Sentence examples for proof containing from inspiring English sources

For note that it seems plausible to assume that no human mathematician will ever be able to comprehend a proof containing 100 million symbols (\(\approx 25000\) pages).

Almost no published proof contains every step; there are just too many.

The remaining part of the proof contains three steps.

This involves assumptions: for example, the assumption that a proof contains premises and a conclusion.

This part of the proof contains a lot of technical details and is very difficult to follow.

Cuntavepanit and Panyanak's proof contains a lemma that uses the existence of Banach limits similar to the Banach limit lemma used in Saejung [7].

The initial proof, contained in [50], is valid for Alexandroff spaces.21 The generalization of the same result to arbitrary MMS satisfying (CD K,infty )) is done in [6].

A dynamic proof for an adaptive logic of inductive generalizations may look as follows: The last column of each line of the proof contains its condition, i.e., a set of abnormalities Υ that encodes the assumptions used to derive the formula in the second column of the line: each υ ∈ Υ is assumed to be false.

Two years back number theorist Dan Goldston at San Jose State University in California and his colleague Cem Yildirim at Bogaziçi University in Istanbul, Turkey, announced a breakthrough--only to quickly learn that their proof contained a fatal error (Science, 4 April 2003, p. 32; 16 May 2003, p. 1066).

Remark 2 explains that if a multi-party transition t is derivable by the rules from the well-behaved set (mathrm{dom}({^{bullet }t}) subseteq S), then its proof contains (k+1) synchronizations at most, each one between a transition (labeled with a sequence) and a singleton-preset transition (labeled with a single action).

The great significance of Banach's principle, and the reason it is possibly one of the most frequently cited fixed point theorems in all of analysis, lies in the fact that its proof contains elements of fundamental importance to the theoretical and practical treatment of mathematical equations.