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It is important that the normalization results in a unique representation of the variant. Before we begin the proof, intuitively, accept that any representation of a variant can be
It is important that the normalization results in a unique representation of the variant. Before we begin the proof, intuitively, accept that any representation of a variant can be
transformed to another representation by removing or adding nucleotides from the reference sequence.
transformed to another representation by adding nucleotides from the reference sequence to either ends of all the alleles at the same time or removing equivalent nucleotides from the ends of
all the alleles at the same time.
Now suppose there are 2 normalized variants A and B. Suppose A is at a different position from B and B is to the right of A (without loss in generality), this is not possible because by the definition of a normalized variant, it is left aligned,
Now suppose there are 2 normalized variants A and B. Suppose A is at a different position from B and B is to the right of A (without loss in generality), this is not possible because by the definition of a normalized variant, it is left aligned,
and if they were at different positions, that means B may be left aligned to A since they represent the same variants leading to a contradiction. So A and B must be at the same position.
and if they were at different positions, that means B may be left aligned to A since they represent the same variants leading to a contradiction. So A and B must be at the same position.
Now, suppose that A and B are of different lengths where B is longer than A, then this is not possible as B is then not parsimonious, so B can be trimmed to the same length as A.
Now, suppose that A and B are at the same position but are of different lengths where B is longer than A (without loss in generality), this is not possible as B is then not parsimonious, so B can be trimmed to the same length as A.
Thus A and B have to be at the same position and have the same length and variant normalization is unique.
Thus A and B have to be at the same position and have the same length and variant normalization is unique.
