mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Analytic.Linear

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 →L[𝕜] F
x : E
⊢ ∀ (m : ℕ), 2 ≤ m → f.fpowerSeries x m = 0

Tactic:
intro m hm
    match m with
    | 0 | 1 => linarith
    | n + 2 => simp [fpowerSeries]

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 (s ∩ Metric.eball x r)
h'' : g x = f x
⊢ ∀ {y : E}, x + y ∈ insert x s → y ∈ Metric.eball 0 r → HasSum (fun n ↦ (p n) fun x ↦ y) (g (x + y))

Tactic:
intro y hy h'y

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f 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 (s ∩ Metric.eball x r)
h'' : g x = f x
y : E
hy : x + y ∈ insert x s
h'y : y ∈ Metric.eball 0 r
⊢ HasSum (fun n ↦ (p n) fun x ↦ y) (g (x + y))
Module: Mathlib.Analysis.Analytic.CPolynomialDef

Goal state before:
pp: case ht
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
x y : E
f : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F := ⋯
⊢ Function.support f ⊆ ⇑changeOriginIndexEquiv ⁻¹' Sigma.fst ⁻¹' {m | m < n}

Tactic:
refine fun s ↦ Not.imp_symm fun hs ↦ ?_
      simp only [preimage_ofPred_eq, changeOriginIndexEquiv_apply_fst, mem_ofPred, not_lt] at hs
      dsimp only [f]
      rw [changeOriginSeriesTerm_bound p hn _ _ _ hs, _root_.zero_apply, _root_.zero_apply]

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
r : ℝ≥0∞
hr : r = 0
hc' : Tendsto (fun n ↦ ENNReal.ofReal ‖c n.succ‖ / ENNReal.ofReal ‖c n‖) atTop (𝓝 0)
h : ∀ᶠ (n : ℕ) in atTop, c n ≠ 0
⊢ Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 0)

Tactic:
refine Tendsto.congr' ?_ <| (tendsto_toReal zero_ne_top).comp 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
r : ℝ≥0∞
hr : r = 0
hc' : Tendsto (fun n ↦ ENNReal.ofReal ‖c n.succ‖ / ENNReal.ofReal ‖c n‖) atTop (𝓝 0)
h : ∀ᶠ (n : ℕ) in atTop, c n ≠ 0
⊢ (ENNReal.toReal ∘ fun n ↦ ENNReal.ofReal ‖c n.succ‖ / ENNReal.ofReal ‖c n‖) =ᶠ[atTop] fun n ↦ ‖c n.succ‖ / ‖c n‖
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
n : ℕ
p : FormalMultilinearSeries 𝕜 E F
c : Composition n
f : F [×c.length]→L[𝕜] G
v : Fin n → E
⊢ (‖f‖ * ∏ i, ‖p (c.blocksFun i)‖) * ∏ i, ∏ j, ‖(v ∘ ⇑(c.embedding i)) j‖ =
    (‖f‖ * ∏ i, ‖p (c.blocksFun i)‖) * ∏ i ∈ Finset.univ.sigma fun x ↦ Finset.univ, ‖v (c.blocksFinEquiv i)‖

Tactic:
Finset.prod_sigma

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
n : ℕ
p : FormalMultilinearSeries 𝕜 E F
c : Composition n
f : F [×c.length]→L[𝕜] G
v : Fin n → E
⊢ (‖f‖ * ∏ i, ‖p (c.blocksFun i)‖) * ∏ i, ∏ j, ‖(v ∘ ⇑(c.embedding i)) j‖ =
    (‖f‖ * ∏ i, ‖p (c.blocksFun i)‖) * ∏ a, ∏ s, ‖v (c.blocksFinEquiv ⟨a, s⟩)‖
Module: Mathlib.Analysis.Analytic.Inverse

Goal state before:
pp: case convert_2.e_f
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)}
C :
  ((p.leftInv i x (Composition.ones (n + 2)).length) fun j ↦ (p 1) fun x ↦ v (Fin.castLE ⋯ j)) =
    (p.leftInv i x (n + 2)) fun j ↦ (p 1) fun x ↦ v j
⊢ (fun a ↦ (p.leftInv i x a.length) (p.applyComposition a (⇑↑i.symm ∘ fun j ↦ (p 1) fun x ↦ v j))) = fun c ↦
    (p.leftInv i x c.length) (p.applyComposition c v)

Tactic:
ext c

Goal state after:
case convert_2.e_f
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)}
C :
  ((p.leftInv i x (Composition.ones (n + 2)).length) fun j ↦ (p 1) fun x ↦ v (Fin.castLE ⋯ j)) =
    (p.leftInv i x (n + 2)) fun j ↦ (p 1) fun x ↦ v j
c : Composition (n + 2)
⊢ (p.leftInv i x c.length) (p.applyComposition c (⇑↑i.symm ∘ fun j ↦ (p 1) fun x ↦ v j)) =
    (p.leftInv i x c.length) (p.applyComposition c 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
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
M N : ℕ
z : E
⊢ (∑ x ∈ (Finset.Ico 0 M).sigma fun n ↦ Fintype.piFinset fun i ↦ Finset.Ico 1 N,
      (q x.fst) fun i ↦ (p (x.snd i)) fun _j ↦ z) =
    ∑ i ∈ compPartialSumTarget 0 M N, (q.compAlongComposition p i.snd) fun _j ↦ z

Tactic:
apply compChangeOfVariables_sum 0 M N

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
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
M N : ℕ
z : E
⊢ ∀ (e : (n : ℕ) × (Fin n → ℕ)) (he : e ∈ compPartialSumSource 0 M N),
    ((q e.fst) fun i ↦ (p (e.snd i)) fun _j ↦ z) =
      (q.compAlongComposition p (compChangeOfVariables 0 M N e he).snd) fun _j ↦ z
Module: Mathlib.Analysis.Calculus.FDeriv.Comp

Goal state before:
pp: case Hg
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' (f p.1 - f p.2)) =o[𝕜; L] fun p ↦ p.1 - p.2

pp: case Hf
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 - f p.2 - f' (p.1 - p.2))) =o[𝕜; L] fun p ↦ p.1 - p.2

Tactic:
case Hg => calc (fun p ↦ g (f p.1) - g (f p.2) - g' (f p.1 - f p.2))
    _ =o[𝕜; L] (fun p ↦ f p.1 - f p.2) :=
      hg.isLittleOTVS.comp_tendsto hL
    _ =O[𝕜; L] (fun p ↦ p.1 - p.2) := hf.isBigOTVS_sub

Goal state after:
case Hf
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 - f p.2 - f' (p.1 - p.2))) =o[𝕜; L] fun p ↦ p.1 - p.2
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 → F
g : F → G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
hgf : HasFPowerSeriesAt (g ∘ f) (q.comp p) x
hf : HasFPowerSeriesAt f p x
hq : 0 < q.radius
⊢ ContinuousAt (fun y ↦ f (x + y)) 0

Tactic:
exact hf.continuousAt.comp_of_eq (by fun_prop) (by simp)

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

Goal state before:
pp: 𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
𝕝 : Type u_8
inst✝¹ : NormedDivisionRing 𝕝
inst✝ : NormedAlgebra 𝕜 𝕝
f : E → 𝕝
z : E
s : Set E
n : ℤ
h₁f : AnalyticWithinAt 𝕜 f s z
h₂f : f z ≠ 0
hn : ¬0 ≤ n
x : E
| (f ^ (- -n)) x

Tactic:
rw [zpow_neg]

Goal state after:
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
𝕝 : Type u_8
inst✝¹ : NormedDivisionRing 𝕝
inst✝ : NormedAlgebra 𝕜 𝕝
f : E → 𝕝
z : E
s : Set E
n : ℤ
h₁f : AnalyticWithinAt 𝕜 f s z
h₂f : f z ≠ 0
hn : ¬0 ≤ n
x : E
| (f ^ (-n))⁻¹ x
Module: Mathlib.Analysis.Analytic.Constructions

Goal state before:
pp: 𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
𝕝 : Type u_8
inst✝¹ : NormedDivisionRing 𝕝
inst✝ : NormedAlgebra 𝕜 𝕝
f : E → 𝕝
z : E
n : ℤ
h₁f : AnalyticAt 𝕜 f z
h₂f : f z ≠ 0
hn : ¬0 ≤ n
x : E
| (f ^ (- -n)) x

Tactic:
rw [zpow_neg]

Goal state after:
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
𝕝 : Type u_8
inst✝¹ : NormedDivisionRing 𝕝
inst✝ : NormedAlgebra 𝕜 𝕝
f : E → 𝕝
z : E
n : ℤ
h₁f : AnalyticAt 𝕜 f z
h₂f : f z ≠ 0
hn : ¬0 ≤ n
x : E
| (f ^ (-n))⁻¹ x
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
f : E → E →L[𝕜] F
pf : FormalMultilinearSeries 𝕜 E (E →L[𝕜] F)
s : Set E
x : E
r : ℝ≥0∞
z : F
hf : HasFPowerSeriesWithinOnBall f pf s x r
⊢ ∀ {y : E},
    x + y ∈ insert x s →
      y ∈ Metric.eball 0 r → HasSum (fun n ↦ (pf.unshift z n) fun x ↦ y) (z + (f (x + y)) (x + y - x))

Tactic:
intro y hy h'y

Goal state after:
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
f : E → E →L[𝕜] F
pf : FormalMultilinearSeries 𝕜 E (E →L[𝕜] F)
s : Set E
x : E
r : ℝ≥0∞
z : F
hf : HasFPowerSeriesWithinOnBall f pf s x r
y : E
hy : x + y ∈ insert x s
h'y : y ∈ Metric.eball 0 r
⊢ HasSum (fun n ↦ (pf.unshift z n) fun x ↦ y) (z + (f (x + y)) (x + y - x))
Module: Mathlib.Analysis.Calculus.FDeriv.Equiv

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
iso : E ≃L[𝕜] F
f : G → E
s : Set G
x : G
hxs : UniqueDiffWithinAt 𝕜 s x
h : ¬DifferentiableWithinAt 𝕜 f s x
this : ¬DifferentiableWithinAt 𝕜 (⇑iso ∘ f) s x
⊢ fderivWithin 𝕜 (⇑iso ∘ f) s x = ↑iso ∘SL fderivWithin 𝕜 f s x

Tactic:
rw [fderivWithin_zero_of_not_differentiableWithinAt h,
      fderivWithin_zero_of_not_differentiableWithinAt this, ContinuousLinearMap.comp_zero]

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

Goal state before:
pp: case refine_2
α : Type u
inst✝ : PseudoMetricSpace α
x y : Completion α
⊢ ∀ (a b : α), dist ↑a ↑b = dist ↑b ↑a

Tactic:
· intro a b
    rw [Completion.dist_eq, Completion.dist_eq, dist_comm]

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
x : E
n : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
h : HasFTaylorSeriesUpToOn n f p s
hn : n ≠ 0
hx : x ∈ s
A : ∀ y ∈ s, f y = (continuousMultilinearCurryFin0 𝕜 E F) (p y 0)
⊢ HasFDerivWithinAt (fun x ↦ p x 0)
    (↑↑(continuousMultilinearCurryFin0 𝕜 E F).symm ∘SL (continuousMultilinearCurryFin1 𝕜 E F) (p x 1)) s x

Tactic:
have : ((0 : ℕ) : ℕ∞) < n := pos_iff_ne_zero.mpr hn

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
p : E → FormalMultilinearSeries 𝕜 E F
h : HasFTaylorSeriesUpToOn n f p s
hn : n ≠ 0
hx : x ∈ s
A : ∀ y ∈ s, f y = (continuousMultilinearCurryFin0 𝕜 E F) (p y 0)
this : ↑↑0 < n
⊢ HasFDerivWithinAt (fun x ↦ p x 0)
    (↑↑(continuousMultilinearCurryFin0 𝕜 E F).symm ∘SL (continuousMultilinearCurryFin1 𝕜 E F) (p x 1)) s x
Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Goal state before:
pp: case fderivWithin
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 + 1)) f p s
⊢ ∀ (m : ℕ), ↑m < ↑n → ∀ x ∈ s, HasFDerivWithinAt (fun x ↦ (p x).shift m) ((p x).shift m.succ).curryLeft s x

Tactic:
intro m (hm : (m : ℕ∞ω) < n) x (hx : x ∈ s)

Goal state after:
case fderivWithin
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 + 1)) f p s
m : ℕ
hm : ↑m < ↑n
x : E
hx : x ∈ s
⊢ HasFDerivWithinAt (fun x ↦ (p x).shift m) ((p x).shift m.succ).curryLeft s x
Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Goal state before:
pp: case step
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
k m : ℕ
hf : HasFTaylorSeriesUpTo n f p
l : ℕ
h : k.le l
ih : ↑l ≤ n → (tsupport fun x ↦ p x l) ⊆ tsupport fun x ↦ p x k
h2 : ↑l.succ ≤ n
hl : ↑l < n
⊢ (tsupport fun x ↦ p x l.succ) ⊆ tsupport fun x ↦ p x l

Tactic:
refine Eq.trans_subset ?_ (tsupport_fderiv_subset 𝕜)

Goal state after:
case step
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
k m : ℕ
hf : HasFTaylorSeriesUpTo n f p
l : ℕ
h : k.le l
ih : ↑l ≤ n → (tsupport fun x ↦ p x l) ⊆ tsupport fun x ↦ p x k
h2 : ↑l.succ ≤ n
hl : ↑l < n
⊢ (tsupport fun x ↦ p x l.succ) = tsupport (fderiv 𝕜 fun x ↦ p x l)
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
m : ℕ
h : ↑m ≤ n
hf : HasFTaylorSeriesUpTo n f p
x : E [×0]→L[𝕜] F
⊢ x.curry0 = 0 ↔ x = 0

Tactic:
exact (continuousMultilinearCurryFin0 _ _ _).map_eq_zero_iff (x := x)

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

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
ι : Type u_2
𝔸' : Type u_3
inst✝¹ : NormedCommRing 𝔸'
inst✝ : NormedAlgebra 𝕜 𝔸'
u : Finset ι
f : ι → 𝕜 → 𝔸'
hd : ∀ i ∈ u, DifferentiableAt 𝕜 (f i) x
⊢ DifferentiableAt 𝕜 (fun x ↦ ∏ i ∈ u, f i x) x

Tactic:
classical
  exact
    (HasDerivAt.fun_finsetProd (fun i hi ↦ DifferentiableAt.hasDerivAt (hd i hi))).differentiableAt

Goal state after:
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) • slope f a b +ᵥ f a = f b

Tactic:
by
  rw [sub_smul_slope, vsub_vadd]

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
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

Tactic:
refine this.mp (Eventually.mono self_mem_nhdsWithin fun z hxz hz ↦ ?_)

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
z : ℝ
hxz : x < z
hz : ‖z - x‖⁻¹ * (‖f z‖ - ‖f x‖) < r
⊢ (z - x)⁻¹ * (‖f z‖ - ‖f x‖) < r
Module: Mathlib.Analysis.Calculus.Deriv.Inv

Goal state before:
pp: 𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
x_ne_zero : x ≠ 0
hxs : UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun x ↦ x⁻¹) s x = -(x ^ 2)⁻¹

Tactic:
rw [DifferentiableAt.derivWithin (differentiableAt_inv x_ne_zero) hxs]
  exact deriv_inv

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
s : Set 𝕜
l : F → E
l' : F →L[𝕜] E
y : F
t : Set F
hl : HasFDerivWithinAt l l' t y
hf : HasDerivWithinAt f f' s x
hst : MapsTo f s t
hy : y = f x
⊢ HasDerivWithinAt (l ∘ f) (l' f') s x

Tactic:
rw [hy] at hl; exact hl.comp_hasDerivWithinAt x hf hst

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

Goal state before:
pp: 𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
x : 𝕜
⊢ deriv (fun x ↦ x ^ m) x = ↑m * x ^ (m - 1)

Tactic:
by_cases H : x ≠ 0 ∨ 0 ≤ m

Goal state after:
case pos
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
x : 𝕜
H : x ≠ 0 ∨ 0 ≤ m
⊢ deriv (fun x ↦ x ^ m) x = ↑m * x ^ (m - 1)

case neg
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
x : 𝕜
H : ¬(x ≠ 0 ∨ 0 ≤ m)
⊢ deriv (fun x ↦ x ^ m) x = ↑m * x ^ (m - 1)
Module: Mathlib.Analysis.Calculus.Deriv.ZPow

Goal state before:
pp: case succ
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
c d : 𝕜
k : ℕ
ihk : (deriv^[k] fun x ↦ (c * x + d)⁻¹) = fun x ↦ (-1) ^ k * ↑k ! * c ^ k * (c * x + d) ^ (-1 - ↑k)
z : 𝕜
⊢ deriv^[1] (fun x ↦ (-1) ^ k * ↑k ! * c ^ k * (c * x + d) ^ (-1 - ↑k)) z =
    (-1) ^ (1 + k) * ↑((1 + k) * k !) * c ^ (1 + k) * (c * z + d) ^ (-1 - ↑(1 + k))

Tactic:
simp only [Int.reduceNeg, iterate_one, deriv_const_mul_field', cast_add, cast_one]

Goal state after:
case succ
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
c d : 𝕜
k : ℕ
ihk : (deriv^[k] fun x ↦ (c * x + d)⁻¹) = fun x ↦ (-1) ^ k * ↑k ! * c ^ k * (c * x + d) ^ (-1 - ↑k)
z : 𝕜
⊢ (-1) ^ k * ↑k ! * c ^ k * deriv (fun x ↦ (c * x + d) ^ (-1 - ↑k)) z =
    (-1) ^ (1 + k) * ↑((1 + k) * k !) * c ^ (1 + k) * (c * z + d) ^ (-1 - (1 + ↑k))
Module: Mathlib.Analysis.Calculus.Deriv.AffineMap

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
a b : E
x : 𝕜
⊢ HasStrictDerivAt (⇑(lineMap a b)) (b - a) x

Tactic:
by
  simpa using (lineMap a b : 𝕜 →ᵃ[𝕜] E).hasStrictDerivAt

Goal state after:
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 C : ℝ
hf : DifferentiableOn ℝ f (Icc a b)
bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C
⊢ ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)

Tactic:
by
  refine norm_image_sub_le_of_norm_deriv_le_segment' ?_ bound
  exact fun x hx => (hf x hx).hasDerivWithinAt

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

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
n✝ : ℕ∞ω
s : Set E
t : Set F
g : F → G
f : E → F
x : E
st : MapsTo f s t
n : ℕ∞
hg : ContDiffWithinAt 𝕜 (↑n) g t (f x)
hf : ContDiffWithinAt 𝕜 (↑n) f s x
m : ℕ
hm : ↑m ≤ n
u : Set E
hu : u ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
hp : HasFTaylorSeriesUpToOn (↑m) f p u
v : Set F
hv : v ∈ 𝓝[insert (f x) t] f x
q : F → FormalMultilinearSeries 𝕜 F G
hq : HasFTaylorSeriesUpToOn (↑m) g q v
w : Set E := insert x s ∩ (u ∩ f ⁻¹' v)
wv : w ⊆ f ⁻¹' v
wu : w ⊆ u
⊢ ∃ u ∈ 𝓝[insert x s] x, ∃ p, HasFTaylorSeriesUpToOn (↑m) (g ∘ f) p u

Tactic:
refine ⟨w, ?_, fun y ↦ (q (f y)).taylorComp (p y), hq.comp (hp.mono wu) wv⟩

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
n✝ : ℕ∞ω
s : Set E
t : Set F
g : F → G
f : E → F
x : E
st : MapsTo f s t
n : ℕ∞
hg : ContDiffWithinAt 𝕜 (↑n) g t (f x)
hf : ContDiffWithinAt 𝕜 (↑n) f s x
m : ℕ
hm : ↑m ≤ n
u : Set E
hu : u ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
hp : HasFTaylorSeriesUpToOn (↑m) f p u
v : Set F
hv : v ∈ 𝓝[insert (f x) t] f x
q : F → FormalMultilinearSeries 𝕜 F G
hq : HasFTaylorSeriesUpToOn (↑m) g q v
w : Set E := insert x s ∩ (u ∩ f ⁻¹' v)
wv : w ⊆ f ⁻¹' v
wu : w ⊆ u
⊢ w ∈ 𝓝[insert x s] x
Module: Mathlib.Analysis.Calculus.ContDiff.Comp

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
x₀ : E
m n : ℕ∞ω
i : ℕ
hf : ContDiffAt 𝕜 n f x₀
hmn : m + ↑i ≤ n
⊢ ContDiffAt 𝕜 m (iteratedFDerivWithin 𝕜 i f univ) x₀

Tactic:
← contDiffWithinAt_univ

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
x₀ : E
m n : ℕ∞ω
i : ℕ
hf : ContDiffWithinAt 𝕜 n f univ x₀
hmn : m + ↑i ≤ n
⊢ ContDiffWithinAt 𝕜 m (iteratedFDerivWithin 𝕜 i f univ) univ x₀