Induction on polynomials

This file contains lemmas dealing with different flavours of induction on polynomials. See also Data/Polynomial/Inductions.lean (with an s!).

The main result is Polynomial.induction_on.

theorem Polynomial.induction_on {R : Type u} [Semiring R] {M : Polynomial R → Prop} (p : Polynomial R) (h_C : (a : R) → M (↑Polynomial.C a)) (h_add : (p q : Polynomial R) → M p → M q → M (p + q)) (h_monomial : (n : ℕ) → (a : R) → M (↑Polynomial.C a * Polynomial.X ^ n) → M (↑Polynomial.C a * Polynomial.X ^ (n + 1))) : M p

theorem Polynomial.induction_on' {R : Type u} [Semiring R] {M : Polynomial R → Prop} (p : Polynomial R) (h_add : (p q : Polynomial R) → M p → M q → M (p + q)) (h_monomial : (n : ℕ) → (a : R) → M (↑(Polynomial.monomial n) a)) : M p

To prove something about polynomials, it suffices to show the condition is closed under taking sums, and it holds for monomials.

theorem Polynomial.span_le_of_C_coeff_mem {R : Type u} [Semiring R] {f : Polynomial R} {I : Ideal (Polynomial R)} (cf : ∀ (i : ℕ), ↑Polynomial.C (Polynomial.coeff f i) ∈ I) : Ideal.span {g | ∃ i, g = ↑Polynomial.C (Polynomial.coeff f i)} ≤ I

If the coefficients of a polynomial belong to an ideal, then that ideal contains the ideal spanned by the coefficients of the polynomial.

theorem Polynomial.mem_span_C_coeff {R : Type u} [Semiring R] {f : Polynomial R} : f ∈ Ideal.span {g | ∃ i, g = ↑Polynomial.C (Polynomial.coeff f i)}

theorem Polynomial.exists_C_coeff_not_mem {R : Type u} [Semiring R] {f : Polynomial R} {I : Ideal (Polynomial R)} : ¬f ∈ I → ∃ i, ¬↑Polynomial.C (Polynomial.coeff f i) ∈ I