X0_2))=X0_2).
% 0.20/0.52 fof(converse_multiplicativity, axiom, ![X0_2, X1_2]: converse(composition(X0_2, X1_2))=composition(converse(X1_2), converse(X0_2))).
% 0.20/0.52 fof(def_top, axiom, ![X0_2]: top=join(X0_2, complement(X0_2))).
% 0.20/0.52 fof(def_zero, axiom, ![X0_2]: zero=meet(X0_2, complement(X0_2))).
% 0.20/0.52 fof(goals, conjecture, ![X0_2, X1_2, X2_2]: (join(X0_2, meet(X1_2, X2_2))=meet(X1_2, X2_2) => (join(X0_2, X1_2)=X1_2 & join(X0_2, X2_2)=X2_2))).
% 0.20/0.52 fof(maddux1_join_commutativity, axiom, ![X0_2, X1_2]: join(X0_2, X1_2)=join(X1_2, X0_2)).
% 0.20/0.52 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)).
% 0.20/0.52 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)))).
% 0.20/0.52 fof(maddux4_definiton_of_meet, axiom, ![X0_2, X1_2]: meet(X0_2, X1_2)=complement(join(complement(X0_2), complement(X1_2)))).
% 0.20/0.52
% 0.20/0.52 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.52 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.52 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.52 fresh(y, y, x1...xn) = u
% 0.20/0.52 C => fresh(s, t, x1...xn) = v
% 0.20/0.52 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.52 variables of u and v.
% 0.20/0.52 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.52 input problem has no model of domain size 1).
% 0.20/0.52
% 0.20/0.52 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.52
% 0.20/0.52 Axiom 1 (converse_idempotence): converse(converse(X)) = X.
% 0.20/0.52 Axiom 2 (composition_identity): composition(X, one) = X.
% 0.20/0.52 Axiom 3 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 0.20/0.52 Axiom 4 (def_zero): zero = meet(X, complement(X)).
% 0.20/0.52 Axiom 5 (def_top): top = join(X, complement(X)).
% 0.20/0.52 Axiom 6 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 0.20/0.52 Axiom 7 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 0.20/0.52 Axiom 8 (goals): join(x0, meet(x1, x2)) = meet(x1, x2).
% 0.20/0.52 Axiom 9 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 0.20/0.52 Axiom 10 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 0.20/0.52 Axiom 11 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 0.20/0.52 Axiom 12 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 0.20/0.52
% 0.20/0.52 Lemma 13: complement(top) = zero.
% 0.20/0.52 Proof:
% 0.20/0.52 complement(top)
% 0.20/0.52 = { by axiom 5 (def_top) }
% 0.20/0.52 complement(join(complement(X), complement(complement(X))))
% 0.20/0.52 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 0.20/0.52 meet(X, complement(X))
% 0.20/0.52 = { by axiom 4 (def_zero) R->L }
% 0.20/0.52 zero
% 0.20/0.52
% 0.20/0.52 Lemma 14: join(X, join(Y, complement(X))) = join(Y, top).
% 0.20/0.52 Proof:
% 0.20/0.52 join(X, join(Y, complement(X)))
% 0.20/0.52 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 0.20/0.52 join(X, join(complement(X), Y))
% 0.20/0.52 = { by axiom 9 (maddux2_join_associativity) }
% 0.20/0.52 join(join(X, complement(X)), Y)
% 0.20/0.52 = { by axiom 5 (def_top) R->L }
% 0.20/0.52 join(top, Y)
% 0.20/0.52 = { by axiom 3 (maddux1_join_commutativity) }
% 0.20/0.52 join(Y, top)
% 0.20/0.52
% 0.20/0.52 Lemma 15: composition(converse(one), X) = X.
% 0.20/0.52 Proof:
% 0.20/0.52 composition(converse(one), X)
% 0.20/0.52 = { by axiom 1 (converse_idempotence) R->L }
% 0.20/0.52 composition(converse(one), converse(converse(X)))
% 0.20/0.52 = { by axiom 6 (converse_multiplicativity) R->L }
% 0.20/0.52 converse(composition(converse(X), one))
% 0.20/0.52 = { by axiom 2 (composition_identity) }
% 0.20/0.52 converse(converse(X))
% 0.20/0.52 = { by axiom 1 (converse_idempotence) }
% 0.20/0.52 X
% 0.20/0.52
% 0.20/0.52 Lemma 16: join(complement(X), complement(X)) = complement(X).
% 0.20/0.52 Proof:
% 0.20/0.52 join(complement(X), complement(X))
% 0.20/0.52 = { by lemma 15 R->L }
% 0.20/0.52 join(complement(X), composition(converse(one), complement(X)))
% 0.20/0.52 = { by lemma 15 R->L }
% 0.20/0.52 join(complement(X), composition(converse(one), complement(composition(converse(one), X))))
% 0.20/0.52 = { by axiom 2 (composition_identity) R->L }
% 0.20/0.52 join(complement(X), composition(converse(one), complement(composition(composition(converse(one), one), X))))
% 0.20/0.52 = { by axiom 7 (composition_associativity) R->L }
% 0.20/0.52 join(complement(X), composition(converse(one), complement(composition(converse(one), composition(one, X)))))
% 0.20/0.52 = { by lemma 15 }
% 0.20/0.52 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 0.20/0.52 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 0.20/0.52 join(composition(converse(one), complement(composition(one, X))), complement(X))
% 0.20/0.52 = { by axiom 11 (converse_cancellativity) }
% 0.20/0.52 complement(X)
% 0.20/0.52
% 0.20/0.52 Lemma 17: join(top, complement(X)) = top.
% 0.20/0.52 Proof:
% 0.20/0.52 join(top, complement(X))
% 0.20/0.52 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 0.20/0.52 join(complement(X), top)
% 0.20/0.52 = { by lemma 14 R->L }
% 0.20/0.52 join(X, join(complement(X), complement(X)))
% 0.20/0.52 = { by lemma 16 }
% 0.20/0.52 join(X, complement(X))
% 0.20/0.52 = { by axiom 5 (def_top) R->L }
% 0.20/0.52 top
% 0.20/0.52
% 0.20/0.52 Lemma 18: join(X, top) = top.
% 0.20/0.52 Proof:
% 0.20/0.52 join(X, top)
% 0.20/0.52 = { by lemma 17 R->L }
% 0.20/0.52 join(X, join(top, complement(X)))
% 0.20/0.52 = { by lemma 14 }
% 0.20/0.52 join(top, top)
% 0.20/0.52 = { by lemma 14 R->L }
% 0.20/0.52 join(join(zero, zero), join(top, complement(join(zero, zero))))
% 0.20/0.52 = { by lemma 17 }
% 0.20/0.52 join(join(zero, zero), top)
% 0.20/0.52 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 0.20/0.52 join(top, join(zero, zero))
% 0.20/0.52 = { by lemma 13 R->L }
% 0.20/0.52 join(top, join(zero, complement(top)))
% 0.20/0.52 = { by lemma 13 R->L }
% 0.20/0.52 join(top, join(complement(top), complement(top)))
% 0.20/0.52 = { by lemma 16 }
% 0.20/0.52 join(top, complement(top))
% 0.20/0.52 = { by axiom 5 (def_top) R->L }
% 0.20/0.52 top
% 0.20/0.52
% 0.20/0.52 Lemma 19: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 0.20/0.52 Proof:
% 0.20/0.52 join(meet(X, Y), complement(join(complement(X), Y)))
% 0.20/0.52 = { by axiom 10 (maddux4_definiton_of_meet) }
% 0.20/0.52 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 0.20/0.52 = { by axiom 12 (maddux3_a_kind_of_de_Morgan) R->L }
% 0.20/0.52 X
% 0.20/0.52
% 0.20/0.52 Lemma 20: join(zero, meet(X, X)) = X.
% 0.20/0.52 Proof:
% 0.20/0.52 join(zero, meet(X, X))
% 0.20/0.52 = { by axiom 10 (maddux4_definiton_of_meet) }
% 0.20/0.52 join(zero, complement(join(complement(X), complement(X))))
% 0.20/0.52 = { by axiom 4 (def_zero) }
% 0.20/0.52 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 0.20/0.52 = { by lemma 19 }
% 0.20/0.52 X
% 0.20/0.52
% 0.20/0.52 Lemma 21: complement(complement(X)) = meet(X, X).
% 0.20/0.52 Proof:
% 0.20/0.52 complement(complement(X))
% 0.20/0.52 = { by lemma 16 R->L }
% 0.20/0.52 complement(join(complement(X), complement(X)))
% 0.20/0.52 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 0.20/0.52 meet(X, X)
% 0.20/0.52
% 0.20/0.52 Lemma 22: meet(Y, X) = meet(X, Y).
% 0.20/0.52 Proof:
% 0.20/0.52 meet(Y, X)
% 0.20/0.52 = { by axiom 10 (maddux4_definiton_of_meet) }
% 0.20/0.52 complement(join(complement(Y), complement(X)))
% 0.20/0.52 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 0.20/0.52 complement(join(complement(X), complement(Y)))
% 0.20/0.52 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 0.20/0.52 meet(X, Y)
% 0.20/0.52
% 0.20/0.52 Lemma 23: complement(join(zero, complement(X))) = meet(X, top).
% 0.20/0.52 Proof:
% 0.20/0.52 complement(join(zero, complement(X)))
% 0.20/0.52 = { by lemma 13 R->L }
% 0.20/0.52 complement(join(complement(top), complement(X)))
% 0.20/0.52 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 0.20/0.52 meet(top, X)
% 0.20/0.52 = { by lemma 22 R->L }
% 0.20/0.52 meet(X, top)
% 0.20/0.52
% 0.20/0.52 Lemma 24: join(X, complement(zero)) = top.
% 0.20/0.52 Proof:
% 0.20/0.52 join(X, complement(zero))
% 0.20/0.52 = { by lemma 20 R->L }
% 0.20/0.52 join(join(zero, meet(X, X)), complement(zero))
% 0.20/0.52 = { by axiom 9 (maddux2_join_associativity) R->L }
% 0.20/0.52 join(zero, join(meet(X, X), complement(zero)))
% 0.20/0.52 = { by lemma 14 }
% 0.20/0.52 join(meet(X, X), top)
% 0.20/0.52 = { by lemma 18 }
% 0.20/0.52 top
% 0.20/0.52
% 0.20/0.52 Lemma 25: join(zero, meet(X, top)) = X.
% 0.20/0.52 Proof:
% 0.20/0.52 join(zero, meet(X, top))
% 0.20/0.52 = { by lemma 24 R->L }
% 0.20/0.52 join(zero, meet(X, join(complement(zero), complement(zero))))
% 0.20/0.52 = { by lemma 16 }
% 0.20/0.52 join(zero, meet(X, complement(zero)))
% 0.20/0.52 = { by lemma 13 R->L }
% 0.20/0.52 join(complement(top), meet(X, complement(zero)))
% 0.20/0.52 = { by lemma 24 R->L }
% 0.20/0.52 join(complement(join(complement(X), complement(zero))), meet(X, complement(zero)))
% 0.20/0.52 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 0.20/0.52 join(meet(X, zero), meet(X, complement(zero)))
% 0.20/0.52 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 0.20/0.52 join(meet(X, complement(zero)), meet(X, zero))
% 0.20/0.52 = { by axiom 10 (maddux4_definiton_of_meet) }
% 0.20/0.52 join(meet(X, complement(zero)), complement(join(complement(X), complement(zero))))
% 0.20/0.52 = { by lemma 19 }
% 0.20/0.52 X
% 0.20/0.52
% 0.20/0.52 Lemma 26: join(zero, complement(X)) = complement(X).
% 0.20/0.52 Proof:
% 0.20/0.52 join(zero, complement(X))
% 0.20/0.52 = { by lemma 20 R->L }
% 0.20/0.52 join(zero, complement(join(zero, meet(X, X))))
% 0.20/0.52 = { by lemma 21 R->L }
% 0.20/0.52 join(zero, complement(join(zero, complement(complement(X)))))
% 0.20/0.52 = { by lemma 23 }
% 0.20/0.52 join(zero, meet(complement(X), top))
% 0.20/0.52 = { by lemma 25 }
% 0.20/0.52 complement(X)
% 0.20/0.52
% 0.20/0.52 Lemma 27: complement(complement(X)) = X.
% 0.20/0.52 Proof:
% 0.20/0.52 complement(complement(X))
% 0.20/0.52 = { by lemma 26 R->L }
% 0.20/0.52 join(zero, complement(complement(X)))
% 0.20/0.52 = { by lemma 21 }
% 0.20/0.52 join(zero, meet(X, X))
% 0.20/0.52 = { by lemma 20 }
% 0.20/0.52 X
% 0.20/0.52
% 0.20/0.52 Lemma 28: meet(X, top) = X.
% 0.20/0.52 Proof:
% 0.20/0.52 meet(X, top)
% 0.20/0.52 = { by lemma 23 R->L }
% 0.20/0.52 complement(join(zero, complement(X)))
% 0.20/0.52 = { by lemma 26 R->L }
% 0.20/0.52 join(zero, complement(join(zero, complement(X))))
% 0.20/0.52 = { by lemma 23 }
% 0.20/0.52 join(zero, meet(X, top))
% 0.20/0.52 = { by lemma 25 }
% 0.20/0.52 X
% 0.20/0.52
% 0.20/0.52 Lemma 29: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
% 0.20/0.52 Proof:
% 0.20/0.52 complement(join(complement(X), meet(Y, Z)))
% 0.20/0.52 = { by lemma 22 }
% 0.20/0.52 complement(join(complement(X), meet(Z, Y)))
% 0.20/0.52 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 0.20/0.52 complement(join(meet(Z, Y), complement(X)))
% 0.20/0.52 = { by axiom 10 (maddux4_definiton_of_meet) }
% 0.20/0.52 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 0.20/0.52 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 0.20/0.52 meet(join(complement(Z), complement(Y)), X)
% 0.20/0.52 = { by lemma 22 R->L }
% 0.20/0.52 meet(X, join(complement(Z), complement(Y)))
% 0.20/0.52 = { by axiom 3 (maddux1_join_commutativity) }
% 0.20/0.52 meet(X, join(complement(Y), complement(Z)))
% 0.20/0.52
% 0.20/0.52 Lemma 30: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 0.20/0.52 Proof:
% 0.20/0.52 join(complement(X), complement(Y))
% 0.20/0.52 = { by lemma 28 R->L }
% 0.20/0.52 meet(join(complement(X), complement(Y)), top)
% 0.20/0.52 = { by lemma 22 R->L }
% 0.20/0.52 meet(top, join(complement(X), complement(Y)))
% 0.20/0.52 = { by lemma 29 R->L }
% 0.20/0.52 complement(join(complement(top), meet(X, Y)))
% 0.20/0.52 = { by lemma 13 }
% 0.20/0.52 complement(join(zero, meet(X, Y)))
% 0.20/0.52 = { by lemma 22 R->L }
% 0.20/0.52 complement(join(zero, meet(Y, X)))
% 0.20/0.52 = { by lemma 27 R->L }
% 0.20/0.52 complement(join(zero, complement(complement(meet(Y, X)))))
% 0.20/0.52 = { by lemma 21 }
% 0.20/0.52 complement(join(zero, meet(meet(Y, X), meet(Y, X))))
% 0.20/0.52 = { by lemma 20 }
% 0.20/0.52 complement(meet(Y, X))
% 0.20/0.52 = { by lemma 22 R->L }
% 0.20/0.52 complement(meet(X, Y))
% 0.20/0.52
% 0.20/0.52 Lemma 31: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 0.20/0.52 Proof:
% 0.20/0.52 complement(meet(X, complement(Y)))
% 0.20/0.52 = { by lemma 22 }
% 0.20/0.52 complement(meet(complement(Y), X))
% 0.20/0.52 = { by lemma 26 R->L }
% 0.20/0.52 complement(meet(join(zero, complement(Y)), X))
% 0.20/0.52 = { by lemma 30 R->L }
% 0.20/0.52 join(complement(join(zero, complement(Y))), complement(X))
% 0.20/0.52 = { by lemma 23 }
% 0.20/0.52 join(meet(Y, top), complement(X))
% 0.20/0.52 = { by lemma 28 }
% 0.20/0.52 join(Y, complement(X))
% 0.20/0.52
% 0.20/0.52 Lemma 32: join(X, meet(X, Y)) = X.
% 0.20/0.52 Proof:
% 0.20/0.52 join(X, meet(X, Y))
% 0.20/0.52 = { by axiom 10 (maddux4_definiton_of_meet) }
% 0.20/0.52 join(X, complement(join(complement(X), complement(Y))))
% 0.20/0.52 = { by lemma 31 R->L }
% 0.20/0.52 complement(meet(join(complement(X), complement(Y)), complement(X)))
% 0.20/0.52 = { by lemma 22 R->L }
% 0.20/0.52 complement(meet(complement(X), join(complement(X), complement(Y))))
% 0.20/0.52 = { by lemma 31 R->L }
% 0.20/0.52 complement(meet(complement(X), complement(meet(Y, complement(complement(X))))))
% 0.20/0.52 = { by lemma 30 R->L }
% 0.20/0.52 complement(meet(complement(X), join(complement(Y), complement(complement(complement(X))))))
% 0.20/0.52 = { by lemma 29 R->L }
% 0.20/0.52 complement(complement(join(complement(complement(X)), meet(Y, complement(complement(X))))))
% 0.20/0.52 = { by lemma 26 R->L }
% 0.20/0.52 complement(join(zero, complement(join(complement(complement(X)), meet(Y, complement(complement(X)))))))
% 0.20/0.52 = { by lemma 13 R->L }
% 0.20/0.52 complement(join(complement(top), complement(join(complement(complement(X)), meet(Y, complement(complement(X)))))))
% 0.20/0.52 = { by lemma 18 R->L }
% 0.20/0.52 complement(join(complement(join(complement(Y), top)), complement(join(complement(complement(X)), meet(Y, complement(complement(X)))))))
% 0.20/0.52 = { by lemma 14 R->L }
% 0.20/0.52 complement(join(complement(join(complement(complement(X)), join(complement(Y), complement(complement(complement(X)))))), complement(join(complement(complement(X)), meet(Y, complement(complement(X)))))))
% 0.20/0.53 = { by lemma 30 }
% 0.20/0.53 complement(join(complement(join(complement(complement(X)), complement(meet(Y, complement(complement(X)))))), complement(join(complement(complement(X)), meet(Y, complement(complement(X)))))))
% 0.20/0.53 = { by lemma 22 R->L }
% 0.20/0.53 complement(join(complement(join(complement(complement(X)), complement(meet(complement(complement(X)), Y)))), complement(join(complement(complement(X)), meet(Y, complement(complement(X)))))))
% 0.20/0.53 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 0.20/0.53 complement(join(meet(complement(X), meet(complement(complement(X)), Y)), complement(join(complement(complement(X)), meet(Y, complement(complement(X)))))))
% 0.20/0.53 = { by lemma 22 R->L }
% 0.20/0.53 complement(join(meet(complement(X), meet(Y, complement(complement(X)))), complement(join(complement(complement(X)), meet(Y, complement(complement(X)))))))
% 0.20/0.53 = { by lemma 19 }
% 0.20/0.53 complement(complement(X))
% 0.20/0.53 = { by lemma 27 }
% 0.20/0.53 X
% 0.20/0.53
% 0.20/0.53 Lemma 33: join(X, meet(Y, X)) = X.
% 0.20/0.53 Proof:
% 0.20/0.53 join(X, meet(Y, X))
% 0.20/0.53 = { by lemma 22 }
% 0.20/0.53 join(X, meet(X, Y))
% 0.20/0.53 = { by lemma 32 }
% 0.20/0.53 X
% 0.20/0.53
% 0.20/0.53 Lemma 34: join(x0, join(X, meet(x1, x2))) = join(X, meet(x1, x2)).
% 0.20/0.53 Proof:
% 0.20/0.53 join(x0, join(X, meet(x1, x2)))
% 0.20/0.53 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 0.20/0.53 join(x0, join(meet(x1, x2), X))
% 0.20/0.53 = { by axiom 9 (maddux2_join_associativity) }
% 0.20/0.53 join(join(x0, meet(x1, x2)), X)
% 0.20/0.53 = { by axiom 8 (goals) }
% 0.20/0.53 join(meet(x1, x2), X)
% 0.20/0.53 = { by axiom 3 (maddux1_join_commutativity) }
% 0.20/0.53 join(X, meet(x1, x2))
% 0.20/0.53
% 0.20/0.53 Goal 1 (goals_1): tuple(join(x0, x1), join(x0, x2)) = tuple(x1, x2).
% 0.20/0.53 Proof:
% 0.20/0.53 tuple(join(x0, x1), join(x0, x2))
% 0.20/0.53 = { by lemma 32 R->L }
% 0.20/0.53 tuple(join(x0, join(x1, meet(x1, x2))), join(x0, x2))
% 0.20/0.53 = { by lemma 34 }
% 0.20/0.53 tuple(join(x1, meet(x1, x2)), join(x0, x2))
% 0.20/0.53 = { by lemma 32 }
% 0.20/0.53 tuple(x1, join(x0, x2))
% 0.20/0.53 = { by lemma 33 R->L }
% 0.20/0.53 tuple(x1, join(x0, join(x2, meet(x1, x2))))
% 0.20/0.53 = { by lemma 34 }
% 0.20/0.53 tuple(x1, join(x2, meet(x1, x2)))
% 0.20/0.53 = { by lemma 33 }
% 0.20/0.53 tuple(x1, x2)
% 0.20/0.53 % SZS output end Proof
% 0.20/0.53
% 0.20/0.53 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------