mathlib-initiative/mathlib-tactics huggingface dataset:
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
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hx : x.1 + x.2 = n
⊢ (n.prod fun s m ↦ a s ^ m) * ((coeff x.1) f * (coeff x.2) g) =
    ((coeff x.1) fun n ↦ (n.prod fun s m ↦ a s ^ m) * (coeff n) f) *
      (coeff x.2) fun n ↦ (n.prod fun s m ↦ a s ^ m) * (coeff n) g

Tactic:
← hx

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
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hx : x.1 + x.2 = n
⊢ ((x.1 + x.2).prod fun s m ↦ a s ^ m) * ((coeff x.1) f * (coeff x.2) g) =
    ((coeff x.1) fun n ↦ (n.prod fun s m ↦ a s ^ m) * (coeff n) f) *
      (coeff x.2) fun n ↦ (n.prod fun s m ↦ a s ^ m) * (coeff n) g
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
s : Set E
x : E
inst✝ : (𝓝[≠] 0).NeBot
r : ℝ≥0∞
hs₀ : 0 ∈ s
h : ∀ (c : 𝕜), c ≠ 0 → x ∈ c • s → r ≤ ‖c‖ₑ
c : 𝕜
⊢ x ∈ c • s → r ≤ ‖c‖ₑ

Tactic:
hc

Goal state after:
𝕜 : Type u_1
inst✝³ : NormedDivisionRing 𝕜
E : Type u_2
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
s : Set E
x : E
inst✝ : (𝓝[≠] 0).NeBot
r : ℝ≥0∞
hs₀ : 0 ∈ s
h : ∀ (c : 𝕜), c ≠ 0 → x ∈ c • s → r ≤ ‖c‖ₑ
c : 𝕜
hc : x ∈ c • s
⊢ r ≤ ‖c‖ₑ
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: case succ.zero
R : Type u_2
inst✝¹ : CommRing R
P : R⟦X⟧
hP : constantCoeff P = 0
inst✝ : Invertible ((coeff 1) P)
⊢ (coeff (0 + 1)) (subst (∑ i, C (P.substInvFun ↑i) * X ^ ↑i) P - X) = 0

Tactic:
rw [map_sub, coeff_subst']
    · rw [finsum_eq_single (a := 1)]
      · simp [substInvFun]
      · rintro (_ | _ | _) _ <;> simp_all [substInvFun, mul_pow, coeff_mul_X_pow']
    · simp [HasSubst, X, substInvFun]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ≥0
c : 𝕜
y : E
hy : ‖y‖₊ ≤ r
⊢ ‖(fun x ↦ c • x) y‖ₑ / ↑r ≤ ‖c‖ₑ

Tactic:
enorm_smul

Goal state after:
𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ≥0
c : 𝕜
y : E
hy : ‖y‖₊ ≤ r
⊢ ‖c‖ₑ * ‖y‖ₑ / ↑r ≤ ‖c‖ₑ
Module: Mathlib.Analysis.Convex.EGauge

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
⊢ ‖x‖ₑ ≤ egauge 𝕜 (ball 0 1) x

Tactic:
simpa using div_le_egauge_ball 𝕜 1 x

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : DivisionSemiring 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : TopologicalSpace E
inst✝² : ContinuousSMul 𝕜 E
s : Set E
x : E
inst✝¹ : (𝓝[≠] 0).NeBot
inst✝ : ContinuousAdd E
h : s ∈ 𝓝 x
⊢ tangentConeAt 𝕜 s x = univ

Tactic:
← s.univ_inter,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁷ : DivisionSemiring 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : TopologicalSpace E
inst✝² : ContinuousSMul 𝕜 E
s : Set E
x : E
inst✝¹ : (𝓝[≠] 0).NeBot
inst✝ : ContinuousAdd E
h : s ∈ 𝓝 x
⊢ tangentConeAt 𝕜 (univ ∩ s) x = univ
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
⊢ Submodule.span ℝ (tangentConeAt ℝ s x) = ⊤

Tactic:
suffices y - x ∈ interior (tangentConeAt ℝ s x) by
    apply (Submodule.span ℝ (tangentConeAt ℝ s x)).eq_top_of_nonempty_interior'
    exact ⟨y - x, interior_mono Submodule.subset_span this⟩

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
⊢ y - x ∈ interior (tangentConeAt ℝ s x)
Module: Mathlib.Analysis.Asymptotics.TVS

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝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
x : α
⊢ ↑C * egauge 𝕜 V (g x) ≤ egauge 𝕜 (id (c⁻¹ • V)) (g x)

Tactic:
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
x : α
⊢ ↑‖c‖₊ * egauge 𝕜 V (g x) ≤ egauge 𝕜 (id (c⁻¹ • V)) (g x)
Module: Mathlib.Analysis.Analytic.ChangeOrigin

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
hp : p (n + 1) = 0
s : { s // s.card = n }
hs : s ∈ Finset.univ
this : p (1 + n) = 0
⊢ p.changeOriginSeriesTerm 1 n ↑s ⋯ = 0

Tactic:
simp [changeOriginSeriesTerm, this]

Goal state after:
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 + y ∈ Metric.eball 0 p.radius

Tactic:
refine mem_eball_zero_iff.2 (lt_of_le_of_lt ?_ h)

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 + y‖ₑ ≤ ↑‖x‖₊ + ↑‖y‖₊
Module: Mathlib.Analysis.Analytic.ConvergenceRadius

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
F : Type u_4
G : Type u_5
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
p : FormalMultilinearSeries 𝕜 F G
u : E →L[𝕜] F
h_zero : 0 < ‖u‖₊
⊢ ‖u‖ₑ ≠ 0

Tactic:
simpa using h_zero

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
F : Type u_4
G : Type u_5
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
p : FormalMultilinearSeries 𝕜 F G
u : E →L[𝕜] F
h_zero : 0 < ‖u‖₊
r : ℝ≥0
hr : ↑r < p.radius
⊢ ‖u‖ₑ ≠ 0

Tactic:
simpa using h_zero

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

Goal state before:
pp: 𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
Ei : Fin n.succ → Type wEi
G : Type wG
G' : Type wG'
inst✝¹⁰ : Fintype ι
inst✝⁹ : Fintype ι'
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁵ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝⁴ : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : Ei 0 →L[𝕜] Ei i.succ [×n]→L[𝕜] G
⊢ ‖{ toFun := curryLeft, map_add' := ⋯, map_smul' := ⋯, invFun := ContinuousLinearMap.uncurryLeft, left_inv := ⋯,
            right_inv := ⋯ }.symm
        f‖ ≤
    ‖f‖

Tactic:
simp only [LinearEquiv.coe_symm_mk]
      exact MultilinearMap.mkContinuous_norm_le _ (norm_nonneg f) _

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

Goal state before:
pp: 𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
Ei : Fin n.succ → Type wEi
G : Type wG
G' : Type wG'
inst✝¹⁰ : Fintype ι
inst✝⁹ : Fintype ι'
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁵ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝⁴ : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
p : Fin (n + 1)
f : Ei p →L[𝕜] Ei (p.succAbove i) [×n]→L[𝕜] G
⊢ ‖{ toFun := curryMid p, map_add' := ⋯, map_smul' := ⋯, invFun := ContinuousLinearMap.uncurryMid p, left_inv := ⋯,
            right_inv := ⋯ }.symm
        f‖ ≤
    ‖f‖

Tactic:
simp only [LinearEquiv.coe_symm_mk]
      exact MultilinearMap.mkContinuous_norm_le _ (norm_nonneg f) _

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

Goal state before:
pp: 𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
Ei : Fin n.succ → Type wEi
G : Type wG
G' : Type wG'
inst✝¹⁰ : Fintype ι
inst✝⁹ : Fintype ι'
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁵ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝⁴ : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : ContinuousMultilinearMap 𝕜 (fun x ↦ G) (ContinuousMultilinearMap 𝕜 (fun x ↦ G) G')
⊢ ‖{ toFun := currySum, map_add' := ⋯, map_smul' := ⋯, invFun := uncurrySum, left_inv := ⋯, right_inv := ⋯ }.symm f‖ ≤
    ‖f‖

Tactic:
simp only [LinearEquiv.coe_symm_mk]
      exact MultilinearMap.mkContinuous_norm_le _ (norm_nonneg f) _

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f g : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : EqOn g f (insert x s ∩ Metric.eball x r)
y : E
hy : x + y ∈ insert x s
h'y : y ∈ Metric.eball 0 r
⊢ x + y ∈ Metric.eball x r

Tactic:
by simpa [edist_eq_enorm_sub] using h'y

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing E
inst✝¹ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
inst✝ : NormOneClass E
hc : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop atTop
r : ℝ≥0
hr : ↑r < (ofScalars E c).radius
this : 0 < r
n : ℕ
⊢ 2 * ↑r⁻¹ ≤ ‖c n.succ‖ / ‖c n‖ → n ≠ 0 → 2 * ‖‖ofScalars E c n‖ * ↑r ^ n‖ ≤ ‖‖ofScalars E c (n + 1)‖ * ↑r ^ (n + 1)‖

Tactic:
hc

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing E
inst✝¹ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
inst✝ : NormOneClass E
hc✝ : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop atTop
r : ℝ≥0
hr : ↑r < (ofScalars E c).radius
this : 0 < r
n : ℕ
hc : 2 * ↑r⁻¹ ≤ ‖c n.succ‖ / ‖c n‖
⊢ n ≠ 0 → 2 * ‖‖ofScalars E c n‖ * ↑r ^ n‖ ≤ ‖‖ofScalars E c (n + 1)‖ * ↑r ^ (n + 1)‖
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✝¹⁵ : CommRing 𝕜
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : AddCommGroup F
inst✝¹² : AddCommGroup G
inst✝¹¹ : Module 𝕜 E
inst✝¹⁰ : Module 𝕜 F
inst✝⁹ : Module 𝕜 G
inst✝⁸ : TopologicalSpace E
inst✝⁷ : TopologicalSpace F
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : ContinuousConstSMul 𝕜 E
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
v : Fin 0 → E
v' : Fin 0 → F
⊢ (q.comp p 0) v = (q 0) v'

Tactic:
let c : Composition 0 := Composition.ones 0
  dsimp [FormalMultilinearSeries.comp]
  have : {c} = (Finset.univ : Finset (Composition 0)) := by
    apply Finset.eq_of_subset_of_card_le <;> simp [Finset.card_univ, composition_card 0]
  rw [← this, Finset.sum_singleton, compAlongComposition_apply]
  symm; congr!

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∞
hf : HasFPowerSeriesWithinOnBall f p s x r
r' : ℝ≥0
hr : ↑r' < r
hr'0 : 0 < r'
a : ℝ
ha : a ∈ Ioo 0 1
C : ℝ
hC : 0 < C
hp : ∀ (n : ℕ), ‖p n‖ ≤ C * a ^ n / ↑r' ^ n
L : E × E → ℝ := fun y ↦ C * (a / ↑r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
y : E × E
hy' : ‖y - (x, x)‖₊ < r'
ys : y ∈ insert x s ×ˢ insert x s
hy : y ∈ Metric.eball x r ×ˢ Metric.eball x r
A : ℕ → F := fun n ↦ ((p n) fun x_1 ↦ y.1 - x) - (p n) fun x_1 ↦ y.2 - x
hA : HasSum (fun n ↦ A (n + 2)) (f y.1 - f y.2 - (p 1) fun x ↦ y.1 - y.2)
B : ℕ → ℝ := fun n ↦ C * (a / ↑r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * ((↑n + 2) * a ^ n)
hAB : ∀ (n : ℕ), ‖A (n + 2)‖ ≤ B n
⊢ ‖a‖ < 1

Tactic:
simp only [Real.norm_eq_abs, abs_of_pos ha.1, ha.2]

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
⊢ p.comp (p.rightInv i x) = id 𝕜 F ((p 0) 0)

Tactic:
ext (n v)

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
v : Fin n → F
⊢ (p.comp (p.rightInv i x) n) v = (id 𝕜 F ((p 0) 0) n) v
Module: Mathlib.Analysis.Analytic.Inverse

Goal state before:
pp: case base
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
hp : 0 < p.radius
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖↑i.symm‖
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : ℕ → ℝ := fun n ↦ ∑ k ∈ Ico 1 n, a ^ k * ‖p.rightInv i x k‖
⊢ ∑ k ∈ ∅, a ^ k * ‖p.rightInv i x k‖ ≤ (I + 1) * a

Tactic:
sum_empty

Goal state after:
case base
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
hp : 0 < p.radius
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖↑i.symm‖
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : ℕ → ℝ := fun n ↦ ∑ k ∈ Ico 1 n, a ^ k * ‖p.rightInv i x k‖
⊢ 0 ≤ (I + 1) * a
Module: Mathlib.Analysis.Analytic.Inverse

Goal state before:
pp: case succ
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
hp : 0 < p.radius
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ⋯
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : ℕ → ℝ := ⋯
⊢ ∀ (n : ℕ), 1 ≤ n → S n ≤ (I + 1) * a → S (n + 1) ≤ (I + 1) * a

Tactic:
· intro n one_le_n hn
      have In : 2 ≤ n + 1 := by lia
      have rSn : r * S n ≤ 1 / 2 :=
        calc
          r * S n ≤ r * ((I + 1) * a) := by gcongr
          _ ≤ 1 / 2 := by rwa [← mul_assoc]
      calc
        S (n + 1) ≤ I * a + I * C * ∑ k ∈ Ico 2 (n + 1), (r * S n) ^ k :=
          radius_rightInv_pos_of_radius_pos_aux2 In p i rpos.le apos.le Cpos.le ple
        _ = I * a + I * C * (((r * S n) ^ 2 - (r * S n) ^ (n + 1)) / (1 - r * S n)) := by
          rw [geom_sum_Ico' _ In]; exact ne_of_lt (rSn.trans_lt (by norm_num))
        _ ≤ I * a + I * C * ((r * S n) ^ 2 / (1 / 2)) := by
          gcongr
          · simp only [sub_le_self_iff]
            positivity
          · linarith only [rSn]
        _ = I * a + 2 * I * C * (r * S n) ^ 2 := by ring
        _ ≤ I * a + 2 * I * C * (r * ((I + 1) * a)) ^ 2 := by gcongr
        _ = (I + 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) * a := by ring
        _ ≤ (I + 1) * a := by gcongr

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
hp : 0 < p.radius
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
⊢ 0 < (p.leftInv i x).radius

Tactic:
leftInv_eq_rightInv _ _ _ h

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
hp : 0 < p.radius
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
⊢ 0 < (p.rightInv i x).radius
Module: Mathlib.Analysis.Analytic.Constructions

Goal state before:
pp: 𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
c : F
e : E
⊢ HasFPowerSeriesOnBall (fun x ↦ c) (constFormalMultilinearSeries 𝕜 E c) e ∞

Tactic:
by
  refine ⟨by simp, WithTop.top_pos, fun _ => hasSum_single 0 fun n hn => ?_⟩
  simp [constFormalMultilinearSeries_apply_of_nonzero hn]

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

Goal state before:
pp: n : ℕ
a : Composition n
b : Composition a.length
i : Fin b.length
j : Fin (b.blocksFun i)
⊢ (a.sigmaCompositionAux b ⟨↑i, ⋯⟩).blocksFun ⟨↑j, ⋯⟩ = a.blocksFun ((b.embedding i) j)

Tactic:
by
  unfold sigmaCompositionAux
  rw [blocksFun, get_eq_getElem, getElem_of_eq (getElem_splitWrtComposition _ _ _ _),
    getElem_drop, getElem_take]; rfl

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

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 → F
f' : E →L[𝕜] F
L : Filter (E × E)
g : F → G
g' : F →L[𝕜] G
L' : Filter (F × F)
hg : HasFDerivAtFilter g g' L'
hf : HasFDerivAtFilter f f' L
hL : Tendsto (Prod.map f f) L L'
⊢ (fun p ↦ (g ∘ f) p.1 - (g ∘ f) p.2 - (g' ∘SL f') (p.1 - p.2)) = fun p ↦
    g (f p.1) - g (f p.2) - g' (f p.1 - f p.2) + g' (f p.1 - f p.2 - f' (p.1 - p.2))

Tactic:
by
      ext; simp

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

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
g : F → G
f : E → F
x : E
s : Set E
t : Set F
hf : DifferentiableWithinAt 𝕜 f s x
h : MapsTo f s t
hxs : UniqueDiffWithinAt 𝕜 s x
v : E
hg : DifferentiableWithinAt 𝕜 g t (f x)
⊢ (fderivWithin 𝕜 g t (f x)) ((fderivWithin 𝕜 f s x) v) = (fderivWithin 𝕜 (g ∘ f) s x) v

Tactic:
rw [fderivWithin_comp x hg hf h hxs, comp_apply]

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

Goal state before:
pp: case inr
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
Fm : ι → Type u_10
inst✝¹ : (i : ι) → NormedAddCommGroup (Fm i)
inst✝ : (i : ι) → NormedSpace 𝕜 (Fm i)
r : ℝ≥0∞
p : (i : ι) → FormalMultilinearSeries 𝕜 E (Fm i)
h : ∀ (i : ι), r ≤ (p i).radius
r' : ℝ≥0
hr'✝ : ↑r' < r
C : ι → ℝ
C_pos : ∀ (i : ι), C i > 0
hC : ∀ (i : ι) (n : ℕ), ‖p i n‖ * ↑r' ^ n ≤ C i
D : ℝ
D_nonneg : D ≥ 0
hD : ∀ (i : ι), C i ≤ D
n : ℕ
hr' : 0 < ↑r' ^ n
⊢ ‖ContinuousMultilinearMap.pi fun i ↦ p i n‖ ≤ D / ↑r' ^ n

Tactic:
ContinuousMultilinearMap.opNorm_pi,

Goal state after:
case inr
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
Fm : ι → Type u_10
inst✝¹ : (i : ι) → NormedAddCommGroup (Fm i)
inst✝ : (i : ι) → NormedSpace 𝕜 (Fm i)
r : ℝ≥0∞
p : (i : ι) → FormalMultilinearSeries 𝕜 E (Fm i)
h : ∀ (i : ι), r ≤ (p i).radius
r' : ℝ≥0
hr'✝ : ↑r' < r
C : ι → ℝ
C_pos : ∀ (i : ι), C i > 0
hC : ∀ (i : ι) (n : ℕ), ‖p i n‖ * ↑r' ^ n ≤ C i
D : ℝ
D_nonneg : D ≥ 0
hD : ∀ (i : ι), C i ≤ D
n : ℕ
hr' : 0 < ↑r' ^ n
⊢ ‖fun i ↦ p i n‖ ≤ D / ↑r' ^ n
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
p : (i : ι) → FormalMultilinearSeries 𝕜 E (Fm i)
hf : ∀ (i : ι), HasFPowerSeriesAt (f i) (p i) e
⊢ HasFPowerSeriesAt (fun x x_1 ↦ f x_1 x) (FormalMultilinearSeries.pi p) e

Tactic:
by
  simp_rw [← hasFPowerSeriesWithinAt_univ] at hf ⊢
  exact HasFPowerSeriesWithinAt.pi hf

Goal state after: