mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Manifold.MFDeriv.Tangent

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : IsManifold I 1 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✝ : IsManifold I' 1 M'
N : Type u_8
f : N → M
g : N → M'
ϕ : (x : N) → TangentSpace I (f x) →L[𝕜] TangentSpace I' (g x)
x₀ x : N
hx : f x ∈ (chartAt H (f x₀)).source
hy : g x ∈ (chartAt H' (g x₀)).source
⊢ inTangentCoordinates I I' f g ϕ x₀ x =
    mfderiv% ↑(extChartAt I' (g x₀)) (g x) ∘SL
      ϕ x ∘SL mfderiv[range ↑I] ↑(extChartAt I (f x₀)).symm (↑(extChartAt I (f x₀)) (f x))

Tactic:
by
  rw [inTangentCoordinates_eq _ _ _ hx hy, tangentBundleCore_coordChange]
  congr
  · have : MDiffAt (extChartAt I' (g x₀)) (g x) := mdifferentiableAt_extChartAt hy
    simp_all [mfderiv]
  · simp only [mfderivWithin, writtenInExtChartAt, modelWithCornersSelf_coe, range_id, inter_univ]
    rw [if_pos]
    · simp [Function.comp_def, OpenPartialHomeomorph.left_inv (chartAt H (f x₀)) hx]
    · apply mdifferentiableWithinAt_extChartAt_symm
      apply (extChartAt I (f x₀)).map_source
      simpa using hx

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable

Goal state before:
pp: case refine_1
𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : FiberBundle F E
inst✝⁶ : (x : B) → AddCommGroup (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
inst✝⁴ : VectorBundle 𝕜 F E
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : ChartedSpace HB B
EB : Type u_8
inst✝¹ : NormedAddCommGroup EB
inst✝ : NormedSpace 𝕜 EB
I : ModelWithCorners 𝕜 EB HB
s : (x : B) → E x
u : Set B
x₀ : B
e : Trivialization F TotalSpace.proj := trivializationAt F E x₀
hs : (MDiffAt[u] fun b ↦ (↑e ⟨b, s b⟩).2) x₀
⊢ (fun b ↦ (↑e ⟨b, (-s) b⟩).2) =ᶠ[𝓝[u] x₀] -fun b ↦ (↑e ⟨b, s b⟩).2

pp: case refine_2
𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : FiberBundle F E
inst✝⁶ : (x : B) → AddCommGroup (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
inst✝⁴ : VectorBundle 𝕜 F E
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : ChartedSpace HB B
EB : Type u_8
inst✝¹ : NormedAddCommGroup EB
inst✝ : NormedSpace 𝕜 EB
I : ModelWithCorners 𝕜 EB HB
s : (x : B) → E x
u : Set B
x₀ : B
e : Trivialization F TotalSpace.proj := trivializationAt F E x₀
hs : (MDiffAt[u] fun b ↦ (↑e ⟨b, s b⟩).2) x₀
⊢ (↑e ⟨x₀, (-s) x₀⟩).2 = (-fun b ↦ (↑e ⟨b, s b⟩).2) x₀

Tactic:
· apply eventually_of_mem (U := e.baseSet)
    · exact mem_nhdsWithin_of_mem_nhds <|
        (e.open_baseSet.mem_nhds <| mem_baseSet_trivializationAt F E x₀)
    · exact fun x hx ↦ (e.linear 𝕜 hx).map_neg ..

Goal state after:
case refine_2
𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : FiberBundle F E
inst✝⁶ : (x : B) → AddCommGroup (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
inst✝⁴ : VectorBundle 𝕜 F E
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : ChartedSpace HB B
EB : Type u_8
inst✝¹ : NormedAddCommGroup EB
inst✝ : NormedSpace 𝕜 EB
I : ModelWithCorners 𝕜 EB HB
s : (x : B) → E x
u : Set B
x₀ : B
e : Trivialization F TotalSpace.proj := trivializationAt F E x₀
hs : (MDiffAt[u] fun b ↦ (↑e ⟨b, s b⟩).2) x₀
⊢ (↑e ⟨x₀, (-s) x₀⟩).2 = (-fun b ↦ (↑e ⟨b, s b⟩).2) x₀
Module: Mathlib.Topology.VectorBundle.Hom

Goal state before:
pp: 𝕜 : Type u_8
F₁ : Type u_9
F₂ : Type u_10
B₁ : Type u_11
B₂ : Type u_12
M : Type u_13
E₁ : B₁ → Type u_14
E₂ : B₂ → Type u_15
inst✝¹⁹ : NontriviallyNormedField 𝕜
inst✝¹⁸ : (x : B₁) → AddCommGroup (E₁ x)
inst✝¹⁷ : (x : B₁) → Module 𝕜 (E₁ x)
inst✝¹⁶ : NormedAddCommGroup F₁
inst✝¹⁵ : NormedSpace 𝕜 F₁
inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝¹³ : (x : B₁) → TopologicalSpace (E₁ x)
inst✝¹² : (x : B₂) → AddCommGroup (E₂ x)
inst✝¹¹ : (x : B₂) → Module 𝕜 (E₂ x)
inst✝¹⁰ : NormedAddCommGroup F₂
inst✝⁹ : NormedSpace 𝕜 F₂
inst✝⁸ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁷ : (x : B₂) → TopologicalSpace (E₂ x)
inst✝⁶ : TopologicalSpace B₁
inst✝⁵ : TopologicalSpace B₂
inst✝⁴ : TopologicalSpace M
inst✝³ : FiberBundle F₁ E₁
inst✝² : VectorBundle 𝕜 F₁ E₁
inst✝¹ : FiberBundle F₂ E₂
inst✝ : VectorBundle 𝕜 F₂ E₂
b₁ : M → B₁
b₂ : M → B₂
m₀ : M
ϕ : (m : M) → E₁ (b₁ m) →L[𝕜] E₂ (b₂ m)
v : (m : M) → E₁ (b₁ m)
s : Set M
hϕ : ContinuousWithinAt (fun m ↦ inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m)) (insert m₀ s) m₀
hv :
  ContinuousWithinAt (fun x ↦ { proj := b₁ x, snd := v x }.proj) (insert m₀ s) m₀ ∧
    ContinuousWithinAt
      (fun x ↦ (↑(trivializationAt F₁ E₁ { proj := b₁ m₀, snd := v m₀ }.proj) { proj := b₁ x, snd := v x }).2)
      (insert m₀ s) m₀
hb₂ : ContinuousWithinAt b₂ (insert m₀ s) m₀
⊢ ∀ᶠ (m : M) in 𝓝[insert m₀ s] m₀, b₁ m ∈ (trivializationAt F₁ E₁ (b₁ m₀)).baseSet

Tactic:
apply hv.1

Goal state after:
𝕜 : Type u_8
F₁ : Type u_9
F₂ : Type u_10
B₁ : Type u_11
B₂ : Type u_12
M : Type u_13
E₁ : B₁ → Type u_14
E₂ : B₂ → Type u_15
inst✝¹⁹ : NontriviallyNormedField 𝕜
inst✝¹⁸ : (x : B₁) → AddCommGroup (E₁ x)
inst✝¹⁷ : (x : B₁) → Module 𝕜 (E₁ x)
inst✝¹⁶ : NormedAddCommGroup F₁
inst✝¹⁵ : NormedSpace 𝕜 F₁
inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝¹³ : (x : B₁) → TopologicalSpace (E₁ x)
inst✝¹² : (x : B₂) → AddCommGroup (E₂ x)
inst✝¹¹ : (x : B₂) → Module 𝕜 (E₂ x)
inst✝¹⁰ : NormedAddCommGroup F₂
inst✝⁹ : NormedSpace 𝕜 F₂
inst✝⁸ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁷ : (x : B₂) → TopologicalSpace (E₂ x)
inst✝⁶ : TopologicalSpace B₁
inst✝⁵ : TopologicalSpace B₂
inst✝⁴ : TopologicalSpace M
inst✝³ : FiberBundle F₁ E₁
inst✝² : VectorBundle 𝕜 F₁ E₁
inst✝¹ : FiberBundle F₂ E₂
inst✝ : VectorBundle 𝕜 F₂ E₂
b₁ : M → B₁
b₂ : M → B₂
m₀ : M
ϕ : (m : M) → E₁ (b₁ m) →L[𝕜] E₂ (b₂ m)
v : (m : M) → E₁ (b₁ m)
s : Set M
hϕ : ContinuousWithinAt (fun m ↦ inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m)) (insert m₀ s) m₀
hv :
  ContinuousWithinAt (fun x ↦ { proj := b₁ x, snd := v x }.proj) (insert m₀ s) m₀ ∧
    ContinuousWithinAt
      (fun x ↦ (↑(trivializationAt F₁ E₁ { proj := b₁ m₀, snd := v m₀ }.proj) { proj := b₁ x, snd := v x }).2)
      (insert m₀ s) m₀
hb₂ : ContinuousWithinAt b₂ (insert m₀ s) m₀
⊢ (trivializationAt F₁ E₁ (b₁ m₀)).baseSet ∈ 𝓝 ((fun x ↦ { proj := b₁ x, snd := v x }.proj) m₀)
Module: Mathlib.Geometry.Manifold.ContMDiffMFDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
m n : WithTop ℕ∞
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
inst✝¹⁰ : ChartedSpace H M
E' : Type u_5
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁷ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : ChartedSpace H' M'
F : Type u_8
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_9
inst✝² : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_10
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
Js : IsManifold J 1 N
Is : IsManifold I 1 M
I's : IsManifold I' 1 M'
x₀ : N
f : N → M → M'
g : N → M
t : Set N
u : Set M
hf : ContMDiffWithinAt (J.prod I) I' n (uncurry f) (t ×ˢ u) (x₀, g x₀)
hg : ContMDiffWithinAt J I m g t x₀
hx₀ : x₀ ∈ t
hu : MapsTo g t u
hmn : m + 1 ≤ n
h'u : UniqueMDiff[u]
t' : Set N := t ∩ g ⁻¹' (extChartAt I (g x₀)).source
ht't : t' ⊆ t
this :
  ContMDiffWithinAt J 𝓘(𝕜, E →L[𝕜] E') m
    (inTangentCoordinates I I' g (fun x ↦ f x (g x)) (fun x ↦ mfderiv[u] (f x) (g x)) x₀) t' x₀
⊢ ContMDiffWithinAt J 𝓘(𝕜, E →L[𝕜] E') m
    (inTangentCoordinates I I' g (fun x ↦ f x (g x)) (fun x ↦ mfderiv[u] (f x) (g x)) x₀) t x₀

Tactic:
apply ContMDiffWithinAt.mono_of_mem_nhdsWithin this
    apply inter_mem self_mem_nhdsWithin
    exact hg.continuousWithinAt.preimage_mem_nhdsWithin (extChartAt_source_mem_nhds (g x₀))

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: case e_6.hV
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁴ : TopologicalSpace H
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
s : Set M
x : M
V W V₁ W₁ : (x : M) → TangentSpace I x
hV : V₁ =ᶠ[𝓝[s] x] V
hxV : V₁ x = V x
hW : W₁ =ᶠ[𝓝[s] x] W
hxW : W₁ x = W x
I1 : NormedAddCommGroup (TangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x)) := ⋯
_I2 : NormedSpace 𝕜 (TangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x)) := inst✝²
⊢ {x_1 |
      (fun x_2 ↦
          mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) V₁ (range ↑I) x_2 =
            mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) V (range ↑I) x_2)
        x_1} ∈
    𝓝[↑(extChartAt I x).symm ⁻¹' s] ↑(extChartAt I x) x

Tactic:
filter_upwards [(continuousAt_extChartAt_symm x).continuousWithinAt.preimage_mem_nhdsWithin''
      hV (by simp)] with y hy

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁴ : TopologicalSpace H
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
s : Set M
x : M
V W V₁ W₁ : (x : M) → TangentSpace I x
hV : V₁ =ᶠ[𝓝[s] x] V
hxV : V₁ x = V x
hW : W₁ =ᶠ[𝓝[s] x] W
hxW : W₁ x = W x
I1 : NormedAddCommGroup (TangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x)) :=
  { toNorm := EventuallyEq.mlieBracketWithin_vectorField_eq._aux_1_1,
    toAddCommGroup := instAddCommGroupTangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x),
    toPseudoMetricSpace := EventuallyEq.mlieBracketWithin_vectorField_eq._aux_1_13, eq_of_dist_eq_zero := ⋯,
    dist_eq := ⋯ }
_I2 : NormedSpace 𝕜 (TangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x)) := inst✝²
y : TangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x)
hy : y ∈ ↑(extChartAt I x).symm ⁻¹' {x | V₁ x = V x}
⊢ mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) V₁ (range ↑I) y =
    mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) V (range ↑I) y
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
x✝¹ : EuclideanHalfSpace n
xval : EuclideanSpace ℝ (Fin n)
xprop : 0 ≤ xval.ofLp 0
x✝ : ⟨xval, xprop⟩ ∈ univ
⊢ update (↑⟨xval, xprop⟩).ofLp 0 (max ((↑⟨xval, xprop⟩).ofLp 0) 0) = (WithLp.equiv 2 ((i : Fin n) → (fun x ↦ ℝ) i)) xval

Tactic:
update_eq_iff

Goal state after:
n : ℕ
inst✝ : NeZero n
x✝¹ : EuclideanHalfSpace n
xval : EuclideanSpace ℝ (Fin n)
xprop : 0 ≤ xval.ofLp 0
x✝ : ⟨xval, xprop⟩ ∈ univ
⊢ max ((↑⟨xval, xprop⟩).ofLp 0) 0 = (WithLp.equiv 2 ((i : Fin n) → (fun x ↦ ℝ) i)) xval 0 ∧
    ∀ (x : Fin n), x ≠ 0 → (↑⟨xval, xprop⟩).ofLp x = (WithLp.equiv 2 ((i : Fin n) → (fun x ↦ ℝ) i)) xval x
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
x✝ : EuclideanSpace ℝ (Fin n)
hx : x✝ ∈ {x | 0 ≤ x.ofLp 0}
⊢ update x✝.ofLp 0 (max (x✝.ofLp 0) 0) = (WithLp.equiv 2 ((i : Fin n) → (fun x ↦ ℝ) i)) x✝

Tactic:
update_eq_iff

Goal state after:
n : ℕ
inst✝ : NeZero n
x✝ : EuclideanSpace ℝ (Fin n)
hx : x✝ ∈ {x | 0 ≤ x.ofLp 0}
⊢ max (x✝.ofLp 0) 0 = (WithLp.equiv 2 ((i : Fin n) → (fun x ↦ ℝ) i)) x✝ 0 ∧
    ∀ (x : Fin n), x ≠ 0 → x✝.ofLp x = (WithLp.equiv 2 ((i : Fin n) → (fun x ↦ ℝ) i)) x✝ x
Module: Mathlib.Geometry.Manifold.Instances.Icc

Goal state before:
pp: case neg
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
H : Type u_2
inst✝² : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
x y : ℝ
h : Fact (x < y)
f : ↑(Icc x y) → M
w : ↑(Icc x y)
hw : ¬MDiffAt f w
⊢ ¬MDiffAt[Icc x y] (f ∘ projIcc x y ⋯) ↑w

Tactic:
· rwa [mdifferentiableWithinAt_comp_projIcc_iff]

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Icc

Goal state before:
pp: case pos
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
H : Type u_2
inst✝² : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
x y : ℝ
h : Fact (x < y)
f : ↑(Icc x y) → M
w : ↑(Icc x y)
hw : MDiffAt f w
⊢ (mfderiv[Icc x y] (f ∘ projIcc x y ⋯) ↑w) 1 = (mfderiv% f w) 1

Tactic:
rw [mfderiv_comp_mfderivWithin (I' := 𝓡∂ 1)]

Goal state after:
case pos
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
H : Type u_2
inst✝² : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
x y : ℝ
h : Fact (x < y)
f : ↑(Icc x y) → M
w : ↑(Icc x y)
hw : MDiffAt f w
⊢ (mfderiv% f (projIcc x y ⋯ ↑w) ∘SL mfderiv[Icc x y] (projIcc x y ⋯) ↑w) 1 = (mfderiv% f w) 1

case pos.hg
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
H : Type u_2
inst✝² : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
x y : ℝ
h : Fact (x < y)
f : ↑(Icc x y) → M
w : ↑(Icc x y)
hw : MDiffAt f w
⊢ MDiffAt f (projIcc x y ⋯ ↑w)

case pos.hf
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
H : Type u_2
inst✝² : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
x y : ℝ
h : Fact (x < y)
f : ↑(Icc x y) → M
w : ↑(Icc x y)
hw : MDiffAt f w
⊢ MDiffAt[Icc x y] (projIcc x y ⋯) ↑w

case pos.hxs
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
H : Type u_2
inst✝² : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
x y : ℝ
h : Fact (x < y)
f : ↑(Icc x y) → M
w : ↑(Icc x y)
hw : MDiffAt f w
⊢ UniqueMDiffAt[Icc x y] ↑w
Module: Mathlib.Geometry.Manifold.IntegralCurve.Basic

Goal state before:
pp: case mp
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
H : Type u_2
inst✝² : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
γ : ℝ → M
v : (x : M) → TangentSpace I x
t₀ : ℝ
h : IsMIntegralCurveAt γ v t₀
⊢ ∃ s ∈ 𝓝 t₀, IsMIntegralCurveOn γ v s

Tactic:
rw [IsMIntegralCurveAt, Filter.eventually_iff_exists_mem] at h

Goal state after:
case mp
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
H : Type u_2
inst✝² : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
γ : ℝ → M
v : (x : M) → TangentSpace I x
t₀ : ℝ
h : ∃ v_1 ∈ 𝓝 t₀, ∀ y ∈ v_1, HasMFDerivAt% γ y (ContinuousLinearMap.smulRight 1 (v (γ y)))
⊢ ∃ s ∈ 𝓝 t₀, IsMIntegralCurveOn γ v s
Module: Mathlib.Geometry.Manifold.ContMDiffMFDeriv

Goal state before:
pp: case ht
𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
n : WithTop ℕ∞
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H 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 : IsManifold I 1 M
inst✝ : IsManifold I' 1 M'
a : TangentBundle I M
b : TangentBundle I' M'
w : H
w' : H'
⊢ UniqueDiffWithinAt 𝕜 (range ↑I') (↑I' w')

Tactic:
exact ModelWithCorners.uniqueDiffWithinAt_image I'

Goal state after:
Module: Mathlib.Geometry.Manifold.Riemannian.PathELength

Goal state before:
pp: case inr
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : (x : M) → ENorm (TangentSpace I x)
a b : ℝ
γ : ℝ → M
inst✝ : ∀ (x : M), ENormSMulClass ℝ (TangentSpace I x)
f : ℝ → ℝ
h✝ : a ≤ b
hf : MonotoneOn f (Icc a b)
h'f : DifferentiableOn ℝ f (Icc a b)
hγ : MDiff[Icc (f a) (f b)] γ
h : a < b
f_im : f '' Icc a b = Icc (f a) (f b)
⊢ pathELength I (γ ∘ f) a b = pathELength I γ (f a) (f b)

Tactic:
simp only [pathELength_eq_lintegral_mfderivWithin_Icc, ← f_im]

Goal state after:
case inr
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : (x : M) → ENorm (TangentSpace I x)
a b : ℝ
γ : ℝ → M
inst✝ : ∀ (x : M), ENormSMulClass ℝ (TangentSpace I x)
f : ℝ → ℝ
h✝ : a ≤ b
hf : MonotoneOn f (Icc a b)
h'f : DifferentiableOn ℝ f (Icc a b)
hγ : MDiff[Icc (f a) (f b)] γ
h : a < b
f_im : f '' Icc a b = Icc (f a) (f b)
⊢ ∫⁻ (t : ℝ) in Icc a b, ‖(mfderiv[Icc a b] (γ ∘ f) t) 1‖ₑ =
    ∫⁻ (t : ℝ) in f '' Icc a b, ‖(mfderiv[f '' Icc a b] γ t) 1‖ₑ
Module: Mathlib.Geometry.Manifold.Riemannian.PathELength

Goal state before:
pp: case inr
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : (x : M) → ENorm (TangentSpace I x)
a b : ℝ
γ : ℝ → M
inst✝ : ∀ (x : M), ENormSMulClass ℝ (TangentSpace I x)
f : ℝ → ℝ
h✝ : a ≤ b
hf : AntitoneOn f (Icc a b)
h'f : DifferentiableOn ℝ f (Icc a b)
hγ : MDiff[Icc (f b) (f a)] γ
h : a < b
f_im : f '' Icc a b = Icc (f b) (f a)
⊢ pathELength I (γ ∘ f) a b = pathELength I γ (f b) (f a)

Tactic:
simp only [pathELength_eq_lintegral_mfderivWithin_Icc, ← f_im]

Goal state after:
case inr
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : (x : M) → ENorm (TangentSpace I x)
a b : ℝ
γ : ℝ → M
inst✝ : ∀ (x : M), ENormSMulClass ℝ (TangentSpace I x)
f : ℝ → ℝ
h✝ : a ≤ b
hf : AntitoneOn f (Icc a b)
h'f : DifferentiableOn ℝ f (Icc a b)
hγ : MDiff[Icc (f b) (f a)] γ
h : a < b
f_im : f '' Icc a b = Icc (f b) (f a)
⊢ ∫⁻ (t : ℝ) in Icc a b, ‖(mfderiv[Icc a b] (γ ∘ f) t) 1‖ₑ =
    ∫⁻ (t : ℝ) in f '' Icc a b, ‖(mfderiv[f '' Icc a b] γ t) 1‖ₑ
Module: Mathlib.Geometry.Manifold.Sheaf.LocallyRingedSpace

Goal state before:
pp: case mpr
𝕜 : Type u
inst✝⁵ : NontriviallyNormedField 𝕜
EM : Type u_1
inst✝⁴ : NormedAddCommGroup EM
inst✝³ : NormedSpace 𝕜 EM
HM : Type u_2
inst✝² : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
M : Type u
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace HM M
x : M
S : TopCat.Presheaf CommRingCat (TopCat.of M) := (smoothSheafCommRing IM 𝓘(𝕜, 𝕜) M 𝕜).presheaf
U : Opens M
hxU : x ∈ U
f : C^∞⟮IM, ↥U; 𝓘(𝕜, 𝕜), 𝕜⟯
hf : (eval IM 𝓘(𝕜, 𝕜) M 𝕜 x) ((CategoryTheory.ConcreteCategory.hom (S.germ U x hxU)) f) ≠ 0
hf' : f ⟨x, hxU⟩ ≠ 0
H : ∀ᶠ (z : ↥U) in 𝓝 ⟨x, hxU⟩, f z ≠ 0
⊢ IsUnit ((CategoryTheory.ConcreteCategory.hom (S.germ U x hxU)) f)

Tactic:
eventually_nhds_iff

Goal state after:
case mpr
𝕜 : Type u
inst✝⁵ : NontriviallyNormedField 𝕜
EM : Type u_1
inst✝⁴ : NormedAddCommGroup EM
inst✝³ : NormedSpace 𝕜 EM
HM : Type u_2
inst✝² : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
M : Type u
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace HM M
x : M
S : TopCat.Presheaf CommRingCat (TopCat.of M) := (smoothSheafCommRing IM 𝓘(𝕜, 𝕜) M 𝕜).presheaf
U : Opens M
hxU : x ∈ U
f : C^∞⟮IM, ↥U; 𝓘(𝕜, 𝕜), 𝕜⟯
hf : (eval IM 𝓘(𝕜, 𝕜) M 𝕜 x) ((CategoryTheory.ConcreteCategory.hom (S.germ U x hxU)) f) ≠ 0
hf' : f ⟨x, hxU⟩ ≠ 0
H : ∃ t, (∀ y ∈ t, f y ≠ 0) ∧ IsOpen[instTopologicalSpaceSubtype] t ∧ ⟨x, hxU⟩ ∈ t
⊢ IsUnit ((CategoryTheory.ConcreteCategory.hom (S.germ U x hxU)) f)
Module: Mathlib.Geometry.Manifold.Submersion

Goal state before:
pp: 𝕜 : Type u_1
H : Type u_7
E : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_11
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
n : ℕ∞ω
inst✝ : IsManifold I n M
⊢ IsSubmersionOfComplement PUnit.{u_15 + 1} I I n id

Tactic:
by
  intro x
  apply IsSubmersionAtOfComplement.mk_of_continuousAt (continuousAt_id)
    (ContinuousLinearEquiv.prodUnique 𝕜 E PUnit).symm
    (chartAt H x) (chartAt H x) (mem_chart_source H x) (mem_chart_source H x)
    (chart_mem_maximalAtlas x) (chart_mem_maximalAtlas x)
  intro y hy
  have : I ((chartAt H x) ((chartAt H x).symm (I.symm y))) = y := by
    rw [(chartAt H x).right_inv (by simp_all), I.right_inv (by simp_all)]
  simpa

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorBundle.LocalFrame

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
F : Type u_5
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝⁵ : TopologicalSpace (TotalSpace F V)
inst✝⁴ : (x : M) → AddCommGroup (V x)
inst✝³ : (x : M) → Module 𝕜 (V x)
inst✝² : (x : M) → TopologicalSpace (V x)
inst✝¹ : FiberBundle F V
ι : Type u_7
s : ι → (x : M) → V x
u : Set M
n : ℕ∞ω
x : M
inst✝ : Finite ι
hs : IsLocalFrameOn I F n s u
hx : x ∈ u
t t' : (x : M) → V x
h : ∀ (i : ι), (hs.coeff i x) (t x) = (hs.coeff i x) (t' x)
this : Fintype ι := Fintype.ofFinite ι
⊢ t x = t' x

Tactic:
calc
    t x = ∑ i, hs.coeff i x (t x) • (s i x) := hs.coeff_sum_eq t hx
    _ = ∑ i, hs.coeff i x (t' x) • (s i x) := by simp [h]
    _ = t' x := (hs.coeff_sum_eq t' hx).symm

Goal state after:
Module: Mathlib.GroupTheory.Commutator.Finite

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : Finite ↑(commutatorSet G)
⊢ Group.rank ↥(closure (commutatorSet G)) ≤ Nat.card ↑(commutatorSet G)

Tactic:
apply Subgroup.rank_closure_finite_le_nat_card

Goal state after:
Module: Mathlib.GroupTheory.SpecificGroups.Dihedral

Goal state before:
pp: case inl
⊢ ∀ (n : ℕ), 0 < n → r 1 ^ n ≠ 1

Tactic:
intro n hn

Goal state after:
case inl
n : ℕ
hn : 0 < n
⊢ r 1 ^ n ≠ 1
Module: Mathlib.GroupTheory.Coprod.Basic

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝¹ : MulOneClass M
inst✝ : MulOneClass N
⊢ Submonoid.closure (Set.range ⇑inl ∪ Set.range ⇑inr) = ⊤

Tactic:
← mrange_mk,

Goal state after:
M : Type u_1
N : Type u_2
inst✝¹ : MulOneClass M
inst✝ : MulOneClass N
⊢ Submonoid.closure (Set.range ⇑inl ∪ Set.range ⇑inr) = MonoidHom.mrange mk
Module: Mathlib.GroupTheory.Coprod.Basic

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝¹ : Monoid M
inst✝ : Monoid N
x : M ∗ N
⊢ (toProd x).1 = fst x

Tactic:
rw [← fst_comp_toProd]; rfl

Goal state after:
Module: Mathlib.GroupTheory.Coxeter.Basic

Goal state before:
pp: B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
this : cs.simple = ⇑cs.mulEquiv.symm ∘ PresentedGroup.of
⊢ Subgroup.closure (⇑cs.mulEquiv.symm.toMonoidHom '' Set.range PresentedGroup.of) = ⊤

Tactic:
← MonoidHom.map_closure,

Goal state after:
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
this : cs.simple = ⇑cs.mulEquiv.symm ∘ PresentedGroup.of
⊢ Subgroup.map cs.mulEquiv.symm.toMonoidHom (Subgroup.closure (Set.range PresentedGroup.of)) = ⊤