mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: R : Type u_2
inst✝¹ : CommRing R
P : R⟦X⟧
inst✝ : Invertible ((coeff 1) P)
⊢ (coeff 1) P.substInv = ⅟((coeff 1) P)

Tactic:
by
  simp [substInv, substInvFun]

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
f : R⟦X⟧
e : Fin 2 →₀ ℕ
⊢ (MvPowerSeries.coeff e) (subst X₀ f * subst X₁ f) = (coeff (e 0)) f * (coeff (e 1)) f

Tactic:
by
  rw [MvPowerSeries.coeff_mul, Finset.sum_eq_single (single 0 (e 0), single 1 (e 1)) ?_ ?_]
  · grind [coeff_subst_single]
  · intro b hb hb'
    by_contra hmul_ne_zero
    rcases ne_zero_and_ne_zero_of_mul hmul_ne_zero with ⟨h0, h1⟩
    simp only [Fin.isValue, coeff_subst_single, ne_eq, ite_eq_right_iff,
      not_forall, exists_prop] at h0 h1
    apply hb'
    rw [Prod.ext_iff, ← mem_antidiagonal.mp hb, h0.1, h1.1]
    simp
  · intro he
    have he' : single 0 (e 0) + single 1 (e 1) = e := by
      ext i
      fin_cases i <;> simp
    exact absurd (mem_antidiagonal.mpr he') he

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : Semiring 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
x : E
s : Set E
𝕜' : Type u_3
inst✝³ : Semiring 𝕜'
inst✝² : SMul 𝕜 𝕜'
inst✝¹ : Module 𝕜' E
inst✝ : IsScalarTower 𝕜 𝕜' E
hs : Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x)) ∧ x ∈ closure[inst✝⁴] s
⊢ ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x)) ⊆ ↑(Submodule.span 𝕜' (tangentConeAt 𝕜' s x))

Tactic:
trans ↑(Submodule.span 𝕜 (tangentConeAt 𝕜' s x))

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁷ : Semiring 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
x : E
s : Set E
𝕜' : Type u_3
inst✝³ : Semiring 𝕜'
inst✝² : SMul 𝕜 𝕜'
inst✝¹ : Module 𝕜' E
inst✝ : IsScalarTower 𝕜 𝕜' E
hs : Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x)) ∧ x ∈ closure[inst✝⁴] s
⊢ ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x)) ⊆ ↑(Submodule.span 𝕜 (tangentConeAt 𝕜' s x))

𝕜 : Type u_1
E : Type u_2
inst✝⁷ : Semiring 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
x : E
s : Set E
𝕜' : Type u_3
inst✝³ : Semiring 𝕜'
inst✝² : SMul 𝕜 𝕜'
inst✝¹ : Module 𝕜' E
inst✝ : IsScalarTower 𝕜 𝕜' E
hs : Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x)) ∧ x ∈ closure[inst✝⁴] s
⊢ ↑(Submodule.span 𝕜 (tangentConeAt 𝕜' s x)) ⊆ ↑(Submodule.span 𝕜' (tangentConeAt 𝕜' s x))
Module: Mathlib.Analysis.Calculus.FDeriv.Basic

Goal state before:
pp: case refine_2
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
f' : E →L[𝕜] F
x₀ : E
hf : HasFDerivAt f f' x₀
C : ℝ
hC₀ : 0 ≤ C
hlip : ∀ᶠ (x : E) in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖
ε : ℝ
ε0 : 0 < ε
⊢ ∀ᶠ (x : E) in 𝓝 0, ‖f' x‖ ≤ (C + ε) * ‖x‖

Tactic:
← map_add_left_nhds_zero x₀,

Goal state after:
case refine_2
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
f' : E →L[𝕜] F
x₀ : E
hf : HasFDerivAt f f' x₀
C : ℝ
hC₀ : 0 ≤ C
hlip : ∀ᶠ (x : E) in map (fun x ↦ x₀ + x) (𝓝 0), ‖f x - f x₀‖ ≤ C * ‖x - x₀‖
ε : ℝ
ε0 : 0 < ε
⊢ ∀ᶠ (x : E) in 𝓝 0, ‖f' x‖ ≤ (C + ε) * ‖x‖
Module: Mathlib.Analysis.Analytic.ConvergenceRadius

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
F : Type u_4
G : Type u_5
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
p : FormalMultilinearSeries 𝕜 F G
u : E →L[𝕜] F
h_zero : 0 < ‖u‖₊
⊢ ‖u‖ₑ ≠ 0

Tactic:
by simpa using h_zero

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
F : Type u_4
G : Type u_5
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
p : FormalMultilinearSeries 𝕜 F G
u : E →L[𝕜] F
h_zero : 0 < ‖u‖₊
r : ℝ≥0
hr : ↑r < p.radius
⊢ ‖u‖ₑ ≠ 0

Tactic:
by simpa using h_zero

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

Goal state before:
pp: case e'_4
𝕜 : Type u_1
E : Type u_3
F : Type u_4
G : Type u_5
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Nontrivial E
p : FormalMultilinearSeries 𝕜 F G
u : E ≃ₗᵢ[𝕜] F
x✝ : Nontrivial F
⊢ p.radius = ‖↑(↑u).symm‖ₑ * p.radius

Tactic:
have : u.toContinuousLinearEquiv.symm.toContinuousLinearMap =
    u.symm.toLinearIsometry.toContinuousLinearMap := rfl

Goal state after:
case e'_4
𝕜 : Type u_1
E : Type u_3
F : Type u_4
G : Type u_5
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Nontrivial E
p : FormalMultilinearSeries 𝕜 F G
u : E ≃ₗᵢ[𝕜] F
x✝ : Nontrivial F
this : ↑(↑u).symm = u.symm.toLinearIsometry.toContinuousLinearMap
⊢ p.radius = ‖↑(↑u).symm‖ₑ * p.radius
Module: Mathlib.Analysis.Normed.Module.Multilinear.Curry

Goal state before:
pp: 𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
Ei : Fin n.succ → Type wEi
G✝ : Type wG
G' : Type wG'
inst✝¹⁴ : Fintype ι
inst✝¹³ : Fintype ι'
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁹ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝⁸ : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝⁷ : NormedAddCommGroup G✝
inst✝⁶ : NormedSpace 𝕜 G✝
inst✝⁵ : NormedAddCommGroup G'
inst✝⁴ : NormedSpace 𝕜 G'
F : Type u_1
G : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : G →L[𝕜] ContinuousMultilinearMap 𝕜 E F
inst : DecidableEq (Option ι)
v : Option ι → G × ((i : ι) → E i)
j : Option ι
x y : G × ((i : ι) → E i)
⊢ ((B (update v j (x + y) none).1) fun i ↦ (update v j (x + y) (some i)).2 i) =
    ((B (update v j x none).1) fun i ↦ (update v j x (some i)).2 i) +
      (B (update v j y none).1) fun i ↦ (update v j y (some i)).2 i

Tactic:
match j with
      | none => simp
      | some j =>
        classical
        have B z : (fun i ↦ (Function.update v (some j) z (some i)).2 i) =
            Function.update (fun (i : ι) ↦ (v i).2 i) j (z.2 j) := by
          ext i
          rcases eq_or_ne i j with rfl | hij
          · simp
          · simp [hij]
        simp [B]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Multilinear.Curry

Goal state before:
pp: 𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
Ei : Fin n.succ → Type wEi
G✝ : Type wG
G' : Type wG'
inst✝¹⁴ : Fintype ι
inst✝¹³ : Fintype ι'
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁹ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝⁸ : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝⁷ : NormedAddCommGroup G✝
inst✝⁶ : NormedSpace 𝕜 G✝
inst✝⁵ : NormedAddCommGroup G'
inst✝⁴ : NormedSpace 𝕜 G'
F : Type u_1
G : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : G →L[𝕜] ContinuousMultilinearMap 𝕜 E F
inst : DecidableEq (Option ι)
v : Option ι → G × ((i : ι) → E i)
j : Option ι
c : 𝕜
x : G × ((i : ι) → E i)
⊢ ((B (update v j (c • x) none).1) fun i ↦ (update v j (c • x) (some i)).2 i) =
    c • (B (update v j x none).1) fun i ↦ (update v j x (some i)).2 i

Tactic:
match j with
      | none => simp
      | some j =>
        classical
        have B z : (fun i ↦ (Function.update v (some j) z (some i)).2 i) =
            Function.update (fun (i : ι) ↦ (v i).2 i) j (z.2 j) := by
          ext i
          rcases eq_or_ne i j with rfl | hij
          · simp
          · simp [hij]
        simp [B]

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

Goal state before:
pp: case inl
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
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
⊢ HasFPowerSeriesWithinAt f p (insert x s) x ↔ HasFPowerSeriesWithinAt f p s x

pp: case inr
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
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x y : E
hy : x ≠ y
⊢ HasFPowerSeriesWithinAt f p (insert y s) x ↔ HasFPowerSeriesWithinAt f p s x

Tactic:
· simp [HasFPowerSeriesWithinAt]

Goal state after:
case inr
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
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x y : E
hy : x ≠ y
⊢ HasFPowerSeriesWithinAt f p (insert y s) x ↔ HasFPowerSeriesWithinAt f p s x
Module: Mathlib.Analysis.Analytic.Composition

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
g : F → G
f : E → F
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
t : Set F
s : Set E
hs : Set.MapsTo f s t
rg : ℝ≥0∞
Hg : HasFPowerSeriesWithinOnBall g q t (f x) rg
rf : ℝ≥0∞
Hf : HasFPowerSeriesWithinOnBall f p s x rf
r : ℝ≥0
r_pos : 0 < r
hr : Summable fun i ↦ ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst
δ : ℝ≥0∞
δpos : 0 < δ
hδ : ∀ {z : E}, z ∈ insert x s ∩ Metric.eball x δ → f z ∈ insert (f x) t ∩ Metric.eball (f x) rg
rf' : ℝ≥0∞ := min rf δ
min_pos : 0 < min rf' ↑r
⊢ HasFPowerSeriesWithinOnBall (g ∘ f) (q.comp p) s x (min rf' ↑r)

Tactic:
refine
    ⟨le_trans (min_le_right rf' r) (FormalMultilinearSeries.le_comp_radius_of_summable q p r hr),
      min_pos, fun {y} h'y hy ↦ ?_⟩

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
g : F → G
f : E → F
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
t : Set F
s : Set E
hs : Set.MapsTo f s t
rg : ℝ≥0∞
Hg : HasFPowerSeriesWithinOnBall g q t (f x) rg
rf : ℝ≥0∞
Hf : HasFPowerSeriesWithinOnBall f p s x rf
r : ℝ≥0
r_pos : 0 < r
hr : Summable fun i ↦ ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst
δ : ℝ≥0∞
δpos : 0 < δ
hδ : ∀ {z : E}, z ∈ insert x s ∩ Metric.eball x δ → f z ∈ insert (f x) t ∩ Metric.eball (f x) rg
rf' : ℝ≥0∞ := min rf δ
min_pos : 0 < min rf' ↑r
y : E
h'y : x + y ∈ insert x s
hy : y ∈ Metric.eball 0 (min rf' ↑r)
⊢ HasSum (fun n ↦ (q.comp p n) fun x ↦ y) ((g ∘ f) (x + y))
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: 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
a : Composition n
b : Composition a.length
j : List ℕ
hj : j ∈ a.blocks.splitWrtComposition b
i : ℕ
hi : i ∈ j
⊢ 1 ≤ i

Tactic:
apply a.one_le_blocks

Goal state after:
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
a : Composition n
b : Composition a.length
j : List ℕ
hj : j ∈ a.blocks.splitWrtComposition b
i : ℕ
hi : i ∈ j
⊢ i ∈ a.blocks
Module: Mathlib.Analysis.Analytic.Within

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E → F
s : Set E
x : E
hf : AnalyticWithinAt 𝕜 f s x
g : E → F
hga : AnalyticAt 𝕜 g x
hfg : ∀ᶠ (x_1 : E) in 𝓝 x, x_1 ∈ insert x s → f x_1 = g x_1
z : E
⊢ (∀ᶠ (x_1 : E) in 𝓝 z, x_1 ∈ insert x s → f x_1 = g x_1) → AnalyticAt 𝕜 g z → z ∈ s → AnalyticWithinAt 𝕜 f s z

Tactic:
hfgz

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E → F
s : Set E
x : E
hf : AnalyticWithinAt 𝕜 f s x
g : E → F
hga : AnalyticAt 𝕜 g x
hfg : ∀ᶠ (x_1 : E) in 𝓝 x, x_1 ∈ insert x s → f x_1 = g x_1
z : E
hfgz : ∀ᶠ (x_1 : E) in 𝓝 z, x_1 ∈ insert x s → f x_1 = g x_1
⊢ AnalyticAt 𝕜 g z → z ∈ s → AnalyticWithinAt 𝕜 f s z
Module: Mathlib.Analysis.Calculus.FDeriv.Comp

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 → F
x : E
s : Set E
g : F → G
t : Set F
y : F
hg : DifferentiableWithinAt 𝕜 g t y
hf : DifferentiableWithinAt 𝕜 f s x
h : MapsTo f s t
hxs : UniqueDiffWithinAt 𝕜 s x
hy : f x = y
⊢ fderivWithin 𝕜 (g ∘ f) s x = fderivWithin 𝕜 g t (f x) ∘SL fderivWithin 𝕜 f s x

Tactic:
subst hy

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
f : E → F
x : E
s : Set E
g : F → G
t : Set F
hf : DifferentiableWithinAt 𝕜 f s x
h : MapsTo f s t
hxs : UniqueDiffWithinAt 𝕜 s x
hg : DifferentiableWithinAt 𝕜 g t (f x)
⊢ fderivWithin 𝕜 (g ∘ f) s x = fderivWithin 𝕜 g t (f x) ∘SL fderivWithin 𝕜 f s x
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
⊢ a.sizeUpTo (b.sizeUpTo i + (j + 1)) = (a.gather b).sizeUpTo i + (a.sigmaCompositionAux b ⟨i, ⋯⟩).sizeUpTo (j + 1)

Tactic:
have : sizeUpTo b i + Nat.succ j = (sizeUpTo b i + j).succ := rfl

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.sizeUpTo (b.sizeUpTo i + (j + 1)) = (a.gather b).sizeUpTo i + (a.sigmaCompositionAux b ⟨i, ⋯⟩).sizeUpTo (j + 1)
Module: Mathlib.Analysis.Analytic.Constructions

Goal state before:
pp: 𝕜 : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
ι : Type u_9
inst✝² : Fintype ι
e : E
Fm : ι → Type u_10
inst✝¹ : (i : ι) → NormedAddCommGroup (Fm i)
inst✝ : (i : ι) → NormedSpace 𝕜 (Fm i)
f : (i : ι) → E → Fm i
hf : ∀ (i : ι), AnalyticWithinAt 𝕜 (f i) Set.univ e
⊢ AnalyticWithinAt 𝕜 (fun x x_1 ↦ f x_1 x) Set.univ e

Tactic:
exact AnalyticWithinAt.pi hf

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

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
ι : Type u_4
inst✝⁹ : Semiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
inst✝⁶ : TopologicalSpace M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : IsTopologicalAddGroup N
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : ContinuousMultilinearMap R (fun x ↦ M) N
v : ι → M
⊢ (alternatization f) v = ∑ σ, Equiv.Perm.sign σ • f (v ∘ ⇑σ)

Tactic:
simp [alternatization, Function.comp_def]

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

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
c : 𝕜
⊢ fderiv 𝕜 (fun x ↦ f (c • x)) x = c • fderiv 𝕜 f (c • x)

Tactic:
rw [← fderivWithin_univ, fderivWithin_comp_smul _ uniqueDiffWithinAt_univ]

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
x : E
c : 𝕜
⊢ c • fderivWithin 𝕜 f (c • univ) (c • x) = c • fderiv 𝕜 f (c • x)
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
f : F [⋀^ι]→L[𝕜] G
g dg : E →L[𝕜] F
v : ι → E
a : ι
⊢ ∀ (j : ι),
    v a = v j →
      a ≠ j →
        (((f.fderivCompContinuousLinearMap fun x ↦ g) ∘SL
                ContinuousLinearMap.pi fun x ↦ ContinuousLinearMap.id 𝕜 (E →L[𝕜] F))
              dg)
            v =
          0

Tactic:
b

Goal state after:
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
f : F [⋀^ι]→L[𝕜] G
g dg : E →L[𝕜] F
v : ι → E
a b : ι
⊢ v a = v b →
    a ≠ b →
      (((f.fderivCompContinuousLinearMap fun x ↦ g) ∘SL
              ContinuousLinearMap.pi fun x ↦ ContinuousLinearMap.id 𝕜 (E →L[𝕜] F))
            dg)
          v =
        0
Module: Mathlib.Topology.MetricSpace.Completion

Goal state before:
pp: case refine_2
α : Type u
inst✝ : PseudoMetricSpace α
x y : Completion α
a : α
⊢ ∀ (b : α), dist ↑a ↑b = dist ↑b ↑a

Tactic:
b

Goal state after:
case refine_2
α : Type u
inst✝ : PseudoMetricSpace α
x y : Completion α
a b : α
⊢ dist ↑a ↑b = dist ↑b ↑a
Module: Mathlib.Topology.MetricSpace.Completion

Goal state before:
pp: case refine_2
α : Type u
inst✝ : PseudoMetricSpace α
x y z : Completion α
a : α
⊢ ∀ (b c : α), dist ↑a ↑c ≤ dist ↑a ↑b + dist ↑b ↑c

Tactic:
b

Goal state after:
case refine_2
α : Type u
inst✝ : PseudoMetricSpace α
x y z : Completion α
a b : α
⊢ ∀ (c : α), dist ↑a ↑c ≤ dist ↑a ↑b + dist ↑b ↑c
Module: Mathlib.Topology.MetricSpace.Completion

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
s : Set (Completion α × Completion α)
ε : ℝ
εpos : ε > 0
hε : ∀ {a b : Completion α}, dist a b < ε → (a, b) ∈ s
r : Set (ℝ × ℝ) := {p | dist p.1 p.2 < ε}
this : r ∈ 𝓤 ℝ
t1 : Set (Completion α × Completion α)
ht1 : t1 ∈ 𝓤 (Completion α)
t2 : Set (Completion α × Completion α)
ht2 : t2 ∈ 𝓤 (Completion α)
ht : t1 ×ˢ t2 ⊆ (fun p ↦ ((p.1.1, p.2.1), p.1.2, p.2.2)) ⁻¹' {x | (dist x.1.1 x.1.2, dist x.2.1 x.2.2) ∈ r}
a : Completion α
⊢ ∀ (b : Completion α), (a, b) ∈ t1 → dist a b < ε

Tactic:
b

Goal state after:
α : Type u
inst✝ : PseudoMetricSpace α
s : Set (Completion α × Completion α)
ε : ℝ
εpos : ε > 0
hε : ∀ {a b : Completion α}, dist a b < ε → (a, b) ∈ s
r : Set (ℝ × ℝ) := {p | dist p.1 p.2 < ε}
this : r ∈ 𝓤 ℝ
t1 : Set (Completion α × Completion α)
ht1 : t1 ∈ 𝓤 (Completion α)
t2 : Set (Completion α × Completion α)
ht2 : t2 ∈ 𝓤 (Completion α)
ht : t1 ×ˢ t2 ⊆ (fun p ↦ ((p.1.1, p.2.1), p.1.2, p.2.2)) ⁻¹' {x | (dist x.1.1 x.1.2, dist x.2.1 x.2.2) ∈ r}
a b : Completion α
⊢ (a, b) ∈ t1 → dist a b < ε
Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Goal state before:
pp: case neg
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
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
h :
  HasFTaylorSeriesUpToOn (↑n) f p s ∧
    (∀ x ∈ s, HasFDerivWithinAt (fun y ↦ p y n) (p x n.succ).curryLeft s x) ∧ ContinuousOn (fun x ↦ p x (n + 1)) s
m : ℕ
hm : ↑m ≤ ↑n + 1
h' : ¬m ≤ n
this : m = n + 1
⊢ ContinuousOn (fun x ↦ p x (n + 1)) s

Tactic:
exact h.2.2

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

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
y : E
⊢ ∀ x_1 ∈ {x}ᶜ, x_1 ∈ insert x s ↔ x_1 ∈ s

Tactic:
by intros; simp_all

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

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
c : F
⊢ deriv (fun x ↦ c + f x) x = deriv f x

Tactic:
simp only [deriv, fderiv_const_add]

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

Goal state before:
pp: case succ
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
h : CPolynomialOn 𝕜 f s
n : ℕ
IH : CPolynomialOn 𝕜 (_root_.iteratedFDeriv 𝕜 n f) s
⊢ CPolynomialOn 𝕜 (_root_.iteratedFDeriv 𝕜 (n + 1) f) s

Tactic:
rw [iteratedFDeriv_succ_eq_comp_left]
    convert! ContinuousLinearMap.comp_cpolynomialOn ?g IH.fderiv
    case g => exact ↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (n + 1) ↦ E) F).symm
    simp

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Slope

Goal state before:
pp: case neg
k : Type u_1
E : Type u_2
inst✝⁷ : Field k
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module k E
inst✝⁴ : LinearOrder k
inst✝³ : IsStrictOrderedRing k
inst✝² : PartialOrder E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : PosSMulMono k E
f : k → E
x y : k
hxy : x ≤ y
hxeqy : ¬x = y
⊢ 0 ≤ slope f x y ↔ f x ≤ f y

Tactic:
refine ⟨fun h ↦ ?_, fun h ↦ smul_nonneg (inv_nonneg.2 (sub_nonneg.2 hxy)) ?_⟩

Goal state after:
case neg.refine_1
k : Type u_1
E : Type u_2
inst✝⁷ : Field k
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module k E
inst✝⁴ : LinearOrder k
inst✝³ : IsStrictOrderedRing k
inst✝² : PartialOrder E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : PosSMulMono k E
f : k → E
x y : k
hxy : x ≤ y
hxeqy : ¬x = y
h : 0 ≤ slope f x y
⊢ f x ≤ f y

case neg.refine_2
k : Type u_1
E : Type u_2
inst✝⁷ : Field k
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module k E
inst✝⁴ : LinearOrder k
inst✝³ : IsStrictOrderedRing k
inst✝² : PartialOrder E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : PosSMulMono k E
f : k → E
x y : k
hxy : x ≤ y
hxeqy : ¬x = y
h : f x ≤ f y
⊢ 0 ≤ f y -ᵥ f x