mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g
i j : σ →₀ ℕ
hij : (i, j) ∈ Finset.antidiagonal d
hi : weightedOrder w f ≤ ↑((weight w) i)
hj : ↑((weight w) j) < weightedOrder w g
⊢ (coeff (i, j).1) f * (coeff (i, j).2) g = 0

Tactic:
coeff_eq_zero_of_lt_weightedOrder w hj,

Goal state after:
case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g
i j : σ →₀ ℕ
hij : (i, j) ∈ Finset.antidiagonal d
hi : weightedOrder w f ≤ ↑((weight w) i)
hj : ↑((weight w) j) < weightedOrder w g
⊢ (coeff (i, j).1) f * 0 = 0
Module: Mathlib.RingTheory.MvPowerSeries.Trunc

Goal state before:
pp: σ : Type u_1
R : Type u_2
n : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
p : MvPolynomial σ R
h : n ≠ 0
⊢ (truncTotal n) ↑p = p ↔ p.totalDegree < n

Tactic:
rw [truncTotal, truncFinset_coe_eq_self_iff, Set.Finite.subset_toFinset,
    MvPolynomial.totalDegree, Finset.sup_lt_iff (bot_lt_iff_ne_bot.mpr h), Set.subset_def]

Goal state after:
σ : Type u_1
R : Type u_2
n : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
p : MvPolynomial σ R
h : n ≠ 0
⊢ (∀ x ∈ ↑p.support, x ∈ {f | degree f < n}) ↔ ∀ b ∈ p.support, (b.sum fun x e ↦ e) < n
Module: Mathlib.Topology.Algebra.TopologicallyNilpotent

Goal state before:
pp: R : Type u_1
inst✝² : TopologicalSpace R
inst✝¹ : Ring R
inst✝ : IsLinearTopology R R
a b : R
ha : IsTopologicallyNilpotent a
hb : IsTopologicallyNilpotent b
h : Commute a b
⊢ IsTopologicallyNilpotent (a + b)

Tactic:
simp only [IsTopologicallyNilpotent, atTop_basis.tendsto_iff IsLinearTopology.hasBasis_ideal,
    true_and]

Goal state after:
R : Type u_1
inst✝² : TopologicalSpace R
inst✝¹ : Ring R
inst✝ : IsLinearTopology R R
a b : R
ha : IsTopologicallyNilpotent a
hb : IsTopologicallyNilpotent b
h : Commute a b
⊢ ∀ (ib : Ideal R), ↑ib ∈ 𝓝 0 → ∃ ia, ∀ x ∈ Set.Ici ia, (a + b) ^ x ∈ ↑ib
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
⊢ DenseRange ((fun x ↦ { toCompletion := ↑((WithVal.equiv (valuation K v)).symm x) }) ∘ ⇑(algebraMap K K))

Tactic:
exact (adicCompletion.ofCompletion_surjective K v).denseRange.comp
    (UniformSpace.Completion.denseRange_coe.comp (WithVal.equiv _).symm.surjective.denseRange
      (UniformSpace.Completion.continuous_coe _))
    (adicCompletion.continuous_ofCompletion K v)

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

Goal state before:
pp: case inr
σ : Type u_1
R : Type u_2
inst✝³ : TopologicalSpace R
inst✝² : Semiring R
ι : Type u_3
f : ι → MvPowerSeries σ R
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrderBot ι
w : σ → ℕ
h : Tendsto (fun i ↦ weightedOrder w (f i)) atTop (𝓝 ⊤)
hempty : Nonempty ι
⊢ ∀ (d : σ →₀ ℕ), Summable fun i ↦ (coeff d) (f i)

Tactic:
simp_rw [ENat.tendsto_nhds_top_iff_natCast_lt, Filter.eventually_atTop] at h

Goal state after:
case inr
σ : Type u_1
R : Type u_2
inst✝³ : TopologicalSpace R
inst✝² : Semiring R
ι : Type u_3
f : ι → MvPowerSeries σ R
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrderBot ι
w : σ → ℕ
hempty : Nonempty ι
h : ∀ (n : ℕ), ∃ a, ∀ (b : ι), a ≤ b → ↑n < weightedOrder w (f b)
⊢ ∀ (d : σ →₀ ℕ), Summable fun i ↦ (coeff d) (f i)
Module: Mathlib.RingTheory.MvPowerSeries.PiTopology

Goal state before:
pp: case inr
σ : Type u_1
R : Type u_2
inst✝³ : TopologicalSpace R
inst✝² : Semiring R
ι : Type u_3
f : ι → MvPowerSeries σ R
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrderBot ι
w : σ → ℕ
hempty : Nonempty ι
h : ∀ (n : ℕ), ∃ a, ∀ (b : ι), a ≤ b → ↑n < weightedOrder w (f b)
d : σ →₀ ℕ
i : ι
hi : ∀ (b : ι), i ≤ b → ↑((Finsupp.weight w) d) < weightedOrder w (f b)
⊢ (support fun i ↦ (coeff d) (f i)) ⊆ Set.Iic i

Tactic:
Function.support_subset_iff,

Goal state after:
case inr
σ : Type u_1
R : Type u_2
inst✝³ : TopologicalSpace R
inst✝² : Semiring R
ι : Type u_3
f : ι → MvPowerSeries σ R
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrderBot ι
w : σ → ℕ
hempty : Nonempty ι
h : ∀ (n : ℕ), ∃ a, ∀ (b : ι), a ≤ b → ↑n < weightedOrder w (f b)
d : σ →₀ ℕ
i : ι
hi : ∀ (b : ι), i ≤ b → ↑((Finsupp.weight w) d) < weightedOrder w (f b)
⊢ ∀ (x : ι), (coeff d) (f x) ≠ 0 → x ∈ Set.Iic i
Module: Mathlib.RingTheory.PowerSeries.Trunc

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
n : ℕ
p : R⟦X⟧
⊢ (trunc n) (X ^ n * p) = 0

Tactic:
by
  ext; simp +contextual [coeff_trunc, coeff_X_pow_mul']

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

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
φ : R⟦X⟧
⊢ X ^ φ.order.toNat ∣ φ

Tactic:
simpa only [X_pow_dvd_iff] using coeff_of_lt_order_toNat

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

Goal state before:
pp: case inr
R : Type u_1
inst✝³ : TopologicalSpace R
inst✝² : Semiring R
ι : Type u_2
f : ι → R⟦X⟧
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrderBot ι
h : Tendsto (fun i ↦ (f i).order) atTop (𝓝 ⊤)
hempty : Nonempty ι
n : ℕ
⊢ Summable fun i ↦ (coeff n) (f i)

Tactic:
simp_rw [ENat.tendsto_nhds_top_iff_natCast_lt, Filter.eventually_atTop] at h

Goal state after:
case inr
R : Type u_1
inst✝³ : TopologicalSpace R
inst✝² : Semiring R
ι : Type u_2
f : ι → R⟦X⟧
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrderBot ι
hempty : Nonempty ι
n : ℕ
h : ∀ (n : ℕ), ∃ a, ∀ (b : ι), a ≤ b → ↑n < (f b).order
⊢ Summable fun i ↦ (coeff n) (f i)
Module: Mathlib.RingTheory.PowerSeries.PiTopology

Goal state before:
pp: case inr
R : Type u_1
inst✝³ : TopologicalSpace R
inst✝² : Semiring R
ι : Type u_2
f : ι → R⟦X⟧
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrderBot ι
hempty : Nonempty ι
n : ℕ
h : ∀ (n : ℕ), ∃ a, ∀ (b : ι), a ≤ b → ↑n < (f b).order
i : ι
hi : ∀ (b : ι), i ≤ b → ↑n < (f b).order
⊢ (support fun i ↦ (coeff n) (f i)) ⊆ Set.Iic i

Tactic:
Function.support_subset_iff,

Goal state after:
case inr
R : Type u_1
inst✝³ : TopologicalSpace R
inst✝² : Semiring R
ι : Type u_2
f : ι → R⟦X⟧
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrderBot ι
hempty : Nonempty ι
n : ℕ
h : ∀ (n : ℕ), ∃ a, ∀ (b : ι), a ≤ b → ↑n < (f b).order
i : ι
hi : ∀ (b : ι), i ≤ b → ↑n < (f b).order
⊢ ∀ (x : ι), (coeff n) (f x) ≠ 0 → x ∈ Set.Iic i
Module: Mathlib.RingTheory.MvPowerSeries.Evaluation

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝³ : CommRing R
inst✝² : UniformSpace R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : UniformSpace S
φ : R →+* S
a : σ → S
f : MvPolynomial σ R
⊢ eval₂ φ a ↑f = MvPolynomial.eval₂ φ a f

Tactic:
by
  have : ∃ p : MvPolynomial σ R, (p : MvPowerSeries σ R) = f := ⟨f, rfl⟩
  rw [eval₂, dif_pos this]
  congr
  rw [← MvPolynomial.coe_inj, this.choose_spec]

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

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝² : CommRing R
inst✝¹ : TopologicalSpace R
inst✝ : IsLinearTopology R R
f : MvPowerSeries σ R
d : σ →₀ ℕ
I : Ideal R
hI : ↑I ∈ 𝓝 0
hf : ∃ a, ∀ (b : ℕ), a ≤ b → (Ideal.Quotient.mk I) (constantCoeff f) ^ b = 0
⊢ ∃ a, ∀ (b : ℕ), a ≤ b → (Ideal.Quotient.mk I) ((coeff d) (f ^ b)) = 0

Tactic:
← coeff_map,

Goal state after:
σ : Type u_1
R : Type u_3
inst✝² : CommRing R
inst✝¹ : TopologicalSpace R
inst✝ : IsLinearTopology R R
f : MvPowerSeries σ R
d : σ →₀ ℕ
I : Ideal R
hI : ↑I ∈ 𝓝 0
hf : ∃ a, ∀ (b : ℕ), a ≤ b → (Ideal.Quotient.mk I) (constantCoeff f) ^ b = 0
⊢ ∃ a, ∀ (b : ℕ), a ≤ b → (coeff d) ((map (Ideal.Quotient.mk I)) (f ^ b)) = 0
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: case zero
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n ↦ rec false (fun x x_1 ↦ true) n
hf_max : ∀ (n : ℕ), f n = true → f_max n = true
g_min : ℕ → Bool := fun n ↦ rec true (fun x x_1 ↦ false) n
hg_min : ∀ (n : ℕ), g_min n = true → g n = true
⊢ cantorFunction c f < cantorFunction c g

Tactic:
apply (cantorFunction_le (le_of_lt h1) h3 hf_max).trans_lt

Goal state after:
case zero
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := ⋯
hf_max : ∀ (n : ℕ), f n = true → f_max n = true
g_min : ℕ → Bool := ⋯
hg_min : ∀ (n : ℕ), g_min n = true → g n = true
⊢ cantorFunction c f_max < cantorFunction c g
Module: Mathlib.LinearAlgebra.Complex.FiniteDimensional

Goal state before:
pp: ⊢ Cardinal.lift.{u, 0} (Module.rank ℝ ℂ) = 2

Tactic:
rw [← finrank_eq_rank, finrank_real_complex, Cardinal.lift_natCast, Nat.cast_ofNat]

Goal state after:
Module: Mathlib.NumberTheory.LSeries.Basic

Goal state before:
pp: f : ℕ → ℂ
hf : f 0 = 0
s : ℂ
n : ℕ
⊢ term f s n = f n * ↑n ^ (-s)

Tactic:
LSeries.term

Goal state after:
f : ℕ → ℂ
hf : f 0 = 0
s : ℂ
n : ℕ
⊢ (if n = 0 then 0 else f n / ↑n ^ s) = f n * ↑n ^ (-s)
Module: Mathlib.NumberTheory.LSeries.Basic

Goal state before:
pp: f g : ℕ → ℂ
s : ℂ
n : ℕ
h : ‖f n‖ ≤ ‖g n‖
⊢ ‖term f s n‖ ≤ ‖term g s n‖

Tactic:
by
  simp only [norm_term_eq]
  split
  · rfl
  · gcongr

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

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
τ : Type u_3
f g : MvPowerSeries τ R
hf : HasSubst f
⊢ HasSubst (f * g)

Tactic:
simp only [HasSubst, map_mul]
  exact (Commute.all _ _).isNilpotent_mul_right hf

Goal state after:
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
q : ℕ
hq : IsPrimePow q
f : PowerSeries R
hf : PowerSeries.constantCoeff f = 1
⊢ ((ofPowerSeries q) f).IsMultiplicative

Tactic:
refine ⟨(ofPowerSeries_apply_one q f).trans hf, fun {m n} hmn ↦ ?_⟩

Goal state after:
R : Type u_1
inst✝ : CommRing R
q : ℕ
hq : IsPrimePow q
f : PowerSeries R
hf : PowerSeries.constantCoeff f = 1
m n : ℕ
hmn : m.Coprime n
⊢ ((ofPowerSeries q) f) (m * n) = ((ofPowerSeries q) f) m * ((ofPowerSeries q) f) n
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝⁴ : CommRing R
τ : Type u_4
S : Type u_5
inst✝³ : CommRing S
inst✝² : Algebra R S
a : σ → MvPowerSeries τ S
f : MvPowerSeries σ R
inst✝¹ : Finite τ
x : σ → ℕ
k : ℕ
inst✝ : Finite σ
h : ∀ (i : σ), constantCoeff (a i) = 0
hx : ∀ (i : σ), k ≤ x i
⊢ (truncTotal k) (subst a f) = (truncTotal k) (subst (fun i ↦ ↑((truncTotal (x i)) (a i))) ↑((truncTotal k) f))

Tactic:
truncTotal_subst_eq_truncTotal_sum_subst_truncTotal_of_le
      (hasSubst_of_constantCoeff_zero h) h hx,

Goal state after:
σ : Type u_1
R : Type u_3
inst✝⁴ : CommRing R
τ : Type u_4
S : Type u_5
inst✝³ : CommRing S
inst✝² : Algebra R S
a : σ → MvPowerSeries τ S
f : MvPowerSeries σ R
inst✝¹ : Finite τ
x : σ → ℕ
k : ℕ
inst✝ : Finite σ
h : ∀ (i : σ), constantCoeff (a i) = 0
hx : ∀ (i : σ), k ≤ x i
⊢ (truncTotal k) (∑ i ∈ range k, subst (fun i ↦ ↑((truncTotal (x i)) (a i))) ((homogeneousComponent i) f)) =
    (truncTotal k) (subst (fun i ↦ ↑((truncTotal (x i)) (a i))) ↑((truncTotal k) f))
Module: Mathlib.Analysis.Convex.EGauge

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NormedDivisionRing 𝕜
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
F : Type u_3
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
U : Set E
V : Set F
hU : Balanced 𝕜 U
hV : Balanced 𝕜 V
a : E
b : F
r : ℝ≥0∞
hr : (∃ c, a ∈ c • U ∧ ‖c‖ₑ < r) ∧ ∃ c, b ∈ c • V ∧ ‖c‖ₑ < r
⊢ ∃ c, (a, b) ∈ (c • U) ×ˢ (c • V) ∧ ‖c‖ₑ < r

Tactic:
rcases hr with ⟨⟨x, hx, hxr⟩, ⟨y, hy, hyr⟩⟩

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NormedDivisionRing 𝕜
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
F : Type u_3
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
U : Set E
V : Set F
hU : Balanced 𝕜 U
hV : Balanced 𝕜 V
a : E
b : F
r : ℝ≥0∞
x : 𝕜
hx : a ∈ x • U
hxr : ‖x‖ₑ < r
y : 𝕜
hy : b ∈ y • V
hyr : ‖y‖ₑ < r
⊢ ∃ c, (a, b) ∈ (c • U) ×ˢ (c • V) ∧ ‖c‖ₑ < r
Module: Mathlib.Analysis.Calculus.TangentCone.DimOne

Goal state before:
pp: case hds
𝕜 : Type u_1
inst✝ : NormedDivisionRing 𝕜
s : Set 𝕜
x : 𝕜
hx : AccPt x (𝓟 s)
y : 𝕜
⊢ ∃ᶠ (n : 𝕜) in 𝓝[≠] x, x + (n - x) ∈ s

Tactic:
simpa [accPt_iff_frequently_nhdsNE] using hx

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Congr

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
F : Type u_3
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 F
inst✝ : TopologicalSpace F
f f₁ : E → F
x : E
h : f₁ =ᶠ[𝓝 x] f
⊢ fderivWithin 𝕜 f₁ univ x = fderivWithin 𝕜 f univ x

Tactic:
h.fderivWithin_eq_of_nhds

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
F : Type u_3
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 F
inst✝ : TopologicalSpace F
f f₁ : E → F
x : E
h : f₁ =ᶠ[𝓝 x] f
⊢ fderivWithin 𝕜 f univ x = fderivWithin 𝕜 f univ x
Module: Mathlib.Analysis.Calculus.TangentCone.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁶ : DivisionSemiring 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace 𝕜
inst✝¹ : (𝓝[≠] 0).NeBot
inst✝ : ContinuousSMul 𝕜 E
x : E
⊢ UniqueDiffWithinAt 𝕜 univ x

Tactic:
by
  rw [uniqueDiffWithinAt_iff, tangentConeAt_univ]
  simp

Goal state after:
Module: Mathlib.Analysis.Asymptotics.TVS

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : Module 𝕜 E
inst✝² : AddCommGroup F
inst✝¹ : TopologicalSpace F
inst✝ : Module 𝕜 F
f₁ f₂ : α → E
g₁ g₂ : α → F
l : Filter α
hf : f₁ =ᶠ[l] f₂
hg : g₁ =ᶠ[l] g₂
⊢ f₁ =O[𝕜; l] g₁ ↔ f₂ =O[𝕜; l] g₂

Tactic:
simp only [isBigOTVS_iff]

Goal state after:
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : Module 𝕜 E
inst✝² : AddCommGroup F
inst✝¹ : TopologicalSpace F
inst✝ : Module 𝕜 F
f₁ f₂ : α → E
g₁ g₂ : α → F
l : Filter α
hf : f₁ =ᶠ[l] f₂
hg : g₁ =ᶠ[l] g₂
⊢ (∀ U ∈ 𝓝 0, ∃ V ∈ 𝓝 0, (fun x ↦ egauge 𝕜 U (f₁ x)) ≤ᶠ[l] fun x ↦ egauge 𝕜 V (g₁ x)) ↔
    ∀ U ∈ 𝓝 0, ∃ V ∈ 𝓝 0, (fun x ↦ egauge 𝕜 U (f₂ x)) ≤ᶠ[l] fun x ↦ egauge 𝕜 V (g₂ x)
Module: Mathlib.Analysis.Analytic.ConvergenceRadius

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
F : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0
this :
  (∃ a ∈ Ioo 0 1, (fun n ↦ ‖p n‖ * ↑r ^ n) =o[atTop] fun x ↦ a ^ x) ↔
    ∃ a < 1, ∃ C, (0 < C ∨ 0 < 1) ∧ ∀ (n : ℕ), |‖p n‖ * ↑r ^ n| ≤ C * a ^ n
t : ℝ≥0
C : ℝ
hC : ∀ (n : ℕ), ‖p n‖ * ↑t ^ n ≤ C
rt : ↑r < ↑t
⊢ ∃ a < 1, ∃ C, (0 < C ∨ 0 < 1) ∧ ∀ (n : ℕ), |‖p n‖ * ↑r ^ n| ≤ C * a ^ n

Tactic:
have : 0 < (t : ℝ) := r.coe_nonneg.trans_lt rt

Goal state after:
𝕜 : Type u_1
E : Type u_3
F : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0
this✝ :
  (∃ a ∈ Ioo 0 1, (fun n ↦ ‖p n‖ * ↑r ^ n) =o[atTop] fun x ↦ a ^ x) ↔
    ∃ a < 1, ∃ C, (0 < C ∨ 0 < 1) ∧ ∀ (n : ℕ), |‖p n‖ * ↑r ^ n| ≤ C * a ^ n
t : ℝ≥0
C : ℝ
hC : ∀ (n : ℕ), ‖p n‖ * ↑t ^ n ≤ C
rt : ↑r < ↑t
this : 0 < ↑t
⊢ ∃ a < 1, ∃ C, (0 < C ∨ 0 < 1) ∧ ∀ (n : ℕ), |‖p n‖ * ↑r ^ n| ≤ C * a ^ n
Module: Mathlib.Analysis.Analytic.ConvergenceRadius

Goal state before:
pp: case mpr
𝕜 : Type u_1
E : Type u_3
F : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
⊢ (∀ (r : ℝ≥0), Summable fun n ↦ ‖p n‖ * ↑r ^ n) → p.radius = ∞

Tactic:
exact p.radius_eq_top_of_summable_norm

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Multilinear.Curry

Goal state before:
pp: 𝕜 : Type u
n : ℕ
Ei : Fin n.succ → Type wEi
G : Type wG
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝² : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : Ei i.castSucc [×n]→L[𝕜] Ei (last n) →L[𝕜] G
m : (i : Fin n.succ) → Ei i
⊢ ‖(f (init m)) (m (last n))‖ ≤ ‖f‖ * ∏ i, ‖m i‖

Tactic:
prod_univ_castSucc,

Goal state after:
𝕜 : Type u
n : ℕ
Ei : Fin n.succ → Type wEi
G : Type wG
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝² : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : Ei i.castSucc [×n]→L[𝕜] Ei (last n) →L[𝕜] G
m : (i : Fin n.succ) → Ei i
⊢ ‖(f (init m)) (m (last n))‖ ≤ ‖f‖ * ((∏ i, ‖m i.castSucc‖) * ‖m (last n)‖)
Module: Mathlib.Analysis.Analytic.CPolynomialDef

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
x : E
m : ℕ
hm : m ∉ Finset.range n
⊢ ((p m) fun x_1 ↦ x) = 0

Tactic:
by rw [Finset.mem_range, not_lt] at hm; rw [hn m hm]; rfl

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f g : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : EqOn g f (s ∩ Metric.eball x r)
h'' : g x = f x
⊢ HasFPowerSeriesWithinOnBall g p s x r

Tactic:
refine ⟨h.r_le, h.r_pos, ?_⟩
  intro y hy h'y
  convert h.hasSum hy h'y
  simp only [mem_insert_iff, add_eq_left] at hy
  rcases hy with rfl | hy
  · simpa using h''
  · apply h'
    refine ⟨hy, ?_⟩
    simpa [edist_eq_enorm_sub] using h'y

Goal state after:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing E
inst✝¹ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
inst✝ : NormOneClass E
r : ℝ≥0∞
hc' : Tendsto (fun n ↦ ENNReal.ofReal ‖c n.succ‖ / ENNReal.ofReal ‖c n‖) atTop (𝓝 r)
hr : 0 < r.toReal
hr' : r ≠ 0 ∧ r ≠ ∞
hr'' : r.toNNReal ≠ 0
⊢ r⁻¹ = ↑r.toNNReal⁻¹

Tactic:
simp [ENNReal.coe_inv hr'', ENNReal.coe_toNNReal (toReal_ne_zero.mp hr.ne.symm).2]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
y : E
hf : HasFPowerSeriesWithinOnBall f p s x r
hy : y ∈ Metric.eball 0 r
h'y : x + y ∈ insert x s
A✝ : Tendsto (fun z ↦ p.partialSum z.1 y) (atTop ×ˢ 𝓝 y) (𝓝 (f (x + y)))
ε : ℝ
εpos : ε > 0
r' : ℝ≥0
yr' : ‖y‖₊ < r'
r'r : ↑r' < r
yr'_2 : ‖y‖ < ↑r'
S : Summable fun n ↦ ‖p n‖ * ↑r' ^ n
k : ℕ
hk : ∑' (n : ℕ), ‖p (n + k)‖ * ↑r' ^ (n + k) < ε / 4
A : ∀ᶠ (z : ℕ × E) in atTop ×ˢ 𝓝 y, dist (p.partialSum k z.2) (p.partialSum k y) < ε / 4
⊢ ∀ᶠ (x : ℕ × E) in atTop ×ˢ 𝓝 y, dist (p.partialSum x.1 x.2 - p.partialSum x.1 y) 0 < ε

Tactic:
have B : ∀ᶠ (z : ℕ × E) in atTop ×ˢ 𝓝 y, ‖z.2‖₊ < r' := by
    suffices ∀ᶠ (z : E) in 𝓝 y, ‖z‖₊ < r' from tendsto_snd this
    have : Metric.ball 0 r' ∈ 𝓝 y := Metric.isOpen_ball.mem_nhds (by simpa using yr'_2)
    filter_upwards [this] with a ha using by simpa [← coe_nnnorm] using ha

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
y : E
hf : HasFPowerSeriesWithinOnBall f p s x r
hy : y ∈ Metric.eball 0 r
h'y : x + y ∈ insert x s
A✝ : Tendsto (fun z ↦ p.partialSum z.1 y) (atTop ×ˢ 𝓝 y) (𝓝 (f (x + y)))
ε : ℝ
εpos : ε > 0
r' : ℝ≥0
yr' : ‖y‖₊ < r'
r'r : ↑r' < r
yr'_2 : ‖y‖ < ↑r'
S : Summable fun n ↦ ‖p n‖ * ↑r' ^ n
k : ℕ
hk : ∑' (n : ℕ), ‖p (n + k)‖ * ↑r' ^ (n + k) < ε / 4
A : ∀ᶠ (z : ℕ × E) in atTop ×ˢ 𝓝 y, dist (p.partialSum k z.2) (p.partialSum k y) < ε / 4
B : ∀ᶠ (z : ℕ × E) in atTop ×ˢ 𝓝 y, ‖z.2‖₊ < r'
⊢ ∀ᶠ (x : ℕ × E) in atTop ×ˢ 𝓝 y, dist (p.partialSum x.1 x.2 - p.partialSum x.1 y) 0 < ε
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
x : E
hf : HasFPowerSeriesAt f p x
n : ℕ
⊢ (fun y ↦ f (x + y) - p.partialSum n y) =O[𝓝 0] fun y ↦ ‖y‖ ^ n

Tactic:
by
  rw [← hasFPowerSeriesWithinAt_univ] at hf
  simpa using hf.isBigO_sub_partialSum_pow n

Goal state after:
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: case inl
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
v0 : Fin 0 → E
⊢ (id 𝕜 F ((p 0) v0)).comp p 0 = p 0

Tactic:
ext v
    simp only [comp_coeff_zero', id_apply_zero]
    congr with i
    exact i.elim0

Goal state after:
Module: Mathlib.Analysis.Analytic.Within

Goal state before:
pp: case mp.refine_1.refine_2.refine_3
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
y : E
ys : y ∈ insert x s
yb : ‖y - x‖ₑ < r
e0 : HasSum (fun n ↦ (p n) fun x_1 ↦ y - x) (p.sum (y - x))
e1 : HasSum (fun n ↦ (p n) fun x_1 ↦ y - x) (f (x + (y - x)))
⊢ f y = (fun y ↦ p.sum (y - x)) y

Tactic:
simp only [add_sub_cancel] at e1

Goal state after:
case mp.refine_1.refine_2.refine_3
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
y : E
ys : y ∈ insert x s
yb : ‖y - x‖ₑ < r
e0 : HasSum (fun n ↦ (p n) fun x_1 ↦ y - x) (p.sum (y - x))
e1 : HasSum (fun n ↦ (p n) fun x_1 ↦ y - x) (f y)
⊢ f y = (fun y ↦ p.sum (y - x)) y
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
m n : ℕ
g : F → G
f : E → F
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
hg : HasFiniteFPowerSeriesAt g q (f x) m
hf : HasFiniteFPowerSeriesAt f p x n
hn : 0 < n
r : ℝ≥0∞
hr : HasFPowerSeriesOnBall (g ∘ f) (q.comp p) x r
i : ℕ
c : Composition i
v : Fin i → E
hc : c.length < m
hi : ∀ (j : Fin c.length), c.blocksFun j < n
⊢ i < m * n

Tactic:
rw [← c.sum_blocksFun]

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
m n : ℕ
g : F → G
f : E → F
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
hg : HasFiniteFPowerSeriesAt g q (f x) m
hf : HasFiniteFPowerSeriesAt f p x n
hn : 0 < n
r : ℝ≥0∞
hr : HasFPowerSeriesOnBall (g ∘ f) (q.comp p) x r
i : ℕ
c : Composition i
v : Fin i → E
hc : c.length < m
hi : ∀ (j : Fin c.length), c.blocksFun j < n
⊢ ∑ i_1, c.blocksFun i_1 < m * n