Some notes for how to modify the current code.

Todo list
---------
> Remove unnecessary/old code


> Remove unnecessary parentheses
> Separate CompTable modifying functions from service-functions.m2 (see discussion below).
> Strategy "Markov basis" --- only subduct on generators of the binomial ideal (rather than a GB), which we can compute quickly using FourTiTwo.
  -- do we have an implementation that would work with just the binomial generators
> Which subductions to export? Anything without compTable
   ? Can we export only some methods ?

> Determine what "subduction" and "%" should do with (RingElement, SAGBIBasis)
  -- currently this "subduction" is not implemented and "%" performs a subduction.

> [DONE - Needs Checking] Implement the correct caching behaviour in subalgebraBasis and sagbi:
  -- A subring is an immutable object - in its cache we store the 'most computationally advanced' SAGBIBasis object
  -- When resuming a subalgebraBasis or sagbi computation from a SAGBIBasis object, use the previously selected options
    * Only adjust the Limit option
  -- Add another option to sagbi / subalgebraBasis: RenewOptions (default: false)
    * if RenewOptions == true then set all the options to ones supplied by the user
    * RenewOptions is false by default because the first priority is to recover the options from the previous computation
  -- matrices cache a Subring which, in turn, cache the SAGBIBasis

> [DONE] When picking up where a computation left off, if the original set of generators was autosubducted then don't autosubduct again.
  -- Could be implemented by setting the AutoSubduce option to false once autosubduction is done

> sort out keys for subrings - remove the quotes around the keys
  ? Can we use unexported symbols for keys in classes ?
  
> [DONE - Needs Checking] sagbi and subalgebraBasis always uses the sagbiBasis object of the subring in the cache of the passed in object
  -- E.g. sagbi M (matrix) --> sagbi M.cache#"Subring".cache."SAGBIBasis" (or something like that)
  -- In the above example, if the matrix has no subring cached then calling sagbi on a matrix makes a subring
  -- methodology: have only 1 subring floating around linked with the latest computation object

> [DONE - Needs Checking] groebner membership test for quotients
  ? Can we just sum the ideal I and the quotienting ideal (lifted to the tensor ring) ?

> Add more asserts!

> [DONE] Add groebnerNormalForm method. It should be a non-exported method that returns an element of the tensor ring
  -- it can be used by groebnerMembershipTest and %
  -- this method may also want to return a projection map from the tensorRing to the ambientRing

> [DONE] copy over the intersection of subrings method to classes.
  -- Add other input options: e.g. intersection(SAGBIBasis, SAGBIBasis), etc.

> [DONE] forceSAGBI - assert that a subring / matrix of generators is a sagbi basis

> intrinsic subductionQuotient
  the subduction quotient is currently implemented in an extrinsic way that uses % to compute the
  normal form and a factorised form (in terms of the generators) of a given element f of the ambient ring.
  Using subduction, we can also construct this factorised form. When a sagbi basis is computed, we must:
  -- Keep track of the factorisation of the new sagbi generators in terms of the subring generators
  Then, given an element f of the ambient ring, when we perform subduction on it with respect to a sagbi basis
  we must keep track of the factorisation of the lead terms that cancel the lead terms of monomials of f
  
  

Discussion
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Ollie: It would be great to have a separate file(s) for subduction/sagbi related functions.
       Currently, service-functions.m2 is getting very crowded and contains many levels of functions:
       E.g. updateComputation passes off to one of three different functions.
       There are at least two ways to solve this:
       1) Have functions that do similar things in one file, such as the four update-related functions above
       2) Have functions on a similar level in one file, such as updateComputation in a higher-level file than what it calls
          We already do this for sagbi, since it sits in one file and calls stuff in service-functions.m2