mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.LSeries.Basic

Goal state before:
pp: f : ℕ → ℂ
s : ℂ
n : ℕ
⊢ (↑n + 1) ^ s * term f s (n + 1) = f (n + 1)

Tactic:
by
  simp [term, natCast_add_one_cpow_ne_zero n _, mul_div_assoc']

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

Goal state before:
pp: 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
ha : HasSubst a
f : MvPowerSeries σ R
⊢ (⨅ i, (a i).order) * f.order ≤ (subst a f).order

Tactic:
refine .trans ?_ (MvPowerSeries.le_weightedOrder_subst _ ha _)

Goal state after:
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
ha : HasSubst a
f : MvPowerSeries σ R
⊢ (⨅ i, (a i).order) * f.order ≤ ⨅ d, ⨅ (_ : (coeff d) f ≠ 0), (Finsupp.weight ((weightedOrder fun x ↦ 1) ∘ a)) d
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
f : R⟦X⟧
⊢ (map (algebraMap R R)) f = f

Tactic:
Algebra.algebraMap_self

Goal state after:
R : Type u_2
inst✝ : CommRing R
f : R⟦X⟧
⊢ (map (RingHom.id R)) f = f
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: 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
⊢ (n.prod fun s m ↦ a s ^ m) * (coeff n) (f * g) =
    (coeff n) ((fun n ↦ (n.prod fun s m ↦ a s ^ m) * (coeff n) f) * fun n ↦ (n.prod fun s m ↦ a s ^ m) * (coeff n) g)

Tactic:
coeff_mul,

Goal state after:
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
⊢ (n.prod fun s m ↦ a s ^ m) * ∑ p ∈ Finset.antidiagonal n, (coeff p.1) f * (coeff p.2) g =
    (coeff n) ((fun n ↦ (n.prod fun s m ↦ a s ^ m) * (coeff n) f) * fun n ↦ (n.prod fun s m ↦ a s ^ m) * (coeff n) g)
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: 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
⊢ (n.prod fun s m ↦ a s ^ m) * ∑ p ∈ Finset.antidiagonal n, (coeff p.1) f * (coeff p.2) g =
    (coeff n) ((fun n ↦ (n.prod fun s m ↦ a s ^ m) * (coeff n) f) * fun n ↦ (n.prod fun s m ↦ a s ^ m) * (coeff n) g)

Tactic:
coeff_mul,

Goal state after:
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
⊢ (n.prod fun s m ↦ a s ^ m) * ∑ p ∈ Finset.antidiagonal n, (coeff p.1) f * (coeff p.2) g =
    ∑ p ∈ Finset.antidiagonal n,
      ((coeff p.1) fun n ↦ (n.prod fun s m ↦ a s ^ m) * (coeff n) f) *
        (coeff p.2) fun n ↦ (n.prod fun s m ↦ a s ^ m) * (coeff n) g
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
a : R
σ : Type u_7
p : MvPowerSeries σ R
hp_lin : ∀ d ∈ Function.support p, Finsupp.degree d = 1
f : R⟦X⟧
hp : HasSubst p
⊢ subst p (subst (a • X) f) = MvPowerSeries.subst (Function.const σ a • MvPowerSeries.X) (subst p f)

Tactic:
subst_comp_subst_apply (HasSubst.smul_X' a) hp

Goal state after:
R : Type u_2
inst✝ : CommRing R
a : R
σ : Type u_7
p : MvPowerSeries σ R
hp_lin : ∀ d ∈ Function.support p, Finsupp.degree d = 1
f : R⟦X⟧
hp : HasSubst p
⊢ subst (subst p (a • X)) f = MvPowerSeries.subst (Function.const σ a • MvPowerSeries.X) (subst p f)
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
x : ℕ → MvPowerSeries σ R
inst✝³ : UniformSpace R
inst✝² : DiscreteUniformity R
inst✝¹ : UniformSpace S
inst✝ : DiscreteUniformity S
hx : Summable x
ha : HasSubst a
⊢ Summable fun i ↦ subst a (x i)

Tactic:
rw [← coe_substAlgHom ha, substAlgHom_eq_aeval ha]
  exact hx.map _ <| continuous_aeval ha.hasEval

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

Goal state before:
pp: case inl.inr
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)
U : (i : ι) → Set (E i)
x : (i : ι) → E i
r : ℝ≥0∞
c : ι → 𝕜
hr₀ : 0 < r
hI : ∅.Finite
hU : ∀ i ∈ ∅, Balanced 𝕜 (U i)
hI₀ : ∅ = univ ∨ (∃ i ∈ ∅, x i ≠ 0) ∨ (𝓝[≠] 0).NeBot
hr : ⨆ i ∈ ∅, egauge 𝕜 (U i) (x i) < r
hc : ∀ i ∈ ∅, x i ∈ c i • U i
hcr : ∀ i ∈ ∅, ‖c i‖ₑ < r
hbot : (𝓝[≠] 0).NeBot
⊢ ∃ c₀, (c₀ ≠ 0 ∨ ∅ = univ) ∧ (∀ i ∈ ∅, ‖c i‖ ≤ ‖c₀‖) ∧ ‖c₀‖ₑ < r

Tactic:
rcases exists_enorm_lt 𝕜 hr₀.ne' with ⟨c₀, hc₀, hc₀r⟩
        exact ⟨c₀, .inl hc₀, by simp, hc₀r⟩

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

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
f : R⟦X⟧
e : Fin 2 →₀ ℕ
⊢ ∑ᶠ (d : Unit →₀ ℕ), (MvPowerSeries.coeff d) f • (MvPowerSeries.coeff e) (d.prod fun s e ↦ (X₀ + X₁) ^ e) =
    ↑((e 0 + e 1).choose (e 0)) * (coeff (e 0 + e 1)) f

Tactic:
Finsupp.prod_pow,

Goal state after:
R : Type u_2
inst✝ : CommRing R
f : R⟦X⟧
e : Fin 2 →₀ ℕ
⊢ ∑ᶠ (d : Unit →₀ ℕ), (MvPowerSeries.coeff d) f • (MvPowerSeries.coeff e) (∏ a, (X₀ + X₁) ^ d a) =
    ↑((e 0 + e 1).choose (e 0)) * (coeff (e 0 + e 1)) f
Module: Mathlib.Analysis.Asymptotics.TVS

Goal state before:
pp: case refine_2
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
ιE : Sort u_7
ιF : Sort u_8
pE : ιE → Prop
pF : ιF → Prop
sE : ιE → Set E
sF : ιF → Set F
hE : (𝓝 0).HasBasis pE sE
hF : (𝓝 0).HasBasis pF sF
x✝¹ : ιE
x✝ : pE x✝¹
⊢ ∀ ⦃s t : Set F⦄,
    s ⊆ t →
      ((fun x ↦ egauge 𝕜 (sE x✝¹) (f x)) ≤ᶠ[l] fun x ↦ egauge 𝕜 t (g x)) →
        (fun x ↦ egauge 𝕜 (sE x✝¹) (f x)) ≤ᶠ[l] fun x ↦ egauge 𝕜 s (g x)

Tactic:
exact fun s t hsub h ↦ h.mono fun x hx ↦ hx.trans <| egauge_anti 𝕜 hsub (g x)

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

Goal state before:
pp: α : Type u_1
β : Type u_2
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
k : β → α
l : Filter β
⊢ f =o[𝕜; map k l] g ↔ (f ∘ k) =o[𝕜; l] (g ∘ k)

Tactic:
simp [isLittleOTVS_iff, EventuallyLE]

Goal state after:
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
f' : E →L[𝕜] F
x : E
⊢ HasFDerivWithinAt f f' ∅ x

Tactic:
by
  simp [HasFDerivWithinAt, hasFDerivAtFilter_iff_isLittleOTVS]

Goal state after:
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
⊢ radius 0 = ∞

Tactic:
← constFormalMultilinearSeries_zero

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
⊢ (constFormalMultilinearSeries 𝕜 E 0).radius = ∞
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : Field 𝕜
inst✝⁴ : Ring E
inst✝³ : Algebra 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalRing E
c : ℕ → 𝕜
inst✝ : Nontrivial E
n : ℕ
⊢ c n = 0 ∨ ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n E = 0 ↔ c n = 0

Tactic:
refine or_iff_left (ContinuousMultilinearMap.ext_iff.1.mt <| not_forall_of_exists_not ?_)

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : Field 𝕜
inst✝⁴ : Ring E
inst✝³ : Algebra 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalRing E
c : ℕ → 𝕜
inst✝ : Nontrivial E
n : ℕ
⊢ ∃ x, ¬(ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n E) x = 0 x
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
⊢ n ≤ k + m

Tactic:
by rw [add_comm]; exact Nat.le_add_of_sub_le hm

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

Goal state before:
pp: case mpr
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
r : ℝ≥0∞
⊢ HasFPowerSeriesOnBall f p x r → HasFPowerSeriesWithinOnBall f p univ x r

Tactic:
· intro h
    exact ⟨h.r_le, h.r_pos, fun {y} _ m => h.hasSum m⟩

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
pf : FormalMultilinearSeries 𝕜 E F
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f pf univ x r
v : Fin 0 → E
⊢ (pf 0) v = f x

Tactic:
exact hf.coeff_zero v

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', ‖y‖₊ < r' ∧ ↑r' < r

Tactic:
simp at hy

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
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
hy : ‖y‖ₑ < r
⊢ ∃ r', ‖y‖₊ < r' ∧ ↑r' < r
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∞
r' : ℝ≥0
hf : HasFPowerSeriesWithinOnBall f p s x r
h : ↑r' < r
⊢ ∃ a ∈ Ioo 0 1,
    ∃ C > 0,
      ∀ y ∈ Metric.ball 0 ↑r',
        ∀ (n : ℕ), x + y ∈ insert x s → ‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / ↑r')) ^ n

Tactic:
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n :=
    p.norm_mul_pow_le_mul_pow_of_lt_radius (h.trans_le hf.r_le)

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∞
r' : ℝ≥0
hf : HasFPowerSeriesWithinOnBall f p s x r
h : ↑r' < r
a : ℝ
ha : a ∈ Ioo 0 1
C : ℝ
hC : C > 0
hp : ∀ (n : ℕ), ‖p n‖ * ↑r' ^ n ≤ C * a ^ n
⊢ ∃ a ∈ Ioo 0 1,
    ∃ C > 0,
      ∀ y ∈ Metric.ball 0 ↑r',
        ∀ (n : ℕ), x + y ∈ insert x s → ‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / ↑r')) ^ n
Module: Mathlib.Analysis.Analytic.Within

Goal state before:
pp: case inr
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
x : E
h : {x} ∈ 𝓝[s] x
t : Set E
ot : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
xt : x ∈ t
r : ℝ
r0 : r > 0
rt : ball x r ⊆ t
y : E
yr : y ∈ eball 0 (ENNReal.ofReal r)
st : ∀ y ∈ t, y ∈ s → y = x
ys : x + y ∈ s
⊢ x + y = x

Tactic:
exact st (x + y) (rt (by simpa using yr)) ys

Goal state after:
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
g : F → G
f : E → F
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
t : Set F
s : Set E
hs : Set.MapsTo f s t
rg : ℝ≥0∞
Hg : HasFPowerSeriesWithinOnBall g q t (f x) rg
rf : ℝ≥0∞
Hf : HasFPowerSeriesWithinOnBall f p s x rf
r : ℝ≥0
r_pos : 0 < r
hr : Summable fun i ↦ ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst
δ : ℝ≥0∞
δpos : 0 < δ
hδ : ∀ {z : E}, z ∈ insert x s ∩ Metric.eball x δ → f z ∈ insert (f x) t ∩ Metric.eball (f x) rg
rf' : ℝ≥0∞ := min rf δ
min_pos : 0 < min rf' ↑r
y : E
h'y : x + y ∈ insert x s
hy : y ∈ Metric.eball 0 (min rf' ↑r)
y_mem : y ∈ Metric.eball 0 rf
fy_mem : f (x + y) ∈ insert (f x) t ∩ Metric.eball (f x) rg
A : Tendsto (fun n ↦ (n, ∑ a ∈ Finset.Ico 1 n, (p a) fun x ↦ y)) atTop (atTop ×ˢ 𝓝 (f (x + y) - f x))
B : Tendsto (fun n ↦ q.partialSum n (∑ a ∈ Finset.Ico 1 n, (p a) fun _b ↦ y)) atTop (𝓝 (g (f (x + y))))
⊢ Tendsto (fun n ↦ ∑ i ∈ compPartialSumTarget 0 n n, (q.compAlongComposition p i.snd) fun _j ↦ y) atTop
    (𝓝 (g (f (x + y))))

Tactic:
simpa [comp_partialSum] using! B

Goal state after:
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
H : Type u_5
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
n : ℕ
a : Composition n
b : Composition a.length
i : Fin (a.gather b).length
⊢ ↑i < (a.blocks.splitWrtComposition b).length

Tactic:
rw [length_splitWrtComposition, ← length_gather]; exact i.2

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

Goal state before:
pp: case zero.succ
n : ℕ
a : Composition n
b : Composition a.length
i : ℕ
IH :
  ∀ (hi : i < b.length),
    0 < b.blocksFun ⟨i, hi⟩ →
      (take (take i b.blocks).sum a.blocks).sum = (take i (List.map sum (a.blocks.splitWrtComposition b))).sum
hi : i + 1 < b.length
hj : 0 < b.blocksFun ⟨i + 1, hi⟩
A : i < b.length
B : i < (List.map sum (a.blocks.splitWrtComposition b)).length
C : 0 < b.blocksFun ⟨i, A⟩
this : take (take i b.blocks).sum a.blocks = take (take i b.blocks).sum (take (take (i + 1) b.blocks).sum a.blocks)
⊢ (take (take (i + 1) b.blocks).sum a.blocks).sum =
    (take (take i b.blocks).sum (take (take (i + 1) b.blocks).sum a.blocks)).sum +
      (a.blocks.splitWrtComposition b)[i].sum

Tactic:
getElem_splitWrtComposition,

Goal state after:
case zero.succ
n : ℕ
a : Composition n
b : Composition a.length
i : ℕ
IH :
  ∀ (hi : i < b.length),
    0 < b.blocksFun ⟨i, hi⟩ →
      (take (take i b.blocks).sum a.blocks).sum = (take i (List.map sum (a.blocks.splitWrtComposition b))).sum
hi : i + 1 < b.length
hj : 0 < b.blocksFun ⟨i + 1, hi⟩
A : i < b.length
B : i < (List.map sum (a.blocks.splitWrtComposition b)).length
C : 0 < b.blocksFun ⟨i, A⟩
this : take (take i b.blocks).sum a.blocks = take (take i b.blocks).sum (take (take (i + 1) b.blocks).sum a.blocks)
⊢ (take (take (i + 1) b.blocks).sum a.blocks).sum =
    (take (take i b.blocks).sum (take (take (i + 1) b.blocks).sum a.blocks)).sum +
      (drop (b.sizeUpTo i) (take (b.sizeUpTo (i + 1)) a.blocks)).sum
Module: Mathlib.Analysis.Analytic.Constructions

Goal state before:
pp: 𝕜 : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
ι : Type u_9
inst✝² : Fintype ι
e : E
Fm : ι → Type u_10
inst✝¹ : (i : ι) → NormedAddCommGroup (Fm i)
inst✝ : (i : ι) → NormedSpace 𝕜 (Fm i)
f : (i : ι) → E → Fm i
s : Set E
r : ℝ≥0∞
p : (i : ι) → FormalMultilinearSeries 𝕜 E (Fm i)
hf : ∀ (i : ι), HasFPowerSeriesWithinOnBall (f i) (p i) s e r
hr : 0 < r
⊢ r ≤ (FormalMultilinearSeries.pi p).radius

Tactic:
apply FormalMultilinearSeries.le_radius_pi (fun i ↦ ?_)
    exact (hf i).r_le

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

Goal state before:
pp: 𝕜 : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
A : Type u_7
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : HasSummableGeomSeries A
⊢ HasFPowerSeriesOnBall (fun x ↦ (1 - x)⁻¹ʳ) (formalMultilinearSeries_geometric 𝕜 A) 0 1

Tactic:
constructor
  · exact one_le_formalMultilinearSeries_geometric_radius 𝕜 A
  · exact one_pos
  · intro y hy
    simp only [Metric.mem_eball, edist_dist, dist_zero_right, ofReal_lt_one] at hy
    simp only [zero_add, NormedRing.inverse_one_sub _ hy, Units.oneSub, Units.inv_mk,
      formalMultilinearSeries_geometric, ContinuousMultilinearMap.mkPiAlgebraFin_apply,
      List.ofFn_const, List.prod_replicate]
    exact (summable_geometric_of_norm_lt_one hy).hasSum

Goal state after:
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
H : Type u_5
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
r : FormalMultilinearSeries 𝕜 G H
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
v : Fin n → E
f : (a : Composition n) × Composition a.length → H :=
  fun c ↦ (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))
g : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=
  fun c ↦ (r c.fst.length) fun i ↦ (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))
a : Composition n
b : Composition a.length
a✝ : ⟨a, b⟩ ∈ Finset.univ
⊢ f ⟨a, b⟩ = g ((sigmaEquivSigmaPi n) ⟨a, b⟩)

Tactic:
dsimp [sigmaEquivSigmaPi]

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
r : FormalMultilinearSeries 𝕜 G H
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
v : Fin n → E
f : (a : Composition n) × Composition a.length → H :=
  fun c ↦ (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))
g : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=
  fun c ↦ (r c.fst.length) fun i ↦ (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))
a : Composition n
b : Composition a.length
a✝ : ⟨a, b⟩ ∈ Finset.univ
⊢ f ⟨a, b⟩ = g ⟨a.gather b, a.sigmaCompositionAux b⟩
Module: Mathlib.Analysis.Calculus.FDeriv.Equiv

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
iso : E ≃L[𝕜] F
f : F → G
x : E
⊢ fderiv 𝕜 (f ∘ ⇑iso) x = fderiv 𝕜 f (iso x) ∘SL ↑iso

Tactic:
by
  rw [← fderivWithin_univ, ← fderivWithin_univ, ← iso.comp_right_fderivWithin, preimage_univ]
  exact uniqueDiffWithinAt_univ

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Alternating.Basic

Goal state before:
pp: 𝕜 : Type u
n : ℕ
E : Type wE
F : Type wF
G : Type wG
ι : Type v
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : SeminormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
g : E →L[𝕜] F
f : F [⋀^ι]→L[𝕜] G
dg : E →L[𝕜] F
v : ι → E
⊢ ‖(({ toFun := fun x ↦ x.fderivCompContinuousLinearMap g, map_add' := ⋯, map_smul' := ⋯ } f) dg) v‖ ≤
    ↑(Fintype.card ι) * ‖g‖ ^ (Fintype.card ι - 1) * ‖f‖ * ‖dg‖ * ∏ i, ‖v i‖

Tactic:
simp? [mul_assoc] says
        simp only [LinearMap.coe_mk, AddHom.coe_mk, fderivCompContinuousLinearMap_apply, mul_assoc]

Goal state after:
𝕜 : Type u
n : ℕ
E : Type wE
F : Type wF
G : Type wG
ι : Type v
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : SeminormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
g : E →L[𝕜] F
f : F [⋀^ι]→L[𝕜] G
dg : E →L[𝕜] F
v : ι → E
⊢ ‖∑ i, f fun j ↦ (Function.update (fun x ↦ g) i dg j) (v j)‖ ≤
    ↑(Fintype.card ι) * (‖g‖ ^ (Fintype.card ι - 1) * (‖f‖ * (‖dg‖ * ∏ i, ‖v i‖)))