unique — Mathlib

English

If two Weierstrass divisions exist for f by g, then their quotient is equal and their remainder is equal.

Русский

Если существуют две деления Вайершстраса для f по g, то их частное и остаток совпадают.

LaTeX

$$If H and H2 are two IsWeierstrassDivision proofs for f by g, then the corresponding quotients and remainders are equal.$$

Lean4

/-- If `q` and `r` are quotient and remainder in the Weierstrass division `f / g`, then they are
equal to `f /ʷ g` and `f %ʷ g`. -/
theorem unique [IsAdicComplete (IsLocalRing.maximalIdeal A) A] {q : A⟦X⟧} {r : A[X]}
    (H : f.IsWeierstrassDivision g q r) : q = f /ʷ g ∧ r = f %ʷ g :=
  H.elim hg (f.isWeierstrassDivision_weierstrassDiv_weierstrassMod hg)

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