TSTP Solution File: GRP181-2 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP181-2 : 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 : n018.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 01:17:35 EDT 2023
% Result : Unsatisfiable 65.44s 8.70s
% Output : Proof 65.66s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : GRP181-2 : TPTP v8.1.2. Bugfixed v1.2.1.
% 0.07/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.35 % Computer : n018.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 : Tue Aug 29 01:28:17 EDT 2023
% 0.13/0.35 % CPUTime :
% 65.44/8.70 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 65.44/8.70
% 65.44/8.70 % SZS status Unsatisfiable
% 65.44/8.70
% 65.66/8.76 % SZS output start Proof
% 65.66/8.76 Axiom 1 (p12_1): inverse(identity) = identity.
% 65.66/8.76 Axiom 2 (left_identity): multiply(identity, X) = X.
% 65.66/8.76 Axiom 3 (idempotence_of_gld): greatest_lower_bound(X, X) = X.
% 65.66/8.76 Axiom 4 (symmetry_of_glb): greatest_lower_bound(X, Y) = greatest_lower_bound(Y, X).
% 65.66/8.76 Axiom 5 (p12_4): greatest_lower_bound(a, c) = greatest_lower_bound(b, c).
% 65.66/8.76 Axiom 6 (symmetry_of_lub): least_upper_bound(X, Y) = least_upper_bound(Y, X).
% 65.66/8.76 Axiom 7 (p12_5): least_upper_bound(a, c) = least_upper_bound(b, c).
% 65.66/8.76 Axiom 8 (p12_2): inverse(inverse(X)) = X.
% 65.66/8.76 Axiom 9 (left_inverse): multiply(inverse(X), X) = identity.
% 65.66/8.76 Axiom 10 (p12_3): inverse(multiply(X, Y)) = multiply(inverse(Y), inverse(X)).
% 65.66/8.76 Axiom 11 (associativity): multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)).
% 65.66/8.76 Axiom 12 (glb_absorbtion): greatest_lower_bound(X, least_upper_bound(X, Y)) = X.
% 65.66/8.76 Axiom 13 (associativity_of_glb): greatest_lower_bound(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(greatest_lower_bound(X, Y), Z).
% 65.66/8.76 Axiom 14 (lub_absorbtion): least_upper_bound(X, greatest_lower_bound(X, Y)) = X.
% 65.66/8.76 Axiom 15 (associativity_of_lub): least_upper_bound(X, least_upper_bound(Y, Z)) = least_upper_bound(least_upper_bound(X, Y), Z).
% 65.66/8.76 Axiom 16 (monotony_glb1): multiply(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(multiply(X, Y), multiply(X, Z)).
% 65.66/8.76 Axiom 17 (monotony_glb2): multiply(greatest_lower_bound(X, Y), Z) = greatest_lower_bound(multiply(X, Z), multiply(Y, Z)).
% 65.66/8.76 Axiom 18 (monotony_lub1): multiply(X, least_upper_bound(Y, Z)) = least_upper_bound(multiply(X, Y), multiply(X, Z)).
% 65.66/8.76 Axiom 19 (monotony_lub2): multiply(least_upper_bound(X, Y), Z) = least_upper_bound(multiply(X, Z), multiply(Y, Z)).
% 65.66/8.76
% 65.66/8.76 Lemma 20: multiply(X, identity) = X.
% 65.66/8.76 Proof:
% 65.66/8.76 multiply(X, identity)
% 65.66/8.76 = { by axiom 8 (p12_2) R->L }
% 65.66/8.76 inverse(inverse(multiply(X, identity)))
% 65.66/8.76 = { by axiom 10 (p12_3) }
% 65.66/8.76 inverse(multiply(inverse(identity), inverse(X)))
% 65.66/8.76 = { by axiom 1 (p12_1) }
% 65.66/8.76 inverse(multiply(identity, inverse(X)))
% 65.66/8.76 = { by axiom 2 (left_identity) }
% 65.66/8.76 inverse(inverse(X))
% 65.66/8.76 = { by axiom 8 (p12_2) }
% 65.66/8.76 X
% 65.66/8.76
% 65.66/8.76 Lemma 21: greatest_lower_bound(c, b) = greatest_lower_bound(c, a).
% 65.66/8.76 Proof:
% 65.66/8.76 greatest_lower_bound(c, b)
% 65.66/8.76 = { by axiom 4 (symmetry_of_glb) R->L }
% 65.66/8.76 greatest_lower_bound(b, c)
% 65.66/8.76 = { by axiom 5 (p12_4) R->L }
% 65.66/8.76 greatest_lower_bound(a, c)
% 65.66/8.76 = { by axiom 4 (symmetry_of_glb) }
% 65.66/8.76 greatest_lower_bound(c, a)
% 65.66/8.76
% 65.66/8.76 Lemma 22: multiply(X, inverse(X)) = identity.
% 65.66/8.76 Proof:
% 65.66/8.76 multiply(X, inverse(X))
% 65.66/8.76 = { by axiom 8 (p12_2) R->L }
% 65.66/8.76 multiply(inverse(inverse(X)), inverse(X))
% 65.66/8.76 = { by axiom 9 (left_inverse) }
% 65.66/8.76 identity
% 65.66/8.76
% 65.66/8.76 Lemma 23: inverse(multiply(inverse(X), Y)) = multiply(inverse(Y), X).
% 65.66/8.76 Proof:
% 65.66/8.76 inverse(multiply(inverse(X), Y))
% 65.66/8.76 = { by axiom 10 (p12_3) }
% 65.66/8.76 multiply(inverse(Y), inverse(inverse(X)))
% 65.66/8.76 = { by axiom 8 (p12_2) }
% 65.66/8.76 multiply(inverse(Y), X)
% 65.66/8.76
% 65.66/8.76 Lemma 24: greatest_lower_bound(X, multiply(Y, X)) = multiply(greatest_lower_bound(Y, identity), X).
% 65.66/8.76 Proof:
% 65.66/8.76 greatest_lower_bound(X, multiply(Y, X))
% 65.66/8.76 = { by axiom 2 (left_identity) R->L }
% 65.66/8.76 greatest_lower_bound(multiply(identity, X), multiply(Y, X))
% 65.66/8.76 = { by axiom 17 (monotony_glb2) R->L }
% 65.66/8.76 multiply(greatest_lower_bound(identity, Y), X)
% 65.66/8.76 = { by axiom 4 (symmetry_of_glb) }
% 65.66/8.76 multiply(greatest_lower_bound(Y, identity), X)
% 65.66/8.76
% 65.66/8.76 Lemma 25: greatest_lower_bound(X, least_upper_bound(Y, X)) = X.
% 65.66/8.76 Proof:
% 65.66/8.76 greatest_lower_bound(X, least_upper_bound(Y, X))
% 65.66/8.76 = { by axiom 6 (symmetry_of_lub) R->L }
% 65.66/8.76 greatest_lower_bound(X, least_upper_bound(X, Y))
% 65.66/8.76 = { by axiom 12 (glb_absorbtion) }
% 65.66/8.76 X
% 65.66/8.76
% 65.66/8.76 Lemma 26: least_upper_bound(X, multiply(Y, X)) = multiply(least_upper_bound(Y, identity), X).
% 65.66/8.76 Proof:
% 65.66/8.76 least_upper_bound(X, multiply(Y, X))
% 65.66/8.76 = { by axiom 2 (left_identity) R->L }
% 65.66/8.76 least_upper_bound(multiply(identity, X), multiply(Y, X))
% 65.66/8.76 = { by axiom 19 (monotony_lub2) R->L }
% 65.66/8.76 multiply(least_upper_bound(identity, Y), X)
% 65.66/8.76 = { by axiom 6 (symmetry_of_lub) }
% 65.66/8.76 multiply(least_upper_bound(Y, identity), X)
% 65.66/8.76
% 65.66/8.76 Lemma 27: multiply(inverse(X), multiply(X, Y)) = Y.
% 65.66/8.76 Proof:
% 65.66/8.76 multiply(inverse(X), multiply(X, Y))
% 65.66/8.76 = { by axiom 11 (associativity) R->L }
% 65.66/8.76 multiply(multiply(inverse(X), X), Y)
% 65.66/8.76 = { by axiom 9 (left_inverse) }
% 65.66/8.76 multiply(identity, Y)
% 65.66/8.76 = { by axiom 2 (left_identity) }
% 65.66/8.76 Y
% 65.66/8.76
% 65.66/8.76 Lemma 28: multiply(X, inverse(multiply(Y, X))) = inverse(Y).
% 65.66/8.76 Proof:
% 65.66/8.76 multiply(X, inverse(multiply(Y, X)))
% 65.66/8.76 = { by axiom 8 (p12_2) R->L }
% 65.66/8.76 multiply(inverse(inverse(X)), inverse(multiply(Y, X)))
% 65.66/8.76 = { by axiom 10 (p12_3) }
% 65.66/8.76 multiply(inverse(inverse(X)), multiply(inverse(X), inverse(Y)))
% 65.66/8.76 = { by lemma 27 }
% 65.66/8.76 inverse(Y)
% 65.66/8.76
% 65.66/8.76 Lemma 29: greatest_lower_bound(identity, multiply(inverse(X), Y)) = multiply(inverse(X), greatest_lower_bound(X, Y)).
% 65.66/8.76 Proof:
% 65.66/8.76 greatest_lower_bound(identity, multiply(inverse(X), Y))
% 65.66/8.76 = { by axiom 9 (left_inverse) R->L }
% 65.66/8.76 greatest_lower_bound(multiply(inverse(X), X), multiply(inverse(X), Y))
% 65.66/8.76 = { by axiom 16 (monotony_glb1) R->L }
% 65.66/8.76 multiply(inverse(X), greatest_lower_bound(X, Y))
% 65.66/8.76
% 65.66/8.76 Lemma 30: multiply(inverse(X), greatest_lower_bound(X, identity)) = greatest_lower_bound(identity, inverse(X)).
% 65.66/8.76 Proof:
% 65.66/8.76 multiply(inverse(X), greatest_lower_bound(X, identity))
% 65.66/8.76 = { by lemma 29 R->L }
% 65.66/8.76 greatest_lower_bound(identity, multiply(inverse(X), identity))
% 65.66/8.76 = { by lemma 24 }
% 65.66/8.76 multiply(greatest_lower_bound(inverse(X), identity), identity)
% 65.66/8.76 = { by lemma 20 }
% 65.66/8.76 greatest_lower_bound(inverse(X), identity)
% 65.66/8.76 = { by axiom 4 (symmetry_of_glb) }
% 65.66/8.77 greatest_lower_bound(identity, inverse(X))
% 65.66/8.77
% 65.66/8.77 Lemma 31: multiply(greatest_lower_bound(X, identity), inverse(X)) = greatest_lower_bound(identity, inverse(X)).
% 65.66/8.77 Proof:
% 65.66/8.77 multiply(greatest_lower_bound(X, identity), inverse(X))
% 65.66/8.77 = { by lemma 24 R->L }
% 65.66/8.77 greatest_lower_bound(inverse(X), multiply(X, inverse(X)))
% 65.66/8.77 = { by lemma 22 }
% 65.66/8.77 greatest_lower_bound(inverse(X), identity)
% 65.66/8.77 = { by axiom 4 (symmetry_of_glb) }
% 65.66/8.77 greatest_lower_bound(identity, inverse(X))
% 65.66/8.77
% 65.66/8.77 Lemma 32: least_upper_bound(identity, multiply(inverse(X), Y)) = multiply(inverse(X), least_upper_bound(X, Y)).
% 65.66/8.77 Proof:
% 65.66/8.77 least_upper_bound(identity, multiply(inverse(X), Y))
% 65.66/8.77 = { by axiom 9 (left_inverse) R->L }
% 65.66/8.77 least_upper_bound(multiply(inverse(X), X), multiply(inverse(X), Y))
% 65.66/8.77 = { by axiom 18 (monotony_lub1) R->L }
% 65.66/8.77 multiply(inverse(X), least_upper_bound(X, Y))
% 65.66/8.77
% 65.66/8.77 Lemma 33: multiply(inverse(X), least_upper_bound(X, inverse(Y))) = least_upper_bound(identity, inverse(multiply(Y, X))).
% 65.66/8.77 Proof:
% 65.66/8.77 multiply(inverse(X), least_upper_bound(X, inverse(Y)))
% 65.66/8.77 = { by lemma 32 R->L }
% 65.66/8.77 least_upper_bound(identity, multiply(inverse(X), inverse(Y)))
% 65.66/8.77 = { by axiom 10 (p12_3) R->L }
% 65.66/8.77 least_upper_bound(identity, inverse(multiply(Y, X)))
% 65.66/8.77
% 65.66/8.77 Lemma 34: greatest_lower_bound(least_upper_bound(X, identity), least_upper_bound(identity, inverse(X))) = multiply(greatest_lower_bound(X, identity), least_upper_bound(identity, inverse(X))).
% 65.66/8.77 Proof:
% 65.66/8.77 greatest_lower_bound(least_upper_bound(X, identity), least_upper_bound(identity, inverse(X)))
% 65.66/8.77 = { by axiom 6 (symmetry_of_lub) R->L }
% 65.66/8.77 greatest_lower_bound(least_upper_bound(X, identity), least_upper_bound(inverse(X), identity))
% 65.66/8.77 = { by lemma 22 R->L }
% 65.66/8.77 greatest_lower_bound(least_upper_bound(X, identity), least_upper_bound(inverse(X), multiply(X, inverse(X))))
% 65.66/8.77 = { by lemma 26 }
% 65.66/8.77 greatest_lower_bound(least_upper_bound(X, identity), multiply(least_upper_bound(X, identity), inverse(X)))
% 65.66/8.77 = { by lemma 20 R->L }
% 65.66/8.77 greatest_lower_bound(multiply(least_upper_bound(X, identity), identity), multiply(least_upper_bound(X, identity), inverse(X)))
% 65.66/8.77 = { by axiom 16 (monotony_glb1) R->L }
% 65.66/8.77 multiply(least_upper_bound(X, identity), greatest_lower_bound(identity, inverse(X)))
% 65.66/8.77 = { by lemma 30 R->L }
% 65.66/8.77 multiply(least_upper_bound(X, identity), multiply(inverse(X), greatest_lower_bound(X, identity)))
% 65.66/8.77 = { by lemma 26 R->L }
% 65.66/8.77 least_upper_bound(multiply(inverse(X), greatest_lower_bound(X, identity)), multiply(X, multiply(inverse(X), greatest_lower_bound(X, identity))))
% 65.66/8.77 = { by axiom 11 (associativity) R->L }
% 65.66/8.77 least_upper_bound(multiply(inverse(X), greatest_lower_bound(X, identity)), multiply(multiply(X, inverse(X)), greatest_lower_bound(X, identity)))
% 65.66/8.77 = { by lemma 22 }
% 65.66/8.77 least_upper_bound(multiply(inverse(X), greatest_lower_bound(X, identity)), multiply(identity, greatest_lower_bound(X, identity)))
% 65.66/8.77 = { by axiom 2 (left_identity) }
% 65.66/8.77 least_upper_bound(multiply(inverse(X), greatest_lower_bound(X, identity)), greatest_lower_bound(X, identity))
% 65.66/8.77 = { by axiom 6 (symmetry_of_lub) }
% 65.66/8.77 least_upper_bound(greatest_lower_bound(X, identity), multiply(inverse(X), greatest_lower_bound(X, identity)))
% 65.66/8.77 = { by lemma 26 }
% 65.66/8.77 multiply(least_upper_bound(inverse(X), identity), greatest_lower_bound(X, identity))
% 65.66/8.77 = { by axiom 6 (symmetry_of_lub) }
% 65.66/8.77 multiply(least_upper_bound(identity, inverse(X)), greatest_lower_bound(X, identity))
% 65.66/8.77 = { by axiom 8 (p12_2) R->L }
% 65.66/8.77 multiply(least_upper_bound(identity, inverse(X)), inverse(inverse(greatest_lower_bound(X, identity))))
% 65.66/8.77 = { by axiom 6 (symmetry_of_lub) R->L }
% 65.66/8.77 multiply(least_upper_bound(inverse(X), identity), inverse(inverse(greatest_lower_bound(X, identity))))
% 65.66/8.77 = { by lemma 26 R->L }
% 65.66/8.77 least_upper_bound(inverse(inverse(greatest_lower_bound(X, identity))), multiply(inverse(X), inverse(inverse(greatest_lower_bound(X, identity)))))
% 65.66/8.77 = { by axiom 10 (p12_3) R->L }
% 65.66/8.77 least_upper_bound(inverse(inverse(greatest_lower_bound(X, identity))), inverse(multiply(inverse(greatest_lower_bound(X, identity)), X)))
% 65.66/8.77 = { by lemma 23 R->L }
% 65.66/8.77 least_upper_bound(inverse(inverse(greatest_lower_bound(X, identity))), inverse(inverse(multiply(inverse(X), greatest_lower_bound(X, identity)))))
% 65.66/8.77 = { by lemma 30 }
% 65.66/8.77 least_upper_bound(inverse(inverse(greatest_lower_bound(X, identity))), inverse(inverse(greatest_lower_bound(identity, inverse(X)))))
% 65.66/8.77 = { by axiom 8 (p12_2) }
% 65.66/8.77 least_upper_bound(greatest_lower_bound(X, identity), inverse(inverse(greatest_lower_bound(identity, inverse(X)))))
% 65.66/8.77 = { by axiom 8 (p12_2) }
% 65.66/8.77 least_upper_bound(greatest_lower_bound(X, identity), greatest_lower_bound(identity, inverse(X)))
% 65.66/8.77 = { by lemma 31 R->L }
% 65.66/8.77 least_upper_bound(greatest_lower_bound(X, identity), multiply(greatest_lower_bound(X, identity), inverse(X)))
% 65.66/8.77 = { by lemma 20 R->L }
% 65.66/8.77 least_upper_bound(multiply(greatest_lower_bound(X, identity), identity), multiply(greatest_lower_bound(X, identity), inverse(X)))
% 65.66/8.77 = { by axiom 18 (monotony_lub1) R->L }
% 65.66/8.77 multiply(greatest_lower_bound(X, identity), least_upper_bound(identity, inverse(X)))
% 65.66/8.77
% 65.66/8.77 Lemma 35: multiply(inverse(least_upper_bound(X, inverse(Y))), X) = greatest_lower_bound(identity, multiply(Y, X)).
% 65.66/8.77 Proof:
% 65.66/8.77 multiply(inverse(least_upper_bound(X, inverse(Y))), X)
% 65.66/8.77 = { by lemma 23 R->L }
% 65.66/8.77 inverse(multiply(inverse(X), least_upper_bound(X, inverse(Y))))
% 65.66/8.77 = { by lemma 33 }
% 65.66/8.77 inverse(least_upper_bound(identity, inverse(multiply(Y, X))))
% 65.66/8.77 = { by lemma 27 R->L }
% 65.66/8.77 inverse(multiply(inverse(greatest_lower_bound(multiply(Y, X), identity)), multiply(greatest_lower_bound(multiply(Y, X), identity), least_upper_bound(identity, inverse(multiply(Y, X))))))
% 65.66/8.77 = { by lemma 23 }
% 65.66/8.77 multiply(inverse(multiply(greatest_lower_bound(multiply(Y, X), identity), least_upper_bound(identity, inverse(multiply(Y, X))))), greatest_lower_bound(multiply(Y, X), identity))
% 65.66/8.77 = { by lemma 34 R->L }
% 65.66/8.77 multiply(inverse(greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), least_upper_bound(identity, inverse(multiply(Y, X))))), greatest_lower_bound(multiply(Y, X), identity))
% 65.66/8.77 = { by axiom 12 (glb_absorbtion) R->L }
% 65.66/8.77 multiply(inverse(greatest_lower_bound(greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), least_upper_bound(identity, inverse(multiply(Y, X)))), least_upper_bound(greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), least_upper_bound(identity, inverse(multiply(Y, X)))), identity))), greatest_lower_bound(multiply(Y, X), identity))
% 65.66/8.77 = { by axiom 13 (associativity_of_glb) R->L }
% 65.66/8.77 multiply(inverse(greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), greatest_lower_bound(least_upper_bound(identity, inverse(multiply(Y, X))), least_upper_bound(greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), least_upper_bound(identity, inverse(multiply(Y, X)))), identity)))), greatest_lower_bound(multiply(Y, X), identity))
% 65.66/8.77 = { by axiom 6 (symmetry_of_lub) }
% 65.66/8.77 multiply(inverse(greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), greatest_lower_bound(least_upper_bound(identity, inverse(multiply(Y, X))), least_upper_bound(identity, greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), least_upper_bound(identity, inverse(multiply(Y, X)))))))), greatest_lower_bound(multiply(Y, X), identity))
% 65.66/8.77 = { by axiom 6 (symmetry_of_lub) R->L }
% 65.66/8.77 multiply(inverse(greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), greatest_lower_bound(least_upper_bound(inverse(multiply(Y, X)), identity), least_upper_bound(identity, greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), least_upper_bound(identity, inverse(multiply(Y, X)))))))), greatest_lower_bound(multiply(Y, X), identity))
% 65.66/8.77 = { by axiom 6 (symmetry_of_lub) R->L }
% 65.66/8.77 multiply(inverse(greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), greatest_lower_bound(least_upper_bound(inverse(multiply(Y, X)), identity), least_upper_bound(identity, greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), least_upper_bound(inverse(multiply(Y, X)), identity)))))), greatest_lower_bound(multiply(Y, X), identity))
% 65.66/8.77 = { by axiom 4 (symmetry_of_glb) R->L }
% 65.66/8.77 multiply(inverse(greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), greatest_lower_bound(least_upper_bound(identity, greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), least_upper_bound(inverse(multiply(Y, X)), identity))), least_upper_bound(inverse(multiply(Y, X)), identity)))), greatest_lower_bound(multiply(Y, X), identity))
% 65.66/8.77 = { by axiom 14 (lub_absorbtion) R->L }
% 65.66/8.77 multiply(inverse(greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), greatest_lower_bound(least_upper_bound(identity, greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), least_upper_bound(inverse(multiply(Y, X)), identity))), least_upper_bound(least_upper_bound(inverse(multiply(Y, X)), identity), greatest_lower_bound(least_upper_bound(inverse(multiply(Y, X)), identity), least_upper_bound(multiply(Y, X), identity)))))), greatest_lower_bound(multiply(Y, X), identity))
% 65.66/8.77 = { by axiom 15 (associativity_of_lub) R->L }
% 65.66/8.77 multiply(inverse(greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), greatest_lower_bound(least_upper_bound(identity, greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), least_upper_bound(inverse(multiply(Y, X)), identity))), least_upper_bound(inverse(multiply(Y, X)), least_upper_bound(identity, greatest_lower_bound(least_upper_bound(inverse(multiply(Y, X)), identity), least_upper_bound(multiply(Y, X), identity))))))), greatest_lower_bound(multiply(Y, X), identity))
% 65.66/8.77 = { by axiom 4 (symmetry_of_glb) }
% 65.66/8.77 multiply(inverse(greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), greatest_lower_bound(least_upper_bound(identity, greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), least_upper_bound(inverse(multiply(Y, X)), identity))), least_upper_bound(inverse(multiply(Y, X)), least_upper_bound(identity, greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), least_upper_bound(inverse(multiply(Y, X)), identity))))))), greatest_lower_bound(multiply(Y, X), identity))
% 65.66/8.77 = { by lemma 25 }
% 65.66/8.77 multiply(inverse(greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), least_upper_bound(identity, greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), least_upper_bound(inverse(multiply(Y, X)), identity))))), greatest_lower_bound(multiply(Y, X), identity))
% 65.66/8.77 = { by axiom 6 (symmetry_of_lub) }
% 65.66/8.78 multiply(inverse(greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), least_upper_bound(identity, greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), least_upper_bound(identity, inverse(multiply(Y, X))))))), greatest_lower_bound(multiply(Y, X), identity))
% 65.66/8.78 = { by lemma 34 }
% 65.66/8.78 multiply(inverse(greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), least_upper_bound(identity, multiply(greatest_lower_bound(multiply(Y, X), identity), least_upper_bound(identity, inverse(multiply(Y, X))))))), greatest_lower_bound(multiply(Y, X), identity))
% 65.66/8.78 = { by axiom 6 (symmetry_of_lub) R->L }
% 65.66/8.78 multiply(inverse(greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), least_upper_bound(identity, multiply(greatest_lower_bound(multiply(Y, X), identity), least_upper_bound(inverse(multiply(Y, X)), identity))))), greatest_lower_bound(multiply(Y, X), identity))
% 65.66/8.78 = { by axiom 18 (monotony_lub1) }
% 65.66/8.78 multiply(inverse(greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), least_upper_bound(identity, least_upper_bound(multiply(greatest_lower_bound(multiply(Y, X), identity), inverse(multiply(Y, X))), multiply(greatest_lower_bound(multiply(Y, X), identity), identity))))), greatest_lower_bound(multiply(Y, X), identity))
% 65.66/8.78 = { by lemma 24 R->L }
% 65.66/8.78 multiply(inverse(greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), least_upper_bound(identity, least_upper_bound(multiply(greatest_lower_bound(multiply(Y, X), identity), inverse(multiply(Y, X))), greatest_lower_bound(identity, multiply(multiply(Y, X), identity)))))), greatest_lower_bound(multiply(Y, X), identity))
% 65.66/8.78 = { by axiom 6 (symmetry_of_lub) R->L }
% 65.66/8.78 multiply(inverse(greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), least_upper_bound(identity, least_upper_bound(greatest_lower_bound(identity, multiply(multiply(Y, X), identity)), multiply(greatest_lower_bound(multiply(Y, X), identity), inverse(multiply(Y, X))))))), greatest_lower_bound(multiply(Y, X), identity))
% 65.66/8.78 = { by axiom 15 (associativity_of_lub) }
% 65.66/8.78 multiply(inverse(greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), least_upper_bound(least_upper_bound(identity, greatest_lower_bound(identity, multiply(multiply(Y, X), identity))), multiply(greatest_lower_bound(multiply(Y, X), identity), inverse(multiply(Y, X)))))), greatest_lower_bound(multiply(Y, X), identity))
% 65.66/8.78 = { by axiom 14 (lub_absorbtion) }
% 65.66/8.78 multiply(inverse(greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), least_upper_bound(identity, multiply(greatest_lower_bound(multiply(Y, X), identity), inverse(multiply(Y, X)))))), greatest_lower_bound(multiply(Y, X), identity))
% 65.66/8.78 = { by lemma 31 }
% 65.66/8.78 multiply(inverse(greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), least_upper_bound(identity, greatest_lower_bound(identity, inverse(multiply(Y, X)))))), greatest_lower_bound(multiply(Y, X), identity))
% 65.66/8.78 = { by axiom 14 (lub_absorbtion) }
% 65.66/8.78 multiply(inverse(greatest_lower_bound(least_upper_bound(multiply(Y, X), identity), identity)), greatest_lower_bound(multiply(Y, X), identity))
% 65.66/8.78 = { by axiom 4 (symmetry_of_glb) }
% 65.66/8.78 multiply(inverse(greatest_lower_bound(identity, least_upper_bound(multiply(Y, X), identity))), greatest_lower_bound(multiply(Y, X), identity))
% 65.66/8.78 = { by lemma 25 }
% 65.66/8.78 multiply(inverse(identity), greatest_lower_bound(multiply(Y, X), identity))
% 65.66/8.78 = { by axiom 1 (p12_1) }
% 65.66/8.78 multiply(identity, greatest_lower_bound(multiply(Y, X), identity))
% 65.66/8.78 = { by axiom 2 (left_identity) }
% 65.66/8.78 greatest_lower_bound(multiply(Y, X), identity)
% 65.66/8.78 = { by axiom 4 (symmetry_of_glb) }
% 65.66/8.78 greatest_lower_bound(identity, multiply(Y, X))
% 65.66/8.78
% 65.66/8.78 Lemma 36: inverse(greatest_lower_bound(identity, multiply(X, Y))) = least_upper_bound(identity, inverse(multiply(X, Y))).
% 65.66/8.78 Proof:
% 65.66/8.78 inverse(greatest_lower_bound(identity, multiply(X, Y)))
% 65.66/8.78 = { by lemma 35 R->L }
% 65.66/8.78 inverse(multiply(inverse(least_upper_bound(Y, inverse(X))), Y))
% 65.66/8.78 = { by lemma 23 }
% 65.66/8.78 multiply(inverse(Y), least_upper_bound(Y, inverse(X)))
% 65.66/8.78 = { by lemma 33 }
% 65.66/8.78 least_upper_bound(identity, inverse(multiply(X, Y)))
% 65.66/8.78
% 65.66/8.78 Lemma 37: greatest_lower_bound(identity, multiply(inverse(X), Y)) = multiply(inverse(X), greatest_lower_bound(Y, X)).
% 65.66/8.78 Proof:
% 65.66/8.78 greatest_lower_bound(identity, multiply(inverse(X), Y))
% 65.66/8.78 = { by lemma 29 }
% 65.66/8.78 multiply(inverse(X), greatest_lower_bound(X, Y))
% 65.66/8.78 = { by axiom 4 (symmetry_of_glb) }
% 65.66/8.78 multiply(inverse(X), greatest_lower_bound(Y, X))
% 65.66/8.78
% 65.66/8.78 Lemma 38: least_upper_bound(identity, multiply(inverse(X), Y)) = multiply(inverse(X), least_upper_bound(Y, X)).
% 65.66/8.78 Proof:
% 65.66/8.78 least_upper_bound(identity, multiply(inverse(X), Y))
% 65.66/8.78 = { by lemma 32 }
% 65.66/8.78 multiply(inverse(X), least_upper_bound(X, Y))
% 65.66/8.78 = { by axiom 6 (symmetry_of_lub) }
% 65.66/8.78 multiply(inverse(X), least_upper_bound(Y, X))
% 65.66/8.78
% 65.66/8.78 Goal 1 (prove_p12): a = b.
% 65.66/8.78 Proof:
% 65.66/8.78 a
% 65.66/8.78 = { by axiom 8 (p12_2) R->L }
% 65.66/8.78 inverse(inverse(a))
% 65.66/8.78 = { by lemma 28 R->L }
% 65.66/8.78 multiply(greatest_lower_bound(c, a), inverse(multiply(inverse(a), greatest_lower_bound(c, a))))
% 65.66/8.78 = { by lemma 37 R->L }
% 65.66/8.78 multiply(greatest_lower_bound(c, a), inverse(greatest_lower_bound(identity, multiply(inverse(a), c))))
% 65.66/8.78 = { by lemma 35 R->L }
% 65.66/8.78 multiply(greatest_lower_bound(c, a), inverse(multiply(inverse(least_upper_bound(c, inverse(inverse(a)))), c)))
% 65.66/8.78 = { by axiom 8 (p12_2) }
% 65.66/8.78 multiply(greatest_lower_bound(c, a), inverse(multiply(inverse(least_upper_bound(c, a)), c)))
% 65.66/8.78 = { by lemma 23 R->L }
% 65.66/8.78 multiply(greatest_lower_bound(c, a), inverse(inverse(multiply(inverse(c), least_upper_bound(c, a)))))
% 65.66/8.78 = { by axiom 6 (symmetry_of_lub) R->L }
% 65.66/8.78 multiply(greatest_lower_bound(c, a), inverse(inverse(multiply(inverse(c), least_upper_bound(a, c)))))
% 65.66/8.78 = { by axiom 7 (p12_5) }
% 65.66/8.78 multiply(greatest_lower_bound(c, a), inverse(inverse(multiply(inverse(c), least_upper_bound(b, c)))))
% 65.66/8.78 = { by lemma 38 R->L }
% 65.66/8.78 multiply(greatest_lower_bound(c, a), inverse(inverse(least_upper_bound(identity, multiply(inverse(c), b)))))
% 65.66/8.78 = { by lemma 23 R->L }
% 65.66/8.78 multiply(greatest_lower_bound(c, a), inverse(inverse(least_upper_bound(identity, inverse(multiply(inverse(b), c))))))
% 65.66/8.78 = { by lemma 36 R->L }
% 65.66/8.78 multiply(greatest_lower_bound(c, a), inverse(inverse(inverse(greatest_lower_bound(identity, multiply(inverse(b), c))))))
% 65.66/8.78 = { by axiom 3 (idempotence_of_gld) R->L }
% 65.66/8.78 multiply(greatest_lower_bound(c, a), inverse(inverse(inverse(greatest_lower_bound(greatest_lower_bound(identity, identity), multiply(inverse(b), c))))))
% 65.66/8.78 = { by axiom 13 (associativity_of_glb) R->L }
% 65.66/8.78 multiply(greatest_lower_bound(c, a), inverse(inverse(inverse(greatest_lower_bound(identity, greatest_lower_bound(identity, multiply(inverse(b), c)))))))
% 65.66/8.78 = { by lemma 37 }
% 65.66/8.78 multiply(greatest_lower_bound(c, a), inverse(inverse(inverse(greatest_lower_bound(identity, multiply(inverse(b), greatest_lower_bound(c, b)))))))
% 65.66/8.78 = { by lemma 21 }
% 65.66/8.78 multiply(greatest_lower_bound(c, a), inverse(inverse(inverse(greatest_lower_bound(identity, multiply(inverse(b), greatest_lower_bound(c, a)))))))
% 65.66/8.78 = { by lemma 36 }
% 65.66/8.78 multiply(greatest_lower_bound(c, a), inverse(inverse(least_upper_bound(identity, inverse(multiply(inverse(b), greatest_lower_bound(c, a)))))))
% 65.66/8.78 = { by lemma 23 }
% 65.66/8.78 multiply(greatest_lower_bound(c, a), inverse(inverse(least_upper_bound(identity, multiply(inverse(greatest_lower_bound(c, a)), b)))))
% 65.66/8.78 = { by lemma 38 }
% 65.66/8.78 multiply(greatest_lower_bound(c, a), inverse(inverse(multiply(inverse(greatest_lower_bound(c, a)), least_upper_bound(b, greatest_lower_bound(c, a))))))
% 65.66/8.78 = { by lemma 21 R->L }
% 65.66/8.78 multiply(greatest_lower_bound(c, a), inverse(inverse(multiply(inverse(greatest_lower_bound(c, a)), least_upper_bound(b, greatest_lower_bound(c, b))))))
% 65.66/8.78 = { by axiom 4 (symmetry_of_glb) R->L }
% 65.66/8.78 multiply(greatest_lower_bound(c, a), inverse(inverse(multiply(inverse(greatest_lower_bound(c, a)), least_upper_bound(b, greatest_lower_bound(b, c))))))
% 65.66/8.78 = { by axiom 14 (lub_absorbtion) }
% 65.66/8.78 multiply(greatest_lower_bound(c, a), inverse(inverse(multiply(inverse(greatest_lower_bound(c, a)), b))))
% 65.66/8.78 = { by lemma 23 }
% 65.66/8.78 multiply(greatest_lower_bound(c, a), inverse(multiply(inverse(b), greatest_lower_bound(c, a))))
% 65.66/8.78 = { by lemma 28 }
% 65.66/8.78 inverse(inverse(b))
% 65.66/8.78 = { by axiom 8 (p12_2) }
% 65.66/8.78 b
% 65.66/8.78 % SZS output end Proof
% 65.66/8.78
% 65.66/8.78 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------