File: Mathlib/Topology/Algebra/Polynomial.lean

full_name: Polynomial.continuous_eval₂
code: @[continuity, fun_prop]
protected theorem continuous_eval₂ [Semiring S] (p : S[X]) (f : S →+* R) :
    Continuous fun x => p.eval₂ f x 
goal: Continuous fun x => Polynomial.eval₂ f x p

full_name: Polynomial.continuous
code: @[continuity, fun_prop]
protected theorem continuous : Continuous fun x => p.eval x 
goal: Continuous fun x => Polynomial.eval x p

full_name: Polynomial.continuousAt
code: @[fun_prop]
protected theorem continuousAt {a : R} : ContinuousAt (fun x => p.eval x) a 
goal: ContinuousAt (fun x => Polynomial.eval x p) a

full_name: Polynomial.continuousWithinAt
code: @[fun_prop]
protected theorem continuousWithinAt {s a} : ContinuousWithinAt (fun x => p.eval x) s a 
goal: ContinuousWithinAt (fun x => Polynomial.eval x p) s a

full_name: Polynomial.continuousOn
code: @[fun_prop]
protected theorem continuousOn {s} : ContinuousOn (fun x => p.eval x) s 
goal: ContinuousOn (fun x => Polynomial.eval x p) s

full_name: Polynomial.continuous_aeval
code: @[continuity, fun_prop]
protected theorem continuous_aeval : Continuous fun x : A => aeval x p 
goal: Continuous fun x => (Polynomial.aeval x) p

full_name: Polynomial.continuousAt_aeval
code: @[fun_prop]
protected theorem continuousAt_aeval {a : A} : ContinuousAt (fun x : A => aeval x p) a 
goal: ContinuousAt (fun x => (Polynomial.aeval x) p) a

full_name: Polynomial.continuousWithinAt_aeval
code: @[fun_prop]
protected theorem continuousWithinAt_aeval {s a} :
    ContinuousWithinAt (fun x : A => aeval x p) s a 
goal: ContinuousWithinAt (fun x => (Polynomial.aeval x) p) s a

full_name: Polynomial.continuousOn_aeval
code: @[fun_prop]
protected theorem continuousOn_aeval {s} : ContinuousOn (fun x : A => aeval x p) s 
goal: ContinuousOn (fun x => (Polynomial.aeval x) p) s

full_name: Polynomial.tendsto_abv_eval₂_atTop
code: theorem tendsto_abv_eval₂_atTop {R S k α : Type*} [Semiring R] [Ring S] [LinearOrderedField k]
    (f : R →+* S) (abv : S → k) [IsAbsoluteValue abv] (p : R[X]) (hd : 0 < degree p)
    (hf : f p.leadingCoeff ≠ 0) {l : Filter α} {z : α → S} (hz : Tendsto (abv ∘ z) l atTop) :
    Tendsto (fun x => abv (p.eval₂ f (z x))) l atTop 
goal: Filter.Tendsto (fun x => abv (Polynomial.eval₂ f (z x) p)) l Filter.atTop

full_name: Polynomial.tendsto_abv_atTop
code: theorem tendsto_abv_atTop {R k α : Type*} [Ring R] [LinearOrderedField k] (abv : R → k)
    [IsAbsoluteValue abv] (p : R[X]) (h : 0 < degree p) {l : Filter α} {z : α → R}
    (hz : Tendsto (abv ∘ z) l atTop) : Tendsto (fun x => abv (p.eval (z x))) l atTop 
goal: Filter.Tendsto (fun x => abv (Polynomial.eval (z x) p)) l Filter.atTop

full_name: Polynomial.tendsto_abv_aeval_atTop
code: theorem tendsto_abv_aeval_atTop {R A k α : Type*} [CommSemiring R] [Ring A] [Algebra R A]
    [LinearOrderedField k] (abv : A → k) [IsAbsoluteValue abv] (p : R[X]) (hd : 0 < degree p)
    (h₀ : algebraMap R A p.leadingCoeff ≠ 0) {l : Filter α} {z : α → A}
    (hz : Tendsto (abv ∘ z) l atTop) : Tendsto (fun x => abv (aeval (z x) p)) l atTop 
goal: Filter.Tendsto (fun x => abv ((Polynomial.aeval (z x)) p)) l Filter.atTop

full_name: Polynomial.tendsto_norm_atTop
code: theorem tendsto_norm_atTop (p : R[X]) (h : 0 < degree p) {l : Filter α} {z : α → R}
    (hz : Tendsto (fun x => ‖z x‖) l atTop) : Tendsto (fun x => ‖p.eval (z x)‖) l atTop 
goal: Filter.Tendsto (fun x => ‖Polynomial.eval (z x) p‖) l Filter.atTop

full_name: Polynomial.exists_forall_norm_le
code: theorem exists_forall_norm_le [ProperSpace R] (p : R[X]) : ∃ x, ∀ y, ‖p.eval x‖ ≤ ‖p.eval y‖ 
goal: ∃ x, ∀ (y : R), ‖Polynomial.eval x p‖ ≤ ‖Polynomial.eval y p‖

full_name: Polynomial.eq_one_of_roots_le
code: theorem eq_one_of_roots_le {p : F[X]} {f : F →+* K} {B : ℝ} (hB : B < 0) (h1 : p.Monic)
    (h2 : Splits f p) (h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B) : p = 1 
goal: p = 1

full_name: Polynomial.coeff_le_of_roots_le
code: theorem coeff_le_of_roots_le {p : F[X]} {f : F →+* K} {B : ℝ} (i : ℕ) (h1 : p.Monic)
    (h2 : Splits f p) (h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B) :
    ‖(map f p).coeff i‖ ≤ B ^ (p.natDegree - i) * p.natDegree.choose i 
goal: ‖(Polynomial.map f p).coeff i‖ ≤ B ^ (p.natDegree - i) * ↑(p.natDegree.choose i)

full_name: Polynomial.coeff_bdd_of_roots_le
code: /-- The coefficients of the monic polynomials of bounded degree with bounded roots are
uniformly bounded. -/
theorem coeff_bdd_of_roots_le {B : ℝ} {d : ℕ} (f : F →+* K) {p : F[X]} (h1 : p.Monic)
    (h2 : Splits f p) (h3 : p.natDegree ≤ d) (h4 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B) (i : ℕ) :
    ‖(map f p).coeff i‖ ≤ max B 1 ^ d * d.choose (d / 2) 
goal: ‖(Polynomial.map f p).coeff i‖ ≤ max B 1 ^ d * ↑(d.choose (d / 2))