mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Manifold.Riemannian.Basic

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
H : Type u_2
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : RiemannianBundle fun x ↦ TangentSpace I x
inst✝¹ : IsManifold I 1 M
inst✝ : IsContinuousRiemannianBundle E fun x ↦ TangentSpace I x
x : M
⊢ ∃ C > 0, ∀ᶠ (y : M) in 𝓝 x, ‖mfderiv[range ↑I] ↑(extChartAt I x).symm (↑(extChartAt I x) y)‖ < C

Tactic:
by
  rcases eventually_norm_symmL_trivializationAt_lt E (fun (x : M) ↦ TangentSpace% x) x
    with ⟨C, C_pos, hC⟩
  refine ⟨C, C_pos, ?_⟩
  have hx : (chartAt H x).source ∈ 𝓝 x := chart_source_mem_nhds H x
  filter_upwards [hC, hx] with y hy h'y
  rw [TangentBundle.symmL_trivializationAt h'y] at hy
  have A : (extChartAt I x).symm (extChartAt I x y) = y :=
    (extChartAt I x).left_inv (by simpa using h'y)
  convert! hy using 3 <;> congr

Goal state after:
Module: Mathlib.Geometry.Manifold.Sheaf.LocallyRingedSpace

Goal state before:
pp: 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
V : Opens M
hUV : V ≤ U
hV₀f : ∀ y ∈ Set.range (Set.inclusion hUV), f y ≠ 0
hV₀ : IsOpen[instTopologicalSpaceSubtype] (Set.range (Set.inclusion hUV))
x₀ : ↑↑V
hxx₀ : Set.inclusion hUV x₀ = ⟨x, hxU⟩
⊢ x ∈ ↑V

Tactic:
convert! x₀.2

Goal state after:
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
V : Opens M
hUV : V ≤ U
hV₀f : ∀ y ∈ Set.range (Set.inclusion hUV), f y ≠ 0
hV₀ : IsOpen[instTopologicalSpaceSubtype] (Set.range (Set.inclusion hUV))
x₀ : ↑↑V
hxx₀ : Set.inclusion hUV x₀ = ⟨x, hxU⟩
⊢ x = ↑x₀
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, 0 < δ ∧ (r' ^ 2)⁻¹ < 1 - δ * C

Tactic:
have A : ∀ᶠ δ in 𝓝[>] (0 : ℝ), 0 < δ := self_mem_nhdsWithin

Goal state after:
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
A : ∀ᶠ (δ : ℝ) in 𝓝[>] 0, 0 < δ
⊢ ∃ δ, 0 < δ ∧ (r' ^ 2)⁻¹ < 1 - δ * C
Module: Mathlib.Geometry.Manifold.Sheaf.LocallyRingedSpace

Goal state before:
pp: case mpr.refine_2
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
V : Opens M
hUV : V ≤ U
hV₀f : ∀ y ∈ Set.range (Set.inclusion hUV), f y ≠ 0
hV₀ : IsOpen[instTopologicalSpaceSubtype] (Set.range (Set.inclusion hUV))
hxV₀ : ⟨x, hxU⟩ ∈ Set.range (Set.inclusion hUV)
hxV : x ∈ ↑V
hVf : ∀ (y : ↥V), f (Set.inclusion hUV y) ≠ 0
g : C^∞⟮IM, ↥V; 𝓘(𝕜, 𝕜), 𝕜⟯ := ⟨(⇑f ∘ Set.inclusion hUV)⁻¹, ?mpr.refine_1⟩
⊢ IsUnit ((CategoryTheory.ConcreteCategory.hom (S.germ U x hxU)) f)

Tactic:
refine ⟨⟨S.germ _ x (hxV) (ContMDiffMap.restrictRingHom IM 𝓘(𝕜) 𝕜 hUV f), S.germ _ x hxV g,
        ?_, ?_⟩, S.germ_res_apply hUV.hom x hxV f⟩
      · rw [← map_mul]
        -- Qualified the name to avoid Lean not finding a `OneHomClass` https://github.com/leanprover-community/mathlib4/pull/8386
        convert! RingHom.map_one _
        apply Subtype.ext
        ext y
        apply mul_inv_cancel₀
        exact hVf y
      · rw [← map_mul]
        -- Qualified the name to avoid Lean not finding a `OneHomClass` https://github.com/leanprover-community/mathlib4/pull/8386
        convert! RingHom.map_one _
        apply Subtype.ext
        ext y
        apply inv_mul_cancel₀
        exact hVf y

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

Goal state before:
pp: case hxs
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
H : Type u_2
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : RiemannianBundle fun x ↦ TangentSpace I x
inst✝¹ : IsManifold I 1 M
inst✝ : IsContinuousRiemannianBundle E fun x ↦ TangentSpace I x
x : M
C : ℝ≥0
C_pos : C > 0
hC : ∀ᶠ (y : E) in 𝓝[range ↑I] ↑(extChartAt I x) x, ‖mfderiv[range ↑I] ↑(extChartAt I x).symm y‖ₑ < ↑C
r : ℝ
r_pos : r > 0
hr :
  ball (↑(extChartAt I x) x) r ∩ range ↑I ⊆
    (extChartAt I x).target ∩ {y | ‖mfderiv[range ↑I] ↑(extChartAt I x).symm y‖ₑ < ↑C}
A : ↑(extChartAt I x) ⁻¹' (ball (↑(extChartAt I x) x) r ∩ range ↑I) ∈ 𝓝 x
y : M
hy : y ∈ ↑(extChartAt I x) ⁻¹' (ball (↑(extChartAt I x) x) r ∩ range ↑I)
h'y : y ∈ (chartAt H x).source
η : ℝ →ᴬ[ℝ] E := ⋯
γ : ℝ → M := ⋯
hη :
  Icc 0 1 ⊆ ⇑η ⁻¹' (extChartAt I x).target ∧ Icc 0 1 ⊆ ⇑η ⁻¹' {y | ‖mfderiv[range ↑I] ↑(extChartAt I x).symm y‖ₑ < ↑C}
η_smooth : ContMDiffOn 𝓘(ℝ, ℝ) 𝓘(ℝ, E) 1 (⇑η) (Icc 0 1)
this : riemannianEDist I x y ≤ pathELength I γ 0 1
t : ℝ
ht : t ∈ Icc 0 1
⊢ UniqueMDiffAt[Icc 0 1] t

Tactic:
uniqueMDiffWithinAt_iff_uniqueDiffWithinAt

Goal state after:
case hxs
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
H : Type u_2
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : RiemannianBundle fun x ↦ TangentSpace I x
inst✝¹ : IsManifold I 1 M
inst✝ : IsContinuousRiemannianBundle E fun x ↦ TangentSpace I x
x : M
C : ℝ≥0
C_pos : C > 0
hC : ∀ᶠ (y : E) in 𝓝[range ↑I] ↑(extChartAt I x) x, ‖mfderiv[range ↑I] ↑(extChartAt I x).symm y‖ₑ < ↑C
r : ℝ
r_pos : r > 0
hr :
  ball (↑(extChartAt I x) x) r ∩ range ↑I ⊆
    (extChartAt I x).target ∩ {y | ‖mfderiv[range ↑I] ↑(extChartAt I x).symm y‖ₑ < ↑C}
A : ↑(extChartAt I x) ⁻¹' (ball (↑(extChartAt I x) x) r ∩ range ↑I) ∈ 𝓝 x
y : M
hy : y ∈ ↑(extChartAt I x) ⁻¹' (ball (↑(extChartAt I x) x) r ∩ range ↑I)
h'y : y ∈ (chartAt H x).source
η : ℝ →ᴬ[ℝ] E := ContinuousAffineMap.lineMap (↑(extChartAt I x) x) (↑(extChartAt I x) y)
γ : ℝ → M := ↑(extChartAt I x).symm ∘ ⇑η
hη :
  Icc 0 1 ⊆ ⇑η ⁻¹' (extChartAt I x).target ∧ Icc 0 1 ⊆ ⇑η ⁻¹' {y | ‖mfderiv[range ↑I] ↑(extChartAt I x).symm y‖ₑ < ↑C}
η_smooth : ContMDiffOn 𝓘(ℝ, ℝ) 𝓘(ℝ, E) 1 (⇑η) (Icc 0 1)
this : riemannianEDist I x y ≤ pathELength I γ 0 1
t : ℝ
ht : t ∈ Icc 0 1
⊢ UniqueDiffWithinAt ℝ (Icc 0 1) t
Module: Mathlib.Geometry.Manifold.Submersion

Goal state before:
pp: case h
𝕜 : Type u_1
E'' : Type u_3
F : Type u_5
H : Type u_7
G : Type u_9
E : Type u
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedAddCommGroup E''
inst✝⁸ : NormedSpace 𝕜 E''
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : TopologicalSpace H
inst✝⁴ : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 E'' G
M : Type u_11
N : Type u_13
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n : ℕ∞ω
f : M → N
h : IsSubmersionOfComplement F I J n f
hM : Nonempty M
inhabited_h : Inhabited M
x : M := ⋯
⊢ IsSubmersionOfComplement ⋯.smallComplement I J n f

Tactic:
exact (IsSubmersionOfComplement.congr_F (h x).smallEquiv).mp h

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

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
F : Type u_5
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝⁴ : TopologicalSpace (TotalSpace F V)
inst✝³ : (x : M) → AddCommGroup (V x)
inst✝² : (x : M) → Module 𝕜 (V x)
inst✝¹ : (x : M) → TopologicalSpace (V x)
inst✝ : FiberBundle F V
ι : Type u_7
s : ι → (x : M) → V x
u : Set M
n : ℕ∞ω
x : M
hs : IsLocalFrameOn I F n s u
hx : x ∉ u
i : ι
⊢ hs.coeff i x = 0

Tactic:
simp [coeff, hx]

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

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
F : Type u_5
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝⁴ : TopologicalSpace (TotalSpace F V)
inst✝³ : (x : M) → AddCommGroup (V x)
inst✝² : (x : M) → Module 𝕜 (V x)
inst✝¹ : (x : M) → TopologicalSpace (V x)
inst✝ : FiberBundle F V
ι : Type u_7
s : ι → (x : M) → V x
u : Set M
n : ℕ∞ω
x : M
hs : IsLocalFrameOn I F n s u
hx : x ∈ u
t : (x : M) → V x
i : ι
⊢ (hs.coeff i x) (t x) = ((hs.toBasisAt hx).repr (t x)) i

Tactic:
simp [coeff, hx]

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

Goal state before:
pp: case hxs
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
⊢ UniqueMDiffAt[Icc 0 t₁] t'

Tactic:
uniqueMDiffWithinAt_iff_uniqueDiffWithinAt

Goal state after:
case hxs
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
t' : ℝ
ht' : t' ∈ Icc 0 t₁
⊢ UniqueDiffWithinAt ℝ (Icc 0 t₁) t'
Module: Mathlib.Geometry.Manifold.VectorBundle.Pullback

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
B' : Type u_3
F : Type u_4
E : B → Type u_5
n : WithTop ℕ∞
inst✝¹⁹ : NontriviallyNormedField 𝕜
inst✝¹⁸ : (x : B) → AddCommMonoid (E x)
inst✝¹⁷ : (x : B) → Module 𝕜 (E x)
inst✝¹⁶ : NormedAddCommGroup F
inst✝¹⁵ : NormedSpace 𝕜 F
inst✝¹⁴ : TopologicalSpace (TotalSpace F E)
inst✝¹³ : (x : B) → TopologicalSpace (E x)
EB : Type u_6
inst✝¹² : NormedAddCommGroup EB
inst✝¹¹ : NormedSpace 𝕜 EB
HB : Type u_7
inst✝¹⁰ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝⁹ : TopologicalSpace B
inst✝⁸ : ChartedSpace HB B
EB' : Type u_8
inst✝⁷ : NormedAddCommGroup EB'
inst✝⁶ : NormedSpace 𝕜 EB'
HB' : Type u_9
inst✝⁵ : TopologicalSpace HB'
IB' : ModelWithCorners 𝕜 EB' HB'
inst✝⁴ : TopologicalSpace B'
inst✝³ : ChartedSpace HB' B'
inst✝² : FiberBundle F E
inst✝¹ : VectorBundle 𝕜 F E
inst✝ : ContMDiffVectorBundle n F E IB
f : C^n⟮IB', B'; IB, B⟯
e : Trivialization F TotalSpace.proj
he : MemTrivializationAtlas e
e' : Trivialization F TotalSpace.proj
he' : MemTrivializationAtlas e'
b : B'
hb : f b ∈ e.baseSet ∩ e'.baseSet
⊢ ↑(Trivialization.coordChangeL 𝕜 (e.pullback f) (e'.pullback f) b) =
    ((fun b ↦ ↑(Trivialization.coordChangeL 𝕜 e e' b)) ∘ ⇑f) b

Tactic:
ext v

Goal state after:
𝕜 : Type u_1
B : Type u_2
B' : Type u_3
F : Type u_4
E : B → Type u_5
n : WithTop ℕ∞
inst✝¹⁹ : NontriviallyNormedField 𝕜
inst✝¹⁸ : (x : B) → AddCommMonoid (E x)
inst✝¹⁷ : (x : B) → Module 𝕜 (E x)
inst✝¹⁶ : NormedAddCommGroup F
inst✝¹⁵ : NormedSpace 𝕜 F
inst✝¹⁴ : TopologicalSpace (TotalSpace F E)
inst✝¹³ : (x : B) → TopologicalSpace (E x)
EB : Type u_6
inst✝¹² : NormedAddCommGroup EB
inst✝¹¹ : NormedSpace 𝕜 EB
HB : Type u_7
inst✝¹⁰ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝⁹ : TopologicalSpace B
inst✝⁸ : ChartedSpace HB B
EB' : Type u_8
inst✝⁷ : NormedAddCommGroup EB'
inst✝⁶ : NormedSpace 𝕜 EB'
HB' : Type u_9
inst✝⁵ : TopologicalSpace HB'
IB' : ModelWithCorners 𝕜 EB' HB'
inst✝⁴ : TopologicalSpace B'
inst✝³ : ChartedSpace HB' B'
inst✝² : FiberBundle F E
inst✝¹ : VectorBundle 𝕜 F E
inst✝ : ContMDiffVectorBundle n F E IB
f : C^n⟮IB', B'; IB, B⟯
e : Trivialization F TotalSpace.proj
he : MemTrivializationAtlas e
e' : Trivialization F TotalSpace.proj
he' : MemTrivializationAtlas e'
b : B'
hb : f b ∈ e.baseSet ∩ e'.baseSet
v : F
⊢ ↑(Trivialization.coordChangeL 𝕜 (e.pullback f) (e'.pullback f) b) v =
    (((fun b ↦ ↑(Trivialization.coordChangeL 𝕜 e e' b)) ∘ ⇑f) b) v
Module: Mathlib.GroupTheory.CommutingProbability

Goal state before:
pp: n : ℕ
hn : Odd n
⊢ (↑n + 3) / (2 * (2 * ↑n)) = (↑n + 3) / (4 * ↑n)

Tactic:
← mul_assoc

Goal state after:
n : ℕ
hn : Odd n
⊢ (↑n + 3) / (2 * 2 * ↑n) = (↑n + 3) / (4 * ↑n)
Module: Mathlib.GroupTheory.PresentedGroup

Goal state before:
pp: case mk.mul
α : Type u_1
rels : Set (FreeGroup α)
H : Subgroup (PresentedGroup rels)
h : ∀ (j : α), of j ∈ H
x : PresentedGroup rels
x✝ y✝ : FreeGroup α
a✝¹ : Quot.mk (⇑(QuotientGroup.leftRel (Subgroup.normalClosure rels))) x✝ ∈ H
a✝ : Quot.mk (⇑(QuotientGroup.leftRel (Subgroup.normalClosure rels))) y✝ ∈ H
⊢ Quot.mk (⇑(QuotientGroup.leftRel (Subgroup.normalClosure rels))) (x✝ * y✝) ∈ H

Tactic:
rename_i h1 h2

Goal state after:
case mk.mul
α : Type u_1
rels : Set (FreeGroup α)
H : Subgroup (PresentedGroup rels)
h : ∀ (j : α), of j ∈ H
x : PresentedGroup rels
x✝ y✝ : FreeGroup α
h1 : Quot.mk (⇑(QuotientGroup.leftRel (Subgroup.normalClosure rels))) x✝ ∈ H
h2 : Quot.mk (⇑(QuotientGroup.leftRel (Subgroup.normalClosure rels))) y✝ ∈ H
⊢ Quot.mk (⇑(QuotientGroup.leftRel (Subgroup.normalClosure rels))) (x✝ * y✝) ∈ H
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
⊢ Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) x ∘SL
      Trivialization.symmL ℝ (trivializationAt F E x) x =
    ContinuousLinearMap.id ℝ F

Tactic:
ext v

Goal state after:
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
v : F
⊢ (Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) x ∘SL
        Trivialization.symmL ℝ (trivializationAt F E x) x)
      v =
    (ContinuousLinearMap.id ℝ F) v
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, 0 < δ ∧ 1 + δ * C < r' ^ 2

Tactic:
have A : ∀ᶠ δ in 𝓝[>] (0 : ℝ), 0 < δ := self_mem_nhdsWithin

Goal state after:
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
A : ∀ᶠ (δ : ℝ) in 𝓝[>] 0, 0 < δ
⊢ ∃ δ, 0 < δ ∧ 1 + δ * C < r' ^ 2
Module: Mathlib.GroupTheory.Coxeter.Basic

Goal state before:
pp: case succ
B : Type u_1
i j : B
n : ℕ
h : ∀ (k : ℕ) (hk : k < n), (alternatingWord i j n)[k] = if Even (n + k) then i else j
⊢ ∀ (k : ℕ) (hk : k < n + 1), (alternatingWord i j (n + 1))[k] = if Even (n + 1 + k) then i else j

Tactic:
grind [CoxeterSystem.alternatingWord_succ']

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

Goal state before:
pp: B : Type u_1
i j : B
p k : ℕ
h' : k < 2 * p → take k (alternatingWord i j (2 * p)) = if Even k then alternatingWord i j k else alternatingWord j i k
h : k + 1 < 2 * p
h_even : ¬Even k
hk : take k (alternatingWord i j (2 * p)) = alternatingWord j i k
⊢ Odd (2 * p + k)

Tactic:
by apply Nat.odd_add.mpr; simp [h_even]

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

Goal state before:
pp: case inl
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
k : ℤ
⊢ 1 * (cs.simple i * cs.simple i') ^ (k * 2 / 2) = 1 * (cs.simple i' * cs.simple i) ^ ((↑(M.M i i') - k) * 2 / 2)

Tactic:
Int.mul_ediv_cancel _ (by simp)

Goal state after:
case inl
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
k : ℤ
⊢ 1 * (cs.simple i * cs.simple i') ^ k = 1 * (cs.simple i' * cs.simple i) ^ ((↑(M.M i i') - k) * 2 / 2)
Module: Mathlib.GroupTheory.Coxeter.Basic

Goal state before:
pp: case inl
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
k : ℤ
⊢ 1 * (cs.simple i * cs.simple i') ^ k = 1 * (cs.simple i' * cs.simple i) ^ ((↑(M.M i i') - k) * 2 / 2)

Tactic:
Int.mul_ediv_cancel _ (by simp)

Goal state after:
case inl
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
k : ℤ
⊢ 1 * (cs.simple i * cs.simple i') ^ k = 1 * (cs.simple i' * cs.simple i) ^ (↑(M.M i i') - k)
Module: Mathlib.GroupTheory.Coxeter.Basic

Goal state before:
pp: case inl
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
k : ℤ
⊢ 1 * (cs.simple i * cs.simple i') ^ k = 1 * (cs.simple i' * cs.simple i) ^ (↑(M.M i i') - k)

Tactic:
Int.mul_ediv_cancel _ (by simp)

Goal state after:
case inl
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
k : ℤ
⊢ 1 * (cs.simple i * cs.simple i') ^ k = 1 * (cs.simple i' * cs.simple i) ^ (↑(M.M i i') - k)
Module: Mathlib.GroupTheory.Coxeter.Length

Goal state before:
pp: B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
w₁ w₂ : W
⊢ cs.length (w₁ * w₂) % 2 = (cs.length w₁ + cs.length w₂) % 2

Tactic:
by
  rw [← ZMod.natCast_eq_natCast_iff', Nat.cast_add]
  simpa only [lengthParity_eq_ofAdd_length, ofAdd_add] using! map_mul cs.lengthParity w₁ w₂

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

Goal state before:
pp: case cons
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i : B
ω : List B
ih : (cs.rightInvSeq ω).length = ω.length
⊢ (cs.rightInvSeq (i :: ω)).length = (i :: ω).length

Tactic:
simpa [rightInvSeq]

Goal state after:
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: case cons.cons
ι : Type u_1
M : ι → Type u_2
inst✝ : (i : ι) → Monoid (M i)
x y : (i : ι) × M i
l : List ((i : ι) × M i)
hi :
  ∀ (hnot1 : ∀ l_1 ∈ y :: l, l_1.snd ≠ 1) (hchain : List.IsChain (fun l l' ↦ l.fst ≠ l'.fst) (y :: l)),
    { toList := y :: l, ne_one := hnot1, chain_ne := hchain } ≠ empty →
      ∃ i j w', w'.toWord.toList = { toList := y :: l, ne_one := hnot1, chain_ne := hchain }.toList
hnot1✝ : ∀ l_1 ∈ x :: y :: l, l_1.snd ≠ 1
hnot1 : x.snd ≠ 1 ∧ ∀ x ∈ y :: l, x.snd ≠ 1
hchain : List.IsChain (fun l l' ↦ l.fst ≠ l'.fst) (x :: y :: l)
h : { toList := x :: y :: l, ne_one := hnot1✝, chain_ne := hchain } ≠ empty
⊢ ∃ i j w', w'.toWord.toList = { toList := x :: y :: l, ne_one := hnot1✝, chain_ne := hchain }.toList

Tactic:
· rw [List.isChain_cons_cons] at hchain
      specialize hi hnot1.2 hchain.2 (by rintro ⟨rfl⟩)
      obtain ⟨i, j, w', hw' : w'.toList = y::l⟩ := hi
      obtain rfl : y = ⟨i, w'.head⟩ := by simpa [hw'] using w'.toList_head?
      refine ⟨x.1, j, append (singleton x.2 hnot1.1) hchain.1 w', ?_⟩
      simpa [toWord] using hw'

Goal state after: