# mathlibdocumentation

data.polynomial.inductions

# Induction on polynomials #

This file contains lemmas dealing with different flavours of induction on polynomials.

noncomputable def polynomial.div_X {R : Type u} [semiring R] (p : polynomial R) :

div_X p returns a polynomial q such that q * X + C (p.coeff 0) = p. It can be used in a semiring where the usual division algorithm is not possible

Equations
@[simp]
theorem polynomial.coeff_div_X {R : Type u} {n : } [semiring R] {p : polynomial R} :
p.div_X.coeff n = p.coeff (n + 1)
theorem polynomial.div_X_mul_X_add {R : Type u} [semiring R] (p : polynomial R) :
@[simp]
theorem polynomial.div_X_C {R : Type u} [semiring R] (a : R) :
.div_X = 0
theorem polynomial.div_X_eq_zero_iff {R : Type u} [semiring R] {p : polynomial R} :
theorem polynomial.div_X_add {R : Type u} [semiring R] {p q : polynomial R} :
(p + q).div_X = p.div_X + q.div_X
theorem polynomial.degree_div_X_lt {R : Type u} [semiring R] {p : polynomial R} (hp0 : p 0) :
noncomputable def polynomial.rec_on_horner {R : Type u} [semiring R] {M : Sort u_1} (p : polynomial R) :
M 0(Π (p : (a : R), p.coeff 0 = 0a 0M pM (p + (Π (p : , p 0M pM (p * polynomial.X))M p

An induction principle for polynomials, valued in Sort* instead of Prop.

Equations
theorem polynomial.degree_pos_induction_on {R : Type u} [semiring R] {P : → Prop} (p : polynomial R) (h0 : 0 < p.degree) (hC : ∀ {a : R}, a 0P ) (hX : ∀ {p : , 0 < p.degreeP pP ) (hadd : ∀ {p : {a : R}, 0 < p.degreeP pP (p + ) :
P p

A property holds for all polynomials of positive degree with coefficients in a semiring R if it holds for

• a * X, with a ∈ R,
• p * X, with p ∈ R[X],
• p + a, with a ∈ R, p ∈ R[X], with appropriate restrictions on each term.

See nat_degree_ne_zero_induction_on for a similar statement involving no explicit multiplication.

theorem polynomial.nat_degree_ne_zero_induction_on {R : Type u} [semiring R] {M : → Prop} {f : polynomial R} (f0 : f.nat_degree 0) (h_C_add : ∀ {a : R} {p : , M pM + p)) (h_add : ∀ {p q : , M pM qM (p + q)) (h_monomial : ∀ {n : } {a : R}, a 0n 0M ( a)) :
M f

A property holds for all polynomials of non-zero nat_degree with coefficients in a semiring R if it holds for

• p + a, with a ∈ R, p ∈ R[X],
• p + q, with p, q ∈ R[X],
• monomials with nonzero coefficient and non-zero exponent, with appropriate restrictions on each term. Note that multiplication is "hidden" in the assumption on monomials, so there is no explicit multiplication in the statement. See degree_pos_induction_on for a similar statement involving more explicit multiplications.