mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case inr
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
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ c : OrderedFinpartition n
k i : Fin c.length
hi : i ∈ univ
j : Fin c.length
hj✝ : j ∈ univ
hij : i ≠ j
h : i ≠ k
hj : j ≠ k
⊢ ∀ (a : Fin (update c.partSize k (c.partSize k + 1) i)) (a_1 : Fin (update c.partSize k (c.partSize k + 1) j)),
    ¬c.emb i (Fin.cast ⋯ a) = c.emb j (Fin.cast ⋯ a_1)

Tactic:
intro a

Goal state after:
case inr
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
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ c : OrderedFinpartition n
k i : Fin c.length
hi : i ∈ univ
j : Fin c.length
hj✝ : j ∈ univ
hij : i ≠ j
h : i ≠ k
hj : j ≠ k
a : Fin (update c.partSize k (c.partSize k + 1) i)
⊢ ∀ (a_1 : Fin (update c.partSize k (c.partSize k + 1) j)), ¬c.emb i (Fin.cast ⋯ a) = c.emb j (Fin.cast ⋯ a_1)
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

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
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) = {0}
i : Fin (c.length - 1)
x✝¹ : i ∈ univ
j : Fin (c.length - 1)
x✝ : j ∈ univ
hij : i ≠ j
⊢ ∀ (a : Fin (c.partSize (Fin.cast ⋯ i.succ))) (a_1 : Fin (c.partSize (Fin.cast ⋯ j.succ))),
    ¬c.emb (Fin.cast ⋯ i.succ) a = c.emb (Fin.cast ⋯ j.succ) a_1

Tactic:
intro a

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) = {0}
i : Fin (c.length - 1)
x✝¹ : i ∈ univ
j : Fin (c.length - 1)
x✝ : j ∈ univ
hij : i ≠ j
a : Fin (c.partSize (Fin.cast ⋯ i.succ))
⊢ ∀ (a_1 : Fin (c.partSize (Fin.cast ⋯ j.succ))), ¬c.emb (Fin.cast ⋯ i.succ) a = c.emb (Fin.cast ⋯ j.succ) a_1
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: case cont
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∞
Hcont : ∀ (m : ℕ), ↑m ≤ n → ContinuousOn (fun x ↦ iteratedFDerivWithin 𝕜 m f s x) s
Hdiff : ∀ (m : ℕ), ↑m < n → DifferentiableOn 𝕜 (fun x ↦ iteratedFDerivWithin 𝕜 m f s x) s
x : E
hx : x ∈ s
m : ℕ
hm : ↑m ≤ n
k : ℕ
hk : ↑k ≤ ↑m
⊢ ContinuousOn (fun x ↦ ftaylorSeriesWithin 𝕜 f s x k) s

Tactic:
exact Hcont k (le_trans (mod_cast hk) (mod_cast hm))

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

Goal state before:
pp: case inl
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
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
i : Fin c.length
x✝¹ : i ∈ univ
h : i ≠ c.index 0
x✝ : c.index 0 ∈ univ
hij : i ≠ c.index 0
⊢ ∀ (a : Fin (update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) i))
    (a_1 : Fin (update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) (c.index 0))),
    ¬c.emb i (Fin.cast ⋯ a) = c.emb (c.index 0) (Fin.cast ⋯ a_1.succ)

Tactic:
intro a

Goal state after:
case inl
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
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
i : Fin c.length
x✝¹ : i ∈ univ
h : i ≠ c.index 0
x✝ : c.index 0 ∈ univ
hij : i ≠ c.index 0
a : Fin (update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) i)
⊢ ∀ (a_1 : Fin (update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) (c.index 0))),
    ¬c.emb i (Fin.cast ⋯ a) = c.emb (c.index 0) (Fin.cast ⋯ a_1.succ)
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case inr
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
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
i : Fin c.length
x✝¹ : i ∈ univ
j : Fin c.length
x✝ : j ∈ univ
hij : i ≠ j
h : i ≠ c.index 0
hj : j ≠ c.index 0
⊢ ∀ (a : Fin (update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) i))
    (a_1 : Fin (update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) j)),
    ¬c.emb i (Fin.cast ⋯ a) = c.emb j (Fin.cast ⋯ a_1)

Tactic:
intro a

Goal state after:
case inr
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
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
i : Fin c.length
x✝¹ : i ∈ univ
j : Fin c.length
x✝ : j ∈ univ
hij : i ≠ j
h : i ≠ c.index 0
hj : j ≠ c.index 0
a : Fin (update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) i)
⊢ ∀ (a_1 : Fin (update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) j)),
    ¬c.emb i (Fin.cast ⋯ a) = c.emb j (Fin.cast ⋯ a_1)
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case emb
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
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n✝ : ℕ
c✝ : OrderedFinpartition n✝
n : ℕ
c : OrderedFinpartition n
o : Option (Fin c.length)
i : Fin c.length
⊢ ∀ (a : Fin c.length),
    c.emb a ≍ fun j ↦
      if h : a = i then ((c.extendMiddle i).emb a (Fin.cast ⋯ j.succ)).pred ⋯
      else ((c.extendMiddle i).emb a (Fin.cast ⋯ j)).pred ⋯

Tactic:
intro a

Goal state after:
case emb
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
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n✝ : ℕ
c✝ : OrderedFinpartition n✝
n : ℕ
c : OrderedFinpartition n
o : Option (Fin c.length)
i a : Fin c.length
⊢ c.emb a ≍ fun j ↦
    if h : a = i then ((c.extendMiddle i).emb a (Fin.cast ⋯ j.succ)).pred ⋯
    else ((c.extendMiddle i).emb a (Fin.cast ⋯ j)).pred ⋯
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case pos.partSize.zero
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
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n✝ : ℕ
c✝ : OrderedFinpartition n✝
n : ℕ
c : OrderedFinpartition (n + 1)
h : range (c.emb 0) = {0}
A : c.length - 1 + 1 = c.length
⊢ { length := c.length - 1 + 1, partSize := cons 1 fun i ↦ c.partSize (Fin.cast ⋯ i.succ), partSize_pos := ⋯,
          emb := fun i ↦
            cases (motive := fun x ↦ Fin (cons 1 (fun i ↦ c.partSize (Fin.cast ⋯ i.succ)) x) → Fin (n + 1)) (fun x ↦ 0)
              (fun m ↦ succ ∘ fun j ↦ (c.emb (Fin.cast ⋯ m.succ) j).pred ⋯) i,
          emb_strictMono := ⋯, parts_strictMono := ⋯, disjoint := ⋯, cover := ⋯ }.partSize
      0 =
    c.partSize ⟨↑0, ⋯⟩

Tactic:
| zero =>

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

Goal state before:
pp: case pos.emb.refine_1.zero
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
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n✝ : ℕ
c✝ : OrderedFinpartition n✝
n : ℕ
c : OrderedFinpartition (n + 1)
h : range (c.emb 0) = {0}
A : c.length - 1 + 1 = c.length
h' : 0 ≍ Fin.cast A 0
⊢ cons 1 (fun i ↦ c.partSize (Fin.cast ⋯ i.succ)) 0 = c.partSize (Fin.cast A 0)

Tactic:
| zero =>

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

Goal state before:
pp: case pos.emb.refine_2.zero
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
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n✝ : ℕ
c✝ : OrderedFinpartition n✝
n : ℕ
c : OrderedFinpartition (n + 1)
h : range (c.emb 0) = {0}
A : c.length - 1 + 1 = c.length
h' : 0 ≍ Fin.cast A 0
j : Fin (cons 1 (fun i ↦ c.partSize (Fin.cast ⋯ i.succ)) 0)
⊢ cases (motive := fun x ↦ Fin (cons 1 (fun i ↦ c.partSize (Fin.cast ⋯ i.succ)) x) → Fin (n + 1)) (fun x ↦ 0)
      (fun m ↦ succ ∘ fun j ↦ (c.emb (Fin.cast ⋯ m.succ) j).pred ⋯) 0 j =
    c.emb (Fin.cast A 0) ⟨↑j, ⋯⟩

Tactic:
| zero =>

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

Goal state before:
pp: case pos
𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : NormedRing 𝔸
inst✝² : NormedAlgebra 𝕂 𝔸
inst✝¹ : CompleteSpace 𝔸
inst✝ : CharZero 𝕂
x : 𝔸
hx : x ∈ Metric.eball 0 0
h : (expSeries 𝕂 𝔸).radius = 0
⊢ AnalyticAt 𝕂 exp x

Tactic:
exact (ENNReal.not_lt_zero hx).elim

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

Goal state before:
pp: 𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : NormedRing 𝔸
inst✝² : NormedAlgebra 𝕂 𝔸
inst✝¹ : CompleteSpace 𝔸
inst✝ : CharZero 𝕂
x : 𝔸
hx : x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
⊢ AnalyticAt 𝕂 exp x

Tactic:
by_cases h : (expSeries 𝕂 𝔸).radius = 0
  · rw [h] at hx; exact (ENNReal.not_lt_zero hx).elim
  · have h := pos_iff_ne_zero.mpr h
    exact (hasFPowerSeriesOnBall_exp_of_radius_pos h).analyticAt_of_mem hx

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

Goal state before:
pp: 𝕂 : Type u_1
𝔸 : Type u_2
𝔹 : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕂
inst✝⁴ : NormedRing 𝔸
inst✝³ : NormedRing 𝔹
inst✝² : NormedAlgebra 𝕂 𝔸
inst✝¹ : CompleteSpace 𝔸
inst✝ : CharZero 𝕂
x : 𝔸
hx : x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
hnx : -x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
⊢ exp x * exp (-x) = 1

Tactic:
← exp_add_of_commute_of_mem_ball (Commute.neg_right <| Commute.refl x) hx hnx,

Goal state after:
𝕂 : Type u_1
𝔸 : Type u_2
𝔹 : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕂
inst✝⁴ : NormedRing 𝔸
inst✝³ : NormedRing 𝔹
inst✝² : NormedAlgebra 𝕂 𝔸
inst✝¹ : CompleteSpace 𝔸
inst✝ : CharZero 𝕂
x : 𝔸
hx : x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
hnx : -x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
⊢ exp (x + -x) = 1
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: case zero
𝔸 : Type u_1
inst✝² : NormedRing 𝔸
inst✝¹ : NormedAlgebra ℚ 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
⊢ exp (0 • x) = exp x ^ 0

Tactic:
| zero =>

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

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
n : ℕ
c : OrderedFinpartition n
f : F [×c.length]→L[𝕜] G
p : (i : Fin c.length) → E [×c.partSize i]→L[𝕜] F
v : Fin n → E
⊢ ‖f (c.applyOrderedFinpartition p v)‖ ≤ ‖f‖ * ((∏ i, ‖p i‖) * ∏ m, ∏ r, ‖v (c.emb m r)‖)

Tactic:
← Finset.prod_mul_distrib

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
n : ℕ
c : OrderedFinpartition n
f : F [×c.length]→L[𝕜] G
p : (i : Fin c.length) → E [×c.partSize i]→L[𝕜] F
v : Fin n → E
⊢ ‖f (c.applyOrderedFinpartition p v)‖ ≤ ‖f‖ * ∏ x, ‖p x‖ * ∏ r, ‖v (c.emb x r)‖
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: case zero
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
s : Set 𝕜
𝕝 : Type u_4
inst✝³ : DivisionSemiring 𝕝
inst✝² : Module 𝕝 F
inst✝¹ : SMulCommClass 𝕜 𝕝 F
inst✝ : ContinuousConstSMul 𝕝 F
c : 𝕝
x : 𝕜
f : 𝕜 → F
⊢ iteratedDerivWithin 0 (c • f) s x = c • iteratedDerivWithin 0 f s x

Tactic:
| zero =>

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

Goal state before:
pp: case zero
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
s : Set 𝕜
𝕜' : Type u_6
inst✝¹ : NormedDivisionRing 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
d : 𝕜'
x : 𝕜
f : 𝕜 → 𝕜'
⊢ iteratedDerivWithin 0 (fun z ↦ f z * d) s x = iteratedDerivWithin 0 f s x * d

Tactic:
| zero =>

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

Goal state before:
pp: case zero
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set 𝕜
f : 𝕜 → F
x : 𝕜
⊢ iteratedDerivWithin 0 (-f) s x = -iteratedDerivWithin 0 f s x

Tactic:
| zero =>

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

Goal state before:
pp: case zero
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set 𝕜
h : UniqueDiffOn 𝕜 s
f : 𝕜 → F
c : 𝕜
hs : Set.MapsTo (fun x ↦ c * x) s s
x : 𝕜
hx : x ∈ s
hf : ContDiffOn 𝕜 (↑0) f s
⊢ iteratedDerivWithin 0 (fun x ↦ f (c * x)) s x = c ^ 0 • iteratedDerivWithin 0 f s (c * x)

Tactic:
| zero =>

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

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
n : ℕ
c : OrderedFinpartition n
f₁ f₂ : F [×c.length]→L[𝕜] G
g₁ g₂ : (i : Fin c.length) → E [×c.partSize i]→L[𝕜] F
⊢ ‖c.compAlongOrderedFinpartition f₁ g₁ - c.compAlongOrderedFinpartition f₁ g₂‖ +
      ‖c.compAlongOrderedFinpartition f₁ g₂ - c.compAlongOrderedFinpartition f₂ g₂‖ ≤
    ‖f₁‖ * ↑c.length * max ‖g₁‖ ‖g₂‖ ^ (c.length - 1) * ‖g₁ - g₂‖ + ‖f₁ - f₂‖ * ∏ i, ‖g₂ i‖

Tactic:
by
    gcongr ?_ + ?_
    · refine ((c.compAlongOrderedFinpartitionL 𝕜 E F G f₁).norm_image_sub_le g₁ g₂).trans ?_
      simp only [Fintype.card_fin]
      gcongr
      apply norm_compAlongOrderedFinpartitionL_apply_le
    · exact c.norm_compAlongOrderedFinpartition_le (f₁ - f₂) g₂

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

Goal state before:
pp: case zero
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
n : ℕ
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
𝔸 : Type u_5
inst✝⁴ : NormedRing 𝔸
inst✝³ : NormedAlgebra 𝕜 𝔸
inst✝² : Module 𝔸 F
inst✝¹ : IsBoundedSMul 𝔸 F
inst✝ : IsScalarTower 𝕜 𝔸 F
f : 𝕜 → 𝔸
g : 𝕜 → F
hf : ContDiffWithinAt 𝕜 (↑0) f s x
hg : ContDiffWithinAt 𝕜 (↑0) g s x
⊢ iteratedDerivWithin 0 (f • g) s x =
    ∑ i ∈ Finset.range (0 + 1), Nat.choose 0 i • iteratedDerivWithin i f s x • iteratedDerivWithin (0 - i) g s x

Tactic:
| zero =>

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Exponential

Goal state before:
pp: 𝕂 : Type u_1
𝕊 : Type u_2
𝔸 : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕂
inst✝⁸ : CharZero 𝕂
inst✝⁷ : NormedCommRing 𝕊
inst✝⁶ : NormedRing 𝔸
inst✝⁵ : NormedSpace 𝕂 𝕊
inst✝⁴ : NormedAlgebra 𝕂 𝔸
inst✝³ : Algebra 𝕊 𝔸
inst✝² : ContinuousSMul 𝕊 𝔸
inst✝¹ : IsScalarTower 𝕂 𝕊 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
t : 𝕊
htx : t • x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
this :
  (fun h ↦ exp (t • x) * (exp ((0 + h) • x) - exp (0 • x) - (ContinuousLinearMap.smulRight 1 x) h)) =ᶠ[𝓝 0] fun h ↦
    exp ((t + h) • x) - exp (t • x) - (exp (t • x) • ContinuousLinearMap.smulRight 1 x) h
⊢ (fun h ↦ exp ((t + h) • x) - exp (t • x) - (exp (t • x) • ContinuousLinearMap.smulRight 1 x) h) =o[𝓝 0] fun h ↦ h

Tactic:
by
    apply (IsLittleO.const_mul_left _ _).congr' this (EventuallyEq.refl _ _)
    rw [← hasFDerivAt_iff_isLittleO_nhds_zero (f := fun u => exp (u • x))
      (f' := (1 : 𝕊 →L[𝕂] 𝕊).smulRight x) (x := 0)]
    have : HasFDerivAt exp (1 : 𝔸 →L[𝕂] 𝔸) ((1 : 𝕊 →L[𝕂] 𝕊).smulRight x 0) := by
      rw [ContinuousLinearMap.smulRight_apply, one_apply_eq_self, zero_smul]
      exact hasFDerivAt_exp_zero_of_radius_pos htx.pos
    exact this.comp 0 ((1 : 𝕊 →L[𝕂] 𝕊).smulRight x).hasFDerivAt

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

Goal state before:
pp: case e_6.inl
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
m : ℕ
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
c : OrderedFinpartition m
e : E
v : Fin m → E
j : Fin c.length
⊢ ∀ (i : ℕ) (him : i < update c.partSize j (c.partSize j + 1) j) (hin : i < c.partSize j + 1),
    (cons e v ∘ (fun i ↦ cases 0 (succ ∘ c.emb j) i) ∘ Fin.cast ⋯) ⟨i, him⟩ = cons e (v ∘ c.emb j) ⟨i, hin⟩

Tactic:
intro a

Goal state after:
case e_6.inl
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
m : ℕ
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
c : OrderedFinpartition m
e : E
v : Fin m → E
j : Fin c.length
a : ℕ
⊢ ∀ (him : a < update c.partSize j (c.partSize j + 1) j) (hin : a < c.partSize j + 1),
    (cons e v ∘ (fun i ↦ cases 0 (succ ∘ c.emb j) i) ∘ Fin.cast ⋯) ⟨a, him⟩ = cons e (v ∘ c.emb j) ⟨a, hin⟩
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
s : Set E
f : E → F
hu : UniqueDiffOn 𝕜 s
x : E
hx : x ∈ s
h : Fin 0 → E
⊢ (iteratedFDerivWithin 𝕜 0 (-f) s x) h = (-iteratedFDerivWithin 𝕜 0 f s x) h

Tactic:
| zero =>

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

Goal state before:
pp: case zero
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
k : ℕ
⊢ iteratedDerivWithin k (fun x ↦ x ^ 0) s x = ↑(Nat.descFactorial 0 k) * x ^ (0 - k)

Tactic:
| zero =>

Goal state after:
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:
| zero =>

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

Goal state before:
pp: case zero
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 : 𝕜
⊢ (iteratedDeriv 0 fun z ↦ f (s + z)) = fun t ↦ iteratedDeriv 0 f (s + t)

Tactic:
| zero =>

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

Goal state before:
pp: case zero
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 : 𝕜
⊢ (iteratedDeriv 0 fun z ↦ f (z + s)) = fun t ↦ iteratedDeriv 0 f (t + s)

Tactic:
| zero =>

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

Goal state before:
pp: case zero
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type uF
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
n : ℕ∞ω
inst✝ : CompleteSpace E
f : OpenPartialHomeomorph E F
f₀' : E ≃L[𝕜] F
a : F
ha : a ∈ f.target
hf₀' : HasFDerivAt (↑f) (↑f₀') (↑f.symm a)
hf : ContDiffAt 𝕜 (↑0) (↑f) (↑f.symm a)
⊢ ContDiffAt 𝕜 (↑0) (↑f.symm) a

Tactic:
| zero =>

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type uF
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
n✝ : ℕ∞ω
inst✝ : CompleteSpace E
f : OpenPartialHomeomorph E F
f₀' : E ≃L[𝕜] F
a : F
ha : a ∈ f.target
hf₀'✝ : HasFDerivAt (↑f) (↑f₀') (↑f.symm a)
n : ℕ
IH : ContDiffAt 𝕜 (↑↑n) (↑f) (↑f.symm a) → ContDiffAt 𝕜 (↑↑n) (↑f.symm) a
hf : ContDiffAt 𝕜 (↑↑n.succ) (↑f) (↑f.symm a)
f' : E → E →L[𝕜] F
hf' : ContDiffAt 𝕜 (↑n) f' (↑f.symm a)
u : Set E
hu : u ∈ 𝓝 (↑f.symm a)
hff' : ∀ x ∈ u, HasFDerivAt (↑f) (f' x) x
eq_f₀' : f' (↑f.symm a) = ↑f₀'
hf₀' : range ContinuousLinearEquiv.toContinuousLinearMap ∈ 𝓝 (f' (↑f.symm a))
⊢ {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑f.symm a)

Tactic:
exact hf'.continuousAt.preimage_mem_nhds hf₀'

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 : AddCommGroup G
inst✝ : SMul R G
C✝ C₁ C₂ C : ConvexCone R G
h₁ : C.Pointed
h₂ : C.Salient
⊢ ∀ (a b : G), a ≤ b → b ≤ a → a = b

Tactic:
intro a

Goal state after:
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 : AddCommGroup G
inst✝ : SMul R G
C✝ C₁ C₂ C : ConvexCone R G
h₁ : C.Pointed
h₂ : C.Salient
a : G
⊢ ∀ (b : G), a ≤ b → b ≤ a → a = b
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: x : ℝ
h : |x| < 1
n : ℕ
F : ℝ → ℝ := fun x ↦ ∑ i ∈ Finset.range n, x ^ (i + 1) / (↑i + 1) + log (1 - x)
F' : ℝ → ℝ := fun x ↦ -x ^ n / (1 - x)
A : ∀ y ∈ Set.Ioo (-1) 1, HasDerivAt F (F' y) y
y : ℝ
hy : y ∈ Set.Icc (-|x|) |x|
this : |y| ≤ |x|
⊢ 1 - |x| ≤ 1 - y

Tactic:
by linarith [hy.2]

Goal state after: