Changes

For the => direction, we prove the contrapositive :  

For the => direction, we prove the contrapositive :  

* If an Indel is left aligned  and right parsimonious then each allele do not end with the same type of nucleotide.

* If an indel is left aligned  and right parsimonious then each allele does not end with the same type of nucleotide.

We first assume an indel is already left aligned and right parsimonious.  Suppose all alleles have a length greater than 1, since the indel is right parsimonious, clearly, each allele do not end with the same type of nucleotide.  Now, suppose that there exists an allele of length 1 and that all the alleles end with a particular nucleotide say 'A'.  This is still considered right parsimonious as there are no superfluous nucleotides to remove without resulting in an empty allele.  It is possible to extend all the alleles one position to the left by copying from a nucleotide on the reference genome, so now we have a superfluous nucleotide on the right side and can remove that nucleotide resulting in a new representation that shifts the Indel to the left by one position where one of the alleles is of length one.  This is left aligning the Indel and thus there is a contradiction, so each allele cannot end with the same type of nucleotide.  

We first assume an indel is already left aligned and right parsimonious.  Suppose all alleles have a length greater than 1, since the indel is right parsimonious; clearly, each allele do not end with the same type of nucleotide.  Now, suppose that there exists an allele of length 1, and that all the alleles end with a particular nucleotide say 'A'.  This is still considered right parsimonious as there are no superfluous nucleotides to remove without resulting in an empty allele.  It is possible to extend all the alleles one position to the left by copying from a nucleotide on the reference genome, so now we have a superfluous nucleotide on the right side.  Trimming off that nucleotide results in a new representation that shifts the indel to the left by one position while retaining the allele lengths of the original representation.  This is left aligning the indel and thus there is a contradiction. Therefore each allele cannot end with the same type of nucleotide.  

For the <= direction :  

For the <= direction :  

* If an Indel is not left aligned or not right parsimonious then each allele ends with the same type of nucleotide.

* If an indel is not left aligned or not right parsimonious then each allele ends with the same type of nucleotide.

Suppose a variant is not left aligned, then it must be possible to extend the alleles one nucleotide to the left and remove one nucleotide from the right to endure that all the alleles remain the same length.  Thus each allele must end with the same type of nucleotide for the removal of the rightmost nucleotide to be possible.

Suppose a variant is not left aligned, then it must be possible to extend the alleles one nucleotide to the left and remove one nucleotide from the right to endure that all the alleles remain the same length.  Thus each allele must end with the same type of nucleotide for the removal of the rightmost nucleotide to be possible.