The traditional proof systems such as natural deduction or sequent calculus turn out to be insufficient for axiomatizing any more or less nontrivial fragments of CoL . This has necessitated developing alternative, more general and flexible methods of proof, such as cirquent calculus.
32.
This is a valid method of proof, because if the original identity is true for general choices of the variables ( say, almost all choices, or choices with the coordinates independent over the rationals ), then it must be true for all values.
33.
All the court needs to decide now, Canaday said, is not " the outer limits or parameters of same-sex harassment, or the methods of proof of such a claim, " but only " whether a same-sex sexual-harassment claim exists as a matter of law ."
34.
The first proof follows Carath�odory's original method of proof from 1913 using properties of Lebesgue measure on the circle : the continuous extension of the inverse function " g " of " f " to " " U " is justified by Fatou's theorem on the boundary behaviour of bounded harmonic functions on the unit disk.
35.
Prior to the first trial date, if it appears to the court that the method of proof indicated by the public prosecutor is obviously insufficient to establish the possibility that the accused is guilty, the court shall, by a ruling, notify the public prosecutor to make it up within a specified time period; if additional evidence is not presented within the specified time period, the court may dismiss the prosecution by a ruling.
36.
The proof of this theorem is a reconstruction theorem, requiring the construction of a Hopf algebra as a minimum, but in most papers done by construction of a nondegenerate pairing of two algebras in the iterated endomorphism algebra tower above B in A, and then a very delicate check that the resulting algebra-coalgebra structure is a Hopf algebra ( see for example the article from 2001 below ); the method of proof is considerably simplified by the 2003 article cited below ( albeit packaged into the definition of Hopf algebroid ).
37.
The 1249 text has articles concerning : the proclaiming of the alleged offence on both sides of the border; the use of pledges to ensure that the plaintiffs and defendants turned up at the day of march; the use of jurors from both kingdoms; the payment of compensation to victims ( such as " manbote " as regards murder ); the method of proof in contested disputes ( usually judicial duel or " wager of battle ", known as " handwarcelle " ); and the guarantee of sanctuary to those who confessed.
38.
I already know a proof of this result using the intermediate value theorem : this is meant to be an alternative method of proof, and unfortunately I can't quite see what I'm meant to be getting at : I don't know quite what it means with'by integrating K analytically'- the only thing I could think to do that they might be referring to is use integration by parts and even then, it seems very illogical to integrate k and differentiate f, when f is already at the correct derivative and K is order 2 ( it's piecewise-quadratic if I've calculated it correctly ) so surely we'd want to differentiate K-a typo, do you think?
39.
Likewise, for the benefit of commoners, Frederick banned trial by ordeal, ordering his judges to use instead " the common methods of proof which have been introduced both by the ancient laws and by our constitutions . " The Constitutions notably used reason and logic to dismiss the superstitious foundations of the ordeal; for example, the use of trial by hot iron was dismissed because people believed " the natural heat of white-hot iron grows hot and, what is even more foolish, grows cold for no good reason at all ", and trial by water was forbidden because of the belief " that the defendant of the crime, who has been established only by his guilty conscience, will not be received by the element of freezing water, when, in fact, it is the retention of sufficient air that prevents him from submerging . " Frederick also banned trial by battle, ordering that more weight be given to the testimony of witnesses, although exceptions to this were granted to knights, and for cases in which no witnesses could be provided.
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