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

Goal state before:
pp: case neg.emb.refine_2
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
⊢ ∀ (i_1 : Fin (c.partSize i)),
    c.emb i i_1 =
      (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 ⋯)
        ⟨↑i_1, ⋯⟩

Tactic:
intro j

Goal state after:
case neg.emb.refine_2
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)
⊢ 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: case empty
𝔸 : Type u_1
inst✝² : NormedRing 𝔸
inst✝¹ : NormedAlgebra ℚ 𝔸
inst✝ : CompleteSpace 𝔸
ι : Type u_3
f : ι → 𝔸
h : (↑∅).Pairwise (Commute on f)
⊢ exp (∑ i ∈ ∅, f i) = ∅.noncommProd (fun i ↦ exp (f i)) ⋯

Tactic:
simp

Goal state after:
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)

Tactic:
simp

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

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
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
A : Disjoint (range (c.emb m)) (range (c.emb (c.index j)))
⊢ c.emb (c.index j) (c.invEmbedding j) = j

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_7
F : Type u_8
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f g : 𝕜 → F
x : 𝕜
s : Set 𝕜
hfg : f =ᶠ[𝓝[insert x s] x] g
⊢ 𝓝[s] x ≤ 𝓝[insert x s] x

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_7
F : Type u_8
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f g : 𝕜 → F
x : 𝕜
s : Set 𝕜
hfg : f =ᶠ[𝓝[insert x s] x] g
⊢ x ∈ insert x s

Tactic:
simp

Goal state after:
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:
simp

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:
simp

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:
simp

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

Goal state before:
pp: β : Type v
inst✝ : NormedField β
f : ℕ → β
hf : CauchySeq f
hf1 : (map f atTop).NeBot
hf2 : ∀ s ∈ uniformity β, ∃ t ∈ map f atTop, t ×ˢ t ⊆ s
ε : ℝ
hε : ε > 0
t : Set β
htsub : t ×ˢ t ⊆ {x | dist x.1 x.2 < ε}
N : ℕ
hN : ∀ (b : ℕ), N ≤ b → f b ∈ t
⊢ ∀ j ≥ N, ‖f j - f N‖ < ε

Tactic:
intro j

Goal state after:
β : Type v
inst✝ : NormedField β
f : ℕ → β
hf : CauchySeq f
hf1 : (map f atTop).NeBot
hf2 : ∀ s ∈ uniformity β, ∃ t ∈ map f atTop, t ×ˢ t ⊆ s
ε : ℝ
hε : ε > 0
t : Set β
htsub : t ×ˢ t ⊆ {x | dist x.1 x.2 < ε}
N : ℕ
hN : ∀ (b : ℕ), N ≤ b → f b ∈ t
j : ℕ
⊢ j ≥ N → ‖f j - f N‖ < ε
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:
simp

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

Tactic:
simp

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:
simp

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

Goal state before:
pp: 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
n : ℕ
IH :
  ∀ {f : 𝕜 → 𝔸} {g : 𝕜 → F},
    ContDiffWithinAt 𝕜 (↑n) f s x →
      ContDiffWithinAt 𝕜 (↑n) g s x →
        iteratedDerivWithin n (f • g) s x =
          ∑ i ∈ Finset.range (n + 1), n.choose i • iteratedDerivWithin i f s x • iteratedDerivWithin (n - i) g s x
f : 𝕜 → 𝔸
g : 𝕜 → F
hf : ContDiffWithinAt 𝕜 (↑(n + 1)) f s x
hg : ContDiffWithinAt 𝕜 (↑(n + 1)) g s x
⊢ ↑(n + 1) ≠ ↑⊤

Tactic:
simp

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

Goal state before:
pp: 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
n : ℕ
IH :
  ∀ {f : 𝕜 → 𝔸} {g : 𝕜 → F},
    ContDiffWithinAt 𝕜 (↑n) f s x →
      ContDiffWithinAt 𝕜 (↑n) g s x →
        iteratedDerivWithin n (f • g) s x =
          ∑ i ∈ Finset.range (n + 1), n.choose i • iteratedDerivWithin i f s x • iteratedDerivWithin (n - i) g s x
f : 𝕜 → 𝔸
g : 𝕜 → F
hf : ContDiffWithinAt 𝕜 (↑(n + 1)) f s x
hg : ContDiffWithinAt 𝕜 (↑(n + 1)) g s x
U : Set 𝕜
hU : U ∈ 𝓝[insert x s] x
H : ∀ y ∈ U, ContDiffWithinAt 𝕜 (↑(n + 1)) f s y ∧ ContDiffWithinAt 𝕜 (↑(n + 1)) g s y
⊢ ∀ i ∈ Finset.range n, 1 ≤ n - i

Tactic:
simp

Goal state after:
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
n : ℕ
IH :
  ∀ {f : 𝕜 → 𝔸} {g : 𝕜 → F},
    ContDiffWithinAt 𝕜 (↑n) f s x →
      ContDiffWithinAt 𝕜 (↑n) g s x →
        iteratedDerivWithin n (f • g) s x =
          ∑ i ∈ Finset.range (n + 1), n.choose i • iteratedDerivWithin i f s x • iteratedDerivWithin (n - i) g s x
f : 𝕜 → 𝔸
g : 𝕜 → F
hf : ContDiffWithinAt 𝕜 (↑(n + 1)) f s x
hg : ContDiffWithinAt 𝕜 (↑(n + 1)) g s x
U : Set 𝕜
hU : U ∈ 𝓝[insert x s] x
H : ∀ y ∈ U, ContDiffWithinAt 𝕜 (↑(n + 1)) f s y ∧ ContDiffWithinAt 𝕜 (↑(n + 1)) g s y
⊢ ∀ i < n, 1 ≤ n - i
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
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
⊢ update c.partSize j (c.partSize j + 1) j = c.partSize j + 1

Tactic:
simp

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
s : Set E
x : E
n✝ : ℕ∞ω
ι : Type u_3
inst✝² : Fintype ι
F' : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (F' i)
inst✝ : (i : ι) → NormedSpace 𝕜 (F' i)
Φ : E → (i : ι) → F' i
pr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj
n : ℕ∞
h : ∀ (i : ι), ContDiffWithinAt 𝕜 (↑n) (fun x ↦ Φ x i) s x
m : ℕ
hm : ↑m ≤ n
⊢ ∃ u ∈ 𝓝[insert x s] x, ∃ p, HasFTaylorSeriesUpToOn (↑m) Φ p u

Tactic:
choose u hux p hp using fun i => h i m hm

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
s : Set E
x : E
n✝ : ℕ∞ω
ι : Type u_3
inst✝² : Fintype ι
F' : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (F' i)
inst✝ : (i : ι) → NormedSpace 𝕜 (F' i)
Φ : E → (i : ι) → F' i
pr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj
n : ℕ∞
h : ∀ (i : ι), ContDiffWithinAt 𝕜 (↑n) (fun x ↦ Φ x i) s x
m : ℕ
hm : ↑m ≤ n
u : ι → Set E
hux : ∀ (i : ι), u i ∈ 𝓝[insert x s] x
p : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)
hp : ∀ (i : ι), HasFTaylorSeriesUpToOn (↑m) (fun x ↦ Φ x i) (p i) (u i)
⊢ ∃ u ∈ 𝓝[insert x s] x, ∃ p, HasFTaylorSeriesUpToOn (↑m) Φ p u
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
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
ha : 0 < update c.partSize j (c.partSize j + 1) j
h'a : 0 < c.partSize j + 1
⊢ (cons e v ∘ (fun i ↦ cases 0 (succ ∘ c.emb j) i) ∘ Fin.cast ⋯) ⟨0, ha⟩ = cons e (v ∘ c.emb j) ⟨0, h'a⟩

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
ι : Type u_3
inst✝³ : Fintype ι
F' : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (F' i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (F' i)
inst✝ : DecidableEq ι
k : ℕ∞ω
x : (i : ι) → F' i
i : ι
⊢ ∀ (i_1 : ι), ContDiff 𝕜 k fun x_1 ↦ update x i x_1 i_1

Tactic:
intro j

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
ι : Type u_3
inst✝³ : Fintype ι
F' : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (F' i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (F' i)
inst✝ : DecidableEq ι
k : ℕ∞ω
x : (i : ι) → F' i
i j : ι
⊢ ContDiff 𝕜 k fun x_1 ↦ update x i x_1 j
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 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
n : ℕ
IH :
  ∀ {f : 𝕜 → 𝔸} {g : 𝕜 → F},
    ContDiffWithinAt 𝕜 (↑n) f s x →
      ContDiffWithinAt 𝕜 (↑n) g s x →
        iteratedDerivWithin n (f • g) s x =
          ∑ i ∈ Finset.range (n + 1), n.choose i • iteratedDerivWithin i f s x • iteratedDerivWithin (n - i) g s x
f : 𝕜 → 𝔸
g : 𝕜 → F
hf : ContDiffWithinAt 𝕜 (↑(n + 1)) f s x
hg : ContDiffWithinAt 𝕜 (↑(n + 1)) g s x
U : Set 𝕜
hU : U ∈ 𝓝[insert x s] x
H : ∀ y ∈ U, ContDiffWithinAt 𝕜 (↑(n + 1)) f s y ∧ ContDiffWithinAt 𝕜 (↑(n + 1)) g s y
⊢ ↑(n + 1) ≠ ↑⊤

Tactic:
simp

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

Goal state before:
pp: x : ℂ
⊢ 1 < 2

Tactic:
simp

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

Goal state before:
pp: case e_6.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
m : ℕ
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
c : OrderedFinpartition m
i : Fin c.length
e : E
v : Fin m → E
j : Fin c.length
hij : j ≠ i
⊢ ∀ (i_1 : ℕ) (him : i_1 < update c.partSize i (c.partSize i + 1) j) (hin : i_1 < c.partSize j),
    (cons e v ∘ succ ∘ c.emb j ∘ Fin.cast ⋯) ⟨i_1, him⟩ = (v ∘ c.emb j) ⟨i_1, hin⟩

Tactic:
simp

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

Goal state before:
pp: 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
x : E
i : ℕ
f g : E → F
hf : ContDiffWithinAt 𝕜 (↑i) f s x
hg : ContDiffWithinAt 𝕜 (↑i) g s x
hu : UniqueDiffOn 𝕜 s
hx : x ∈ s
⊢ ↑i ≠ ∞

Tactic:
simp

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

Goal state before:
pp: 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
n : ℕ
IH :
  ∀ {f : 𝕜 → 𝔸} {g : 𝕜 → F},
    ContDiffWithinAt 𝕜 (↑n) f s x →
      ContDiffWithinAt 𝕜 (↑n) g s x →
        iteratedDerivWithin n (f • g) s x =
          ∑ i ∈ Finset.range (n + 1), n.choose i • iteratedDerivWithin i f s x • iteratedDerivWithin (n - i) g s x
f : 𝕜 → 𝔸
g : 𝕜 → F
hf : ContDiffWithinAt 𝕜 (↑(n + 1)) f s x
hg : ContDiffWithinAt 𝕜 (↑(n + 1)) g s x
U : Set 𝕜
hU : U ∈ 𝓝[insert x s] x
H : ∀ y ∈ U, ContDiffWithinAt 𝕜 (↑(n + 1)) f s y ∧ ContDiffWithinAt 𝕜 (↑(n + 1)) g s y
y : 𝕜
hfy : ContDiffWithinAt 𝕜 (↑(n + 1)) f s y
hgy : ContDiffWithinAt 𝕜 (↑(n + 1)) g s y
⊢ f y • derivWithin g s y + derivWithin f s y • g y = (f • derivWithin g s + derivWithin f s • g) y

Tactic:
simp

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

Goal state before:
pp: 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
n : ℕ
IH :
  ∀ {f : 𝕜 → 𝔸} {g : 𝕜 → F},
    ContDiffWithinAt 𝕜 (↑n) f s x →
      ContDiffWithinAt 𝕜 (↑n) g s x →
        iteratedDerivWithin n (f • g) s x =
          ∑ i ∈ Finset.range (n + 1), n.choose i • iteratedDerivWithin i f s x • iteratedDerivWithin (n - i) g s x
f : 𝕜 → 𝔸
g : 𝕜 → F
hf : ContDiffWithinAt 𝕜 (↑(n + 1)) f s x
hg : ContDiffWithinAt 𝕜 (↑(n + 1)) g s x
U : Set 𝕜
hU : U ∈ 𝓝[insert x s] x
H : ∀ y ∈ U, ContDiffWithinAt 𝕜 (↑(n + 1)) f s y ∧ ContDiffWithinAt 𝕜 (↑(n + 1)) g s y
y : 𝕜
hfy : ContDiffWithinAt 𝕜 (↑(n + 1)) f s y
hgy : ContDiffWithinAt 𝕜 (↑(n + 1)) g s y
⊢ ↑(n + 1) ≠ 0

Tactic:
simp

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
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:
simp

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

Goal state before:
pp: case succ.zero
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
i : ℕ
IH : ∀ (k : ℕ), iteratedDerivWithin k (fun x ↦ x ^ i) s x = ↑(i.descFactorial k) * x ^ (i - k)
⊢ iteratedDerivWithin 0 (fun x ↦ x ^ (i + 1)) s x = ↑((i + 1).descFactorial 0) * x ^ (i + 1 - 0)

Tactic:
simp

Goal state after:
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
i : ℕ
IH : ∀ (k : ℕ), iteratedDerivWithin k (fun x ↦ x ^ i) s x = ↑(i.descFactorial k) * x ^ (i - k)
k : ℕ
this : (i + 1).descFactorial (k + 1) = (k + 1) * i.descFactorial k + i.descFactorial (k + 1)
hik : i ≤ k
⊢ ∀ x ∈ Finset.range k, k + 1 - x ≠ 0

Tactic:
simp

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
i : ℕ
IH : ∀ (k : ℕ), iteratedDerivWithin k (fun x ↦ x ^ i) s x = ↑(i.descFactorial k) * x ^ (i - k)
k : ℕ
this : (i + 1).descFactorial (k + 1) = (k + 1) * i.descFactorial k + i.descFactorial (k + 1)
hik : i ≤ k
⊢ ∀ x < k, ¬k + 1 - x = 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
i : ℕ
IH : ∀ (k : ℕ), iteratedDerivWithin k (fun x ↦ x ^ i) s x = ↑(i.descFactorial k) * x ^ (i - k)
k : ℕ
this : (i + 1).descFactorial (k + 1) = (k + 1) * i.descFactorial k + i.descFactorial (k + 1)
hik : i ≤ k
⊢ ∀ x ∈ Finset.range k, k + 1 - x ≠ 1

Tactic:
simp

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
i : ℕ
IH : ∀ (k : ℕ), iteratedDerivWithin k (fun x ↦ x ^ i) s x = ↑(i.descFactorial k) * x ^ (i - k)
k : ℕ
this : (i + 1).descFactorial (k + 1) = (k + 1) * i.descFactorial k + i.descFactorial (k + 1)
hik : i ≤ k
⊢ ∀ x < k, ¬k + 1 - x = 1
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
i : ℕ
IH : ∀ (k : ℕ), iteratedDerivWithin k (fun x ↦ x ^ i) s x = ↑(i.descFactorial k) * x ^ (i - k)
k : ℕ
this : (i + 1).descFactorial (k + 1) = (k + 1) * i.descFactorial k + i.descFactorial (k + 1)
hik : k < i
⊢ ∀ x ∈ Finset.range k, k + 1 - x ≠ 0

Tactic:
simp

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
i : ℕ
IH : ∀ (k : ℕ), iteratedDerivWithin k (fun x ↦ x ^ i) s x = ↑(i.descFactorial k) * x ^ (i - k)
k : ℕ
this : (i + 1).descFactorial (k + 1) = (k + 1) * i.descFactorial k + i.descFactorial (k + 1)
hik : k < i
⊢ ∀ x < k, ¬k + 1 - x = 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
i : ℕ
IH : ∀ (k : ℕ), iteratedDerivWithin k (fun x ↦ x ^ i) s x = ↑(i.descFactorial k) * x ^ (i - k)
k : ℕ
this : (i + 1).descFactorial (k + 1) = (k + 1) * i.descFactorial k + i.descFactorial (k + 1)
hik : k < i
⊢ ∀ x ∈ Finset.range k, k + 1 - x ≠ 1

Tactic:
simp

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
i : ℕ
IH : ∀ (k : ℕ), iteratedDerivWithin k (fun x ↦ x ^ i) s x = ↑(i.descFactorial k) * x ^ (i - k)
k : ℕ
this : (i + 1).descFactorial (k + 1) = (k + 1) * i.descFactorial k + i.descFactorial (k + 1)
hik : k < i
⊢ ∀ x < k, ¬k + 1 - x = 1
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: case empty
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
ι : Type u_3
f : ι → E → F
i : ℕ
x : E
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
h : ∀ j ∈ ∅, ContDiffWithinAt 𝕜 (↑i) (f j) s x
⊢ iteratedFDerivWithin 𝕜 i (fun x ↦ ∑ j ∈ ∅, f j x) s x = ∑ j ∈ ∅, iteratedFDerivWithin 𝕜 i (f j) s x

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.Order.ExtendFrom

Goal state before:
pp: case neg.hf.inr.inr
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : DenselyOrdered α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
f : α → β
a b : α
la lb : β
inst✝ : RegularSpace β
hf : ContinuousOn f (Ioo a b)
ha : Tendsto f (𝓝[>] a) (𝓝 la)
hb : Tendsto f (𝓝[<] b) (𝓝 lb)
hab : a ≠ b
x : α
x_in : x ∈ Icc a b
h : x ∈ Ioo a b
⊢ ∃ y, Tendsto f (𝓝[Ioo a b] x) (𝓝 y)

Tactic:
exact ⟨f x, hf x h⟩

Goal state after:
Module: Mathlib.Topology.Order.ExtendFrom

Goal state before:
pp: case neg.hB
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : DenselyOrdered α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
f : α → β
a b : α
la : β
inst✝ : RegularSpace β
hf : ContinuousOn f (Ioo a b)
ha : Tendsto f (𝓝[>] a) (𝓝 la)
hab : a < b
⊢ Ico a b ⊆ closure[inst✝⁵] (Ioo a b)

Tactic:
rw [closure_Ioo hab.ne]

Goal state after:
case neg.hB
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : DenselyOrdered α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
f : α → β
a b : α
la : β
inst✝ : RegularSpace β
hf : ContinuousOn f (Ioo a b)
ha : Tendsto f (𝓝[>] a) (𝓝 la)
hab : a < b
⊢ Ico a b ⊆ Icc a b
Module: Mathlib.Topology.Order.ExtendFrom

Goal state before:
pp: case neg.hf.inr
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : DenselyOrdered α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
f : α → β
a b : α
la : β
inst✝ : RegularSpace β
hf : ContinuousOn f (Ioo a b)
ha : Tendsto f (𝓝[>] a) (𝓝 la)
hab : a < b
x : α
x_in : x ∈ Ico a b
h : x ∈ Ioo a b
⊢ ∃ y, Tendsto f (𝓝[Ioo a b] x) (𝓝 y)

Tactic:
exact ⟨f x, hf x h⟩

Goal state after: