TSTP Solution File: MVA011-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : MVA011-1 : TPTP v8.1.2. Released v8.1.0.
% 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 : n002.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 09:32:30 EDT 2023
% Result : Unsatisfiable 29.77s 4.14s
% Output : Proof 30.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.12 % Problem : MVA011-1 : TPTP v8.1.2. Released v8.1.0.
% 0.10/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.33 % Computer : n002.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Sun Aug 27 05:49:03 EDT 2023
% 0.12/0.33 % CPUTime :
% 29.77/4.14 Command-line arguments: --ground-connectedness --complete-subsets
% 29.77/4.14
% 29.77/4.14 % SZS status Unsatisfiable
% 29.77/4.14
% 30.19/4.19 % SZS output start Proof
% 30.19/4.19 Axiom 1 (right_monoid_unit): op(X, unit) = X.
% 30.19/4.19 Axiom 2 (left_monoid_unit): op(unit, X) = X.
% 30.19/4.19 Axiom 3 (idempotence_of_meet): meet(X, X) = X.
% 30.19/4.19 Axiom 4 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 30.19/4.19 Axiom 5 (idempotence_of_join): join(X, X) = X.
% 30.19/4.19 Axiom 6 (commutativity_of_join): join(X, Y) = join(Y, X).
% 30.19/4.19 Axiom 7 (monoid_associativity): op(op(X, Y), Z) = op(X, op(Y, Z)).
% 30.19/4.19 Axiom 8 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 30.19/4.19 Axiom 9 (absorption_b): meet(join(X, Y), X) = X.
% 30.19/4.19 Axiom 10 (absorption_a): join(meet(X, Y), X) = X.
% 30.19/4.19 Axiom 11 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 30.19/4.19 Axiom 12 (generalized_mv_algebra_a): join(X, Y) = rd(X, ld(join(X, Y), X)).
% 30.19/4.19 Axiom 13 (residual_d): meet(rd(join(op(X, Y), Z), Y), X) = X.
% 30.19/4.19 Axiom 14 (residual_c): meet(ld(X, join(op(X, Y), Z)), Y) = Y.
% 30.19/4.19 Axiom 15 (residual_a): join(op(X, meet(ld(X, Y), Z)), Y) = Y.
% 30.19/4.19 Axiom 16 (residual_b): join(op(meet(X, rd(Y, Z)), Z), Y) = Y.
% 30.19/4.19 Axiom 17 (definition_of_at): at(X, Y) = op(op(X, ld(X, unit)), ld(ld(Y, unit), unit)).
% 30.19/4.19
% 30.19/4.19 Lemma 18: meet(X, ld(Y, join(op(Y, X), Z))) = X.
% 30.19/4.19 Proof:
% 30.19/4.19 meet(X, ld(Y, join(op(Y, X), Z)))
% 30.19/4.19 = { by axiom 4 (commutativity_of_meet) R->L }
% 30.19/4.19 meet(ld(Y, join(op(Y, X), Z)), X)
% 30.19/4.20 = { by axiom 14 (residual_c) }
% 30.19/4.20 X
% 30.19/4.20
% 30.19/4.20 Lemma 19: meet(X, ld(Y, op(Y, X))) = X.
% 30.19/4.20 Proof:
% 30.19/4.20 meet(X, ld(Y, op(Y, X)))
% 30.19/4.20 = { by axiom 5 (idempotence_of_join) R->L }
% 30.19/4.20 meet(X, ld(Y, join(op(Y, X), op(Y, X))))
% 30.19/4.20 = { by lemma 18 }
% 30.19/4.20 X
% 30.19/4.20
% 30.19/4.20 Lemma 20: meet(unit, ld(X, X)) = unit.
% 30.19/4.20 Proof:
% 30.19/4.20 meet(unit, ld(X, X))
% 30.19/4.20 = { by axiom 1 (right_monoid_unit) R->L }
% 30.19/4.20 meet(unit, ld(X, op(X, unit)))
% 30.19/4.20 = { by lemma 19 }
% 30.19/4.20 unit
% 30.19/4.20
% 30.19/4.20 Lemma 21: join(X, meet(X, Y)) = X.
% 30.19/4.20 Proof:
% 30.19/4.20 join(X, meet(X, Y))
% 30.19/4.20 = { by axiom 6 (commutativity_of_join) R->L }
% 30.19/4.20 join(meet(X, Y), X)
% 30.19/4.20 = { by axiom 10 (absorption_a) }
% 30.19/4.20 X
% 30.19/4.20
% 30.19/4.20 Lemma 22: join(X, meet(Y, X)) = X.
% 30.19/4.20 Proof:
% 30.19/4.20 join(X, meet(Y, X))
% 30.19/4.20 = { by axiom 4 (commutativity_of_meet) R->L }
% 30.19/4.20 join(X, meet(X, Y))
% 30.19/4.20 = { by lemma 21 }
% 30.19/4.20 X
% 30.19/4.20
% 30.19/4.20 Lemma 23: join(unit, ld(X, X)) = ld(X, X).
% 30.19/4.20 Proof:
% 30.19/4.20 join(unit, ld(X, X))
% 30.19/4.20 = { by axiom 6 (commutativity_of_join) R->L }
% 30.19/4.20 join(ld(X, X), unit)
% 30.19/4.20 = { by lemma 20 R->L }
% 30.19/4.20 join(ld(X, X), meet(unit, ld(X, X)))
% 30.19/4.20 = { by lemma 22 }
% 30.19/4.20 ld(X, X)
% 30.19/4.20
% 30.19/4.20 Lemma 24: meet(X, rd(join(op(X, Y), Z), Y)) = X.
% 30.19/4.20 Proof:
% 30.19/4.20 meet(X, rd(join(op(X, Y), Z), Y))
% 30.19/4.20 = { by axiom 4 (commutativity_of_meet) R->L }
% 30.19/4.20 meet(rd(join(op(X, Y), Z), Y), X)
% 30.19/4.20 = { by axiom 13 (residual_d) }
% 30.19/4.20 X
% 30.19/4.20
% 30.19/4.20 Lemma 25: meet(X, rd(op(X, Y), Y)) = X.
% 30.19/4.20 Proof:
% 30.19/4.20 meet(X, rd(op(X, Y), Y))
% 30.19/4.20 = { by axiom 5 (idempotence_of_join) R->L }
% 30.19/4.20 meet(X, rd(join(op(X, Y), op(X, Y)), Y))
% 30.19/4.20 = { by lemma 24 }
% 30.19/4.20 X
% 30.19/4.20
% 30.19/4.20 Lemma 26: meet(unit, rd(X, X)) = unit.
% 30.19/4.20 Proof:
% 30.19/4.20 meet(unit, rd(X, X))
% 30.19/4.20 = { by axiom 2 (left_monoid_unit) R->L }
% 30.19/4.20 meet(unit, rd(op(unit, X), X))
% 30.19/4.20 = { by lemma 25 }
% 30.19/4.20 unit
% 30.19/4.20
% 30.19/4.20 Lemma 27: join(unit, rd(X, X)) = rd(X, X).
% 30.19/4.20 Proof:
% 30.19/4.20 join(unit, rd(X, X))
% 30.19/4.20 = { by axiom 6 (commutativity_of_join) R->L }
% 30.19/4.20 join(rd(X, X), unit)
% 30.19/4.20 = { by lemma 26 R->L }
% 30.19/4.20 join(rd(X, X), meet(unit, rd(X, X)))
% 30.19/4.20 = { by lemma 22 }
% 30.19/4.20 rd(X, X)
% 30.19/4.20
% 30.19/4.20 Lemma 28: join(X, op(Y, meet(ld(Y, X), Z))) = X.
% 30.19/4.20 Proof:
% 30.19/4.20 join(X, op(Y, meet(ld(Y, X), Z)))
% 30.19/4.20 = { by axiom 6 (commutativity_of_join) R->L }
% 30.19/4.20 join(op(Y, meet(ld(Y, X), Z)), X)
% 30.19/4.20 = { by axiom 15 (residual_a) }
% 30.19/4.20 X
% 30.19/4.20
% 30.19/4.20 Lemma 29: join(X, op(Y, ld(Y, X))) = X.
% 30.19/4.20 Proof:
% 30.19/4.20 join(X, op(Y, ld(Y, X)))
% 30.19/4.20 = { by axiom 3 (idempotence_of_meet) R->L }
% 30.19/4.20 join(X, op(Y, meet(ld(Y, X), ld(Y, X))))
% 30.19/4.20 = { by lemma 28 }
% 30.19/4.20 X
% 30.19/4.20
% 30.19/4.20 Lemma 30: ld(unit, X) = X.
% 30.19/4.20 Proof:
% 30.19/4.20 ld(unit, X)
% 30.19/4.20 = { by lemma 22 R->L }
% 30.19/4.20 join(ld(unit, X), meet(X, ld(unit, X)))
% 30.19/4.20 = { by axiom 2 (left_monoid_unit) R->L }
% 30.19/4.20 join(ld(unit, X), meet(X, ld(unit, op(unit, X))))
% 30.19/4.20 = { by lemma 19 }
% 30.19/4.20 join(ld(unit, X), X)
% 30.19/4.20 = { by axiom 6 (commutativity_of_join) }
% 30.19/4.20 join(X, ld(unit, X))
% 30.19/4.20 = { by axiom 2 (left_monoid_unit) R->L }
% 30.19/4.20 join(X, op(unit, ld(unit, X)))
% 30.19/4.20 = { by lemma 29 }
% 30.19/4.20 X
% 30.19/4.20
% 30.19/4.20 Lemma 31: op(X, op(ld(X, unit), ld(ld(Y, unit), unit))) = at(X, Y).
% 30.19/4.20 Proof:
% 30.19/4.20 op(X, op(ld(X, unit), ld(ld(Y, unit), unit)))
% 30.19/4.20 = { by axiom 7 (monoid_associativity) R->L }
% 30.19/4.20 op(op(X, ld(X, unit)), ld(ld(Y, unit), unit))
% 30.19/4.20 = { by axiom 17 (definition_of_at) R->L }
% 30.19/4.20 at(X, Y)
% 30.19/4.20
% 30.19/4.20 Lemma 32: op(X, ld(X, unit)) = at(X, unit).
% 30.19/4.20 Proof:
% 30.19/4.20 op(X, ld(X, unit))
% 30.19/4.20 = { by axiom 1 (right_monoid_unit) R->L }
% 30.19/4.20 op(X, op(ld(X, unit), unit))
% 30.19/4.20 = { by lemma 30 R->L }
% 30.19/4.20 op(X, op(ld(X, unit), ld(unit, unit)))
% 30.19/4.20 = { by lemma 30 R->L }
% 30.19/4.20 op(X, op(ld(X, unit), ld(ld(unit, unit), unit)))
% 30.19/4.20 = { by lemma 31 }
% 30.19/4.20 at(X, unit)
% 30.19/4.20
% 30.19/4.20 Lemma 33: meet(X, join(X, Y)) = X.
% 30.19/4.20 Proof:
% 30.19/4.20 meet(X, join(X, Y))
% 30.19/4.20 = { by axiom 4 (commutativity_of_meet) R->L }
% 30.19/4.20 meet(join(X, Y), X)
% 30.19/4.20 = { by axiom 9 (absorption_b) }
% 30.19/4.20 X
% 30.19/4.20
% 30.19/4.20 Lemma 34: meet(X, join(Y, X)) = X.
% 30.19/4.20 Proof:
% 30.19/4.20 meet(X, join(Y, X))
% 30.19/4.20 = { by axiom 6 (commutativity_of_join) R->L }
% 30.19/4.20 meet(X, join(X, Y))
% 30.19/4.20 = { by lemma 33 }
% 30.19/4.20 X
% 30.19/4.20
% 30.19/4.20 Lemma 35: meet(X, op(Y, ld(Y, X))) = op(Y, ld(Y, X)).
% 30.19/4.20 Proof:
% 30.19/4.20 meet(X, op(Y, ld(Y, X)))
% 30.19/4.20 = { by axiom 4 (commutativity_of_meet) R->L }
% 30.19/4.20 meet(op(Y, ld(Y, X)), X)
% 30.19/4.20 = { by lemma 29 R->L }
% 30.19/4.20 meet(op(Y, ld(Y, X)), join(X, op(Y, ld(Y, X))))
% 30.19/4.20 = { by lemma 34 }
% 30.19/4.20 op(Y, ld(Y, X))
% 30.19/4.20
% 30.19/4.20 Lemma 36: meet(unit, at(X, unit)) = at(X, unit).
% 30.19/4.20 Proof:
% 30.19/4.20 meet(unit, at(X, unit))
% 30.19/4.20 = { by lemma 32 R->L }
% 30.19/4.20 meet(unit, op(X, ld(X, unit)))
% 30.19/4.20 = { by lemma 35 }
% 30.19/4.20 op(X, ld(X, unit))
% 30.19/4.20 = { by lemma 32 }
% 30.19/4.20 at(X, unit)
% 30.19/4.20
% 30.19/4.20 Lemma 37: rd(at(X, unit), at(X, unit)) = unit.
% 30.19/4.20 Proof:
% 30.19/4.20 rd(at(X, unit), at(X, unit))
% 30.19/4.20 = { by lemma 36 R->L }
% 30.19/4.20 rd(at(X, unit), meet(unit, at(X, unit)))
% 30.19/4.20 = { by lemma 30 R->L }
% 30.19/4.20 rd(at(X, unit), ld(unit, meet(unit, at(X, unit))))
% 30.19/4.20 = { by lemma 36 R->L }
% 30.19/4.20 rd(meet(unit, at(X, unit)), ld(unit, meet(unit, at(X, unit))))
% 30.19/4.20 = { by lemma 21 R->L }
% 30.19/4.20 rd(meet(unit, at(X, unit)), ld(join(unit, meet(unit, at(X, unit))), meet(unit, at(X, unit))))
% 30.19/4.20 = { by axiom 6 (commutativity_of_join) R->L }
% 30.19/4.20 rd(meet(unit, at(X, unit)), ld(join(meet(unit, at(X, unit)), unit), meet(unit, at(X, unit))))
% 30.19/4.20 = { by axiom 12 (generalized_mv_algebra_a) R->L }
% 30.19/4.20 join(meet(unit, at(X, unit)), unit)
% 30.19/4.20 = { by axiom 6 (commutativity_of_join) }
% 30.19/4.20 join(unit, meet(unit, at(X, unit)))
% 30.19/4.20 = { by lemma 21 }
% 30.19/4.20 unit
% 30.19/4.20
% 30.19/4.20 Lemma 38: join(X, op(meet(Y, rd(X, Z)), Z)) = X.
% 30.19/4.20 Proof:
% 30.19/4.20 join(X, op(meet(Y, rd(X, Z)), Z))
% 30.19/4.20 = { by axiom 6 (commutativity_of_join) R->L }
% 30.19/4.20 join(op(meet(Y, rd(X, Z)), Z), X)
% 30.19/4.20 = { by axiom 16 (residual_b) }
% 30.19/4.20 X
% 30.19/4.20
% 30.19/4.20 Lemma 39: join(X, op(rd(X, Y), Y)) = X.
% 30.19/4.20 Proof:
% 30.19/4.20 join(X, op(rd(X, Y), Y))
% 30.19/4.20 = { by axiom 3 (idempotence_of_meet) R->L }
% 30.19/4.20 join(X, op(meet(rd(X, Y), rd(X, Y)), Y))
% 30.19/4.20 = { by lemma 38 }
% 30.19/4.20 X
% 30.19/4.20
% 30.19/4.20 Lemma 40: meet(X, op(rd(X, Y), Y)) = op(rd(X, Y), Y).
% 30.19/4.20 Proof:
% 30.19/4.20 meet(X, op(rd(X, Y), Y))
% 30.19/4.20 = { by axiom 4 (commutativity_of_meet) R->L }
% 30.19/4.20 meet(op(rd(X, Y), Y), X)
% 30.19/4.20 = { by lemma 39 R->L }
% 30.19/4.20 meet(op(rd(X, Y), Y), join(X, op(rd(X, Y), Y)))
% 30.19/4.20 = { by lemma 34 }
% 30.19/4.20 op(rd(X, Y), Y)
% 30.19/4.20
% 30.19/4.20 Lemma 41: meet(X, meet(join(X, Y), Z)) = meet(X, Z).
% 30.19/4.20 Proof:
% 30.19/4.20 meet(X, meet(join(X, Y), Z))
% 30.19/4.20 = { by axiom 8 (associativity_of_meet) R->L }
% 30.19/4.20 meet(meet(X, join(X, Y)), Z)
% 30.19/4.20 = { by lemma 33 }
% 30.19/4.20 meet(X, Z)
% 30.19/4.20
% 30.19/4.20 Lemma 42: meet(X, ld(Y, join(Z, op(Y, X)))) = X.
% 30.19/4.20 Proof:
% 30.19/4.20 meet(X, ld(Y, join(Z, op(Y, X))))
% 30.19/4.20 = { by axiom 6 (commutativity_of_join) R->L }
% 30.19/4.20 meet(X, ld(Y, join(op(Y, X), Z)))
% 30.19/4.20 = { by lemma 18 }
% 30.19/4.20 X
% 30.19/4.20
% 30.19/4.20 Lemma 43: op(rd(X, X), at(X, unit)) = at(X, unit).
% 30.19/4.20 Proof:
% 30.19/4.20 op(rd(X, X), at(X, unit))
% 30.19/4.20 = { by lemma 32 R->L }
% 30.19/4.20 op(rd(X, X), op(X, ld(X, unit)))
% 30.19/4.20 = { by axiom 7 (monoid_associativity) R->L }
% 30.19/4.20 op(op(rd(X, X), X), ld(X, unit))
% 30.19/4.20 = { by lemma 40 R->L }
% 30.19/4.20 op(meet(X, op(rd(X, X), X)), ld(X, unit))
% 30.19/4.20 = { by lemma 30 R->L }
% 30.19/4.20 op(meet(X, ld(unit, op(rd(X, X), X))), ld(X, unit))
% 30.19/4.20 = { by lemma 27 R->L }
% 30.19/4.20 op(meet(X, ld(unit, op(join(unit, rd(X, X)), X))), ld(X, unit))
% 30.19/4.20 = { by lemma 33 R->L }
% 30.19/4.20 op(meet(X, ld(meet(unit, join(unit, rd(X, X))), op(join(unit, rd(X, X)), X))), ld(X, unit))
% 30.19/4.20 = { by lemma 25 R->L }
% 30.19/4.20 op(meet(X, ld(meet(unit, meet(join(unit, rd(X, X)), rd(op(join(unit, rd(X, X)), X), X))), op(join(unit, rd(X, X)), X))), ld(X, unit))
% 30.19/4.20 = { by lemma 41 }
% 30.19/4.20 op(meet(X, ld(meet(unit, rd(op(join(unit, rd(X, X)), X), X)), op(join(unit, rd(X, X)), X))), ld(X, unit))
% 30.19/4.20 = { by lemma 38 R->L }
% 30.19/4.20 op(meet(X, ld(meet(unit, rd(op(join(unit, rd(X, X)), X), X)), join(op(join(unit, rd(X, X)), X), op(meet(unit, rd(op(join(unit, rd(X, X)), X), X)), X)))), ld(X, unit))
% 30.19/4.20 = { by lemma 42 }
% 30.19/4.20 op(X, ld(X, unit))
% 30.19/4.20 = { by lemma 32 }
% 30.19/4.20 at(X, unit)
% 30.19/4.20
% 30.19/4.20 Lemma 44: join(X, rd(op(X, Y), Y)) = rd(op(X, Y), Y).
% 30.19/4.20 Proof:
% 30.19/4.20 join(X, rd(op(X, Y), Y))
% 30.19/4.20 = { by axiom 6 (commutativity_of_join) R->L }
% 30.19/4.20 join(rd(op(X, Y), Y), X)
% 30.19/4.20 = { by lemma 25 R->L }
% 30.19/4.20 join(rd(op(X, Y), Y), meet(X, rd(op(X, Y), Y)))
% 30.19/4.20 = { by lemma 22 }
% 30.19/4.20 rd(op(X, Y), Y)
% 30.19/4.20
% 30.19/4.20 Lemma 45: rd(X, X) = unit.
% 30.19/4.20 Proof:
% 30.19/4.20 rd(X, X)
% 30.19/4.20 = { by lemma 27 R->L }
% 30.19/4.20 join(unit, rd(X, X))
% 30.19/4.20 = { by axiom 6 (commutativity_of_join) R->L }
% 30.19/4.20 join(rd(X, X), unit)
% 30.19/4.20 = { by lemma 37 R->L }
% 30.19/4.20 join(rd(X, X), rd(at(X, unit), at(X, unit)))
% 30.19/4.20 = { by lemma 43 R->L }
% 30.19/4.20 join(rd(X, X), rd(op(rd(X, X), at(X, unit)), at(X, unit)))
% 30.19/4.20 = { by lemma 44 }
% 30.19/4.20 rd(op(rd(X, X), at(X, unit)), at(X, unit))
% 30.19/4.20 = { by lemma 43 }
% 30.19/4.20 rd(at(X, unit), at(X, unit))
% 30.19/4.20 = { by lemma 37 }
% 30.19/4.20 unit
% 30.19/4.20
% 30.19/4.20 Lemma 46: join(X, op(meet(Y, meet(Z, rd(X, W))), W)) = X.
% 30.19/4.20 Proof:
% 30.19/4.20 join(X, op(meet(Y, meet(Z, rd(X, W))), W))
% 30.19/4.20 = { by axiom 8 (associativity_of_meet) R->L }
% 30.19/4.20 join(X, op(meet(meet(Y, Z), rd(X, W)), W))
% 30.19/4.20 = { by lemma 38 }
% 30.19/4.20 X
% 30.19/4.20
% 30.19/4.20 Lemma 47: meet(X, rd(join(Y, op(X, Z)), Z)) = X.
% 30.19/4.20 Proof:
% 30.19/4.20 meet(X, rd(join(Y, op(X, Z)), Z))
% 30.19/4.20 = { by axiom 6 (commutativity_of_join) R->L }
% 30.19/4.20 meet(X, rd(join(op(X, Z), Y), Z))
% 30.19/4.20 = { by lemma 24 }
% 30.19/4.20 X
% 30.19/4.20
% 30.19/4.20 Lemma 48: meet(X, rd(Y, meet(Z, ld(X, Y)))) = X.
% 30.19/4.20 Proof:
% 30.19/4.20 meet(X, rd(Y, meet(Z, ld(X, Y))))
% 30.19/4.20 = { by axiom 4 (commutativity_of_meet) R->L }
% 30.19/4.20 meet(X, rd(Y, meet(ld(X, Y), Z)))
% 30.19/4.20 = { by lemma 28 R->L }
% 30.19/4.20 meet(X, rd(join(Y, op(X, meet(ld(X, Y), Z))), meet(ld(X, Y), Z)))
% 30.19/4.20 = { by lemma 47 }
% 30.19/4.20 X
% 30.19/4.20
% 30.19/4.20 Lemma 49: join(X, op(Y, op(Z, ld(op(Y, Z), X)))) = X.
% 30.19/4.20 Proof:
% 30.19/4.20 join(X, op(Y, op(Z, ld(op(Y, Z), X))))
% 30.19/4.20 = { by axiom 7 (monoid_associativity) R->L }
% 30.19/4.20 join(X, op(op(Y, Z), ld(op(Y, Z), X)))
% 30.19/4.20 = { by lemma 29 }
% 30.19/4.20 X
% 30.19/4.20
% 30.19/4.20 Lemma 50: join(X, ld(Y, join(Z, op(Y, X)))) = ld(Y, join(Z, op(Y, X))).
% 30.19/4.20 Proof:
% 30.19/4.20 join(X, ld(Y, join(Z, op(Y, X))))
% 30.19/4.20 = { by axiom 6 (commutativity_of_join) R->L }
% 30.19/4.20 join(X, ld(Y, join(op(Y, X), Z)))
% 30.19/4.20 = { by axiom 6 (commutativity_of_join) R->L }
% 30.19/4.20 join(ld(Y, join(op(Y, X), Z)), X)
% 30.19/4.20 = { by lemma 18 R->L }
% 30.19/4.20 join(ld(Y, join(op(Y, X), Z)), meet(X, ld(Y, join(op(Y, X), Z))))
% 30.19/4.20 = { by lemma 22 }
% 30.19/4.20 ld(Y, join(op(Y, X), Z))
% 30.19/4.20 = { by axiom 6 (commutativity_of_join) }
% 30.19/4.20 ld(Y, join(Z, op(Y, X)))
% 30.19/4.20
% 30.19/4.20 Lemma 51: op(ld(X, X), ld(X, Y)) = ld(X, Y).
% 30.19/4.20 Proof:
% 30.19/4.20 op(ld(X, X), ld(X, Y))
% 30.19/4.20 = { by lemma 46 R->L }
% 30.19/4.20 join(op(ld(X, X), ld(X, Y)), op(meet(unit, meet(ld(X, X), rd(op(ld(X, X), ld(X, Y)), ld(X, Y)))), ld(X, Y)))
% 30.19/4.20 = { by lemma 25 }
% 30.19/4.20 join(op(ld(X, X), ld(X, Y)), op(meet(unit, ld(X, X)), ld(X, Y)))
% 30.19/4.20 = { by lemma 20 }
% 30.19/4.20 join(op(ld(X, X), ld(X, Y)), op(unit, ld(X, Y)))
% 30.19/4.20 = { by axiom 2 (left_monoid_unit) }
% 30.19/4.20 join(op(ld(X, X), ld(X, Y)), ld(X, Y))
% 30.19/4.21 = { by axiom 6 (commutativity_of_join) }
% 30.19/4.21 join(ld(X, Y), op(ld(X, X), ld(X, Y)))
% 30.19/4.21 = { by lemma 48 R->L }
% 30.19/4.21 join(ld(X, Y), op(ld(X, X), ld(meet(X, rd(op(X, join(unit, ld(X, X))), meet(unit, ld(X, op(X, join(unit, ld(X, X))))))), Y)))
% 30.19/4.21 = { by lemma 41 R->L }
% 30.19/4.21 join(ld(X, Y), op(ld(X, X), ld(meet(X, rd(op(X, join(unit, ld(X, X))), meet(unit, meet(join(unit, ld(X, X)), ld(X, op(X, join(unit, ld(X, X)))))))), Y)))
% 30.19/4.21 = { by lemma 19 }
% 30.19/4.21 join(ld(X, Y), op(ld(X, X), ld(meet(X, rd(op(X, join(unit, ld(X, X))), meet(unit, join(unit, ld(X, X))))), Y)))
% 30.19/4.21 = { by lemma 33 }
% 30.19/4.21 join(ld(X, Y), op(ld(X, X), ld(meet(X, rd(op(X, join(unit, ld(X, X))), unit)), Y)))
% 30.19/4.21 = { by lemma 23 }
% 30.19/4.21 join(ld(X, Y), op(ld(X, X), ld(meet(X, rd(op(X, ld(X, X)), unit)), Y)))
% 30.19/4.21 = { by lemma 22 R->L }
% 30.19/4.21 join(ld(X, Y), op(ld(X, X), ld(meet(X, join(rd(op(X, ld(X, X)), unit), meet(op(X, ld(X, X)), rd(op(X, ld(X, X)), unit)))), Y)))
% 30.19/4.21 = { by axiom 1 (right_monoid_unit) R->L }
% 30.19/4.21 join(ld(X, Y), op(ld(X, X), ld(meet(X, join(rd(op(X, ld(X, X)), unit), meet(op(X, ld(X, X)), rd(op(op(X, ld(X, X)), unit), unit)))), Y)))
% 30.19/4.21 = { by lemma 25 }
% 30.19/4.21 join(ld(X, Y), op(ld(X, X), ld(meet(X, join(rd(op(X, ld(X, X)), unit), op(X, ld(X, X)))), Y)))
% 30.19/4.21 = { by axiom 6 (commutativity_of_join) }
% 30.19/4.21 join(ld(X, Y), op(ld(X, X), ld(meet(X, join(op(X, ld(X, X)), rd(op(X, ld(X, X)), unit))), Y)))
% 30.19/4.21 = { by axiom 1 (right_monoid_unit) R->L }
% 30.19/4.21 join(ld(X, Y), op(ld(X, X), ld(meet(X, join(op(X, ld(X, X)), op(rd(op(X, ld(X, X)), unit), unit))), Y)))
% 30.19/4.21 = { by lemma 39 }
% 30.19/4.21 join(ld(X, Y), op(ld(X, X), ld(meet(X, op(X, ld(X, X))), Y)))
% 30.19/4.21 = { by lemma 35 }
% 30.19/4.21 join(ld(X, Y), op(ld(X, X), ld(op(X, ld(X, X)), Y)))
% 30.19/4.21 = { by axiom 6 (commutativity_of_join) R->L }
% 30.19/4.21 join(op(ld(X, X), ld(op(X, ld(X, X)), Y)), ld(X, Y))
% 30.19/4.21 = { by lemma 49 R->L }
% 30.19/4.21 join(op(ld(X, X), ld(op(X, ld(X, X)), Y)), ld(X, join(Y, op(X, op(ld(X, X), ld(op(X, ld(X, X)), Y))))))
% 30.19/4.21 = { by lemma 50 }
% 30.19/4.21 ld(X, join(Y, op(X, op(ld(X, X), ld(op(X, ld(X, X)), Y)))))
% 30.19/4.21 = { by lemma 49 }
% 30.19/4.21 ld(X, Y)
% 30.19/4.21
% 30.19/4.21 Lemma 52: ld(X, X) = unit.
% 30.19/4.21 Proof:
% 30.19/4.21 ld(X, X)
% 30.19/4.21 = { by lemma 23 R->L }
% 30.19/4.21 join(unit, ld(X, X))
% 30.19/4.21 = { by axiom 6 (commutativity_of_join) R->L }
% 30.19/4.21 join(ld(X, X), unit)
% 30.19/4.21 = { by lemma 45 R->L }
% 30.19/4.21 join(ld(X, X), rd(ld(X, Y), ld(X, Y)))
% 30.19/4.21 = { by lemma 51 R->L }
% 30.19/4.21 join(ld(X, X), rd(op(ld(X, X), ld(X, Y)), ld(X, Y)))
% 30.19/4.21 = { by lemma 44 }
% 30.19/4.21 rd(op(ld(X, X), ld(X, Y)), ld(X, Y))
% 30.19/4.21 = { by lemma 51 }
% 30.19/4.21 rd(ld(X, Y), ld(X, Y))
% 30.19/4.21 = { by lemma 45 }
% 30.19/4.21 unit
% 30.19/4.21
% 30.19/4.21 Lemma 53: ld(ld(X, unit), unit) = at(unit, X).
% 30.19/4.21 Proof:
% 30.19/4.21 ld(ld(X, unit), unit)
% 30.19/4.21 = { by axiom 2 (left_monoid_unit) R->L }
% 30.19/4.21 op(unit, ld(ld(X, unit), unit))
% 30.19/4.21 = { by axiom 2 (left_monoid_unit) R->L }
% 30.19/4.21 op(unit, op(unit, ld(ld(X, unit), unit)))
% 30.19/4.21 = { by lemma 30 R->L }
% 30.19/4.21 op(unit, op(ld(unit, unit), ld(ld(X, unit), unit)))
% 30.19/4.21 = { by lemma 31 }
% 30.19/4.21 at(unit, X)
% 30.19/4.21
% 30.19/4.21 Lemma 54: meet(X, rd(Y, ld(X, Y))) = X.
% 30.19/4.21 Proof:
% 30.19/4.21 meet(X, rd(Y, ld(X, Y)))
% 30.19/4.21 = { by lemma 29 R->L }
% 30.19/4.21 meet(X, rd(join(Y, op(X, ld(X, Y))), ld(X, Y)))
% 30.19/4.21 = { by lemma 47 }
% 30.19/4.21 X
% 30.19/4.21
% 30.19/4.21 Lemma 55: meet(op(X, Y), ld(rd(unit, X), Y)) = op(X, Y).
% 30.19/4.21 Proof:
% 30.19/4.21 meet(op(X, Y), ld(rd(unit, X), Y))
% 30.19/4.21 = { by lemma 46 R->L }
% 30.19/4.21 meet(op(X, Y), ld(rd(unit, X), join(Y, op(meet(op(rd(unit, X), X), meet(unit, rd(Y, Y))), Y))))
% 30.19/4.21 = { by lemma 26 }
% 30.19/4.21 meet(op(X, Y), ld(rd(unit, X), join(Y, op(meet(op(rd(unit, X), X), unit), Y))))
% 30.19/4.21 = { by axiom 4 (commutativity_of_meet) }
% 30.19/4.21 meet(op(X, Y), ld(rd(unit, X), join(Y, op(meet(unit, op(rd(unit, X), X)), Y))))
% 30.19/4.21 = { by lemma 40 }
% 30.19/4.21 meet(op(X, Y), ld(rd(unit, X), join(Y, op(op(rd(unit, X), X), Y))))
% 30.19/4.21 = { by axiom 7 (monoid_associativity) }
% 30.19/4.21 meet(op(X, Y), ld(rd(unit, X), join(Y, op(rd(unit, X), op(X, Y)))))
% 30.19/4.21 = { by lemma 42 }
% 30.19/4.21 op(X, Y)
% 30.19/4.21
% 30.19/4.21 Lemma 56: meet(rd(unit, X), ld(X, unit)) = rd(unit, X).
% 30.19/4.21 Proof:
% 30.19/4.21 meet(rd(unit, X), ld(X, unit))
% 30.19/4.21 = { by lemma 33 R->L }
% 30.19/4.21 meet(rd(unit, X), ld(X, meet(unit, join(unit, op(X, rd(unit, X))))))
% 30.19/4.21 = { by lemma 52 R->L }
% 30.19/4.21 meet(rd(unit, X), ld(X, meet(ld(rd(unit, X), rd(unit, X)), join(unit, op(X, rd(unit, X))))))
% 30.19/4.21 = { by lemma 55 R->L }
% 30.19/4.21 meet(rd(unit, X), ld(X, meet(ld(rd(unit, X), rd(unit, X)), join(unit, meet(op(X, rd(unit, X)), ld(rd(unit, X), rd(unit, X)))))))
% 30.19/4.21 = { by axiom 4 (commutativity_of_meet) R->L }
% 30.19/4.21 meet(rd(unit, X), ld(X, meet(ld(rd(unit, X), rd(unit, X)), join(unit, meet(ld(rd(unit, X), rd(unit, X)), op(X, rd(unit, X)))))))
% 30.19/4.21 = { by axiom 6 (commutativity_of_join) R->L }
% 30.19/4.21 meet(rd(unit, X), ld(X, meet(ld(rd(unit, X), rd(unit, X)), join(meet(ld(rd(unit, X), rd(unit, X)), op(X, rd(unit, X))), unit))))
% 30.19/4.21 = { by lemma 23 R->L }
% 30.19/4.21 meet(rd(unit, X), ld(X, meet(join(unit, ld(rd(unit, X), rd(unit, X))), join(meet(ld(rd(unit, X), rd(unit, X)), op(X, rd(unit, X))), unit))))
% 30.19/4.21 = { by axiom 6 (commutativity_of_join) R->L }
% 30.19/4.21 meet(rd(unit, X), ld(X, meet(join(unit, ld(rd(unit, X), rd(unit, X))), join(unit, meet(ld(rd(unit, X), rd(unit, X)), op(X, rd(unit, X)))))))
% 30.19/4.21 = { by axiom 6 (commutativity_of_join) R->L }
% 30.19/4.21 meet(rd(unit, X), ld(X, meet(join(ld(rd(unit, X), rd(unit, X)), unit), join(unit, meet(ld(rd(unit, X), rd(unit, X)), op(X, rd(unit, X)))))))
% 30.19/4.21 = { by axiom 4 (commutativity_of_meet) R->L }
% 30.19/4.21 meet(rd(unit, X), ld(X, meet(join(unit, meet(ld(rd(unit, X), rd(unit, X)), op(X, rd(unit, X)))), join(ld(rd(unit, X), rd(unit, X)), unit))))
% 30.19/4.21 = { by lemma 21 R->L }
% 30.19/4.21 meet(rd(unit, X), ld(X, meet(join(unit, meet(ld(rd(unit, X), rd(unit, X)), op(X, rd(unit, X)))), join(join(ld(rd(unit, X), rd(unit, X)), meet(ld(rd(unit, X), rd(unit, X)), op(X, rd(unit, X)))), unit))))
% 30.19/4.21 = { by axiom 11 (associativity_of_join) }
% 30.19/4.21 meet(rd(unit, X), ld(X, meet(join(unit, meet(ld(rd(unit, X), rd(unit, X)), op(X, rd(unit, X)))), join(ld(rd(unit, X), rd(unit, X)), join(meet(ld(rd(unit, X), rd(unit, X)), op(X, rd(unit, X))), unit)))))
% 30.19/4.21 = { by axiom 6 (commutativity_of_join) }
% 30.19/4.21 meet(rd(unit, X), ld(X, meet(join(unit, meet(ld(rd(unit, X), rd(unit, X)), op(X, rd(unit, X)))), join(ld(rd(unit, X), rd(unit, X)), join(unit, meet(ld(rd(unit, X), rd(unit, X)), op(X, rd(unit, X))))))))
% 30.19/4.21 = { by lemma 34 }
% 30.19/4.21 meet(rd(unit, X), ld(X, join(unit, meet(ld(rd(unit, X), rd(unit, X)), op(X, rd(unit, X))))))
% 30.19/4.21 = { by axiom 4 (commutativity_of_meet) }
% 30.19/4.21 meet(rd(unit, X), ld(X, join(unit, meet(op(X, rd(unit, X)), ld(rd(unit, X), rd(unit, X))))))
% 30.19/4.21 = { by lemma 55 }
% 30.19/4.21 meet(rd(unit, X), ld(X, join(unit, op(X, rd(unit, X)))))
% 30.19/4.21 = { by lemma 42 }
% 30.19/4.21 rd(unit, X)
% 30.19/4.21
% 30.19/4.21 Lemma 57: meet(X, at(unit, X)) = X.
% 30.19/4.21 Proof:
% 30.19/4.21 meet(X, at(unit, X))
% 30.19/4.21 = { by lemma 53 R->L }
% 30.19/4.21 meet(X, ld(ld(X, unit), unit))
% 30.19/4.21 = { by lemma 54 R->L }
% 30.19/4.21 meet(meet(X, rd(unit, ld(X, unit))), ld(ld(X, unit), unit))
% 30.19/4.21 = { by axiom 8 (associativity_of_meet) }
% 30.19/4.21 meet(X, meet(rd(unit, ld(X, unit)), ld(ld(X, unit), unit)))
% 30.19/4.21 = { by lemma 56 }
% 30.19/4.21 meet(X, rd(unit, ld(X, unit)))
% 30.19/4.21 = { by lemma 54 }
% 30.19/4.21 X
% 30.19/4.21
% 30.19/4.21 Lemma 58: join(X, op(Y, op(ld(Y, unit), X))) = X.
% 30.19/4.21 Proof:
% 30.19/4.21 join(X, op(Y, op(ld(Y, unit), X)))
% 30.19/4.21 = { by axiom 7 (monoid_associativity) R->L }
% 30.19/4.21 join(X, op(op(Y, ld(Y, unit)), X))
% 30.19/4.21 = { by lemma 32 }
% 30.19/4.21 join(X, op(at(Y, unit), X))
% 30.19/4.21 = { by lemma 36 R->L }
% 30.19/4.21 join(X, op(meet(unit, at(Y, unit)), X))
% 30.19/4.21 = { by lemma 25 R->L }
% 30.19/4.21 join(X, op(meet(meet(unit, rd(op(unit, X), X)), at(Y, unit)), X))
% 30.19/4.21 = { by axiom 8 (associativity_of_meet) }
% 30.19/4.21 join(X, op(meet(unit, meet(rd(op(unit, X), X), at(Y, unit))), X))
% 30.19/4.21 = { by axiom 4 (commutativity_of_meet) R->L }
% 30.19/4.21 join(X, op(meet(unit, meet(at(Y, unit), rd(op(unit, X), X))), X))
% 30.19/4.21 = { by lemma 32 R->L }
% 30.19/4.21 join(X, op(meet(unit, meet(op(Y, ld(Y, unit)), rd(op(unit, X), X))), X))
% 30.19/4.21 = { by axiom 8 (associativity_of_meet) R->L }
% 30.19/4.21 join(X, op(meet(meet(unit, op(Y, ld(Y, unit))), rd(op(unit, X), X)), X))
% 30.19/4.21 = { by lemma 35 }
% 30.19/4.21 join(X, op(meet(op(Y, ld(Y, unit)), rd(op(unit, X), X)), X))
% 30.19/4.21 = { by lemma 32 }
% 30.19/4.21 join(X, op(meet(at(Y, unit), rd(op(unit, X), X)), X))
% 30.19/4.21 = { by axiom 4 (commutativity_of_meet) }
% 30.19/4.21 join(X, op(meet(rd(op(unit, X), X), at(Y, unit)), X))
% 30.19/4.21 = { by axiom 2 (left_monoid_unit) }
% 30.19/4.21 join(X, op(meet(rd(X, X), at(Y, unit)), X))
% 30.19/4.21 = { by axiom 4 (commutativity_of_meet) }
% 30.19/4.21 join(X, op(meet(at(Y, unit), rd(X, X)), X))
% 30.19/4.21 = { by lemma 38 }
% 30.19/4.21 X
% 30.19/4.21
% 30.19/4.21 Lemma 59: join(ld(X, Y), op(ld(X, unit), Y)) = ld(X, Y).
% 30.19/4.21 Proof:
% 30.19/4.21 join(ld(X, Y), op(ld(X, unit), Y))
% 30.19/4.21 = { by axiom 6 (commutativity_of_join) R->L }
% 30.19/4.21 join(op(ld(X, unit), Y), ld(X, Y))
% 30.19/4.21 = { by lemma 58 R->L }
% 30.19/4.21 join(op(ld(X, unit), Y), ld(X, join(Y, op(X, op(ld(X, unit), Y)))))
% 30.19/4.21 = { by lemma 50 }
% 30.19/4.21 ld(X, join(Y, op(X, op(ld(X, unit), Y))))
% 30.19/4.21 = { by lemma 58 }
% 30.19/4.21 ld(X, Y)
% 30.19/4.21
% 30.19/4.21 Goal 1 (goal): at(unit, ld(x, unit)) = ld(x, unit).
% 30.19/4.21 Proof:
% 30.19/4.21 at(unit, ld(x, unit))
% 30.19/4.21 = { by lemma 48 R->L }
% 30.19/4.21 meet(at(unit, ld(x, unit)), rd(unit, meet(x, ld(at(unit, ld(x, unit)), unit))))
% 30.19/4.21 = { by lemma 53 R->L }
% 30.19/4.21 meet(at(unit, ld(x, unit)), rd(unit, meet(x, ld(ld(ld(ld(x, unit), unit), unit), unit))))
% 30.19/4.21 = { by lemma 53 }
% 30.19/4.21 meet(at(unit, ld(x, unit)), rd(unit, meet(x, at(unit, ld(ld(x, unit), unit)))))
% 30.19/4.21 = { by lemma 52 R->L }
% 30.19/4.21 meet(at(unit, ld(x, unit)), rd(unit, meet(x, at(unit, ld(ld(x, unit), ld(x, x))))))
% 30.19/4.21 = { by lemma 59 R->L }
% 30.19/4.21 meet(at(unit, ld(x, unit)), rd(unit, meet(x, at(unit, ld(ld(x, unit), join(ld(x, x), op(ld(x, unit), x)))))))
% 30.19/4.21 = { by lemma 50 R->L }
% 30.19/4.21 meet(at(unit, ld(x, unit)), rd(unit, meet(x, at(unit, join(x, ld(ld(x, unit), join(ld(x, x), op(ld(x, unit), x))))))))
% 30.19/4.21 = { by lemma 59 }
% 30.19/4.21 meet(at(unit, ld(x, unit)), rd(unit, meet(x, at(unit, join(x, ld(ld(x, unit), ld(x, x)))))))
% 30.19/4.21 = { by lemma 52 }
% 30.19/4.21 meet(at(unit, ld(x, unit)), rd(unit, meet(x, at(unit, join(x, ld(ld(x, unit), unit))))))
% 30.19/4.21 = { by lemma 53 }
% 30.19/4.21 meet(at(unit, ld(x, unit)), rd(unit, meet(x, at(unit, join(x, at(unit, x))))))
% 30.19/4.21 = { by lemma 41 R->L }
% 30.19/4.21 meet(at(unit, ld(x, unit)), rd(unit, meet(x, meet(join(x, at(unit, x)), at(unit, join(x, at(unit, x)))))))
% 30.19/4.21 = { by lemma 57 }
% 30.19/4.21 meet(at(unit, ld(x, unit)), rd(unit, meet(x, join(x, at(unit, x)))))
% 30.19/4.21 = { by lemma 33 }
% 30.19/4.21 meet(at(unit, ld(x, unit)), rd(unit, x))
% 30.19/4.21 = { by lemma 56 R->L }
% 30.19/4.21 meet(at(unit, ld(x, unit)), meet(rd(unit, x), ld(x, unit)))
% 30.19/4.21 = { by axiom 4 (commutativity_of_meet) }
% 30.19/4.21 meet(at(unit, ld(x, unit)), meet(ld(x, unit), rd(unit, x)))
% 30.19/4.21 = { by lemma 52 R->L }
% 30.19/4.21 meet(at(unit, ld(x, unit)), meet(ld(x, unit), rd(ld(x, x), x)))
% 30.19/4.21 = { by lemma 59 R->L }
% 30.19/4.21 meet(at(unit, ld(x, unit)), meet(ld(x, unit), rd(join(ld(x, x), op(ld(x, unit), x)), x)))
% 30.19/4.21 = { by lemma 47 }
% 30.19/4.22 meet(at(unit, ld(x, unit)), ld(x, unit))
% 30.19/4.22 = { by axiom 4 (commutativity_of_meet) }
% 30.19/4.22 meet(ld(x, unit), at(unit, ld(x, unit)))
% 30.19/4.22 = { by lemma 57 }
% 30.19/4.22 ld(x, unit)
% 30.19/4.22 % SZS output end Proof
% 30.19/4.22
% 30.19/4.22 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------