Although not suggested by natural language, and in contrast to usage in other areas, Proof of Principle and Proof of Concept are not synonymous in drug development.

* Proof Markup Language, a markup language for provenance exchange

Proof: If we have a description in, we can convert it into a description in our optimal language by first describing as a computer program ( part 1 ), and then using the original description as input to that program ( part 2 ).

: Proof: Consider a language L from PSPACE.

* A Proof of the Queueing Formula L = λ W, Sigman, K., Columbia University

: Proof: If we assume, w. l. o. g., Q < sub > A </ sub >∩ Q < sub > B </ sub > is empty then L ( A )∪ L ( B ) is recognized by the Büchi automaton ( Q < sub > A </ sub >∪ Q < sub > B </ sub >, Σ, Δ < sub > A </ sub >∪ Δ < sub > B </ sub >, I < sub > A </ sub >∪ I < sub > B </ sub >, F < sub > A </ sub >∪ F < sub > B </ sub >).

: Proof: The Büchi automaton A '=( Q ', Σ, Δ ', I ', F ') recognizes L ( A )∩ L ( B ), where

Proof: Let p < sub > 1 </ sub >⊆ p < sub > 2 </ sub > be prime ideals of A and let q < sub > 2 </ sub > be a prime ideal of B such that q < sub > 2 </ sub > ∩ A

: Proof: Assume the contrary, and let C be the largest cardinal number.

: Proof: Let S be a set, and let T be the union of the elements of S. Then every element of S is a subset of T, and hence is of cardinality less than or equal to the cardinality of T. Cantor's theorem then implies that every element of S is of cardinality strictly less than the cardinality of 2 < sup > lTl </ sup >.

: Proof: Apply Statement 2 to N = M / Σ < sub > i </ sub > Rm < sub > i </ sub >.

: Proof: Let A be infinite RE.

Proof for the existence of a common Germanic goddess once known as * Fraujon does not exist, but scholars have commented that this may simply be due to lack of evidence.

Lessing outlined the concept of the religious " Proof of Power ": How can miracles continue to be used as a base for Christianity when we have no proof of miracles?

#* Proof: suppose that p is composite, hence can be written with a and Then is prime, but and contradicting statement 1.

* ( German ), reprinted in English translation as " Proof that every set can be well-ordered ", van Heijenoort 1976, pp. 139 – 141.

Proof: Let d be the position of the leftmost ( most significant ) nonzero bit in the binary representation of s, and choose k such that the dth bit of x < sub > k </ sub > is also nonzero.

Proof of this must be supplied with one's application for naturalisation.

Proof: Since p is a prime number the only possible values of gcd ( p < sup > k </ sup >, m ) are 1, p, p < sup > 2 </ sup >, ..., p < sup > k </ sup >, and the only way for gcd ( p < sup > k </ sup >, m ) to not equal 1 is for m to be a multiple of p. The multiples of p that are less than or equal to p < sup > k </ sup > are p, 2p, 3p, ..., p < sup > k − 1 </ sup > p = p < sup > k </ sup >, and there are p < sup > k − 1 </ sup > of them.

* Proof by verbosity, sometimes colloquially referred to as argumentum verbosum-a rhetorical technique that tries to persuade by overwhelming those considering an argument with such a volume of material that the argument sounds plausible, superficially appears to be well-researched, and it is so laborious to untangle and check supporting facts that the argument might be allowed to slide by unchallenged.

This informal analysis can be formalized to make a rigorous proof of the incompleteness theorem, as described in the section " Proof sketch for the first theorem " below.

Proof beyond a reasonable doubt, therefore, is proof of such a convincing character that you would be willing to rely and act upon it without hesitation in the most important of your own affairs.

In the shortest of them ( 43 pages as of 2009 ), which he titles " Apology for the Proof of the Riemann Hypothesis " ( using the word " apology " in the rarely used sense of apologia ), he claims to use his tools on the theory of Hilbert spaces of entire functions to prove the Riemann Hypothesis for Dirichlet L-functions ( thus proving GRH ) and a similar statement for the Euler zeta function, and even to be able to assert that zeros are simple.

Proof that competitors have shared prices can be used as part of the evidence of an illegal price fixing agreement.

1922: Proof by Abraham Fraenkel that the axiom of choice cannot be proved from the axioms of Zermelo's set theory with urelements.

Proof theory is important in philosophical logic, where the primary interest is in the idea of a proof-theoretic semantics, an idea which depends upon technical ideas in structural proof theory to be feasible.

As there is no explicit Federal requirement for a two person train crew the Utah Transit Authority's FrontRunner service was originally planned to be operated by an operator only with revenue collection being handled by a Proof of Payment system.

Proof of two way contacts, either in the form of QSL cards or via digital entry into Logbook of The World ( LoTW ), must be submitted to qualify.

When a second child is cut, the node itself needs to be cut from its parent and becomes the root of a new tree ( see Proof of degree bounds, below ).

: Proof: Let D be the midpoint of AC.

Proof by exhaustion, also known as proof by cases, perfect induction, or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases and each case is checked to see if the proposition in question holds.

Proof coins of the early 19th century even appear to be scratched, but it was part of the production process.

Proof coins through 1955 could be ordered individually from the mint.

Proof: Let ( v, λ ) be an eigenvector-eigenvalue pair for a matrix A.

* Final Proof ; or, the Value of Evidence ( short story collection ; 1898 )

Proof: By symmetry, it suffices to prove that there is some constant c such that for all bitstrings s

Generally this is in a very small detail, such as the number of leaves on the ear of corn on the recent US Wisconsin state quarter: File: 2004 WI Proof. png.

* Composer John Massari would go on to work on such films as Killer Klowns From Outer Space, Skeletons, Retro Puppet Master and such television shows as POV Murder and Prison Break: Proof of Innocence.

In fact, Thurston resolved so many outstanding problems in foliation theory in such a short period of time that, according to Thurston, it led to a kind of exodus from the field, where advisors counselled students against going into foliation theory because Thurston was " cleaning out the subject " ( see " On Proof and Progress in Mathematics ", especially section 6 ).

Proof was provided in the form of unconnected bits of information such as pictures drawn by patients, heavy metal album covers, historical folklore about Satan worshippers, and pictures of mutilated animals.

# is a polynomial ( of degree ≤ n − 1 ) in .< ref > Proof: When A is normal, use Lagrange's interpolation formula to construct a polynomial P such that, where are the eigenvalues of A .</ ref >

Proof of guilt had been established by magical means such as the Ordeal.

One such free community, Collectors Proof enables manufacturers and users alike to associate unique identification numbers to virtually any item so that each new owner can update its chain of custody.

After leaving the U. S. Attorney's office, Turow became a novelist, writing legal thrillers such as The Burden of Proof, Presumed Innocent, Pleading Guilty, and Personal Injuries, which Time magazine named as the Best Fiction Novel of 1999.

Fusion shred guitarist, and session musician Greg Howe has released solo albums such as Introspection ( 1993 ), Uncertain Terms ( 1994 ), Parallax ( 1995 ), Five ( 1996 ), Ascend ( 1999 ), Hyperacuity ( 2000 ), Extraction ( 2003 ) with electric bassist Victor Wooten and drummer Dennis Chambers, and Sound Proof ( 2008 ).

Proof of legal acquisition, such as a bill of sale or purchase receipt, is contributory.

It spawned two spin-offs, CSI: Miami and CSI: NY, and inspired shows such as Body of Proof, Bones, Castle, Cold Case, Criminal Minds, Crossing Jordan, NCIS, The Mentalist and Without A Trace.

Songs such as " Shatter Proof " and the early short films of Parker and Stone were considered to be a part of the new DVD, but these additions were ultimately rejected.

She has appeared in films such as Kids, Men in Black II, 25th Hour, Sin City, Clerks II, Rent, Death Proof, The Rundown, Eagle Eye, Alexander, Seven Pounds, Percy Jackson and the Olympians: The Lightning Thief, and Unstoppable.

Also included are some original pieces such as " Fate to Despair ", " Awake ", and " Proof of Blood ".

On television she has appeared in the 2006 made-for-TV movie Proof of Lies and episodes of such drama series as What About Brian and Private Practice.

Proof of such a property interest would not, of course, entitle him to reinstatement.

He continued to appear in such high-profile television movies as Skokie ( 1981 ), Split Image ( 1982 ), Day One, ( 1989 ), A Killing in a Small Town ( 1990 ) opposite Barbara Hershey, In Broad Daylight ( 1991 ), Scott Turow's The Burden of Proof and the miniseries A Season in Purgatory.

The Birmingham pen trade prospered and areas such as the Gun Quarter and Jewellery Quarter existed with high standards of production set out by institutions such as the Birmingham Proof House and Birmingham Assay Office.