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

Goal state before:
pp: σ : Type u_1
R : Type u_2
S : Type u_3
T : Type u_4
inst✝² : Semiring R
inst✝¹ : Semiring S
inst✝ : Semiring T
f : R →+* S
g : S →+* T
n : σ →₀ ℕ
⊢ f ((coeff n) 1) = (coeff n) 1

Tactic:
coeff_one,

Goal state after:
σ : Type u_1
R : Type u_2
S : Type u_3
T : Type u_4
inst✝² : Semiring R
inst✝¹ : Semiring S
inst✝ : Semiring T
f : R →+* S
g : S →+* T
n : σ →₀ ℕ
⊢ f (if n = 0 then 1 else 0) = (coeff n) 1
Module: Mathlib.RingTheory.PowerSeries.Basic

Goal state before:
pp: case inr.succ
R : Type u_2
inst✝ : CommSemiring R
φ : R⟦X⟧
n' : ℕ
ih : n' > 0 → (coeff 1) (φ ^ n') = ↑n' * (coeff 1) φ * constantCoeff φ ^ (n' - 1)
hn : n' + 1 > 0
⊢ (coeff 1) (φ ^ (n' + 1)) = ↑(n' + 1) * (coeff 1) φ * constantCoeff φ ^ (n' + 1 - 1)

Tactic:
| succ n' ih =>

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f : MvPowerSeries σ R
n : ℕ∞
h : ∀ (d : σ →₀ ℕ), ↑((weight w) d) < n → (coeff d) f = 0
⊢ n ≤ weightedOrder w f

Tactic:
by
  cases n
  · rw [top_le_iff, weightedOrder_eq_top_iff]
    ext d; exact h d (ENat.natCast_lt_top _)
  · apply nat_le_weightedOrder;
    simpa only [ENat.some_eq_natCast, Nat.cast_lt] using h

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
⊢ min (weightedOrder w f) (weightedOrder w g) ≤ weightedOrder w (f + g)

Tactic:
apply le_weightedOrder w
  simp +contextual only
    [coeff_eq_zero_of_lt_weightedOrder w, lt_min_iff, map_add, add_zero,
      imp_true_iff]

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
n : ℕ
⊢ f.order = ↑n ↔ (∃ d, (coeff d) f ≠ 0 ∧ degree d = n) ∧ ∀ (d : σ →₀ ℕ), degree d < n → (coeff d) f = 0

Tactic:
simp_rw [degree_eq_weight_one]
  exact weightedOrder_eq_nat _

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: case ih
R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : IsDedekindDomain R
K : Type u_2
S : Type u_3
inst✝⁴ : Field K
inst✝³ : CommSemiring S
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
v : HeightOneSpectrum R
inst✝ : Algebra S K
r : S
x : WithVal (valuation K v)
⊢ r • ↑x = (Completion.coeRingHom.comp (algebraMap S (WithVal (valuation K v)))) r * ↑x

Tactic:
simp [Algebra.smul_def, Completion.algebraMap_def, WithVal.algebraMap_right_apply,
        Completion.coeRingHom]

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

Goal state before:
pp: case inr
σ : Type u_3
R : Type u_4
inst✝³ : TopologicalSpace R
inst✝² : CommSemiring R
ι : Type u_5
f : ι → MvPowerSeries σ R
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrderBot ι
w : σ → ℕ
hempty : Nonempty ι
d : σ →₀ ℕ
h : ∀ (n : ℕ), ∃ a, ∀ (b : ι), a ≤ b → ↑n < weightedOrder w (f b)
i : ι
hi : ∀ (b : ι), i ≤ b → ↑((Finsupp.weight w) d) < weightedOrder w (f b)
s : Finset ι
hs : (coeff d) (∏ i ∈ s, f i) ≠ 0
⊢ ↑s ⊆ Set.Iio i

Tactic:
contrapose hs

Goal state after:
case inr
σ : Type u_3
R : Type u_4
inst✝³ : TopologicalSpace R
inst✝² : CommSemiring R
ι : Type u_5
f : ι → MvPowerSeries σ R
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrderBot ι
w : σ → ℕ
hempty : Nonempty ι
d : σ →₀ ℕ
h : ∀ (n : ℕ), ∃ a, ∀ (b : ι), a ≤ b → ↑n < weightedOrder w (f b)
i : ι
hi : ∀ (b : ι), i ≤ b → ↑((Finsupp.weight w) d) < weightedOrder w (f b)
s : Finset ι
hs : ¬↑s ⊆ Set.Iio i
⊢ (coeff d) (∏ i ∈ s, f i) = 0
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: case neg
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
a : adicCompletion K v
ha : 1 < Valued.v a
⊢ ∃ b ∈ R⁰, a * ↑b ∈ adicCompletionIntegers K v

Tactic:
obtain ⟨ϖ, hϖ⟩ := intValuation_exists_uniformizer v

Goal state after:
case neg
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
a : adicCompletion K v
ha : 1 < Valued.v a
ϖ : R
hϖ : v.intValuation ϖ = exp (-1)
⊢ ∃ b ∈ R⁰, a * ↑b ∈ adicCompletionIntegers K v
Module: Mathlib.RingTheory.PowerSeries.PiTopology

Goal state before:
pp: case inr
R : Type u_1
inst✝³ : TopologicalSpace R
inst✝² : CommSemiring R
ι : Type u_2
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrderBot ι
f : ι → R⟦X⟧
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
s : Finset ι
hs : (coeff n) (∏ i ∈ s, f i) ≠ 0
⊢ ↑s ⊆ Set.Iio i

Tactic:
contrapose hs

Goal state after:
case inr
R : Type u_1
inst✝³ : TopologicalSpace R
inst✝² : CommSemiring R
ι : Type u_2
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrderBot ι
f : ι → R⟦X⟧
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
s : Finset ι
hs : ¬↑s ⊆ Set.Iio i
⊢ (coeff n) (∏ i ∈ s, f i) = 0
Module: Mathlib.RingTheory.PowerSeries.Order

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
φ ψ : R⟦X⟧
⊢ (φ * ψ).order = φ.order + ψ.order

Tactic:
by
  apply le_antisymm _ (le_order_mul _ _)
  by_cases! h : φ = 0 ∨ ψ = 0
  · rcases h with h | h <;> simp [h]
  · rw [← coe_toNat_order h.1, ← coe_toNat_order h.2, ← ENat.natCast_add]
    apply order_le
    rw [coeff_mul, Finset.sum_eq_single_of_mem ⟨φ.order.toNat, ψ.order.toNat⟩ (by simp)]
    · exact mul_ne_zero (coeff_order h.1) (coeff_order h.2)
    · intro ij hij h
      rcases trichotomy_of_add_eq_add (mem_antidiagonal.mp hij) with h' | h' | h'
      · exact False.elim (h (by simp [Prod.ext_iff, h'.1, h'.2]))
      · rw [coeff_of_lt_order_toNat ij.1 h', zero_mul]
      · rw [coeff_of_lt_order_toNat ij.2 h', mul_zero]

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

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
f g : R⟦X⟧
⊢ (f * g).divXPowOrder = f.divXPowOrder * g.divXPowOrder

Tactic:
by_cases! h : f = 0 ∨ g = 0
  · rcases h with (h | h) <;> simp [h]
  apply X_pow_mul_cancel (k := f.order.toNat + g.order.toNat)
  calc
    _ = X ^ ((f * g).order.toNat) * (f * g).divXPowOrder := by
        rw [order_mul, ENat.toNat_add (order_eq_top.not.mpr h.1) (order_eq_top.not.mpr h.2)]
    _ = f * g := by
        simp [X_pow_order_mul_divXPowOrder]
    _ = (X ^ f.order.toNat * f.divXPowOrder) * (X ^ g.order.toNat * g.divXPowOrder) := by
        simp [X_pow_order_mul_divXPowOrder]
    _ = f.divXPowOrder * g.divXPowOrder * X ^ (g.order.toNat + f.order.toNat) := by
        rw [mul_assoc, X_pow_mul, X_pow_mul, ← mul_assoc, mul_assoc, ← pow_add]
    _ = X ^ (f.order.toNat + g.order.toNat) * (f.divXPowOrder * g.divXPowOrder) := by
        rw [X_pow_mul, add_comm]

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝¹⁰ : CommRing R
inst✝⁹ : UniformSpace R
S : Type u_3
inst✝⁸ : CommRing S
inst✝⁷ : UniformSpace S
φ : R →+* S
a : σ → S
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : IsUniformAddGroup R
inst✝⁴ : IsUniformAddGroup S
inst✝³ : CompleteSpace S
inst✝² : T2Space S
inst✝¹ : IsTopologicalRing S
inst✝ : IsLinearTopology S S
hφ : Continuous[inst✝⁹.toTopologicalSpace, inst✝⁷.toTopologicalSpace] ⇑φ
ha : HasEval a
f : MvPowerSeries σ R
⊢ Continuous[instTopologicalSpace R, inst✝⁷.toTopologicalSpace] (eval₂ φ a)

Tactic:
exact continuous_eval₂ hφ ha

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

Goal state before:
pp: f : ℕ → ℂ
x : ℝ
s : ℂ
hs : x < s.re
C : ℝ
hC : ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C * ↑n ^ (x - 1)
hC₀ : 0 ≤ C
⊢ LSeriesSummable f s

Tactic:
have hsum : Summable fun n : ℕ ↦ ‖(C : ℂ) / n ^ (s + (1 - x))‖ := by
    simp_rw [div_eq_mul_inv, norm_mul, ← cpow_neg]
    have hsx : -s.re + x - 1 < -1 := by linarith only [hs]
    refine Summable.mul_left _ <|
      Summable.of_norm_bounded_eventually_nat (g := fun n ↦ (n : ℝ) ^ (-s.re + x - 1)) ?_ ?_
    · simpa
    · simp only [norm_norm, Filter.eventually_atTop]
      refine ⟨1, fun n hn ↦ le_of_eq ?_⟩
      simp only [norm_natCast_cpow_of_pos hn, add_re, sub_re, neg_re, ofReal_re, one_re]
      ring_nf

Goal state after:
f : ℕ → ℂ
x : ℝ
s : ℂ
hs : x < s.re
C : ℝ
hC : ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C * ↑n ^ (x - 1)
hC₀ : 0 ≤ C
hsum : Summable fun n ↦ ‖↑C / ↑n ^ (s + (1 - ↑x))‖
⊢ LSeriesSummable f s
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: A : Type u_1
inst✝ : CommRing A
f : A⟦X⟧
hf : HasSubst f
n : ℕ
⊢ ∀ᶠ (m : ℕ) in Filter.atTop, ∀ n' ≤ n, (coeff n') (f ^ m) = 0

Tactic:
by
  obtain ⟨k, hk⟩ := id hf
  refine Filter.eventually_of_mem (Filter.Ici_mem_atTop (k * (n + 1))) fun m hm n' hn' ↦
    coeff_of_lt_order _ ?_
  obtain ⟨m, rfl⟩ := le_iff_exists_add.mp (Set.mem_Ici.mp hm)
  grw [pow_add, ← order_mul_ge, pow_mul, ← le_order_pow_of_constantCoeff_eq_zero _
    (by rwa [map_pow]), ← _root_.le_add_right le_rfl, Nat.cast_lt]
  lia

Goal state after:
Module: Mathlib.Analysis.Convex.EGauge

Goal state before:
pp: case inr
𝕜 : Type u_1
inst✝² : NormedDivisionRing 𝕜
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
c : 𝕜
s : Set E
x : E
a : 𝕜
y : E
hy : y ∈ s
hxy : (fun x ↦ a • x) y = c • x
hc : c ≠ 0
⊢ (fun x ↦ (c⁻¹ * a) • x) y = x

Tactic:
simp only [mul_smul, hxy, inv_smul_smul₀ hc]

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

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝ : CommRing R
a : σ → R
f : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ ∀ b ∈ ⋯.toFinset, b ≠ n → (coeff b) f * (coeff n) (b.prod fun s e ↦ (a s • X s) ^ e) = 0

pp: σ : Type u_1
R : Type u_3
inst✝ : CommRing R
a : σ → R
f : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ n ∉ ⋯.toFinset → (coeff n) f * (coeff n) (n.prod fun s e ↦ (a s • X s) ^ e) = 0

Tactic:
· intro b hb hbn
    rw [← monomial_eq, coeff_monomial, if_neg (Ne.symm hbn), mul_zero]

Goal state after:
σ : Type u_1
R : Type u_3
inst✝ : CommRing R
a : σ → R
f : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ n ∉ ⋯.toFinset → (coeff n) f * (coeff n) (n.prod fun s e ↦ (a s • X s) ^ e) = 0
Module: Mathlib.Analysis.Convex.EGauge

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : (i : ι) → AddCommGroup (E i)
inst✝ : (i : ι) → Module 𝕜 (E i)
I : Set ι
hI : I.Finite
U : (i : ι) → Set (E i)
hU : ∀ i ∈ I, Balanced 𝕜 (U i)
x : (i : ι) → E i
hI₀ : I = univ ∨ (∃ i ∈ I, x i ≠ 0) ∨ (𝓝[≠] 0).NeBot
⊢ egauge 𝕜 (I.pi U) x = ⨆ i ∈ I, egauge 𝕜 (U i) (x i)

Tactic:
refine le_antisymm ?_ (iSup₂_le fun i hi ↦ le_egauge_pi hi _ _)
  refine le_of_forall_gt fun r hr ↦ ?_
  have : ∀ i ∈ I, ∃ c : 𝕜, x i ∈ c • U i ∧ ‖c‖ₑ < r := fun i hi ↦
    egauge_lt_iff.mp <| (le_iSup₂ i hi).trans_lt hr
  choose! c hc hcr using this
  obtain ⟨c₀, hc₀, hc₀I, hc₀r⟩ :
      ∃ c₀ : 𝕜, (c₀ ≠ 0 ∨ I = univ) ∧ (∀ i ∈ I, ‖c i‖ ≤ ‖c₀‖) ∧ ‖c₀‖ₑ < r := by
    have hr₀ : 0 < r := hr.bot_lt
    rcases I.eq_empty_or_nonempty with rfl | hIne
    · obtain hι | hbot : IsEmpty ι ∨ (𝓝[≠] (0 : 𝕜)).NeBot := by simpa [@eq_comm _ ∅] using hI₀
      · use 0
        simp [@eq_comm _ ∅, hι, hr₀]
      · rcases exists_enorm_lt 𝕜 hr₀.ne' with ⟨c₀, hc₀, hc₀r⟩
        exact ⟨c₀, .inl hc₀, by simp, hc₀r⟩
    · obtain ⟨i₀, hi₀I, hc_max⟩ : ∃ i₀ ∈ I, IsMaxOn (‖c ·‖ₑ) I i₀ :=
        exists_max_image _ (‖c ·‖ₑ) hI hIne
      by_cases! H : c i₀ ≠ 0 ∨ I = univ
      · exact ⟨c i₀, H, fun i hi ↦ by simpa [enorm] using! hc_max hi, hcr _ hi₀I⟩
      · have hc0 (i : ι) (hi : i ∈ I) : c i = 0 := by simpa [H] using hc_max hi
        have heg0 (i : ι) (hi : i ∈ I) : x i = 0 :=
          zero_smul_set_subset (α := 𝕜) (U i) (hc0 i hi ▸ hc i hi)
        have : (𝓝[≠] (0 : 𝕜)).NeBot := (hI₀.resolve_left H.2).resolve_left (by simpa)
        rcases exists_enorm_lt 𝕜 hr₀.ne' with ⟨c₁, hc₁, hc₁r⟩
        refine ⟨c₁, .inl hc₁, fun i hi ↦ ?_, hc₁r⟩
        simp [hc0 i hi]
  refine egauge_lt_iff.2 ⟨c₀, ?_, hc₀r⟩
  rw [smul_set_pi₀' hc₀]
  intro i hi
  exact (hU i hi).smul_mono (hc₀I i hi) (hc i hi)

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
f : α → E
g : α → F
l : Filter α
⊢ f =o[𝕜; l] g ↔ ∀ U ∈ 𝓝 0, ∃ V ∈ 𝓝 0, Tendsto (fun x ↦ egauge 𝕜 U (f x) / egauge 𝕜 V (g x)) l (𝓝 0)

Tactic:
by
  simp only [isLittleOTVS_iff, ← ENNReal.coe_zero, ENNReal.nhds_coe, ← NNReal.bot_eq_zero,
    (nhds_bot_basis_Iic.map _).tendsto_right_iff]
  simp +contextual [ENNReal.div_le_iff_le_mul, pos_iff_ne_zero, EventuallyLE]

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

Goal state before:
pp: case mp
α : Type u_1
𝕜 : Type u_3
E : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : TopologicalSpace E
inst✝¹ : Module 𝕜 E
l : Filter α
f : α → E
inst✝ : ContinuousSMul 𝕜 E
hf : f =o[𝕜; l] 1
⊢ Tendsto f l (𝓝 0)

Tactic:
rw [(basis_sets _).isLittleOTVS_iff nhds_basis_ball] at hf

Goal state after:
case mp
α : Type u_1
𝕜 : Type u_3
E : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : TopologicalSpace E
inst✝¹ : Module 𝕜 E
l : Filter α
f : α → E
inst✝ : ContinuousSMul 𝕜 E
hf :
  ∀ i ∈ 𝓝 0, ∃ j, 0 < j ∧ ∀ (ε : ℝ≥0), ε ≠ 0 → ∀ᶠ (x : α) in l, egauge 𝕜 (id i) (f x) ≤ ↑ε * egauge 𝕜 (ball 0 j) (1 x)
⊢ Tendsto f l (𝓝 0)
Module: Mathlib.Analysis.Asymptotics.TVS

Goal state before:
pp: case mp
α : Type u_1
𝕜 : Type u_3
E : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : TopologicalSpace E
inst✝¹ : Module 𝕜 E
l : Filter α
f : α → E
inst✝ : ContinuousSMul 𝕜 E
hf :
  ∀ i ∈ 𝓝 0, ∃ j, 0 < j ∧ ∀ (ε : ℝ≥0), ε ≠ 0 → ∀ᶠ (x : α) in l, egauge 𝕜 (id i) (f x) ≤ ↑ε * egauge 𝕜 (ball 0 j) (1 x)
U : Set E
hU : U ∈ 𝓝 0
hUb : Balanced 𝕜 U
r : ℝ≥0
hr₀ : 0 < ↑r
hr : ∀ (ε : ℝ≥0), ε ≠ 0 → ∀ᶠ (x : α) in l, egauge 𝕜 (id U) (f x) ≤ ↑ε * egauge 𝕜 (ball 0 ↑r) (1 x)
⊢ ∀ᶠ (x : α) in l, f x ∈ id U

Tactic:
norm_cast at hr₀

Goal state after:
case mp
α : Type u_1
𝕜 : Type u_3
E : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : TopologicalSpace E
inst✝¹ : Module 𝕜 E
l : Filter α
f : α → E
inst✝ : ContinuousSMul 𝕜 E
hf :
  ∀ i ∈ 𝓝 0, ∃ j, 0 < j ∧ ∀ (ε : ℝ≥0), ε ≠ 0 → ∀ᶠ (x : α) in l, egauge 𝕜 (id i) (f x) ≤ ↑ε * egauge 𝕜 (ball 0 j) (1 x)
U : Set E
hU : U ∈ 𝓝 0
hUb : Balanced 𝕜 U
r : ℝ≥0
hr : ∀ (ε : ℝ≥0), ε ≠ 0 → ∀ᶠ (x : α) in l, egauge 𝕜 (id U) (f x) ≤ ↑ε * egauge 𝕜 (ball 0 ↑r) (1 x)
hr₀ : 0 < r
⊢ ∀ᶠ (x : α) in l, f x ∈ id U
Module: Mathlib.Analysis.Asymptotics.TVS

Goal state before:
pp: case refine_1
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
f : α → E
g : α → F
l : Filter α
c : 𝕜
hc : 1 < ‖c‖₊
hc₀ : 0 < ‖c‖₊
h :
  ∀ (i : ℝ),
    0 < i →
      ∃ j, 0 < j ∧ ∀ (ε : ℝ≥0), ε ≠ 0 → ∀ᶠ (x : α) in l, egauge 𝕜 (ball 0 i) (f x) ≤ ↑ε * egauge 𝕜 (ball 0 j) (g x)
ε : ℝ≥0
hε : 0 < ↑ε
δ : ℝ≥0
hδ₀ : 0 < ↑δ
hδ : ∀ (ε_1 : ℝ≥0), ε_1 ≠ 0 → ∀ᶠ (x : α) in l, egauge 𝕜 (ball 0 ↑ε) (f x) ≤ ↑ε_1 * egauge 𝕜 (ball 0 ↑δ) (g x)
⊢ ∀ᶠ (x : α) in l, ‖f x‖ ≤ ↑ε * ‖g x‖

Tactic:
norm_cast at hε hδ₀

Goal state after:
case refine_1
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
f : α → E
g : α → F
l : Filter α
c : 𝕜
hc : 1 < ‖c‖₊
hc₀ : 0 < ‖c‖₊
h :
  ∀ (i : ℝ),
    0 < i →
      ∃ j, 0 < j ∧ ∀ (ε : ℝ≥0), ε ≠ 0 → ∀ᶠ (x : α) in l, egauge 𝕜 (ball 0 i) (f x) ≤ ↑ε * egauge 𝕜 (ball 0 j) (g x)
ε δ : ℝ≥0
hδ : ∀ (ε_1 : ℝ≥0), ε_1 ≠ 0 → ∀ᶠ (x : α) in l, egauge 𝕜 (ball 0 ↑ε) (f x) ≤ ↑ε_1 * egauge 𝕜 (ball 0 ↑δ) (g x)
hε : 0 < ε
hδ₀ : 0 < δ
⊢ ∀ᶠ (x : α) in l, ‖f x‖ ≤ ↑ε * ‖g x‖
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✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
f : α → E
g : α → F
l : Filter α
c : 𝕜
hc : 1 < ‖c‖₊
hc₀ : 0 < ‖c‖₊
h :
  ∀ (i : ℝ),
    0 < i →
      ∃ j, 0 < j ∧ ∀ (ε : ℝ≥0), ε ≠ 0 → ∀ᶠ (x : α) in l, egauge 𝕜 (ball 0 i) (f x) ≤ ↑ε * egauge 𝕜 (ball 0 j) (g x)
ε δ : ℝ≥0
hδ : ∀ (ε_1 : ℝ≥0), ε_1 ≠ 0 → ∀ᶠ (x : α) in l, egauge 𝕜 (ball 0 ↑ε) (f x) ≤ ↑ε_1 * egauge 𝕜 (ball 0 ↑δ) (g x)
hε : 0 < ε
hδ₀ : 0 < δ
x : α
hx : egauge 𝕜 (ball 0 ↑ε) (f x) ≤ ↑(δ / ‖c‖₊) * egauge 𝕜 (ball 0 ↑δ) (g x)
⊢ ↑δ / ↑‖c‖₊ * (↑‖c‖₊ * ↑‖g x‖₊ / ↑δ) = ↑δ / ↑δ * (↑‖c‖₊ / ↑‖c‖₊) * ↑‖g x‖₊

Tactic:
simp only [div_eq_mul_inv]; ring

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

Goal state before:
pp: case mp
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
f : α → E
g : α → F
l : Filter α
c : 𝕜
hc : 1 < ‖c‖₊
h : ∀ (i : ℝ), 0 < i → ∃ j, 0 < j ∧ ∀ᶠ (x : α) in l, egauge 𝕜 (ball 0 i) (f x) ≤ egauge 𝕜 (ball 0 j) (g x)
r : ℝ≥0
hr₀ : 0 < ↑r
hr : ∀ᶠ (x : α) in l, egauge 𝕜 (ball 0 1) (f x) ≤ egauge 𝕜 (ball 0 ↑r) (g x)
⊢ ∃ c, ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖

Tactic:
norm_cast at hr₀

Goal state after:
case mp
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
f : α → E
g : α → F
l : Filter α
c : 𝕜
hc : 1 < ‖c‖₊
h : ∀ (i : ℝ), 0 < i → ∃ j, 0 < j ∧ ∀ᶠ (x : α) in l, egauge 𝕜 (ball 0 i) (f x) ≤ egauge 𝕜 (ball 0 j) (g x)
r : ℝ≥0
hr : ∀ᶠ (x : α) in l, egauge 𝕜 (ball 0 1) (f x) ≤ egauge 𝕜 (ball 0 ↑r) (g x)
hr₀ : 0 < r
⊢ ∃ c, ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖
Module: Mathlib.Analysis.Asymptotics.TVS

Goal state before:
pp: case mpr
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
f : α → E
g : α → F
l : Filter α
c : 𝕜
hc : 1 < ‖c‖₊
hc₀ : 0 < ‖c‖₊
C : ℝ≥0
hC : ∀ᶠ (x : α) in l, ‖f x‖ ≤ ↑C * ‖g x‖
hC₀ : 0 < C
r : ℝ≥0
hr₀ : 0 < ↑r
⊢ ∃ j, 0 < j ∧ ∀ᶠ (x : α) in l, egauge 𝕜 (ball 0 ↑r) (f x) ≤ egauge 𝕜 (ball 0 j) (g x)

Tactic:
norm_cast at hr₀

Goal state after:
case mpr
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
f : α → E
g : α → F
l : Filter α
c : 𝕜
hc : 1 < ‖c‖₊
hc₀ : 0 < ‖c‖₊
C : ℝ≥0
hC : ∀ᶠ (x : α) in l, ‖f x‖ ≤ ↑C * ‖g x‖
hC₀ : 0 < C
r : ℝ≥0
hr₀ : 0 < r
⊢ ∃ j, 0 < j ∧ ∀ᶠ (x : α) in l, egauge 𝕜 (ball 0 ↑r) (f x) ≤ egauge 𝕜 (ball 0 j) (g x)
Module: Mathlib.Analysis.Analytic.ConvergenceRadius

Goal state before:
pp: case mp
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
h : p.radius = ∞
r : ℝ≥0
⊢ Summable fun n ↦ ‖p n‖ * ↑r ^ n

Tactic:
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius
      (show (r : ℝ≥0∞) < p.radius from h.symm ▸ ENNReal.coe_lt_top)

Goal state after:
case mp
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
h : p.radius = ∞
r : ℝ≥0
a : ℝ
ha : a ∈ Ioo 0 1
C : ℝ
hp : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C * a ^ n
⊢ Summable fun n ↦ ‖p n‖ * ↑r ^ n
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 : ContinuousMultilinearMap 𝕜 Ei G
m : (i : Fin n.succ) → Ei i
⊢ f.curryRight.uncurryRight m = f m

Tactic:
rw [uncurryRight_apply, curryRight_apply, snoc_init_self]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
r : ℝ≥0
hr : r ≠ 0
hc : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 ↑r)
r' : ℝ≥0
hr' : ↑r' < ↑r⁻¹
⊢ ↑r' ≤ (ofScalars E c).radius

Tactic:
rw [coe_lt_coe, NNReal.lt_inv_iff_mul_lt hr] at hr'

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
r : ℝ≥0
hr : r ≠ 0
hc : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 ↑r)
r' : ℝ≥0
hr' : r' * r < 1
⊢ ↑r' ≤ (ofScalars E c).radius
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
h : HasFPowerSeriesWithinAt f p s x
h' : g =ᶠ[𝓝[s] x] f
h'' : g x = f x
⊢ HasFPowerSeriesWithinAt g p s x

Tactic:
by
  rcases h with ⟨r, hr⟩
  obtain ⟨ε, εpos, hε⟩ : ∃ ε > 0, Metric.eball x ε ∩ s ⊆ {y | g y = f y} :=
    EMetric.mem_nhdsWithin_iff.1 h'
  let r' := min r ε
  refine ⟨r', ?_⟩
  have := hr.of_le (r' := r') (by simp [r', εpos, hr.r_pos]) (min_le_left _ _)
  apply this.congr _ h''
  intro z hz
  exact hε ⟨Metric.eball_subset_eball (min_le_right _ _) hz.2, hz.1⟩

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
hc✝ : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop atTop
r : ℝ≥0
hr : ↑r < (ofScalars E c).radius
this : 0 < r
n : ℕ
hc : 2 * ↑r⁻¹ ≤ ‖c n.succ‖ / ‖c n‖
hn : n ≠ 0
⊢ ↑r ^ n / (↑r ^ n * ↑r) = ↑r⁻¹

Tactic:
div_mul_cancel_left₀,

Goal state after:
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
hc✝ : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop atTop
r : ℝ≥0
hr : ↑r < (ofScalars E c).radius
this : 0 < r
n : ℕ
hc : 2 * ↑r⁻¹ ≤ ‖c n.succ‖ / ‖c n‖
hn : n ≠ 0
⊢ (↑r)⁻¹ = ↑r⁻¹

case ha
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
hc✝ : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop atTop
r : ℝ≥0
hr : ↑r < (ofScalars E c).radius
this : 0 < r
n : ℕ
hc : 2 * ↑r⁻¹ ≤ ‖c n.succ‖ / ‖c n‖
hn : n ≠ 0
⊢ ↑r ^ n ≠ 0
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
s : Set E
x : E
hf : AnalyticWithinAt 𝕜 f s x
hs : g =ᶠ[𝓝[s] x] f
hx : g x = f x
⊢ AnalyticWithinAt 𝕜 g s x

Tactic:
by
  rcases hf with ⟨p, hp⟩
  exact ⟨p, hp.congr hs hx⟩

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
s : Set E
x : E
hf : AnalyticWithinAt 𝕜 f s x
hs : g =ᶠ[𝓝[insert x s] x] f
⊢ AnalyticWithinAt 𝕜 g s x

Tactic:
apply hf.congr_of_eventuallyEq (nhdsWithin_mono x (subset_insert x s) hs)
  apply mem_of_mem_nhdsWithin (mem_insert x s) hs

Goal state after: