For the frequent case of propositional logic, the problem is decidable but Co-NP-complete, and hence only exponential-time algorithms are believed to exist for general proof tasks.

For the proof of the equivalence of the four approaches the reader is referred to mathematical expositions like or.

( For groups of low 2-rank the proof of this breaks down, because theorems such as the signalizer functor theorem only work for groups with elementary abelian subgroups of rank at least 3.

For all his successes, the nature of his proof stirred up more trouble than Hilbert could have imagined at the time.

For example, the proof that the column rank of a matrix over a field equals its row rank yields for matrices over division rings only that the left column rank equals its right row rank: it does not make sense to speak about the rank of a matrix over a division ring.

For the developers, Freeciv 1. 0 was a successful proof of concept, but a rather boring game, so they went back to XPilot.

For years the statue had been used by historians as proof of Hadrian's love of Hellenic culture.

Others have independently come to the same conclusion, and publication of independent analyses may be forthcoming [...] For scientists to label sight reports and questionable photographs as ' proof ' of such an extraordinary record is delving into ' faith-based ' ornithology and doing a disservice to science.

For example, the old instructions on burden of proof in civil cases read:

( For a proof and an interactive simulation see reference below, Algorithm for Computing the LCM.

For libertarians, that diversity of practice within a framework of common principles is proof of the vitality of those principles and of their flexibility and strength.

For example, Russell's paradox suggests a proof that the class of all sets which do not contain themselves is proper, and the Burali-Forti paradox suggests that the class of all ordinal numbers is proper.

( For example, the Coq proof assistant system features a tactic for Presburger arithmetic.

For other examples, see proof that the square root of 2 is not rational and Cantor's diagonal argument.

For instance, one report from Burma stated that red pandas were still fairly common in some areas, and was accompanied by a photograph of a " red panda " as proof.

For example, in an axiomatic system, proof of soundness amounts to verifying the validity of the axioms and that the rules of inference preserve validity ( or the weaker property, truth ).

For contemporary Marxists, the existence of persistent unemployment is proof of the inability of capitalism to ensure full employment.

For the first three years of clad production, in lieu of proof sets, specimen sets were specially sold as " Special Mint Sets " minted at the San Francisco mint in 1965, 1966, and 1967 ( Deep Cameo versions of these spectacular coins are highly valued because of their rarity ).

“ For man, and especially the Proficient, is not the Couplement of Soul and body: the proof is that man can be disengaged from the body and disdain its nominal goods .” ( Enneads I. 4. 14 ) The human who has achieved happiness will not be bothered by sickness, discomfort, etc., as his focus is on the greatest things.

For a complete proof, see the reciprocity ( electromagnetism ) article.

For his part, Ernst Zermelo in his ( 1908 ) A new proof of the possibility of a well-ordering ( published at the same time he published " the first axiomatic set theory ") laid claim to prior discovery of the antinomy in Cantor's naive set theory.

For example, the credit card network does not require a personal identification number for authentication of the claimed identity ; and a small transaction usually does not even require a signature of the authenticated person for proof of authorization of the transaction.

For example, Euclid provides an elaborate proof of the Pythagorean theorem, by using addition of areas instead of the much simpler proof from similar triangles, which relies on ratios of line segments.

For example, let's consider a standard 283 cubic inch Chevy Aj.

For an example let's dream up an engine that has a final combustion chamber volume of 5 cubic inches and a cylinder volume of 45 cubic inches.

For the sake of the example ( and this is a gross simplification ), let's assume that he values this particular risk at 5 % per annum ( we could perform a more precise probabilistic analysis of the risk, but that is beyond the scope of this article ).

For example, " There's some furniture in the room " can be restated, with a change of focus, to " There are some pieces of furniture in the room "; and " let's have some fun " can be refocused as " Let's have a bit of fun ".

For example, in the 1997 federal election, Bloc Québécois MP Suzanne Tremblay attacked Charest by saying, " First, let's recall who Jean Charest really is ... his real name is John, that's what's on his birth certificate, not Jean.

For example, let's say 500 grains is a pile and 1, 000 grains is a heap.

We're hoping that fans will figure ' For God's sake, let's just call him Hawkman and get him in the Justice League as Hawkman ,' and the editors will relent.

For the sake of simplicity, let's assume it is now 2.

For the sake of explanation let's label the prisoners in line order A B and C. Thus B can see A ( and A's hat color ) and C can see A and B.

For example, let's say A holds 100 shares of a Petroleum Company.

For example, let's say A wants to invest in company X listed in Country C. However, A is not allowed to invest in Country C due to capital control regulations.

For now let's just consider the case of the phonons of the quantum harmonic oscillator, which are bosons, because fermions are more complicated.

Linux kernel developer Ingo Molnár called atime " perhaps the most stupid Unix design idea of all times ," adding: " hink about this a bit: ' For every file that is read from the disk, let's do a ... write to the disk!

For example, " I run " counts as one word, and " rhinoceros " and " let's talk " both count as two.

This section was not so well received, though it did introduce the " Art For ' em " sketch featuring pretentious artists-cum-critics Dominic Bel Geddes and Daniel Cakebread, and concluding each week with the memorable catchphrase " let's roll on the floor !".

For a simple example, let's consider a simulation showing a character walking across a field, jumping over any rocks it encounters.

For example, let's say that a eurozone card is used for a withdrawal in the UK.

2. For a body consisting of 2 particles ( ex. diatomic molecule ) in 3-D plane with constant distance between them ( let's say d ) we can show it's degree of freedom to be 5.

For example, let's assume there is a fabric with just one node, namely a server's FC-compatible HBA.