mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.PowerSeries.Trunc

Goal state before:
pp: R : Type u_2
inst✝ : CommSemiring R
n m : ℕ
f : R⟦X⟧
h : n < m
⊢ (coeff n) ↑((trunc m) f) = (coeff n) f

Tactic:
by
  rwa [coeff_coe, coeff_trunc, if_pos]

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

Goal state before:
pp: R : Type u_1
inst✝¹ : TopologicalSpace R
inst✝ : Semiring R
ι : Type u_2
f : ι → R⟦X⟧
u : Filter ι
g : R⟦X⟧
d : Unit →₀ ℕ
⊢ Tendsto (fun i ↦ (MvPowerSeries.coeff d) (f i)) u (𝓝 ((MvPowerSeries.coeff d) g)) ↔
    Tendsto (fun i ↦ (coeff ((Finsupp.uniqueLinearEquiv ℕ ℕ ()).toEquiv d)) (f i)) u
      (𝓝 ((coeff ((Finsupp.uniqueLinearEquiv ℕ ℕ ()).toEquiv d)) g))

Tactic:
simp only [LinearEquiv.coe_toEquiv, Finsupp.uniqueLinearEquiv_apply, coeff]

Goal state after:
R : Type u_1
inst✝¹ : TopologicalSpace R
inst✝ : Semiring R
ι : Type u_2
f : ι → R⟦X⟧
u : Filter ι
g : R⟦X⟧
d : Unit →₀ ℕ
⊢ Tendsto (fun i ↦ (MvPowerSeries.coeff d) (f i)) u (𝓝 ((MvPowerSeries.coeff d) g)) ↔
    Tendsto (fun i ↦ (MvPowerSeries.coeff (Finsupp.single () (d ()))) (f i)) u
      (𝓝 ((MvPowerSeries.coeff (Finsupp.single () (d ()))) g))
Module: Mathlib.RingTheory.PowerSeries.Order

Goal state before:
pp: R : Type u_2
inst✝ : Semiring R
φ : R⟦X⟧
hφ : φ ≠ 0
n : ℕ
ho : φ.order = ↑n
hn : φ.order.toNat = n
ψ : R⟦X⟧
H : φ = X ^ (φ.order.toNat + 1) * ψ
this : (coeff n) φ = (coeff n) (X ^ (φ.order.toNat + 1) * ψ)
⊢ False

Tactic:
rw [X_pow_mul, coeff_mul_of_lt_order, ← hn] at this

Goal state after:
R : Type u_2
inst✝ : Semiring R
φ : R⟦X⟧
hφ : φ ≠ 0
n : ℕ
ho : φ.order = ↑n
hn : φ.order.toNat = n
ψ : R⟦X⟧
H : φ = X ^ (φ.order.toNat + 1) * ψ
this : (coeff φ.order.toNat) φ = 0
⊢ False

R : Type u_2
inst✝ : Semiring R
φ : R⟦X⟧
hφ : φ ≠ 0
n : ℕ
ho : φ.order = ↑n
hn : φ.order.toNat = n
ψ : R⟦X⟧
H : φ = X ^ (φ.order.toNat + 1) * ψ
this : (coeff n) φ = (coeff n) (ψ * X ^ (φ.order.toNat + 1))
⊢ ↑n < (X ^ (φ.order.toNat + 1)).order
Module: Mathlib.RingTheory.PowerSeries.PiTopology

Goal state before:
pp: R : Type u_1
inst✝¹ : TopologicalSpace R
inst✝ : CommSemiring R
f : R⟦X⟧
⊢ Tendsto (fun d ↦ ↑((trunc d) f)) atTop (𝓝 f)

Tactic:
rw [tendsto_iff_coeff_tendsto]
  intro d
  exact tendsto_atTop_of_eventually_const fun n (hdn : d < n) ↦ (by simp [coeff_trunc, hdn])

Goal state after:
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
r : R
⊢ eval₂ φ a (C r) = φ r

Tactic:
by
  rw [← coe_C, eval₂_coe, MvPolynomial.eval₂_C]

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝²¹ : CommRing R
inst✝²⁰ : UniformSpace R
S : Type u_3
inst✝¹⁹ : CommRing S
inst✝¹⁸ : UniformSpace S
a : σ → S
inst✝¹⁷ : IsTopologicalSemiring R
inst✝¹⁶ : IsUniformAddGroup R
inst✝¹⁵ : IsUniformAddGroup S
inst✝¹⁴ : CompleteSpace S
inst✝¹³ : T2Space S
inst✝¹² : IsTopologicalRing S
inst✝¹¹ : IsLinearTopology S S
inst✝¹⁰ : Algebra R S
inst✝⁹ : ContinuousSMul R S
ha : HasEval a
T : Type u_4
inst✝⁸ : CommRing T
inst✝⁷ : UniformSpace T
inst✝⁶ : IsUniformAddGroup T
inst✝⁵ : IsTopologicalRing T
inst✝⁴ : IsLinearTopology T T
inst✝³ : T2Space T
inst✝² : Algebra R T
inst✝¹ : ContinuousSMul R T
inst✝ : CompleteSpace T
ε : S →ₐ[R] T
hε : Continuous[inst✝¹⁸.toTopologicalSpace, inst✝⁷.toTopologicalSpace] ⇑ε
⊢ ε.comp (aeval ha) = aeval ⋯

Tactic:
by
  apply DFunLike.ext'
  simp only [AlgHom.coe_comp, coe_aeval ha]
  rw [← RingHom.coe_coe,
    comp_eval₂ (continuous_algebraMap R S) ha (show Continuous (ε : S →+* T) from hε), coe_aeval]
  congr!
  simp only [AlgHom.comp_algebraMap_of_tower]

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

Goal state before:
pp: c : ℝ
f : ℕ → Bool
⊢ cantorFunctionAux c f 0 = bif f 0 then 1 else 0

Tactic:
cases h : f 0 <;> simp [h]

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

Goal state before:
pp: σ : Type u_1
τ : Type u_4
S : Type u_5
inst✝² : CommRing S
a : σ → MvPowerSeries τ S
υ : Type u_7
T : Type u_8
inst✝¹ : CommRing T
inst✝ : Algebra S T
b : τ → MvPowerSeries υ T
ha : HasSubst a
hb : HasSubst b
this : UniformSpace S := ⊥
⊢ ∀ (d : υ →₀ ℕ), {s | (coeff d) ((substAlgHom hb) (a s)) ≠ 0}.Finite

Tactic:
let : UniformSpace T := ⊥

Goal state after:
σ : Type u_1
τ : Type u_4
S : Type u_5
inst✝² : CommRing S
a : σ → MvPowerSeries τ S
υ : Type u_7
T : Type u_8
inst✝¹ : CommRing T
inst✝ : Algebra S T
b : τ → MvPowerSeries υ T
ha : HasSubst a
hb : HasSubst b
this✝ : UniformSpace S := ⊥
this : UniformSpace T := ⊥
⊢ ∀ (d : υ →₀ ℕ), {s | (coeff d) ((substAlgHom hb) (a s)) ≠ 0}.Finite
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝⁶ : CommRing R
τ : Type u_4
S : Type u_5
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
a : σ → MvPowerSeries τ S
υ : Type u_7
T : Type u_8
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
b : τ → MvPowerSeries υ T
ha : HasSubst a
hb : HasSubst b
this✝ : UniformSpace R := ⊥
this : UniformSpace S := ⊥
⊢ (AlgHom.restrictScalars R (substAlgHom hb)).comp (substAlgHom ha) = substAlgHom ⋯

Tactic:
let : UniformSpace T := ⊥

Goal state after:
σ : Type u_1
R : Type u_3
inst✝⁶ : CommRing R
τ : Type u_4
S : Type u_5
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
a : σ → MvPowerSeries τ S
υ : Type u_7
T : Type u_8
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
b : τ → MvPowerSeries υ T
ha : HasSubst a
hb : HasSubst b
this✝¹ : UniformSpace R := ⊥
this✝ : UniformSpace S := ⊥
this : UniformSpace T := ⊥
⊢ (AlgHom.restrictScalars R (substAlgHom hb)).comp (substAlgHom ha) = substAlgHom ⋯
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: A : Type u_1
inst✝ : CommRing A
f : A⟦X⟧
hf : HasSubst f
n k : ℕ
hk : MvPowerSeries.constantCoeff f ^ k = 0
n' : ℕ
hn' : n' ≤ n
m : ℕ
hm : k * (n + 1) + m ∈ Set.Ici (k * (n + 1))
⊢ ↑n' < (f ^ (k * (n + 1)) * f ^ m).order

Tactic:
← order_mul_ge,

Goal state after:
A : Type u_1
inst✝ : CommRing A
f : A⟦X⟧
hf : HasSubst f
n k : ℕ
hk : MvPowerSeries.constantCoeff f ^ k = 0
n' : ℕ
hn' : n' ≤ n
m : ℕ
hm : k * (n + 1) + m ∈ Set.Ici (k * (n + 1))
⊢ ↑n' < (f ^ (k * (n + 1))).order + (f ^ m).order
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: R : Type u_2
inst✝² : CommRing R
S : Type u_4
inst✝¹ : CommRing S
inst✝ : Algebra R S
b : S⟦X⟧
hb : HasSubst b
f : R⟦X⟧
e : ℕ
⊢ (coeff e) (subst b f) = ∑ᶠ (d : ℕ), (coeff d) f • (coeff e) (b ^ d)

Tactic:
by
  simp [PowerSeries.coeff, coeff_subst hb]

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
h : c • x ∈ s
hc : c ≠ 0
⊢ egauge 𝕜 s x ≤ ‖c‖ₑ⁻¹

Tactic:
· exact (egauge_le_of_smul_mem_of_ne h hc).trans ENNReal.coe_inv_le

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
this✝¹ : UniformSpace R := ⊥
this✝ : IsUniformInducing DFunLike.coe
n : ℕ
s : Set ι
hs : s ∈ cofinite
hs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, (f y) i = 1 i
t : Finset ι := ⋯.toFinset
u v : Finset ι
hu : ∀ i ∈ u \ t, i ∈ s
hv : ∀ i ∈ v \ t, i ∈ s
this : ∀ k ≤ n, (∏ x ∈ u \ t, f x) k = (∏ x ∈ v \ t, f x) k
k : ℕ × ℕ
hk : k ∈ n.divisorsAntidiagonal
⊢ (∏ x ∈ u \ t, f x) k.1 * (∏ x ∈ t, f x) k.2 = (∏ x ∈ v \ t, f x) k.1 * (∏ x ∈ t, f x) k.2

Tactic:
rw [this k.1 (Nat.divisor_le (Nat.fst_mem_divisors_of_mem_antidiagonal hk))]

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

Goal state before:
pp: case neg
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
r : ℝ≥0∞
hr : ⨆ i ∈ I, egauge 𝕜 (U i) (x i) < r
c : ι → 𝕜
hc : ∀ i ∈ I, x i ∈ c i • U i
hcr : ∀ i ∈ I, ‖c i‖ₑ < r
hr₀ : 0 < r
hIne : I.Nonempty
i₀ : ι
hi₀I : i₀ ∈ I
hc_max : IsMaxOn (fun x ↦ ‖c x‖ₑ) I i₀
H : c i₀ = 0 ∧ I ≠ univ
hc0 : ∀ i ∈ I, c i = 0
heg0 : ∀ i ∈ I, x i = 0
this : (𝓝[≠] 0).NeBot
⊢ ∃ c₀, (c₀ ≠ 0 ∨ I = univ) ∧ (∀ i ∈ I, ‖c i‖ ≤ ‖c₀‖) ∧ ‖c₀‖ₑ < r

Tactic:
rcases exists_enorm_lt 𝕜 hr₀.ne' with ⟨c₁, hc₁, hc₁r⟩

Goal state after:
case neg
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
r : ℝ≥0∞
hr : ⨆ i ∈ I, egauge 𝕜 (U i) (x i) < r
c : ι → 𝕜
hc : ∀ i ∈ I, x i ∈ c i • U i
hcr : ∀ i ∈ I, ‖c i‖ₑ < r
hr₀ : 0 < r
hIne : I.Nonempty
i₀ : ι
hi₀I : i₀ ∈ I
hc_max : IsMaxOn (fun x ↦ ‖c x‖ₑ) I i₀
H : c i₀ = 0 ∧ I ≠ univ
hc0 : ∀ i ∈ I, c i = 0
heg0 : ∀ i ∈ I, x i = 0
this : (𝓝[≠] 0).NeBot
c₁ : 𝕜
hc₁ : c₁ ≠ 0
hc₁r : ‖c₁‖ₑ < r
⊢ ∃ c₀, (c₀ ≠ 0 ∨ I = univ) ∧ (∀ i ∈ I, ‖c i‖ ≤ ‖c₀‖) ∧ ‖c₀‖ₑ < r
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: R : Type u_2
inst✝¹ : CommRing R
P : R⟦X⟧
inst✝ : Invertible ((coeff 1) P)
⊢ HasSubst P.substInv

Tactic:
by simp [HasSubst, ← constantCoeff.eq_def]

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

Goal state before:
pp: case h
𝕜 : Type u_1
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : SMul 𝕜 E
inst✝¹ : TopologicalSpace E
s t : Set E
x : E
inst✝ : ContinuousAdd E
h : 𝓝[s] x ≤ 𝓝[t] x
y : E
hy : ClusterPt y (⊤ • 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0)
⊢ Tendsto (fun x_1 ↦ x + x_1) (𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0) (𝓝[s] x)

Tactic:
refine .inf ?_ (mapsTo_preimage _ _).tendsto

Goal state after:
case h
𝕜 : Type u_1
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : SMul 𝕜 E
inst✝¹ : TopologicalSpace E
s t : Set E
x : E
inst✝ : ContinuousAdd E
h : 𝓝[s] x ≤ 𝓝[t] x
y : E
hy : ClusterPt y (⊤ • 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0)
⊢ Tendsto (fun x_1 ↦ x + x_1) (𝓝 0) (𝓝 x)
Module: Mathlib.Analysis.Calculus.TangentCone.Real

Goal state before:
pp: E : Type u_1
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousSMul ℝ E
s : Set E
x : E
inst✝ : IsTopologicalAddGroup E
conv : Convex ℝ s
hx : x ∈ closure[inst✝²] s
y : E
hy : y ∈ interior s
⊢ y - x ∈ interior (tangentConeAt ℝ s x)

Tactic:
rw [mem_interior_iff_mem_nhds]

Goal state after:
E : Type u_1
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousSMul ℝ E
s : Set E
x : E
inst✝ : IsTopologicalAddGroup E
conv : Convex ℝ s
hx : x ∈ closure[inst✝²] s
y : E
hy : y ∈ interior s
⊢ tangentConeAt ℝ s x ∈ 𝓝 (y - x)
Module: Mathlib.Analysis.Calculus.Deriv.Basic

Goal state before:
pp: 𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : 𝕜 → F
f' : F
x : 𝕜
inst✝ : PartialOrder 𝕜
⊢ HasDerivWithinAt f f' (Iio x) x ↔ HasDerivWithinAt f f' (Iic x) x

Tactic:
← Iic_sdiff_right,

Goal state after:
𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : 𝕜 → F
f' : F
x : 𝕜
inst✝ : PartialOrder 𝕜
⊢ HasDerivWithinAt f f' (Iic x \ {x}) x ↔ HasDerivWithinAt f f' (Iic x) 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
s : Set F
x : 𝕜
⊢ deriv f x ∈ s ↔ DifferentiableAt 𝕜 f x ∧ deriv f x ∈ s ∨ ¬DifferentiableAt 𝕜 f x ∧ 0 ∈ s

Tactic:
by_cases hx : DifferentiableAt 𝕜 f x

Goal state after:
case pos
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
s : Set F
x : 𝕜
hx : DifferentiableAt 𝕜 f x
⊢ deriv f x ∈ s ↔ DifferentiableAt 𝕜 f x ∧ deriv f x ∈ s ∨ ¬DifferentiableAt 𝕜 f x ∧ 0 ∈ s

case neg
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
s : Set F
x : 𝕜
hx : ¬DifferentiableAt 𝕜 f x
⊢ deriv f x ∈ s ↔ DifferentiableAt 𝕜 f x ∧ deriv f x ∈ s ∨ ¬DifferentiableAt 𝕜 f x ∧ 0 ∈ s
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
t : Set 𝕜
s : Set F
x : 𝕜
⊢ derivWithin f t x ∈ s ↔
    DifferentiableWithinAt 𝕜 f t x ∧ derivWithin f t x ∈ s ∨ ¬DifferentiableWithinAt 𝕜 f t x ∧ 0 ∈ s

Tactic:
by
  by_cases hx : DifferentiableWithinAt 𝕜 f t x <;>
    simp [derivWithin_zero_of_not_differentiableWithinAt, *]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
F : Type u_3
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : TopologicalSpace F
f : E → F
f' : E →L[𝕜] F
x : E
s : Set E
inst✝ : T1Space E
y : E
⊢ HasFDerivWithinAt f f' (insert y s) x ↔ HasFDerivWithinAt f f' s x

Tactic:
rcases eq_or_ne x y with (rfl | h)
  · apply hasFDerivWithinAt_insert_self
  · refine ⟨.of_insert, fun hf => hf.mono_of_mem_nhdsWithin ?_⟩
    simp_rw [nhdsWithin_insert_of_ne h, self_mem_nhdsWithin]

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

Goal state before:
pp: case h.h.h.h.h.h.hq
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
l : Filter α
f : α → E
g : α → F
c : α → 𝕜
h : ∀ U ∈ 𝓝 0, ∃ V ∈ 𝓝 0, ∀ (ε : ℝ≥0), ε ≠ 0 → (fun x ↦ egauge 𝕜 U (f x)) ≤ᶠ[l] fun x ↦ ↑ε * egauge 𝕜 V (g x)
U : Set E
hU : U ∈ 𝓝 0
V : Set F
hV : V ∈ 𝓝 0
ε : ℝ≥0
hε : ε ≠ 0
x : α
hx : egauge 𝕜 U (f x) ≤ ↑ε * egauge 𝕜 V (g x)
⊢ egauge 𝕜 U (c x • f x) ≤ ↑ε * egauge 𝕜 V (c x • g x)

Tactic:
rw [egauge_smul_right, egauge_smul_right, mul_left_comm]

Goal state after:
case h.h.h.h.h.h.hq
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
l : Filter α
f : α → E
g : α → F
c : α → 𝕜
h : ∀ U ∈ 𝓝 0, ∃ V ∈ 𝓝 0, ∀ (ε : ℝ≥0), ε ≠ 0 → (fun x ↦ egauge 𝕜 U (f x)) ≤ᶠ[l] fun x ↦ ↑ε * egauge 𝕜 V (g x)
U : Set E
hU : U ∈ 𝓝 0
V : Set F
hV : V ∈ 𝓝 0
ε : ℝ≥0
hε : ε ≠ 0
x : α
hx : egauge 𝕜 U (f x) ≤ ↑ε * egauge 𝕜 V (g x)
⊢ ‖c x‖ₑ * egauge 𝕜 U (f x) ≤ ‖c x‖ₑ * (↑ε * egauge 𝕜 V (g x))

case h.h.h.h.h.h.hq.h
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
l : Filter α
f : α → E
g : α → F
c : α → 𝕜
h : ∀ U ∈ 𝓝 0, ∃ V ∈ 𝓝 0, ∀ (ε : ℝ≥0), ε ≠ 0 → (fun x ↦ egauge 𝕜 U (f x)) ≤ᶠ[l] fun x ↦ ↑ε * egauge 𝕜 V (g x)
U : Set E
hU : U ∈ 𝓝 0
V : Set F
hV : V ∈ 𝓝 0
ε : ℝ≥0
hε : ε ≠ 0
x : α
hx : egauge 𝕜 U (f x) ≤ ↑ε * egauge 𝕜 V (g x)
⊢ c x = 0 → V.Nonempty

case h.h.h.h.h.h.hq.h
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
l : Filter α
f : α → E
g : α → F
c : α → 𝕜
h : ∀ U ∈ 𝓝 0, ∃ V ∈ 𝓝 0, ∀ (ε : ℝ≥0), ε ≠ 0 → (fun x ↦ egauge 𝕜 U (f x)) ≤ᶠ[l] fun x ↦ ↑ε * egauge 𝕜 V (g x)
U : Set E
hU : U ∈ 𝓝 0
V : Set F
hV : V ∈ 𝓝 0
ε : ℝ≥0
hε : ε ≠ 0
x : α
hx : egauge 𝕜 U (f x) ≤ ↑ε * egauge 𝕜 V (g x)
⊢ c x = 0 → U.Nonempty
Module: Mathlib.Analysis.Calculus.FDeriv.Basic

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 : E → F
s : Set (E →L[𝕜] F)
x : E
⊢ fderiv 𝕜 f x ∈ s ↔ DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ s ∨ ¬DifferentiableAt 𝕜 f x ∧ 0 ∈ s

Tactic:
by_cases hx : DifferentiableAt 𝕜 f x

Goal state after:
case pos
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 : E → F
s : Set (E →L[𝕜] F)
x : E
hx : DifferentiableAt 𝕜 f x
⊢ fderiv 𝕜 f x ∈ s ↔ DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ s ∨ ¬DifferentiableAt 𝕜 f x ∧ 0 ∈ s

case neg
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 : E → F
s : Set (E →L[𝕜] F)
x : E
hx : ¬DifferentiableAt 𝕜 f x
⊢ fderiv 𝕜 f x ∈ s ↔ DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ s ∨ ¬DifferentiableAt 𝕜 f x ∧ 0 ∈ s
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 > 0, ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖) →
    ∀ (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)

Tactic:
rintro ⟨C, hC₀, hC⟩ r 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
hC₀ : C > 0
hC : ∀ᶠ (x : α) in l, ‖f x‖ ≤ C * ‖g x‖
r : ℝ
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.ChangeOrigin

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
p : FormalMultilinearSeries 𝕜 E F
x y : E
h : ↑‖x‖₊ + ↑‖y‖₊ < p.radius
x_mem_ball : x ∈ Metric.eball 0 p.radius
y_mem_ball : y ∈ Metric.eball 0 (p.changeOrigin x).radius
x_add_y_mem_ball : x + y ∈ Metric.eball 0 p.radius
f : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=
  fun s ↦ ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 ↦ x) fun x ↦ y
hsf : Summable f
k : ℕ
this : HasSum (fun n ↦ (p.changeOriginSeries k n) fun x_1 ↦ x) (p.changeOrigin (0 + x) k)
⊢ HasSum (fun c ↦ (p.changeOriginSeriesTerm k c.fst ↑c.snd ⋯) fun x_1 ↦ x) (p.changeOrigin x k)

Tactic:
zero_add

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
p : FormalMultilinearSeries 𝕜 E F
x y : E
h : ↑‖x‖₊ + ↑‖y‖₊ < p.radius
x_mem_ball : x ∈ Metric.eball 0 p.radius
y_mem_ball : y ∈ Metric.eball 0 (p.changeOrigin x).radius
x_add_y_mem_ball : x + y ∈ Metric.eball 0 p.radius
f : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=
  fun s ↦ ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 ↦ x) fun x ↦ y
hsf : Summable f
k : ℕ
this : HasSum (fun n ↦ (p.changeOriginSeries k n) fun x_1 ↦ x) (p.changeOrigin x k)
⊢ HasSum (fun c ↦ (p.changeOriginSeriesTerm k c.fst ↑c.snd ⋯) fun x_1 ↦ x) (p.changeOrigin x k)
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : Ring E
inst✝² : Algebra 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : IsTopologicalRing E
c : ℕ → 𝕜
n : ℕ
⊢ ((ofScalars E c n) fun x ↦ 0) = Pi.single 0 (c 0 • 1) n

Tactic:
rw [ofScalars]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : Ring E
inst✝² : Algebra 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : IsTopologicalRing E
c : ℕ → 𝕜
n : ℕ
⊢ ((c n • ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n E) fun x ↦ 0) = Pi.single 0 (c 0 • 1) 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
y : E
x✝ : y ∈ Metric.eball 0 ∞
⊢ HasSum (fun n ↦ (p n) fun x ↦ y) (p.sum (0 + y))

Tactic:
zero_add

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
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
y : E
x✝ : y ∈ Metric.eball 0 ∞
⊢ HasSum (fun n ↦ (p n) fun x ↦ y) (p.sum y)
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
k : ℕ
x : E
x✝ : Fin k → E
m : ℕ
hm : n - k ≤ m
⊢ ((p.changeOriginSeries k m) fun x_1 ↦ x) = 0

Tactic:
p.changeOriginSeries_finite_of_finite hn k (by rw [add_comm]; exact Nat.le_add_of_sub_le hm),

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
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
k : ℕ
x : E
x✝ : Fin k → E
m : ℕ
hm : n - k ≤ m
⊢ (0 fun x_1 ↦ x) = 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 : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
⊢ HasFPowerSeriesWithinOnBall f p (insert x s) x r ↔ HasFPowerSeriesWithinOnBall f p s x r

Tactic:
by
  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩  <;>
  exact ⟨h.r_le, h.r_pos, fun {y} ↦ by simpa only [insert_idem] using h.hasSum (y := y)⟩

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

Goal state before:
pp: case neg
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∞
hr : r = 0
hc' : Tendsto (fun n ↦ ENNReal.ofReal ‖c n.succ‖ / ENNReal.ofReal ‖c n‖) atTop (𝓝 0)
h : ¬∀ᶠ (n : ℕ) in atTop, c n ≠ 0
⊢ ∀ᶠ (n : ℕ) in atTop, ofScalars E c n = 0

Tactic:
simp only [eventually_atTop, not_exists, not_forall, not_not] at h ⊢

Goal state after:
case neg
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∞
hr : r = 0
hc' : Tendsto (fun n ↦ ENNReal.ofReal ‖c n.succ‖ / ENNReal.ofReal ‖c n‖) atTop (𝓝 0)
h : ∀ (x : ℕ), ∃ x_1, ∃ (_ : x ≤ x_1), c x_1 = 0
⊢ ∃ a, ∀ (b : ℕ), a ≤ b → ofScalars E c b = 0