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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : WithTop ℕ∞
x : E
t : ℝ
ht : t ≠ 0
h : ContDiffAt ℝ n (fun x ↦ ‖x‖) x
h1 : ContDiffAt ℝ n (fun y ↦ t⁻¹ • y) (t • x)
h2 : ContDiffAt ℝ n (fun y ↦ |t| * ‖y‖) (t⁻¹ • t • x)
y : E
⊢ ‖y‖ = |t| * ‖t⁻¹ • y‖

Tactic:
norm_smul,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : WithTop ℕ∞
x : E
t : ℝ
ht : t ≠ 0
h : ContDiffAt ℝ n (fun x ↦ ‖x‖) x
h1 : ContDiffAt ℝ n (fun y ↦ t⁻¹ • y) (t • x)
h2 : ContDiffAt ℝ n (fun y ↦ |t| * ‖y‖) (t⁻¹ • t • x)
y : E
⊢ ‖y‖ = |t| * (‖t⁻¹‖ * ‖y‖)
Module: Mathlib.Analysis.Calculus.FDeriv.Norm

Goal state before:
pp: case e'_11
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : StrongDual ℝ E
x : E
t : ℝ
ht : t ≠ 0
h : HasStrictFDerivAt (fun x ↦ ‖x‖) f x
h1 : HasStrictFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)
h2 : HasStrictFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) (t⁻¹ • t • x)
x✝ : E
⊢ ‖x✝‖ = |t| * ‖t⁻¹ • x✝‖

Tactic:
norm_smul,

Goal state after:
case e'_11
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : StrongDual ℝ E
x : E
t : ℝ
ht : t ≠ 0
h : HasStrictFDerivAt (fun x ↦ ‖x‖) f x
h1 : HasStrictFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)
h2 : HasStrictFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) (t⁻¹ • t • x)
x✝ : E
⊢ ‖x✝‖ = |t| * (‖t⁻¹‖ * ‖x✝‖)
Module: Mathlib.Analysis.Calculus.FDeriv.Norm

Goal state before:
pp: case e'_11
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : StrongDual ℝ E
x : E
t : ℝ
ht : t ≠ 0
h : HasFDerivAt (fun x ↦ ‖x‖) f x
h1 : HasFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)
h2 : HasFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) (t⁻¹ • t • x)
x✝ : E
⊢ ‖x✝‖ = |t| * ‖t⁻¹ • x✝‖

Tactic:
norm_smul,

Goal state after:
case e'_11
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : StrongDual ℝ E
x : E
t : ℝ
ht : t ≠ 0
h : HasFDerivAt (fun x ↦ ‖x‖) f x
h1 : HasFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)
h2 : HasFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) (t⁻¹ • t • x)
x✝ : E
⊢ ‖x✝‖ = |t| * (‖t⁻¹‖ * ‖x✝‖)
Module: Mathlib.Analysis.Calculus.ImplicitFunction.ProdDomain

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E₁ : Type u_2
inst✝⁸ : NormedAddCommGroup E₁
inst✝⁷ : NormedSpace 𝕜 E₁
inst✝⁶ : CompleteSpace E₁
E₂ : Type u_3
inst✝⁵ : NormedAddCommGroup E₂
inst✝⁴ : NormedSpace 𝕜 E₂
inst✝³ : CompleteSpace E₂
F : Type u_4
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
u : E₁ × E₂
f : E₁ × E₂ → F
f'u : E₁ × E₂ →L[𝕜] F
dfu : HasStrictFDerivAt f f'u u
if₂u : (f'u ∘SL ContinuousLinearMap.inr 𝕜 E₁ E₂).IsInvertible
this : Tendsto (dfu.implicitFunctionOfProdDomain if₂u) (𝓝 u.1) (𝓝 (dfu.implicitFunctionOfProdDomain if₂u u.1))
⊢ Tendsto (dfu.implicitFunctionOfProdDomain if₂u) (𝓝 u.1) (𝓝 u.2)

Tactic:
rwa [(dfu.eventually_apply_eq_iff_implicitFunctionOfProdDomain if₂u).self_of_nhds.mp rfl] at this

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

Goal state before:
pp: a b : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hab : a < b
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioo a b, g' x ≠ 0
hfa : Tendsto f (𝓝[>] a) (𝓝 0)
hga : Tendsto g (𝓝[>] a) (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[>] a) l
sub : ∀ x ∈ Ioo a b, Ioo a x ⊆ Ioo a b
hg : ∀ x ∈ Ioo a b, g x ≠ 0
c : ℝ → ℝ
hc : ∀ x ∈ Ioo a b, c x ∈ Ioo a x ∧ f x * g' (c x) = g x * f' (c x)
this : ∀ x ∈ Ioo a b, ((fun x' ↦ f' x' / g' x') ∘ c) x = f x / g x
cmp : ∀ x ∈ Ioo a b, a < c x ∧ c x < x
⊢ Tendsto ((fun x' ↦ f' x' / g' x') ∘ c) (𝓝[Ioo a b] a) l

Tactic:
apply hdiv.comp

Goal state after:
a b : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hab : a < b
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioo a b, g' x ≠ 0
hfa : Tendsto f (𝓝[>] a) (𝓝 0)
hga : Tendsto g (𝓝[>] a) (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[>] a) l
sub : ∀ x ∈ Ioo a b, Ioo a x ⊆ Ioo a b
hg : ∀ x ∈ Ioo a b, g x ≠ 0
c : ℝ → ℝ
hc : ∀ x ∈ Ioo a b, c x ∈ Ioo a x ∧ f x * g' (c x) = g x * f' (c x)
this : ∀ x ∈ Ioo a b, ((fun x' ↦ f' x' / g' x') ∘ c) x = f x / g x
cmp : ∀ x ∈ Ioo a b, a < c x ∧ c x < x
⊢ Tendsto c (𝓝[Ioo a b] a) (𝓝[>] a)
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
m : ℝ≥0
hm : ↑m < ENNReal.ofReal |A.det|
mpos : 0 < m
hA : A.det ≠ 0
B : E ≃L[ℝ] E := A.toContinuousLinearEquivOfDetNeZero hA
I : ENNReal.ofReal |(↑B.symm).det| < ↑m⁻¹
this : ∀ᶠ (δ : ℝ≥0) in 𝓝[>] 0, ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g (↑B.symm) t δ → μ (g '' t) ≤ ↑m⁻¹ * μ t
δ₀ : ℝ≥0
h : ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g (↑B.symm) t δ₀ → μ (g '' t) ≤ ↑m⁻¹ * μ t
h' : 0 < δ₀
⊢ ∃ δ, 0 < δ ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g (↑B.symm) t δ → μ (g '' t) ≤ ↑m⁻¹ * μ t

Tactic:
exact ⟨δ₀, h', h⟩

Goal state after:
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: case inr
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
m : ℝ≥0
hm : ↑m < ENNReal.ofReal |A.det|
mpos : 0 < m
hA : A.det ≠ 0
B : E ≃L[ℝ] E := A.toContinuousLinearEquivOfDetNeZero hA
I : ENNReal.ofReal |(↑B.symm).det| < ↑m⁻¹
δ₀ : ℝ≥0
δ₀pos : 0 < δ₀
hδ₀ : ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g (↑B.symm) t δ₀ → μ (g '' t) ≤ ↑m⁻¹ * μ t
L1 : ∀ᶠ (δ : ℝ≥0) in 𝓝 0, Subsingleton E ∨ δ < ‖↑B.symm‖₊⁻¹
H : ‖↑B.symm‖₊ ≠ 0
⊢ Tendsto (fun δ ↦ ‖↑B.symm‖₊ * (‖↑B.symm‖₊⁻¹ - δ)⁻¹ * δ) (𝓝 0) (𝓝 (‖↑B.symm‖₊ * (‖↑B.symm‖₊⁻¹ - 0)⁻¹ * 0))

Tactic:
refine Tendsto.mul (tendsto_const_nhds.mul ?_) tendsto_id

Goal state after:
case inr
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
m : ℝ≥0
hm : ↑m < ENNReal.ofReal |A.det|
mpos : 0 < m
hA : A.det ≠ 0
B : E ≃L[ℝ] E := A.toContinuousLinearEquivOfDetNeZero hA
I : ENNReal.ofReal |(↑B.symm).det| < ↑m⁻¹
δ₀ : ℝ≥0
δ₀pos : 0 < δ₀
hδ₀ : ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g (↑B.symm) t δ₀ → μ (g '' t) ≤ ↑m⁻¹ * μ t
L1 : ∀ᶠ (δ : ℝ≥0) in 𝓝 0, Subsingleton E ∨ δ < ‖↑B.symm‖₊⁻¹
H : ‖↑B.symm‖₊ ≠ 0
⊢ Tendsto (fun δ ↦ (‖↑B.symm‖₊⁻¹ - δ)⁻¹) (𝓝 0) (𝓝 (‖↑B.symm‖₊⁻¹ - 0)⁻¹)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: a b : ℝ
f f' g : ℝ → ℝ
h : ∀ x ∈ [[a, b]], HasDerivAt f (f' x) x
h' : ContinuousOn f' [[a, b]]
hg : ContinuousOn g (f '' [[a, b]])
⊢ ∫ (x : ℝ) in a..b, (g ∘ f) x * f' x = ∫ (x : ℝ) in f a..f b, g x

Tactic:
simpa [mul_comm] using integral_deriv_smul_comp' h h' hg

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: case hf
E : Type u_1
f : ℝ → E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hf : ContDiff ℝ 1 f
h2f : HasCompactSupport f
b : ℝ
this : ∀ x ∈ Ioi b, HasDerivAt f (deriv f x) x
⊢ Tendsto f atTop (𝓝 0)

Tactic:
rw [hasCompactSupport_iff_eventuallyEq, Filter.coclosedCompact_eq_cocompact] at h2f

Goal state after:
case hf
E : Type u_1
f : ℝ → E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hf : ContDiff ℝ 1 f
h2f : f =ᶠ[cocompact ℝ] 0
b : ℝ
this : ∀ x ∈ Ioi b, HasDerivAt f (deriv f x) x
⊢ Tendsto f atTop (𝓝 0)
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: case hf
E : Type u_1
f : ℝ → E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hf : ContDiff ℝ 1 f
h2f : HasCompactSupport f
b : ℝ
this : ∀ x ∈ Iio b, HasDerivAt f (deriv f x) x
⊢ Tendsto f atBot (𝓝 0)

Tactic:
rw [hasCompactSupport_iff_eventuallyEq, Filter.coclosedCompact_eq_cocompact] at h2f

Goal state after:
case hf
E : Type u_1
f : ℝ → E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hf : ContDiff ℝ 1 f
h2f : f =ᶠ[cocompact ℝ] 0
b : ℝ
this : ∀ x ∈ Iio b, HasDerivAt f (deriv f x) x
⊢ Tendsto f atBot (𝓝 0)
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
a : ℝ
hf : ContinuousOn f (Ici a)
hft : Tendsto f atTop atTop
hff' : ∀ x ∈ Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioi a)
hg1 : IntegrableOn g (f '' Ici a) volume
hg2 : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) (Ici a) volume
eq : ∀ (b : ℝ), a < b → ∫ (x : ℝ) in a..b, f' x • (g ∘ f) x = ∫ (u : ℝ) in f a..f b, g u
⊢ ∫ (x : ℝ) in Ioi a, f' x • (g ∘ f) x = ∫ (u : ℝ) in Ioi (f a), g u

Tactic:
integrableOn_Ici_iff_integrableOn_Ioi

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
a : ℝ
hf : ContinuousOn f (Ici a)
hft : Tendsto f atTop atTop
hff' : ∀ x ∈ Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioi a)
hg1 : IntegrableOn g (f '' Ici a) volume
hg2 : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) (Ioi a) volume
eq : ∀ (b : ℝ), a < b → ∫ (x : ℝ) in a..b, f' x • (g ∘ f) x = ∫ (u : ℝ) in f a..f b, g u
⊢ ∫ (x : ℝ) in Ioi a, f' x • (g ∘ f) x = ∫ (u : ℝ) in Ioi (f a), g u
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: f f' g : ℝ → ℝ
a : ℝ
hf : ContinuousOn f (Ici a)
hft : Tendsto f atTop atTop
hff' : ∀ x ∈ Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioi a)
hg1 : IntegrableOn g (f '' Ici a) volume
hg2 : IntegrableOn (fun x ↦ (g ∘ f) x * f' x) (Ici a) volume
hg2' : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) (Ici a) volume
⊢ ∫ (x : ℝ) in Ioi a, (g ∘ f) x * f' x = ∫ (u : ℝ) in Ioi (f a), g u

Tactic:
simpa [mul_comm] using integral_deriv_smul_comp_Ioi hf hft hff' hg_cont hg1 hg2'

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Iio a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Iio a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Iic a) volume
h_zero : Tendsto (u * v) (𝓝[Iic a \ {a}] a) (𝓝 a')
h_infty : Tendsto (u * v) atBot (𝓝 b')
f : ℝ → A := Function.update (u * v) a a'
hderiv : ∀ x ∈ Iio a, HasDerivAt f (u' x * v x + u x * v' x) x
htendsto : Tendsto f atBot (𝓝 b')
⊢ ∫ (x : ℝ) in Iic a, u' x * v x + u x * v' x = a' - b'

Tactic:
simpa using integral_Iic_of_hasDerivAt_of_tendsto
    (continuousWithinAt_update_same.mpr h_zero) hderiv huv htendsto

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

Goal state before:
pp: E : Type u_1
ι : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
U : Set ℂ
φ : Filter ι
F : ι → ℂ → E
f : ℂ → E
inst✝¹ : CompleteSpace E
inst✝ : φ.NeBot
hf : TendstoLocallyUniformlyOn F f φ U
hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U
hU : IsOpen U
x : ℂ
hx : x ∈ U
K : Set ℂ
hKU : K ⊆ U
hKx : K ∈ 𝓝 x
hK : IsCompact K
δ : ℝ
left✝¹ : δ > 0
left✝ : cthickening δ K ⊆ U
h1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K
h2 : interior K ⊆ U
h3 : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) (interior K)
h4 : TendstoLocallyUniformlyOn F f φ (interior K)
h5 : TendstoLocallyUniformlyOn (deriv ∘ F) (cderiv δ f) φ (interior K)
⊢ DifferentiableWithinAt ℂ f U x

Tactic:
have h6 : ∀ x ∈ interior K, HasDerivAt f (cderiv δ f x) x := fun x h =>
    hasDerivAt_of_tendsto_locally_uniformly_on' isOpen_interior h5 h3 (fun _ => h4.tendsto_at) h

Goal state after:
E : Type u_1
ι : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
U : Set ℂ
φ : Filter ι
F : ι → ℂ → E
f : ℂ → E
inst✝¹ : CompleteSpace E
inst✝ : φ.NeBot
hf : TendstoLocallyUniformlyOn F f φ U
hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U
hU : IsOpen U
x : ℂ
hx : x ∈ U
K : Set ℂ
hKU : K ⊆ U
hKx : K ∈ 𝓝 x
hK : IsCompact K
δ : ℝ
left✝¹ : δ > 0
left✝ : cthickening δ K ⊆ U
h1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K
h2 : interior K ⊆ U
h3 : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) (interior K)
h4 : TendstoLocallyUniformlyOn F f φ (interior K)
h5 : TendstoLocallyUniformlyOn (deriv ∘ F) (cderiv δ f) φ (interior K)
h6 : ∀ x ∈ interior K, HasDerivAt f (cderiv δ f x) x
⊢ DifferentiableWithinAt ℂ f U x
Module: Mathlib.Geometry.Manifold.ChartedSpace

Goal state before:
pp: H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝ : TopologicalSpace H
c : ChartedSpaceCore H M
e✝ e : PartialEquiv M H
he : e ∈ c.atlas
⊢ ContinuousOn e.invFun e.target

Tactic:
by
      let : TopologicalSpace M := c.toTopologicalSpace
      apply continuousOn_isOpen_of_generateFrom
      intro t ht
      simp only [exists_prop, mem_iUnion, mem_singleton_iff] at ht
      rcases ht with ⟨e', e'_atlas, s, s_open, ts⟩
      rw [ts]
      let f := e.symm.trans e'
      have : IsOpen (f ⁻¹' s ∩ f.source) := by
        simpa [f, inter_comm] using (continuousOn_open_iff (c.open_source e e' he e'_atlas)).1
          (c.continuousOn_toFun e e' he e'_atlas) s s_open
      have A : e' ∘ e.symm ⁻¹' s ∩ (e.target ∩ e.symm ⁻¹' e'.source) =
          e.target ∩ (e' ∘ e.symm ⁻¹' s ∩ e.symm ⁻¹' e'.source) := by
        rw [← inter_assoc, ← inter_assoc]
        congr 1
        exact inter_comm _ _
      simpa [f, PartialEquiv.trans_source, preimage_inter, preimage_comp.symm, A] using this

Goal state after:
Module: Mathlib.Geometry.Manifold.HasGroupoid

Goal state before:
pp: H : Type u
M : Type u_2
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
U V : Opens M
hUV : U ≤ V
x : ↥U
⊢ ↑(chartAt H (inclusion hUV x)).symm =ᶠ[𝓝 (↑(chartAt H (inclusion hUV x)) (Set.inclusion hUV x))]
    inclusion hUV ∘ ↑(chartAt H x).symm

Tactic:
set e := chartAt H (x : M)
  have heUx_nhds : (e.subtypeRestr ⟨x⟩).target ∈ 𝓝 (e x) := by
    apply (e.subtypeRestr ⟨x⟩).open_target.mem_nhds
    exact e.map_subtype_source ⟨x⟩ (mem_chart_source _ _)
  exact Filter.eventuallyEq_of_mem heUx_nhds <| e.subtypeRestr_symm_eqOn_of_le ⟨x⟩
    ⟨Opens.inclusion hUV x⟩ hUV

Goal state after:
Module: Mathlib.Geometry.Manifold.HasGroupoid

Goal state before:
pp: H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝⁶ : TopologicalSpace H
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : TopologicalSpace M'
inst✝² : TopologicalSpace M''
G : StructureGroupoid H
inst✝¹ : ChartedSpace H M'
inst✝ : ChartedSpace H M''
e : Structomorph G M M'
e' : Structomorph G M' M''
c : OpenPartialHomeomorph M H
c' : OpenPartialHomeomorph M'' H
hc : c ∈ atlas H M
hc' : c' ∈ atlas H M''
x : H
hx : x ∈ (c.symm ≫ₕ (e.trans e'.toHomeomorph).toOpenPartialHomeomorph ≫ₕ c').source
f₁ : OpenPartialHomeomorph M M' := e.toOpenPartialHomeomorph
f₂ : OpenPartialHomeomorph M' M'' := e'.toOpenPartialHomeomorph
f : OpenPartialHomeomorph M M'' := (e.trans e'.toHomeomorph).toOpenPartialHomeomorph
feq : f = f₁ ≫ₕ f₂
y : M' := ↑(c.symm ≫ₕ f₁) x
g : OpenPartialHomeomorph M' H := chartAt H y
hg₁ : chartAt H y ∈ atlas H M'
hg₂ : y ∈ (chartAt H y).source
s : Set H := (c.symm ≫ₕ f₁).source ∩ ↑(c.symm ≫ₕ f₁) ⁻¹' g.source
open_s : IsOpen[inst✝⁶] s
this : x ∈ s
F₁ : OpenPartialHomeomorph H H := (c.symm ≫ₕ f₁ ≫ₕ g) ≫ₕ g.symm ≫ₕ f₂ ≫ₕ c'
A : F₁ ∈ G
F₂ : OpenPartialHomeomorph H H := (c.symm ≫ₕ f ≫ₕ c').restr s
⊢ F₁ ≈ c.symm ≫ₕ f₁ ≫ₕ (g ≫ₕ g.symm) ≫ₕ f₂ ≫ₕ c'

Tactic:
by
            simp only [F₁, trans_assoc, _root_.refl]

Goal state after:
Module: Mathlib.Geometry.Manifold.IsManifold.ExtChartAt

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
⊢ (f.toPartialEquiv ≫ I.toPartialEquiv).source = f.source

Tactic:
PartialEquiv.trans_source,

Goal state after:
𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
⊢ f.source ∩ ↑f.toPartialEquiv ⁻¹' I.source = f.source
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
⊢ I.target = range ↑I

Tactic:
← image_univ,

Goal state after:
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
⊢ I.target = ↑I '' univ
Module: Mathlib.Geometry.Manifold.IsManifold.ExtChartAt

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
s : Set M
y : M
hy : y ∈ f.source
⊢ 𝓝[↑(f.extend I) '' ((f.extend I).source ∩ s)] ↑(f.extend I) y = 𝓝[↑(f.extend I).symm ⁻¹' s ∩ range ↑I] ↑(f.extend I) y

Tactic:
nhdsWithin_inter,

Goal state after:
𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
s : Set M
y : M
hy : y ∈ f.source
⊢ 𝓝[↑(f.extend I) '' ((f.extend I).source ∩ s)] ↑(f.extend I) y =
    𝓝[↑(f.extend I).symm ⁻¹' s] ↑(f.extend I) y ⊓ 𝓝[range ↑I] ↑(f.extend I) y
Module: Mathlib.Geometry.Manifold.IsManifold.ExtChartAt

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
e e' : OpenPartialHomeomorph M H
⊢ ((e.extend I).symm ≫ e'.extend I).source = ↑I '' (e.symm ≫ₕ e').source

Tactic:
PartialEquiv.trans_source,

Goal state after:
𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
e e' : OpenPartialHomeomorph M H
⊢ (e.extend I).symm.source ∩ ↑(e.extend I).symm ⁻¹' (e'.extend I).source = ↑I '' (e.symm ≫ₕ e').source
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
ι : Type v
inst✝³ : Fintype ι
E : ι → Type w
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)
H : ι → Type u'
inst✝ : (i : ι) → TopologicalSpace (H i)
I : (i : ι) → ModelWithCorners 𝕜 (E i) (H i)
⊢ if h : IsRCLikeNormedField 𝕜 then Convex ℝ (range (Pi.map fun i ↦ ↑(I i).toPartialEquiv))
  else range (Pi.map fun i ↦ ↑(I i).toPartialEquiv) = univ

Tactic:
Set.range_piMap

Goal state after:
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
ι : Type v
inst✝³ : Fintype ι
E : ι → Type w
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)
H : ι → Type u'
inst✝ : (i : ι) → TopologicalSpace (H i)
I : (i : ι) → ModelWithCorners 𝕜 (E i) (H i)
⊢ if h : IsRCLikeNormedField 𝕜 then Convex ℝ (univ.pi fun i ↦ range ↑(I i).toPartialEquiv)
  else (univ.pi fun i ↦ range ↑(I i).toPartialEquiv) = univ
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
ι : Type v
inst✝³ : Fintype ι
E : ι → Type w
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)
H : ι → Type u'
inst✝ : (i : ι) → TopologicalSpace (H i)
I : (i : ι) → ModelWithCorners 𝕜 (E i) (H i)
⊢ (interior (range (Pi.map fun i ↦ ↑(I i).toPartialEquiv))).Nonempty

Tactic:
Set.range_piMap

Goal state after:
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
ι : Type v
inst✝³ : Fintype ι
E : ι → Type w
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)
H : ι → Type u'
inst✝ : (i : ι) → TopologicalSpace (H i)
I : (i : ι) → ModelWithCorners 𝕜 (E i) (H i)
⊢ (interior (univ.pi fun i ↦ range ↑(I i).toPartialEquiv)).Nonempty
Module: Mathlib.Geometry.Manifold.ContMDiff.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
E' : Type u_5
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁷ : TopologicalSpace M'
E'' : Type u_8
inst✝⁶ : NormedAddCommGroup E''
inst✝⁵ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝⁴ : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝³ : TopologicalSpace M''
inst✝² : ChartedSpace H M
inst✝¹ : ChartedSpace H' M'
inst✝ : ChartedSpace H'' M''
f : M → M'
s : Set M
n : ℕ∞ω
g : M' → M''
x : M
y : M'
hg : ContMDiffAt I' I'' n g y
hf : ContMDiffWithinAt I I' n f s x
hx : f x = y
⊢ ContMDiffWithinAt I I'' n (g ∘ f) s x

Tactic:
subst hx; exact hg.comp_contMDiffWithinAt x hf

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁸ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁷ : TopologicalSpace M
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H M
inst✝¹ : ChartedSpace H' M'
f : M → M'
n : ℕ∞ω
inst✝ : DiscreteTopology M
⊢ ContMDiff I I' n f

Tactic:
by
  intro x
  -- f is locally constant, and constant functions are smooth.
  apply contMDiff_const (c := f x).contMDiffAt.congr_of_eventuallyEq
  simp [EventuallyEq]

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝² : TopologicalSpace M'
inst✝¹ : ChartedSpace H M
inst✝ : ChartedSpace H' M'
f : M → M'
n : ℕ∞ω
ι : Type u_11
s : ι → Set M
hf : ∀ (i : ι), ContMDiffOn I I' n f (s i)
hs : ∀ (i : ι), IsOpen[inst✝⁶] (s i)
⊢ ContMDiffOn I I' n f (⋃ i, s i)

Tactic:
rintro x ⟨si, ⟨i, rfl⟩, hxsi⟩
  exact (hf i).contMDiffAt ((hs i).mem_nhds hxsi) |>.contMDiffWithinAt

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.Monoid

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
n : ℕ∞ω
H : Type u_2
inst✝⁶ : TopologicalSpace H
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝³ : Monoid G
inst✝² : TopologicalSpace G
inst✝¹ : ChartedSpace H G
inst✝ : ContMDiffMul I n G
⊢ ContMDiff I I n fun a ↦ a ^ 0

Tactic:
simp only [pow_zero, contMDiff_const]

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.Structures

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
n : ℕ∞ω
x : 𝕜 × 𝕜
y : 𝕜
⊢ ContDiffOn 𝕜 n (fun p ↦ p.1 * p.2) Set.univ

Tactic:
rw [contDiffOn_univ]

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
n : ℕ∞ω
x : 𝕜 × 𝕜
y : 𝕜
⊢ ContDiff 𝕜 n fun p ↦ p.1 * p.2