TSTP Solution File: GRP167-2 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP167-2 : 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 : 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 01:17:27 EDT 2023
% Result : Unsatisfiable 0.18s 0.59s
% Output : Proof 0.18s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : GRP167-2 : 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
% 0.12/0.34 % Computer : n001.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Mon Aug 28 20:11:06 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.18/0.59 Command-line arguments: --no-flatten-goal
% 0.18/0.59
% 0.18/0.59 % SZS status Unsatisfiable
% 0.18/0.59
% 0.18/0.61 % SZS output start Proof
% 0.18/0.61 Axiom 1 (lat4_1): inverse(identity) = identity.
% 0.18/0.61 Axiom 2 (lat4_2): inverse(inverse(X)) = X.
% 0.18/0.61 Axiom 3 (symmetry_of_glb): greatest_lower_bound(X, Y) = greatest_lower_bound(Y, X).
% 0.18/0.61 Axiom 4 (lat4_5): negative_part(X) = greatest_lower_bound(X, identity).
% 0.18/0.61 Axiom 5 (symmetry_of_lub): least_upper_bound(X, Y) = least_upper_bound(Y, X).
% 0.18/0.61 Axiom 6 (lat4_4): positive_part(X) = least_upper_bound(X, identity).
% 0.18/0.61 Axiom 7 (left_identity): multiply(identity, X) = X.
% 0.18/0.61 Axiom 8 (left_inverse): multiply(inverse(X), X) = identity.
% 0.18/0.61 Axiom 9 (glb_absorbtion): greatest_lower_bound(X, least_upper_bound(X, Y)) = X.
% 0.18/0.61 Axiom 10 (lub_absorbtion): least_upper_bound(X, greatest_lower_bound(X, Y)) = X.
% 0.18/0.61 Axiom 11 (lat4_3): inverse(multiply(X, Y)) = multiply(inverse(Y), inverse(X)).
% 0.18/0.61 Axiom 12 (associativity): multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)).
% 0.18/0.61 Axiom 13 (lat4_6): least_upper_bound(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(least_upper_bound(X, Y), least_upper_bound(X, Z)).
% 0.18/0.61 Axiom 14 (monotony_glb1): multiply(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(multiply(X, Y), multiply(X, Z)).
% 0.18/0.61 Axiom 15 (monotony_glb2): multiply(greatest_lower_bound(X, Y), Z) = greatest_lower_bound(multiply(X, Z), multiply(Y, Z)).
% 0.18/0.61 Axiom 16 (lat4_7): greatest_lower_bound(X, least_upper_bound(Y, Z)) = least_upper_bound(greatest_lower_bound(X, Y), greatest_lower_bound(X, Z)).
% 0.18/0.61 Axiom 17 (monotony_lub1): multiply(X, least_upper_bound(Y, Z)) = least_upper_bound(multiply(X, Y), multiply(X, Z)).
% 0.18/0.61 Axiom 18 (monotony_lub2): multiply(least_upper_bound(X, Y), Z) = least_upper_bound(multiply(X, Z), multiply(Y, Z)).
% 0.18/0.61
% 0.18/0.61 Lemma 19: multiply(X, identity) = X.
% 0.18/0.61 Proof:
% 0.18/0.61 multiply(X, identity)
% 0.18/0.61 = { by axiom 2 (lat4_2) R->L }
% 0.18/0.61 inverse(inverse(multiply(X, identity)))
% 0.18/0.61 = { by axiom 11 (lat4_3) }
% 0.18/0.61 inverse(multiply(inverse(identity), inverse(X)))
% 0.18/0.61 = { by axiom 1 (lat4_1) }
% 0.18/0.61 inverse(multiply(identity, inverse(X)))
% 0.18/0.61 = { by axiom 7 (left_identity) }
% 0.18/0.61 inverse(inverse(X))
% 0.18/0.61 = { by axiom 2 (lat4_2) }
% 0.18/0.61 X
% 0.18/0.61
% 0.18/0.61 Lemma 20: greatest_lower_bound(identity, X) = negative_part(X).
% 0.18/0.61 Proof:
% 0.18/0.61 greatest_lower_bound(identity, X)
% 0.18/0.61 = { by axiom 3 (symmetry_of_glb) R->L }
% 0.18/0.61 greatest_lower_bound(X, identity)
% 0.18/0.61 = { by axiom 4 (lat4_5) R->L }
% 0.18/0.61 negative_part(X)
% 0.18/0.61
% 0.18/0.61 Lemma 21: least_upper_bound(identity, X) = positive_part(X).
% 0.18/0.61 Proof:
% 0.18/0.61 least_upper_bound(identity, X)
% 0.18/0.61 = { by axiom 5 (symmetry_of_lub) R->L }
% 0.18/0.61 least_upper_bound(X, identity)
% 0.18/0.61 = { by axiom 6 (lat4_4) R->L }
% 0.18/0.61 positive_part(X)
% 0.18/0.61
% 0.18/0.61 Lemma 22: negative_part(positive_part(X)) = identity.
% 0.18/0.61 Proof:
% 0.18/0.61 negative_part(positive_part(X))
% 0.18/0.61 = { by lemma 20 R->L }
% 0.18/0.61 greatest_lower_bound(identity, positive_part(X))
% 0.18/0.61 = { by lemma 21 R->L }
% 0.18/0.61 greatest_lower_bound(identity, least_upper_bound(identity, X))
% 0.18/0.61 = { by axiom 9 (glb_absorbtion) }
% 0.18/0.61 identity
% 0.18/0.61
% 0.18/0.61 Lemma 23: multiply(inverse(X), least_upper_bound(X, Y)) = positive_part(multiply(inverse(X), Y)).
% 0.18/0.61 Proof:
% 0.18/0.61 multiply(inverse(X), least_upper_bound(X, Y))
% 0.18/0.61 = { by axiom 5 (symmetry_of_lub) R->L }
% 0.18/0.61 multiply(inverse(X), least_upper_bound(Y, X))
% 0.18/0.61 = { by axiom 17 (monotony_lub1) }
% 0.18/0.61 least_upper_bound(multiply(inverse(X), Y), multiply(inverse(X), X))
% 0.18/0.61 = { by axiom 8 (left_inverse) }
% 0.18/0.61 least_upper_bound(multiply(inverse(X), Y), identity)
% 0.18/0.61 = { by axiom 6 (lat4_4) R->L }
% 0.18/0.61 positive_part(multiply(inverse(X), Y))
% 0.18/0.61
% 0.18/0.61 Lemma 24: multiply(inverse(X), positive_part(X)) = positive_part(inverse(X)).
% 0.18/0.61 Proof:
% 0.18/0.61 multiply(inverse(X), positive_part(X))
% 0.18/0.61 = { by axiom 6 (lat4_4) }
% 0.18/0.61 multiply(inverse(X), least_upper_bound(X, identity))
% 0.18/0.61 = { by lemma 23 }
% 0.18/0.62 positive_part(multiply(inverse(X), identity))
% 0.18/0.62 = { by lemma 19 }
% 0.18/0.62 positive_part(inverse(X))
% 0.18/0.62
% 0.18/0.62 Lemma 25: greatest_lower_bound(X, multiply(Y, X)) = multiply(negative_part(Y), X).
% 0.18/0.62 Proof:
% 0.18/0.62 greatest_lower_bound(X, multiply(Y, X))
% 0.18/0.62 = { by axiom 3 (symmetry_of_glb) R->L }
% 0.18/0.62 greatest_lower_bound(multiply(Y, X), X)
% 0.18/0.62 = { by axiom 7 (left_identity) R->L }
% 0.18/0.62 greatest_lower_bound(multiply(Y, X), multiply(identity, X))
% 0.18/0.62 = { by axiom 15 (monotony_glb2) R->L }
% 0.18/0.62 multiply(greatest_lower_bound(Y, identity), X)
% 0.18/0.62 = { by axiom 4 (lat4_5) R->L }
% 0.18/0.62 multiply(negative_part(Y), X)
% 0.18/0.62
% 0.18/0.62 Lemma 26: least_upper_bound(negative_part(X), negative_part(Y)) = negative_part(least_upper_bound(X, Y)).
% 0.18/0.62 Proof:
% 0.18/0.62 least_upper_bound(negative_part(X), negative_part(Y))
% 0.18/0.62 = { by axiom 5 (symmetry_of_lub) R->L }
% 0.18/0.62 least_upper_bound(negative_part(Y), negative_part(X))
% 0.18/0.62 = { by lemma 20 R->L }
% 0.18/0.62 least_upper_bound(negative_part(Y), greatest_lower_bound(identity, X))
% 0.18/0.62 = { by lemma 20 R->L }
% 0.18/0.62 least_upper_bound(greatest_lower_bound(identity, Y), greatest_lower_bound(identity, X))
% 0.18/0.62 = { by axiom 16 (lat4_7) R->L }
% 0.18/0.62 greatest_lower_bound(identity, least_upper_bound(Y, X))
% 0.18/0.62 = { by lemma 20 }
% 0.18/0.62 negative_part(least_upper_bound(Y, X))
% 0.18/0.62 = { by axiom 5 (symmetry_of_lub) }
% 0.18/0.62 negative_part(least_upper_bound(X, Y))
% 0.18/0.62
% 0.18/0.62 Lemma 27: multiply(inverse(X), multiply(X, Y)) = Y.
% 0.18/0.62 Proof:
% 0.18/0.62 multiply(inverse(X), multiply(X, Y))
% 0.18/0.62 = { by axiom 12 (associativity) R->L }
% 0.18/0.62 multiply(multiply(inverse(X), X), Y)
% 0.18/0.62 = { by axiom 8 (left_inverse) }
% 0.18/0.62 multiply(identity, Y)
% 0.18/0.62 = { by axiom 7 (left_identity) }
% 0.18/0.62 Y
% 0.18/0.62
% 0.18/0.62 Lemma 28: negative_part(least_upper_bound(X, inverse(X))) = positive_part(greatest_lower_bound(X, inverse(X))).
% 0.18/0.62 Proof:
% 0.18/0.62 negative_part(least_upper_bound(X, inverse(X)))
% 0.18/0.62 = { by axiom 5 (symmetry_of_lub) R->L }
% 0.18/0.62 negative_part(least_upper_bound(inverse(X), X))
% 0.18/0.62 = { by lemma 26 R->L }
% 0.18/0.62 least_upper_bound(negative_part(inverse(X)), negative_part(X))
% 0.18/0.62 = { by lemma 27 R->L }
% 0.18/0.62 multiply(inverse(inverse(negative_part(inverse(X)))), multiply(inverse(negative_part(inverse(X))), least_upper_bound(negative_part(inverse(X)), negative_part(X))))
% 0.18/0.62 = { by lemma 23 }
% 0.18/0.62 multiply(inverse(inverse(negative_part(inverse(X)))), positive_part(multiply(inverse(negative_part(inverse(X))), negative_part(X))))
% 0.18/0.62 = { by axiom 2 (lat4_2) }
% 0.18/0.62 multiply(negative_part(inverse(X)), positive_part(multiply(inverse(negative_part(inverse(X))), negative_part(X))))
% 0.18/0.62 = { by axiom 4 (lat4_5) }
% 0.18/0.62 multiply(negative_part(inverse(X)), positive_part(multiply(inverse(negative_part(inverse(X))), greatest_lower_bound(X, identity))))
% 0.18/0.62 = { by axiom 8 (left_inverse) R->L }
% 0.18/0.62 multiply(negative_part(inverse(X)), positive_part(multiply(inverse(negative_part(inverse(X))), greatest_lower_bound(X, multiply(inverse(X), X)))))
% 0.18/0.62 = { by lemma 25 }
% 0.18/0.62 multiply(negative_part(inverse(X)), positive_part(multiply(inverse(negative_part(inverse(X))), multiply(negative_part(inverse(X)), X))))
% 0.18/0.62 = { by lemma 27 }
% 0.18/0.62 multiply(negative_part(inverse(X)), positive_part(X))
% 0.18/0.62 = { by lemma 25 R->L }
% 0.18/0.62 greatest_lower_bound(positive_part(X), multiply(inverse(X), positive_part(X)))
% 0.18/0.62 = { by lemma 24 }
% 0.18/0.62 greatest_lower_bound(positive_part(X), positive_part(inverse(X)))
% 0.18/0.62 = { by axiom 3 (symmetry_of_glb) R->L }
% 0.18/0.62 greatest_lower_bound(positive_part(inverse(X)), positive_part(X))
% 0.18/0.62 = { by lemma 21 R->L }
% 0.18/0.62 greatest_lower_bound(positive_part(inverse(X)), least_upper_bound(identity, X))
% 0.18/0.62 = { by lemma 21 R->L }
% 0.18/0.62 greatest_lower_bound(least_upper_bound(identity, inverse(X)), least_upper_bound(identity, X))
% 0.18/0.62 = { by axiom 13 (lat4_6) R->L }
% 0.18/0.62 least_upper_bound(identity, greatest_lower_bound(inverse(X), X))
% 0.18/0.62 = { by lemma 21 }
% 0.18/0.62 positive_part(greatest_lower_bound(inverse(X), X))
% 0.18/0.62 = { by axiom 3 (symmetry_of_glb) }
% 0.18/0.62 positive_part(greatest_lower_bound(X, inverse(X)))
% 0.18/0.62
% 0.18/0.62 Goal 1 (prove_lat4): a = multiply(positive_part(a), negative_part(a)).
% 0.18/0.62 Proof:
% 0.18/0.62 a
% 0.18/0.62 = { by axiom 2 (lat4_2) R->L }
% 0.18/0.62 inverse(inverse(a))
% 0.18/0.62 = { by lemma 27 R->L }
% 0.18/0.62 multiply(inverse(inverse(positive_part(a))), multiply(inverse(positive_part(a)), inverse(inverse(a))))
% 0.18/0.62 = { by axiom 11 (lat4_3) R->L }
% 0.18/0.62 multiply(inverse(inverse(positive_part(a))), inverse(multiply(inverse(a), positive_part(a))))
% 0.18/0.62 = { by axiom 2 (lat4_2) }
% 0.18/0.62 multiply(positive_part(a), inverse(multiply(inverse(a), positive_part(a))))
% 0.18/0.62 = { by lemma 24 }
% 0.18/0.62 multiply(positive_part(a), inverse(positive_part(inverse(a))))
% 0.18/0.62 = { by lemma 19 R->L }
% 0.18/0.62 multiply(positive_part(a), multiply(inverse(positive_part(inverse(a))), identity))
% 0.18/0.62 = { by axiom 10 (lub_absorbtion) R->L }
% 0.18/0.62 multiply(positive_part(a), multiply(inverse(positive_part(inverse(a))), least_upper_bound(identity, greatest_lower_bound(identity, least_upper_bound(a, inverse(a))))))
% 0.18/0.62 = { by lemma 21 }
% 0.18/0.62 multiply(positive_part(a), multiply(inverse(positive_part(inverse(a))), positive_part(greatest_lower_bound(identity, least_upper_bound(a, inverse(a))))))
% 0.18/0.62 = { by lemma 20 }
% 0.18/0.62 multiply(positive_part(a), multiply(inverse(positive_part(inverse(a))), positive_part(negative_part(least_upper_bound(a, inverse(a))))))
% 0.18/0.62 = { by lemma 28 }
% 0.18/0.62 multiply(positive_part(a), multiply(inverse(positive_part(inverse(a))), positive_part(positive_part(greatest_lower_bound(a, inverse(a))))))
% 0.18/0.62 = { by axiom 6 (lat4_4) }
% 0.18/0.62 multiply(positive_part(a), multiply(inverse(positive_part(inverse(a))), least_upper_bound(positive_part(greatest_lower_bound(a, inverse(a))), identity)))
% 0.18/0.62 = { by lemma 22 R->L }
% 0.18/0.62 multiply(positive_part(a), multiply(inverse(positive_part(inverse(a))), least_upper_bound(positive_part(greatest_lower_bound(a, inverse(a))), negative_part(positive_part(greatest_lower_bound(a, inverse(a)))))))
% 0.18/0.62 = { by axiom 4 (lat4_5) }
% 0.18/0.62 multiply(positive_part(a), multiply(inverse(positive_part(inverse(a))), least_upper_bound(positive_part(greatest_lower_bound(a, inverse(a))), greatest_lower_bound(positive_part(greatest_lower_bound(a, inverse(a))), identity))))
% 0.18/0.62 = { by axiom 10 (lub_absorbtion) }
% 0.18/0.62 multiply(positive_part(a), multiply(inverse(positive_part(inverse(a))), positive_part(greatest_lower_bound(a, inverse(a)))))
% 0.18/0.62 = { by lemma 28 R->L }
% 0.18/0.62 multiply(positive_part(a), multiply(inverse(positive_part(inverse(a))), negative_part(least_upper_bound(a, inverse(a)))))
% 0.18/0.62 = { by axiom 7 (left_identity) R->L }
% 0.18/0.62 multiply(positive_part(a), multiply(multiply(identity, inverse(positive_part(inverse(a)))), negative_part(least_upper_bound(a, inverse(a)))))
% 0.18/0.62 = { by lemma 22 R->L }
% 0.18/0.62 multiply(positive_part(a), multiply(multiply(negative_part(positive_part(inverse(a))), inverse(positive_part(inverse(a)))), negative_part(least_upper_bound(a, inverse(a)))))
% 0.18/0.62 = { by lemma 25 R->L }
% 0.18/0.62 multiply(positive_part(a), multiply(greatest_lower_bound(inverse(positive_part(inverse(a))), multiply(positive_part(inverse(a)), inverse(positive_part(inverse(a))))), negative_part(least_upper_bound(a, inverse(a)))))
% 0.18/0.62 = { by axiom 2 (lat4_2) R->L }
% 0.18/0.62 multiply(positive_part(a), multiply(greatest_lower_bound(inverse(positive_part(inverse(a))), multiply(inverse(inverse(positive_part(inverse(a)))), inverse(positive_part(inverse(a))))), negative_part(least_upper_bound(a, inverse(a)))))
% 0.18/0.62 = { by axiom 8 (left_inverse) }
% 0.18/0.62 multiply(positive_part(a), multiply(greatest_lower_bound(inverse(positive_part(inverse(a))), identity), negative_part(least_upper_bound(a, inverse(a)))))
% 0.18/0.62 = { by axiom 4 (lat4_5) R->L }
% 0.18/0.62 multiply(positive_part(a), multiply(negative_part(inverse(positive_part(inverse(a)))), negative_part(least_upper_bound(a, inverse(a)))))
% 0.18/0.62 = { by lemma 26 R->L }
% 0.18/0.62 multiply(positive_part(a), multiply(negative_part(inverse(positive_part(inverse(a)))), least_upper_bound(negative_part(a), negative_part(inverse(a)))))
% 0.18/0.62 = { by lemma 19 R->L }
% 0.18/0.62 multiply(positive_part(a), multiply(negative_part(inverse(positive_part(inverse(a)))), least_upper_bound(negative_part(a), negative_part(multiply(inverse(a), identity)))))
% 0.18/0.62 = { by axiom 4 (lat4_5) }
% 0.18/0.62 multiply(positive_part(a), multiply(negative_part(inverse(positive_part(inverse(a)))), least_upper_bound(negative_part(a), greatest_lower_bound(multiply(inverse(a), identity), identity))))
% 0.18/0.62 = { by axiom 8 (left_inverse) R->L }
% 0.18/0.62 multiply(positive_part(a), multiply(negative_part(inverse(positive_part(inverse(a)))), least_upper_bound(negative_part(a), greatest_lower_bound(multiply(inverse(a), identity), multiply(inverse(a), a)))))
% 0.18/0.62 = { by axiom 14 (monotony_glb1) R->L }
% 0.18/0.62 multiply(positive_part(a), multiply(negative_part(inverse(positive_part(inverse(a)))), least_upper_bound(negative_part(a), multiply(inverse(a), greatest_lower_bound(identity, a)))))
% 0.18/0.62 = { by axiom 3 (symmetry_of_glb) }
% 0.18/0.62 multiply(positive_part(a), multiply(negative_part(inverse(positive_part(inverse(a)))), least_upper_bound(negative_part(a), multiply(inverse(a), greatest_lower_bound(a, identity)))))
% 0.18/0.62 = { by axiom 4 (lat4_5) R->L }
% 0.18/0.62 multiply(positive_part(a), multiply(negative_part(inverse(positive_part(inverse(a)))), least_upper_bound(negative_part(a), multiply(inverse(a), negative_part(a)))))
% 0.18/0.62 = { by axiom 5 (symmetry_of_lub) R->L }
% 0.18/0.62 multiply(positive_part(a), multiply(negative_part(inverse(positive_part(inverse(a)))), least_upper_bound(multiply(inverse(a), negative_part(a)), negative_part(a))))
% 0.18/0.62 = { by axiom 7 (left_identity) R->L }
% 0.18/0.62 multiply(positive_part(a), multiply(negative_part(inverse(positive_part(inverse(a)))), least_upper_bound(multiply(inverse(a), negative_part(a)), multiply(identity, negative_part(a)))))
% 0.18/0.62 = { by axiom 18 (monotony_lub2) R->L }
% 0.18/0.62 multiply(positive_part(a), multiply(negative_part(inverse(positive_part(inverse(a)))), multiply(least_upper_bound(inverse(a), identity), negative_part(a))))
% 0.18/0.62 = { by axiom 6 (lat4_4) R->L }
% 0.18/0.62 multiply(positive_part(a), multiply(negative_part(inverse(positive_part(inverse(a)))), multiply(positive_part(inverse(a)), negative_part(a))))
% 0.18/0.62 = { by lemma 25 R->L }
% 0.18/0.62 multiply(positive_part(a), greatest_lower_bound(multiply(positive_part(inverse(a)), negative_part(a)), multiply(inverse(positive_part(inverse(a))), multiply(positive_part(inverse(a)), negative_part(a)))))
% 0.18/0.63 = { by lemma 27 }
% 0.18/0.63 multiply(positive_part(a), greatest_lower_bound(multiply(positive_part(inverse(a)), negative_part(a)), negative_part(a)))
% 0.18/0.63 = { by axiom 3 (symmetry_of_glb) }
% 0.18/0.63 multiply(positive_part(a), greatest_lower_bound(negative_part(a), multiply(positive_part(inverse(a)), negative_part(a))))
% 0.18/0.63 = { by lemma 25 }
% 0.18/0.63 multiply(positive_part(a), multiply(negative_part(positive_part(inverse(a))), negative_part(a)))
% 0.18/0.63 = { by lemma 22 }
% 0.18/0.63 multiply(positive_part(a), multiply(identity, negative_part(a)))
% 0.18/0.63 = { by axiom 7 (left_identity) }
% 0.18/0.63 multiply(positive_part(a), negative_part(a))
% 0.18/0.63 % SZS output end Proof
% 0.18/0.63
% 0.18/0.63 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------