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refactor(Algebra/Polynomial): make coeff return a Finsupp #41544

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refactor(Algebra/Polynomial): make coeff return a Finsupp #41544

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5 changes: 2 additions & 3 deletions Mathlib/Algebra/Polynomial/Basic.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -174,7 +174,6 @@ theorem ofFinsupp_smul {S : Type*} [SMulZeroClass S R] (a : S) (b) :
(⟨a • b⟩ : R[X]) = (a • ⟨b⟩ : R[X]) :=
rfl

set_option linter.style.whitespace false in -- manual alignment is not recognised
@[simp]
theorem ofFinsupp_pow (a) (n : ℕ) : (⟨a ^ n⟩ : R[X]) = ⟨a⟩ ^ n := by
change _ = npowRec n _
Expand Down Expand Up @@ -560,13 +559,13 @@ theorem X_pow_mul_monomial (k n : ℕ) (r : R) : X ^ k * monomial n r = monomial
rw [X_pow_mul, monomial_mul_X_pow]

/-- `coeff p n` (often denoted `p.coeff n`) is the coefficient of `X^n` in `p`. -/
def coeff : R[X] → ℕ → R
def coeff : R[X] → ℕ →₀ R
| ⟨p⟩ => p.coeff

@[simp]
theorem coeff_ofFinsupp (p) : coeff (⟨p⟩ : R[X]) = p.coeff := by rw [coeff]

theorem coeff_injective : Injective (coeff : R[X] → ℕ → R) := by rintro ⟨p⟩ ⟨q⟩; simp [coeff]
theorem coeff_injective : Injective (coeff : R[X] → ℕ →₀ R) := by rintro ⟨p⟩ ⟨q⟩; simp [coeff]

@[simp]
theorem coeff_inj : p.coeff = q.coeff ↔ p = q :=
Expand Down
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