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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
φ : MvPowerSeries σ R
⊢ X s ∣ φ ↔ ∀ (m : σ →₀ ℕ), m s = 0 → (coeff m) φ = 0

Tactic:
by
  rw [← pow_one (X s : MvPowerSeries σ R), X_pow_dvd_iff]
  constructor <;> intro h m hm
  · exact h m (hm.symm ▸ zero_lt_one)
  · exact h m (Nat.eq_zero_of_le_zero <| Nat.le_of_succ_le_succ hm)

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

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝¹ : CommSemiring R
inst✝ : DecidableEq σ
f : MvPowerSeries σ R
n : ℕ
d : σ →₀ ℕ
⊢ f ^ n = ∏ x ∈ range n, f

Tactic:
Finset.prod_const,

Goal state after:
σ : Type u_1
R : Type u_3
inst✝¹ : CommSemiring R
inst✝ : DecidableEq σ
f : MvPowerSeries σ R
n : ℕ
d : σ →₀ ℕ
⊢ f ^ n = f ^ #(range n)
Module: Mathlib.RingTheory.MvPowerSeries.Basic

Goal state before:
pp: σ : Type u_4
R : Type u_5
inst✝ : CommSemiring R
e : σ →₀ ℕ
r : R
⊢ (MvPowerSeries.monomial e) (r ^ e.sum fun x n ↦ n) = e.prod fun s e ↦ (r • MvPowerSeries.X s) ^ e

Tactic:
rw [MvPowerSeries.prod_smul_X_eq_smul_monomial_one, ← map_smul, smul_eq_mul, mul_one]
  simp only [Finsupp.sum, Finsupp.prod, Finset.prod_pow_eq_pow_sum]

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

Goal state before:
pp: case ind
σ : Type u_1
R : Type u_2
n : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
p : MvPowerSeries σ R
k : ℕ
ih : ∀ m ≤ k, ∀ {x : σ →₀ ℕ}, degree x < n → MvPolynomial.coeff x ((truncTotal n) p ^ m) = (coeff x) (p ^ m)
x : σ →₀ ℕ
h : degree x < n
⊢ MvPolynomial.coeff x ((truncTotal n) p ^ k.succ) = (coeff x) (p ^ k.succ)

Tactic:
Nat.succ_eq_add_one,

Goal state after:
case ind
σ : Type u_1
R : Type u_2
n : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
p : MvPowerSeries σ R
k : ℕ
ih : ∀ m ≤ k, ∀ {x : σ →₀ ℕ}, degree x < n → MvPolynomial.coeff x ((truncTotal n) p ^ m) = (coeff x) (p ^ m)
x : σ →₀ ℕ
h : degree x < n
⊢ MvPolynomial.coeff x ((truncTotal n) p ^ (k + 1)) = (coeff x) (p ^ (k + 1))
Module: Mathlib.RingTheory.MvPowerSeries.Trunc

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

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

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
p : ℕ
a : R
f : MvPowerSeries σ R
⊢ (fun d ↦ if (weight w) d = p then (coeff d) (a • f) else 0) =
    (RingHom.id R) a • fun d ↦ if (weight w) d = p then (coeff d) f else 0

Tactic:
by
    ext d
    simp only [map_smul,
      smul_eq_mul, RingHom.id_apply, coeff_apply, mul_ite, mul_zero]

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

Goal state before:
pp: case right
R : Type u_1
inst✝ : Semiring R
φ ψ : R⟦X⟧
H : φ.order < ψ.order
i : ℕ
hi : ↑i < φ.order
⊢ 0 + (coeff i) ψ = 0

Tactic:
coeff_of_lt_order i (lt_trans hi H),

Goal state after:
case right
R : Type u_1
inst✝ : Semiring R
φ ψ : R⟦X⟧
H : φ.order < ψ.order
i : ℕ
hi : ↑i < φ.order
⊢ 0 + 0 = 0
Module: Mathlib.Topology.Algebra.LinearTopology

Goal state before:
pp: R : Type u_1
M : Type u_3
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : TopologicalSpace M
inst✝ : IsLinearTopology R M
ι : Type u_4
f : Filter ι
a : ι → R
m : ι → M
ha : ∀ (i : Submodule R M), ↑i ∈ 𝓝 0 → ∀ᶠ (x : ι) in f, m x ∈ ↑i
⊢ ∀ (i : Submodule R M), ↑i ∈ 𝓝 0 → ∀ᶠ (x : ι) in f, (a • m) x ∈ ↑i

Tactic:
intro I hI

Goal state after:
R : Type u_1
M : Type u_3
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : TopologicalSpace M
inst✝ : IsLinearTopology R M
ι : Type u_4
f : Filter ι
a : ι → R
m : ι → M
ha : ∀ (i : Submodule R M), ↑i ∈ 𝓝 0 → ∀ᶠ (x : ι) in f, m x ∈ ↑i
I : Submodule R M
hI : ↑I ∈ 𝓝 0
⊢ ∀ᶠ (x : ι) in f, (a • m) x ∈ ↑I
Module: Mathlib.RingTheory.MvPowerSeries.Evaluation

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : UniformSpace R
S : Type u_3
inst✝⁴ : CommRing S
inst✝³ : UniformSpace S
φ : R →+* S
a : σ → S
inst✝² : IsUniformAddGroup R
inst✝¹ : IsUniformAddGroup S
inst✝ : IsLinearTopology S S
hφ : Continuous[inst✝⁵.toTopologicalSpace, inst✝³.toTopologicalSpace] ⇑φ
ha : HasEval a
⊢ ∀ (i : Ideal S), ↑i ∈ 𝓝 0 → ∀ᶠ (x : MvPolynomial σ R) in 𝓝 0, (eval₂Hom φ a) x ∈ ↑i

Tactic:
intro I hI

Goal state after:
σ : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : UniformSpace R
S : Type u_3
inst✝⁴ : CommRing S
inst✝³ : UniformSpace S
φ : R →+* S
a : σ → S
inst✝² : IsUniformAddGroup R
inst✝¹ : IsUniformAddGroup S
inst✝ : IsLinearTopology S S
hφ : Continuous[inst✝⁵.toTopologicalSpace, inst✝³.toTopologicalSpace] ⇑φ
ha : HasEval a
I : Ideal S
hI : ↑I ∈ 𝓝 0
⊢ ∀ᶠ (x : MvPolynomial σ R) in 𝓝 0, (eval₂Hom φ a) x ∈ ↑I
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
ε : MvPowerSeries σ R → S
hε : Continuous[instTopologicalSpace R, inst✝⁷.toTopologicalSpace] ε
h : ∀ (p : MvPolynomial σ R), ε ↑p = MvPolynomial.eval₂ φ a p
⊢ ε = eval₂ φ a

Tactic:
rw [← coe_eval₂Hom hφ ha]
  exact (toMvPowerSeries_isDenseInducing.extend_unique h hε).symm

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✝ : CommRing S
⊢ HasSubst fun x ↦ 0

Tactic:
by
  let : UniformSpace S := ⊥
  simpa [hasSubst_iff_hasEval_of_discreteTopology] using! HasEval.zero

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
e : τ →₀ ℕ
⊢ (coeff e) (subst a f) = ∑ᶠ (d : σ →₀ ℕ), (coeff d) f • (coeff e) (d.prod fun s e ↦ a s ^ e)

Tactic:
by
  let : UniformSpace R := ⊥
  let : UniformSpace S := ⊥
  have := ((hasSum_aeval ha.hasEval f).map (coeff e) (continuous_coeff S e))
  simp [← coe_substAlgHom ha, substAlgHom, ← this.tsum_eq,
    tsum_eq_finsum (coeff_subst_finite ha f e)]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : NNNorm 𝕜
E : Type u_2
inst✝ : SMul 𝕜 E
x : E
⊢ egauge 𝕜 ∅ x = ∞

Tactic:
simp [egauge]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : NNNorm 𝕜
E : Type u_2
inst✝ : SMul 𝕜 E
s : Set E
x : E
⊢ egauge 𝕜 s x = ∞ ↔ ∀ (c : 𝕜), x ∉ c • s

Tactic:
simp [egauge]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NormedDivisionRing 𝕜
E : Type u_2
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
x : E
inst✝ : (𝓝[≠] 0).NeBot
⊢ egauge 𝕜 univ x = 0

Tactic:
by
  rw [egauge_eq_zero_iff]
  refine (frequently_iff_neBot.2 ‹_›).mono fun c hc ↦ ?_
  simp_all [smul_set_univ₀]

Goal state after:
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:
by
  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 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
i : ι
hi : i ∈ I
⊢ ‖c i‖ ≤ ‖c₁‖

Tactic:
simp [hc0 i hi]

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
⊢ (MvPowerSeries.coeff (single 0 (e 0), single 1 (e 1)).1) (subst X₀ f) *
      (MvPowerSeries.coeff (single 0 (e 0), single 1 (e 1)).2) (subst X₁ f) =
    (coeff (e 0)) f * (coeff (e 1)) f

Tactic:
grind [coeff_subst_single]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : SMul 𝕜 E
inst✝¹ : TopologicalSpace E
s : Set E
x : E
inst✝ : ContinuousAdd E
y : E
hy : y ∈ tangentConeAt 𝕜 s x
ι : Type (max u_1 u_2)
l : Filter ι
hl : l.NeBot
d : ι → E
hd : Tendsto d l (𝓝 0)
hds : ∀ᶠ (n : ι) in l, x + d n ∈ s
⊢ Tendsto (fun x_1 ↦ x + d x_1) l (𝓝 x)

Tactic:
simpa using tendsto_const_nhds.add hd

Goal state after:
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.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 t : Set E
hf : DifferentiableOn 𝕜 f s
hf' : DifferentiableOn 𝕜 f t
hst : s ∪ t = univ
hs : IsOpen[inst✝³] s
ht : IsOpen[inst✝³] t
⊢ Differentiable 𝕜 f

Tactic:
by
  rw [← differentiableOn_univ, ← hst]
  exact hf.union_of_isOpen hf' hs ht

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

Goal state before:
pp: case mpr
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
⊢ (fun x ↦ egauge 𝕜 U (f x)) ≤ᶠ[l] fun x ↦ ↑ε * egauge 𝕜 (ball 0 1) (1 x)

Tactic:
filter_upwards [hf ((set_smul_mem_nhds_zero_iff hc₀).2 hU)] with a ha

Goal state after:
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)
⊢ egauge 𝕜 U (f a) ≤ ↑ε * egauge 𝕜 (ball 0 1) (1 a)
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
inst✝ : ContinuousSMul 𝕜 E
f : α → 𝕜
g : α → E
h : g =o[𝕜; l] f
⊢ Tendsto (fun x ↦ (f x)⁻¹ • g x) l (𝓝 0)

Tactic:
← isLittleOTVS_one (𝕜 := 𝕜),

Goal state after:
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
inst✝ : ContinuousSMul 𝕜 E
f : α → 𝕜
g : α → E
h : g =o[𝕜; l] f
⊢ (fun x ↦ (f x)⁻¹ • g x) =o[𝕜; l] 1
Module: Mathlib.Analysis.Calculus.FormalMultilinearSeries

Goal state before:
pp: 𝕜 : Type u
E : Type v
F : Type w
inst✝¹⁰ : Semiring 𝕜
inst✝⁹ : AddCommMonoid E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : TopologicalSpace E
inst✝⁶ : ContinuousAdd E
inst✝⁵ : ContinuousConstSMul 𝕜 E
inst✝⁴ : AddCommMonoid F
inst✝³ : Module 𝕜 F
inst✝² : TopologicalSpace F
inst✝¹ : ContinuousAdd F
inst✝ : ContinuousConstSMul 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
hp : p ≠ 0
⊢ p.order = 0 ↔ p 0 ≠ 0

Tactic:
by
  simp [order_eq_zero_iff', hp]

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

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

Tactic:
i.prod_univ_succAbove,

Goal state after:
𝕜 : Type u
n : ℕ
Ei : Fin n.succ → Type wEi
G : Type wG
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝² : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
i : Fin (n + 1)
f : Ei i →L[𝕜] Ei (i.succAbove j) [×n]→L[𝕜] G
m : (i : Fin n.succ) → Ei i
⊢ ‖(f (m i)) (i.removeNth m)‖ ≤ ‖f‖ * (‖m i‖ * ∏ i_1, ‖m (i.succAbove i_1)‖)
Module: Mathlib.Analysis.Analytic.CPolynomialDef

Goal state before:
pp: case e_6
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 : HasFiniteFPowerSeriesOnBall f pf x 1 r
y : E
hy : y ∈ Metric.eball x r
⊢ ∀ (x_1 : Fin 0), y - x = x - x

Tactic:
simp only [IsEmpty.forall_iff]

Goal state after:
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 c' : ℕ → 𝕜
n✝ : ℕ
x✝ : Fin n✝ → E
⊢ (ofScalars E (c + c') n✝) x✝ = ((ofScalars E c + ofScalars E c') n✝) x✝

Tactic:
simp [ofScalars, add_smul]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
n : ℕ
v : Fin 1 → E
⊢ ((p.leftInv i x).comp p 1) v = (id 𝕜 E x 1) v

Tactic:
simp only [leftInv_coeff_one, comp_coeff_one, h, id_apply_one, ContinuousLinearEquiv.coe_apply,
      ContinuousLinearEquiv.symm_apply_apply, continuousMultilinearCurryFin1_symm_apply]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
n✝ n : ℕ
v : Fin (n + 2) → E
A : univ = {c | c.length < n + 2}.toFinset ∪ {Composition.ones (n + 2)}
⊢ ((p.leftInv i x).comp p (n + 2)) v = (id 𝕜 E x (n + 2)) v

Tactic:
have B :
      Disjoint ({c | Composition.length c < n + 2} : Set (Composition (n + 2))).toFinset
        {Composition.ones (n + 2)} := by
      simp

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
n✝ n : ℕ
v : Fin (n + 2) → E
A : univ = {c | c.length < n + 2}.toFinset ∪ {Composition.ones (n + 2)}
B : Disjoint {c | c.length < n + 2}.toFinset {Composition.ones (n + 2)}
⊢ ((p.leftInv i x).comp p (n + 2)) v = (id 𝕜 E x (n + 2)) v
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))))
C :
  Tendsto (fun n ↦ ∑ i ∈ compPartialSumTarget 0 n n, (q.compAlongComposition p i.snd) fun _j ↦ y) atTop
    (𝓝 (g (f (x + y))))
n : ℕ
c : Composition n
⊢ ‖q.compAlongComposition p c‖ * ∏ _j, ‖y‖ ≤ ‖q.compAlongComposition p c‖ * ↑r ^ n

Tactic:
Finset.prod_const,

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
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))))
C :
  Tendsto (fun n ↦ ∑ i ∈ compPartialSumTarget 0 n n, (q.compAlongComposition p i.snd) fun _j ↦ y) atTop
    (𝓝 (g (f (x + y))))
n : ℕ
c : Composition n
⊢ ‖q.compAlongComposition p c‖ * ‖y‖ ^ Finset.univ.card ≤ ‖q.compAlongComposition p c‖ * ↑r ^ n
Module: Mathlib.Analysis.Analytic.Inverse

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
f : E → G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
hq : 0 < q.radius
hp : 0 < p.radius
r0 : ℝ≥0∞
h0 : HasFPowerSeriesOnBall f (q.comp p) x r0
r1 : NNReal
r1_pos : 0 < r1
hr1 : Summable fun i ↦ ‖q.compAlongComposition p i.snd‖₊ * r1 ^ i.fst
r : ℝ≥0∞ := ⋯
this : Metric.eball 0 r ∈ 𝓝 0
y : E
hy : y ∈ Metric.eball 0 r
hy0 : y ∈ Metric.eball 0 r0
n : ℕ
c : Composition n
⊢ ∏ _j, ‖y‖ ≤ ↑r1 ^ n

Tactic:
Finset.prod_const,

Goal state after:
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
f : E → G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
hq : 0 < q.radius
hp : 0 < p.radius
r0 : ℝ≥0∞
h0 : HasFPowerSeriesOnBall f (q.comp p) x r0
r1 : NNReal
r1_pos : 0 < r1
hr1 : Summable fun i ↦ ‖q.compAlongComposition p i.snd‖₊ * r1 ^ i.fst
r : ℝ≥0∞ := min r0 ↑r1
this : Metric.eball 0 r ∈ 𝓝 0
y : E
hy : y ∈ Metric.eball 0 r
hy0 : y ∈ Metric.eball 0 r0
n : ℕ
c : Composition n
⊢ ‖y‖ ^ #univ ≤ ↑r1 ^ n
Module: Mathlib.Analysis.Analytic.Within

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E → F
s : Set E
x : E
h : AnalyticWithinAt 𝕜 f s x
g : E → F
h'g : EqOn f g (insert x s)
hg : AnalyticAt 𝕜 g x
u : Set E := insert x s ∩ {y | AnalyticAt 𝕜 g y}
⊢ ∃ u ∈ 𝓝[insert x s] x, AnalyticOn 𝕜 f u

Tactic:
refine ⟨u, ?_, ?_⟩

Goal state after:
case refine_1
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E → F
s : Set E
x : E
h : AnalyticWithinAt 𝕜 f s x
g : E → F
h'g : EqOn f g (insert x s)
hg : AnalyticAt 𝕜 g x
u : Set E := insert x s ∩ {y | AnalyticAt 𝕜 g y}
⊢ u ∈ 𝓝[insert x s] x

case refine_2
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E → F
s : Set E
x : E
h : AnalyticWithinAt 𝕜 f s x
g : E → F
h'g : EqOn f g (insert x s)
hg : AnalyticAt 𝕜 g x
u : Set E := insert x s ∩ {y | AnalyticAt 𝕜 g y}
⊢ AnalyticOn 𝕜 f u
Module: Mathlib.Analysis.Analytic.Within

Goal state before:
pp: case refine_1
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E → F
s : Set E
x : E
h : AnalyticWithinAt 𝕜 f s x
g : E → F
h'g : EqOn f g (insert x s)
hg : AnalyticAt 𝕜 g x
u : Set E := insert x s ∩ {y | AnalyticAt 𝕜 g y}
⊢ u ∈ 𝓝[insert x s] x

Tactic:
exact inter_mem_nhdsWithin _ ((isOpen_analyticAt 𝕜 g).mem_nhds hg)

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 : ℕ
i : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i))
⊢ (ofFn fun j ↦ (i.snd j).blocks).flatten.sum = n

Tactic:
by simp [sum_ofFn, Composition.blocks_sum, Composition.sum_blocksFun]

Goal state after: