mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Manifold.VectorBundle.LocalFrame

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
F : Type u_5
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝⁶ : TopologicalSpace (TotalSpace F V)
inst✝⁵ : (x : M) → AddCommGroup (V x)
inst✝⁴ : (x : M) → Module 𝕜 (V x)
inst✝³ : (x : M) → TopologicalSpace (V x)
inst✝² : FiberBundle F V
ι : Type u_7
s : ι → (x : M) → V x
u : Set M
n : ℕ∞ω
x : M
t t' : (x : M) → V x
inst✝¹ : VectorBundle 𝕜 F V
inst✝ : FiniteDimensional 𝕜 F
hs : IsLocalFrameOn I F n s u
hx : x ∈ u
⊢ t x = t' x ↔ ∀ (i : ι), (hs.coeff i x) (t x) = (hs.coeff i x) (t' x)

Tactic:
let := fintypeOfFiniteDimensional hs hx

Goal state after:
𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
F : Type u_5
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝⁶ : TopologicalSpace (TotalSpace F V)
inst✝⁵ : (x : M) → AddCommGroup (V x)
inst✝⁴ : (x : M) → Module 𝕜 (V x)
inst✝³ : (x : M) → TopologicalSpace (V x)
inst✝² : FiberBundle F V
ι : Type u_7
s : ι → (x : M) → V x
u : Set M
n : ℕ∞ω
x : M
t t' : (x : M) → V x
inst✝¹ : VectorBundle 𝕜 F V
inst✝ : FiniteDimensional 𝕜 F
hs : IsLocalFrameOn I F n s u
hx : x ∈ u
this : Fintype ι := hs.fintypeOfFiniteDimensional hx
⊢ t x = t' x ↔ ∀ (i : ι), (hs.coeff i x) (t x) = (hs.coeff i x) (t' x)
Module: Mathlib.Geometry.Manifold.Riemannian.Basic

Goal state before:
pp: E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : RiemannianBundle fun x ↦ TangentSpace I x
inst✝² : IsManifold I 1 M
inst✝¹ : IsContinuousRiemannianBundle E fun x ↦ TangentSpace I x
inst✝ : RegularSpace M
x : M
s : Set M
hs : s ∈ 𝓝 x
C : ℝ≥0
C_pos : C > 0
hC✝ : ∀ᶠ (y : M) in 𝓝 x, ‖mfderiv% ↑(extChartAt I x) y‖ₑ < ↑C
u : Set M
u_mem : u ∈ 𝓝 x
u_closed : IsClosed[inst✝⁵] u
us : u ⊆ s
hu : u ⊆ {y | ‖mfderiv% ↑(extChartAt I x) y‖ₑ < ↑C}
uc : u ⊆ (extChartAt I x).source
uc' : u ⊆ (chartAt H x).source
v : Set M
hv : v ⊆ u
v_mem : v ∈ 𝓝 x
v_open : IsOpen[inst✝⁵] v
r : ℝ≥0
hr : ball (↑(extChartAt I x) x) ↑r ⊆ ↑(extChartAt I x).symm ⁻¹' v
r_pos : 0 < r
y : M
hy : y ∈ {y | riemannianEDist I x y < ↑(r / C)}
γ : ℝ → M
hγx : γ 0 = x
hγy : γ 1 = y
γ_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ
hγ : pathELength I γ 0 1 < ↑(r / C)
A : Set ℝ := ⋯
zero_mem : 0 ∈ A
A_closed : IsClosed (A ∩ Icc 0 1)
t₁ : ℝ
ht₁0 : 0 ≤ t₁
ht₁1 : t₁ < 1
t₁_mem : Icc 0 t₁ ⊆ A
γ' : ℝ → E := ⋯
hC : ContMDiffOn 𝓘(ℝ, ℝ) 𝓘(ℝ, E) 1 γ' (Icc 0 t₁)
h't' : 0 < t₁
t' : ℝ
ht' : t' ∈ Icc 0 t₁
this✝ : mfderiv[Icc 0 t₁] γ' t' = mfderiv% ↑(extChartAt I x) (γ t') ∘SL mfderiv[Icc 0 t₁] γ t'
this : (mfderiv[Icc 0 t₁] γ' t') 1 = (mfderiv% ↑(extChartAt I x) (γ t')) ((mfderiv[Icc 0 t₁] γ t') 1)
⊢ ‖mfderiv% ↑(extChartAt I x) (γ t')‖ₑ * ‖(mfderiv[Icc 0 t₁] γ t') 1‖ₑ ≤ ↑C * ‖(mfderiv[Icc 0 t₁] γ t') 1‖ₑ

Tactic:
gcongr

Goal state after:
case h₁
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : RiemannianBundle fun x ↦ TangentSpace I x
inst✝² : IsManifold I 1 M
inst✝¹ : IsContinuousRiemannianBundle E fun x ↦ TangentSpace I x
inst✝ : RegularSpace M
x : M
s : Set M
hs : s ∈ 𝓝 x
C : ℝ≥0
C_pos : C > 0
hC✝ : ∀ᶠ (y : M) in 𝓝 x, ‖mfderiv% ↑(extChartAt I x) y‖ₑ < ↑C
u : Set M
u_mem : u ∈ 𝓝 x
u_closed : IsClosed[inst✝⁵] u
us : u ⊆ s
hu : u ⊆ {y | ‖mfderiv% ↑(extChartAt I x) y‖ₑ < ↑C}
uc : u ⊆ (extChartAt I x).source
uc' : u ⊆ (chartAt H x).source
v : Set M
hv : v ⊆ u
v_mem : v ∈ 𝓝 x
v_open : IsOpen[inst✝⁵] v
r : ℝ≥0
hr : ball (↑(extChartAt I x) x) ↑r ⊆ ↑(extChartAt I x).symm ⁻¹' v
r_pos : 0 < r
y : M
hy : y ∈ {y | riemannianEDist I x y < ↑(r / C)}
γ : ℝ → M
hγx : γ 0 = x
hγy : γ 1 = y
γ_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ
hγ : pathELength I γ 0 1 < ↑(r / C)
A : Set ℝ := ⋯
zero_mem : 0 ∈ A
A_closed : IsClosed (A ∩ Icc 0 1)
t₁ : ℝ
ht₁0 : 0 ≤ t₁
ht₁1 : t₁ < 1
t₁_mem : Icc 0 t₁ ⊆ A
γ' : ℝ → E := ⋯
hC : ContMDiffOn 𝓘(ℝ, ℝ) 𝓘(ℝ, E) 1 γ' (Icc 0 t₁)
h't' : 0 < t₁
t' : ℝ
ht' : t' ∈ Icc 0 t₁
this✝ : mfderiv[Icc 0 t₁] γ' t' = mfderiv% ↑(extChartAt I x) (γ t') ∘SL mfderiv[Icc 0 t₁] γ t'
this : (mfderiv[Icc 0 t₁] γ' t') 1 = (mfderiv% ↑(extChartAt I x) (γ t')) ((mfderiv[Icc 0 t₁] γ t') 1)
⊢ ‖mfderiv% ↑(extChartAt I x) (γ t')‖ₑ ≤ ↑C
Module: Mathlib.Geometry.Manifold.VectorBundle.LocalFrame

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁴ : TopologicalSpace M
inst✝¹³ : ChartedSpace H M
F : Type u_5
inst✝¹² : NormedAddCommGroup F
inst✝¹¹ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝¹⁰ : TopologicalSpace (TotalSpace F V)
inst✝⁹ : (x : M) → AddCommGroup (V x)
inst✝⁸ : (x : M) → Module 𝕜 (V x)
inst✝⁷ : (x : M) → TopologicalSpace (V x)
inst✝⁶ : FiberBundle F V
inst✝⁵ : VectorBundle 𝕜 F V
inst✝⁴ : ContMDiffVectorBundle 1 F V I
e : Trivialization F TotalSpace.proj
inst✝³ : MemTrivializationAtlas e
ι : Type u_7
b : Basis ι 𝕜 F
s : (x : M) → V x
k : ℕ∞ω
x : M
inst✝² : FiniteDimensional 𝕜 F
inst✝¹ : CompleteSpace 𝕜
inst✝ : ContMDiffVectorBundle k F V I
hxe : x ∈ e.baseSet
hs : ContMDiffAt I (I.prod 𝓘(𝕜, F)) k (fun x ↦ ⟨x, s x⟩) x
i : ι
aux : M → 𝕜 := fun x ↦ (b.repr (↑e ((fun x ↦ ⟨x, s x⟩) x)).2) i
h₁ : ContMDiffAt I 𝓘(𝕜, F) k (fun x ↦ (↑e ((fun x ↦ ⟨x, s x⟩) x)).2) x
⊢ ContMDiffAt I 𝓘(𝕜, 𝕜) k (fun x ↦ (b.repr (↑e ⟨x, s x⟩).2) i) x

Tactic:
let breprl : F →ₗ[𝕜] 𝕜 :=
    { toFun v := b.repr v i
      map_add' m m' := by simp
      map_smul' m x := by simp }

Goal state after:
𝕜 : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁴ : TopologicalSpace M
inst✝¹³ : ChartedSpace H M
F : Type u_5
inst✝¹² : NormedAddCommGroup F
inst✝¹¹ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝¹⁰ : TopologicalSpace (TotalSpace F V)
inst✝⁹ : (x : M) → AddCommGroup (V x)
inst✝⁸ : (x : M) → Module 𝕜 (V x)
inst✝⁷ : (x : M) → TopologicalSpace (V x)
inst✝⁶ : FiberBundle F V
inst✝⁵ : VectorBundle 𝕜 F V
inst✝⁴ : ContMDiffVectorBundle 1 F V I
e : Trivialization F TotalSpace.proj
inst✝³ : MemTrivializationAtlas e
ι : Type u_7
b : Basis ι 𝕜 F
s : (x : M) → V x
k : ℕ∞ω
x : M
inst✝² : FiniteDimensional 𝕜 F
inst✝¹ : CompleteSpace 𝕜
inst✝ : ContMDiffVectorBundle k F V I
hxe : x ∈ e.baseSet
hs : ContMDiffAt I (I.prod 𝓘(𝕜, F)) k (fun x ↦ ⟨x, s x⟩) x
i : ι
aux : M → 𝕜 := fun x ↦ (b.repr (↑e ((fun x ↦ ⟨x, s x⟩) x)).2) i
h₁ : ContMDiffAt I 𝓘(𝕜, F) k (fun x ↦ (↑e ((fun x ↦ ⟨x, s x⟩) x)).2) x
breprl : F →ₗ[𝕜] 𝕜 := { toFun := fun v ↦ (b.repr v) i, map_add' := ⋯, map_smul' := ⋯ }
⊢ ContMDiffAt I 𝓘(𝕜, 𝕜) k (fun x ↦ (b.repr (↑e ⟨x, s x⟩).2) i) x
Module: Mathlib.Geometry.Manifold.Riemannian.Basic

Goal state before:
pp: E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : RiemannianBundle fun x ↦ TangentSpace I x
inst✝² : IsManifold I 1 M
inst✝¹ : IsContinuousRiemannianBundle E fun x ↦ TangentSpace I x
inst✝ : RegularSpace M
x : M
s : Set M
hs : s ∈ 𝓝 x
C : ℝ≥0
C_pos : C > 0
hC✝ : ∀ᶠ (y : M) in 𝓝 x, ‖mfderiv% ↑(extChartAt I x) y‖ₑ < ↑C
u : Set M
u_mem : u ∈ 𝓝 x
u_closed : IsClosed[inst✝⁵] u
us : u ⊆ s
hu : u ⊆ {y | ‖mfderiv% ↑(extChartAt I x) y‖ₑ < ↑C}
uc : u ⊆ (extChartAt I x).source
uc' : u ⊆ (chartAt H x).source
v : Set M
hv : v ⊆ u
v_mem : v ∈ 𝓝 x
v_open : IsOpen[inst✝⁵] v
r : ℝ≥0
hr : ball (↑(extChartAt I x) x) ↑r ⊆ ↑(extChartAt I x).symm ⁻¹' v
r_pos : 0 < r
y : M
hy : y ∈ {y | riemannianEDist I x y < ↑(r / C)}
γ : ℝ → M
hγx : γ 0 = x
hγy : γ 1 = y
γ_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ
hγ : pathELength I γ 0 1 < ↑(r / C)
A : Set ℝ := γ ⁻¹' u
zero_mem : 0 ∈ A
A_closed : IsClosed (A ∩ Icc 0 1)
t₁ : ℝ
ht₁0 : 0 ≤ t₁
ht₁1 : t₁ < 1
t₁_mem : Icc 0 t₁ ⊆ A
γ' : ℝ → E := ↑(extChartAt I x) ∘ γ
hC : ContMDiffOn 𝓘(ℝ, ℝ) 𝓘(ℝ, E) 1 γ' (Icc 0 t₁)
h't' : 0 < t₁
⊢ ↑C * pathELength I γ 0 t₁ ≤ ↑C * pathELength I γ 0 1

Tactic:
gcongr

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹³ : TopologicalSpace M
inst✝¹² : ChartedSpace H M
F : Type u_5
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝⁹ : TopologicalSpace (TotalSpace F V)
inst✝⁸ : (x : M) → AddCommGroup (V x)
inst✝⁷ : (x : M) → Module 𝕜 (V x)
inst✝⁶ : (x : M) → TopologicalSpace (V x)
inst✝⁵ : FiberBundle F V
inst✝⁴ : VectorBundle 𝕜 F V
inst✝³ : ContMDiffVectorBundle 1 F V I
e : Trivialization F TotalSpace.proj
inst✝² : MemTrivializationAtlas e
ι : Type u_7
b : Basis ι 𝕜 F
s : (x : M) → V x
x : M
inst✝¹ : FiniteDimensional 𝕜 F
inst✝ : CompleteSpace 𝕜
hxe : x ∈ e.baseSet
hs : MDiffAt (T% s) x
i : ι
aux : M → 𝕜 := fun x ↦ (b.repr (↑e ((fun x ↦ ⟨x, s x⟩) x)).2) i
h₁ : (MDiffAt fun x ↦ (↑e ((fun x ↦ ⟨x, s x⟩) x)).2) x
⊢ (MDiffAt fun x ↦ (b.repr (↑e ⟨x, s x⟩).2) i) x

Tactic:
let breprl : F →ₗ[𝕜] 𝕜 :=
    { toFun v := b.repr v i
      map_add' m m' := by simp
      map_smul' m x := by simp }

Goal state after:
𝕜 : Type u_1
inst✝¹⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹³ : TopologicalSpace M
inst✝¹² : ChartedSpace H M
F : Type u_5
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝⁹ : TopologicalSpace (TotalSpace F V)
inst✝⁸ : (x : M) → AddCommGroup (V x)
inst✝⁷ : (x : M) → Module 𝕜 (V x)
inst✝⁶ : (x : M) → TopologicalSpace (V x)
inst✝⁵ : FiberBundle F V
inst✝⁴ : VectorBundle 𝕜 F V
inst✝³ : ContMDiffVectorBundle 1 F V I
e : Trivialization F TotalSpace.proj
inst✝² : MemTrivializationAtlas e
ι : Type u_7
b : Basis ι 𝕜 F
s : (x : M) → V x
x : M
inst✝¹ : FiniteDimensional 𝕜 F
inst✝ : CompleteSpace 𝕜
hxe : x ∈ e.baseSet
hs : MDiffAt (T% s) x
i : ι
aux : M → 𝕜 := fun x ↦ (b.repr (↑e ((fun x ↦ ⟨x, s x⟩) x)).2) i
h₁ : (MDiffAt fun x ↦ (↑e ((fun x ↦ ⟨x, s x⟩) x)).2) x
breprl : F →ₗ[𝕜] 𝕜 := { toFun := fun v ↦ (b.repr v) i, map_add' := ⋯, map_smul' := ⋯ }
⊢ (MDiffAt fun x ↦ (b.repr (↑e ⟨x, s x⟩).2) i) x
Module: Mathlib.Geometry.Manifold.VectorBundle.Tensoriality

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
F : Type u_5
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝⁶ : TopologicalSpace (TotalSpace F V)
inst✝⁵ : (x : M) → AddCommGroup (V x)
inst✝⁴ : (x : M) → Module 𝕜 (V x)
inst✝³ : (x : M) → TopologicalSpace (V x)
inst✝² : FiberBundle F V
A : Type u_9
inst✝¹ : AddCommGroup A
inst✝ : Module 𝕜 A
Φ : ((x : M) → V x) → A
x : M
hΦ : TensorialAt I F Φ x
σ σ' : (x : M) → V x
hσ : MDiffAt (T% σ) x
hσ' : MDiffAt (T% σ') x
hσσ' : ∀ᶠ (x' : M) in 𝓝 x, σ x' = σ' x'
ψ : M → 𝕜 := fun x' ↦ if σ x' = σ' x' then 1 else 0
hψx : ψ x = 1
this : ∀ (x' : M), (ψ • σ) x' = (ψ • σ') x'
hψ' : MDiffAt ψ x
⊢ Φ (ψ • σ) = Φ (ψ • σ')

Tactic:
by rw [funext this]

Goal state after:
Module: Mathlib.Topology.VectorBundle.Riemannian

Goal state before:
pp: B : Type u_1
inst✝⁷ : TopologicalSpace B
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
E : B → Type u_3
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (x : B) → NormedAddCommGroup (E x)
inst✝² : (x : B) → InnerProductSpace ℝ (E x)
inst✝¹ : FiberBundle F E
inst✝ : VectorBundle ℝ F E
h : IsContinuousRiemannianBundle F E
x : B
r : ℝ
hr : 1 < r
r' : ℝ
hr' : 1 < r'
r'r : r' < r
h'x : x ∈ (trivializationAt F E x).baseSet
G : E x ≃L[ℝ] F := Trivialization.continuousLinearEquivAt ℝ (trivializationAt F E x) x h'x
C : ℝ := ‖↑G‖ ^ 2
δ : ℝ
δpos : 0 < δ
hδ : (r' ^ 2)⁻¹ < 1 - δ * C
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_cont :
  Continuous[inst✝⁷, topologicalSpaceTotalSpace (RingHom.id ℝ) F E (F →L[ℝ] ℝ) fun x ↦ E x →L[ℝ] ℝ] fun x ↦
    TotalSpace.mk' (F →L[ℝ] F →L[ℝ] ℝ) x (g x)
hg : ∀ (x : B) (v w : E x), ⟪v, w⟫ = ((g x) v) w
g' : B → F →L[ℝ] F →L[ℝ] ℝ := fun y ↦ inCoordinates F E (F →L[ℝ] ℝ) (fun x ↦ E x →L[ℝ] ℝ) x y x y (g y)
hg' : ContinuousAt g' x
this✝ : ∀ᶠ (y : B) in 𝓝 x, dist (g' y) (g' x) < δ
y : B
hy : dist (g' y) (g' x) < δ
h'y : y ∈ (trivializationAt F E x).baseSet
this : ‖g' x - g' y‖ ≤ δ
v : E y
w : F := (Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) y) v
hgy : ((g y) v) v = ((g' y) w) w
hgx : ((g x) ((trivializationAt F E x).symm x w)) ((trivializationAt F E x).symm x w) = ((g' x) w) w
⊢ ‖g' x - g' y‖ * (‖w‖ * ‖w‖) + ((g' y) w) w ≤ δ * (‖w‖ * ‖w‖) + ((g' y) w) w

Tactic:
gcongr

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

Goal state before:
pp: case intro
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I ∞ M
inst✝¹ : T2Space M
ι : Type u_1
inst✝ : Finite ι
f : SmoothBumpCovering ι I M
val✝ : Fintype ι
F : Type := EuclideanSpace ℝ (Fin (finrank ℝ (ι → E × ℝ)))
⊢ ∃ n e, ContMDiff I (𝓡 n) ∞ e ∧ Injective e ∧ ∀ (x : M), Injective ⇑(mfderiv% e x)

Tactic:
let : IsNoetherian ℝ (E × ℝ) := IsNoetherian.iff_fg.2 inferInstance

Goal state after:
case intro
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I ∞ M
inst✝¹ : T2Space M
ι : Type u_1
inst✝ : Finite ι
f : SmoothBumpCovering ι I M
val✝ : Fintype ι
F : Type := EuclideanSpace ℝ (Fin (finrank ℝ (ι → E × ℝ)))
this : IsNoetherian ℝ (E × ℝ) := IsNoetherian.iff_fg.mpr inferInstance
⊢ ∃ n e, ContMDiff I (𝓡 n) ∞ e ∧ Injective e ∧ ∀ (x : M), Injective ⇑(mfderiv% e x)
Module: Mathlib.GroupTheory.GroupAction.CardCommute

Goal state before:
pp: G : Type u_3
inst✝ : Group G
⊢ Nat.card { p // Commute p.1 p.2 } = Nat.card (ConjClasses G) * Nat.card G

Tactic:
by
  classical
  rcases fintypeOrInfinite G; swap
  · rw [mul_comm, Nat.card_eq_zero_of_infinite, Nat.card_eq_zero_of_infinite, zero_mul]
  simp only [Nat.card_eq_fintype_card]
  calc card { p : G × G // Commute p.1 p.2 }
      _ = card ((a : G) × { b // Commute a b }) :=
            card_congr (Equiv.subtypeProdEquivSigmaSubtype Commute)
      _ = ∑ i, card { b // Commute i b } := card_sigma
      _ = ∑ x, card (MulAction.fixedBy G x) :=
            sum_equiv ConjAct.toConjAct.toEquiv (fun a ↦ card { b // Commute a b })
              (fun g ↦ card (MulAction.fixedBy G g))
              fun g ↦ card_congr' <| congr_arg _ <| funext fun h ↦ mul_inv_eq_iff_eq_mul.symm.eq
      _ = card (Quotient (MulAction.orbitRel (ConjAct G) G)) * card (ConjAct G) :=
             MulAction.sum_card_fixedBy_eq_card_orbits_mul_card_group _ _
      _ = card (ConjClasses G) * card G := by
             congr 1; apply card_congr'; congr; ext
             exact (Setoid.comm' _).trans isConj_iff.symm

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

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝¹ : MulOneClass M
inst✝ : MulOneClass N
motive : M ∗ N → Prop
m : M ∗ N
one : motive 1
inl_mul : ∀ (m : M) (x : M ∗ N), motive x → motive (inl m * x)
inr_mul : ∀ (n : N) (x : M ∗ N), motive x → motive (inr n * x)
⊢ motive m

Tactic:
rcases mk_surjective m with ⟨x, rfl⟩

Goal state after:
M : Type u_1
N : Type u_2
inst✝¹ : MulOneClass M
inst✝ : MulOneClass N
motive : M ∗ N → Prop
one : motive 1
inl_mul : ∀ (m : M) (x : M ∗ N), motive x → motive (inl m * x)
inr_mul : ∀ (n : N) (x : M ∗ N), motive x → motive (inr n * x)
x : FreeMonoid (M ⊕ N)
⊢ motive (mk x)
Module: Mathlib.Topology.VectorBundle.Riemannian

Goal state before:
pp: B : Type u_1
inst✝⁷ : TopologicalSpace B
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
E : B → Type u_3
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (x : B) → NormedAddCommGroup (E x)
inst✝² : (x : B) → InnerProductSpace ℝ (E x)
inst✝¹ : FiberBundle F E
inst✝ : VectorBundle ℝ F E
h : IsContinuousRiemannianBundle F E
x y : B
hy :
  ‖Trivialization.symmL ℝ (trivializationAt F E x) x ∘SL
        Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) y‖ <
    2
A :
  Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) x ∘SL
      Trivialization.symmL ℝ (trivializationAt F E x) x =
    ContinuousLinearMap.id ℝ F
this :
  Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) y =
    ContinuousLinearMap.id ℝ F ∘SL Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) y
⊢ ‖Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) x‖ *
      ‖Trivialization.symmL ℝ (trivializationAt F E x) x ∘SL
          Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) y‖ <
    (1 + ‖Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) x‖) * 2

Tactic:
gcongr

Goal state after:
case h₁
B : Type u_1
inst✝⁷ : TopologicalSpace B
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
E : B → Type u_3
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (x : B) → NormedAddCommGroup (E x)
inst✝² : (x : B) → InnerProductSpace ℝ (E x)
inst✝¹ : FiberBundle F E
inst✝ : VectorBundle ℝ F E
h : IsContinuousRiemannianBundle F E
x y : B
hy :
  ‖Trivialization.symmL ℝ (trivializationAt F E x) x ∘SL
        Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) y‖ <
    2
A :
  Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) x ∘SL
      Trivialization.symmL ℝ (trivializationAt F E x) x =
    ContinuousLinearMap.id ℝ F
this :
  Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) y =
    ContinuousLinearMap.id ℝ F ∘SL Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) y
⊢ ‖Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) x‖ <
    1 + ‖Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) x‖
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: ι : Type u_1
M : ι → Type u_2
inst✝¹ : (i : ι) → Monoid (M i)
N : Type u_3
inst✝ : Monoid N
i : ι
⊢ empty.fstIdx ≠ some i

Tactic:
by tauto

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

Goal state before:
pp: B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
j : ℕ
⊢ (cs.leftInvSeq ω).getD j 1 * (cs.leftInvSeq ω).getD j 1 = 1

Tactic:
simp_rw [getD_leftInvSeq, mul_assoc]

Goal state after:
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
j : ℕ
⊢ cs.wordProd (take j ω) *
      ((Option.map cs.simple ω[j]?).getD 1 *
        ((cs.wordProd (take j ω))⁻¹ *
          (cs.wordProd (take j ω) * ((Option.map cs.simple ω[j]?).getD 1 * (cs.wordProd (take j ω))⁻¹)))) =
    1
Module: Mathlib.GroupTheory.Descent

Goal state before:
pp: G : Type u_1
inst✝¹ : CommGroup G
n : ℕ
h : G → ℝ
a b c₀ : ℝ
c : G → ℝ
ha : 0 ≤ a
H₀ : a < b
H₁ : (powMonoidHom n).range.FiniteIndex
H₂ : ∀ (g x : G), h x ≤ a * h (g * x) + c g
H₃ : ∀ (x : G), b * h x - c₀ ≤ h (x ^ n)
inst✝ : Northcott h
f : G →* G := powMonoidHom n
q : G → G ⧸ f.range := QuotientGroup.mk
qi : G ⧸ f.range → G := Function.surjInv ⋯
s : Set G := Set.range qi
g : G
hg₁ : g ∈ s
hg₂ : ∀ b ∈ s, c b ≤ c g
H₁' : s * ↑f.range = Set.univ
⊢ Group.FG G

Tactic:
let c' : ℝ := max c₀ (c g)

Goal state after:
G : Type u_1
inst✝¹ : CommGroup G
n : ℕ
h : G → ℝ
a b c₀ : ℝ
c : G → ℝ
ha : 0 ≤ a
H₀ : a < b
H₁ : (powMonoidHom n).range.FiniteIndex
H₂ : ∀ (g x : G), h x ≤ a * h (g * x) + c g
H₃ : ∀ (x : G), b * h x - c₀ ≤ h (x ^ n)
inst✝ : Northcott h
f : G →* G := powMonoidHom n
q : G → G ⧸ f.range := QuotientGroup.mk
qi : G ⧸ f.range → G := Function.surjInv ⋯
s : Set G := Set.range qi
g : G
hg₁ : g ∈ s
hg₂ : ∀ b ∈ s, c b ≤ c g
H₁' : s * ↑f.range = Set.univ
c' : ℝ := max c₀ (c g)
⊢ Group.FG G
Module: Mathlib.GroupTheory.Descent

Goal state before:
pp: G : Type u_1
inst✝¹ : CommGroup G
h : G → ℝ
C : ℝ
H : ∀ (x y : G), |h (x * y) + h (x / y) - 2 * (h x + h y)| ≤ C
inst✝ : Northcott h
⊢ Finite ↥(torsion G)

Tactic:
have H' x : 4 * h x - (h 1 + C) ≤ h (x ^ 2) := by grind [pow_two, div_self']
  exact finite_torsion_of_descent (b := 4) (by norm_num) H'

Goal state after:
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: M : Type u_1
inst✝ : AddCommMonoid M
α : Type u_2
x y : DivisibleHull M
f : M → ℕ+ → M → ℕ+ → α
h : ∀ (m n m' n' : M) (s t s' t' : ℕ+), mk m s = mk m' s' → mk n t = mk n' t' → f m s n t = f m' s' n' t'
p q p' q' : M × ↥(nonZeroDivisors ℕ)
heq : p ≈ p'
heq' : q ≈ q'
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
obtain ⟨u, hu⟩ := heq'
        exact mk_eq_mk.mpr ⟨↑ⁿ.symm u, hu⟩

Goal state after:
Module: Mathlib.GroupTheory.FiniteAbelian.Duality

Goal state before:
pp: G : Type u_1
M : Type u_2
inst✝² : CommGroup G
inst✝¹ : Finite G
inst✝ : CommMonoid M
hM : HasEnoughRootsOfUnity M (Monoid.exponent G)
H : Subgroup G
this✝¹ : Fintype ↥H
this✝ : HasEnoughRootsOfUnity M (Monoid.exponent ↥H)
this : HasEnoughRootsOfUnity M (Monoid.exponent (G ⧸ H))
⊢ Nat.card ↥(domRestrictHom H Mˣ).ker = Nat.card (G ⧸ H)

Tactic:
← card_monoidHom_of_hasEnoughRootsOfUnity (G ⧸ H) M,

Goal state after:
G : Type u_1
M : Type u_2
inst✝² : CommGroup G
inst✝¹ : Finite G
inst✝ : CommMonoid M
hM : HasEnoughRootsOfUnity M (Monoid.exponent G)
H : Subgroup G
this✝¹ : Fintype ↥H
this✝ : HasEnoughRootsOfUnity M (Monoid.exponent ↥H)
this : HasEnoughRootsOfUnity M (Monoid.exponent (G ⧸ H))
⊢ Nat.card ↥(domRestrictHom H Mˣ).ker = Nat.card (G ⧸ H →* Mˣ)
Module: Mathlib.GroupTheory.Transfer

Goal state before:
pp: G : Type u_1
inst✝² : Group G
H : Subgroup G
A : Type u_2
inst✝¹ : CommGroup A
ϕ : ↥H →* A
inst✝ : H.FiniteIndex
g : G
key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g₀⁻¹ * g ^ k * g₀ = g ^ k
⊢ ϕ.transfer g = ϕ ⟨g ^ H.index, ⋯⟩

Tactic:
by
  classical
    let := H.fintypeQuotientOfFiniteIndex
    change ∀ (k g₀) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑(⟨g₀⁻¹ * g ^ k * g₀, hk⟩ : H) = g ^ k at key
    rw [transfer_eq_prod_quotient_orbitRel_zpowers_quot, ← Finset.prod_map_toList,
      ← Function.comp_def ϕ, List.prod_map_hom]
    refine congrArg ϕ (Subtype.coe_injective ?_)
    dsimp only
    rw [H.coe_mk, ← (zpowers g).coe_mk g (mem_zpowers g), ← (zpowers g).coe_pow,
      index_eq_sum_minimalPeriod H g, ← Finset.prod_pow_eq_pow_sum, ← Finset.prod_map_toList]
    simp only [Subgroup.val_list_prod, List.map_map]
    congr 2
    funext
    apply key

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.CyclicallyReduced

Goal state before:
pp: case refine_2
α : Type u
L₁ L₂ L₃ : List (α × Bool)
h₁ : IsCyclicallyReduced L₂
h₂ : IsReduced (L₁ ++ L₂ ++ L₃)
n✝ n : ℕ
hn : n + 1 ≠ 0
h : ¬L₂ = []
h' : (replicate (n + 1) L₂).flatten ≠ []
⊢ IsReduced ((replicate n L₂).flatten ++ L₂)

Tactic:
rw [← flatten_concat, ← replicate_succ']

Goal state after:
case refine_2
α : Type u
L₁ L₂ L₃ : List (α × Bool)
h₁ : IsCyclicallyReduced L₂
h₂ : IsReduced (L₁ ++ L₂ ++ L₃)
n✝ n : ℕ
hn : n + 1 ≠ 0
h : ¬L₂ = []
h' : (replicate (n + 1) L₂).flatten ≠ []
⊢ IsReduced (replicate (n + 1) L₂).flatten