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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
inst✝¹⁴ : NormedField 𝕜
inst✝¹³ : AddCommGroup E
inst✝¹² : Module 𝕜 E
inst✝¹¹ : TopologicalSpace E
inst✝¹⁰ : ContinuousSMul 𝕜 E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜 F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : IsTopologicalAddGroup F
inst✝⁵ : ContinuousConstSMul 𝕜 F
inst✝⁴ : AddCommGroup G
inst✝³ : Module 𝕜 G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
x✝¹ : 𝕜
x✝ : F →L[𝕜] G
⊢ (let __LinearMap := (compContinuousAlternatingMapₗ 𝕜 E F G) (x✝¹ • x✝);
    { toLinearMap := __LinearMap, cont := ⋯ }) =
    (RingHom.id 𝕜) x✝¹ •
      let __LinearMap := (compContinuousAlternatingMapₗ 𝕜 E F G) x✝;
      { toLinearMap := __LinearMap, cont := ⋯ }

Tactic:
ext; simp

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

Goal state before:
pp: 𝕜 : Type u
E : Type wE
F : Type wF
G : Type wG
ι : Type v
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : SeminormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : SeminormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : IsEmpty ι
⊢ fderivCompContinuousLinearMap = 0

Tactic:
ext; simp

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Goal state before:
pp: case succ
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s t : Set E
f f₁ : E → F
x : E
h : f₁ =ᶠ[𝓝[s] x] f
ht : t ⊆ s
n : ℕ
ihn : iteratedFDerivWithin 𝕜 n f₁ t =ᶠ[𝓝[s] x] iteratedFDerivWithin 𝕜 n f t
⊢ iteratedFDerivWithin 𝕜 (n + 1) f₁ t =ᶠ[𝓝[s] x] iteratedFDerivWithin 𝕜 (n + 1) f t

Tactic:
have : fderivWithin 𝕜 _ t =ᶠ[𝓝[s] x] fderivWithin 𝕜 _ t := ihn.fderivWithin' ht
    refine this.mono fun y hy => ?_
    simp only [iteratedFDerivWithin_succ_eq_comp_left, hy, (· ∘ ·)]

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
n : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
H : HasFTaylorSeriesUpToOn n f p univ
⊢ ∀ (x : E), (p x 0).curry0 = f x

Tactic:
simpa using H.zero_eq

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
n : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
H : HasFTaylorSeriesUpTo n f p
⊢ ∀ x ∈ univ, (p x 0).curry0 = f x

Tactic:
simpa using H.zero_eq

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Goal state before:
pp: case zero
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
⊢ tsupport (iteratedFDeriv 𝕜 0 f) ⊆ tsupport f

Tactic:
rw [iteratedFDeriv_zero_eq_comp]
    exact closure_minimal ((support_comp_subset (map_zero _) _).trans subset_closure)
      isClosed_closure

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

Goal state before:
pp: case inr
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : HasFPowerSeriesWithinOnBall f p s x r
y : E
hy✝ : y ∈ insert x s ∩ Metric.eball x r
Z : DifferentiableWithinAt 𝕜 f (insert y s) y
hy : y ≠ x
⊢ s ∈ 𝓝[insert x s] y

Tactic:
rw [nhdsWithin_insert_of_ne hy]

Goal state after:
case inr
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : HasFPowerSeriesWithinOnBall f p s x r
y : E
hy✝ : y ∈ insert x s ∩ Metric.eball x r
Z : DifferentiableWithinAt 𝕜 f (insert y s) y
hy : y ≠ x
⊢ s ∈ 𝓝[s] y
Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : E
f : E → F
⊢ ‖iteratedFDerivWithin 𝕜 1 f univ x‖ = ‖fderivWithin 𝕜 f univ x‖

Tactic:
exact norm_iteratedFDerivWithin_one f uniqueDiffWithinAt_univ

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

Goal state before:
pp: case refine_1
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : HasFPowerSeriesWithinOnBall f p s x r
hu : UniqueDiffOn 𝕜 (insert x s)
⊢ HasFPowerSeriesWithinOnBall (fun z ↦ (continuousMultilinearCurryFin1 𝕜 E F) (p.changeOrigin (z - x) 1)) p.derivSeries
    s x r

Tactic:
refine continuousMultilinearCurryFin1 𝕜 E F
      |>.toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesWithinOnBall ?_
    apply HasFPowerSeriesOnBall.hasFPowerSeriesWithinOnBall
    simpa using! ((p.hasFPowerSeriesOnBall_changeOrigin 1
      (h.r_pos.trans_le h.r_le)).mono h.r_pos h.r_le).comp_sub x

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

Goal state before:
pp: case refine_2
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
s : Set E
inst✝ : CompleteSpace F
n : WithTop ℕ∞
h : AnalyticOnNhd 𝕜 f s
m : ℕ
_hm : ↑m ≤ n
x : E
hx : x ∈ s
⊢ ContinuousWithinAt (fun x ↦ ftaylorSeries 𝕜 f x m) s x

Tactic:
apply (DifferentiableAt.continuousAt (𝕜 := 𝕜) ?_).continuousWithinAt

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
s : Set E
inst✝ : CompleteSpace F
n : WithTop ℕ∞
h : AnalyticOnNhd 𝕜 f s
m : ℕ
_hm : ↑m ≤ n
x : E
hx : x ∈ s
⊢ DifferentiableAt 𝕜 (fun x ↦ ftaylorSeries 𝕜 f x m) x
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
x : 𝕜
𝕜' : Type u_2
inst✝⁴ : NormedRing 𝕜'
inst✝³ : NormedAlgebra 𝕜 𝕜'
inst✝² : Module 𝕜' F
inst✝¹ : IsBoundedSMul 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
c : 𝕜 → 𝕜'
c' : 𝕜'
hc : HasStrictDerivAt c c' x
f : F
this : HasStrictDerivAt (c • fun x ↦ f) (c x • 0 + c' • f) x
⊢ HasStrictDerivAt (fun y ↦ c y • f) (c' • f) x

Tactic:
rwa [smul_zero, zero_add] at this

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
x : 𝕜
s : Set 𝕜
𝕜' : Type u_2
inst✝⁴ : NormedRing 𝕜'
inst✝³ : NormedAlgebra 𝕜 𝕜'
inst✝² : Module 𝕜' F
inst✝¹ : IsBoundedSMul 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
c : 𝕜 → 𝕜'
c' : 𝕜'
hc : HasDerivWithinAt c c' s x
f : F
this : HasDerivWithinAt (c • fun x ↦ f) (c x • 0 + c' • f) s x
⊢ HasDerivWithinAt (fun y ↦ c y • f) (c' • f) s x

Tactic:
rwa [smul_zero, zero_add] at this

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

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
y : E
c : ℝ
c_pos : c > 0
t : Set E
t_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
δ : ℝ
δ_pos : δ > 0
δε : Metric.ball 0 δ ⊆ t
n : ℕ
p : E [×n + 1]→L[𝕜] F
ht : ∀ y ∈ t, ‖p fun x ↦ y‖ ≤ c * ‖‖y‖ ^ (n + 1 + 1)‖
hy : 0 < ‖y‖
ε : ℝ
ε_pos : 0 < ε
h₀ : 0 < c * ‖y‖ ^ (n.succ + 1)
k : 𝕜
k_pos : 0 < ‖k‖
k_norm : ‖k‖ < min (δ * ‖y‖⁻¹) (ε * (c * ‖y‖ ^ (n.succ + 1))⁻¹)
⊢ ‖k • y‖ < δ

Tactic:
rw [norm_smul]

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
y : E
c : ℝ
c_pos : c > 0
t : Set E
t_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
δ : ℝ
δ_pos : δ > 0
δε : Metric.ball 0 δ ⊆ t
n : ℕ
p : E [×n + 1]→L[𝕜] F
ht : ∀ y ∈ t, ‖p fun x ↦ y‖ ≤ c * ‖‖y‖ ^ (n + 1 + 1)‖
hy : 0 < ‖y‖
ε : ℝ
ε_pos : 0 < ε
h₀ : 0 < c * ‖y‖ ^ (n.succ + 1)
k : 𝕜
k_pos : 0 < ‖k‖
k_norm : ‖k‖ < min (δ * ‖y‖⁻¹) (ε * (c * ‖y‖ ^ (n.succ + 1))⁻¹)
⊢ ‖k‖ * ‖y‖ < δ
Module: Mathlib.LinearAlgebra.AffineSpace.Slope

Goal state before:
pp: case pos
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a b c : k
hab : ¬a = b
hbc : b = c
⊢ ((b - a) / (c - a)) • slope f a b + ((c - b) / (c - a)) • slope f b c = slope f a c

pp: case neg
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a b c : k
hab : ¬a = b
hbc : ¬b = c
⊢ ((b - a) / (c - a)) • slope f a b + ((c - b) / (c - a)) • slope f b c = slope f a c

Tactic:
· subst hbc
    simp [sub_ne_zero.2 (Ne.symm hab)]

Goal state after:
case neg
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a b c : k
hab : ¬a = b
hbc : ¬b = c
⊢ ((b - a) / (c - a)) • slope f a b + ((c - b) / (c - a)) • slope f b c = slope f a c
Module: Mathlib.Analysis.Calculus.DSlope

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
a b : 𝕜
f : 𝕜 → E
h : b ≠ a
⊢ dslope (fun x ↦ (x - a) • f x) a b = f b

Tactic:
rw [dslope_of_ne _ h, slope_sub_smul _ h.symm]

Goal state after:
Module: Mathlib.Analysis.Calculus.DSlope

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
a b : 𝕜
s : Set 𝕜
h : b ≠ a
⊢ ContinuousWithinAt (dslope f a) s b ↔ ContinuousWithinAt f s b

Tactic:
refine ⟨ContinuousWithinAt.of_dslope, fun hc => ?_⟩
  classical
  simp only [dslope, continuousWithinAt_update_of_ne h]
  exact ((continuousWithinAt_id.sub continuousWithinAt_const).inv₀ (sub_ne_zero.2 h)).smul
    (hc.sub continuousWithinAt_const)

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

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type w
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
h₂ : 𝕜' → 𝕜'
h₂' y : 𝕜'
f : E → 𝕜'
f' : E →L[𝕜] 𝕜'
t : Set 𝕜'
x : E
hh : HasDerivWithinAt h₂ h₂' t y
hf : HasFDerivAt f f' x
ht : ∀ᶠ (x' : E) in 𝓝 x, f x' ∈ t
hy : y = f x
⊢ HasFDerivAt (h₂ ∘ f) (h₂' • f') x

Tactic:
by
  subst y; exact hh.comp_hasFDerivAt x hf ht

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

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → F
f' : F
x : 𝕜
l : F → E
l' : F →L[𝕜] E
y : F
hl : HasStrictFDerivAt l l' y
hf : HasStrictDerivAt f f' x
hy : y = f x
⊢ HasStrictDerivAt (l ∘ f) (l' f') x

Tactic:
by
  rw [hy] at hl; exact hl.comp_hasStrictDerivAt x hf

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

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → F
x : 𝕜
s : Set 𝕜
l : F → E
y : F
t : Set F
hl : DifferentiableWithinAt 𝕜 l t y
hf : DifferentiableWithinAt 𝕜 f s x
hs : MapsTo f s t
hy : y = f x
⊢ derivWithin (l ∘ f) s x = (fderivWithin 𝕜 l t (f x)) (derivWithin f s x)

Tactic:
by
  rw [hy] at hl; exact fderivWithin_comp_derivWithin x hl hf hs

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

Goal state before:
pp: case zero
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
s : Set E
⊢ iteratedFDerivWithin 𝕜 0 0 s = 0

Tactic:
ext; simp

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : ℕ∞ω
h : ContDiffWithinAt 𝕜 n f s x
hn : n ≠ 0
⊢ DifferentiableWithinAt 𝕜 f (insert x s) x

Tactic:
rcases contDiffWithinAt_nat.1 (h.of_le <| ENat.one_le_iff_ne_zero_withTop.mpr hn)
    with ⟨u, hu, p, H⟩

Goal state after:
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : ℕ∞ω
h : ContDiffWithinAt 𝕜 n f s x
hn : n ≠ 0
u : Set E
hu : u ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
H : HasFTaylorSeriesUpToOn (↑1) f p u
⊢ DifferentiableWithinAt 𝕜 f (insert x s) x
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: case mp.refine_2.refine_2
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : ℕ∞ω
hn : n ≠ ∞
h'n : n + 1 ≠ ∞
h : ContDiffWithinAt 𝕜 (n + 1) f s x
u : Set E
hu : u ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
Hp : HasFTaylorSeriesUpToOn (n + 1) f p u
H'p : n + 1 = ω → ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ p x i) u
⊢ HasFTaylorSeriesUpToOn n (fun y ↦ (continuousMultilinearCurryFin1 𝕜 E F) (p y 1)) (fun y ↦ (p y).shift) u ∧
    (n = ω → ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ (fun y ↦ (p y).shift) x i) u)

Tactic:
rw [hasFTaylorSeriesUpToOn_succ_iff_right] at Hp
      refine ⟨Hp.2.2, ?_⟩
      rintro rfl i
      change AnalyticOn 𝕜
        (fun x ↦ (continuousMultilinearCurryRightEquiv' 𝕜 i E F) (p x (i + 1))) u
      apply (LinearIsometryEquiv.analyticOnNhd _ _).comp_analyticOn
        ?_ (Set.mapsTo_univ _ _)
      exact H'p rfl _

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

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 : E → F
f : F → G
h : E → G
f' : F →L[𝕜] G
h' : E →L[𝕜] G
f'symm : G →L[𝕜] F
lE : Filter (E × E)
lF : Filter (F × F)
hg : Tendsto (Prod.map g g) lE lF
hf : HasFDerivAtFilter f f' lF
hh : HasFDerivAtFilter h h' lE
hcomp : Prod.map (f ∘ g) (f ∘ g) =ᶠ[lE] Prod.map h h
hf'symm : Function.LeftInverse ⇑f'symm ⇑f'
⊢ HasFDerivAtFilter g (f'symm ∘SL h') lE

Tactic:
by
  apply of_comp_aux (f := f) (f' := f') <;> try assumption
  · exact Topology.IsEmbedding.of_leftInverse hf'symm (map_continuous _) (map_continuous _)
  · refine f'symm.isBigOTVS_comp.congr_left ?_
    simp [hf'symm _]

Goal state after:
Module: Mathlib.Topology.OpenPartialHomeomorph.IsImage

Goal state before:
pp: X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
s : Set X
t : Set Y
⊢ e.IsImage s t ↔ e.target ∩ ↑e.symm ⁻¹' (e.source ∩ s) = e.target ∩ t

Tactic:
by
  rw [iff_symm_preimage_eq, ← image_source_inter_eq, ← image_source_inter_eq']

Goal state after:
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : 𝕜 → F
s : Set 𝕜
x : 𝕜
⊢ iteratedDerivWithin (n + 1) f s x = iteratedDerivWithin n (derivWithin f s) s x

Tactic:
iteratedDerivWithin_eq_iterate,

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : 𝕜 → F
s : Set 𝕜
x : 𝕜
⊢ (fun g ↦ derivWithin g s)^[n + 1] f x = iteratedDerivWithin n (derivWithin f s) s x
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: n : ℕ
c : OrderedFinpartition n
i : Fin c.length
this : (c.extendMiddle i).emb i 0 = 0
| 0

Tactic:
rw [← (c.extendMiddle i).emb_invEmbedding 0]

Goal state after:
n : ℕ
c : OrderedFinpartition n
i : Fin c.length
this : (c.extendMiddle i).emb i 0 = 0
| (c.extendMiddle i).emb ((c.extendMiddle i).index 0) ((c.extendMiddle i).invEmbedding 0)
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
n : ℕ∞ω
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
⊢ ContDiffOn 𝕜 (n + 1) f s ↔ DifferentiableOn 𝕜 f s ∧ (n = ω → AnalyticOn 𝕜 f s) ∧ ContDiffOn 𝕜 n (fderiv 𝕜 f) s

Tactic:
by
  rw [contDiffOn_succ_iff_fderivWithin hs.uniqueDiffOn,
    contDiffOn_congr fun x hx ↦ fderivWithin_of_isOpen hs hx]

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
n : ℕ
⊢ ContDiff 𝕜 (↑n) f ↔
    (∀ m ≤ n, Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] fun x ↦ iteratedFDeriv 𝕜 m f x) ∧
      ∀ m < n, Differentiable 𝕜 fun x ↦ iteratedFDeriv 𝕜 m f x

Tactic:
by
  rw [← WithTop.coe_natCast, contDiff_iff_continuous_differentiable]
  simp

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: 𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁵ : Field 𝕂
inst✝⁴ : Ring 𝔸
inst✝³ : Algebra 𝕂 𝔸
inst✝² : TopologicalSpace 𝔸
inst✝¹ : IsTopologicalRing 𝔸
inst✝ : CharZero 𝕂
⊢ exp = fun x ↦ ∑' (n : ℕ), (↑n !)⁻¹ • x ^ n

Tactic:
rw [exp_eq_expSeries_sum 𝕂]

Goal state after:
𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁵ : Field 𝕂
inst✝⁴ : Ring 𝔸
inst✝³ : Algebra 𝕂 𝔸
inst✝² : TopologicalSpace 𝔸
inst✝¹ : IsTopologicalRing 𝔸
inst✝ : CharZero 𝕂
⊢ (expSeries 𝕂 𝔸).sum = fun x ↦ ∑' (n : ℕ), (↑n !)⁻¹ • x ^ n
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: case zero
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
i : ℕ
a : 𝕜
hf : ContDiff 𝕜 (↑0) f
⊢ (iteratedFDeriv 𝕜 0 fun z ↦ f (a • z)) = fun x ↦ a ^ 0 • iteratedFDeriv 𝕜 0 f (a • x)

Tactic:
ext; simp

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: f f' : ℝ → ℝ
a b : ℝ
hab : a < b
g g' : ℝ → ℝ
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
⊢ ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c

Tactic:
let h x := (lgb - lga) * f x - (lfb - lfa) * g x

Goal state after:
f f' : ℝ → ℝ
a b : ℝ
hab : a < b
g g' : ℝ → ℝ
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
h : ℝ → ℝ := fun x ↦ (lgb - lga) * f x - (lfb - lfa) * g x
⊢ ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ContinuousWithinAt f (Ici a) a
hdiff : Tendsto (slope f a) (𝓝[>] a) (𝓝 (derivWithin f (Ioi a) a))
⊢ ∃ i, a < i ∧ ∀ ⦃x : ℝ⦄, x ∈ Ioo a i → ∀ x_1 ∈ Ioc a x, max (derivWithin f (Ioi a) a + 1) 0 < derivWithin f (Ioi a) x_1

Tactic:
(nhdsGT_basis a).tendsto_left_iff

Goal state after:
f : ℝ → ℝ
a : ℝ
hf : ∀ t ∈ atTop, ∃ i, a < i ∧ MapsTo (derivWithin f (Ioi a)) (Ioo a i) t
hcont_at_a : ContinuousWithinAt f (Ici a) a
hdiff : Tendsto (slope f a) (𝓝[>] a) (𝓝 (derivWithin f (Ioi a) a))
⊢ ∃ i, a < i ∧ ∀ ⦃x : ℝ⦄, x ∈ Ioo a i → ∀ x_1 ∈ Ioc a x, max (derivWithin f (Ioi a) a + 1) 0 < derivWithin f (Ioi a) x_1
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: case inl
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : DifferentiableOn ℝ f (interior D)
C : ℝ
hf'_ge : ∀ x ∈ interior D, C ≤ deriv f x
x : ℝ
hx : x ∈ D
y : ℝ
hy : y ∈ D
hxy : x ≤ y
hxy' : x = y
⊢ C * (y - x) ≤ f y - f x

Tactic:
hxy',

Goal state after:
case inl
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : DifferentiableOn ℝ f (interior D)
C : ℝ
hf'_ge : ∀ x ∈ interior D, C ≤ deriv f x
x : ℝ
hx : x ∈ D
y : ℝ
hy : y ∈ D
hxy : x ≤ y
hxy' : x = y
⊢ C * (y - y) ≤ f y - f y
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

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
f : E → F
ε : ℝ
hε : 0 < ε
x : E
hx : DifferentiableAt 𝕜 f x
δ : ℝ := ε / 2 / 2
R : ℝ
R_pos : R > 0
hR : ∀ y ∈ ball x R, ‖f y - f x - (fderiv 𝕜 f x) (y - x)‖ ≤ δ * ‖y - x‖
r : ℝ
hr : r ∈ Ioo 0 R
this : r ∈ Ioc (r / 2) r
y : E
hy : y ∈ ball x r
z : E
hz : z ∈ ball x r
⊢ ‖f z - f y - (fderiv 𝕜 f x) (z - y)‖ = ‖f z - f x - (fderiv 𝕜 f x) (z - x) - (f y - f x - (fderiv 𝕜 f x) (y - x))‖

Tactic:
by
      simp only [map_sub]; abel_nf

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: 𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
C₁ C₂ : ConvexCone R M
x : M
x✝ : ConvexCone R M
⊢ 0 + x✝ = x✝

Tactic:
ext; simp

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: 𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
C₁ C₂ : ConvexCone R M
x : M
x✝ : ConvexCone R M
⊢ x✝ + 0 = x✝

Tactic:
ext; simp

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

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
f : E → F
K : Set (E →L[𝕜] F)
x : E
hx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}
⊢ ∀ (i : ℕ), x ∈ ⋃ n, ⋂ p, ⋂ (_ : p ≥ n), ⋂ q, ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ i)

Tactic:
intro e

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
f : E → F
K : Set (E →L[𝕜] F)
x : E
hx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}
e : ℕ
⊢ x ∈ ⋃ n, ⋂ p, ⋂ (_ : p ≥ n), ⋂ q, ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)