TSTP Solution File: GRP183-3 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP183-3 : TPTP v8.1.2. Bugfixed v1.2.1.
% 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 : n001.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 01:17:37 EDT 2023
% Result : Unsatisfiable 0.19s 0.78s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : GRP183-3 : TPTP v8.1.2. Bugfixed v1.2.1.
% 0.00/0.13 % 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 : n001.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 : Mon Aug 28 20:22:20 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.19/0.78 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.19/0.78
% 0.19/0.78 % SZS status Unsatisfiable
% 0.19/0.78
% 0.19/0.81 % SZS output start Proof
% 0.19/0.81 Axiom 1 (symmetry_of_lub): least_upper_bound(X, Y) = least_upper_bound(Y, X).
% 0.19/0.81 Axiom 2 (left_identity): multiply(identity, X) = X.
% 0.19/0.81 Axiom 3 (symmetry_of_glb): greatest_lower_bound(X, Y) = greatest_lower_bound(Y, X).
% 0.19/0.81 Axiom 4 (left_inverse): multiply(inverse(X), X) = identity.
% 0.19/0.81 Axiom 5 (lub_absorbtion): least_upper_bound(X, greatest_lower_bound(X, Y)) = X.
% 0.19/0.81 Axiom 6 (associativity_of_lub): least_upper_bound(X, least_upper_bound(Y, Z)) = least_upper_bound(least_upper_bound(X, Y), Z).
% 0.19/0.81 Axiom 7 (associativity): multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)).
% 0.19/0.81 Axiom 8 (glb_absorbtion): greatest_lower_bound(X, least_upper_bound(X, Y)) = X.
% 0.19/0.81 Axiom 9 (associativity_of_glb): greatest_lower_bound(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(greatest_lower_bound(X, Y), Z).
% 0.19/0.81 Axiom 10 (monotony_lub1): multiply(X, least_upper_bound(Y, Z)) = least_upper_bound(multiply(X, Y), multiply(X, Z)).
% 0.19/0.81 Axiom 11 (monotony_lub2): multiply(least_upper_bound(X, Y), Z) = least_upper_bound(multiply(X, Z), multiply(Y, Z)).
% 0.19/0.81 Axiom 12 (monotony_glb1): multiply(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(multiply(X, Y), multiply(X, Z)).
% 0.19/0.81 Axiom 13 (monotony_glb2): multiply(greatest_lower_bound(X, Y), Z) = greatest_lower_bound(multiply(X, Z), multiply(Y, Z)).
% 0.19/0.81
% 0.19/0.81 Lemma 14: multiply(inverse(X), multiply(X, Y)) = Y.
% 0.19/0.81 Proof:
% 0.19/0.81 multiply(inverse(X), multiply(X, Y))
% 0.19/0.81 = { by axiom 7 (associativity) R->L }
% 0.19/0.81 multiply(multiply(inverse(X), X), Y)
% 0.19/0.81 = { by axiom 4 (left_inverse) }
% 0.19/0.81 multiply(identity, Y)
% 0.19/0.81 = { by axiom 2 (left_identity) }
% 0.19/0.81 Y
% 0.19/0.81
% 0.19/0.81 Lemma 15: multiply(inverse(inverse(X)), Y) = multiply(X, Y).
% 0.19/0.81 Proof:
% 0.19/0.81 multiply(inverse(inverse(X)), Y)
% 0.19/0.81 = { by lemma 14 R->L }
% 0.19/0.81 multiply(inverse(inverse(X)), multiply(inverse(X), multiply(X, Y)))
% 0.19/0.81 = { by lemma 14 }
% 0.19/0.81 multiply(X, Y)
% 0.19/0.81
% 0.19/0.81 Lemma 16: multiply(inverse(inverse(X)), identity) = X.
% 0.19/0.81 Proof:
% 0.19/0.81 multiply(inverse(inverse(X)), identity)
% 0.19/0.81 = { by axiom 4 (left_inverse) R->L }
% 0.19/0.81 multiply(inverse(inverse(X)), multiply(inverse(X), X))
% 0.19/0.81 = { by lemma 14 }
% 0.19/0.81 X
% 0.19/0.81
% 0.19/0.81 Lemma 17: multiply(X, identity) = X.
% 0.19/0.81 Proof:
% 0.19/0.81 multiply(X, identity)
% 0.19/0.81 = { by lemma 15 R->L }
% 0.19/0.81 multiply(inverse(inverse(X)), identity)
% 0.19/0.81 = { by lemma 16 }
% 0.19/0.81 X
% 0.19/0.81
% 0.19/0.81 Lemma 18: multiply(X, inverse(X)) = identity.
% 0.19/0.81 Proof:
% 0.19/0.81 multiply(X, inverse(X))
% 0.19/0.81 = { by lemma 15 R->L }
% 0.19/0.81 multiply(inverse(inverse(X)), inverse(X))
% 0.19/0.81 = { by axiom 4 (left_inverse) }
% 0.19/0.81 identity
% 0.19/0.81
% 0.19/0.81 Lemma 19: multiply(X, least_upper_bound(identity, Y)) = least_upper_bound(X, multiply(X, Y)).
% 0.19/0.81 Proof:
% 0.19/0.81 multiply(X, least_upper_bound(identity, Y))
% 0.19/0.81 = { by axiom 10 (monotony_lub1) }
% 0.19/0.81 least_upper_bound(multiply(X, identity), multiply(X, Y))
% 0.19/0.81 = { by lemma 17 }
% 0.19/0.81 least_upper_bound(X, multiply(X, Y))
% 0.19/0.81
% 0.19/0.81 Lemma 20: multiply(least_upper_bound(X, identity), Y) = least_upper_bound(Y, multiply(X, Y)).
% 0.19/0.81 Proof:
% 0.19/0.81 multiply(least_upper_bound(X, identity), Y)
% 0.19/0.81 = { by axiom 1 (symmetry_of_lub) R->L }
% 0.19/0.81 multiply(least_upper_bound(identity, X), Y)
% 0.19/0.81 = { by axiom 11 (monotony_lub2) }
% 0.19/0.81 least_upper_bound(multiply(identity, Y), multiply(X, Y))
% 0.19/0.81 = { by axiom 2 (left_identity) }
% 0.19/0.81 least_upper_bound(Y, multiply(X, Y))
% 0.19/0.81
% 0.19/0.81 Lemma 21: multiply(least_upper_bound(identity, Y), X) = least_upper_bound(X, multiply(Y, X)).
% 0.19/0.81 Proof:
% 0.19/0.81 multiply(least_upper_bound(identity, Y), X)
% 0.19/0.81 = { by axiom 1 (symmetry_of_lub) R->L }
% 0.19/0.81 multiply(least_upper_bound(Y, identity), X)
% 0.19/0.81 = { by lemma 20 }
% 0.19/0.81 least_upper_bound(X, multiply(Y, X))
% 0.19/0.81
% 0.19/0.81 Lemma 22: greatest_lower_bound(X, multiply(X, Y)) = multiply(X, greatest_lower_bound(Y, identity)).
% 0.19/0.81 Proof:
% 0.19/0.81 greatest_lower_bound(X, multiply(X, Y))
% 0.19/0.81 = { by lemma 17 R->L }
% 0.19/0.81 greatest_lower_bound(multiply(X, identity), multiply(X, Y))
% 0.19/0.81 = { by axiom 12 (monotony_glb1) R->L }
% 0.19/0.81 multiply(X, greatest_lower_bound(identity, Y))
% 0.19/0.81 = { by axiom 3 (symmetry_of_glb) }
% 0.19/0.81 multiply(X, greatest_lower_bound(Y, identity))
% 0.19/0.81
% 0.19/0.81 Lemma 23: greatest_lower_bound(X, least_upper_bound(Y, X)) = X.
% 0.19/0.81 Proof:
% 0.19/0.81 greatest_lower_bound(X, least_upper_bound(Y, X))
% 0.19/0.81 = { by axiom 1 (symmetry_of_lub) }
% 0.19/0.81 greatest_lower_bound(X, least_upper_bound(X, Y))
% 0.19/0.81 = { by axiom 8 (glb_absorbtion) }
% 0.19/0.81 X
% 0.19/0.81
% 0.19/0.81 Lemma 24: least_upper_bound(X, greatest_lower_bound(Y, X)) = X.
% 0.19/0.81 Proof:
% 0.19/0.81 least_upper_bound(X, greatest_lower_bound(Y, X))
% 0.19/0.81 = { by axiom 3 (symmetry_of_glb) R->L }
% 0.19/0.81 least_upper_bound(X, greatest_lower_bound(X, Y))
% 0.19/0.81 = { by axiom 5 (lub_absorbtion) }
% 0.19/0.81 X
% 0.19/0.81
% 0.19/0.81 Lemma 25: multiply(least_upper_bound(inverse(X), identity), X) = least_upper_bound(X, identity).
% 0.19/0.81 Proof:
% 0.19/0.81 multiply(least_upper_bound(inverse(X), identity), X)
% 0.19/0.81 = { by axiom 1 (symmetry_of_lub) }
% 0.19/0.81 multiply(least_upper_bound(identity, inverse(X)), X)
% 0.19/0.81 = { by lemma 21 }
% 0.19/0.81 least_upper_bound(X, multiply(inverse(X), X))
% 0.19/0.81 = { by axiom 4 (left_inverse) }
% 0.19/0.81 least_upper_bound(X, identity)
% 0.19/0.81
% 0.19/0.81 Lemma 26: least_upper_bound(X, greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X))) = greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X)).
% 0.19/0.81 Proof:
% 0.19/0.81 least_upper_bound(X, greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X)))
% 0.19/0.81 = { by axiom 1 (symmetry_of_lub) R->L }
% 0.19/0.81 least_upper_bound(greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X)), X)
% 0.19/0.81 = { by lemma 23 R->L }
% 0.19/0.81 least_upper_bound(greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X)), greatest_lower_bound(X, least_upper_bound(Z, X)))
% 0.19/0.81 = { by axiom 3 (symmetry_of_glb) R->L }
% 0.19/0.81 least_upper_bound(greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X)), greatest_lower_bound(least_upper_bound(Z, X), X))
% 0.19/0.81 = { by lemma 23 R->L }
% 0.19/0.81 least_upper_bound(greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X)), greatest_lower_bound(least_upper_bound(Z, X), greatest_lower_bound(X, least_upper_bound(Y, X))))
% 0.19/0.81 = { by axiom 3 (symmetry_of_glb) R->L }
% 0.19/0.81 least_upper_bound(greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X)), greatest_lower_bound(least_upper_bound(Z, X), greatest_lower_bound(least_upper_bound(Y, X), X)))
% 0.19/0.81 = { by axiom 9 (associativity_of_glb) }
% 0.19/0.81 least_upper_bound(greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X)), greatest_lower_bound(greatest_lower_bound(least_upper_bound(Z, X), least_upper_bound(Y, X)), X))
% 0.19/0.81 = { by axiom 3 (symmetry_of_glb) R->L }
% 0.19/0.81 least_upper_bound(greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X)), greatest_lower_bound(greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X)), X))
% 0.19/0.81 = { by axiom 3 (symmetry_of_glb) }
% 0.19/0.81 least_upper_bound(greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X)), greatest_lower_bound(X, greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X))))
% 0.19/0.81 = { by lemma 24 }
% 0.19/0.81 greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X))
% 0.19/0.81
% 0.19/0.81 Lemma 27: multiply(X, greatest_lower_bound(least_upper_bound(X, identity), least_upper_bound(inverse(X), identity))) = multiply(least_upper_bound(X, identity), greatest_lower_bound(identity, X)).
% 0.19/0.81 Proof:
% 0.19/0.81 multiply(X, greatest_lower_bound(least_upper_bound(X, identity), least_upper_bound(inverse(X), identity)))
% 0.19/0.81 = { by axiom 3 (symmetry_of_glb) }
% 0.19/0.81 multiply(X, greatest_lower_bound(least_upper_bound(inverse(X), identity), least_upper_bound(X, identity)))
% 0.19/0.81 = { by lemma 25 R->L }
% 0.19/0.81 multiply(X, greatest_lower_bound(least_upper_bound(inverse(X), identity), multiply(least_upper_bound(inverse(X), identity), X)))
% 0.19/0.81 = { by lemma 22 }
% 0.19/0.81 multiply(X, multiply(least_upper_bound(inverse(X), identity), greatest_lower_bound(X, identity)))
% 0.19/0.81 = { by axiom 3 (symmetry_of_glb) }
% 0.19/0.81 multiply(X, multiply(least_upper_bound(inverse(X), identity), greatest_lower_bound(identity, X)))
% 0.19/0.81 = { by lemma 14 R->L }
% 0.19/0.81 multiply(X, multiply(least_upper_bound(inverse(X), identity), multiply(inverse(inverse(X)), multiply(inverse(X), greatest_lower_bound(identity, X)))))
% 0.19/0.81 = { by lemma 15 }
% 0.19/0.81 multiply(X, multiply(least_upper_bound(inverse(X), identity), multiply(X, multiply(inverse(X), greatest_lower_bound(identity, X)))))
% 0.19/0.81 = { by axiom 1 (symmetry_of_lub) }
% 0.19/0.81 multiply(X, multiply(least_upper_bound(identity, inverse(X)), multiply(X, multiply(inverse(X), greatest_lower_bound(identity, X)))))
% 0.19/0.81 = { by lemma 21 }
% 0.19/0.81 multiply(X, least_upper_bound(multiply(X, multiply(inverse(X), greatest_lower_bound(identity, X))), multiply(inverse(X), multiply(X, multiply(inverse(X), greatest_lower_bound(identity, X))))))
% 0.19/0.81 = { by lemma 14 }
% 0.19/0.81 multiply(X, least_upper_bound(multiply(X, multiply(inverse(X), greatest_lower_bound(identity, X))), multiply(inverse(X), greatest_lower_bound(identity, X))))
% 0.19/0.81 = { by axiom 1 (symmetry_of_lub) }
% 0.19/0.81 multiply(X, least_upper_bound(multiply(inverse(X), greatest_lower_bound(identity, X)), multiply(X, multiply(inverse(X), greatest_lower_bound(identity, X)))))
% 0.19/0.81 = { by lemma 20 R->L }
% 0.19/0.81 multiply(X, multiply(least_upper_bound(X, identity), multiply(inverse(X), greatest_lower_bound(identity, X))))
% 0.19/0.81 = { by axiom 3 (symmetry_of_glb) R->L }
% 0.19/0.81 multiply(X, multiply(least_upper_bound(X, identity), multiply(inverse(X), greatest_lower_bound(X, identity))))
% 0.19/0.81 = { by axiom 12 (monotony_glb1) }
% 0.19/0.81 multiply(X, multiply(least_upper_bound(X, identity), greatest_lower_bound(multiply(inverse(X), X), multiply(inverse(X), identity))))
% 0.19/0.81 = { by axiom 4 (left_inverse) }
% 0.19/0.81 multiply(X, multiply(least_upper_bound(X, identity), greatest_lower_bound(identity, multiply(inverse(X), identity))))
% 0.19/0.81 = { by axiom 2 (left_identity) R->L }
% 0.19/0.81 multiply(X, multiply(least_upper_bound(X, identity), greatest_lower_bound(multiply(identity, identity), multiply(inverse(X), identity))))
% 0.19/0.81 = { by axiom 13 (monotony_glb2) R->L }
% 0.19/0.81 multiply(X, multiply(least_upper_bound(X, identity), multiply(greatest_lower_bound(identity, inverse(X)), identity)))
% 0.19/0.81 = { by axiom 3 (symmetry_of_glb) }
% 0.19/0.81 multiply(X, multiply(least_upper_bound(X, identity), multiply(greatest_lower_bound(inverse(X), identity), identity)))
% 0.19/0.81 = { by lemma 17 }
% 0.19/0.81 multiply(X, multiply(least_upper_bound(X, identity), greatest_lower_bound(inverse(X), identity)))
% 0.19/0.81 = { by axiom 3 (symmetry_of_glb) }
% 0.19/0.81 multiply(X, multiply(least_upper_bound(X, identity), greatest_lower_bound(identity, inverse(X))))
% 0.19/0.81 = { by lemma 20 }
% 0.19/0.81 multiply(X, least_upper_bound(greatest_lower_bound(identity, inverse(X)), multiply(X, greatest_lower_bound(identity, inverse(X)))))
% 0.19/0.81 = { by axiom 10 (monotony_lub1) }
% 0.19/0.81 least_upper_bound(multiply(X, greatest_lower_bound(identity, inverse(X))), multiply(X, multiply(X, greatest_lower_bound(identity, inverse(X)))))
% 0.19/0.81 = { by lemma 20 R->L }
% 0.19/0.81 multiply(least_upper_bound(X, identity), multiply(X, greatest_lower_bound(identity, inverse(X))))
% 0.19/0.81 = { by axiom 3 (symmetry_of_glb) R->L }
% 0.19/0.81 multiply(least_upper_bound(X, identity), multiply(X, greatest_lower_bound(inverse(X), identity)))
% 0.19/0.81 = { by lemma 15 R->L }
% 0.19/0.81 multiply(least_upper_bound(X, identity), multiply(inverse(inverse(X)), greatest_lower_bound(inverse(X), identity)))
% 0.19/0.81 = { by axiom 12 (monotony_glb1) }
% 0.19/0.81 multiply(least_upper_bound(X, identity), greatest_lower_bound(multiply(inverse(inverse(X)), inverse(X)), multiply(inverse(inverse(X)), identity)))
% 0.19/0.81 = { by axiom 4 (left_inverse) }
% 0.19/0.81 multiply(least_upper_bound(X, identity), greatest_lower_bound(identity, multiply(inverse(inverse(X)), identity)))
% 0.19/0.81 = { by lemma 16 }
% 0.19/0.81 multiply(least_upper_bound(X, identity), greatest_lower_bound(identity, X))
% 0.19/0.81
% 0.19/0.81 Lemma 28: least_upper_bound(X, multiply(greatest_lower_bound(least_upper_bound(Y, identity), least_upper_bound(Z, identity)), X)) = multiply(greatest_lower_bound(least_upper_bound(Y, identity), least_upper_bound(Z, identity)), X).
% 0.19/0.81 Proof:
% 0.19/0.81 least_upper_bound(X, multiply(greatest_lower_bound(least_upper_bound(Y, identity), least_upper_bound(Z, identity)), X))
% 0.19/0.81 = { by lemma 21 R->L }
% 0.19/0.81 multiply(least_upper_bound(identity, greatest_lower_bound(least_upper_bound(Y, identity), least_upper_bound(Z, identity))), X)
% 0.19/0.81 = { by lemma 26 }
% 0.19/0.81 multiply(greatest_lower_bound(least_upper_bound(Y, identity), least_upper_bound(Z, identity)), X)
% 0.19/0.81
% 0.19/0.81 Goal 1 (prove_20x): greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)) = identity.
% 0.19/0.81 Proof:
% 0.19/0.81 greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))
% 0.19/0.81 = { by lemma 17 R->L }
% 0.19/0.81 multiply(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)), identity)
% 0.19/0.81 = { by lemma 18 R->L }
% 0.19/0.81 multiply(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)), multiply(least_upper_bound(a, identity), inverse(least_upper_bound(a, identity))))
% 0.19/0.81 = { by axiom 7 (associativity) R->L }
% 0.19/0.81 multiply(multiply(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)), least_upper_bound(a, identity)), inverse(least_upper_bound(a, identity)))
% 0.19/0.81 = { by lemma 28 R->L }
% 0.19/0.81 multiply(least_upper_bound(least_upper_bound(a, identity), multiply(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)), least_upper_bound(a, identity))), inverse(least_upper_bound(a, identity)))
% 0.19/0.81 = { by lemma 24 R->L }
% 0.19/0.81 multiply(least_upper_bound(least_upper_bound(least_upper_bound(a, identity), greatest_lower_bound(a, least_upper_bound(a, identity))), multiply(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)), least_upper_bound(a, identity))), inverse(least_upper_bound(a, identity)))
% 0.19/0.81 = { by axiom 8 (glb_absorbtion) }
% 0.19/0.81 multiply(least_upper_bound(least_upper_bound(least_upper_bound(a, identity), a), multiply(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)), least_upper_bound(a, identity))), inverse(least_upper_bound(a, identity)))
% 0.19/0.81 = { by axiom 1 (symmetry_of_lub) }
% 0.19/0.81 multiply(least_upper_bound(least_upper_bound(a, least_upper_bound(a, identity)), multiply(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)), least_upper_bound(a, identity))), inverse(least_upper_bound(a, identity)))
% 0.19/0.81 = { by axiom 6 (associativity_of_lub) R->L }
% 0.19/0.81 multiply(least_upper_bound(a, least_upper_bound(least_upper_bound(a, identity), multiply(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)), least_upper_bound(a, identity)))), inverse(least_upper_bound(a, identity)))
% 0.19/0.81 = { by lemma 28 }
% 0.19/0.81 multiply(least_upper_bound(a, multiply(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)), least_upper_bound(a, identity))), inverse(least_upper_bound(a, identity)))
% 0.19/0.81 = { by axiom 1 (symmetry_of_lub) }
% 0.19/0.81 multiply(least_upper_bound(a, multiply(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)), least_upper_bound(identity, a))), inverse(least_upper_bound(a, identity)))
% 0.19/0.82 = { by lemma 19 }
% 0.19/0.82 multiply(least_upper_bound(a, least_upper_bound(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)), multiply(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)), a))), inverse(least_upper_bound(a, identity)))
% 0.19/0.82 = { by axiom 3 (symmetry_of_glb) R->L }
% 0.19/0.82 multiply(least_upper_bound(a, least_upper_bound(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)), multiply(greatest_lower_bound(least_upper_bound(inverse(a), identity), least_upper_bound(a, identity)), a))), inverse(least_upper_bound(a, identity)))
% 0.19/0.82 = { by axiom 13 (monotony_glb2) }
% 0.19/0.82 multiply(least_upper_bound(a, least_upper_bound(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)), greatest_lower_bound(multiply(least_upper_bound(inverse(a), identity), a), multiply(least_upper_bound(a, identity), a)))), inverse(least_upper_bound(a, identity)))
% 0.19/0.82 = { by lemma 25 }
% 0.19/0.82 multiply(least_upper_bound(a, least_upper_bound(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)), greatest_lower_bound(least_upper_bound(a, identity), multiply(least_upper_bound(a, identity), a)))), inverse(least_upper_bound(a, identity)))
% 0.19/0.82 = { by lemma 22 }
% 0.19/0.82 multiply(least_upper_bound(a, least_upper_bound(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)), multiply(least_upper_bound(a, identity), greatest_lower_bound(a, identity)))), inverse(least_upper_bound(a, identity)))
% 0.19/0.82 = { by axiom 3 (symmetry_of_glb) }
% 0.19/0.82 multiply(least_upper_bound(a, least_upper_bound(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)), multiply(least_upper_bound(a, identity), greatest_lower_bound(identity, a)))), inverse(least_upper_bound(a, identity)))
% 0.19/0.82 = { by lemma 27 R->L }
% 0.19/0.82 multiply(least_upper_bound(a, least_upper_bound(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)), multiply(a, greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))), inverse(least_upper_bound(a, identity)))
% 0.19/0.82 = { by axiom 6 (associativity_of_lub) }
% 0.19/0.82 multiply(least_upper_bound(least_upper_bound(a, greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), multiply(a, greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), inverse(least_upper_bound(a, identity)))
% 0.19/0.82 = { by axiom 1 (symmetry_of_lub) R->L }
% 0.19/0.82 multiply(least_upper_bound(least_upper_bound(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)), a), multiply(a, greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), inverse(least_upper_bound(a, identity)))
% 0.19/0.82 = { by lemma 26 R->L }
% 0.19/0.82 multiply(least_upper_bound(least_upper_bound(least_upper_bound(identity, greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), a), multiply(a, greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), inverse(least_upper_bound(a, identity)))
% 0.19/0.82 = { by axiom 6 (associativity_of_lub) R->L }
% 0.19/0.82 multiply(least_upper_bound(least_upper_bound(identity, least_upper_bound(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)), a)), multiply(a, greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), inverse(least_upper_bound(a, identity)))
% 0.19/0.82 = { by axiom 1 (symmetry_of_lub) R->L }
% 0.19/0.82 multiply(least_upper_bound(least_upper_bound(identity, least_upper_bound(a, greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), multiply(a, greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), inverse(least_upper_bound(a, identity)))
% 0.19/0.82 = { by axiom 6 (associativity_of_lub) }
% 0.19/0.82 multiply(least_upper_bound(least_upper_bound(least_upper_bound(identity, a), greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), multiply(a, greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), inverse(least_upper_bound(a, identity)))
% 0.19/0.82 = { by axiom 1 (symmetry_of_lub) R->L }
% 0.19/0.82 multiply(least_upper_bound(least_upper_bound(least_upper_bound(a, identity), greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), multiply(a, greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), inverse(least_upper_bound(a, identity)))
% 0.19/0.82 = { by axiom 3 (symmetry_of_glb) }
% 0.19/0.82 multiply(least_upper_bound(least_upper_bound(least_upper_bound(a, identity), greatest_lower_bound(least_upper_bound(inverse(a), identity), least_upper_bound(a, identity))), multiply(a, greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), inverse(least_upper_bound(a, identity)))
% 0.19/0.82 = { by lemma 24 }
% 0.19/0.82 multiply(least_upper_bound(least_upper_bound(a, identity), multiply(a, greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), inverse(least_upper_bound(a, identity)))
% 0.19/0.82 = { by lemma 27 }
% 0.19/0.82 multiply(least_upper_bound(least_upper_bound(a, identity), multiply(least_upper_bound(a, identity), greatest_lower_bound(identity, a))), inverse(least_upper_bound(a, identity)))
% 0.19/0.82 = { by lemma 19 R->L }
% 0.19/0.82 multiply(multiply(least_upper_bound(a, identity), least_upper_bound(identity, greatest_lower_bound(identity, a))), inverse(least_upper_bound(a, identity)))
% 0.19/0.82 = { by axiom 5 (lub_absorbtion) }
% 0.19/0.82 multiply(multiply(least_upper_bound(a, identity), identity), inverse(least_upper_bound(a, identity)))
% 0.19/0.82 = { by lemma 17 }
% 0.19/0.82 multiply(least_upper_bound(a, identity), inverse(least_upper_bound(a, identity)))
% 0.19/0.82 = { by lemma 18 }
% 0.19/0.82 identity
% 0.19/0.82 % SZS output end Proof
% 0.19/0.82
% 0.19/0.82 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------