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

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
x✝ : 𝕜
⊢ (fderivWithin 𝕜 f univ x✝) 1 = (fderiv 𝕜 f x✝) 1

Tactic:
fderivWithin_univ

Goal state after:
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
x✝ : 𝕜
⊢ (fderiv 𝕜 f x✝) 1 = (fderiv 𝕜 f x✝) 1
Module: Mathlib.Analysis.Calculus.Deriv.Basic

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
x : 𝕜
s t : Set 𝕜
ht : t ∈ 𝓝 x
⊢ (fderivWithin 𝕜 f (s ∩ t) x) 1 = (fderivWithin 𝕜 f s x) 1

Tactic:
fderivWithin_inter ht

Goal state after:
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
x : 𝕜
s t : Set 𝕜
ht : t ∈ 𝓝 x
⊢ (fderivWithin 𝕜 f s x) 1 = (fderivWithin 𝕜 f s x) 1
Module: Mathlib.Analysis.Calculus.FDeriv.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
F : Type u_3
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 F
inst✝ : TopologicalSpace F
f : E → F
f' : E →L[𝕜] F
x : E
s : Set E
⊢ HasFDerivWithinAt f f' (insert x s) x ↔ HasFDerivWithinAt f f' s x

Tactic:
simp_rw [hasFDerivWithinAt_iff_isLittleOTVS]
  apply isLittleOTVS_insert
  simp only [sub_self, map_zero]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
F : Type u_3
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 F
inst✝ : TopologicalSpace F
f : E → F
f' : E →L[𝕜] F
x : E
hf : HasStrictFDerivAt f f' x
⊢ (fun x' ↦ f x' - f x - f' (x' - x)) =o[𝕜; 𝓝 x] fun x' ↦ x' - x

Tactic:
by
    simpa only using! hf.isLittleOTVS.comp_tendsto (tendsto_id.prodMk_nhds tendsto_const_nhds)

Goal state after:
Module: Mathlib.Analysis.Asymptotics.TVS

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
F : Type u_5
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : AddCommGroup F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : Module 𝕜 F
l : Filter α
g : α → F
ι : Type u_7
E : ι → Type u_8
inst✝³ : (i : ι) → AddCommGroup (E i)
inst✝² : (i : ι) → Module 𝕜 (E i)
inst✝¹ : (i : ι) → TopologicalSpace (E i)
inst✝ : ∀ (i : ι), ContinuousSMul 𝕜 (E i)
f : (i : ι) → α → E i
h : ∀ (i : ι), f i =O[𝕜; l] g
this✝ :
  (𝓝 0).HasBasis (fun If ↦ If.1.Finite ∧ ∀ i ∈ If.1, If.2 i ∈ 𝓝 0 ∧ Balanced 𝕜 (If.2 i)) fun If ↦
    If.1.pi fun i ↦ id (If.2 i)
I : Set ι
U : (i : ι) → Set (E i)
hIf : I.Finite
hU : ∀ x ∈ I, U x ∈ 𝓝 0
Ub : ∀ x ∈ I, Balanced 𝕜 (U x)
this : ∀ i ∈ I, ∀ᶠ (V : Set F) in (𝓝 0).smallSets, ∀ᶠ (x : α) in l, egauge 𝕜 (U i) (f i x) ≤ egauge 𝕜 V (g x)
V : Set F
hV₀ : V ∈ 𝓝 0
hV : ∀ i ∈ I, ∀ᶠ (x : α) in l, egauge 𝕜 (U i) (f i x) ≤ egauge 𝕜 V (g x)
⊢ ∃ j ∈ 𝓝 0, ∀ᶠ (x : α) in l, (egauge 𝕜 (I.pi fun i ↦ U i) fun i ↦ f i x) ≤ egauge 𝕜 j (g x)

Tactic:
use V, hV₀

Goal state after:
case right
α : Type u_1
𝕜 : Type u_3
F : Type u_5
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : AddCommGroup F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : Module 𝕜 F
l : Filter α
g : α → F
ι : Type u_7
E : ι → Type u_8
inst✝³ : (i : ι) → AddCommGroup (E i)
inst✝² : (i : ι) → Module 𝕜 (E i)
inst✝¹ : (i : ι) → TopologicalSpace (E i)
inst✝ : ∀ (i : ι), ContinuousSMul 𝕜 (E i)
f : (i : ι) → α → E i
h : ∀ (i : ι), f i =O[𝕜; l] g
this✝ :
  (𝓝 0).HasBasis (fun If ↦ If.1.Finite ∧ ∀ i ∈ If.1, If.2 i ∈ 𝓝 0 ∧ Balanced 𝕜 (If.2 i)) fun If ↦
    If.1.pi fun i ↦ id (If.2 i)
I : Set ι
U : (i : ι) → Set (E i)
hIf : I.Finite
hU : ∀ x ∈ I, U x ∈ 𝓝 0
Ub : ∀ x ∈ I, Balanced 𝕜 (U x)
this : ∀ i ∈ I, ∀ᶠ (V : Set F) in (𝓝 0).smallSets, ∀ᶠ (x : α) in l, egauge 𝕜 (U i) (f i x) ≤ egauge 𝕜 V (g x)
V : Set F
hV₀ : V ∈ 𝓝 0
hV : ∀ i ∈ I, ∀ᶠ (x : α) in l, egauge 𝕜 (U i) (f i x) ≤ egauge 𝕜 V (g x)
⊢ ∀ᶠ (x : α) in l, (egauge 𝕜 (I.pi fun i ↦ U i) fun i ↦ f i x) ≤ egauge 𝕜 V (g x)
Module: Mathlib.Analysis.Asymptotics.TVS

Goal state before:
pp: case h₂
α : Type u_1
𝕜 : Type u_3
E : Type u_4
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : TopologicalSpace E
inst✝² : Module 𝕜 E
l : Filter α
f : α → E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul 𝕜 E
x : E
h : Tendsto (fun x_1 ↦ f x_1 - x) l (𝓝 0)
U : Set E
hU₀ : U ∈ 𝓝 0
hUb : Balanced 𝕜 U
r : ℝ≥0
hr₀ : 0 < r
hr₁ : r ≤ 1
hr : ↑r ≤ (egauge 𝕜 U x)⁻¹
a : α
ha : f a - x ∈ U
⊢ egauge 𝕜 U x ≤ (↑r)⁻¹

Tactic:
rwa [ENNReal.le_inv_iff_le_inv]

Goal state after:
Module: Mathlib.Analysis.Asymptotics.TVS

Goal state before:
pp: case refine_1
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
f : α → E
g : α → F
l : Filter α
c : 𝕜
hc : 1 < ‖c‖₊
hc₀ : 0 < ‖c‖₊
h :
  ∀ (i : ℝ),
    0 < i →
      ∃ j, 0 < j ∧ ∀ (ε : ℝ≥0), ε ≠ 0 → ∀ᶠ (x : α) in l, egauge 𝕜 (ball 0 i) (f x) ≤ ↑ε * egauge 𝕜 (ball 0 j) (g x)
δ : ℝ
hδ₀ : 0 < δ
ε : ℝ≥0
hε : 0 < ↑ε
hδ : ∀ (ε_1 : ℝ≥0), ε_1 ≠ 0 → ∀ᶠ (x : α) in l, egauge 𝕜 (ball 0 ↑ε) (f x) ≤ ↑ε_1 * egauge 𝕜 (ball 0 δ) (g x)
⊢ ∀ᶠ (x : α) in l, ‖f x‖ ≤ ↑ε * ‖g x‖

Tactic:
lift δ to ℝ≥0 using hδ₀.le

Goal state after:
case refine_1
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
f : α → E
g : α → F
l : Filter α
c : 𝕜
hc : 1 < ‖c‖₊
hc₀ : 0 < ‖c‖₊
h :
  ∀ (i : ℝ),
    0 < i →
      ∃ j, 0 < j ∧ ∀ (ε : ℝ≥0), ε ≠ 0 → ∀ᶠ (x : α) in l, egauge 𝕜 (ball 0 i) (f x) ≤ ↑ε * egauge 𝕜 (ball 0 j) (g x)
ε : ℝ≥0
hε : 0 < ↑ε
δ : ℝ≥0
hδ₀ : 0 < ↑δ
hδ : ∀ (ε_1 : ℝ≥0), ε_1 ≠ 0 → ∀ᶠ (x : α) in l, egauge 𝕜 (ball 0 ↑ε) (f x) ≤ ↑ε_1 * egauge 𝕜 (ball 0 ↑δ) (g x)
⊢ ∀ᶠ (x : α) in l, ‖f x‖ ≤ ↑ε * ‖g x‖
Module: Mathlib.Analysis.Analytic.ChangeOrigin

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
p : FormalMultilinearSeries 𝕜 E F
k l : ℕ
⊢ ∑ a, ‖p.changeOriginSeriesTerm k l ↑a ⋯‖₊ = ∑' (x : { s // s.card = l }), ‖p (k + l)‖₊

Tactic:
by
    simp_rw [tsum_fintype, nnnorm_changeOriginSeriesTerm (p := p) (k := k) (l := l)]

Goal state after:
Module: Mathlib.Analysis.Analytic.ConvergenceRadius

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
F : Type u_4
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
𝕜' : Type u_6
E' : Type u_7
F' : Type u_8
inst✝⁴ : NontriviallyNormedField 𝕜'
inst✝³ : NormedAddCommGroup E'
inst✝² : NormedSpace 𝕜' E'
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace 𝕜' F'
p : FormalMultilinearSeries 𝕜 E F
q : FormalMultilinearSeries 𝕜' E' F'
h : ∀ (n : ℕ), ‖p n‖ ≤ ‖q n‖
⊢ q.radius ≤ p.radius

Tactic:
apply le_of_forall_nnreal_lt (fun r hr ↦ ?_)

Goal state after:
𝕜 : Type u_1
E : Type u_3
F : Type u_4
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
𝕜' : Type u_6
E' : Type u_7
F' : Type u_8
inst✝⁴ : NontriviallyNormedField 𝕜'
inst✝³ : NormedAddCommGroup E'
inst✝² : NormedSpace 𝕜' E'
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace 𝕜' F'
p : FormalMultilinearSeries 𝕜 E F
q : FormalMultilinearSeries 𝕜' E' F'
h : ∀ (n : ℕ), ‖p n‖ ≤ ‖q n‖
r : ℝ≥0
hr : ↑r < q.radius
⊢ ↑r ≤ p.radius
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: case neg.refine_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
r : ℝ≥0
hr : r ≠ 0
hc : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 ↑r)
r' : ℝ≥0
hr' : r' * r < 1
hrz : ¬r' = 0
⊢ Summable fun n ↦ ‖‖c n‖ * ↑r' ^ n‖

Tactic:
refine summable_of_ratio_test_tendsto_lt_one hr' ?_ ?_
    · refine (hc.eventually_ne (NNReal.coe_ne_zero.mpr hr)).mp (Eventually.of_forall ?_)
      simp_all
    · simp_rw [norm_norm]
      exact tendsto_succ_norm_div_norm c hrz hc

Goal state after:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing E
inst✝¹ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
inst✝ : NormOneClass E
r : ℝ≥0∞
hr : r = 0
hc' : Tendsto (fun n ↦ ENNReal.ofReal ‖c n.succ‖ / ENNReal.ofReal ‖c n‖) atTop (𝓝 0)
h : ∀ᶠ (n : ℕ) in atTop, c n ≠ 0
⊢ (ENNReal.toReal ∘ fun n ↦ ENNReal.ofReal ‖c n.succ‖ / ENNReal.ofReal ‖c n‖) =ᶠ[atTop] fun n ↦ ‖c n.succ‖ / ‖c n‖

Tactic:
filter_upwards [h]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing E
inst✝¹ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
inst✝ : NormOneClass E
r : ℝ≥0∞
hr : r = 0
hc' : Tendsto (fun n ↦ ENNReal.ofReal ‖c n.succ‖ / ENNReal.ofReal ‖c n‖) atTop (𝓝 0)
h : ∀ᶠ (n : ℕ) in atTop, c n ≠ 0
⊢ ∀ (a : ℕ),
    c a ≠ 0 → (ENNReal.toReal ∘ fun n ↦ ENNReal.ofReal ‖c n.succ‖ / ENNReal.ofReal ‖c n‖) a = ‖c a.succ‖ / ‖c a‖
Module: Mathlib.Analysis.Analytic.Basic

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
s : Set E
x : E
hf : AnalyticAt 𝕜 f x
⊢ AnalyticWithinAt 𝕜 f s x

Tactic:
rw [← analyticWithinAt_univ] at hf

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
s : Set E
x : E
hf : AnalyticWithinAt 𝕜 f univ x
⊢ AnalyticWithinAt 𝕜 f s x
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: case h₀
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
p : FormalMultilinearSeries 𝕜 E F
x : E
n : ℕ
b : Composition n
a✝ : b ∈ Finset.univ
hb : b ≠ Composition.ones n
k : ℕ
hk : k ∈ b.blocks
lt_k : 1 < k
i : Fin b.blocks.length
hi : b.blocks[i] = k
j : Fin b.length := ⟨↑i, ⋯⟩
A : 1 < b.blocksFun j
v : Fin n → E
⊢ (id 𝕜 E x (b.blocksFun j)) (v ∘ ⇑(b.embedding j)) = 0

Tactic:
rw [id_apply_of_one_lt _ _ _ A, _root_.zero_apply]

Goal state after:
Module: Mathlib.Analysis.Analytic.Inverse

Goal state before:
pp: case hf
n : ℕ
p : ℕ → ℝ
hp : ∀ (k : ℕ), 0 ≤ p k
r a : ℝ
hr : 0 ≤ r
ha : 0 ≤ a
⊢ ∀ i ∈ compPartialSumTarget 2 (n + 1) n,
    (i ∉ (Ico 2 (n + 1)).sigma fun k ↦ {c | 1 < c.length}.toFinset) →
      0 ≤ ∏ j, r * (a ^ i.snd.blocksFun j * p (i.snd.blocksFun j))

Tactic:
intro x _ _

Goal state after:
case hf
n : ℕ
p : ℕ → ℝ
hp : ∀ (k : ℕ), 0 ≤ p k
r a : ℝ
hr : 0 ≤ r
ha : 0 ≤ a
x : (i : ℕ) × Composition i
a✝¹ : x ∈ compPartialSumTarget 2 (n + 1) n
a✝ : x ∉ (Ico 2 (n + 1)).sigma fun k ↦ {c | 1 < c.length}.toFinset
⊢ 0 ≤ ∏ j, r * (a ^ x.snd.blocksFun j * p (x.snd.blocksFun j))
Module: Mathlib.Analysis.Analytic.Inverse

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
f : E → G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
hq : 0 < q.radius
hp : 0 < p.radius
r0 : ℝ≥0∞
h0 : HasFPowerSeriesOnBall f (q.comp p) x r0
r1 : NNReal
r1_pos : 0 < r1
hr1 : Summable fun i ↦ ‖q.compAlongComposition p i.snd‖₊ * r1 ^ i.fst
r : ℝ≥0∞ := min r0 ↑r1
⊢ 0 < ↑r1

Tactic:
by exact_mod_cast r1_pos

Goal state after:
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: n : ℕ
a : Composition n
b : Composition a.length
i : Fin b.length
⊢ (a.blocks.splitWrtComposition b)[↑i].length = b.blocksFun i

Tactic:
getElem_map_rev List.length,

Goal state after:
n : ℕ
a : Composition n
b : Composition a.length
i : Fin b.length
⊢ (List.map List.length (a.blocks.splitWrtComposition b))[↑i] = b.blocksFun i
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: case zero
n : ℕ
a : Composition n
b : Composition a.length
i : ℕ
hi : i < b.length
hj : 0 < b.blocksFun ⟨i, hi⟩
⊢ (take (take i b.blocks).sum a.blocks).sum = (take i (List.map sum (a.blocks.splitWrtComposition b))).sum

Tactic:
induction i with

Goal state after:
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: case succ
n : ℕ
a : Composition n
b : Composition a.length
i : ℕ
hi : i < b.length
j : ℕ
IHj :
  j < b.blocksFun ⟨i, hi⟩ →
    a.sizeUpTo (b.sizeUpTo i + j) = (a.gather b).sizeUpTo i + (a.sigmaCompositionAux b ⟨i, ⋯⟩).sizeUpTo j
hj : j + 1 < b.blocksFun ⟨i, hi⟩
A : j < b.blocksFun ⟨i, hi⟩
B : j < (a.sigmaCompositionAux b ⟨i, ⋯⟩).length
C : b.sizeUpTo i + j < b.sizeUpTo (i + 1)
D : b.sizeUpTo i + j < a.length
this : b.sizeUpTo i + j.succ = (b.sizeUpTo i + j).succ
⊢ a.sizeUpTo (b.sizeUpTo i + (j + 1)) = (a.gather b).sizeUpTo i + (a.sigmaCompositionAux b ⟨i, ⋯⟩).sizeUpTo (j + 1)

Tactic:
rw [this, sizeUpTo_succ _ D, IHj A, sizeUpTo_succ _ B]

Goal state after:
case succ
n : ℕ
a : Composition n
b : Composition a.length
i : ℕ
hi : i < b.length
j : ℕ
IHj :
  j < b.blocksFun ⟨i, hi⟩ →
    a.sizeUpTo (b.sizeUpTo i + j) = (a.gather b).sizeUpTo i + (a.sigmaCompositionAux b ⟨i, ⋯⟩).sizeUpTo j
hj : j + 1 < b.blocksFun ⟨i, hi⟩
A : j < b.blocksFun ⟨i, hi⟩
B : j < (a.sigmaCompositionAux b ⟨i, ⋯⟩).length
C : b.sizeUpTo i + j < b.sizeUpTo (i + 1)
D : b.sizeUpTo i + j < a.length
this : b.sizeUpTo i + j.succ = (b.sizeUpTo i + j).succ
⊢ (a.gather b).sizeUpTo i + (a.sigmaCompositionAux b ⟨i, ⋯⟩).sizeUpTo j + a.blocks[b.sizeUpTo i + j] =
    (a.gather b).sizeUpTo i + ((a.sigmaCompositionAux b ⟨i, ⋯⟩).sizeUpTo j + (a.sigmaCompositionAux b ⟨i, ⋯⟩).blocks[j])
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: case e'_3
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
n✝ n : ℕ
c : Composition n
d : (i : Fin c.length) → Composition (c.blocksFun i)
this : List.map sum (ofFn fun i ↦ (d i).blocks) = c.blocks
e_2✝ :
  ({ blocks := (ofFn fun j ↦ (d j).blocks).flatten, blocks_pos := ⋯, blocks_sum := ⋯ }.gather
        { blocks := ofFn fun j ↦ (d j).length, blocks_pos := ⋯, blocks_sum := ⋯ }).length =
    c.length
⊢ (fun i ↦
      ({ blocks := (ofFn fun j ↦ (d j).blocks).flatten, blocks_pos := ⋯, blocks_sum := ⋯ }.sigmaCompositionAux
          { blocks := ofFn fun j ↦ (d j).length, blocks_pos := ⋯, blocks_sum := ⋯ } i).blocks) ≍
    fun i ↦ (d i).blocks

Tactic:
Fin.heq_fun_iff

Goal state after:
case e'_3
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
n✝ n : ℕ
c : Composition n
d : (i : Fin c.length) → Composition (c.blocksFun i)
this : List.map sum (ofFn fun i ↦ (d i).blocks) = c.blocks
e_2✝ :
  ({ blocks := (ofFn fun j ↦ (d j).blocks).flatten, blocks_pos := ⋯, blocks_sum := ⋯ }.gather
        { blocks := ofFn fun j ↦ (d j).length, blocks_pos := ⋯, blocks_sum := ⋯ }).length =
    c.length
⊢ ∀
    (i :
      Fin
        ({ blocks := (ofFn fun j ↦ (d j).blocks).flatten, blocks_pos := ⋯, blocks_sum := ⋯ }.gather
            { blocks := ofFn fun j ↦ (d j).length, blocks_pos := ⋯, blocks_sum := ⋯ }).length),
    ({ blocks := (ofFn fun j ↦ (d j).blocks).flatten, blocks_pos := ⋯, blocks_sum := ⋯ }.sigmaCompositionAux
          { blocks := ofFn fun j ↦ (d j).length, blocks_pos := ⋯, blocks_sum := ⋯ } i).blocks =
      (d ⟨↑i, ⋯⟩).blocks

case e'_3.h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
n✝ n : ℕ
c : Composition n
d : (i : Fin c.length) → Composition (c.blocksFun i)
this : List.map sum (ofFn fun i ↦ (d i).blocks) = c.blocks
e_2✝ :
  ({ blocks := (ofFn fun j ↦ (d j).blocks).flatten, blocks_pos := ⋯, blocks_sum := ⋯ }.gather
        { blocks := ofFn fun j ↦ (d j).length, blocks_pos := ⋯, blocks_sum := ⋯ }).length =
    c.length
⊢ ({ blocks := (ofFn fun j ↦ (d j).blocks).flatten, blocks_pos := ⋯, blocks_sum := ⋯ }.gather
        { blocks := ofFn fun j ↦ (d j).length, blocks_pos := ⋯, blocks_sum := ⋯ }).length =
    c.length
Module: Mathlib.Analysis.Analytic.Constructions

Goal state before:
pp: 𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
𝕝 : Type u_8
inst✝¹ : NormedDivisionRing 𝕝
inst✝ : NormedAlgebra 𝕜 𝕝
f : E → 𝕝
z : E
s : Set E
n : ℤ
h₁f : AnalyticWithinAt 𝕜 f s z
h₂f : f z ≠ 0
⊢ AnalyticWithinAt 𝕜 (f ^ n) s z

Tactic:
by
  by_cases hn : 0 ≤ n
  · exact zpow_nonneg h₁f hn
  · rw [(Int.eq_neg_comm.mp rfl : n = -(-n))]
    conv => arg 2; intro x; rw [zpow_neg]
    exact (h₁f.zpow_nonneg (by linarith)).inv (zpow_ne_zero (-n) h₂f)

Goal state after:
Module: Mathlib.Analysis.Analytic.Constructions

Goal state before:
pp: 𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
𝕝 : Type u_8
inst✝¹ : NormedDivisionRing 𝕝
inst✝ : NormedAlgebra 𝕜 𝕝
f : E → 𝕝
z : E
n : ℤ
h₁f : AnalyticAt 𝕜 f z
h₂f : f z ≠ 0
⊢ AnalyticAt 𝕜 (f ^ n) z

Tactic:
by
  by_cases hn : 0 ≤ n
  · exact zpow_nonneg h₁f hn
  · rw [(Int.eq_neg_comm.mp rfl : n = -(-n))]
    conv => arg 2; intro x; rw [zpow_neg]
    exact (h₁f.zpow_nonneg (by linarith)).inv (zpow_ne_zero (-n) h₂f)

Goal state after:
Module: Mathlib.Analysis.Analytic.Constructions

Goal state before:
pp: case neg
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
α : Type u_9
A : Type u_10
inst✝¹ : NormedCommRing A
inst✝ : NormedAlgebra 𝕜 A
f : α → E → A
c : E
h : ∀ (a : α), AnalyticAt 𝕜 (f a) c
hf : ¬(Function.mulSupport f).Finite
⊢ AnalyticAt 𝕜 (∏ᶠ (n : α), f n) c

Tactic:
· rw [finprod_of_infinite_mulSupport hf]
    apply analyticAt_const

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

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
iso : E ≃L[𝕜] F
f : G → E
s : Set G
x : G
H : DifferentiableWithinAt 𝕜 (⇑iso ∘ f) s x
⊢ DifferentiableWithinAt 𝕜 f s x

Tactic:
have : DifferentiableWithinAt 𝕜 (iso.symm ∘ iso ∘ f) s x :=
    iso.symm.differentiable.differentiableAt.comp_differentiableWithinAt x H

Goal state after:
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
iso : E ≃L[𝕜] F
f : G → E
s : Set G
x : G
H : DifferentiableWithinAt 𝕜 (⇑iso ∘ f) s x
this : DifferentiableWithinAt 𝕜 (⇑iso.symm ∘ ⇑iso ∘ f) s x
⊢ DifferentiableWithinAt 𝕜 f s x
Module: Mathlib.Analysis.Calculus.FDeriv.Add

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
x : E
s : Set E
a : E
⊢ fderivWithin 𝕜 (fun x ↦ f (x + a)) s x = fderivWithin 𝕜 f (a +ᵥ s) (x + a)

Tactic:
simp only [add_comm _ a, fderivWithin_comp_add_left]

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Alternating.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹² : Semiring R
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : Module R M
inst✝⁹ : TopologicalSpace M
inst✝⁸ : AddCommMonoid M'
inst✝⁷ : Module R M'
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : AddCommMonoid N'
inst✝¹ : Module R N'
inst✝ : TopologicalSpace N'
n : ℕ
f✝ g : M [⋀^ι]→L[R] N
e : M ≃L[R] M'
f : M [⋀^ι]→L[R] N
⊢ (fun f ↦ f.compContinuousLinearMap ↑e) ((fun f ↦ f.compContinuousLinearMap ↑e.symm) f) = f

Tactic:
ext; simp [Function.comp_def]

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Alternating.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹² : Semiring R
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : Module R M
inst✝⁹ : TopologicalSpace M
inst✝⁸ : AddCommMonoid M'
inst✝⁷ : Module R M'
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : AddCommMonoid N'
inst✝¹ : Module R N'
inst✝ : TopologicalSpace N'
n : ℕ
f✝ g : M [⋀^ι]→L[R] N
e : M ≃L[R] M'
f : M' [⋀^ι]→L[R] N
⊢ (fun f ↦ f.compContinuousLinearMap ↑e.symm) ((fun f ↦ f.compContinuousLinearMap ↑e) f) = f

Tactic:
ext; simp [Function.comp_def]

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Alternating.Topology

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
ι : Type u_4
inst✝¹⁴ : NormedField 𝕜
inst✝¹³ : AddCommGroup E
inst✝¹² : Module 𝕜 E
inst✝¹¹ : TopologicalSpace E
inst✝¹⁰ : AddCommGroup F
inst✝⁹ : Module 𝕜 F
inst✝⁸ : UniformSpace F
inst✝⁷ : IsUniformAddGroup F
𝕜' : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜'
inst✝⁵ : NormedAlgebra 𝕜' 𝕜
inst✝⁴ : Module 𝕜' E
inst✝³ : IsScalarTower 𝕜' 𝕜 E
inst✝² : Module 𝕜' F
inst✝¹ : IsScalarTower 𝕜' 𝕜 F
inst✝ : ContinuousSMul 𝕜 E
⊢ IsUniformEmbedding (restrictScalars 𝕜')

Tactic:
rw [← isUniformEmbedding_toContinuousMultilinearMap.of_comp_iff]

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
ι : Type u_4
inst✝¹⁴ : NormedField 𝕜
inst✝¹³ : AddCommGroup E
inst✝¹² : Module 𝕜 E
inst✝¹¹ : TopologicalSpace E
inst✝¹⁰ : AddCommGroup F
inst✝⁹ : Module 𝕜 F
inst✝⁸ : UniformSpace F
inst✝⁷ : IsUniformAddGroup F
𝕜' : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜'
inst✝⁵ : NormedAlgebra 𝕜' 𝕜
inst✝⁴ : Module 𝕜' E
inst✝³ : IsScalarTower 𝕜' 𝕜 E
inst✝² : Module 𝕜' F
inst✝¹ : IsScalarTower 𝕜' 𝕜 F
inst✝ : ContinuousSMul 𝕜 E
⊢ IsUniformEmbedding (toContinuousMultilinearMap ∘ restrictScalars 𝕜')
Module: Mathlib.Analysis.Normed.Module.Alternating.Basic

Goal state before:
pp: 𝕜 : Type u
n : ℕ
E : Type wE
F : Type wF
G : Type wG
ι : Type v
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : SeminormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
g : E →L[𝕜] F
c : 𝕜
f : F [⋀^ι]→L[𝕜] G
⊢ (c • f).fderivCompContinuousLinearMap g = (RingHom.id 𝕜) c • f.fderivCompContinuousLinearMap g

Tactic:
ext; simp [Finset.smul_sum]

Goal state after: