mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.Valued.WithVal

Goal state before:
pp: Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
R : Type u_3
inst✝ : Ring R
v : Valuation R Γ₀
b : WithVal v
⊢ (WithZero.map' ↑(valueGroupEquiv v)) (if h : (valuation v) b = 0 then 0 else ↑⟨Units.mk0 ((valuation v) b) ⋯, ⋯⟩) =
    if h : (valuation v) b = 0 then 0 else ↑⟨Units.mk0 ((valuation v) b) ⋯, ⋯⟩

Tactic:
by_cases hb : v b.ofVal = 0 <;> simp [hb]

Goal state after:
Module: Mathlib.RingTheory.Valuation.Discrete.IsDiscreteValuationRing

Goal state before:
pp: A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : IsDiscreteValuationRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
v : Valuation K (WithZero (Multiplicative ℤ)) := valuation K (maximalIdeal A)
π : K := ⋯.choose
hπ : v π = ↑(ofAdd (-1))
⊢ Nontrivial ↥(MonoidWithZeroHom.ofClass (valuation K (maximalIdeal A))).valueGroup

Tactic:
Subgroup.nontrivial_iff_exists_ne_one

Goal state after:
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : IsDiscreteValuationRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
v : Valuation K (WithZero (Multiplicative ℤ)) := valuation K (maximalIdeal A)
π : K := ⋯.choose
hπ : v π = ↑(ofAdd (-1))
⊢ ∃ x ∈ (MonoidWithZeroHom.ofClass (valuation K (maximalIdeal A))).valueGroup, x ≠ 1
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝² : DivisionRing K
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : Valued K Γ₀
x : K
x_ne : x ≠ 0
⊢ ∃ γ, {y | v.restrict (y - 0) < ↑γ} ⊆ {k | v k < v x}

Tactic:
set γ' := Units.mk0 ((ValueGroup₀.restrict₀ _) x) (v.restrict.ne_zero_iff.mpr x_ne) with hdef

Goal state after:
K : Type u_1
inst✝² : DivisionRing K
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : Valued K Γ₀
x : K
x_ne : x ≠ 0
γ' : (ofClass v).ValueGroup₀ˣ := Units.mk0 ((restrict₀ (ofClass v)) x) ⋯
hdef : γ' = Units.mk0 ((restrict₀ (ofClass v)) x) ⋯
⊢ ∃ γ, {y | v.restrict (y - 0) < ↑γ} ⊆ {k | v k < v x}
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case l₂
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
| 𝓝 1

Tactic:
← one_mul (1 : hat K)

Goal state after:
case l₂
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
| 𝓝 (1 * 1)
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case mpr.inr
G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : MulArchimedean G
a b : G
h : b = |a|ₘ ∧ 1 < b
this : ∀ {a : G}, b = |a|ₘ ∧ 1 < b → 1 ≤ a → IsLeast {y | y ∈ closure {a} ∧ 1 < y} b
ha : ¬1 ≤ a
⊢ b = |a⁻¹|ₘ ∧ 1 < b

Tactic:
rwa [mabs_inv]

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

Goal state before:
pp: case insert
σ : Type u_1
R : Type u_3
inst✝² : CommSemiring R
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq σ
f : ι → MvPowerSeries σ R
a : ι
s : Finset ι
ha : a ∉ s
ih : ∀ (d : σ →₀ ℕ), (coeff d) (∏ j ∈ s, f j) = ∑ l ∈ s.finsuppAntidiag d, ∏ i ∈ s, (coeff (l i)) (f i)
d : σ →₀ ℕ
⊢ ∀ (a_1 b : σ →₀ ℕ),
    a_1 + b = d →
      ∀ (a_3 b_1 : σ →₀ ℕ),
        a_3 + b_1 = d →
          (a_1 = a_3 → ¬b = b_1) →
            ∀ a_6 ∈ s.finsuppAntidiag b, ∀ x ∈ s.finsuppAntidiag b_1, ¬x.update a a_3 = a_6.update a a_1

Tactic:
rintro u v rfl u' v' huv h k - l - hkl

Goal state after:
case insert
σ : Type u_1
R : Type u_3
inst✝² : CommSemiring R
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq σ
f : ι → MvPowerSeries σ R
a : ι
s : Finset ι
ha : a ∉ s
ih : ∀ (d : σ →₀ ℕ), (coeff d) (∏ j ∈ s, f j) = ∑ l ∈ s.finsuppAntidiag d, ∏ i ∈ s, (coeff (l i)) (f i)
u v u' v' : σ →₀ ℕ
huv : u' + v' = u + v
h : u = u' → ¬v = v'
k l : ι →₀ σ →₀ ℕ
hkl : l.update a u' = k.update a u
⊢ False
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
r : R
⊢ (valuation K v) ↑r ≤ 1

Tactic:
rw [valuation_of_algebraMap]

Goal state after:
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
r : R
⊢ v.intValuation r ≤ 1
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
r : R
⊢ (valuation K v) ↑r < 1 ↔ v.asIdeal ∣ Ideal.span {r}

Tactic:
rw [valuation_of_algebraMap]

Goal state after:
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
r : R
⊢ v.intValuation r < 1 ↔ v.asIdeal ∣ Ideal.span {r}
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
r : R
⊢ (valuation K v) ↑r < 1 ↔ r ∈ v.asIdeal

Tactic:
rw [valuation_of_algebraMap]

Goal state after:
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
r : R
⊢ v.intValuation r < 1 ↔ r ∈ v.asIdeal
Module: Mathlib.Data.Finset.Finsupp

Goal state before:
pp: case neg
ι : Type u_1
α : Type u_2
inst✝ : Zero α
s : Finset ι
f : ι →₀ α
t : ι →₀ Finset α
ht : t.support ⊆ s
i : ι
h : (i ∈ f.support → i ∈ s) ∧ (i ∈ s → f i ∈ t i)
hi : i ∉ s
⊢ 0 ∈ 0

Tactic:
exact zero_mem_zero

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

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
a b : R
hv : v.intValuation b ≤ v.intValuation a
γ : Multiplicative ℤ
ha : ¬a = 0
hvaz : v.intValuation a ≠ 0
hγz : ↑γ ≠ 0
n : ℕ
hna : exp (-↑n) < v.intValuation a
hnγ : exp (-↑n) < ↑γ
⊢ emultiplicity v.asIdeal (Ideal.span {a}) ≤ ↑n

Tactic:
grw [← exp_le_intValuation_iff_emultiplicity_le, hna]

Goal state after:
Module: Mathlib.Topology.Instances.ENat

Goal state before:
pp: α : Type u_1
l : Filter α
f : α → ℕ∞
⊢ (∀ (i : ℕ∞), i ≠ ⊤ → Tendsto f l (𝓟 (Ioi i))) ↔ ∀ (n : ℕ), ∀ᶠ (a : α) in l, ↑n < f a

Tactic:
tendsto_principal,

Goal state after:
α : Type u_1
l : Filter α
f : α → ℕ∞
⊢ (∀ (i : ℕ∞), i ≠ ⊤ → ∀ᶠ (a : α) in l, f a ∈ Ioi i) ↔ ∀ (n : ℕ), ∀ᶠ (a : α) in l, ↑n < f a
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
n : (WithZero (Multiplicative ℤ))ˣ
⊢ (∃ a, ¬Valued.v a = 0 ∧ ∃ x, Valued.v a * ↑n = Valued.v x) ↔
    ∃ a, ¬(valuation K v) a = 0 ∧ ∃ x, (valuation K v) a * ↑n = (valuation K v) x

Tactic:
refine ⟨fun ⟨a, ha0, x, hx⟩ ↦ ?_, fun ⟨a, ha0, x, hx⟩ ↦
    ⟨↑a, by simpa using ha0, ↑x, by simpa using hx⟩⟩

Goal state after:
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
n : (WithZero (Multiplicative ℤ))ˣ
x✝ : ∃ a, ¬Valued.v a = 0 ∧ ∃ x, Valued.v a * ↑n = Valued.v x
a : adicCompletion K v
ha0 : ¬Valued.v a = 0
x : adicCompletion K v
hx : Valued.v a * ↑n = Valued.v x
⊢ ∃ a, ¬(valuation K v) a = 0 ∧ ∃ x, (valuation K v) a * ↑n = (valuation K v) x
Module: Mathlib.RingTheory.PowerSeries.Order

Goal state before:
pp: case coe
R : Type u_1
inst✝ : Semiring R
φ : R⟦X⟧
n : ℕ
h : ∀ (i : ℕ), ↑i < ↑n → (coeff i) φ = 0
⊢ ↑n ≤ φ.order

Tactic:
convert! nat_le_order φ n _
    simpa using h

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

Goal state before:
pp: R : Type u_2
inst✝ : CommSemiring R
n : ℕ
f : R[X]
hn : f.natDegree < n
m : ℕ
h : n ≤ m
⊢ 0 = (coeff m) ↑f

Tactic:
rw [coeff_coe]

Goal state after:
R : Type u_2
inst✝ : CommSemiring R
n : ℕ
f : R[X]
hn : f.natDegree < n
m : ℕ
h : n ≤ m
⊢ 0 = f.coeff m
Module: Mathlib.RingTheory.MvPowerSeries.LinearTopology

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Ring R
Jd : TwoSidedIdeal R × (σ →₀ ℕ)
f g : MvPowerSeries σ R
hg : g ∈ {f | ∀ e ≤ Jd.2, (coeff e) f ∈ Jd.1}
e : σ →₀ ℕ
he : e ≤ Jd.2
⊢ ∑ p ∈ Finset.antidiagonal e, (coeff p.1) f * (coeff p.2) g ∈ Jd.1

Tactic:
apply sum_mem

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Ring R
Jd : TwoSidedIdeal R × (σ →₀ ℕ)
f g : MvPowerSeries σ R
hg : g ∈ {f | ∀ e ≤ Jd.2, (coeff e) f ∈ Jd.1}
e : σ →₀ ℕ
he : e ≤ Jd.2
⊢ ∀ c ∈ Finset.antidiagonal e, (coeff c.1) f * (coeff c.2) g ∈ Jd.1
Module: Mathlib.RingTheory.MvPowerSeries.LinearTopology

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Ring R
Jd : TwoSidedIdeal R × (σ →₀ ℕ)
f g : MvPowerSeries σ R
hf : f ∈ {f | ∀ e ≤ Jd.2, (coeff e) f ∈ Jd.1}
e : σ →₀ ℕ
he : e ≤ Jd.2
⊢ ∑ p ∈ Finset.antidiagonal e, (coeff p.1) f * (coeff p.2) g ∈ Jd.1

Tactic:
apply sum_mem

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Ring R
Jd : TwoSidedIdeal R × (σ →₀ ℕ)
f g : MvPowerSeries σ R
hf : f ∈ {f | ∀ e ≤ Jd.2, (coeff e) f ∈ Jd.1}
e : σ →₀ ℕ
he : e ≤ Jd.2
⊢ ∀ c ∈ Finset.antidiagonal e, (coeff c.1) f * (coeff c.2) g ∈ Jd.1
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
this : ∃ p, ↑p = ↑f
⊢ eval₂ φ a ↑f = MvPolynomial.eval₂ φ a f

Tactic:
eval₂,

Goal state after:
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
this : ∃ p, ↑p = ↑f
⊢ (if H : ∃ p, ↑p = ↑f then MvPolynomial.eval₂ φ a H.choose else ⋯.extend (MvPolynomial.eval₂ φ a) ↑f) =
    MvPolynomial.eval₂ φ a f
Module: Mathlib.RingTheory.PowerSeries.Evaluation

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
S : Type u_2
inst✝² : CommRing S
φ : R →+* S
a : S
inst✝¹ : UniformSpace R
inst✝ : UniformSpace S
⊢ eval₂ φ a X = a

Tactic:
by
  rw [← Polynomial.coe_X, eval₂_coe, Polynomial.eval₂_X]

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

Goal state before:
pp: R : Type u_1
inst✝¹⁰ : CommRing R
S : Type u_2
inst✝⁹ : CommRing S
φ : R →+* S
a : S
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformSpace S
inst✝⁶ : IsUniformAddGroup R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : IsUniformAddGroup S
inst✝³ : T2Space S
inst✝² : CompleteSpace S
inst✝¹ : IsTopologicalRing S
inst✝ : IsLinearTopology S S
hφ : Continuous ⇑φ
ha : HasEval a
ε : R⟦X⟧ → S
hε : Continuous ε
h : ∀ (p : Polynomial R), ε ↑p = Polynomial.eval₂ φ a p
⊢ ε = eval₂ φ a

Tactic:
refine MvPowerSeries.eval₂_unique hφ (hasEval ha) hε (fun p ↦ ?_)

Goal state after:
R : Type u_1
inst✝¹⁰ : CommRing R
S : Type u_2
inst✝⁹ : CommRing S
φ : R →+* S
a : S
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformSpace S
inst✝⁶ : IsUniformAddGroup R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : IsUniformAddGroup S
inst✝³ : T2Space S
inst✝² : CompleteSpace S
inst✝¹ : IsTopologicalRing S
inst✝ : IsLinearTopology S S
hφ : Continuous ⇑φ
ha : HasEval a
ε : R⟦X⟧ → S
hε : Continuous ε
h : ∀ (p : Polynomial R), ε ↑p = Polynomial.eval₂ φ a p
p : MvPolynomial Unit R
⊢ ε ↑p = MvPolynomial.eval₂ φ (fun x ↦ a) p
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
τ : Type u_4
S : Type u_5
inst✝ : CommRing S
a : σ → MvPowerSeries τ R
ha : HasSubst a
h : R →+* S
d : τ →₀ ℕ
⊢ {s | (coeff d) ((map h) (a s)) ≠ 0} ⊆ {s | (coeff d) (a s) ≠ 0}

Tactic:
by grind [coeff_map]

Goal state after:
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: a b : ℝ
h1 : #↑((fun x ↦ x - a) '' Ioo a b) ≤ #↑(Ioo a b)
h : 0 < b - a
h2 : #↑(Inv.inv '' Ioo 0 (b - a)) ≤ #↑(Ioo 0 (b - a))
⊢ 𝔠 ≤ #↑(Ioo 0 (b - a))

Tactic:
refine le_trans ?_ h2

Goal state after:
a b : ℝ
h1 : #↑((fun x ↦ x - a) '' Ioo a b) ≤ #↑(Ioo a b)
h : 0 < b - a
h2 : #↑(Inv.inv '' Ioo 0 (b - a)) ≤ #↑(Ioo 0 (b - a))
⊢ 𝔠 ≤ #↑(Inv.inv '' Ioo 0 (b - a))
Module: Mathlib.NumberTheory.LSeries.Basic

Goal state before:
pp: s : ℂ
hs : s ≠ 0
f : ℕ → ℂ
n : ℕ
⊢ term f s n = f n / ↑n ^ s

Tactic:
rcases eq_or_ne n 0 with rfl | hn
  · rw [term_zero, Nat.cast_zero, zero_cpow hs, div_zero]
  · rw [term_of_ne_zero hn]

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

Goal state before:
pp: case neg
σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : σ → MvPowerSeries τ S
w : τ → ℕ
ha : HasSubst a
f : MvPowerSeries σ R
d : τ →₀ ℕ
hd : ↑((Finsupp.weight w) d) < ⨅ d, ⨅ (_ : (coeff d) f ≠ 0), (Finsupp.weight (weightedOrder w ∘ a)) d
x : σ →₀ ℕ
hfx : ¬(coeff x) f = 0
⊢ ↑((Finsupp.weight w) d) < weightedOrder w (x.prod fun s e ↦ a s ^ e)

Tactic:
refine hd.trans_le (((biInf_le _ hfx).trans ?_).trans (le_weightedOrder_prod ..))

Goal state after:
case neg
σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : σ → MvPowerSeries τ S
w : τ → ℕ
ha : HasSubst a
f : MvPowerSeries σ R
d : τ →₀ ℕ
hd : ↑((Finsupp.weight w) d) < ⨅ d, ⨅ (_ : (coeff d) f ≠ 0), (Finsupp.weight (weightedOrder w ∘ a)) d
x : σ →₀ ℕ
hfx : ¬(coeff x) f = 0
⊢ (Finsupp.weight (weightedOrder w ∘ a)) x ≤ ∑ i ∈ x.support, weightedOrder w ((fun s e ↦ a s ^ e) i (x i))
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
τ : Type u_3
⊢ HasSubst 0

Tactic:
by
  rw [hasSubst_iff]
  exact MvPowerSeries.HasSubst.zero

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

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

Tactic:
classical
  ext d
  by_cases hd : d.degree < k
  · rw [coeff_truncTotal _ hd, coeff_truncTotal _ hd, coeff_subst ha, coeff_subst, finsum_congr]
    · intro n
      simp_rw [Finsupp.prod, coeff_prod]
      congr! 3 with l hl i hi
      obtain ⟨hl₁, -⟩ := mem_finsuppAntidiag.mp hl
      have : (l i).degree ≤ d.degree :=
        hl₁ ▸ Finsupp.degree_mono (single_le_sum_of_canonicallyOrdered hi)
      exact_mod_cast (coeff_truncTotal_pow _ (by nlinarith [hx i])).symm
    · exact ha.truncTotal
  simp_rw [coeff_truncTotal_eq_zero _ (not_lt.mp hd)]

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

Goal state before:
pp: f : ℕ → ℂ
x : ℝ
s : ℂ
hs : x < s.re
h : f =O[atTop] fun n ↦ ↑n ^ (x - 1)
C : ℝ
hC : ∀ᶠ (x_1 : ℕ) in atTop, ‖f x_1‖ ≤ C * ‖↑x_1 ^ (x - 1)‖
m : ℕ
hm : ∀ (b : ℕ), m ≤ b → ‖f b‖ ≤ C * ‖↑b ^ (x - 1)‖
C' : ℝ := max C ((insert 0 (image (fun n ↦ ‖f n‖ / ↑n ^ (x - 1)) (range m))).max' ⋯)
⊢ LSeriesSummable f s

Tactic:
have hC'₀ : 0 ≤ C' := (le_max' _ _ (mem_insert.mpr (Or.inl rfl))).trans <| le_max_right ..

Goal state after:
f : ℕ → ℂ
x : ℝ
s : ℂ
hs : x < s.re
h : f =O[atTop] fun n ↦ ↑n ^ (x - 1)
C : ℝ
hC : ∀ᶠ (x_1 : ℕ) in atTop, ‖f x_1‖ ≤ C * ‖↑x_1 ^ (x - 1)‖
m : ℕ
hm : ∀ (b : ℕ), m ≤ b → ‖f b‖ ≤ C * ‖↑b ^ (x - 1)‖
C' : ℝ := max C ((insert 0 (image (fun n ↦ ‖f n‖ / ↑n ^ (x - 1)) (range m))).max' ⋯)
hC'₀ : 0 ≤ C'
⊢ LSeriesSummable f s
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: R : Type u_2
inst✝² : CommRing R
τ : Type u_3
S : Type u_4
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : MvPowerSeries τ S
ha : HasSubst a
p : Polynomial R
⊢ subst a ↑p = (Polynomial.aeval a) p

Tactic:
rw [← coe_substAlgHom ha, substAlgHom_coe]

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

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
f : ι → ArithmeticFunction R
hf : ∀ (n : ℕ), ∀ᶠ (i : ι) in cofinite, (f i) n = 1 n
this✝ : UniformSpace R := ⊥
this : IsUniformInducing DFunLike.coe
⊢ ∀ (i : ℕ),
    Tendsto
      ((fun a ↦ (a.1 i, a.2 i)) ∘ fun p ↦
        ((DFunLike.coe ∘ fun s ↦ ∏ b ∈ s, f b) p.1, (DFunLike.coe ∘ fun s ↦ ∏ b ∈ s, f b) p.2))
      (atTop ×ˢ atTop) (𝓟 SetRel.id)

Tactic:
tendsto_principal,

Goal state after:
ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
f : ι → ArithmeticFunction R
hf : ∀ (n : ℕ), ∀ᶠ (i : ι) in cofinite, (f i) n = 1 n
this✝ : UniformSpace R := ⊥
this : IsUniformInducing DFunLike.coe
⊢ ∀ (i : ℕ),
    ∀ᶠ (a : Finset ι × Finset ι) in atTop ×ˢ atTop,
      ((fun a ↦ (a.1 i, a.2 i)) ∘ fun p ↦
            ((DFunLike.coe ∘ fun s ↦ ∏ b ∈ s, f b) p.1, (DFunLike.coe ∘ fun s ↦ ∏ b ∈ s, f b) p.2))
          a ∈
        SetRel.id
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: case e_a.e_a
σ : Type u_1
A : Type u_2
inst✝²¹ : CommSemiring A
R✝ : Type u_3
inst✝²⁰ : CommRing R✝
inst✝¹⁹ : Algebra A R✝
τ : Type u_4
S : Type u_5
inst✝¹⁸ : CommRing S
inst✝¹⁷ : Algebra A S
inst✝¹⁶ : Algebra R✝ S
inst✝¹⁵ : IsScalarTower A R✝ S
a✝¹ a✝ : σ → MvPowerSeries τ S
T✝ : Type u_6
inst✝¹⁴ : CommRing T✝
inst✝¹³ : UniformSpace T✝
inst✝¹² : T2Space T✝
inst✝¹¹ : CompleteSpace T✝
inst✝¹⁰ : IsUniformAddGroup T✝
inst✝⁹ : IsTopologicalRing T✝
inst✝⁸ : IsLinearTopology T✝ T✝
inst✝⁷ : Algebra R✝ T✝
ε : MvPowerSeries τ S →ₐ[R✝] T✝
inst✝⁶ : Algebra S T✝
inst✝⁵ : IsScalarTower R✝ S T✝
υ : Type u_7
T : Type u_8
inst✝⁴ : CommRing T
inst✝³ : Algebra R✝ T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R✝ S T
b : τ → MvPowerSeries υ T
R : Type u_9
inst✝ : CommSemiring R
a : σ → R
f g : MvPowerSeries σ R
n : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hx : x.1 + x.2 = n
⊢ ∏ x_1 ∈ x.1.support ∪ x.2.support, a x_1 ^ (x.1 + x.2) x_1 =
    ∏ x_1 ∈ x.1.support ∪ x.2.support, a x_1 ^ x.1 x_1 * a x_1 ^ x.2 x_1

Tactic:
apply Finset.prod_congr rfl

Goal state after:
case e_a.e_a
σ : Type u_1
A : Type u_2
inst✝²¹ : CommSemiring A
R✝ : Type u_3
inst✝²⁰ : CommRing R✝
inst✝¹⁹ : Algebra A R✝
τ : Type u_4
S : Type u_5
inst✝¹⁸ : CommRing S
inst✝¹⁷ : Algebra A S
inst✝¹⁶ : Algebra R✝ S
inst✝¹⁵ : IsScalarTower A R✝ S
a✝¹ a✝ : σ → MvPowerSeries τ S
T✝ : Type u_6
inst✝¹⁴ : CommRing T✝
inst✝¹³ : UniformSpace T✝
inst✝¹² : T2Space T✝
inst✝¹¹ : CompleteSpace T✝
inst✝¹⁰ : IsUniformAddGroup T✝
inst✝⁹ : IsTopologicalRing T✝
inst✝⁸ : IsLinearTopology T✝ T✝
inst✝⁷ : Algebra R✝ T✝
ε : MvPowerSeries τ S →ₐ[R✝] T✝
inst✝⁶ : Algebra S T✝
inst✝⁵ : IsScalarTower R✝ S T✝
υ : Type u_7
T : Type u_8
inst✝⁴ : CommRing T
inst✝³ : Algebra R✝ T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R✝ S T
b : τ → MvPowerSeries υ T
R : Type u_9
inst✝ : CommSemiring R
a : σ → R
f g : MvPowerSeries σ R
n : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hx : x.1 + x.2 = n
⊢ ∀ x_1 ∈ x.1.support ∪ x.2.support, a x_1 ^ (x.1 + x.2) x_1 = a x_1 ^ x.1 x_1 * a x_1 ^ x.2 x_1
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_9
inst✝ : CommSemiring R
x : MvPowerSeries σ R
n : σ →₀ ℕ
h : n = 0
⊢ ((coeff n) fun n ↦ (n.prod fun s m ↦ 0 ^ m) * (coeff n) x) = constantCoeff x

pp: case neg
σ : Type u_1
R : Type u_9
inst✝ : CommSemiring R
x : MvPowerSeries σ R
n : σ →₀ ℕ
h : ¬n = 0
⊢ ((coeff n) fun n ↦ (n.prod fun s m ↦ 0 ^ m) * (coeff n) x) = 0

Tactic:
· simp [h, coeff_apply, ← @coeff_zero_eq_constantCoeff_apply, coeff_apply]

Goal state after:
case neg
σ : Type u_1
R : Type u_9
inst✝ : CommSemiring R
x : MvPowerSeries σ R
n : σ →₀ ℕ
h : ¬n = 0
⊢ ((coeff n) fun n ↦ (n.prod fun s m ↦ 0 ^ m) * (coeff n) x) = 0
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✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : Module 𝕜 E
inst✝³ : AddCommGroup F
inst✝² : TopologicalSpace F
inst✝¹ : Module 𝕜 F
l : Filter α
f : α → E
g : α → F
inst✝ : ContinuousConstSMul 𝕜 F
ι : Sort u_7
p : ι → Prop
U : ι → Set E
hb : (𝓝 0).HasBasis p U
h : ∀ (i : ι), p i → ∃ C, ∃ V ∈ 𝓝 0, (fun x ↦ egauge 𝕜 (U i) (f x)) ≤ᶠ[l] fun x ↦ ↑C * egauge 𝕜 V (g x)
i : ι
hi : p i
C : ℝ≥0
V : Set F
hV₀ : V ∈ 𝓝 0
hV : (fun x ↦ egauge 𝕜 (U i) (f x)) ≤ᶠ[l] fun x ↦ ↑C * egauge 𝕜 V (g x)
c : 𝕜
hc : C < ‖c‖₊
⊢ ∃ j ∈ 𝓝 0, ∀ᶠ (x : α) in l, egauge 𝕜 (U i) (f x) ≤ egauge 𝕜 (id j) (g x)

Tactic:
have hc₀ : c ≠ 0 := by rintro rfl; simp at hc

Goal state after:
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : Module 𝕜 E
inst✝³ : AddCommGroup F
inst✝² : TopologicalSpace F
inst✝¹ : Module 𝕜 F
l : Filter α
f : α → E
g : α → F
inst✝ : ContinuousConstSMul 𝕜 F
ι : Sort u_7
p : ι → Prop
U : ι → Set E
hb : (𝓝 0).HasBasis p U
h : ∀ (i : ι), p i → ∃ C, ∃ V ∈ 𝓝 0, (fun x ↦ egauge 𝕜 (U i) (f x)) ≤ᶠ[l] fun x ↦ ↑C * egauge 𝕜 V (g x)
i : ι
hi : p i
C : ℝ≥0
V : Set F
hV₀ : V ∈ 𝓝 0
hV : (fun x ↦ egauge 𝕜 (U i) (f x)) ≤ᶠ[l] fun x ↦ ↑C * egauge 𝕜 V (g x)
c : 𝕜
hc : C < ‖c‖₊
hc₀ : c ≠ 0
⊢ ∃ j ∈ 𝓝 0, ∀ᶠ (x : α) in l, egauge 𝕜 (U i) (f x) ≤ egauge 𝕜 (id j) (g x)
Module: Mathlib.Analysis.Calculus.Deriv.Basic

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
x : 𝕜
⊢ toSpanSingleton 𝕜 (deriv f x) = fderiv 𝕜 f x

Tactic:
simp only [deriv, ContinuousLinearMap.toSpanSingleton_apply_map_one]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : TopologicalSpace E
F : Type u_3
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜 F
inst✝⁴ : TopologicalSpace F
f : E → F
f' : E →L[𝕜] F
x : E
inst✝³ : ContinuousAdd E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : ContinuousAdd F
inst✝ : ContinuousSMul 𝕜 F
hf : HasFDerivAt f f' x
v : E
α : Type u_4
c : α → 𝕜
l : Filter α
hc : Tendsto (fun n ↦ ‖c n‖) l atTop
⊢ Tendsto (fun n ↦ c n • (f (x + (c n)⁻¹ • v) - f x)) l (𝓝 (f' v))

Tactic:
refine (hasFDerivWithinAt_univ.2 hf).lim ?_ (.of_forall fun _ ↦ mem_univ _) ?_
  · rw [tendsto_norm_atTop_iff_cobounded] at hc
    simpa using (tendsto_inv₀_cobounded.comp hc).smul (tendsto_const_nhds (x := v))
  · refine tendsto_nhds_of_eventually_eq ?_
    refine (eventually_ne_of_tendsto_norm_atTop hc (0 : 𝕜)).mono fun y hy => ?_
    simp [hy]

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

Goal state before:
pp: 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
c : 𝕜
hc : 1 < ‖c‖
⊢ ∃ ε, 0 < ε ∧ ↑ε * ↑‖c‖₊ / ↑r < 1

Tactic:
by
      apply Eventually.exists_gt
      refine Continuous.tendsto' ?_ _ _ (by simp) |>.eventually_lt_const zero_lt_one
      fun_prop (disch := intros; first | apply ENNReal.coe_ne_top | positivity)

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

Goal state before:
pp: 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 : Tendsto f l (𝓝 0)
U : Set E
hU : U ∈ 𝓝 0
ε : ℝ≥0
hε : ε ≠ 0
c : 𝕜
hcε : ‖c‖ < ↑ε
hc₀ : c ≠ 0
a : α
ha : a ∈ f ⁻¹' (c • U)
⊢ ↑ε ≤ ↑ε * egauge 𝕜 (ball 0 1) 1

Tactic:
apply le_mul_of_one_le_right'
        simpa using! le_egauge_ball_one 𝕜 (1 : 𝕜)

Goal state after: