mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Polynomial.CauchyBound

Goal state before:
pp: K : Type u_1
inst✝ : NormedDivisionRing K
p : K[X]
hp : p ≠ 0
a : K
h : ‖p.leadingCoeff‖₊ * ‖a‖₊ ^ p.natDegree ≤ ‖∑ i ∈ range p.natDegree, p.coeff i * a ^ i‖₊
pld : ‖p.leadingCoeff‖₊ ≠ 0
⊢ (∑ x ∈ range p.natDegree, ‖p.leadingCoeff‖₊ * (p.cauchyBound - 1) * ‖a‖₊ ^ x) / ‖p.leadingCoeff‖₊ =
    (p.cauchyBound - 1) * ∑ x ∈ range p.natDegree, ‖a‖₊ ^ x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
⊢ (fun x ↦ eval x P / eval x Q) ~[atBot] fun x ↦ P.leadingCoeff / Q.leadingCoeff * x ^ (↑P.natDegree - ↑Q.natDegree)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Polynomial.Fourier

Goal state before:
pp: p : ℂ[X]
this :
  ∑ x ∈ p.support, ‖p.coeff x‖ ^ 2 =
    ∫ (t : AddCircle (2 * π)), ‖↑(ContinuousMap.toAEEqFun haarAddCircle (toAddCircle p)) t‖ ^ 2 ∂haarAddCircle
⊢ ∫ (t : AddCircle (2 * π)), ‖(toAddCircle p) t‖ ^ 2 ∂haarAddCircle =
    ∫ (t : AddCircle (2 * π)), ‖↑(ContinuousMap.toAEEqFun haarAddCircle (toAddCircle p)) t‖ ^ 2 ∂haarAddCircle

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : P.degree < Q.degree
⊢ Tendsto (fun x ↦ eval x P / eval x Q) atBot (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Polynomial.CauchyBound

Goal state before:
pp: K : Type u_1
inst✝ : NormedDivisionRing K
p : K[X]
hp : p ≠ 0
a : K
h : p.IsRoot a
⊢ ‖a‖₊ < p.cauchyBound

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E → E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ x : E
a r L K : ℝ≥0
hf : IsPicardLindelof f t₀ x₀ a r L K
hx : x ∈ closedBall x₀ ↑r
α : FunSpace t₀ x₀ r L
m : ℕ
C : ℝ≥0
hm : LipschitzWith C (next hf hx)^[m]
n : ℕ
⊢ dist α ((next hf hx)^[m]^[n] α) ≤
    ((∑ i ∈ Finset.range m, (↑K * max (tmax - ↑t₀) (↑t₀ - tmin)) ^ i / ↑i !) * ∑ i ∈ Finset.range n, ↑C ^ i) *
      dist α (next hf hx α)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hQ : Q ≠ 0
h : Tendsto (fun x ↦ eval x P / eval x Q) atBot (𝓝 0)
⊢ Q.comp (-X) ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hQ : Q ≠ 0
⊢ Tendsto (fun x ↦ eval x P / eval x Q) atBot (𝓝 0) ↔ P.degree < Q.degree

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : P.degree = Q.degree
⊢ Tendsto (fun x ↦ eval x P / eval x Q) atBot (𝓝 (P.leadingCoeff / Q.leadingCoeff))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℝ → E → E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ : E
a r L K : ℝ≥0
inst✝ : CompleteSpace E
hf : IsPicardLindelof f t₀ x₀ a r L K
m : ℕ
C : ℝ≥0
h : ∀ (x : E) (hx : x ∈ closedBall x₀ ↑r), ContractingWith C (next hf hx)^[m]
L' : ℝ := (∑ i ∈ Finset.range m, (↑K * max (tmax - ↑t₀) (↑t₀ - tmin)) ^ i / ↑i !) * ↑(1 - C)⁻¹
⊢ 0 ≤ L'

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : (Q.comp (-X)).degree < (P.comp (-X)).degree
hQ : Q ≠ 0
⊢ Q.comp (-X) ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : Q.degree < P.degree
hQ : Q ≠ 0
⊢ Tendsto (fun x ↦ |eval x P / eval x Q|) atBot atTop

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Polynomial.Fourier

Goal state before:
pp: p : ℂ[X]
⊢ ∑ i ∈ p.support, ‖p.coeff i‖ ^ 2 = Real.circleAverage (fun θ ↦ ‖eval θ p‖ ^ 2) 0 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
h : P.degree < Q.degree
⊢ (fun x ↦ eval x P) =o[atTop] fun x ↦ eval x Q

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
h : P.degree < Q.degree
⊢ (fun x ↦ eval x P) =o[atBot] fun x ↦ eval x Q

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.GaussNorm

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
v : R → ℝ
c : ℝ
f : R⟦X⟧
⊢ gaussNorm v c f = ⨆ i, v ((coeff i) f) * c ^ i

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.GaussNorm

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
v : R → ℝ
c : ℝ
f : R⟦X⟧
h : HasGaussNorm v c f
this :
  (Set.range fun t ↦ v ((coeff t) f) * c ^ t) =
    Set.range fun t ↦ v ((MvPowerSeries.coeff t) f) * t.prod fun x x2 ↦ c ^ x2
⊢ MvPowerSeries.HasGaussNorm v (fun x ↦ c) f

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.GaussNorm

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
v : R → ℝ
c : ℝ
f : R⟦X⟧
h : HasGaussNorm v c f
x✝ : ℝ
⊢ (x✝ ∈ Set.range fun t ↦ v ((coeff t) f) * c ^ t) ↔
    x✝ ∈ Set.range fun t ↦ v ((MvPowerSeries.coeff t) f) * t.prod fun x x2 ↦ c ^ x2

Tactic:
simp only [Set.mem_range, Finsupp.prod_pow, Finset.univ_unique, PUnit.default_eq_unit,
    Finset.prod_singleton]

Goal state after:
R : Type u_1
inst✝ : Semiring R
v : R → ℝ
c : ℝ
f : R⟦X⟧
h : HasGaussNorm v c f
x✝ : ℝ
⊢ (∃ y, v ((coeff y) f) * c ^ y = x✝) ↔ ∃ y, v ((MvPowerSeries.coeff y) f) * c ^ y PUnit.unit = x✝
Module: Mathlib.RingTheory.PowerSeries.GaussNorm

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
v : R → ℝ
c : ℝ
f : R⟦X⟧
h : HasGaussNorm v c f
⊢ MvPowerSeries.HasGaussNorm v (fun x ↦ c) f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
h : P.degree ≤ Q.degree
⊢ (fun x ↦ eval x P) =O[atTop] fun x ↦ eval x Q

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.GaussNorm

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
v : R → ℝ
c : ℝ
f : R⟦X⟧
hbd : HasGaussNorm v c f
t : ℕ
⊢ v ((coeff t) f) * c ^ t ≤ gaussNorm v c f

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.GaussNorm

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
v : R → ℝ
c : ℝ
f : R⟦X⟧
vZero : v 0 = 0
vNonneg : ∀ (a : R), v a ≥ 0
h_eq_zero : ∀ (x : R), v x = 0 → x = 0
hc : 0 < c
hbd : HasGaussNorm v c f
⊢ ∀ (i : Unit), 0 < c

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.GaussNorm

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
v : R → ℝ
c : ℝ
f g : R⟦X⟧
hc : 0 ≤ c
vNonneg : ∀ (a : R), v a ≥ 0
hv : ∀ (x y : R), v (x + y) ≤ max (v x) (v y)
hbfd : HasGaussNorm v c f
hbgd : HasGaussNorm v c g
x✝ : Unit
⊢ 0 x✝ ≤ (fun x ↦ c) x✝

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
h : P.degree ≤ Q.degree
⊢ (fun x ↦ eval x P) =O[atBot] fun x ↦ eval x Q

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: R : Type u_2
inst✝¹ : NormedRing R
inst✝ : NormMulClass R
P : R[X]
⊢ (fun x ↦ eval x P) ~[cobounded R] fun x ↦ P.leadingCoeff * x ^ P.natDegree

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: R : Type u_2
inst✝¹ : NormedRing R
inst✝ : NormMulClass R
P Q : R[X]
h : P.degree < Q.degree
⊢ (fun x ↦ eval x P) =o[cobounded R] fun x ↦ eval x Q

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℝ → E → E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ : E
a r L K : ℝ≥0
inst✝ : CompleteSpace E
hf : IsPicardLindelof f t₀ x₀ a r L K
⊢ ∃ L',
    ∀ (x y : E) (hx : x ∈ closedBall x₀ ↑r) (hy : y ∈ closedBall x₀ ↑r) (α β : FunSpace t₀ x₀ r L),
      IsFixedPt (next hf hx) α → IsFixedPt (next hf hy) β → dist α β ≤ ↑L' * dist x y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: R : Type u_2
inst✝¹ : NormedRing R
inst✝ : NormMulClass R
P Q : R[X]
h : P.degree ≤ Q.degree
⊢ (fun x ↦ eval x P) =O[cobounded R] fun x ↦ eval x Q

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℝ → E → E
α : ℝ → E
u : Set E
t₀ tmin tmax : ℝ
ht₀ : t₀ ∈ Icc tmin tmax
hf : ContinuousOn (uncurry f) (Icc tmin tmax ×ˢ u)
hα : ContinuousOn α (Icc tmin tmax)
hmem : ∀ t ∈ Icc tmin tmax, α t ∈ u
x₀ : E
t : ℝ
ht : t ∈ Icc tmin tmax
⊢ HasDerivWithinAt (picard f t₀ x₀ α) (f t (α t)) (Icc tmin tmax) t

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: P Q : ℤ[X]
h : Q.degree < P.degree
o : (fun x ↦ eval x Q) =o[cofinite] fun x ↦ eval x P
nr : ∀ᶠ (x : ℤ) in cofinite, ¬P.IsRoot x
⊢ ∀ (x : ℤ), ¬P.IsRoot x ↔ 0 < ‖eval x P‖

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussNorm

Goal state before:
pp: R : Type u_1
F : Type u_2
inst✝¹ : Semiring R
inst✝ : FunLike F R ℝ
v : F
c : ℝ
⊢ gaussNorm v c 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: P Q : ℤ[X]
h : Q.degree < P.degree
⊢ {x | |eval x P| ≤ |eval x Q|}.Finite

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussNorm

Goal state before:
pp: R : Type u_1
F : Type u_2
inst✝² : Semiring R
inst✝¹ : FunLike F R ℝ
v : F
c : ℝ
p : R[X]
inst✝ : ZeroHomClass F R ℝ
⊢ ∃ i, gaussNorm v c p = v (p.coeff i) * c ^ i

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussNorm

Goal state before:
pp: R : Type u_1
F : Type u_2
inst✝² : Semiring R
inst✝¹ : FunLike F R ℝ
v : F
c : ℝ
inst✝ : ZeroHomClass F R ℝ
r : R
⊢ gaussNorm v c (C r) = v r

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussNorm

Goal state before:
pp: R : Type u_1
F : Type u_2
inst✝² : Semiring R
inst✝¹ : FunLike F R ℝ
v : F
c : ℝ
inst✝ : ZeroHomClass F R ℝ
n : ℕ
r : R
⊢ gaussNorm v c ((monomial n) r) = v r * c ^ n

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussNorm

Goal state before:
pp: R : Type u_1
F : Type u_2
inst✝² : Semiring R
inst✝¹ : FunLike F R ℝ
v : F
c : ℝ
inst✝ : NonnegHomClass F R ℝ
p : R[X]
h : p.support.Nonempty
hc : 0 ≤ c
⊢ 0 ≤ p.support.sup' h fun i ↦ v (p.coeff i) * c ^ i

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussNorm

Goal state before:
pp: R : Type u_1
F : Type u_2
inst✝² : Semiring R
inst✝¹ : FunLike F R ℝ
v : F
c : ℝ
p : R[X]
inst✝ : ZeroHomClass F R ℝ
f : ↥p.support → ℝ := fun i ↦ v (p.coeff ↑i) * c ^ ↑i
⊢ (f '' ⊤ ∪ {0}).Finite

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℝ → E → E
α : ℝ → E
u : Set E
t₀ t : ℝ
hf : ContinuousOn (uncurry f) (uIcc t₀ t ×ˢ u)
hα : ∀ t' ∈ uIcc t₀ t, HasDerivWithinAt α (f t' (α t')) (uIcc t₀ t) t'
hmap : MapsTo α (uIcc t₀ t) u
⊢ picard f t₀ (α t₀) α t = α t

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussNorm

Goal state before:
pp: R : Type u_1
F : Type u_2
inst✝² : Semiring R
inst✝¹ : FunLike F R ℝ
v : F
c : ℝ
p : R[X]
inst✝ : ZeroHomClass F R ℝ
f : ↥p.support → ℝ := fun i ↦ v (p.coeff ↑i) * c ^ ↑i
h_fin : (f '' ⊤ ∪ {0}).Finite
⊢ BddAbove {x | ∃ i, v (p.coeff i) * c ^ i = x}

Tactic:
refine Set.Finite.bddAbove <| Set.Finite.subset h_fin fun _ ↦ ?_

Goal state after:
R : Type u_1
F : Type u_2
inst✝² : Semiring R
inst✝¹ : FunLike F R ℝ
v : F
c : ℝ
p : R[X]
inst✝ : ZeroHomClass F R ℝ
f : ↥p.support → ℝ := fun i ↦ v (p.coeff ↑i) * c ^ ↑i
h_fin : (f '' ⊤ ∪ {0}).Finite
x✝ : ℝ
⊢ x✝ ∈ {x | ∃ i, v (p.coeff i) * c ^ i = x} → x✝ ∈ f '' ⊤ ∪ {0}
Module: Mathlib.RingTheory.Polynomial.GaussNorm

Goal state before:
pp: R : Type u_1
F : Type u_2
inst✝² : Semiring R
inst✝¹ : FunLike F R ℝ
v : F
c : ℝ
p : R[X]
inst✝ : ZeroHomClass F R ℝ
⊢ BddAbove {x | ∃ i, v (p.coeff i) * c ^ i = x}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: P Q : ℤ[X]
mQ : Q.Monic
h : {a | eval a Q ∣ eval a P}.Infinite
⊢ Q ∣ P

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℝ → E → E
α : ℝ → E
u : Set E
t₀ tmin tmax : ℝ
ht₀ : t₀ ∈ Icc tmin tmax
hα : ContinuousOn α (Icc tmin tmax)
hmem : ∀ t ∈ Icc tmin tmax, α t ∈ u
x₀ : E
heqon : ∀ t ∈ Icc tmin tmax, α t = picard f t₀ x₀ α t
hlt : tmin < tmax
this : ∀ t ∈ Icc tmin tmax, HasDerivWithinAt (picard f t₀ x₀ α) (f t (α t)) (Icc tmin tmax) t
n : ℕ
hn : ContDiffOn ℝ (↑n) (uncurry f) (Icc tmin tmax ×ˢ u) → ContDiffOn ℝ (↑n) (picard f t₀ x₀ α) (Icc tmin tmax)
hf : ContDiffOn ℝ (↑n + 1) (uncurry f) (Icc tmin tmax ×ˢ u)
⊢ ↑n = ⊤ → AnalyticOn ℝ (picard f t₀ x₀ α) (Icc tmin tmax)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussNorm

Goal state before:
pp: R : Type u_1
F : Type u_2
inst✝³ : Semiring R
inst✝² : FunLike F R ℝ
v : F
c : ℝ
p : R[X]
inst✝¹ : ZeroHomClass F R ℝ
inst✝ : NonnegHomClass F R ℝ
hc : 0 ≤ c
⊢ PowerSeries.gaussNorm (⇑v) c ↑p = gaussNorm v c p

Tactic:
by_cases hp : p = 0
  · simp [hp]
  · simp only [PowerSeries.gaussNorm_eq, coeff_coe, gaussNorm, support_nonempty, ne_eq, hp,
      not_false_eq_true, ↓reduceDIte]
    apply le_antisymm
    · apply ciSup_le
      intro n
      by_cases h : n ∈ p.support
      · exact Finset.le_sup' (fun j ↦ v (p.coeff j) * c ^ j) h
      · simp_all [sup'_nonneg_of_ne_zero v (support_nonempty.mpr hp) hc]
    · obtain ⟨i, hi⟩ := exists_eq_gaussNorm v c p
      simp only [gaussNorm, support_nonempty.mpr hp, ↓reduceDIte] at hi
      rw [hi]
      exact le_ciSup (aux_bdd v p) i

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussNorm

Goal state before:
pp: R : Type u_1
F : Type u_2
inst✝³ : Semiring R
inst✝² : FunLike F R ℝ
v : F
c : ℝ
p : R[X]
inst✝¹ : ZeroHomClass F R ℝ
inst✝ : NonnegHomClass F R ℝ
hc : 0 ≤ c
⊢ PowerSeries.gaussNorm (⇑v) c ↑p = gaussNorm v c p

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussNorm

Goal state before:
pp: R : Type u_1
F : Type u_2
inst✝³ : Semiring R
inst✝² : FunLike F R ℝ
v : F
c : ℝ
p : R[X]
inst✝¹ : ZeroHomClass F R ℝ
inst✝ : NonnegHomClass F R ℝ
h_eq_zero : ∀ (x : R), v x = 0 → x = 0
hc : 0 < c
⊢ v 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussNorm

Goal state before:
pp: R : Type u_1
F : Type u_2
inst✝³ : Semiring R
inst✝² : FunLike F R ℝ
v : F
c : ℝ
p : R[X]
inst✝¹ : ZeroHomClass F R ℝ
inst✝ : NonnegHomClass F R ℝ
h_eq_zero : ∀ (x : R), v x = 0 → x = 0
hc : 0 < c
⊢ ∀ (a : R), v a ≥ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussNorm

Goal state before:
pp: R : Type u_1
F : Type u_2
inst✝³ : Semiring R
inst✝² : FunLike F R ℝ
v : F
c : ℝ
p : R[X]
inst✝¹ : ZeroHomClass F R ℝ
inst✝ : NonnegHomClass F R ℝ
h_eq_zero : ∀ (x : R), v x = 0 → x = 0
hc : 0 < c
⊢ PowerSeries.HasGaussNorm (⇑v) c ↑p

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.GaussNorm

Goal state before:
pp: R : Type u_1
σ : Type u_2
v : R → ℝ
c : σ → ℝ
inst✝ : Semiring R
vZero : v 0 = 0
⊢ gaussNorm v c 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussNorm

Goal state before:
pp: R : Type u_1
F : Type u_2
inst✝³ : Semiring R
inst✝² : FunLike F R ℝ
v : F
c : ℝ
p : R[X]
inst✝¹ : ZeroHomClass F R ℝ
inst✝ : NonnegHomClass F R ℝ
h_eq_zero : ∀ (x : R), v x = 0 → x = 0
hc : 0 < c
⊢ gaussNorm v c p = 0 ↔ p = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.GaussNorm

Goal state before:
pp: R : Type u_1
σ : Type u_2
v : R → ℝ
c : σ → ℝ
f : MvPowerSeries σ R
inst✝ : Semiring R
hbd : HasGaussNorm v c f
t : σ →₀ ℕ
⊢ (v ((coeff t) f) * t.prod fun x1 x2 ↦ c x1 ^ x2) ≤ gaussNorm v c f

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussNorm

Goal state before:
pp: R : Type u_1
F : Type u_2
inst✝² : Semiring R
inst✝¹ : FunLike F R ℝ
v : F
c : ℝ
p : R[X]
inst✝ : NonnegHomClass F R ℝ
hc : 0 ≤ c
⊢ 0 ≤ gaussNorm v c p

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussNorm

Goal state before:
pp: R : Type u_1
F : Type u_2
inst✝³ : Semiring R
inst✝² : FunLike F R ℝ
v : F
c : ℝ
p : R[X]
inst✝¹ : ZeroHomClass F R ℝ
inst✝ : NonnegHomClass F R ℝ
hc : 0 ≤ c
i : ℕ
⊢ v (p.coeff i) * c ^ i ≤ gaussNorm v c p

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussNorm

Goal state before:
pp: R : Type u_1
F : Type u_2
inst✝³ : Semiring R
inst✝² : FunLike F R ℝ
v : F
p : R[X]
inst✝¹ : ZeroHomClass F R ℝ
inst✝ : NonnegHomClass F R ℝ
⊢ (fun i ↦ v (p.coeff i) * 0 ^ i) = fun i ↦ if i = 0 then v (p.coeff 0) else 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℝ → E → E
α : ℝ → E
u : Set E
t₀ tmin tmax : ℝ
ht₀ : t₀ ∈ Icc tmin tmax
n : ℕ
hf : ContDiffOn ℝ (↑n) (uncurry f) (Icc tmin tmax ×ˢ u)
hα : ContinuousOn α (Icc tmin tmax)
hmem : ∀ t ∈ Icc tmin tmax, α t ∈ u
x₀ : E
heqon : ∀ t ∈ Icc tmin tmax, α t = picard f t₀ x₀ α t
⊢ ContDiffOn ℝ (↑n) (picard f t₀ x₀ α) (Icc tmin tmax)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℝ → E → E
α : ℝ → E
u : Set E
t₀ tmin tmax : ℝ
ht₀ : t₀ ∈ Icc tmin tmax
n : ℕ∞
hf : ContDiffOn ℝ (↑n) (uncurry f) (Icc tmin tmax ×ˢ u)
hα : ContinuousOn α (Icc tmin tmax)
hmem : ∀ t ∈ Icc tmin tmax, α t ∈ u
x₀ : E
heqon : ∀ t ∈ Icc tmin tmax, α t = picard f t₀ x₀ α t
⊢ ContDiffOn ℝ (↑n) (picard f t₀ x₀ α) (Icc tmin tmax)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussNorm

Goal state before:
pp: R : Type u_1
F : Type u_2
inst✝³ : Semiring R
inst✝² : FunLike F R ℝ
v : F
p : R[X]
inst✝¹ : ZeroHomClass F R ℝ
inst✝ : NonnegHomClass F R ℝ
⊢ gaussNorm v 0 p = v (p.coeff 0)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℝ → E → E
α : ℝ → E
u : Set E
tmin tmax : ℝ
n : ℕ∞
hf : ContDiffOn ℝ (↑n) (uncurry f) (Icc tmin tmax ×ˢ u)
hα : ∀ t ∈ Icc tmin tmax, HasDerivWithinAt α (f t (α t)) (Icc tmin tmax) t
hmem : MapsTo α (Icc tmin tmax) u
hlt : tmin < tmax
t₀ : ℝ := (tmin + tmax) / 2
h : t₀ = (tmin + tmax) / 2
⊢ tmin ≤ t₀

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℝ → E → E
α : ℝ → E
u : Set E
tmin tmax : ℝ
n : ℕ∞
hf : ContDiffOn ℝ (↑n) (uncurry f) (Icc tmin tmax ×ˢ u)
hα : ∀ t ∈ Icc tmin tmax, HasDerivWithinAt α (f t (α t)) (Icc tmin tmax) t
hmem : MapsTo α (Icc tmin tmax) u
hlt : tmin < tmax
t₀ : ℝ := (tmin + tmax) / 2
h : t₀ = (tmin + tmax) / 2
⊢ t₀ ≤ tmax

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussNorm

Goal state before:
pp: R : Type u_1
F : Type u_2
inst✝³ : Semiring R
inst✝² : FunLike F R ℝ
v : F
c : ℝ
inst✝¹ : ZeroHomClass F R ℝ
inst✝ : NonnegHomClass F R ℝ
p : R[X]
hc : 0 ≤ c
⊢ {i | gaussNorm v c p = v (p.coeff i) * c ^ i}.Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussNorm

Goal state before:
pp: R : Type u_1
F : Type u_2
inst✝³ : Semiring R
inst✝² : FunLike F R ℝ
v : F
c : ℝ
inst✝¹ : ZeroHomClass F R ℝ
inst✝ : NonnegHomClass F R ℝ
p : R[X]
hc : 0 ≤ c
⊢ ∃ i, gaussNorm v c p = v (p.coeff i) * c ^ i ∧ ∀ j < i, v (p.coeff j) * c ^ j < gaussNorm v c p

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℝ → E → E
α : ℝ → E
u : Set E
tmin tmax : ℝ
n : ℕ∞
hf : ContDiffOn ℝ (↑n) (uncurry f) (Icc tmin tmax ×ˢ u)
hα : ∀ t ∈ Icc tmin tmax, HasDerivWithinAt α (f t (α t)) (Icc tmin tmax) t
hmem : MapsTo α (Icc tmin tmax) u
hlt : tmin < tmax
t₀ : ℝ := (tmin + tmax) / 2
h : t₀ = (tmin + tmax) / 2
ht₀ : t₀ ∈ Icc tmin tmax
⊢ ∀ t ∈ Icc tmin tmax, α t = picard f t₀ (α t₀) α t

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.GaussNorm

Goal state before:
pp: R : Type u_1
σ : Type u_2
v : R → ℝ
c : σ → ℝ
f : MvPowerSeries σ R
inst✝ : Semiring R
vNonneg : ∀ (a : R), v a ≥ 0
⊢ 0 ≤ gaussNorm v c f

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.GaussNorm

Goal state before:
pp: R : Type u_1
σ : Type u_2
v : R → ℝ
c : σ → ℝ
f : MvPowerSeries σ R
inst✝ : Semiring R
vZero : v 0 = 0
vNonneg : ∀ (a : R), v a ≥ 0
h_eq_zero : ∀ (x : R), v x = 0 → x = 0
hc : ∀ (i : σ), 0 < c i
hbd : HasGaussNorm v c f
hf : f = 0
⊢ gaussNorm v c f = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussNorm

Goal state before:
pp: R : Type u_1
F : Type u_2
inst✝³ : Semiring R
inst✝² : FunLike F R ℝ
v : F
inst✝¹ : ZeroHomClass F R ℝ
inst✝ : NonnegHomClass F R ℝ
hna : IsNonarchimedean ⇑v
c : ℝ
hc : 0 ≤ c
p q : R[X]
h✝¹ : p ≠ 0
h✝ : q ≠ 0
hpq : p + q ≠ 0
i : ℕ
a✝ : i ∈ (p + q).support
⊢ v ((p + q).coeff i) * c ^ i ≤ max (v (p.coeff i)) (v (q.coeff i)) * c ^ i

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Polynomial.Norm

Goal state before:
pp: A : Type u_1
inst✝ : SeminormedRing A
p : A[X]
⊢ p.supNorm = if hp : p.support.Nonempty then p.support.sup' hp (norm ∘ p.coeff) else 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.GaussNorm

Goal state before:
pp: R : Type u_1
σ : Type u_2
v : R → ℝ
c : σ → ℝ
f : MvPowerSeries σ R
inst✝ : Semiring R
vZero : v 0 = 0
vNonneg : ∀ (a : R), v a ≥ 0
h_eq_zero : ∀ (x : R), v x = 0 → x = 0
hc : ∀ (i : σ), 0 < c i
hbd : HasGaussNorm v c f
hf : f ≠ 0
n : σ →₀ ℕ
hn : (coeff n) f ≠ 0
⊢ 0 < ∏ i ∈ n.support, c i ^ n i

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Polynomial.Norm

Goal state before:
pp: A : Type u_1
inst✝ : SeminormedRing A
p : A[X]
⊢ 0 ≤ p.supNorm

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussNorm

Goal state before:
pp: R : Type u_1
F : Type u_2
inst✝³ : Semiring R
inst✝² : FunLike F R ℝ
v : F
inst✝¹ : ZeroHomClass F R ℝ
inst✝ : NonnegHomClass F R ℝ
hna : IsNonarchimedean ⇑v
c : ℝ
hc : 0 ≤ c
p q : R[X]
h✝¹ : p ≠ 0
h✝ : q ≠ 0
hpq : p + q ≠ 0
i : ℕ
a✝ : i ∈ (p + q).support
⊢ max (v (p.coeff i)) (v (q.coeff i)) * c ^ i = max (v (p.coeff i) * c ^ i) (v (q.coeff i) * c ^ i)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Polynomial.Norm

Goal state before:
pp: A : Type u_1
inst✝ : SeminormedRing A
n : ℕ
a : A
⊢ ((monomial n) a).supNorm = ‖a‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Polynomial.Norm

Goal state before:
pp: A : Type u_1
inst✝¹ : SeminormedRing A
inst✝ : NormOneClass A
⊢ X.supNorm = 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Polynomial.Norm

Goal state before:
pp: A : Type u_1
inst✝ : SeminormedRing A
p : A[X]
i : ℕ
⊢ 0 ≤ 1

Tactic:
by

Goal state after: