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

Goal state before:
pp: case mpr.refine_2.refine_1
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
y : E
hy : y ∈ v ∩ u
⊢ HasFDerivWithinAt (fun y ↦ (p' y).unshift (f y) 0) ((p' y).unshift (f y) 1).curryLeft (v ∩ u) y

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:
· change
          HasFDerivWithinAt (fun z => (continuousMultilinearCurryFin0 𝕜 E F).symm (f z))
            (FormalMultilinearSeries.unshift (p' y) (f y) 1).curryLeft (v ∩ u) y
        rw [← Function.comp_def _ f, LinearIsometryEquiv.comp_hasFDerivWithinAt_iff']
        convert! (f'_eq_deriv y hy.2).mono inter_subset_right
        rw [← Hp'.zero_eq y hy.1]
        ext z
        change ((p' y 0) (init (@cons 0 (fun _ => E) z 0))) (@cons 0 (fun _ => E) z 0 (last 0)) =
          ((p' y 0) 0) z
        congr
        norm_num [eq_iff_true_of_subsingleton]

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

Goal state before:
pp: case succ.zero
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ c : OrderedFinpartition n
i : Fin c.length
hi : i.succ ∈ univ
hj : 0 ∈ univ
a✝ :
  i.castSucc ∈ univ →
    i.castSucc ≠ 0 →
      0 < i.castSucc →
        (Disjoint on fun m ↦
            range
              (cases (motive := fun x ↦ Fin (cons 1 c.partSize x) → Fin (n + 1)) (fun x ↦ 0) (fun m ↦ succ ∘ c.emb m)
                m))
          i.castSucc 0
hij : i.succ ≠ 0
h : 0 < i.succ
⊢ (Disjoint on fun m ↦
      range (cases (motive := fun x ↦ Fin (cons 1 c.partSize x) → Fin (n + 1)) (fun x ↦ 0) (fun m ↦ succ ∘ c.emb m) m))
    i.succ 0

Tactic:
simp only [onFun, cases_succ, cases_zero]
        apply Set.disjoint_iff_forall_ne.2
        simp only [mem_range, comp_apply, exists_prop', cons_zero, ne_eq, and_imp,
          Nonempty.forall, forall_const, forall_eq', forall_exists_index, forall_apply_eq_imp_iff]
        exact fun _ ↦ succ_ne_zero _

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
hs : UniqueDiffOn 𝕜 s
h : ContDiffAt 𝕜 (↑n) f x
hx : x ∈ s
⊢ iteratedDerivWithin n f s x = iteratedDeriv n f x

Tactic:
rw [iteratedDerivWithin, iteratedDeriv, iteratedFDerivWithin_eq_iteratedFDeriv hs h hx]

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
hx : x ∈ s
⊢ ContDiffWithinAt 𝕜 0 f s x ↔ ∃ u ∈ 𝓝[s] x, ContinuousOn f (s ∩ u)

Tactic:
by
  constructor
  · intro h
    obtain ⟨u, H, p, hp⟩ := h 0 le_rfl
    refine ⟨u, ?_, ?_⟩
    · simpa [hx] using H
    · simp only [Nat.cast_zero, hasFTaylorSeriesUpToOn_zero_iff] at hp
      exact hp.1.mono inter_subset_right
  · rintro ⟨u, H, hu⟩
    rw [← contDiffWithinAt_inter' H]
    have h' : x ∈ s ∩ u := ⟨hx, mem_of_mem_nhdsWithin hx H⟩
    exact (contDiffOn_zero.mpr hu).contDiffWithinAt h'

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
n : ℕ∞ω
h : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
⊢ ContDiffOn 𝕜 n f s

Tactic:
suffices ContDiffOn 𝕜 ω f s from this.of_le le_top

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
s : Set E
f : E → F
n : ℕ∞ω
h : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
⊢ ContDiffOn 𝕜 ω f s
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
n : ℕ∞ω
h : ∀ (m : ℕ), AnalyticOn 𝕜 (iteratedFDerivWithin 𝕜 m f s) s
⊢ ContDiffOn 𝕜 n f s

Tactic:
suffices ContDiffOn 𝕜 ω f s from this.of_le le_top

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
s : Set E
f : E → F
n : ℕ∞ω
h : ∀ (m : ℕ), AnalyticOn 𝕜 (iteratedFDerivWithin 𝕜 m f s) s
⊢ ContDiffOn 𝕜 ω f s
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case e'_4.inr
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
i j : Fin c.length
hij : i < j
hj : j ≠ c.index 0
⊢ ↑(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)

Tactic:
simp only [hj, ↓reduceDIte, ne_eq, not_false_eq_true, update_of_ne, cast_mk, val_pred]
        apply Nat.sub_add_cancel
        have : c.emb j ⟨c.partSize j - 1, Nat.sub_one_lt_of_lt (c.partSize_pos j)⟩
            ≠ c.emb (c.index 0) 0 := c.emb_ne_emb_of_ne hj
        simp only [c.emb_zero, ne_eq, ← val_eq_val, val_zero] at this
        lia

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

Goal state before:
pp: 𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : NormedRing 𝔸
inst✝² : NormedAlgebra 𝕂 𝔸
inst✝¹ : CompleteSpace 𝔸
inst✝ : CharZero 𝕂
x y : 𝔸
hxy : Commute x y
hx : x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
hy : y ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
⊢ (fun x ↦ ∑' (n : ℕ), (↑n !)⁻¹ • x ^ n) (x + y) =
    (fun x ↦ ∑' (n : ℕ), (↑n !)⁻¹ • x ^ n) x * (fun x ↦ ∑' (n : ℕ), (↑n !)⁻¹ • x ^ n) y

Tactic:
tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm
      (norm_expSeries_summable_of_mem_ball' x hx) (norm_expSeries_summable_of_mem_ball' y hy)

Goal state after:
𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : NormedRing 𝔸
inst✝² : NormedAlgebra 𝕂 𝔸
inst✝¹ : CompleteSpace 𝔸
inst✝ : CharZero 𝕂
x y : 𝔸
hxy : Commute x y
hx : x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
hy : y ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
⊢ (fun x ↦ ∑' (n : ℕ), (↑n !)⁻¹ • x ^ n) (x + y) =
    ∑' (n : ℕ), ∑ kl ∈ Finset.antidiagonal n, (↑kl.1!)⁻¹ • x ^ kl.1 * (↑kl.2!)⁻¹ • y ^ kl.2
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)

Tactic:
← norm_div,

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.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
⊢ j ≥ N → ‖f j - f N‖ < ε

Tactic:
hj

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
hj : j ≥ N
⊢ ‖f j - f N‖ < ε
Module: Mathlib.Analysis.SpecialFunctions.ExpDeriv

Goal state before:
pp: ⊢ AnalyticOnNhd ℝ rexp univ

Tactic:
rw [Real.exp_eq_exp_ℝ]

Goal state after:
⊢ AnalyticOnNhd ℝ NormedSpace.exp univ
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
f' : StrongDual ℝ E
a : E
h : IsLocalMin f a
hf : HasFDerivAt f f' a
⊢ f' = 0

Tactic:
by
  ext y
  apply (h.on univ).hasFDerivWithinAt_eq_zero hf.hasFDerivWithinAt <;>
      rw [posTangentConeAt_univ] <;>
    apply mem_univ

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

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

Tactic:
apply contDiffAt_zero.2

Goal state after:
case zero
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type uF
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
n : ℕ∞ω
inst✝ : CompleteSpace E
f : OpenPartialHomeomorph E F
f₀' : E ≃L[𝕜] F
a : F
ha : a ∈ f.target
hf₀' : HasFDerivAt (↑f) (↑f₀') (↑f.symm a)
hf : ContDiffAt 𝕜 (↑0) (↑f) (↑f.symm a)
⊢ ∃ u ∈ 𝓝 a, ContinuousOn (↑f.symm) u
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: f f' : ℝ → ℝ
hf : ∀ (x : ℝ), HasDerivAt f (f' x) x
hf' : ∀ (x : ℝ), 0 < f' x
x : ℝ
⊢ 0 < deriv f x

Tactic:
(hf _).deriv

Goal state after:
f f' : ℝ → ℝ
hf : ∀ (x : ℝ), HasDerivAt f (f' x) x
hf' : ∀ (x : ℝ), 0 < f' x
x : ℝ
⊢ 0 < f' x
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: f f' : ℝ → ℝ
hf : ∀ (x : ℝ), HasDerivAt f (f' x) x
hf' : 0 ≤ f'
x : ℝ
⊢ 0 ≤ deriv f x

Tactic:
(hf _).deriv

Goal state after:
f f' : ℝ → ℝ
hf : ∀ (x : ℝ), HasDerivAt f (f' x) x
hf' : 0 ≤ f'
x : ℝ
⊢ 0 ≤ f' x
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: f f' : ℝ → ℝ
hf : ∀ (x : ℝ), HasDerivAt f (f' x) x
hf' : ∀ (x : ℝ), f' x < 0
x : ℝ
⊢ deriv f x < 0

Tactic:
(hf _).deriv

Goal state after:
f f' : ℝ → ℝ
hf : ∀ (x : ℝ), HasDerivAt f (f' x) x
hf' : ∀ (x : ℝ), f' x < 0
x : ℝ
⊢ f' x < 0
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: f f' : ℝ → ℝ
hf : ∀ (x : ℝ), HasDerivAt f (f' x) x
hf' : f' ≤ 0
x : ℝ
⊢ deriv f x ≤ 0

Tactic:
(hf _).deriv

Goal state after:
f f' : ℝ → ℝ
hf : ∀ (x : ℝ), HasDerivAt f (f' x) x
hf' : f' ≤ 0
x : ℝ
⊢ f' x ≤ 0
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: R : Type u_2
M : Type u_4
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommMonoid M
inst✝ : SMul R M
C : ConvexCone R M
⊢ ↑C = ∅ ↔ C = ⊥

Tactic:
by rw [← coe_bot (R := R)]; norm_cast

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

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
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
e p q e' p' q' : ℕ
hp : n e ≤ p
hq : n e ≤ q
hp' : n e' ≤ p'
⊢ n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e

Tactic:
hq'

Goal state after:
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
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
e p q e' p' q' : ℕ
hp : n e ≤ p
hq : n e ≤ q
hp' : n e' ≤ p'
hq' : n e' ≤ q'
⊢ e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
c : ℝ≥0
s : Set α
hs : MeasurableSet s
int_f : ∫⁻ (x : α), ↑((piecewise s hs (const α c) (const α 0)) x) ∂μ ≠ ∞
ε : ℝ≥0∞
ε0 : ε ≠ 0
hc : ¬c = 0
μs_lt_top : μ s < ∞
this : 0 < ε / ↑c
F : Set α
Fs : F ⊆ s
F_closed : IsClosed F
μF : μ s < μ F + ε / ↑c
⊢ ↑c * μ s ≤ ↑c * (μ F + ε / ↑c)

Tactic:
grw [μF]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: case neg.refine_2
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
c : ℝ≥0
s : Set α
hs : MeasurableSet s
int_f : ∫⁻ (x : α), ↑((piecewise s hs (const α c) (const α 0)) x) ∂μ ≠ ∞
ε : ℝ≥0∞
ε0 : ε ≠ 0
hc : ¬c = 0
μs_lt_top : μ s < ∞
this : 0 < ε / ↑c
F : Set α
Fs : F ⊆ s
F_closed : IsClosed F
μF : μ s < μ F + ε / ↑c
⊢ ↑c * μ s ≤ ↑c * μ F + ε

Tactic:
calc
        (c : ℝ≥0∞) * μ s ≤ c * (μ F + ε / c) := by grw [μF]
        _ = c * μ F + ε := by
          simp_rw [mul_add]
          rw [ENNReal.mul_div_cancel _ ENNReal.coe_ne_top]
          simpa using hc

Goal state after:
Module: Mathlib.Analysis.Convex.Cone.Extension

Goal state before:
pp: case refine_2.inr.inr
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : PointedCone ℝ E
f : E →ₗ.[ℝ] ℝ
nonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x
dense : ∀ (y : E), ∃ x, ↑x + y ∈ s
hdom : f.domain ≠ ⊤
y : E
hy : y ∉ f.domain
c : ℝ
le_c : ∀ (x : ↥f.domain), -↑x - y ∈ s → ↑f x ≤ c
c_le : ∀ (x : ↥f.domain), ↑x + y ∈ s → c ≤ ↑f x
x : E
hx : x ∈ f.domain
r : ℝ
hy' : r • y ∈ (LinearPMap.mkSpanSingleton y (-c) ⋯).domain
hz : x + r • y ∈ (f.supSpanSingleton y (-c) hy).domain
hzs : x + r • y ∈ s
hr : 0 < r
this : c ≤ ↑f (r⁻¹ • ⟨x, hx⟩)
⊢ (RingHom.id ℝ) r • c ≤ ↑f ⟨x, hx⟩

Tactic:
rwa [← mul_le_mul_iff_right₀ hr, f.map_smul, smul_eq_mul, ← mul_assoc, mul_inv_cancel₀ hr.ne',
        one_mul] at this

Goal state after:
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: case inl
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
p : Seminorm ℝ E
x : E
hp : {r | 0 < r ∧ x ∈ r • p.ball 0 1} = ∅
⊢ gauge (p.ball 0 1) x = p x

Tactic:
gauge,

Goal state after:
case inl
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
p : Seminorm ℝ E
x : E
hp : {r | 0 < r ∧ x ∈ r • p.ball 0 1} = ∅
⊢ sInf {r | 0 < r ∧ x ∈ r • p.ball 0 1} = p x
Module: Mathlib.Analysis.LocallyConvex.SeparatingDual

Goal state before:
pp: case mp
R : Type u_1
V : Type u_2
inst✝⁹ : Field R
inst✝⁸ : AddCommGroup V
inst✝⁷ : TopologicalSpace R
inst✝⁶ : TopologicalSpace V
inst✝⁵ : IsTopologicalRing R
inst✝⁴ : Module R V
inst✝³ : SeparatingDual R V
W : Type u_3
inst✝² : AddCommGroup W
inst✝¹ : Module R W
inst✝ : FiniteDimensional R W
f : W →ₗ[R] V
⊢ Surjective (⇑f.dualMap ∘ ContinuousLinearMap.toLinearMap) → Injective ⇑f

Tactic:
intro hf

Goal state after:
case mp
R : Type u_1
V : Type u_2
inst✝⁹ : Field R
inst✝⁸ : AddCommGroup V
inst✝⁷ : TopologicalSpace R
inst✝⁶ : TopologicalSpace V
inst✝⁵ : IsTopologicalRing R
inst✝⁴ : Module R V
inst✝³ : SeparatingDual R V
W : Type u_3
inst✝² : AddCommGroup W
inst✝¹ : Module R W
inst✝ : FiniteDimensional R W
f : W →ₗ[R] V
hf : Surjective (⇑f.dualMap ∘ ContinuousLinearMap.toLinearMap)
⊢ Injective ⇑f
Module: Mathlib.Analysis.LocallyConvex.SeparatingDual

Goal state before:
pp: case mpr
R : Type u_1
V : Type u_2
inst✝⁹ : Field R
inst✝⁸ : AddCommGroup V
inst✝⁷ : TopologicalSpace R
inst✝⁶ : TopologicalSpace V
inst✝⁵ : IsTopologicalRing R
inst✝⁴ : Module R V
inst✝³ : SeparatingDual R V
W : Type u_3
inst✝² : AddCommGroup W
inst✝¹ : Module R W
inst✝ : FiniteDimensional R W
f : W →ₗ[R] V
⊢ Injective ⇑f → Surjective (⇑f.dualMap ∘ ContinuousLinearMap.toLinearMap)

Tactic:
intro hf

Goal state after:
case mpr
R : Type u_1
V : Type u_2
inst✝⁹ : Field R
inst✝⁸ : AddCommGroup V
inst✝⁷ : TopologicalSpace R
inst✝⁶ : TopologicalSpace V
inst✝⁵ : IsTopologicalRing R
inst✝⁴ : Module R V
inst✝³ : SeparatingDual R V
W : Type u_3
inst✝² : AddCommGroup W
inst✝¹ : Module R W
inst✝ : FiniteDimensional R W
f : W →ₗ[R] V
hf : Injective ⇑f
⊢ Surjective (⇑f.dualMap ∘ ContinuousLinearMap.toLinearMap)
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

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
f : E → F
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : CompleteSpace F
⊢ Measurable (fderiv 𝕜 f)

Tactic:
refine measurable_of_isClosed fun s hs => ?_
  have :
    fderiv 𝕜 f ⁻¹' s =
      { x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ s } ∪
        { x | ¬DifferentiableAt 𝕜 f x } ∩ { _x | (0 : E →L[𝕜] F) ∈ s } :=
    Set.ext fun x => mem_preimage.trans fderiv_mem_iff
  rw [this]
  exact
    (measurableSet_of_differentiableAt_of_isComplete _ _ hs.isComplete).union
      ((measurableSet_of_differentiableAt _ _).compl.inter (MeasurableSet.const _))

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: case inr
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : CompleteSpace F
inst✝¹ : MeasurableSpace 𝕜
inst✝ : OpensMeasurableSpace 𝕜
h : SecondCountableTopologyEither 𝕜 F
f : 𝕜 → F
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
hF : SecondCountableTopology F
⊢ StronglyMeasurable (deriv f)

Tactic:
· exact (measurable_deriv f).stronglyMeasurable

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
r x : ℝ
hr : 0 < r
ε : ℝ
L₁ L₂ : F
h₁ : x ∈ A f L₁ r ε
h₂ : x ∈ A f L₂ r ε
⊢ ‖f (x + r / 2) - f x - (x + r / 2 - x) • L₂‖ + ‖f (x + r / 2) - f x - (x + r / 2 - x) • L₁‖ ≤ ε * r + ε * r

Tactic:
by
      apply add_le_add
      · apply le_of_mem_A h₂ <;> simp [(half_pos hr).le]
      · apply le_of_mem_A h₁ <;> simp [(half_pos hr).le]

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
x : ℝ
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
e p q e' p' q' : ℕ
hp : n e ≤ p
hq : n e ≤ q
hp' : n e' ≤ p'
⊢ n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * (1 / 2) ^ e

Tactic:
hq'

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
x : ℝ
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
e p q e' p' q' : ℕ
hp : n e ≤ p
hq : n e ≤ q
hp' : n e' ≤ p'
hq' : n e' ≤ q'
⊢ e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * (1 / 2) ^ e
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
pp: case e_f
X : Type u_1
inst✝¹ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
inst✝ : RCLike 𝕜
f : X → 𝕜
hf : Integrable f μ
⊢ (fun a ↦ ↑(re (f a)) + I • ↑(im (f a))) = fun x ↦ f x

Tactic:
ext1 x

Goal state after:
case e_f
X : Type u_1
inst✝¹ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
inst✝ : RCLike 𝕜
f : X → 𝕜
hf : Integrable f μ
x : X
⊢ ↑(re (f x)) + I • ↑(im (f x)) = f x
Module: Mathlib.Analysis.SpecialFunctions.NonIntegrable

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
f : ℝ → E
g : ℝ → F
k : Set ℝ
l : Filter ℝ
inst✝¹ : l.NeBot
inst✝ : TendstoIxxClass Icc l l
hl : k ∈ l
hd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x
hf : Tendsto (fun x ↦ ‖f x‖) l atTop
hfg : deriv f =O[l] g
a : E →ₗᵢ[ℝ] UniformSpace.Completion E := UniformSpace.Completion.toComplₗᵢ
f' : ℝ → UniformSpace.Completion E := ⇑a ∘ f
h'd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f' x
h'f : Tendsto (fun x ↦ ‖f' x‖) l atTop
x : ℝ
hx : DifferentiableAt ℝ f x
⊢ ‖deriv f' x‖ = ‖deriv f x‖

Tactic:
have : deriv f' x = a (deriv f x) := by
      rw [fderiv_comp_deriv x _ hx]
      · have : fderiv ℝ a (f x) = a.toContinuousLinearMap := a.toContinuousLinearMap.fderiv
        simp only [this]
        rfl
      · exact a.toContinuousLinearMap.differentiableAt

Goal state after:
E : Type u_1
F : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
f : ℝ → E
g : ℝ → F
k : Set ℝ
l : Filter ℝ
inst✝¹ : l.NeBot
inst✝ : TendstoIxxClass Icc l l
hl : k ∈ l
hd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x
hf : Tendsto (fun x ↦ ‖f x‖) l atTop
hfg : deriv f =O[l] g
a : E →ₗᵢ[ℝ] UniformSpace.Completion E := UniformSpace.Completion.toComplₗᵢ
f' : ℝ → UniformSpace.Completion E := ⇑a ∘ f
h'd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f' x
h'f : Tendsto (fun x ↦ ‖f' x‖) l atTop
x : ℝ
hx : DifferentiableAt ℝ f x
this : deriv f' x = a (deriv f x)
⊢ ‖deriv f' x‖ = ‖deriv f x‖