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TagSetDelayed

Defines a delayed upvalue assignment associated with a tag symbol. The tag /: lhs := rhs shorthand is the usual way to write it.

$ wo 'TagSetDelayed[g, f[g[x_]], 1 + 2]; f[g[5]]'
3

Upvalues attached to Plus, Times and Power also apply to the arithmetic shorthands that expand to those heads: a - b is Plus[a, Times[-1, b]], a / b is Times[a, Power[b, -1]] and -a is Times[-1, a].

$ wo 'mytag /: mytag[a_] + mytag[b_] := mytag[a + b]; mytag /: i_Integer mytag[a_] := mytag[a i]; mytag[3] - mytag[3]'
mytag[0]

$ wo 'mytag /: i_Integer mytag[a_] := mytag[a i]; -mytag[3]'
mytag[-3]

$ wo 'q /: q[a_] q[b_] := q[a b]; q /: q[a_]^n_Integer := q[a^n]; q[6] / q[3]'
q[2]

Together they implement modular arithmetic, where m[a, n] stands for a modulo n:

$ wo 'm /: m[a_, n_] + m[b_, n_] := m[Mod[a + b, n], n]; m /: c_Integer m[a_, n_] := m[Mod[c a, n], n]; m[3, 7] - m[6, 7]'
m[4, 7]