quotientSpanXSubCAlgEquiv — Mathlib

English

Same monotonicity fact for descPochhammer on [n−1, ∞).

Русский

Та же факт монотонности(descPochhammer) на [n−1, ∞).

LaTeX

$$$\\text{MonotoneOn}\\Bigl((\\operatorname{descPochhammer} \\ S\\, n).\\mathrm{eval},\, \\mathrm{Set.Ici}(n-1)\\Bigr)$$$

Lean4

/-- For a commutative ring $R$, evaluating a polynomial at an element $x \in R$ induces an
isomorphism of $R$-algebras $R[X] / \langle X - x \rangle \cong R$. -/
noncomputable def quotientSpanXSubCAlgEquiv (x : R) : (R[X] ⧸ Ideal.span ({X - C x} : Set R[X])) ≃ₐ[R] R :=
  let e :=
    RingHom.quotientKerEquivOfRightInverse
      (fun x => by exact eval_C : Function.RightInverse (fun a : R => (C a : R[X])) (@aeval R R _ _ _ x))
  (Ideal.quotientEquivAlgOfEq R (ker_evalRingHom x).symm).trans { e with commutes' := fun r => e.apply_symm_apply r }

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