While the word problem for monoids is undecidable in general, having a decision procedure for some finitely presented monoid of interest has numerous applications. This paper presents a toolbox for the Rocq proof assistant that can be used to verify the decidability of the word problem for a given monoid and, in some cases, to produce the corresponding decision procedure. As this verification can be computationally intensive, the toolbox heavily relies on proofs by reflection guided by an external oracle. This approach has been successfully used on several large presentations from the literature, as well as on a database of one million 1-relation monoids. The huge size of this database forced some unusual considerations onto the Rocq formalization, so that the formal proofs could be checked in a reasonable amount of time.

Funding: Supported by EPSRC funding (EP/N509759/1).

Funding: Engineering and Physical Sciences Research Council (EP/V520123/1).

A method for detection and removal of random spike noise in magnetic resonance (MR) raw data (k-space data) is described. This method would reduce or eliminate the corduroy-type and higher than usual level artifacts in MR images resulting from random spike noise in k-space data. The method described involves applying a spatially varying threshold to be k-space data. Any data point that has a magnitude greater than that of the threshold value at that location will be replaced by a local complex average of the neighboring data points or some other suitable data replacement scheme.