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Part1

A and B are any arrays including empty. The zero in the third rule is the leftmost one.

HNF(x)=x+1

HNF(A,a,b)=HNF(A,a-1,HNF(A,a,b-1))

HNF(A,a,0,B)=HNF(A,a-1,HNF(A,a-1,a,B),B)

HNF(0,A)=HNF(A)

Examples:

HNF(3,2,0)=HNF(3,1,HNF(3,1,2))
HNF(3,0,0,3)=HNF(2,HNF(2,3,0,3),0,3)=HNF(2,HNF(2,2,HNF(2,2,3,3),3),0,3)
HNF(0,2)=HNF(2)=3

Part2(rows)

The rule in part 1 is for the rightmost(last) row. The zero in the second rule is the leftmost one.

A can include (1) in the last and B can include it in the first.

HNF(Ax(1)y)=HNF(Ax-1(1)HNF(Ax-1(1)y-1))

HNF(Ax(1)0B)=HNF(Ax-1(1)HNF(Ax-1(1)xB)B)

HNF(A0)=HNF(0)

Examples:

HNF(3(1)1)=HNF( 2 (1) HNF(3(1)0) )
HNF(3(1)0)=HNF( 2 (1) HNF(2(1)3) )
HNF(3(1)0(1)0(1)3)=HNF( 2 (1) HNF(2(1)3(1)0(1)3) (1) 0 (1) 3 )

Part3(dimensions)

Part4(superdimensions)

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