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

Goal state before:
pp: case neg.emb.refine_2.inl.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✝
n : ℕ
c : OrderedFinpartition (n + 1)
h : ¬range (c.emb 0) = {0}
B : c.partSize (c.index 0) - 1 + 1 = c.partSize (c.index 0)
j : Fin (c.partSize (c.index 0))
hj : j ≠ 0
⊢ c.emb (c.index 0) j = cases 0 (succ ∘ fun j ↦ (c.emb (c.index 0) (Fin.cast ⋯ j.succ)).pred ⋯) (Fin.cast ⋯ ⟨↑j, ⋯⟩)

Tactic:
·

Goal state after:
case neg.emb.refine_2.inl.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✝
n : ℕ
c : OrderedFinpartition (n + 1)
h : ¬range (c.emb 0) = {0}
B : c.partSize (c.index 0) - 1 + 1 = c.partSize (c.index 0)
j : Fin (c.partSize (c.index 0))
hj : j ≠ 0
⊢ c.emb (c.index 0) j = cases 0 (succ ∘ fun j ↦ (c.emb (c.index 0) (Fin.cast ⋯ j.succ)).pred ⋯) (Fin.cast ⋯ ⟨↑j, ⋯⟩)
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: 𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : CharZero 𝕂
inst✝² : ContinuousSMul ℚ 𝕂
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
this : ∀ {n : ℕ}, ↑n ! ≠ 0
⊢ Tendsto (fun n ↦ ‖(↑n.succ !)⁻¹‖ / ‖(↑n !)⁻¹‖) atTop (𝓝 0)

pp: 𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : CharZero 𝕂
inst✝² : ContinuousSMul ℚ 𝕂
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
this : ∀ {n : ℕ}, ↑n ! ≠ 0
n : ℕ
⊢ (↑n !)⁻¹ ≠ 0

Tactic:
·

Goal state after:
𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : CharZero 𝕂
inst✝² : ContinuousSMul ℚ 𝕂
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
this : ∀ {n : ℕ}, ↑n ! ≠ 0
⊢ Tendsto (fun n ↦ ‖(↑n.succ !)⁻¹‖ / ‖(↑n !)⁻¹‖) atTop (𝓝 0)
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case neg.emb.refine_2.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✝
n : ℕ
c : OrderedFinpartition (n + 1)
h : ¬range (c.emb 0) = {0}
B : c.partSize (c.index 0) - 1 + 1 = c.partSize (c.index 0)
i : Fin c.length
j : Fin (c.partSize i)
hi : i ≠ c.index 0
⊢ c.emb i j =
    (if h_1 : i = c.index 0 then
        (fun i ↦ cases 0 (succ ∘ fun j ↦ (c.emb (c.index 0) (Fin.cast ⋯ j.succ)).pred ⋯) i) ∘ Fin.cast ⋯
      else
        succ ∘
          (fun j ↦
              if h : i = c.index 0 then (c.emb i (Fin.cast ⋯ j.succ)).pred ⋯ else (c.emb i (Fin.cast ⋯ j)).pred ⋯) ∘
            Fin.cast ⋯)
      ⟨↑j, ⋯⟩

Tactic:
·

Goal state after:
case neg.emb.refine_2.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✝
n : ℕ
c : OrderedFinpartition (n + 1)
h : ¬range (c.emb 0) = {0}
B : c.partSize (c.index 0) - 1 + 1 = c.partSize (c.index 0)
i : Fin c.length
j : Fin (c.partSize i)
hi : i ≠ c.index 0
⊢ c.emb i j =
    (if h_1 : i = c.index 0 then
        (fun i ↦ cases 0 (succ ∘ fun j ↦ (c.emb (c.index 0) (Fin.cast ⋯ j.succ)).pred ⋯) i) ∘ Fin.cast ⋯
      else
        succ ∘
          (fun j ↦
              if h : i = c.index 0 then (c.emb i (Fin.cast ⋯ j.succ)).pred ⋯ else (c.emb i (Fin.cast ⋯ j)).pred ⋯) ∘
            Fin.cast ⋯)
      ⟨↑j, ⋯⟩
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: 𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : CharZero 𝕂
inst✝² : ContinuousSMul ℚ 𝕂
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
this : ∀ {n : ℕ}, ↑n ! ≠ 0
n : ℕ
⊢ (↑n !)⁻¹ ≠ 0

Tactic:
·

Goal state after:
𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : CharZero 𝕂
inst✝² : ContinuousSMul ℚ 𝕂
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
this : ∀ {n : ℕ}, ↑n ! ≠ 0
n : ℕ
⊢ (↑n !)⁻¹ ≠ 0
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case pos
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
n : ℕ
c : OrderedFinpartition n
p : (i : Fin c.length) → E [×c.partSize i]→L[𝕜] F
j : Fin n
v : Fin n → E
z : E
m : Fin c.length
h : m = c.index j
⊢ c.applyOrderedFinpartition p (update v j z) m =
    update (c.applyOrderedFinpartition p v) (c.index j)
      ((p (c.index j)) (update (v ∘ c.emb (c.index j)) (c.invEmbedding j) z)) m

pp: case neg
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
n : ℕ
c : OrderedFinpartition n
p : (i : Fin c.length) → E [×c.partSize i]→L[𝕜] F
j : Fin n
v : Fin n → E
z : E
m : Fin c.length
h : ¬m = c.index j
⊢ c.applyOrderedFinpartition p (update v j z) m =
    update (c.applyOrderedFinpartition p v) (c.index j)
      ((p (c.index j)) (update (v ∘ c.emb (c.index j)) (c.invEmbedding j) z)) m

Tactic:
·

Goal state after:
case pos
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
n : ℕ
c : OrderedFinpartition n
p : (i : Fin c.length) → E [×c.partSize i]→L[𝕜] F
j : Fin n
v : Fin n → E
z : E
m : Fin c.length
h : m = c.index j
⊢ c.applyOrderedFinpartition p (update v j z) m =
    update (c.applyOrderedFinpartition p v) (c.index j)
      ((p (c.index j)) (update (v ∘ c.emb (c.index j)) (c.invEmbedding j) z)) m
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case pos.e_6
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
n : ℕ
c : OrderedFinpartition n
p : (i : Fin c.length) → E [×c.partSize i]→L[𝕜] F
j : Fin n
v : Fin n → E
z : E
m : Fin c.length
h : m = c.index j
⊢ update v j z ∘ c.emb (c.index j) = update v (c.emb (c.index j) (c.invEmbedding j)) z ∘ c.emb (c.index j)

pp: case pos.e_6.hf
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
n : ℕ
c : OrderedFinpartition n
p : (i : Fin c.length) → E [×c.partSize i]→L[𝕜] F
j : Fin n
v : Fin n → E
z : E
m : Fin c.length
h : m = c.index j
⊢ Injective (c.emb (c.index j))

Tactic:
·

Goal state after:
case pos.e_6
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
n : ℕ
c : OrderedFinpartition n
p : (i : Fin c.length) → E [×c.partSize i]→L[𝕜] F
j : Fin n
v : Fin n → E
z : E
m : Fin c.length
h : m = c.index j
⊢ update v j z ∘ c.emb (c.index j) = update v (c.emb (c.index j) (c.invEmbedding j)) z ∘ c.emb (c.index j)
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case pos.e_6.hf
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
n : ℕ
c : OrderedFinpartition n
p : (i : Fin c.length) → E [×c.partSize i]→L[𝕜] F
j : Fin n
v : Fin n → E
z : E
m : Fin c.length
h : m = c.index j
⊢ Injective (c.emb (c.index j))

Tactic:
·

Goal state after:
case pos.e_6.hf
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
n : ℕ
c : OrderedFinpartition n
p : (i : Fin c.length) → E [×c.partSize i]→L[𝕜] F
j : Fin n
v : Fin n → E
z : E
m : Fin c.length
h : m = c.index j
⊢ Injective (c.emb (c.index j))
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case neg
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
n : ℕ
c : OrderedFinpartition n
p : (i : Fin c.length) → E [×c.partSize i]→L[𝕜] F
j : Fin n
v : Fin n → E
z : E
m : Fin c.length
h : ¬m = c.index j
⊢ c.applyOrderedFinpartition p (update v j z) m =
    update (c.applyOrderedFinpartition p v) (c.index j)
      ((p (c.index j)) (update (v ∘ c.emb (c.index j)) (c.invEmbedding j) z)) m

Tactic:
·

Goal state after:
case neg
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
n : ℕ
c : OrderedFinpartition n
p : (i : Fin c.length) → E [×c.partSize i]→L[𝕜] F
j : Fin n
v : Fin n → E
z : E
m : Fin c.length
h : ¬m = c.index j
⊢ c.applyOrderedFinpartition p (update v j z) m =
    update (c.applyOrderedFinpartition p v) (c.index j)
      ((p (c.index j)) (update (v ∘ c.emb (c.index j)) (c.invEmbedding j) z)) m
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case pos
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
n : ℕ
c : OrderedFinpartition n
p : (i : Fin c.length) → E [×c.partSize i]→L[𝕜] F
m : Fin c.length
v : Fin n → E
q : E [×c.partSize m]→L[𝕜] F
d : Fin c.length
h : d = m
⊢ c.applyOrderedFinpartition (update p m q) v d = update (c.applyOrderedFinpartition p v) m (q (v ∘ c.emb m)) d

pp: case neg
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
n : ℕ
c : OrderedFinpartition n
p : (i : Fin c.length) → E [×c.partSize i]→L[𝕜] F
m : Fin c.length
v : Fin n → E
q : E [×c.partSize m]→L[𝕜] F
d : Fin c.length
h : ¬d = m
⊢ c.applyOrderedFinpartition (update p m q) v d = update (c.applyOrderedFinpartition p v) m (q (v ∘ c.emb m)) d

Tactic:
·

Goal state after:
case pos
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
n : ℕ
c : OrderedFinpartition n
p : (i : Fin c.length) → E [×c.partSize i]→L[𝕜] F
m : Fin c.length
v : Fin n → E
q : E [×c.partSize m]→L[𝕜] F
d : Fin c.length
h : d = m
⊢ c.applyOrderedFinpartition (update p m q) v d = update (c.applyOrderedFinpartition p v) m (q (v ∘ c.emb m)) d
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case neg
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
n : ℕ
c : OrderedFinpartition n
p : (i : Fin c.length) → E [×c.partSize i]→L[𝕜] F
m : Fin c.length
v : Fin n → E
q : E [×c.partSize m]→L[𝕜] F
d : Fin c.length
h : ¬d = m
⊢ c.applyOrderedFinpartition (update p m q) v d = update (c.applyOrderedFinpartition p v) m (q (v ∘ c.emb m)) d

Tactic:
·

Goal state after:
case neg
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
n : ℕ
c : OrderedFinpartition n
p : (i : Fin c.length) → E [×c.partSize i]→L[𝕜] F
m : Fin c.length
v : Fin n → E
q : E [×c.partSize m]→L[𝕜] F
d : Fin c.length
h : ¬d = m
⊢ c.applyOrderedFinpartition (update p m q) v d = update (c.applyOrderedFinpartition p v) m (q (v ∘ c.emb m)) d
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: case inl
𝔸 : Type u_1
inst✝² : NormedDivisionRing 𝔸
inst✝¹ : NormedAlgebra ℚ 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
n : ℕ
⊢ exp (↑n • x) = exp x ^ ↑n

pp: case inr
𝔸 : Type u_1
inst✝² : NormedDivisionRing 𝔸
inst✝¹ : NormedAlgebra ℚ 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
n : ℕ
⊢ exp (-↑n • x) = exp x ^ (-↑n)

Tactic:
·

Goal state after:
case inl
𝔸 : Type u_1
inst✝² : NormedDivisionRing 𝔸
inst✝¹ : NormedAlgebra ℚ 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
n : ℕ
⊢ exp (↑n • x) = exp x ^ ↑n
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: case inr
𝔸 : Type u_1
inst✝² : NormedDivisionRing 𝔸
inst✝¹ : NormedAlgebra ℚ 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
n : ℕ
⊢ exp (-↑n • x) = exp x ^ (-↑n)

Tactic:
·

Goal state after:
case inr
𝔸 : Type u_1
inst✝² : NormedDivisionRing 𝔸
inst✝¹ : NormedAlgebra ℚ 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
n : ℕ
⊢ exp (-↑n • x) = exp x ^ (-↑n)
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case h₁
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‖ ≤
    ‖f₁‖ * ↑c.length * max ‖g₁‖ ‖g₂‖ ^ (c.length - 1) * ‖g₁ - g₂‖

pp: case h₂
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 g₂ - c.compAlongOrderedFinpartition f₂ g₂‖ ≤ ‖f₁ - f₂‖ * ∏ i, ‖g₂ i‖

Tactic:
·

Goal state after:
case h₁
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‖ ≤
    ‖f₁‖ * ↑c.length * max ‖g₁‖ ‖g₂‖ ^ (c.length - 1) * ‖g₁ - g₂‖
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case h₂
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 g₂ - c.compAlongOrderedFinpartition f₂ g₂‖ ≤ ‖f₁ - f₂‖ * ∏ i, ‖g₂ i‖

Tactic:
·

Goal state after:
case h₂
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 g₂ - c.compAlongOrderedFinpartition f₂ g₂‖ ≤ ‖f₁ - f₂‖ * ∏ i, ‖g₂ i‖
Module: Mathlib.Topology.MetricSpace.CauSeqFilter

Goal state before:
pp: case left
β : Type v
inst✝ : NormedField β
f : CauSeq β norm
s : Set (β × β)
hs : s ∈ uniformity β
ε : ℝ
hε : ε > 0
hεs : ∀ ⦃a b : β⦄, dist a b < ε → (a, b) ∈ s
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, ‖↑f j - ↑f k‖ < ε
⊢ ∃ a, ∀ (b : ℕ), a ≤ b → b ∈ ↑f ⁻¹' ↑f '' {n | n ≥ N}

pp: case right
β : Type v
inst✝ : NormedField β
f : CauSeq β norm
s : Set (β × β)
hs : s ∈ uniformity β
ε : ℝ
hε : ε > 0
hεs : ∀ ⦃a b : β⦄, dist a b < ε → (a, b) ∈ s
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, ‖↑f j - ↑f k‖ < ε
⊢ (↑f '' {n | n ≥ N}) ×ˢ (↑f '' {n | n ≥ N}) ⊆ s

Tactic:
·

Goal state after:
case left
β : Type v
inst✝ : NormedField β
f : CauSeq β norm
s : Set (β × β)
hs : s ∈ uniformity β
ε : ℝ
hε : ε > 0
hεs : ∀ ⦃a b : β⦄, dist a b < ε → (a, b) ∈ s
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, ‖↑f j - ↑f k‖ < ε
⊢ ∃ a, ∀ (b : ℕ), a ≤ b → b ∈ ↑f ⁻¹' ↑f '' {n | n ≥ N}
Module: Mathlib.Topology.MetricSpace.CauSeqFilter

Goal state before:
pp: case right
β : Type v
inst✝ : NormedField β
f : CauSeq β norm
s : Set (β × β)
hs : s ∈ uniformity β
ε : ℝ
hε : ε > 0
hεs : ∀ ⦃a b : β⦄, dist a b < ε → (a, b) ∈ s
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, ‖↑f j - ↑f k‖ < ε
⊢ (↑f '' {n | n ≥ N}) ×ˢ (↑f '' {n | n ≥ N}) ⊆ s

Tactic:
·

Goal state after:
case right
β : Type v
inst✝ : NormedField β
f : CauSeq β norm
s : Set (β × β)
hs : s ∈ uniformity β
ε : ℝ
hε : ε > 0
hεs : ∀ ⦃a b : β⦄, dist a b < ε → (a, b) ∈ s
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, ‖↑f j - ↑f k‖ < ε
⊢ (↑f '' {n | n ≥ N}) ×ˢ (↑f '' {n | n ≥ N}) ⊆ s
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case refine_1
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
α : Type u_5
H : Type u_6
inst✝ : NormedAddCommGroup H
l : Filter α
p₁ p₂ : α → FormalMultilinearSeries 𝕜 F G
q₁ q₂ : α → FormalMultilinearSeries 𝕜 E F
f : α → H
n : ℕ
hp_bdd : ∀ k ≤ n, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖p₁ x k‖
hpf : ∀ k ≤ n, (fun a ↦ p₁ a k - p₂ a k) =O[l] f
hq₁_bdd : ∀ k ≤ n, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖q₁ x k‖
hq₂_bdd : ∀ k ≤ n, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖q₂ x k‖
hqf : ∀ k ≤ n, (fun a ↦ q₁ a k - q₂ a k) =O[l] f
c : OrderedFinpartition n
x✝ : c ∈ Finset.univ
⊢ (fun x ↦
      ‖p₁ x c.length‖ * ↑c.length * max ‖fun m ↦ q₁ x (c.partSize m)‖ ‖fun m ↦ q₂ x (c.partSize m)‖ ^ (c.length - 1) *
        ‖(fun m ↦ q₁ x (c.partSize m)) - fun m ↦ q₂ x (c.partSize m)‖) =O[l]
    f

pp: case refine_2
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
α : Type u_5
H : Type u_6
inst✝ : NormedAddCommGroup H
l : Filter α
p₁ p₂ : α → FormalMultilinearSeries 𝕜 F G
q₁ q₂ : α → FormalMultilinearSeries 𝕜 E F
f : α → H
n : ℕ
hp_bdd : ∀ k ≤ n, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖p₁ x k‖
hpf : ∀ k ≤ n, (fun a ↦ p₁ a k - p₂ a k) =O[l] f
hq₁_bdd : ∀ k ≤ n, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖q₁ x k‖
hq₂_bdd : ∀ k ≤ n, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖q₂ x k‖
hqf : ∀ k ≤ n, (fun a ↦ q₁ a k - q₂ a k) =O[l] f
c : OrderedFinpartition n
x✝ : c ∈ Finset.univ
⊢ (fun x ↦ ‖p₁ x c.length - p₂ x c.length‖ * ∏ i, ‖q₂ x (c.partSize i)‖) =O[l] f

Tactic:
·

Goal state after:
case refine_1
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
α : Type u_5
H : Type u_6
inst✝ : NormedAddCommGroup H
l : Filter α
p₁ p₂ : α → FormalMultilinearSeries 𝕜 F G
q₁ q₂ : α → FormalMultilinearSeries 𝕜 E F
f : α → H
n : ℕ
hp_bdd : ∀ k ≤ n, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖p₁ x k‖
hpf : ∀ k ≤ n, (fun a ↦ p₁ a k - p₂ a k) =O[l] f
hq₁_bdd : ∀ k ≤ n, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖q₁ x k‖
hq₂_bdd : ∀ k ≤ n, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖q₂ x k‖
hqf : ∀ k ≤ n, (fun a ↦ q₁ a k - q₂ a k) =O[l] f
c : OrderedFinpartition n
x✝ : c ∈ Finset.univ
⊢ (fun x ↦
      ‖p₁ x c.length‖ * ↑c.length * max ‖fun m ↦ q₁ x (c.partSize m)‖ ‖fun m ↦ q₂ x (c.partSize m)‖ ^ (c.length - 1) *
        ‖(fun m ↦ q₁ x (c.partSize m)) - fun m ↦ q₂ x (c.partSize m)‖) =O[l]
    f
Module: Mathlib.Analysis.SpecialFunctions.Exponential

Goal state before:
pp: 𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : NormedRing 𝔸
inst✝² : CharZero 𝕂
inst✝¹ : NormedAlgebra 𝕂 𝔸
inst✝ : CompleteSpace 𝔸
h : 0 < (expSeries 𝕂 𝔸).radius
⊢ HasStrictFDerivAt exp 1 0

Tactic:
convert! (hasFPowerSeriesAt_exp_zero_of_radius_pos h).hasStrictFDerivAt
  ext x
  change x = expSeries 𝕂 𝔸 1 fun _ => x
  simp [expSeries_apply_eq, Nat.factorial]

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
⊢ iteratedDerivWithin 0 id s x = if 0 = 0 then x else if 0 = 1 then 1 else 0

pp: case succ
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
n : ℕ
⊢ iteratedDerivWithin (n + 1) id s x = if n + 1 = 0 then x else if n + 1 = 1 then 1 else 0

Tactic:
·

Goal state after:
case zero
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
⊢ iteratedDerivWithin 0 id s x = if 0 = 0 then x else if 0 = 1 then 1 else 0
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: case succ
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
n : ℕ
⊢ iteratedDerivWithin (n + 1) id s x = if n + 1 = 0 then x else if n + 1 = 1 then 1 else 0

Tactic:
·

Goal state after:
case succ
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
n : ℕ
⊢ iteratedDerivWithin (n + 1) id s x = if n + 1 = 0 then x else if n + 1 = 1 then 1 else 0
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: case succ
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
n : ℕ
⊢ iteratedDerivWithin n (fun x ↦ 1) s x = if n + 1 = 0 then x else if n + 1 = 1 then 1 else 0

pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
n : ℕ
⊢ Set.EqOn (derivWithin id s) (fun x ↦ 1) s

Tactic:
·

Goal state after:
case succ
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
n : ℕ
⊢ iteratedDerivWithin n (fun x ↦ 1) s x = if n + 1 = 0 then x else if n + 1 = 1 then 1 else 0
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
n : ℕ
⊢ Set.EqOn (derivWithin id s) (fun x ↦ 1) s

Tactic:
·

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
n : ℕ
⊢ Set.EqOn (derivWithin id s) (fun x ↦ 1) s