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

Goal state before:
pp: case inr.mpr
𝕜 : Type u_1
𝕜' : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NontriviallyNormedField 𝕜'
inst✝¹ : NormedAlgebra 𝕜 𝕜'
inst✝ : IsRCLikeNormedField 𝕜
f g : 𝕜 → 𝕜'
s : Set 𝕜
hf : DifferentiableOn 𝕜 f s
hg : DifferentiableOn 𝕜 g s
hs2 : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hsc : IsPreconnected s
hgn : ∀ x ∈ s, g x ≠ 0
hfn : ∀ x ∈ s, f x ≠ 0
t : 𝕜
ht : t ∈ s
⊢ (∃ z, z ≠ 0 ∧ EqOn f (z • g) s) → EqOn (logDeriv f) (logDeriv g) s

Tactic:
rintro ⟨z, hz0, hz⟩ x hx
      simp [logDeriv_apply, hz.deriv hs2 hx, hz hx, deriv_const_smul _
        (hg.differentiableAt (hs2.mem_nhds hx)), mul_div_mul_left (deriv g x) (g x) hz0]

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 : ℝ
f' : ℝ → E
C : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x
bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C
g : ℝ → E := fun x ↦ f x - f a
hg : ContinuousOn g (Icc a b)
hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x
B : ℝ → ℝ := fun x ↦ C * (x - a)
hB : ∀ (x : ℝ), HasDerivAt B C x
⊢ 0 ≤ C * 0

Tactic:
mul_zero

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
f' : ℝ → E
C : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x
bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C
g : ℝ → E := fun x ↦ f x - f a
hg : ContinuousOn g (Icc a b)
hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x
B : ℝ → ℝ := fun x ↦ C * (x - a)
hB : ∀ (x : ℝ), HasDerivAt B C x
⊢ 0 ≤ 0
Module: Mathlib.Analysis.Calculus.MeanValue

Goal state before:
pp: 𝕜 : Type u_3
G : Type u_4
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 𝕜
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : E → G
C : ℝ≥0
hf : Differentiable 𝕜 f
bound : ∀ (x : E), ‖fderiv 𝕜 f x‖₊ ≤ C
⊢ LipschitzWith C f

Tactic:
let : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
  let A : NormedSpace ℝ E := .restrictScalars ℝ 𝕜 E
  rw [← lipschitzOnWith_univ]
  exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ

Goal state after:
Module: Mathlib.Analysis.Calculus.MeanValue

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
𝕜 : Type u_3
G : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f g : E → G
s : Set E
x : E
hs : Convex ℝ s
hf : DifferentiableOn 𝕜 f s
hg : DifferentiableOn 𝕜 g s
hs' : UniqueDiffOn 𝕜 s
hf' : EqOn (fderivWithin 𝕜 f s) (fderivWithin 𝕜 g s) s
hx : x ∈ s
hfgx : f x = g x
y : E
hy : y ∈ s
z : E
hz : z ∈ s
⊢ fderivWithin 𝕜 (f - g) s z = 0

Tactic:
fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz),

Goal state after:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
𝕜 : Type u_3
G : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f g : E → G
s : Set E
x : E
hs : Convex ℝ s
hf : DifferentiableOn 𝕜 f s
hg : DifferentiableOn 𝕜 g s
hs' : UniqueDiffOn 𝕜 s
hf' : EqOn (fderivWithin 𝕜 f s) (fderivWithin 𝕜 g s) s
hx : x ∈ s
hfgx : f x = g x
y : E
hy : y ∈ s
z : E
hz : z ∈ s
⊢ fderivWithin 𝕜 f s z - fderivWithin 𝕜 g s z = 0
Module: Mathlib.Analysis.Calculus.ContDiff.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E → F
x : E
n : ℕ∞ω
e : F ≃L[𝕜] G
⊢ ContDiffAt 𝕜 n (⇑e ∘ f) x ↔ ContDiffAt 𝕜 n f x

Tactic:
by
  simp only [← contDiffWithinAt_univ, e.comp_contDiffWithinAt_iff]

Goal state after:
Module: Mathlib.Analysis.Calculus.MeanValue

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
𝕜 : Type u_3
G : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E → G
s : Set E
f' : E → E →L[𝕜] G
x₀ : E
n : ℕ
hs : Convex ℝ s
hx₀s : x₀ ∈ s
hff' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf' : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : E) in 𝓝[s] x₀, ‖f' x‖ ≤ c * ‖‖x - x₀‖ ^ n‖
⊢ ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : E) in 𝓝[s] x₀, ‖f x - f x₀‖ ≤ c * ‖‖x - x₀‖ ^ (n + 1)‖

Tactic:
intro c

Goal state after:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
𝕜 : Type u_3
G : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E → G
s : Set E
f' : E → E →L[𝕜] G
x₀ : E
n : ℕ
hs : Convex ℝ s
hx₀s : x₀ ∈ s
hff' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf' : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : E) in 𝓝[s] x₀, ‖f' x‖ ≤ c * ‖‖x - x₀‖ ^ n‖
c : ℝ
⊢ 0 < c → ∀ᶠ (x : E) in 𝓝[s] x₀, ‖f x - f x₀‖ ≤ c * ‖‖x - x₀‖ ^ (n + 1)‖
Module: Mathlib.Analysis.Calculus.ContDiff.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E → F
n : ℕ∞ω
e : G ≃L[𝕜] E
⊢ ContDiff 𝕜 n (f ∘ ⇑e) ↔ ContDiff 𝕜 n f

Tactic:
rw [← contDiffOn_univ, ← contDiffOn_univ, ← preimage_univ]

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
f : E → F
n : ℕ∞ω
e : G ≃L[𝕜] E
⊢ ContDiffOn 𝕜 n (f ∘ ⇑e) (?m.75 ⁻¹' univ) ↔ ContDiffOn 𝕜 n f univ

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
f : E → F
n : ℕ∞ω
e : G ≃L[𝕜] E
⊢ Type ?u.68

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
f : E → F
n : ℕ∞ω
e : G ≃L[𝕜] E
⊢ G → ?m.74
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

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
h : ContDiffWithinAt 𝕜 n f s x
t : Set E
hst : s =ᶠ[𝓝 x] t
⊢ ContDiffWithinAt 𝕜 n f t x

Tactic:
by
  rw [← nhdsWithin_eq_iff_eventuallyEq] at hst
  apply h.mono_of_mem_nhdsWithin <| hst ▸ self_mem_nhdsWithin

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

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
s : Set 𝕜
hs : UniqueDiffOn 𝕜 s
⊢ ContDiffOn 𝕜 ∞ f s ↔ DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 ∞ (derivWithin f s) s

Tactic:
rw [show ∞ = ∞ + 1 by rfl, contDiffOn_succ_iff_derivWithin hs]

Goal state after:
𝕜 : Type u_1
F : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
s : Set 𝕜
hs : UniqueDiffOn 𝕜 s
⊢ DifferentiableOn 𝕜 f s ∧ (∞ = ω → AnalyticOn 𝕜 f s) ∧ ContDiffOn 𝕜 ∞ (derivWithin f s) s ↔
    DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 (∞ + 1) (derivWithin f s) s
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
s : Set E
x₀ : E
m : ℕ∞ω
f : E → F → G
g : E → F
t : Set F
ht : ∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)
hgt : t ∈ 𝓝[g '' s] g x₀
hg : ContDiffWithinAt 𝕜 ω g s x₀
this : ∀ (k : ℕ), ↑k ≤ ω → ContDiffWithinAt 𝕜 (↑k) (fun x ↦ fderivWithin 𝕜 (f x) t (g x)) s x₀
hf : ContDiffWithinAt 𝕜 ω (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)
hmn : ω + 1 ≤ ω
⊢ ContDiffWithinAt 𝕜 ω (fun x ↦ fderivWithin 𝕜 (f x) t (g x)) s x₀

Tactic:
obtain ⟨v, hv, -, f', hvf', hf'⟩ := hf.hasFDerivWithinAt_nhds (by simp) hg hgt

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
x₀ : E
m : ℕ∞ω
f : E → F → G
g : E → F
t : Set F
ht : ∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)
hgt : t ∈ 𝓝[g '' s] g x₀
hg : ContDiffWithinAt 𝕜 ω g s x₀
this : ∀ (k : ℕ), ↑k ≤ ω → ContDiffWithinAt 𝕜 (↑k) (fun x ↦ fderivWithin 𝕜 (f x) t (g x)) s x₀
hf : ContDiffWithinAt 𝕜 ω (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)
hmn : ω + 1 ≤ ω
v : Set E
hv : v ∈ 𝓝[insert x₀ s] x₀
f' : E → F →L[𝕜] G
hvf' : ∀ x ∈ v, HasFDerivWithinAt (f x) (f' x) t (g x)
hf' : ContDiffWithinAt 𝕜 ω (fun x ↦ f' x) s x₀
⊢ ContDiffWithinAt 𝕜 ω (fun x ↦ fderivWithin 𝕜 (f x) t (g x)) s x₀
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 (iteratedFDeriv 𝕜 i f) x₀

Tactic:
rw [← iteratedFDerivWithin_univ, ← contDiffWithinAt_univ] at *

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
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: case mpr.refine_2.refine_2
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : ℕ∞ω
hn : n ≠ ∞
h'n : n + 1 ≠ ∞
u : Set E
hu : u ∈ 𝓝[insert x s] x
hf : n = ω → AnalyticOn 𝕜 f u
f' : E → E →L[𝕜] F
f'_eq_deriv : ∀ x ∈ u, HasFDerivWithinAt f (f' x) u x
Hf' : ContDiffWithinAt 𝕜 n f' u x
v : Set E
hv : v ∈ 𝓝[insert x u] x
p' : E → FormalMultilinearSeries 𝕜 E (E →L[𝕜] F)
Hp' : HasFTaylorSeriesUpToOn n f' p' v
p'_an : n = ω → ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ p' x i) v
⊢ HasFTaylorSeriesUpToOn n (fun x ↦ (continuousMultilinearCurryFin1 𝕜 E F) ((p' x).unshift (f x) 1))
    (fun x ↦ ((p' x).unshift (f x)).shift) (v ∩ u)

Tactic:
convert! (Hp'.mono inter_subset_left).congr fun x hx => Hp'.zero_eq x hx.1 using 1
        · ext x y
          change p' x 0 (init (@snoc 0 (fun _ : Fin 1 => E) 0 y)) y = p' x 0 0 y
          rw [init_snoc]
        · ext x k v y
          change p' x k (init (@snoc k (fun _ : Fin k.succ => E) v y))
            (@snoc k (fun _ : Fin k.succ => E) v y (last k)) = p' x k v y
          rw [snoc_last, init_snoc]

Goal state after:
Module: Mathlib.Topology.OpenPartialHomeomorph.IsImage

Goal state before:
pp: case left
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e e' : OpenPartialHomeomorph X Y
h : EqOn (↑e) (↑e') (e.source ∩ e'.source)
⊢ (e.restr e'.source).source = e'.source ∩ e.source

Tactic:
rw [e.restr_source' _ e'.open_source]

Goal state after:
case left
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e e' : OpenPartialHomeomorph X Y
h : EqOn (↑e) (↑e') (e.source ∩ e'.source)
⊢ e.source ∩ e'.source = e'.source ∩ e.source
Module: Mathlib.Topology.OpenPartialHomeomorph.IsImage

Goal state before:
pp: case right
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e e' : OpenPartialHomeomorph X Y
h : EqOn (↑e) (↑e') (e.source ∩ e'.source)
⊢ EqOn (↑(e.restr e'.source)) (↑(e'.restr e.source)) (e.restr e'.source).source

Tactic:
rw [e.restr_source' _ e'.open_source]

Goal state after:
case right
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e e' : OpenPartialHomeomorph X Y
h : EqOn (↑e) (↑e') (e.source ∩ e'.source)
⊢ EqOn (↑(e.restr e'.source)) (↑(e'.restr e.source)) (e.source ∩ e'.source)
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Defs

Goal state before:
pp: 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
x : 𝕜
⊢ iteratedDeriv n f x = deriv^[n] f x

Tactic:
rw [← iteratedDerivWithin_univ]

Goal state after:
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
x : 𝕜
⊢ iteratedDerivWithin n f univ x = deriv^[n] f x
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case 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✝ c : OrderedFinpartition n
k : Fin c.length
i : Fin n
⊢ ∃ m y,
    (if h : m = k then (fun i ↦ cases 0 (succ ∘ c.emb k) i) ∘ Fin.cast ⋯ else succ ∘ c.emb m ∘ Fin.cast ⋯) y = i.succ

Tactic:
rcases eq_or_ne (c.index i) k with rfl | hi

Goal state after:
case refine_2.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✝ c : OrderedFinpartition n
i : Fin n
⊢ ∃ m y,
    (if h : m = c.index i then (fun i_1 ↦ cases 0 (succ ∘ c.emb (c.index i)) i_1) ∘ Fin.cast ⋯
        else succ ∘ c.emb m ∘ Fin.cast ⋯)
        y =
      i.succ

case 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✝ c : OrderedFinpartition n
k : Fin c.length
i : Fin n
hi : c.index i ≠ k
⊢ ∃ m y,
    (if h : m = k then (fun i ↦ cases 0 (succ ∘ c.emb k) i) ∘ Fin.cast ⋯ else succ ∘ c.emb m ∘ Fin.cast ⋯) y = i.succ
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 j : Fin c.length
hij : i < j
⊢ ↑(if h : i = c.index 0 then
          (c.emb i (Fin.cast ⋯ ⟨update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) i - 1, ⋯⟩.succ)).pred ⋯
        else (c.emb i (Fin.cast ⋯ ⟨update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) i - 1, ⋯⟩)).pred ⋯) +
      1 <
    ↑(if h : j = c.index 0 then
          (c.emb j (Fin.cast ⋯ ⟨update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) j - 1, ⋯⟩.succ)).pred ⋯
        else (c.emb j (Fin.cast ⋯ ⟨update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) j - 1, ⋯⟩)).pred ⋯) +
      1

Tactic:
convert! Fin.lt_def.1 (c.parts_strictMono hij)

Goal state after:
case e'_3
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 j : Fin c.length
hij : i < j
⊢ ↑(if h : i = c.index 0 then
          (c.emb i (Fin.cast ⋯ ⟨update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) i - 1, ⋯⟩.succ)).pred ⋯
        else (c.emb i (Fin.cast ⋯ ⟨update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) i - 1, ⋯⟩)).pred ⋯) +
      1 =
    ↑((fun m ↦ c.emb m ⟨c.partSize m - 1, ⋯⟩) i)

case e'_4
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 j : Fin c.length
hij : i < j
⊢ ↑(if h : j = c.index 0 then
          (c.emb j (Fin.cast ⋯ ⟨update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) j - 1, ⋯⟩.succ)).pred ⋯
        else (c.emb j (Fin.cast ⋯ ⟨update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) j - 1, ⋯⟩)).pred ⋯) +
      1 =
    ↑((fun m ↦ c.emb m ⟨c.partSize m - 1, ⋯⟩) j)
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✝
n : ℕ
c : OrderedFinpartition n
o : Option (Fin c.length)
⊢ (fun c ↦ if h : range (c.emb 0) = {0} then ⟨c.eraseLeft h, none⟩ else ⟨c.eraseMiddle h, some (c.index 0)⟩)
      ((fun c ↦ c.fst.extend c.snd) ⟨c, o⟩) =
    ⟨c, o⟩

Tactic:
match o with
    | none =>
      simp only [extend, range_extendLeft_zero, ↓reduceDIte, Sigma.mk.inj_iff, heq_eq_eq,
        and_true]
      rfl
    | some i =>
      simp only [extend, range_emb_extendMiddle_ne_singleton_zero, ↓reduceDIte,
        Sigma.mk.inj_iff, heq_eq_eq, and_true, eraseMiddle,
        index_extendMiddle_zero]
      ext
      · rfl
      · simp only [heq_eq_eq, index_extendMiddle_zero]
        ext j
        rcases eq_or_ne i j with rfl | hij
        · simp [extendMiddle]
        · simp [hij.symm, extendMiddle]
      · refine HEq.symm (hfunext rfl ?_)
        simp only [heq_eq_eq, forall_eq']
        intro a
        rcases eq_or_ne a i with rfl | hij
        · refine (Fin.heq_fun_iff ?_).mpr ?_
          · rw [index_extendMiddle_zero]
            simp [extendMiddle]
          · simp [extendMiddle]
        · refine (Fin.heq_fun_iff ?_).mpr ?_
          · rw [index_extendMiddle_zero]
            simp [extendMiddle]
          · simp [extendMiddle, hij]

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

Goal state before:
pp: 𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : Field 𝕂
inst✝³ : Ring 𝔸
inst✝² : Algebra 𝕂 𝔸
inst✝¹ : TopologicalSpace 𝔸
inst✝ : IsTopologicalRing 𝔸
n : ℕ
⊢ ((expSeries 𝕂 𝔸 n) fun x ↦ 0) = Pi.single 0 1 n

Tactic:
expSeries_apply_eq

Goal state after:
𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : Field 𝕂
inst✝³ : Ring 𝔸
inst✝² : Algebra 𝕂 𝔸
inst✝¹ : TopologicalSpace 𝔸
inst✝ : IsTopologicalRing 𝔸
n : ℕ
⊢ (↑n !)⁻¹ • 0 ^ n = Pi.single 0 1 n
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: 𝔸 : Type u_2
inst✝² : NormedCommRing 𝔸
inst✝¹ : NormedAlgebra ℚ 𝔸
inst✝ : CompleteSpace 𝔸
ι : Type u_3
s : Finset ι
f : ι → 𝔸
⊢ exp (∑ i ∈ s, f i) = ∏ i ∈ s, exp (f i)

Tactic:
rw [exp_sum_of_commute, Finset.noncommProd_eq_prod]

Goal state after:
case h
𝔸 : Type u_2
inst✝² : NormedCommRing 𝔸
inst✝¹ : NormedAlgebra ℚ 𝔸
inst✝ : CompleteSpace 𝔸
ι : Type u_3
s : Finset ι
f : ι → 𝔸
⊢ (↑s).Pairwise (Function.onFun Commute f)
Module: Mathlib.Analysis.SpecialFunctions.ExpDeriv

Goal state before:
pp: x : ℂ
⊢ HasDerivAt cexp (cexp x) x

Tactic:
hasDerivAt_iff_isLittleO_nhds_zero

Goal state after:
x : ℂ
⊢ (fun h ↦ cexp (x + h) - cexp x - h • cexp x) =o[𝓝 0] fun h ↦ h
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:
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.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝕜' : Type u_6
inst✝¹ : NormedDivisionRing 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
n : ℕ
c : 𝕜'
f : 𝕜 → 𝕜'
⊢ iteratedDeriv n (fun x ↦ c * f x) x = c * iteratedDeriv n f x

Tactic:
simpa only [iteratedDerivWithin_univ] using
    iteratedDerivWithin_const_mul_field (s := .univ) c f

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

Goal state before:
pp: case hx
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✝ : T2Space β
hab : a < b
ha : Tendsto f (𝓝[>] a) (𝓝 la)
⊢ a ∈ closure[inst✝⁵] (Ioo a b)

Tactic:
closure_Ioo hab.ne

Goal state after:
case hx
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✝ : T2Space β
hab : a < b
ha : Tendsto f (𝓝[>] a) (𝓝 la)
⊢ a ∈ Icc a b
Module: Mathlib.Topology.Order.ExtendFrom

Goal state before:
pp: case hx
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
lb : β
inst✝ : T2Space β
hab : a < b
hb : Tendsto f (𝓝[<] b) (𝓝 lb)
⊢ b ∈ closure[inst✝⁵] (Ioo a b)

Tactic:
closure_Ioo hab.ne

Goal state after:
case hx
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
lb : β
inst✝ : T2Space β
hab : a < b
hb : Tendsto f (𝓝[<] b) (𝓝 lb)
⊢ b ∈ Icc a b