Proofs and Refutations (450 bytes)
1: '''Proof and Refutations''' is a book by the [[philosopher]] [[Imre Lakato...

List of mathematical proofs (5269 bytes)
4: *[[Bertrand's postulate]] and [[Proof of Bertrand's postulate|a proof]]
7: ...em]] and [[Proofs of Fermat's little theorem|some proofs]]
8: *[[Gödel's completeness theorem]] and [[Original proof of Gödel's completeness theorem|...
9: *[[Mathematical induction]] and [[Proof of mathematical induction|a proof]]
64: ...demann-Weierstrass theorem|Transcendence of ''e'' and π]] (as corollaries of Lindemann-Weierstrass)

Proofs of Fermat's little theorem (12066 bytes)
1: This is a collection of [[mathematical proof|proofs]] of [[Fermat's little theorem]]:
3: for every [[prime number]] ''p'' and every [[integer]] ''a''.
13: ...' If ''a'' is relatively prime to ''p'' and ''x'' and ''y'' are integers such that ''xa'' = ''ya'' (mod...
18: ... numbers are different modulo ''p'' by the lemma, and none of them is zero modulo ''p'' (again by the l...
26: ... sides. This is allowed by the lemma, since ''p'' and (''p'' − 1)! can have no factor in common, ...

Nonconstructive proof (2909 bytes)
3: ... therefore been shown to be logically impossible, and yet an actual example of the thing has not been f...
5: ...ccurate, as both constructive and nonconstructive proofs can be used to prove existence. See [[existence t...
7: == Some examples of nonconstructive proofs ==
9: ...There exist [[irrational number]]s <math>a</math> and <math>b</math> such that <math>a^b</math> is rati...
11: *Recall that <math>\sqrt{2}</math> is irrational, and 2 is rational. Consider the number <math>q = \sq...

Imre Lakatos (8325 bytes)
1: ...[1974]]) was a [[philosopher]] of [[mathematics]] and of [[science]].
4: ...He received a degree in mathematics, [[physics]], and [[philosophy]] from the [[University of Debrecen]...
6: ...arian Workers' Party| Hungarian communist party]] and was imprisoned on charges of [[revisionism]] from...
8: ...mmunist, his political views had shifted markedly and he was involved with at least one dissident stude...
10: ...niversity of Cambridge]]. The book ''[[Proofs and Refutations]]'', published after his death, is based on this ...

Interactive theorem proving (416 bytes)
1: ...h for proofs, the details of which are stored in, and some steps provided by, a [[computer]].

Compactness theorem (2342 bytes)
2: ...tisfiable, i.e., has a [[model theory|model]], if and only if every finite [[subset]] of it is satisfia...
8: ...s negation ~''S'', together with the field axioms and the infinite series of sentences 1+1 &ne; 0, 1+1+...
10: ...with uncountably many natural numbers. The [[nonstandard analysis]] is another example where infinite n...
12: == Proofs ==
14: ...be proven from it. Since proofs are always finite and therefore involve only finitely many of the given...

Proof of concept (1098 bytes)
1: A '''proof of concept''' is a short and/or incomplete realization of a certain method or ...
7: ...luded proofs of concept of the animation of cloth and of human facial expressions. These techniques we...
9: ...ng toward the camera from a long distance. These proofs of concept demonstrated ways for the team to acco...

Analytic proof (1195 bytes)
3: ...both the principal premise of an elimination rule and the conclusion of an introduction rule;
4: * In Gentzen's [[sequent calculus]] the analytic proofs are those that do not use the [[cut-elimination|c...
6: ...possible to extend both calculi so that there are proofs that satisfy the condition but are not analytic: ...

Proof net (684 bytes)
1: ...alculi such as the [[natural deduction]] calculus and the [[sequent calculus]]; by this means the forma...
12: * ''Proofs and Types''. Girard J-Y, Lafont Y, and Taylor P. Cambridge Press, 1989.

Diagram chasing (680 bytes)
3: ...ma]], the [[snake lemma]], the [[zig-zag lemma]], and the [[nine lemma]].
9: [[Category:Homological algebra]][[Category:Proofs]]

Proof by exhaustion (2573 bytes)
1: ...be proved is split into a finite number of cases, and each case is proved separately. A proof by exhaus...
6: ...n]]''' of [[Eudoxus of Cnidus]] was a geometrical and essentially rigorous way of calculating mathemati...
16: ...iple of 3 then the cube of n is a multiple of 27, and so certainly a multiple of 9.
20: [To complete the proof, the claims in cases 2 and 3 can be proved using simple algebra.]
25: ...cases were checked by a computer program, not by hand. The shortest known proof of the four colour theo...

Sefer Zadok (420 bytes)
1: ...w]]ish scholars of that time period, generally in refutations of their beliefs.

Natural proof (1047 bytes)
1: ...v]] and [[Steven Rudich]], to classify the set of proofs that will fail to prove separations between [[com...
12: * A. A. Razborov and S. Rudich. Natural proofs. In Proceedings, 26th ACM Symposium on Theory of ...

Three forms of mathematical induction (1116 bytes)
1: Proofs that a subset of { 1, 2, 3, ... } is in fact the ...
4: ...stantial part of the proof is the case ''n'' = 2, and the case ''n'' = 2 is relied on in the trivial in...
5: ...s form works not only when the values of <i>k</i> and <i>n</i> are natural numbers, but also when they ...
9: [[Category:Mathematical logic]][[Category:Proofs]]

Insolubilia (2143 bytes)
3: ...paradox, including [[St. Augustine]], [[Cicero]], and the quotation of [[Epimenides]] appearing the [[E...
5: ...he difficulties raised by these statements. [[Alexander Neckham]], writing later in the twelfth century...
7: ...the problems are seen as fundamentally insoluble, and central to the foundations of logic.

Coq (1049 bytes)
1: ...dles mathematical assertions, checks mechanically proofs of these assertions,
2: helps to find formal proofs, and extracts a certified program from the constructiv...
4: ... de Lyon]]). The team leaders are Pr Gilles Dowek and Pr Christine Paulin-Mohring. Coq is written in th...
6: ...in [[French language|French]] - and [[Thierry Coquand]] (along with [[Gérard Huet]]) developed the afor...

Hypergeometric function identities (2298 bytes)
1: ... hand, there exist now several algorithms to find and prove all hypergeometric identities.
21: ...s of sums over hypergeometric terms, the definite and indefinite sums. A definite sum is of the form
28: == Proofs ==
29: ...ertificate which anyone could use to easily check and prove the correctness of the identity.
35: ... written by [[Marko Petkovsek]], [[Herbert Wilf]] and [[Doron Zeilberger]] describing the three main ap...

On the Number of Primes Less Than a Given Magnitude (2539 bytes)
1: ...hods; all of these have become essential concepts and tools of modern [[analytic number theory]].
6: ...or ''x'' &ge; 0, which is defined by ''J''(0) = 0 and ''J''(''x'') jumps by 1/''n'' at each prime power...
8: Among the proofs and sketches of proofs:
9: *Two proofs of the [[functional equation (number theory)|func...
11: ...f &xi;(''s'') whose imaginary part lies between 0 and ''T''

System F (1299 bytes)
2: ...endently by the [[logician]] [[Jean-Yves Girard]] and the [[computer scientist]] [[John C. Reynolds]]. ...
5: ... calculus]] has variables ranging over functions, and binders for them,
6: ...da calculus has variables ranging over ''types'', and binders for them.
23: ....uk/~pt/stable/Proofs+Types.pdf Textbook ''Proofs and Types'']
24: ...p://www.cs.man.ac.uk/~pt/stable/prot.pdf ''Proofs and Types - New link'']

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