TSTP Solution File: GRP167-1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP167-1 : 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 : n029.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 01:17:27 EDT 2023
% Result : Unsatisfiable 0.18s 0.53s
% Output : Proof 1.51s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : GRP167-1 : TPTP v8.1.2. Bugfixed v1.2.1.
% 0.03/0.13 % 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 : n029.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 : Mon Aug 28 21:05:41 EDT 2023
% 0.12/0.33 % CPUTime :
% 0.18/0.53 Command-line arguments: --no-flatten-goal
% 0.18/0.53
% 0.18/0.53 % SZS status Unsatisfiable
% 0.18/0.53
% 0.18/0.54 % SZS output start Proof
% 0.18/0.54 Axiom 1 (left_identity): multiply(identity, X) = X.
% 0.18/0.54 Axiom 2 (symmetry_of_glb): greatest_lower_bound(X, Y) = greatest_lower_bound(Y, X).
% 0.18/0.54 Axiom 3 (lat4_2): negative_part(X) = greatest_lower_bound(X, identity).
% 0.18/0.54 Axiom 4 (symmetry_of_lub): least_upper_bound(X, Y) = least_upper_bound(Y, X).
% 0.18/0.54 Axiom 5 (lat4_1): positive_part(X) = least_upper_bound(X, identity).
% 0.18/0.54 Axiom 6 (left_inverse): multiply(inverse(X), X) = identity.
% 0.18/0.54 Axiom 7 (associativity): multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)).
% 0.18/0.54 Axiom 8 (glb_absorbtion): greatest_lower_bound(X, least_upper_bound(X, Y)) = X.
% 0.18/0.54 Axiom 9 (lub_absorbtion): least_upper_bound(X, greatest_lower_bound(X, Y)) = X.
% 0.18/0.54 Axiom 10 (monotony_glb1): multiply(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(multiply(X, Y), multiply(X, Z)).
% 0.18/0.54 Axiom 11 (monotony_glb2): multiply(greatest_lower_bound(X, Y), Z) = greatest_lower_bound(multiply(X, Z), multiply(Y, Z)).
% 0.18/0.54 Axiom 12 (lat4_3): least_upper_bound(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(least_upper_bound(X, Y), least_upper_bound(X, Z)).
% 0.18/0.54 Axiom 13 (monotony_lub1): multiply(X, least_upper_bound(Y, Z)) = least_upper_bound(multiply(X, Y), multiply(X, Z)).
% 0.18/0.54 Axiom 14 (monotony_lub2): multiply(least_upper_bound(X, Y), Z) = least_upper_bound(multiply(X, Z), multiply(Y, Z)).
% 0.18/0.54 Axiom 15 (lat4_4): greatest_lower_bound(X, least_upper_bound(Y, Z)) = least_upper_bound(greatest_lower_bound(X, Y), greatest_lower_bound(X, Z)).
% 0.18/0.54
% 0.18/0.54 Lemma 16: multiply(inverse(X), multiply(X, Y)) = Y.
% 0.18/0.54 Proof:
% 1.51/0.54 multiply(inverse(X), multiply(X, Y))
% 1.51/0.54 = { by axiom 7 (associativity) R->L }
% 1.51/0.54 multiply(multiply(inverse(X), X), Y)
% 1.51/0.54 = { by axiom 6 (left_inverse) }
% 1.51/0.54 multiply(identity, Y)
% 1.51/0.54 = { by axiom 1 (left_identity) }
% 1.51/0.54 Y
% 1.51/0.54
% 1.51/0.54 Lemma 17: multiply(inverse(inverse(X)), Y) = multiply(X, Y).
% 1.51/0.54 Proof:
% 1.51/0.54 multiply(inverse(inverse(X)), Y)
% 1.51/0.54 = { by lemma 16 R->L }
% 1.51/0.54 multiply(inverse(inverse(X)), multiply(inverse(X), multiply(X, Y)))
% 1.51/0.54 = { by lemma 16 }
% 1.51/0.54 multiply(X, Y)
% 1.51/0.54
% 1.51/0.54 Lemma 18: multiply(inverse(inverse(X)), identity) = X.
% 1.51/0.54 Proof:
% 1.51/0.54 multiply(inverse(inverse(X)), identity)
% 1.51/0.54 = { by axiom 6 (left_inverse) R->L }
% 1.51/0.54 multiply(inverse(inverse(X)), multiply(inverse(X), X))
% 1.51/0.54 = { by lemma 16 }
% 1.51/0.54 X
% 1.51/0.54
% 1.51/0.54 Lemma 19: multiply(X, identity) = X.
% 1.51/0.54 Proof:
% 1.51/0.54 multiply(X, identity)
% 1.51/0.54 = { by lemma 17 R->L }
% 1.51/0.54 multiply(inverse(inverse(X)), identity)
% 1.51/0.54 = { by lemma 18 }
% 1.51/0.54 X
% 1.51/0.54
% 1.51/0.54 Lemma 20: greatest_lower_bound(identity, X) = negative_part(X).
% 1.51/0.54 Proof:
% 1.51/0.54 greatest_lower_bound(identity, X)
% 1.51/0.54 = { by axiom 2 (symmetry_of_glb) R->L }
% 1.51/0.54 greatest_lower_bound(X, identity)
% 1.51/0.54 = { by axiom 3 (lat4_2) R->L }
% 1.51/0.54 negative_part(X)
% 1.51/0.54
% 1.51/0.54 Lemma 21: least_upper_bound(identity, X) = positive_part(X).
% 1.51/0.54 Proof:
% 1.51/0.54 least_upper_bound(identity, X)
% 1.51/0.54 = { by axiom 4 (symmetry_of_lub) R->L }
% 1.51/0.54 least_upper_bound(X, identity)
% 1.51/0.54 = { by axiom 5 (lat4_1) R->L }
% 1.51/0.54 positive_part(X)
% 1.51/0.54
% 1.51/0.54 Lemma 22: negative_part(positive_part(X)) = identity.
% 1.51/0.54 Proof:
% 1.51/0.54 negative_part(positive_part(X))
% 1.51/0.54 = { by lemma 20 R->L }
% 1.51/0.54 greatest_lower_bound(identity, positive_part(X))
% 1.51/0.54 = { by lemma 21 R->L }
% 1.51/0.54 greatest_lower_bound(identity, least_upper_bound(identity, X))
% 1.51/0.54 = { by axiom 8 (glb_absorbtion) }
% 1.51/0.54 identity
% 1.51/0.54
% 1.51/0.54 Lemma 23: multiply(inverse(X), negative_part(X)) = negative_part(inverse(X)).
% 1.51/0.54 Proof:
% 1.51/0.54 multiply(inverse(X), negative_part(X))
% 1.51/0.54 = { by axiom 3 (lat4_2) }
% 1.51/0.54 multiply(inverse(X), greatest_lower_bound(X, identity))
% 1.51/0.54 = { by axiom 2 (symmetry_of_glb) R->L }
% 1.51/0.54 multiply(inverse(X), greatest_lower_bound(identity, X))
% 1.51/0.54 = { by axiom 10 (monotony_glb1) }
% 1.51/0.54 greatest_lower_bound(multiply(inverse(X), identity), multiply(inverse(X), X))
% 1.51/0.54 = { by axiom 6 (left_inverse) }
% 1.51/0.54 greatest_lower_bound(multiply(inverse(X), identity), identity)
% 1.51/0.54 = { by axiom 3 (lat4_2) R->L }
% 1.51/0.54 negative_part(multiply(inverse(X), identity))
% 1.51/0.54 = { by lemma 19 }
% 1.51/0.54 negative_part(inverse(X))
% 1.51/0.54
% 1.51/0.54 Lemma 24: greatest_lower_bound(X, multiply(Y, X)) = multiply(negative_part(Y), X).
% 1.51/0.54 Proof:
% 1.51/0.54 greatest_lower_bound(X, multiply(Y, X))
% 1.51/0.54 = { by axiom 2 (symmetry_of_glb) R->L }
% 1.51/0.54 greatest_lower_bound(multiply(Y, X), X)
% 1.51/0.54 = { by axiom 1 (left_identity) R->L }
% 1.51/0.54 greatest_lower_bound(multiply(Y, X), multiply(identity, X))
% 1.51/0.54 = { by axiom 11 (monotony_glb2) R->L }
% 1.51/0.54 multiply(greatest_lower_bound(Y, identity), X)
% 1.51/0.54 = { by axiom 3 (lat4_2) R->L }
% 1.51/0.54 multiply(negative_part(Y), X)
% 1.51/0.54
% 1.51/0.54 Lemma 25: least_upper_bound(negative_part(X), negative_part(Y)) = negative_part(least_upper_bound(X, Y)).
% 1.51/0.54 Proof:
% 1.51/0.54 least_upper_bound(negative_part(X), negative_part(Y))
% 1.51/0.54 = { by axiom 4 (symmetry_of_lub) R->L }
% 1.51/0.54 least_upper_bound(negative_part(Y), negative_part(X))
% 1.51/0.54 = { by lemma 20 R->L }
% 1.51/0.54 least_upper_bound(negative_part(Y), greatest_lower_bound(identity, X))
% 1.51/0.54 = { by lemma 20 R->L }
% 1.51/0.54 least_upper_bound(greatest_lower_bound(identity, Y), greatest_lower_bound(identity, X))
% 1.51/0.54 = { by axiom 15 (lat4_4) R->L }
% 1.51/0.54 greatest_lower_bound(identity, least_upper_bound(Y, X))
% 1.51/0.54 = { by lemma 20 }
% 1.51/0.54 negative_part(least_upper_bound(Y, X))
% 1.51/0.54 = { by axiom 4 (symmetry_of_lub) }
% 1.51/0.54 negative_part(least_upper_bound(X, Y))
% 1.51/0.54
% 1.51/0.54 Lemma 26: multiply(inverse(X), least_upper_bound(X, Y)) = positive_part(multiply(inverse(X), Y)).
% 1.51/0.54 Proof:
% 1.51/0.54 multiply(inverse(X), least_upper_bound(X, Y))
% 1.51/0.54 = { by axiom 4 (symmetry_of_lub) R->L }
% 1.51/0.54 multiply(inverse(X), least_upper_bound(Y, X))
% 1.51/0.54 = { by axiom 13 (monotony_lub1) }
% 1.51/0.54 least_upper_bound(multiply(inverse(X), Y), multiply(inverse(X), X))
% 1.51/0.54 = { by axiom 6 (left_inverse) }
% 1.51/0.54 least_upper_bound(multiply(inverse(X), Y), identity)
% 1.51/0.54 = { by axiom 5 (lat4_1) R->L }
% 1.51/0.54 positive_part(multiply(inverse(X), Y))
% 1.51/0.54
% 1.51/0.54 Lemma 27: multiply(inverse(negative_part(inverse(X))), negative_part(X)) = X.
% 1.51/0.54 Proof:
% 1.51/0.54 multiply(inverse(negative_part(inverse(X))), negative_part(X))
% 1.51/0.54 = { by axiom 3 (lat4_2) }
% 1.51/0.54 multiply(inverse(negative_part(inverse(X))), greatest_lower_bound(X, identity))
% 1.51/0.54 = { by axiom 6 (left_inverse) R->L }
% 1.51/0.54 multiply(inverse(negative_part(inverse(X))), greatest_lower_bound(X, multiply(inverse(X), X)))
% 1.51/0.54 = { by lemma 24 }
% 1.51/0.54 multiply(inverse(negative_part(inverse(X))), multiply(negative_part(inverse(X)), X))
% 1.51/0.54 = { by lemma 16 }
% 1.51/0.54 X
% 1.51/0.54
% 1.51/0.54 Lemma 28: negative_part(least_upper_bound(X, inverse(X))) = positive_part(greatest_lower_bound(X, inverse(X))).
% 1.51/0.54 Proof:
% 1.51/0.54 negative_part(least_upper_bound(X, inverse(X)))
% 1.51/0.54 = { by axiom 4 (symmetry_of_lub) R->L }
% 1.51/0.54 negative_part(least_upper_bound(inverse(X), X))
% 1.51/0.54 = { by lemma 25 R->L }
% 1.51/0.54 least_upper_bound(negative_part(inverse(X)), negative_part(X))
% 1.51/0.54 = { by lemma 16 R->L }
% 1.51/0.54 multiply(inverse(inverse(negative_part(inverse(X)))), multiply(inverse(negative_part(inverse(X))), least_upper_bound(negative_part(inverse(X)), negative_part(X))))
% 1.51/0.54 = { by lemma 26 }
% 1.51/0.54 multiply(inverse(inverse(negative_part(inverse(X)))), positive_part(multiply(inverse(negative_part(inverse(X))), negative_part(X))))
% 1.51/0.54 = { by lemma 17 }
% 1.51/0.54 multiply(negative_part(inverse(X)), positive_part(multiply(inverse(negative_part(inverse(X))), negative_part(X))))
% 1.51/0.54 = { by lemma 27 }
% 1.51/0.54 multiply(negative_part(inverse(X)), positive_part(X))
% 1.51/0.54 = { by lemma 24 R->L }
% 1.51/0.54 greatest_lower_bound(positive_part(X), multiply(inverse(X), positive_part(X)))
% 1.51/0.54 = { by axiom 5 (lat4_1) }
% 1.51/0.55 greatest_lower_bound(positive_part(X), multiply(inverse(X), least_upper_bound(X, identity)))
% 1.51/0.55 = { by lemma 26 }
% 1.51/0.55 greatest_lower_bound(positive_part(X), positive_part(multiply(inverse(X), identity)))
% 1.51/0.55 = { by lemma 19 }
% 1.51/0.55 greatest_lower_bound(positive_part(X), positive_part(inverse(X)))
% 1.51/0.55 = { by axiom 2 (symmetry_of_glb) R->L }
% 1.51/0.55 greatest_lower_bound(positive_part(inverse(X)), positive_part(X))
% 1.51/0.55 = { by lemma 21 R->L }
% 1.51/0.55 greatest_lower_bound(positive_part(inverse(X)), least_upper_bound(identity, X))
% 1.51/0.55 = { by lemma 21 R->L }
% 1.51/0.55 greatest_lower_bound(least_upper_bound(identity, inverse(X)), least_upper_bound(identity, X))
% 1.51/0.55 = { by axiom 12 (lat4_3) R->L }
% 1.51/0.55 least_upper_bound(identity, greatest_lower_bound(inverse(X), X))
% 1.51/0.55 = { by lemma 21 }
% 1.51/0.55 positive_part(greatest_lower_bound(inverse(X), X))
% 1.51/0.55 = { by axiom 2 (symmetry_of_glb) }
% 1.51/0.55 positive_part(greatest_lower_bound(X, inverse(X)))
% 1.51/0.55
% 1.51/0.55 Goal 1 (prove_lat4): a = multiply(positive_part(a), negative_part(a)).
% 1.51/0.55 Proof:
% 1.51/0.55 a
% 1.51/0.55 = { by lemma 27 R->L }
% 1.51/0.55 multiply(inverse(negative_part(inverse(a))), negative_part(a))
% 1.51/0.55 = { by axiom 1 (left_identity) R->L }
% 1.51/0.55 multiply(inverse(multiply(identity, negative_part(inverse(a)))), negative_part(a))
% 1.51/0.55 = { by lemma 22 R->L }
% 1.51/0.55 multiply(inverse(multiply(negative_part(positive_part(inverse(inverse(a)))), negative_part(inverse(a)))), negative_part(a))
% 1.51/0.55 = { by lemma 24 R->L }
% 1.51/0.55 multiply(inverse(greatest_lower_bound(negative_part(inverse(a)), multiply(positive_part(inverse(inverse(a))), negative_part(inverse(a))))), negative_part(a))
% 1.51/0.55 = { by axiom 2 (symmetry_of_glb) R->L }
% 1.51/0.55 multiply(inverse(greatest_lower_bound(multiply(positive_part(inverse(inverse(a))), negative_part(inverse(a))), negative_part(inverse(a)))), negative_part(a))
% 1.51/0.55 = { by lemma 16 R->L }
% 1.51/0.55 multiply(inverse(greatest_lower_bound(multiply(positive_part(inverse(inverse(a))), negative_part(inverse(a))), multiply(inverse(positive_part(inverse(inverse(a)))), multiply(positive_part(inverse(inverse(a))), negative_part(inverse(a)))))), negative_part(a))
% 1.51/0.55 = { by lemma 24 }
% 1.51/0.55 multiply(inverse(multiply(negative_part(inverse(positive_part(inverse(inverse(a))))), multiply(positive_part(inverse(inverse(a))), negative_part(inverse(a))))), negative_part(a))
% 1.51/0.55 = { by axiom 5 (lat4_1) }
% 1.51/0.55 multiply(inverse(multiply(negative_part(inverse(positive_part(inverse(inverse(a))))), multiply(least_upper_bound(inverse(inverse(a)), identity), negative_part(inverse(a))))), negative_part(a))
% 1.51/0.55 = { by axiom 14 (monotony_lub2) }
% 1.51/0.55 multiply(inverse(multiply(negative_part(inverse(positive_part(inverse(inverse(a))))), least_upper_bound(multiply(inverse(inverse(a)), negative_part(inverse(a))), multiply(identity, negative_part(inverse(a)))))), negative_part(a))
% 1.51/0.55 = { by axiom 1 (left_identity) }
% 1.51/0.55 multiply(inverse(multiply(negative_part(inverse(positive_part(inverse(inverse(a))))), least_upper_bound(multiply(inverse(inverse(a)), negative_part(inverse(a))), negative_part(inverse(a))))), negative_part(a))
% 1.51/0.55 = { by axiom 4 (symmetry_of_lub) }
% 1.51/0.55 multiply(inverse(multiply(negative_part(inverse(positive_part(inverse(inverse(a))))), least_upper_bound(negative_part(inverse(a)), multiply(inverse(inverse(a)), negative_part(inverse(a)))))), negative_part(a))
% 1.51/0.55 = { by lemma 23 }
% 1.51/0.55 multiply(inverse(multiply(negative_part(inverse(positive_part(inverse(inverse(a))))), least_upper_bound(negative_part(inverse(a)), negative_part(inverse(inverse(a)))))), negative_part(a))
% 1.51/0.55 = { by lemma 25 }
% 1.51/0.55 multiply(inverse(multiply(negative_part(inverse(positive_part(inverse(inverse(a))))), negative_part(least_upper_bound(inverse(a), inverse(inverse(a)))))), negative_part(a))
% 1.51/0.55 = { by lemma 23 R->L }
% 1.51/0.55 multiply(inverse(multiply(multiply(inverse(positive_part(inverse(inverse(a)))), negative_part(positive_part(inverse(inverse(a))))), negative_part(least_upper_bound(inverse(a), inverse(inverse(a)))))), negative_part(a))
% 1.51/0.55 = { by lemma 22 }
% 1.51/0.55 multiply(inverse(multiply(multiply(inverse(positive_part(inverse(inverse(a)))), identity), negative_part(least_upper_bound(inverse(a), inverse(inverse(a)))))), negative_part(a))
% 1.51/0.55 = { by lemma 19 }
% 1.51/0.55 multiply(inverse(multiply(inverse(positive_part(inverse(inverse(a)))), negative_part(least_upper_bound(inverse(a), inverse(inverse(a)))))), negative_part(a))
% 1.51/0.55 = { by lemma 28 }
% 1.51/0.55 multiply(inverse(multiply(inverse(positive_part(inverse(inverse(a)))), positive_part(greatest_lower_bound(inverse(a), inverse(inverse(a)))))), negative_part(a))
% 1.51/0.55 = { by axiom 9 (lub_absorbtion) R->L }
% 1.51/0.55 multiply(inverse(multiply(inverse(positive_part(inverse(inverse(a)))), least_upper_bound(positive_part(greatest_lower_bound(inverse(a), inverse(inverse(a)))), greatest_lower_bound(positive_part(greatest_lower_bound(inverse(a), inverse(inverse(a)))), identity)))), negative_part(a))
% 1.51/0.55 = { by axiom 3 (lat4_2) R->L }
% 1.51/0.55 multiply(inverse(multiply(inverse(positive_part(inverse(inverse(a)))), least_upper_bound(positive_part(greatest_lower_bound(inverse(a), inverse(inverse(a)))), negative_part(positive_part(greatest_lower_bound(inverse(a), inverse(inverse(a)))))))), negative_part(a))
% 1.51/0.55 = { by lemma 22 }
% 1.51/0.55 multiply(inverse(multiply(inverse(positive_part(inverse(inverse(a)))), least_upper_bound(positive_part(greatest_lower_bound(inverse(a), inverse(inverse(a)))), identity))), negative_part(a))
% 1.51/0.55 = { by axiom 5 (lat4_1) R->L }
% 1.51/0.55 multiply(inverse(multiply(inverse(positive_part(inverse(inverse(a)))), positive_part(positive_part(greatest_lower_bound(inverse(a), inverse(inverse(a))))))), negative_part(a))
% 1.51/0.55 = { by lemma 28 R->L }
% 1.51/0.55 multiply(inverse(multiply(inverse(positive_part(inverse(inverse(a)))), positive_part(negative_part(least_upper_bound(inverse(a), inverse(inverse(a))))))), negative_part(a))
% 1.51/0.55 = { by lemma 20 R->L }
% 1.51/0.55 multiply(inverse(multiply(inverse(positive_part(inverse(inverse(a)))), positive_part(greatest_lower_bound(identity, least_upper_bound(inverse(a), inverse(inverse(a))))))), negative_part(a))
% 1.51/0.55 = { by lemma 21 R->L }
% 1.51/0.55 multiply(inverse(multiply(inverse(positive_part(inverse(inverse(a)))), least_upper_bound(identity, greatest_lower_bound(identity, least_upper_bound(inverse(a), inverse(inverse(a))))))), negative_part(a))
% 1.51/0.55 = { by axiom 9 (lub_absorbtion) }
% 1.51/0.55 multiply(inverse(multiply(inverse(positive_part(inverse(inverse(a)))), identity)), negative_part(a))
% 1.51/0.55 = { by lemma 19 }
% 1.51/0.55 multiply(inverse(inverse(positive_part(inverse(inverse(a))))), negative_part(a))
% 1.51/0.55 = { by lemma 19 R->L }
% 1.51/0.55 multiply(inverse(inverse(positive_part(multiply(inverse(inverse(a)), identity)))), negative_part(a))
% 1.51/0.55 = { by lemma 18 }
% 1.51/0.55 multiply(inverse(inverse(positive_part(a))), negative_part(a))
% 1.51/0.55 = { by lemma 17 }
% 1.51/0.55 multiply(positive_part(a), negative_part(a))
% 1.51/0.55 % SZS output end Proof
% 1.51/0.55
% 1.51/0.55 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------