mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.FDeriv.Prod

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
n : ℕ
F' : Fin n.succ → Type u_6
inst✝¹ : (i : Fin n.succ) → NormedAddCommGroup (F' i)
inst✝ : (i : Fin n.succ) → NormedSpace 𝕜 (F' i)
φ : E → F' 0
φs : E → (i : Fin n) → F' i.succ
φ' : E →L[𝕜] (i : Fin n.succ) → F' i
l : Filter (E × E)
⊢ (HasFDerivAtFilter (fun x ↦ Fin.cons (φ x) (φs x) 0)
        { toFun := fun x ↦ φ' x 0, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ } l ∧
      ∀ (i : Fin n),
        HasFDerivAtFilter (fun x ↦ Fin.cons (φ x) (φs x) i.succ)
          { toFun := fun x ↦ φ' x i.succ, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ } l) ↔
    HasFDerivAtFilter φ { toFun := fun x ↦ φ' x 0, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ } l ∧
      ∀ (i : Fin n),
        HasFDerivAtFilter (fun x ↦ φs x i) { toFun := fun x ↦ φ' x i.succ, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ } l

Tactic:
simp only [Fin.cons_zero, Fin.cons_succ]

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

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
x : E
s : Set E
n : ℕ
F' : Fin n.succ → Type u_6
inst✝¹ : (i : Fin n.succ) → NormedAddCommGroup (F' i)
inst✝ : (i : Fin n.succ) → NormedSpace 𝕜 (F' i)
φ : E → F' 0
φs : E → (i : Fin n) → F' i.succ
⊢ (DifferentiableWithinAt 𝕜 (fun x ↦ Fin.cons (φ x) (φs x) 0) s x ∧
      ∀ (i : Fin n), DifferentiableWithinAt 𝕜 (fun x ↦ Fin.cons (φ x) (φs x) i.succ) s x) ↔
    DifferentiableWithinAt 𝕜 φ s x ∧ ∀ (i : Fin n), DifferentiableWithinAt 𝕜 (fun x ↦ φs x i) s x

Tactic:
simp only [Fin.cons_zero, Fin.cons_succ]

Goal state after:
Module: Mathlib.Topology.MetricSpace.Completion

Goal state before:
pp: case refine_1
α : Type u
inst✝ : PseudoMetricSpace α
s : Set (Completion α × Completion α)
hs : s ∈ 𝓤 (Completion α)
t : Set (Completion α × Completion α)
ht : t ∈ 𝓤 (Completion α)
tclosed : IsClosed t
ts : t ⊆ s
A : {x | (↑x.1, ↑x.2) ∈ t} ∈ 𝓤 α
ε : ℝ
εpos : ε > 0
hε : ∀ ⦃a b : α⦄, dist a b < ε → (a, b) ∈ {x | (↑x.1, ↑x.2) ∈ t}
x y : Completion α
hxy : dist x y < ε
⊢ IsClosed {x | ε ≤ dist x.1 x.2 ∨ (x.1, x.2) ∈ t}

Tactic:
have : { x : Completion α × Completion α | ε ≤ dist x.fst x.snd ∨ (x.fst, x.snd) ∈ t } =
               { p : Completion α × Completion α | ε ≤ dist p.1 p.2 } ∪ t := by ext; simp
        rw [this]
        apply IsClosed.union _ tclosed
        exact isClosed_le continuous_const Completion.uniformContinuous_dist.continuous

Goal state after:
Module: Mathlib.Topology.MetricSpace.Completion

Goal state before:
pp: case refine_1
α : Type u
inst✝ : PseudoMetricSpace α
s : Set (Completion α × Completion α)
hs : s ∈ 𝓤 (Completion α)
t : Set (Completion α × Completion α)
ht : t ∈ 𝓤 (Completion α)
tclosed : IsClosed t
ts : t ⊆ s
A : {x | (↑x.1, ↑x.2) ∈ t} ∈ 𝓤 α
ε : ℝ
εpos : ε > 0
hε : ∀ ⦃a b : α⦄, dist a b < ε → (a, b) ∈ {x | (↑x.1, ↑x.2) ∈ t}
x y : Completion α
hxy : dist x y < ε
⊢ IsClosed {x | ε ≤ dist x.1 x.2 ∨ (x.1, x.2) ∈ t}

pp: case refine_2
α : Type u
inst✝ : PseudoMetricSpace α
s : Set (Completion α × Completion α)
hs : s ∈ 𝓤 (Completion α)
t : Set (Completion α × Completion α)
ht : t ∈ 𝓤 (Completion α)
tclosed : IsClosed t
ts : t ⊆ s
A : {x | (↑x.1, ↑x.2) ∈ t} ∈ 𝓤 α
ε : ℝ
εpos : ε > 0
hε : ∀ ⦃a b : α⦄, dist a b < ε → (a, b) ∈ {x | (↑x.1, ↑x.2) ∈ t}
x y : Completion α
hxy : dist x y < ε
⊢ ∀ (a b : α), ε ≤ dist ↑a ↑b ∨ (↑a, ↑b) ∈ t

Tactic:
· have : { x : Completion α × Completion α | ε ≤ dist x.fst x.snd ∨ (x.fst, x.snd) ∈ t } =
               { p : Completion α × Completion α | ε ≤ dist p.1 p.2 } ∪ t := by ext; simp
        rw [this]
        apply IsClosed.union _ tclosed
        exact isClosed_le continuous_const Completion.uniformContinuous_dist.continuous

Goal state after:
case refine_2
α : Type u
inst✝ : PseudoMetricSpace α
s : Set (Completion α × Completion α)
hs : s ∈ 𝓤 (Completion α)
t : Set (Completion α × Completion α)
ht : t ∈ 𝓤 (Completion α)
tclosed : IsClosed t
ts : t ⊆ s
A : {x | (↑x.1, ↑x.2) ∈ t} ∈ 𝓤 α
ε : ℝ
εpos : ε > 0
hε : ∀ ⦃a b : α⦄, dist a b < ε → (a, b) ∈ {x | (↑x.1, ↑x.2) ∈ t}
x y : Completion α
hxy : dist x y < ε
⊢ ∀ (a b : α), ε ≤ dist ↑a ↑b ∨ (↑a, ↑b) ∈ t
Module: Mathlib.Analysis.Calculus.FDeriv.Prod

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
x : E
n : ℕ
F' : Fin n.succ → Type u_6
inst✝¹ : (i : Fin n.succ) → NormedAddCommGroup (F' i)
inst✝ : (i : Fin n.succ) → NormedSpace 𝕜 (F' i)
φ : E → F' 0
φs : E → (i : Fin n) → F' i.succ
⊢ (DifferentiableAt 𝕜 (fun x ↦ Fin.cons (φ x) (φs x) 0) x ∧
      ∀ (i : Fin n), DifferentiableAt 𝕜 (fun x ↦ Fin.cons (φ x) (φs x) i.succ) x) ↔
    DifferentiableAt 𝕜 φ x ∧ ∀ (i : Fin n), DifferentiableAt 𝕜 (fun x ↦ φs x i) x

Tactic:
simp only [Fin.cons_zero, Fin.cons_succ]

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

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
s : Set E
n : ℕ
F' : Fin n.succ → Type u_6
inst✝¹ : (i : Fin n.succ) → NormedAddCommGroup (F' i)
inst✝ : (i : Fin n.succ) → NormedSpace 𝕜 (F' i)
φ : E → F' 0
φs : E → (i : Fin n) → F' i.succ
⊢ (DifferentiableOn 𝕜 (fun x ↦ Fin.cons (φ x) (φs x) 0) s ∧
      ∀ (i : Fin n), DifferentiableOn 𝕜 (fun x ↦ Fin.cons (φ x) (φs x) i.succ) s) ↔
    DifferentiableOn 𝕜 φ s ∧ ∀ (i : Fin n), DifferentiableOn 𝕜 (fun x ↦ φs x i) s

Tactic:
simp only [Fin.cons_zero, Fin.cons_succ]

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

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
n : ℕ
F' : Fin n.succ → Type u_6
inst✝¹ : (i : Fin n.succ) → NormedAddCommGroup (F' i)
inst✝ : (i : Fin n.succ) → NormedSpace 𝕜 (F' i)
φ : E → F' 0
φs : E → (i : Fin n) → F' i.succ
⊢ ((Differentiable 𝕜 fun x ↦ Fin.cons (φ x) (φs x) 0) ∧
      ∀ (i : Fin n), Differentiable 𝕜 fun x ↦ Fin.cons (φ x) (φs x) i.succ) ↔
    Differentiable 𝕜 φ ∧ ∀ (i : Fin n), Differentiable 𝕜 fun x ↦ φs x i

Tactic:
simp only [Fin.cons_zero, Fin.cons_succ]

Goal state after:
Module: Mathlib.Topology.MetricSpace.Completion

Goal state before:
pp: case ih
α : Type u
β : Type v
inst✝⁵ : PseudoMetricSpace α
M : Type u_1
inst✝⁴ : Zero M
inst✝³ : Zero α
inst✝² : SMul M α
inst✝¹ : PseudoMetricSpace M
inst✝ : IsBoundedSMul M α
c₁ c₂ : M
x : α
⊢ dist (c₁ • x) (c₂ • x) ≤ dist c₁ c₂ * dist (↑x) 0

Tactic:
← coe_zero,

Goal state after:
case ih
α : Type u
β : Type v
inst✝⁵ : PseudoMetricSpace α
M : Type u_1
inst✝⁴ : Zero M
inst✝³ : Zero α
inst✝² : SMul M α
inst✝¹ : PseudoMetricSpace M
inst✝ : IsBoundedSMul M α
c₁ c₂ : M
x : α
⊢ dist (c₁ • x) (c₂ • x) ≤ dist c₁ c₂ * dist ↑x ↑0
Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Goal state before:
pp: case neg
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
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
h :
  HasFTaylorSeriesUpToOn (↑n) f p s ∧
    (∀ x ∈ s, HasFDerivWithinAt (fun y ↦ p y n) (p x n.succ).curryLeft s x) ∧ ContinuousOn (fun x ↦ p x (n + 1)) s
m : ℕ
hm : ↑m ≤ ↑n + 1
h' : ¬m ≤ n
⊢ ContinuousOn (fun x ↦ p x m) s

Tactic:
have : m = n + 1 := le_antisymm (mod_cast hm) (not_le.1 h')
        rw [this]
        exact h.2.2

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

Goal state before:
pp: case 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
N : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
hN : ∞ ≤ N
h :
  (∀ x ∈ s, (p x 0).curry0 = f x) ∧
    (∀ x ∈ s, HasFDerivWithinAt (fun y ↦ p y 0) (p x 1).curryLeft s x) ∧
      HasFTaylorSeriesUpToOn N (fun x ↦ (continuousMultilinearCurryFin1 𝕜 E F) (p x 1)) (fun x ↦ (p x).shift) s
⊢ HasFTaylorSeriesUpToOn N f p s

Tactic:
apply (hasFTaylorSeriesUpToOn_top_iff_add hN 1).2 (fun n ↦ ?_)
    rw [hasFTaylorSeriesUpToOn_succ_nat_iff_right]
    exact ⟨h.1, h.2.1, (h.2.2).of_le (m := n) (natCast_le_of_coe_top_le_withTop hN n)⟩

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
s : Set E
f : E → F
hs : UniqueDiffOn 𝕜 s
n : ℕ
IH :
  ∀ {x : E},
    x ∈ s →
      ∀ (m : Fin (n + 1) → E),
        (iteratedFDerivWithin 𝕜 (n + 1) f s x) m =
          ((iteratedFDerivWithin 𝕜 n (fun y ↦ fderivWithin 𝕜 f s y) s x) (init m)) (m (last n))
x : E
hx : x ∈ s
m : Fin (n + 1 + 1) → E
I : (E [×n]→L[𝕜] E →L[𝕜] F) ≃ₗᵢ[𝕜] E [×n.succ]→L[𝕜] F := (continuousMultilinearCurryRightEquiv' 𝕜 n E F).symm
A : ∀ y ∈ s, iteratedFDerivWithin 𝕜 n.succ f s y = (⇑I ∘ iteratedFDerivWithin 𝕜 n (fun y ↦ fderivWithin 𝕜 f s y) s) y
⊢ (((fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n (fun y ↦ fderivWithin 𝕜 f s y) s) s x) (m 0)) (init (tail m)))
      (tail m (last n)) =
    (((fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n (fun y ↦ fderivWithin 𝕜 f s y) s) s x) (init m 0)) (tail (init m)))
      (m (last (n + 1)))

Tactic:
tail_init_eq_init_tail

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
hs : UniqueDiffOn 𝕜 s
n : ℕ
IH :
  ∀ {x : E},
    x ∈ s →
      ∀ (m : Fin (n + 1) → E),
        (iteratedFDerivWithin 𝕜 (n + 1) f s x) m =
          ((iteratedFDerivWithin 𝕜 n (fun y ↦ fderivWithin 𝕜 f s y) s x) (init m)) (m (last n))
x : E
hx : x ∈ s
m : Fin (n + 1 + 1) → E
I : (E [×n]→L[𝕜] E →L[𝕜] F) ≃ₗᵢ[𝕜] E [×n.succ]→L[𝕜] F := (continuousMultilinearCurryRightEquiv' 𝕜 n E F).symm
A : ∀ y ∈ s, iteratedFDerivWithin 𝕜 n.succ f s y = (⇑I ∘ iteratedFDerivWithin 𝕜 n (fun y ↦ fderivWithin 𝕜 f s y) s) y
⊢ (((fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n (fun y ↦ fderivWithin 𝕜 f s y) s) s x) (m 0)) (init (tail m)))
      (tail m (last n)) =
    (((fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n (fun y ↦ fderivWithin 𝕜 f s y) s) s x) (init m 0)) (init (tail m)))
      (m (last (n + 1)))
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 : E
⊢ p x 0 = (continuousMultilinearCurryFin0 𝕜 E F).symm (f x)

Tactic:
rw [← h.zero_eq x]

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
f : E → F
n : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
h : HasFTaylorSeriesUpTo n f p
x : E
⊢ p x 0 = (continuousMultilinearCurryFin0 𝕜 E F).symm (p x 0).curry0
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 : HasDerivAt c c' x
f : F
⊢ HasDerivAt (fun y ↦ c y • f) (c' • f) x

Tactic:
rw [← hasDerivWithinAt_univ] at *
  exact hc.smul_const f

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

Goal state before:
pp: case hf'
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
hr : HasFPowerSeriesWithinOnBall f p s x r
h : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
y : E
hy : x + y ∈ insert x s
h'y : y ∈ Metric.eball 0 r
⊢ HasFDerivWithinAt f (fderivWithin 𝕜 f s (x + y)) (insert x s) (x + y)

Tactic:
insert_eq_of_mem hx

Goal state after:
case hf'
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
hr : HasFPowerSeriesWithinOnBall f p s x r
h : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
y : E
hy : x + y ∈ s
h'y : y ∈ Metric.eball 0 r
⊢ HasFDerivWithinAt f (fderivWithin 𝕜 f s (x + y)) s (x + y)
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: case hu
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
hr : HasFPowerSeriesWithinOnBall f p s x r
h : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
y : E
hy : x + y ∈ insert x s
h'y : y ∈ Metric.eball 0 r
⊢ UniqueDiffOn 𝕜 (insert x s)

Tactic:
insert_eq_of_mem hx

Goal state after:
case hu
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
hr : HasFPowerSeriesWithinOnBall f p s x r
h : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
y : E
hy : x + y ∈ insert x s
h'y : y ∈ Metric.eball 0 r
⊢ UniqueDiffOn 𝕜 s
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: case refine_1
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
n : WithTop ℕ∞
h : AnalyticOn 𝕜 f s
hu : UniqueDiffOn 𝕜 s
m : ℕ
_hm : ↑m < n
x : E
hx : x ∈ s
this :
  HasFDerivWithinAt (iteratedFDerivWithin 𝕜 m f s) (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 m f s) (insert x s) x)
    (insert x s) x
⊢ HasFDerivWithinAt (fun x ↦ ftaylorSeriesWithin 𝕜 f s x m) (ftaylorSeriesWithin 𝕜 f s x m.succ).curryLeft s x

Tactic:
insert_eq_of_mem hx

Goal state after:
case refine_1
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
n : WithTop ℕ∞
h : AnalyticOn 𝕜 f s
hu : UniqueDiffOn 𝕜 s
m : ℕ
_hm : ↑m < n
x : E
hx : x ∈ s
this : HasFDerivWithinAt (iteratedFDerivWithin 𝕜 m f s) (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 m f s) s x) s x
⊢ HasFDerivWithinAt (fun x ↦ ftaylorSeriesWithin 𝕜 f s x m) (ftaylorSeriesWithin 𝕜 f s x m.succ).curryLeft s x
Module: Mathlib.LinearAlgebra.AffineSpace.Slope

Goal state before:
pp: 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 : k
⊢ (b - a)⁻¹ • -(f a -ᵥ f b) = (a - b)⁻¹ • (f a -ᵥ f b)

Tactic:
smul_neg,

Goal state after:
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 : k
⊢ -((b - a)⁻¹ • (f a -ᵥ f b)) = (a - b)⁻¹ • (f a -ᵥ f b)
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: case neg.h
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
n : ℕ
x v : (i : ι) → E i
m : Fin n → (i : ι) → E i
e : Fin n.succ ↪ ι
k : ι
hkf : k ∉ range ⇑((Equiv.embeddingFinSucc n ι) e).fst
hke : ⟨k, ⋯⟩ = ((Equiv.embeddingFinSucc n ι) e).snd
⊢ ↑((Equiv.embeddingFinSucc n ι) e).snd ∈ range ⇑e

Tactic:
Equiv.embeddingFinSucc_snd e

Goal state after:
case neg.h
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
n : ℕ
x v : (i : ι) → E i
m : Fin n → (i : ι) → E i
e : Fin n.succ ↪ ι
k : ι
hkf : k ∉ range ⇑((Equiv.embeddingFinSucc n ι) e).fst
hke : ⟨k, ⋯⟩ = ((Equiv.embeddingFinSucc n ι) e).snd
⊢ e 0 ∈ range ⇑e
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: case zero
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
x : E
r : ℝ≥0∞
y : E
F : Type (max u v)
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
p : FormalMultilinearSeries 𝕜 E F
f : E → F
h : HasFPowerSeriesOnBall f p x r
⊢ (0! • (p 0) fun x ↦ y) = (iteratedFDeriv 𝕜 0 f x) fun x ↦ y

Tactic:
rw [factorial_zero, one_smul, h.iteratedFDeriv_zero_apply_diag]

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

Goal state before:
pp: case zero
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
f : E → F
x : E
r : ℝ≥0∞
h : HasFPowerSeriesOnBall f p x r
y : E
inst✝ : CompleteSpace F
⊢ (0! • (p 0) fun x ↦ y) = (iteratedFDeriv 𝕜 0 f x) fun x ↦ y

Tactic:
rw [factorial_zero, one_smul, h.iteratedFDeriv_zero_apply_diag]

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 : ContinuousWithinAt (dslope f a) s b
this : ContinuousWithinAt (fun x ↦ (x - a) • dslope f a x + f a) s b
⊢ ContinuousWithinAt f s b

Tactic:
simpa only [sub_smul_dslope, sub_add_cancel] using this

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

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
f' : E
x r : ℝ
hf : HasDerivWithinAt f f' (Ici x) x
hr : ‖f'‖ < r
⊢ ∃ᶠ (z : ℝ) in 𝓝[>] x, (z - x)⁻¹ * (‖f z‖ - ‖f x‖) < r

Tactic:
have := (hf.Ioi_of_Ici.limsup_slope_norm_le hr).frequently

Goal state after:
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
f' : E
x r : ℝ
hf : HasDerivWithinAt f f' (Ici x) x
hr : ‖f'‖ < r
this : ∃ᶠ (x_1 : ℝ) in 𝓝[>] x, ‖x_1 - x‖⁻¹ * (‖f x_1‖ - ‖f x‖) < r
⊢ ∃ᶠ (z : ℝ) in 𝓝[>] x, (z - x)⁻¹ * (‖f z‖ - ‖f x‖) < r
Module: Mathlib.Analysis.Analytic.IsolatedZeros

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
p : FormalMultilinearSeries 𝕜 𝕜 E
hp : HasFPowerSeriesAt f p z₀
⊢ (∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0) ∨ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z ≠ 0

Tactic:
by_cases h : p = 0

Goal state after:
case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
p : FormalMultilinearSeries 𝕜 𝕜 E
hp : HasFPowerSeriesAt f p z₀
h : p = 0
⊢ (∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0) ∨ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z ≠ 0

case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
p : FormalMultilinearSeries 𝕜 𝕜 E
hp : HasFPowerSeriesAt f p z₀
h : ¬p = 0
⊢ (∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0) ∨ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z ≠ 0
Module: Mathlib.Analysis.Analytic.IsolatedZeros

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
n : ℕ
g : 𝕜 → E
hg_an : AnalyticAt 𝕜 g z₀
hg_eq : ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • g z
hg_ne : ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0
z : 𝕜
hf_eq : f z = (z - z₀) ^ n • g z
hf0 : f z = 0
hz : z ∈ {z₀}ᶜ
⊢ g z = 0

Tactic:
rwa [hf0, eq_comm, smul_eq_zero_iff_right] at hf_eq

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
n : ℕ
g : 𝕜 → E
hg_an : AnalyticAt 𝕜 g z₀
hg_eq : ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • g z
hg_ne : ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0
z : 𝕜
hf_eq : (z - z₀) ^ n • g z = 0
hf0 : f z = 0
hz : z ∈ {z₀}ᶜ
⊢ (z - z₀) ^ n ≠ 0
Module: Mathlib.Analysis.Calculus.MeanValue

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
f' : ℝ → E
C : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x
bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C
g : ℝ → E := fun x ↦ f x - f a
hg : ContinuousOn g (Icc a b)
hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x
⊢ ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)

Tactic:
let B x := C * (x - a)

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
f' : ℝ → E
C : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x
bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C
g : ℝ → E := fun x ↦ f x - f a
hg : ContinuousOn g (Icc a b)
hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x
B : ℝ → ℝ := fun x ↦ C * (x - a)
⊢ ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)
Module: Mathlib.Analysis.Calculus.ContDiff.Basic

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
s : Set E
x : E
n : ℕ∞ω
f : E → F
g : F →L[𝕜] G
hf : ContDiffWithinAt 𝕜 n f s x
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
i : ℕ
hi : ↑i ≤ n
U : Set E
hU : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
hxU : x ∈ U
hfU : ContDiffOn 𝕜 (↑i) f (insert x s ∩ U)
⊢ iteratedFDerivWithin 𝕜 i (⇑g ∘ f) (s ∩ U) x = g.compContinuousMultilinearMap (iteratedFDerivWithin 𝕜 i f (s ∩ U) x)

Tactic:
insert_eq_of_mem hx

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
s : Set E
x : E
n : ℕ∞ω
f : E → F
g : F →L[𝕜] G
hf : ContDiffWithinAt 𝕜 n f s x
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
i : ℕ
hi : ↑i ≤ n
U : Set E
hU : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
hxU : x ∈ U
hfU : ContDiffOn 𝕜 (↑i) f (s ∩ U)
⊢ iteratedFDerivWithin 𝕜 i (⇑g ∘ f) (s ∩ U) x = g.compContinuousMultilinearMap (iteratedFDerivWithin 𝕜 i f (s ∩ U) x)