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

Goal state before:
pp: 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
h : EqOn (logDeriv f) (logDeriv g) s
⊢ f t * (g t)⁻¹ ≠ 0

Tactic:
grind

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

Goal state before:
pp: 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
h : EqOn (logDeriv f) (logDeriv g) s
y : 𝕜
hy : y ∈ s
z : 𝕜
hz : z ∈ s
⊢ g⁻¹ = (fun x ↦ x⁻¹) ∘ g

Tactic:
rfl

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

Goal state before:
pp: case zero_eq
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
f : E → F
n : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
hf : HasFTaylorSeriesUpToOn n f p s
g : G →ᴬ[𝕜] E
A : (m : ℕ) → (E [×m]→L[𝕜] F) → G [×m]→L[𝕜] F := fun m h ↦ h.compContinuousLinearMap fun x ↦ g.contLinear
hA : ∀ (m : ℕ), IsBoundedLinearMap 𝕜 (A m)
x : G
hx : x ∈ ⇑g ⁻¹' s
⊢ ((p (g x) 0) fun x ↦ 0) = (p (g x) 0) 0

Tactic:
rfl

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

Goal state before:
pp: 𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
H : Type u_5
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
f : G → H
f' : G → G →L[𝕜] H
x : G
hder : ∀ᶠ (y : G) in 𝓝 x, HasFDerivAt f (f' y) y
hcont : ContinuousAt f' x
c : ℝ
hc : 0 < c
ε : ℝ
ε0 : ε > 0
hε : ball x ε ⊆ {x | (fun y ↦ HasFDerivAt f (f' y) y) x} ∩ f' ⁻¹' ball (f' x) c
a b : G
h : a ∈ ball x ε ∧ b ∈ ball x ε
hf' : ∀ x' ∈ ball x ε, ‖f' x' - f' x‖ ≤ c
this : NormedSpace ℝ G := NormedSpace.restrictScalars ℝ 𝕜 G
⊢ ∀ x_1 ∈ ball x ε, HasFDerivWithinAt f (f' x_1) (ball x ε) x_1

Tactic:
exact fun y hy => (hε hy).1.hasFDerivWithinAt

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

Goal state before:
pp: case zero_eq
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
f : E → F
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ∞ω
hf : HasFTaylorSeriesUpToOn n f p s
g : E → G
q : E → FormalMultilinearSeries 𝕜 E G
hg : HasFTaylorSeriesUpToOn n g q s
L : (m : ℕ) → (E [×m]→L[𝕜] F) × (E [×m]→L[𝕜] G) ≃ₗᵢ[𝕜] E [×m]→L[𝕜] F × G :=
  fun m ↦ ContinuousMultilinearMap.prodL 𝕜 (fun x ↦ E) F G
x : E
hx : x ∈ s
⊢ ((p x 0).prod (q x 0)).curry0 = ((p x 0).curry0, (q x 0).curry0)

Tactic:
rfl

Goal state after:
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
⊢ (∀ (m : ℕ), ↑m ≤ ⊤ → ContDiffWithinAt 𝕜 (↑m) f s x) ↔ ∀ (n : ℕ), ContDiffWithinAt 𝕜 (↑n) f s x

Tactic:
simp only [forall_prop_of_true, le_top]

Goal state after:
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 f₁ : E → F
x : E
n✝ : ℕ∞ω
h₁ : f₁ =ᶠ[𝓝[s] x] f
hx : f₁ x = f x
n : ℕ∞
h : ContDiffWithinAt 𝕜 (↑n) f s x
⊢ ContDiffWithinAt 𝕜 (↑n) f₁ s x

Tactic:
intro m hm
    let ⟨u, hu, p, H⟩ := h m hm
    exact ⟨{ x ∈ u | f₁ x = f x }, Filter.inter_mem hu (mem_nhdsWithin_insert.2 ⟨hx, h₁⟩), p,
      (H.mono (sep_subset _ _)).congr fun _ ↦ And.right⟩

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
⊢ ∞ = ∞ + 1

Tactic:
rfl

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 : IsOpen s
⊢ ∞ = ∞ + 1

Tactic:
rfl

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

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
f' : F
s : Set 𝕜
x : 𝕜
h : HasDerivWithinAt f f' s x
hf' : f' ≠ 0
x✝¹ x✝ : 𝕜
⊢ ‖x✝¹ - x✝‖ ≤ ‖x✝¹ - x✝‖

Tactic:
rfl

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

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
f' : F
s : Set 𝕜
x : 𝕜
c : F
h : HasDerivWithinAt f f' s x
hf' : f' ≠ 0
x✝¹ x✝ : 𝕜
⊢ ‖x✝¹ - x✝‖ ≤ ‖x✝¹ - x✝‖

Tactic:
rfl

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

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
f' : F
s : Set 𝕜
x : 𝕜
h : HasDerivWithinAt f f' s x
hf' : f' ≠ 0
t : Set F
ht : ¬AccPt (f x) (𝓟 t)
x✝¹ x✝ : 𝕜
⊢ ‖x✝¹ - x✝‖ ≤ ‖x✝¹ - x✝‖

Tactic:
rfl

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

Goal state before:
pp: X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
s : Set X
hs : IsOpen[inst✝¹] s
⊢ (e.restr s).source = e.source ∩ s

Tactic:
grind

Goal state after:
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
hn : n ≠ ∞
⊢ ContDiffWithinAt 𝕜 n f s x ↔ ∃ u ∈ 𝓝[insert x s] x, ContDiffOn 𝕜 n f u

Tactic:
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
  · rcases h.contDiffOn le_rfl (by simp [hn]) with ⟨u, hu, h'u⟩
    exact ⟨u, hu, h'u.2⟩
  · rcases h with ⟨u, u_mem, hu⟩
    have : x ∈ u := mem_of_mem_nhdsWithin (mem_insert x s) u_mem
    exact (hu x this).mono_of_mem_nhdsWithin (nhdsWithin_mono _ (subset_insert x s) u_mem)

Goal state after:
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
s : Set 𝕜
x : 𝕜
⊢ iteratedDerivWithin n f s x =
    (⇑(ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F).symm ∘ iteratedFDerivWithin 𝕜 n f s) x

Tactic:
rfl

Goal state after:
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
s : Set 𝕜
x : 𝕜
⊢ (fun g ↦ derivWithin g s)^[n + 1] f x = (fun g ↦ derivWithin g s)^[n] (derivWithin f s) x

Tactic:
rfl

Goal state after:
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
f : 𝕜 → F
s : Set 𝕜
n : ℕ
hs : UniqueDiffOn 𝕜 s
⊢ ((∀ m ≤ n + 1, ContinuousOn (iteratedDerivWithin m f s) s) ∧
      ∀ m < n + 1, DifferentiableOn 𝕜 (iteratedDerivWithin m f s) s) ↔
    ((∀ m ≤ n, ContinuousOn (iteratedDerivWithin m f s) s) ∧
        ∀ m < n, DifferentiableOn 𝕜 (iteratedDerivWithin m f s) s) ∧
      DifferentiableOn 𝕜 (iteratedDerivWithin n f s) s ∧ ContinuousOn (iteratedDerivWithin (n + 1) f s) s

Tactic:
grind

Goal state after:
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 = (⇑(ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F).symm ∘ iteratedFDeriv 𝕜 n f) x

Tactic:
rfl

Goal state after:
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
f : 𝕜 → F
n : ℕ
⊢ ContDiff 𝕜 (↑n) f ↔
    (∀ m ≤ n,
        Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace,
            PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
          (iteratedDeriv m f)) ∧
      ∀ m < n, Differentiable 𝕜 (iteratedDeriv m f)

Tactic:
rw [← WithTop.coe_natCast, contDiff_iff_iteratedDeriv]
  simp

Goal state after:
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
f : 𝕜 → F
n : ℕ
⊢ ((∀ m ≤ n + 1,
        Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace,
            PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
          (iteratedDeriv m f)) ∧
      ∀ m < n + 1, Differentiable 𝕜 (iteratedDeriv m f)) ↔
    ((∀ m ≤ n,
          Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace,
              PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
            (iteratedDeriv m f)) ∧
        ∀ m < n, Differentiable 𝕜 (iteratedDeriv m f)) ∧
      Differentiable 𝕜 (iteratedDeriv n f) ∧
        Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace,
            PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
          (iteratedDeriv (n + 1) f)

Tactic:
grind

Goal state after:
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
n : ℕ∞ω
f : E → F
s : Set E
a : E
h : ContDiffWithinAt 𝕜 n f s a
hs : UniqueDiffOn 𝕜 s
ha : a ∈ s
m : ℕ
hm : ↑m ≤ n
⊢ ∀ᶠ (t : Set E) in (𝓝[s] a).smallSets, HasFTaylorSeriesUpToOn (↑m) f (ftaylorSeriesWithin 𝕜 f s) t

Tactic:
rcases h.contDiffOn' hm (by simp) with ⟨U, hUo, haU, hfU⟩

Goal state after:
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ∞ω
f : E → F
s : Set E
a : E
h : ContDiffWithinAt 𝕜 n f s a
hs : UniqueDiffOn 𝕜 s
ha : a ∈ s
m : ℕ
hm : ↑m ≤ n
U : Set E
hUo : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
haU : a ∈ U
hfU : ContDiffOn 𝕜 (↑m) f (insert a s ∩ U)
⊢ ∀ᶠ (t : Set E) in (𝓝[s] a).smallSets, HasFTaylorSeriesUpToOn (↑m) f (ftaylorSeriesWithin 𝕜 f s) t
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
f : E → F
x : E
⊢ ContDiffAt 𝕜 1 f x ↔ ∃ f', ∃ u ∈ 𝓝 x, ContinuousOn f' u ∧ ∀ x ∈ u, HasFDerivAt f (f' x) x

Tactic:
show (1 : ℕ∞ω) = (0 : ℕ) + 1 from rfl

Goal state after:
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
x : E
⊢ ContDiffAt 𝕜 (↑0 + 1) f x ↔ ∃ f', ∃ u ∈ 𝓝 x, ContinuousOn f' u ∧ ∀ x ∈ u, HasFDerivAt f (f' x) x
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)
⊢ c.extendLeft.eraseLeft ⋯ = c

Tactic:
rfl

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

Goal state before:
pp: case length
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
⊢ { length := (c.extendMiddle i).length,
        partSize :=
          update (c.extendMiddle i).partSize ((c.extendMiddle i).index 0)
            ((c.extendMiddle i).partSize ((c.extendMiddle i).index 0) - 1),
        partSize_pos := ⋯,
        emb := fun i_1 j ↦
          if h : i_1 = i then ((c.extendMiddle i).emb i_1 (Fin.cast ⋯ j.succ)).pred ⋯
          else ((c.extendMiddle i).emb i_1 (Fin.cast ⋯ j)).pred ⋯,
        emb_strictMono := ⋯, parts_strictMono := ⋯, disjoint := ⋯, cover := ⋯ }.length =
    c.length

Tactic:
rfl

Goal state after:
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
f : E → F
⊢ ContDiff 𝕜 ∞ f ↔ Differentiable 𝕜 f ∧ ContDiff 𝕜 ∞ (fderiv 𝕜 f)

Tactic:
rw [← ENat.coe_top_add_one, contDiff_succ_iff_fderiv]

Goal state after:
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
⊢ Differentiable 𝕜 f ∧ (∞ = ω → AnalyticOnNhd 𝕜 f univ) ∧ ContDiff 𝕜 ∞ (fderiv 𝕜 f) ↔
    Differentiable 𝕜 f ∧ ContDiff 𝕜 (∞ + 1) (fderiv 𝕜 f)
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: case neg
𝔸 : Type u_2
inst✝² : Ring 𝔸
inst✝¹ : TopologicalSpace 𝔸
inst✝ : IsTopologicalRing 𝔸
h✝ : ¬Nonempty (Algebra ℚ 𝔸)
⊢ 1 = 1

Tactic:
rfl

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

Goal state before:
pp: case pos.partSize.succ
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
i : Fin (c.length - 1)
a✝ :
  { 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
      i.castSucc =
    c.partSize ⟨↑i.castSucc, ⋯⟩
⊢ c.partSize (Fin.cast ⋯ i.succ) = c.partSize ⟨↑i + 1, ⋯⟩

Tactic:
rfl

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 : DivisionRing 𝔸
inst✝² : Algebra 𝕂 𝔸
inst✝¹ : TopologicalSpace 𝔸
inst✝ : IsTopologicalRing 𝔸
x : 𝔸
n : ℕ
⊢ ((expSeries 𝕂 𝔸 n) fun x_1 ↦ x) = (↑n !)⁻¹ • x ^ n

Tactic:
expSeries_apply_eq,

Goal state after:
𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : Field 𝕂
inst✝³ : DivisionRing 𝔸
inst✝² : Algebra 𝕂 𝔸
inst✝¹ : TopologicalSpace 𝔸
inst✝ : IsTopologicalRing 𝔸
x : 𝔸
n : ℕ
⊢ (↑n !)⁻¹ • x ^ n = (↑n !)⁻¹ • x ^ n
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case neg.length
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)
⊢ { length := c.length,
        partSize :=
          update (update c.partSize (c.index 0) (c.partSize (c.index 0) - 1)) (c.index 0)
            (update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) (c.index 0) + 1),
        partSize_pos := ⋯,
        emb := fun m ↦
          if h_1 : m = 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 : m = c.index 0 then (c.emb m (Fin.cast ⋯ j.succ)).pred ⋯ else (c.emb m (Fin.cast ⋯ j)).pred ⋯) ∘
                Fin.cast ⋯,
        emb_strictMono := ⋯, parts_strictMono := ⋯, disjoint := ⋯, cover := ⋯ }.length =
    c.length

Tactic:
rfl

Goal state after:
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
j' : Fin (c.partSize (c.index 0) - 1) := (Fin.cast ⋯ j).pred ⋯
this : j = Fin.cast B j'.succ
⊢ c.emb (c.index 0) (Fin.cast B j'.succ) = c.emb (c.index 0) ⟨↑j' + 1, ⋯⟩

Tactic:
rfl

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

Goal state before:
pp: 𝕂 : Type u_1
𝕂' : Type u_2
𝔸 : Type u_3
inst✝⁶ : Field 𝕂
inst✝⁵ : Field 𝕂'
inst✝⁴ : Ring 𝔸
inst✝³ : Algebra 𝕂 𝔸
inst✝² : Algebra 𝕂' 𝔸
inst✝¹ : TopologicalSpace 𝔸
inst✝ : IsTopologicalRing 𝔸
n : ℕ
x : 𝔸
⊢ ((expSeries 𝕂 𝔸 n) fun x_1 ↦ x) = (expSeries 𝕂' 𝔸 n) fun x_1 ↦ x

Tactic:
expSeries_apply_eq,

Goal state after:
𝕂 : Type u_1
𝕂' : Type u_2
𝔸 : Type u_3
inst✝⁶ : Field 𝕂
inst✝⁵ : Field 𝕂'
inst✝⁴ : Ring 𝔸
inst✝³ : Algebra 𝕂 𝔸
inst✝² : Algebra 𝕂' 𝔸
inst✝¹ : TopologicalSpace 𝔸
inst✝ : IsTopologicalRing 𝔸
n : ℕ
x : 𝔸
⊢ (↑n !)⁻¹ • x ^ n = (expSeries 𝕂' 𝔸 n) fun x_1 ↦ x
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: 𝕂 : Type u_1
𝕂' : Type u_2
𝔸 : Type u_3
inst✝⁶ : Field 𝕂
inst✝⁵ : Field 𝕂'
inst✝⁴ : Ring 𝔸
inst✝³ : Algebra 𝕂 𝔸
inst✝² : Algebra 𝕂' 𝔸
inst✝¹ : TopologicalSpace 𝔸
inst✝ : IsTopologicalRing 𝔸
n : ℕ
x : 𝔸
⊢ (↑n !)⁻¹ • x ^ n = (expSeries 𝕂' 𝔸 n) fun x_1 ↦ x

Tactic:
expSeries_apply_eq,

Goal state after:
𝕂 : Type u_1
𝕂' : Type u_2
𝔸 : Type u_3
inst✝⁶ : Field 𝕂
inst✝⁵ : Field 𝕂'
inst✝⁴ : Ring 𝔸
inst✝³ : Algebra 𝕂 𝔸
inst✝² : Algebra 𝕂' 𝔸
inst✝¹ : TopologicalSpace 𝔸
inst✝ : IsTopologicalRing 𝔸
n : ℕ
x : 𝔸
⊢ (↑n !)⁻¹ • x ^ n = (↑n !)⁻¹ • x ^ n
Module: Mathlib.Topology.MetricSpace.CauSeqFilter

Goal state before:
pp: β : Type v
inst✝ : NormedField β
f : CauSeq β norm
s : Set (β × β)
hs : s ∈ uniformity β
⊢ ∃ t ∈ map (↑f) atTop, t ×ˢ t ⊆ s

Tactic:
rcases mem_uniformity_dist.1 hs with ⟨ε, ⟨hε, hεs⟩⟩

Goal state after:
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
⊢ ∃ t ∈ map (↑f) atTop, t ×ˢ t ⊆ s
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: 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
k : Fin (n + 1)
⊢ ↑k ≤ n

Tactic:
grind

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

Goal state before:
pp: case e'_1.e
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
s : Set E
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
P' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)
n : ℕ∞ω
x✝ : Set E
⊢ HasFTaylorSeriesUpToOn n Φ P' x✝ ↔
    HasFTaylorSeriesUpToOn n (fun x i ↦ Φ x i)
      (fun x m ↦ ContinuousMultilinearMap.pi fun i ↦ (ContinuousLinearMap.proj i).compContinuousMultilinearMap (P' x m))
      x✝

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
s : Set E
a y : E
h : IsLocalMaxOn f s a
hy : y ∈ posTangentConeAt s a
hf : ¬DifferentiableWithinAt ℝ f s a
⊢ 0 y ≤ 0

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
s : Set E
a y : E
h : IsLocalMaxOn f s a
hy : y ∈ posTangentConeAt s a
hy' : -y ∈ posTangentConeAt s a
hf : ¬DifferentiableWithinAt ℝ f s a
⊢ 0 y = 0

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
s : Set E
a y : E
h : IsLocalMinOn f s a
hy : y ∈ posTangentConeAt s a
hf : ¬DifferentiableWithinAt ℝ f s a
⊢ 0 ≤ 0 y

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
s : Set E
a y : E
h : IsLocalMinOn f s a
hy : y ∈ posTangentConeAt s a
hy' : -y ∈ posTangentConeAt s a
hf : ¬DifferentiableWithinAt ℝ f s a
⊢ 0 y = 0

Tactic:
rfl

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

Goal state before:
pp: case neg.hf
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
⊢ ∃ y, Tendsto f (𝓝[Ioo a b] x) (𝓝 y)

Tactic:
rcases eq_left_or_mem_Ioo_of_mem_Ico x_in with (rfl | h)

Goal state after:
case neg.hf.inl
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
b : α
la : β
inst✝ : RegularSpace β
x : α
hf : ContinuousOn f (Ioo x b)
ha : Tendsto f (𝓝[>] x) (𝓝 la)
hab : x < b
x_in : x ∈ Ico x b
⊢ ∃ y, Tendsto f (𝓝[Ioo x b] x) (𝓝 y)

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)