Set attribute of a function to be a specific equation that it fulfills

Question

I would like to define a function f(x,y,z) by its attribute, namely by the specific equation f(b,c,d)f(a,b+c,d)f(a,b,c)=f(a+b,c,d)f(a,b,c+d) for all a,b,c,d. Is it pssible to set this specific attribute to f?

Will this be used in simplifying expressions? You could perhaps use your constraint (expressed as an equation, using ==) as an Assumption within Simplify. Can you provide some more context? — MarcoB, Feb 09 '17 at 17:10

score 3 · Accepted Answer · answered Feb 09 '17 at 19:19

Use UpSet

f /: f[b_, c_, d_] f[a_, b_ + c_, d_] f[a_, b_, c_] = 
  f[a + b, c, d] f[a, b, c + d];

f[b, c, d] f[a, b + c, d] f[a, b, c]

(*  f[a, b, c + d] f[a + b, c, d]  *)

f[y, x, t] f[z, y + x, t] f[z, y, x]

(*  f[z, y, t + x] f[y + z, x, t]  *)

However, this won't work for numeric inputs that cause the additions to be carried out

f[2, 3, t] f[z, 2 + 3, t] f[z, 2, 3]

(*  f[2, 3, t] f[z, 2, 3] f[z, 5, t]  *)

Set attribute of a function to be a specific equation that it fulfills

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