datatype tree = L | N of tree * tree

datatype herp
  = H0
  | H1 of tree
  | H2 of tree * tree
  | H3 of tree * tree * tree
  | H4 of tree * tree * tree * tree
  | H5 of tree * tree * tree * tree * tree
  | H6 of tree * tree * tree * tree * tree * tree
  | H8 of tree * tree * tree * tree * tree * tree * tree * tree

datatype derp
  = D0
  | D2 of tree * tree
  | D3 of tree * tree * tree
  | D4 of tree * tree * tree * tree
  | D5 of tree * tree * tree * tree * tree
  | D6 of tree * tree * tree * tree * tree * tree
  | D7 of tree * tree * tree * tree * tree * tree * tree
  | D8 of tree * tree * tree * tree * tree * tree * tree * tree

fun foo L = H0
  | foo (N (a,L)) = H1 a
  | foo (N (a,N (b,L))) = H2 (a,b)
  | foo (N (a,N (b,N (c,L)))) = H3 (a,b,c)
  | foo (N (a,N (b,N (c,N (d,L))))) = H4 (a,b,c,d)
  | foo (N (a,N (b,N (c,N (d,N (e,L)))))) = H5 (a,b,c,d,e)
  | foo (N (a,N (b,N (c,N (d,N (e,N (f,L))))))) = H6 (a,b,c,d,e,f)
  | foo (N (a,N (b,N (c,N (d,N (e,N (f,N (g,h)))))))) = H8 (a,b,c,d,e,f,g,h)

fun foo' H0 = L
  | foo' (H1 a) = N (a,L)
  | foo' (H2 (a,b)) = N (a,N (b,L))
  | foo' (H3 (a,b,c)) = N (a,N (b,N (c,L)))
  | foo' (H4 (a,b,c,d)) = N (a,N (b,N (c,N (d,L))))
  | foo' (H5 (a,b,c,d,e)) = N (a,N (b,N (c,N (d,N (e,L)))))
  | foo' (H6 (a,b,c,d,e,f)) = N (a,N (b,N (c,N (d,N (e,N (f,L))))))
  | foo' (H8 (a,b,c,d,e,f,g,h)) = N (a,N (b,N (c,N (d,N (e,N (f,N (g,h)))))))

fun bar H0 = D0
  | bar (H1 a) = D2 (L,a)
  | bar (H2 (a,b)) = D2 (N (L,a),b)
  | bar (H3 xs) = D3 xs
  | bar (H4 xs) = D4 xs
  | bar (H5 xs) = D5 xs
  | bar (H6 (L,L,a,b,c,d)) = D2 (N (N (a,b),c),d)
  | bar (H6 (L,N(a,b),c,d,e,f)) = D6 (a,b,c,d,e,f)
  | bar (H6 (N(a,b),c,d,e,f,g)) = D7 (a,b,c,d,e,f,g)
  | bar (H8 xs) = D8 xs

fun bar' D0 = H0
  | bar' (D2 (L,a)) = H1 a
  | bar' (D2 (N (L,a),b)) = H2 (a,b)
  | bar' (D2 (N (N (a,b),c),d)) = H6 (L,L,a,b,c,d)
  | bar' (D3 xs) = H3 xs
  | bar' (D4 xs) = H4 xs
  | bar' (D5 xs) = H5 xs
  | bar' (D6 (a,b,c,d,e,f)) = H6 (L,N(a,b),c,d,e,f)
  | bar' (D7 (a,b,c,d,e,f,g)) = H6 (N(a,b),c,d,e,f,g)
  | bar' (D8 xs) = H8 xs

fun qux D0 = (L,L,L,L,L,L,L)
  | qux (D2 (a,b)) = (L,L,L,L,L,L,N(a,b))
  | qux (D3 (a,b,c)) = (L,L,L,L,L,N(a,b),c)
  | qux (D4 (a,b,c,d)) = (L,L,L,L,N(a,b),c,d)
  | qux (D5 (a,b,c,d,e)) = (L,L,L,N(a,b),c,d,e)
  | qux (D6 (a,b,c,d,e,f)) = (L,L,N(a,b),c,d,e,f)
  | qux (D7 (a,b,c,d,e,f,g)) = (L,N(a,b),c,d,e,f,g)
  | qux (D8 (a,b,c,d,e,f,g,h)) = (N(a,b),c,d,e,f,g,h)

fun qux' (L,L,L,L,L,L,L) = D0
  | qux' (L,L,L,L,L,L,N(a,b)) = D2 (a,b)
  | qux' (L,L,L,L,L,N(a,b),c) = D3 (a,b,c)
  | qux' (L,L,L,L,N(a,b),c,d) = D4 (a,b,c,d)
  | qux' (L,L,L,N(a,b),c,d,e) = D5 (a,b,c,d,e)
  | qux' (L,L,N(a,b),c,d,e,f) = D6 (a,b,c,d,e,f)
  | qux' (L,N(a,b),c,d,e,f,g) = D7 (a,b,c,d,e,f,g)
  | qux' (N(a,b),c,d,e,f,g,h) = D8 (a,b,c,d,e,f,g,h)

val fwd = qux o bar o foo
val bwd = foo' o bar' o qux'