mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Analytic.Constructions

Goal state before:
pp: case mpr
𝕜 : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
𝕝 : Type u_8
inst✝⁴ : NormedDivisionRing 𝕝
inst✝³ : NormedAlgebra 𝕜 𝕝
inst✝² : Module 𝕝 F
inst✝¹ : IsBoundedSMul 𝕝 F
inst✝ : IsScalarTower 𝕜 𝕝 F
f : E → 𝕝
g : E → F
z : E
h₁f : AnalyticAt 𝕜 f z
h₂f : f z ≠ 0
hprod : AnalyticAt 𝕜 (fun z ↦ f z • g z) z
⊢ AnalyticAt 𝕜 g z

Tactic:
analyticAt_congr (g := (f⁻¹ • f) • g),

Goal state after:
case mpr
𝕜 : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
𝕝 : Type u_8
inst✝⁴ : NormedDivisionRing 𝕝
inst✝³ : NormedAlgebra 𝕜 𝕝
inst✝² : Module 𝕝 F
inst✝¹ : IsBoundedSMul 𝕝 F
inst✝ : IsScalarTower 𝕜 𝕝 F
f : E → 𝕝
g : E → F
z : E
h₁f : AnalyticAt 𝕜 f z
h₂f : f z ≠ 0
hprod : AnalyticAt 𝕜 (fun z ↦ f z • g z) z
⊢ AnalyticAt 𝕜 ((f⁻¹ • f) • g) z

case mpr
𝕜 : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
𝕝 : Type u_8
inst✝⁴ : NormedDivisionRing 𝕝
inst✝³ : NormedAlgebra 𝕜 𝕝
inst✝² : Module 𝕝 F
inst✝¹ : IsBoundedSMul 𝕝 F
inst✝ : IsScalarTower 𝕜 𝕝 F
f : E → 𝕝
g : E → F
z : E
h₁f : AnalyticAt 𝕜 f z
h₂f : f z ≠ 0
hprod : AnalyticAt 𝕜 (fun z ↦ f z • g z) z
⊢ g =ᶠ[𝓝 z] (f⁻¹ • f) • g
Module: Mathlib.Analysis.Analytic.Constructions

Goal state before:
pp: 𝕜 : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
𝕝 : Type u_8
inst✝⁴ : NormedDivisionRing 𝕝
inst✝³ : NormedAlgebra 𝕜 𝕝
inst✝² : Module 𝕝 F
inst✝¹ : IsBoundedSMul 𝕝 F
inst✝ : IsScalarTower 𝕜 𝕝 F
f : E → 𝕝
g : E → F
z : E
h₁f : AnalyticAt 𝕜 f z
h₂f : f z ≠ 0
hprod : AnalyticAt 𝕜 (fun z ↦ f z • g z) z
y : E
hy : ¬f y = 0
⊢ g y = ((f⁻¹ • f) • g) y

Tactic:
simp [hy]

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
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
⊢ AnalyticAt 𝕜 (∏ᶠ (n : α), f n) c

Tactic:
by_cases hf : (Function.mulSupport f).Finite

Goal state after:
case pos
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

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
Module: Mathlib.Analysis.Calculus.FDeriv.Bilinear

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
b : E × F → G
h : IsBoundedBilinearMap 𝕜 b
p : E × F
T : Type (max u_3 u_2) := (E × F) × E × F
⊢ (fun x ↦ (h.deriv (x.1 - x.2)) (x.2.1, x.1.2)) =o[𝓝 0] fun x ↦ x.1 - x.2

Tactic:
refine h.toContinuousLinearMap.deriv₂.isBoundedBilinearMap.isBigO_comp.trans_isLittleO ?_

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
b : E × F → G
h : IsBoundedBilinearMap 𝕜 b
p : E × F
T : Type (max u_3 u_2) := (E × F) × E × F
⊢ (fun x ↦ ‖x.1 - x.2‖ * ‖(x.2.1, x.1.2)‖) =o[𝓝 0] fun x ↦ x.1 - x.2
Module: Mathlib.Analysis.Calculus.FDeriv.Pow

Goal state before:
pp: 𝕜 : Type u_1
𝔸 : Type u_2
E : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedRing 𝔸
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAlgebra 𝕜 𝔸
inst✝ : NormedSpace 𝕜 E
f : E → 𝔸
f' : E →L[𝕜] 𝔸
x : E
n : ℕ
⊢ ∑ x_1 ∈ Finset.range (n + 1), (f x * f x ^ (n - x_1)) •> f' <• f x ^ x_1 + f' <• f x ^ (n + 1) =
    ∑ x_1 ∈ Finset.range (n + 1), f x ^ (n + 1 - x_1) •> f' <• f x ^ x_1 + f' <• f x ^ (n + 1)

Tactic:
← pow_succ'

Goal state after:
𝕜 : Type u_1
𝔸 : Type u_2
E : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedRing 𝔸
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAlgebra 𝕜 𝔸
inst✝ : NormedSpace 𝕜 E
f : E → 𝔸
f' : E →L[𝕜] 𝔸
x : E
n : ℕ
⊢ ∑ x_1 ∈ Finset.range (n + 1), f x ^ (n - x_1 + 1) •> f' <• f x ^ x_1 + f' <• f x ^ (n + 1) =
    ∑ x_1 ∈ Finset.range (n + 1), f x ^ (n + 1 - x_1) •> f' <• f x ^ x_1 + f' <• f x ^ (n + 1)
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝔸 : Type u_3
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸
c : 𝕜 → 𝔸
c' : 𝔸
hc : HasDerivWithinAt c c' univ x
d : 𝔸
⊢ HasDerivWithinAt (fun y ↦ c y * d) (c' * d) univ x

Tactic:
exact hc.mul_const d

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

Goal state before:
pp: case neg
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝕜' : Type u_2
inst✝¹ : NormedDivisionRing 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
u : 𝕜 → 𝕜'
v : 𝕜'
hu : ¬DifferentiableAt 𝕜 u x
⊢ deriv (fun y ↦ u y * v) x = deriv u x * v

Tactic:
rw [deriv_zero_of_not_differentiableAt hu, zero_mul]

Goal state after:
case neg
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝕜' : Type u_2
inst✝¹ : NormedDivisionRing 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
u : 𝕜 → 𝕜'
v : 𝕜'
hu : ¬DifferentiableAt 𝕜 u x
⊢ deriv (fun y ↦ u y * v) x = 0
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: case inl.refine_2
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
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
h : HasFiniteFPowerSeriesOnBall f p x 0 r
n : ℕ
a : Fin n → E
⊢ (continuousMultilinearCurryFin1 𝕜 E F) (0 a) = 0

Tactic:
exact map_zero _

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

Goal state before:
pp: k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a b r : k
hab : a ≠ b
⊢ r * a - r * b = a - (a - r * a) - r * b

Tactic:
sub_sub_cancel

Goal state after:
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a b r : k
hab : a ≠ b
⊢ r * a - r * b = r * a - r * b
Module: Mathlib.Analysis.Calculus.Deriv.Slope

Goal state before:
pp: 𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
inst✝ : OrderTopology 𝕜
g : 𝕜 → 𝕜
g' : 𝕜
hx : AccPt x (𝓟 s)
hd : HasDerivWithinAt g g' s x
hg : MonotoneOn g s
this✝ : (𝓝[s \ {x}] x).NeBot
h'g : MonotoneOn g (insert x s)
this : Tendsto (slope g x) (𝓝[s \ {x}] x) (𝓝 g')
y : 𝕜
hy : y ∈ s ∧ ¬y = x
⊢ y ∈ insert x s

Tactic:
simp [hy]

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

Goal state before:
pp: 𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
x : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
inst✝ : OrderTopology 𝕜
g : 𝕜 → 𝕜
g' : 𝕜
hd : HasDerivWithinAt g g' univ x
hg : Monotone g
⊢ AccPt x (𝓟 univ)

Tactic:
exact PerfectSpace.univ_preperfect _ (mem_univ _)

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

Goal state before:
pp: 𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
x : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
inst✝ : OrderTopology 𝕜
g : 𝕜 → 𝕜
g' : 𝕜
hd : HasDerivWithinAt g g' univ x
hg : Antitone g
⊢ AccPt x (𝓟 univ)

Tactic:
exact PerfectSpace.univ_preperfect _ (mem_univ _)

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: α : Type u_1
ι : Type u_2
f : ι → α → ℂ
p : Filter ι
g : α → ℂ
hf : TendstoUniformly f g p
⊢ TendstoUniformly (fun n x ↦ (f n x).im) (fun y ↦ (g y).im) p

Tactic:
by
  apply UniformContinuous.comp_tendstoUniformly uniformContinuous_im hf

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

Goal state before:
pp: 𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
x : 𝕜
𝕜' : Type u_1
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
h : 𝕜 → 𝕜'
h' : 𝕜'
g₁ : 𝕜' → F
g₁' : F
y : 𝕜'
L : Filter 𝕜
L' : Filter 𝕜'
hg : HasDerivAtFilter g₁ g₁' (L' ×ˢ pure y)
hh : HasDerivAtFilter h h' (L ×ˢ pure x)
hy : y = h x
hL : Tendsto h L L'
⊢ Tendsto h (pure x) (pure y)

Tactic:
simp [hy]

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

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type w
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
h₂ : 𝕜' → 𝕜'
h₂' y : 𝕜'
f : E → 𝕜'
f' : E →L[𝕜] 𝕜'
x : E
L' : Filter 𝕜'
L'' : Filter E
hh₂ : HasDerivAtFilter h₂ h₂' (L' ×ˢ pure y)
hf : HasFDerivAtFilter f f' (L'' ×ˢ pure x)
hL : Tendsto f L'' L'
hy : y = f x
⊢ Tendsto f (pure x) (pure y)

Tactic:
simp [hy]

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

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
h : 𝕜 → 𝕜'
h₂ : 𝕜' → 𝕜'
h' h₂' y : 𝕜'
L : Filter 𝕜
L' : Filter 𝕜'
hh₂ : HasDerivAtFilter h₂ h₂' (L' ×ˢ pure y)
hh : HasDerivAtFilter h h' (L ×ˢ pure x)
hL : Tendsto h L L'
hy : y = h x
⊢ Tendsto h (pure x) (pure y)

Tactic:
simp [hy]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f g : 𝕜 → E
z₀ : 𝕜
U : Set 𝕜
hf : AnalyticOnNhd 𝕜 f U
hg : AnalyticOnNhd 𝕜 g U
hU : IsPreconnected U
h₀ : z₀ ∈ U
hfg : ∃ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = g z
⊢ EqOn f g U

Tactic:
have hfg' : ∃ᶠ z in 𝓝[≠] z₀, (f - g) z = 0 :=
    hfg.mono fun z h => by rw [Pi.sub_apply, h, sub_self]
  simpa [sub_eq_zero] using! fun z hz =>
    (hf.sub hg).eqOn_zero_of_preconnected_of_frequently_eq_zero hU h₀ hfg' hz

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
n : ℕ∞ω
inst✝ : Subsingleton F
⊢ ContDiff 𝕜 n f

Tactic:
rw [Subsingleton.elim f fun _ => 0]

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
n : ℕ∞ω
inst✝ : Subsingleton F
⊢ ContDiff 𝕜 n fun x ↦ 0
Module: Mathlib.Analysis.Calculus.ContDiff.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
x : E
n : ℕ∞ω
inst✝ : Subsingleton F
⊢ ContDiffAt 𝕜 n f x

Tactic:
rw [Subsingleton.elim f fun _ => 0]

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
x : E
n : ℕ∞ω
inst✝ : Subsingleton F
⊢ ContDiffAt 𝕜 n (fun x ↦ 0) x
Module: Mathlib.Analysis.Calculus.ContDiff.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : ℕ∞ω
inst✝ : Subsingleton F
⊢ ContDiffWithinAt 𝕜 n f s x

Tactic:
rw [Subsingleton.elim f fun _ => 0]

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : ℕ∞ω
inst✝ : Subsingleton F
⊢ ContDiffWithinAt 𝕜 n (fun x ↦ 0) s x
Module: Mathlib.Analysis.Calculus.ContDiff.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
s : Set E
f : E → F
n : ℕ∞ω
inst✝ : Subsingleton F
⊢ ContDiffOn 𝕜 n f s

Tactic:
rw [Subsingleton.elim f fun _ => 0]

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
s : Set E
f : E → F
n : ℕ∞ω
inst✝ : Subsingleton F
⊢ ContDiffOn 𝕜 n (fun x ↦ 0) 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
x : E
n : ℕ∞ω
hn : n ≠ ∞
h : ContDiffWithinAt 𝕜 n f s x
⊢ n = ∞ → n = ω

Tactic:
by simp [hn]

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
h : ContDiffWithinAt 𝕜 n f s x
hn : n ≠ ∞
⊢ n = ∞ → n = ω

Tactic:
by simp [hn]

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
hn : n ≠ ∞
h : ContDiffOn 𝕜 (n + 1) f s
x : E
hx : x ∈ s
u : Set E
hu : u ∈ 𝓝[insert x s] x
f_an : n = ω → AnalyticOn 𝕜 f u
f' : E → E →L[𝕜] F
hf' : ∀ x ∈ u, HasFDerivWithinAt f (f' x) u x
Hf' : ContDiffWithinAt 𝕜 n f' u x
⊢ n = ∞ → n = ω

Tactic:
by simp [hn]

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

Goal state before:
pp: case fderivWithin
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 : ContDiffOn 𝕜 n f s
hs : UniqueDiffOn 𝕜 s
m : ℕ
hm : ↑m < n
x : E
hx : x ∈ s
this : ↑(m + 1) ≤ n
u : Set E
hu : u ∈ 𝓝[s] x
p : E → FormalMultilinearSeries 𝕜 E F
Hp : HasFTaylorSeriesUpToOn (↑(m + 1)) f p u
o : Set E
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
xo : x ∈ o
ho : o ∩ s ⊆ u
⊢ HasFDerivWithinAt (fun x ↦ ftaylorSeriesWithin 𝕜 f s x m) (ftaylorSeriesWithin 𝕜 f s x m.succ).curryLeft s x

Tactic:
rw [inter_comm] at ho

Goal state after:
case fderivWithin
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 : ContDiffOn 𝕜 n f s
hs : UniqueDiffOn 𝕜 s
m : ℕ
hm : ↑m < n
x : E
hx : x ∈ s
this : ↑(m + 1) ≤ n
u : Set E
hu : u ∈ 𝓝[s] x
p : E → FormalMultilinearSeries 𝕜 E F
Hp : HasFTaylorSeriesUpToOn (↑(m + 1)) f p u
o : Set E
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
xo : x ∈ o
ho : s ∩ o ⊆ u
⊢ HasFDerivWithinAt (fun x ↦ ftaylorSeriesWithin 𝕜 f s x m) (ftaylorSeriesWithin 𝕜 f s x m.succ).curryLeft s x
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: case cont
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 : ContDiffOn 𝕜 n f s
hs : UniqueDiffOn 𝕜 s
m : ℕ
hm : ↑m ≤ n
x : E
hx : x ∈ s
u : Set E
hu : u ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
Hp : HasFTaylorSeriesUpToOn (↑m) f p u
o : Set E
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
xo : x ∈ o
ho : o ∩ s ⊆ u
⊢ ∃ t,
    IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t ∧
      x ∈ t ∧ ContinuousOn (fun x ↦ ftaylorSeriesWithin 𝕜 f s x m) (s ∩ t)

Tactic:
rw [inter_comm] at ho

Goal state after:
case cont
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 : ContDiffOn 𝕜 n f s
hs : UniqueDiffOn 𝕜 s
m : ℕ
hm : ↑m ≤ n
x : E
hx : x ∈ s
u : Set E
hu : u ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
Hp : HasFTaylorSeriesUpToOn (↑m) f p u
o : Set E
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
xo : x ∈ o
ho : s ∩ o ⊆ u
⊢ ∃ t,
    IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t ∧
      x ∈ t ∧ ContinuousOn (fun x ↦ ftaylorSeriesWithin 𝕜 f s x m) (s ∩ t)
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
i : Fin c.length
a b : Fin (update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) i)
hab : a < b
⊢ (fun j ↦ if h : i = c.index 0 then (c.emb i (Fin.cast ⋯ j.succ)).pred ⋯ else (c.emb i (Fin.cast ⋯ j)).pred ⋯) a <
    (fun j ↦ if h : i = c.index 0 then (c.emb i (Fin.cast ⋯ j.succ)).pred ⋯ else (c.emb i (Fin.cast ⋯ j)).pred ⋯) b

Tactic:
rcases eq_or_ne i (c.index 0) with rfl | hi
    · simp only [↓reduceDIte, Nat.succ_eq_add_one, pred_lt_pred_iff]
      exact (c.emb_strictMono _).comp (cast_strictMono _) (by simpa using hab)
    · simp only [hi, ↓reduceDIte, pred_lt_pred_iff, Nat.succ_eq_add_one]
      exact (c.emb_strictMono _).comp (cast_strictMono _) hab

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

Goal state before:
pp: case e'_4.inl
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
i : Fin c.length
hij : i < c.index 0
⊢ ↑(if h : c.index 0 = c.index 0 then
          (c.emb (c.index 0)
                (Fin.cast ⋯ ⟨update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) (c.index 0) - 1, ⋯⟩.succ)).pred
            ⋯
        else
          (c.emb (c.index 0)
                (Fin.cast ⋯ ⟨update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) (c.index 0) - 1, ⋯⟩)).pred
            ⋯) +
      1 =
    ↑((fun m ↦ c.emb m ⟨c.partSize m - 1, ⋯⟩) (c.index 0))

Tactic:
simp only [↓reduceDIte, update_self, succ_mk, cast_mk, val_pred]
        have A := c.one_lt_partSize_index_zero hc
        rw [Nat.sub_add_cancel]
        · congr; lia
        · rw [Order.one_le_iff_pos]
          conv_lhs => rw [show (0 : ℕ) = c.emb (c.index 0) 0 by simp [emb_zero]]
          rw [← lt_def]
          apply c.emb_strictMono
          simp [lt_def]

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

Goal state before:
pp: case mpr
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
n : ℕ∞
⊢ HasFTaylorSeriesUpTo (↑n) f (ftaylorSeries 𝕜 f) → ContDiff 𝕜 (↑n) f

Tactic:
exact fun h ↦ ⟨ftaylorSeries 𝕜 f, h⟩

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ExpDeriv

Goal state before:
pp: n : ℕ
⊢ deriv^[n + 1] cexp = cexp

Tactic:
rw [iterate_succ_apply, deriv_exp, iter_deriv_exp n]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ExpDeriv

Goal state before:
pp: n : ℕ
⊢ deriv^[n + 1] rexp = rexp

Tactic:
rw [iterate_succ_apply, deriv_exp, iter_deriv_exp n]

Goal state after:
Module: Mathlib.Topology.ExtendFrom

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : RegularSpace Y
f : X → Y
A B : Set X
hB : B ⊆ closure[inst✝²] A
hf : ∀ x ∈ B, ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)
⊢ ContinuousOn (extendFrom A f) B

Tactic:
set φ := extendFrom A f

Goal state after:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : RegularSpace Y
f : X → Y
A B : Set X
hB : B ⊆ closure[inst✝²] A
hf : ∀ x ∈ B, ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)
φ : X → Y := extendFrom A f
⊢ ContinuousOn φ B
Module: Mathlib.Topology.ExtendFrom

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : RegularSpace Y
f : X → Y
A B : Set X
hB : B ⊆ closure[inst✝²] A
hf : ∀ x ∈ B, ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)
φ : X → Y := extendFrom A f
x : X
x_in : x ∈ B
V' : Set Y
V'_in : V' ∈ 𝓝 (φ x)
V'_closed : IsClosed[inst✝¹] V'
V : Set X
V_in : V ∈ 𝓝 x
V_op : IsOpen[inst✝²] V
hV : V ∩ A ⊆ f ⁻¹' V'
y : X
hyV : y ∈ V
hyB : y ∈ B
this : (𝓝[A] y).NeBot
limy : Tendsto f (𝓝[A] y) (𝓝 (φ y))
hVy : V ∈ 𝓝 y
⊢ φ y ∈ V'

Tactic:
have : V ∩ A ∈ 𝓝[A] y := by simpa only [inter_comm] using inter_mem_nhdsWithin A hVy

Goal state after:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : RegularSpace Y
f : X → Y
A B : Set X
hB : B ⊆ closure[inst✝²] A
hf : ∀ x ∈ B, ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)
φ : X → Y := extendFrom A f
x : X
x_in : x ∈ B
V' : Set Y
V'_in : V' ∈ 𝓝 (φ x)
V'_closed : IsClosed[inst✝¹] V'
V : Set X
V_in : V ∈ 𝓝 x
V_op : IsOpen[inst✝²] V
hV : V ∩ A ⊆ f ⁻¹' V'
y : X
hyV : y ∈ V
hyB : y ∈ B
this✝ : (𝓝[A] y).NeBot
limy : Tendsto f (𝓝[A] y) (𝓝 (φ y))
hVy : V ∈ 𝓝 y
this : V ∩ A ∈ 𝓝[A] y
⊢ φ y ∈ V'
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F : Type uF
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n✝ : ℕ∞ω
𝕜' : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜'
inst✝⁴ : NormedAlgebra 𝕜 𝕜'
inst✝³ : NormedSpace 𝕜' E
inst✝² : IsScalarTower 𝕜 𝕜' E
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
n : ℕ∞
h : ContDiffWithinAt 𝕜' (↑n) f s x
m : ℕ
hm : ↑m ≤ n
u : Set E
u_mem : u ∈ 𝓝[insert x s] x
p' : E → FormalMultilinearSeries 𝕜' E F
hp' : HasFTaylorSeriesUpToOn (↑m) f p' u
⊢ ∃ u ∈ 𝓝[insert x s] x, ∃ p, HasFTaylorSeriesUpToOn (↑m) f p u

Tactic:
exact ⟨u, u_mem, _, hp'.restrictScalars _⟩

Goal state after:
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✝ : Module R M
⊢ Pointed 0

Tactic:
by rw [Pointed, mem_zero]

Goal state after: