HOL (Gordon and Melham 1993) is another higher-order proof development system primarily used as an aid in the development of hardware and software safety-critical systems.
The uniform Gaussian estimates and Schauder estimates in Theorem 1.4 applied to (8.8) yield the higher order estimates and conclude the proof.
Furthermore, we introduce the matrix representation of some results for higher-order Daehee numbers and polynomials obtained by Kim et al. [8] in terms of Stirling numbers, Nörlund numbers, and Bernoulli numbers of higher order and give simple and short proofs of these results.
Given that our model shows good predictive capability, it serves not only as an informational tool for yeast biology, but also as a proof of principle for higher order system models.
The synthesisable subset of higher order logic can be extended using additional proof-based tools that transform definitions into the subset.
Under such circumstances it is a higher order effect (the interaction) that is assumed zero until proof to the contrary is available.
Logic, or at least proof-theory, is focused on formal proof systems: intuitionistic predicate calculus, classical predicate calculus, arithmetics, higher order calculi, and a wealth of similar consistent and structured sets of process-building rules.
A crucial ingredient of the proof is developing an approach to separate the higher order linear effect and the boundary nonlinear effect so that the kernels of the nonlinear problem are captured.
The key of the proof is to derive the suitable estimates for the higher order derivatives of the bilinear terms.
First of all, we are not claiming that they have high weak order uniformly in the stepsize, and we have no such proof apart from in the linear case when the higher order is inherited directly from the underlying deterministic extrapolation methods.
To extend the results to th-order fuzzy differential equation, we can follow the proof of Theorem 4.2 to get the same results for derivatives of higher order.
