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Proof theory is a major branch [ 1] of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed ...
Modern proof theory treats proofs as inductively defined data structures, not requiring an assumption that axioms are "true" in any sense. This allows parallel mathematical theories as formal models of a given intuitive concept, based on alternate sets of axioms, for example Axiomatic set theory and Non-Euclidean geometry.
Indeed, the above proof that the law of excluded middle implies proof by contradiction can be repurposed to show that a decidable proposition is ¬¬-stable. A typical example of a decidable proposition is a statement that can be checked by direct computation, such as " n {\displaystyle n} is prime" or " a {\displaystyle a} divides b ...
This is perhaps the simplest known proof, requiring the least mathematical background. It is an attractive example of a combinatorial proof (a proof that involves counting a collection of objects in two different ways). The proof given here is an adaptation of Golomb's proof. [1] To keep things simple, let us assume that a is a positive integer.
Proof by infinite descent. In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction [ 1] used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number ...
Lagrange's theorem (group theory) Lagrange's theorem (number theory) Liouville's theorem (complex analysis) Markov's inequality (proof of a generalization) Mean value theorem. Multivariate normal distribution (to do) Holomorphic functions are analytic. Pythagorean theorem. Quadratic equation.
A propositional proof system P p-simulates Q (written as P ≤ pQ) when there is a polynomial-time function F such that P ( F ( x )) = Q ( x) for every x. [1] That is, given a Q -proof x, we can find in polynomial time a P -proof of the same tautology. If P ≤ pQ and Q ≤ pP, the proof systems P and Q are p-equivalent.
e. In mathematics, an impossibility theorem is a theorem that demonstrates a problem or general set of problems cannot be solved. These are also known as proofs of impossibility, negative proofs, or negative results. Impossibility theorems often resolve decades or centuries of work spent looking for a solution by proving there is no solution.