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

Goal state before:
pp: 𝕜 : Type u_1
E'' : Type u_3
H : Type u_7
G : Type u_9
E : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup E''
inst✝⁶ : NormedSpace 𝕜 E''
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 g : M → N
x : M
hf : IsSubmersionAt I J n f x
hfg : f =ᶠ[𝓝 x] g
⊢ NormedAddCommGroup hf.complement

Tactic:
infer_instance

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

Goal state before:
pp: 𝕜 : Type u_1
E'' : Type u_3
H : Type u_7
G : Type u_9
E : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup E''
inst✝⁶ : NormedSpace 𝕜 E''
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 g : M → N
x : M
hf : IsSubmersionAt I J n f x
hfg : f =ᶠ[𝓝 x] g
⊢ NormedSpace 𝕜 hf.complement

Tactic:
infer_instance

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

Goal state before:
pp: 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 : IsEmpty M
⊢ NormedAddCommGroup PUnit.{u + 1}

Tactic:
infer_instance

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

Goal state before:
pp: 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 : IsEmpty M
⊢ NormedSpace 𝕜 PUnit.{u + 1}

Tactic:
infer_instance

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

Goal state before:
pp: 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 := default
⊢ NormedAddCommGroup ⋯.smallComplement

Tactic:
infer_instance

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

Goal state before:
pp: 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 := default
⊢ NormedSpace 𝕜 ⋯.smallComplement

Tactic:
infer_instance

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

Goal state before:
pp: 𝕜 : Type u_1
E'' : Type u_3
H : Type u_7
G : Type u_9
E : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup E''
inst✝⁶ : NormedSpace 𝕜 E''
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 : IsSubmersion I J n f
x : M
⊢ NormedAddCommGroup h.complement

Tactic:
infer_instance

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

Goal state before:
pp: 𝕜 : Type u_1
E'' : Type u_3
H : Type u_7
G : Type u_9
E : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup E''
inst✝⁶ : NormedSpace 𝕜 E''
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 : IsSubmersion I J n f
x : M
⊢ NormedSpace 𝕜 h.complement

Tactic:
infer_instance

Goal state after:
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
⊢ NormedAddCommGroup PUnit.{u + 1}

Tactic:
infer_instance

Goal state after:
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
⊢ NormedSpace 𝕜 PUnit.{u + 1}

Tactic:
infer_instance

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

Goal state before:
pp: case hs
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
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
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : IsManifold I (minSmoothness 𝕜 3) M
inst✝ : CompleteSpace E
U V W : (x : M) → TangentSpace I x
s : Set M
x : M
hs : UniqueMDiff[s]
h's : x ∈ closure[inst✝³] (interior s)
hx : x ∈ s
hU : ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, U x⟩) s x
hV : ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, V x⟩) s x
hW : ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, W x⟩) s x
A : minSmoothness 𝕜 2 + 1 ≤ minSmoothness 𝕜 3
s_inter_mem : s ∩ (extChartAt I x).source ∈ 𝓝[s] x
pre_mem : ↑(extChartAt I x) ⁻¹' ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) ∈ 𝓝[s] x
U' : (x : E) → TangentSpace 𝓘(𝕜, E) x := ⋯
V' : (x : E) → TangentSpace 𝓘(𝕜, E) x := ⋯
W' : (x : E) → TangentSpace 𝓘(𝕜, E) x := ⋯
J0U :
  ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, U' x⟩)
    ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) x)
J0V :
  ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, V' x⟩)
    ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) x)
J0W :
  ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, W' x⟩)
    ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) x)
J1U :
  ∀ᶠ (y : M) in 𝓝[s] x,
    ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, U' x⟩)
      ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) y)
J1V :
  ∀ᶠ (y : M) in 𝓝[s] x,
    ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, V' x⟩)
      ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) y)
J1W :
  ∀ᶠ (y : M) in 𝓝[s] x,
    ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, W' x⟩)
      ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) y)
JU : U =ᶠ[𝓝[s] x] mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) U'
JV : V =ᶠ[𝓝[s] x] mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) V'
JW : W =ᶠ[𝓝[s] x] mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) W'
this :
  ∀ᶠ (y : M) in 𝓝[s] x,
    mlieBracketWithin I (mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) V') (mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) W') s y =
      mpullback I 𝓘(𝕜, E) (↑(extChartAt I x))
        (mlieBracketWithin 𝓘(𝕜, E) V' W' ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s)) y
⊢ UniqueMDiff[(extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s]

Tactic:
exact hs.uniqueMDiffOn_target_inter x

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

Goal state before:
pp: case hs
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
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
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : IsManifold I (minSmoothness 𝕜 3) M
inst✝ : CompleteSpace E
U V W : (x : M) → TangentSpace I x
s : Set M
x : M
hs : UniqueMDiff[s]
h's : x ∈ closure[inst✝³] (interior s)
hx : x ∈ s
hU : ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, U x⟩) s x
hV : ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, V x⟩) s x
hW : ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, W x⟩) s x
A : minSmoothness 𝕜 2 + 1 ≤ minSmoothness 𝕜 3
s_inter_mem : s ∩ (extChartAt I x).source ∈ 𝓝[s] x
pre_mem : ↑(extChartAt I x) ⁻¹' ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) ∈ 𝓝[s] x
U' : (x : E) → TangentSpace 𝓘(𝕜, E) x := ⋯
V' : (x : E) → TangentSpace 𝓘(𝕜, E) x := ⋯
W' : (x : E) → TangentSpace 𝓘(𝕜, E) x := ⋯
J0U :
  ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, U' x⟩)
    ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) x)
J0V :
  ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, V' x⟩)
    ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) x)
J0W :
  ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, W' x⟩)
    ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) x)
J1U :
  ∀ᶠ (y : M) in 𝓝[s] x,
    ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, U' x⟩)
      ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) y)
J1V :
  ∀ᶠ (y : M) in 𝓝[s] x,
    ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, V' x⟩)
      ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) y)
J1W :
  ∀ᶠ (y : M) in 𝓝[s] x,
    ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, W' x⟩)
      ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) y)
JU : U =ᶠ[𝓝[s] x] mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) U'
JV : V =ᶠ[𝓝[s] x] mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) V'
JW : W =ᶠ[𝓝[s] x] mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) W'
this✝ :
  ∀ᶠ (y : M) in 𝓝[s] x,
    mlieBracketWithin I (mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) V') (mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) W') s y =
      mpullback I 𝓘(𝕜, E) (↑(extChartAt I x))
        (mlieBracketWithin 𝓘(𝕜, E) V' W' ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s)) y
this :
  ∀ᶠ (y : M) in 𝓝[s] x,
    mlieBracketWithin I (mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) U') (mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) V') s y =
      mpullback I 𝓘(𝕜, E) (↑(extChartAt I x))
        (mlieBracketWithin 𝓘(𝕜, E) U' V' ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s)) y
⊢ UniqueMDiff[(extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s]

Tactic:
exact hs.uniqueMDiffOn_target_inter x

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Torsion

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
x : M
inst✝³ : IsManifold I 2 M
inst✝² : CompleteSpace E
cov : ((x : M) → TangentSpace I x) → (x : M) → TangentSpace I x →L[𝕜] TangentSpace I x
inst✝¹ : CompleteSpace 𝕜
inst✝ : FiniteDimensional 𝕜 E
hcov : IsCovariantDerivativeOn E cov
X₀ Y₀ : TangentSpace I x
⊢ ((hcov.torsion x) X₀) Y₀ = -((hcov.torsion x) Y₀) X₀

Tactic:
by
  simp only [torsion_apply_eq_extend, neg_sub]
  rw [VectorField.mlieBracket_swap]
  dsimp
  module

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

Goal state before:
pp: ι : Type uι
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
inst✝ : Fintype ι
s : Set M
f : SmoothBumpCovering ι I M s
x : M
hx : x ∈ s
y : M
a✝ : y ∈ s
h :
  ∀ (x_1 : ι),
    (↑(f.toFun x_1) x • ↑(extChartAt I (f.c x_1)) x, ↑(f.toFun x_1) x) =
      (↑(f.toFun x_1) y • ↑(extChartAt I (f.c x_1)) y, ↑(f.toFun x_1) y)
h₁ : ↑(extChartAt I (f.c (f.ind x hx))) x = ↑(extChartAt I (f.c (f.ind x hx))) y
h₂ : 1 = ↑(f.toFun (f.ind x hx)) y
this : y ∈ (extChartAt I (f.c (f.ind x hx))).source
⊢ x = y

Tactic:
exact (extChartAt I (f.c _)).injOn (f.mem_extChartAt_ind_source x hx) this h₁

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

Goal state before:
pp: 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
inst✝ : CompactSpace M
ι : Type uM
f : SmoothBumpCovering ι I M
⊢ ∃ n e, ContMDiff I (𝓡 n) ∞ e ∧ IsClosedEmbedding e ∧ ∀ (x : M), Injective ⇑(mfderiv% e x)

Tactic:
have := f.fintype

Goal state after:
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
inst✝ : CompactSpace M
ι : Type uM
f : SmoothBumpCovering ι I M
this : Fintype ι
⊢ ∃ n e, ContMDiff I (𝓡 n) ∞ e ∧ IsClosedEmbedding e ∧ ∀ (x : M), Injective ⇑(mfderiv% e x)
Module: Mathlib.GroupTheory.GroupAction.CardCommute

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : Group α
inst✝⁴ : MulAction α β
inst✝³ : Fintype α
inst✝² : Fintype β
inst✝¹ : Fintype (Quotient (orbitRel α β))
inst✝ : (b : β) → Fintype ↥(stabilizer α b)
φ : Quotient (orbitRel α β) → β
hφ : LeftInverse Quotient.mk'' φ
ω : Quotient (orbitRel α β)
⊢ Nonempty ↥(stabilizer α (φ ω))

Tactic:
infer_instance

Goal state after:
Module: Mathlib.GroupTheory.GroupAction.CardCommute

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : Mul α
inst✝ : Infinite α
⊢ Injective fun a ↦ ⟨(a, a), ⋯⟩

Tactic:
intro

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : Mul α
inst✝ : Infinite α
a₁✝ : α
⊢ ∀ ⦃a₂ : α⦄, (fun a ↦ ⟨(a, a), ⋯⟩) a₁✝ = (fun a ↦ ⟨(a, a), ⋯⟩) a₂ → a₁✝ = a₂
Module: Mathlib.GroupTheory.CommutingProbability

Goal state before:
pp: case pos
G : Type u_2
inst✝ : Group G
h : ↑(Nat.card G) = 0
⊢ ↑(Nat.card (ConjClasses G)) * 0 / 0 = ↑(Nat.card (ConjClasses G)) / 0

Tactic:
div_zero,

Goal state after:
case pos
G : Type u_2
inst✝ : Group G
h : ↑(Nat.card G) = 0
⊢ 0 = ↑(Nat.card (ConjClasses G)) / 0
Module: Mathlib.GroupTheory.CommutingProbability

Goal state before:
pp: case pos
n : ℕ
h0 : n = 0
⊢ commProb (DihedralGroup 0) = 0

Tactic:
apply commProb_eq_zero_of_infinite

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

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝¹ : Monoid M
inst✝ : Monoid N
⊢ (MonoidHom.snd M N).comp toProd = snd

Tactic:
rw [← fst_prod_snd, MonoidHom.snd_comp_prod]

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

Goal state before:
pp: B : Type u_1
B' : Type u_2
M : CoxeterMatrix B
e : B ≃ B'
⊢ Surjective ⇑↑(FreeGroup.freeGroupCongr e)

Tactic:
by simpa using (FreeGroup.freeGroupCongr e).surjective

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 '' Set.range PresentedGroup.of) = ⊤

Tactic:
← MulEquiv.coe_toMonoidHom,

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.closure (⇑cs.mulEquiv.symm.toMonoidHom '' Set.range PresentedGroup.of) = ⊤
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
⊢ Set.range cs.simple = Set.range cs.simple ∪ (Set.range cs.simple)⁻¹

Tactic:
inv_range,

Goal state after:
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
⊢ Set.range cs.simple = Set.range cs.simple ∪ Set.range fun i ↦ (cs.simple i)⁻¹
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: ι : Type u_1
M : ι → Type u_2
inst✝² : (i : ι) → Monoid (M i)
N : Type u_3
inst✝¹ : Monoid N
G : ι → Type u_4
inst✝ : (i : ι) → Group (G i)
m : CoprodI G
⊢ MulOpposite.unop ((lift fun i ↦ (MonoidHom.op of).comp (MulEquiv.inv' (G i)).toMonoidHom) m) * m = 1

Tactic:
induction m using CoprodI.induction_on with
      | one => rw [map_one, MulOpposite.unop_one, one_mul]
      | of m ih =>
        change of _⁻¹ * of _ = 1
        rw [← of.map_mul, inv_mul_cancel, of.map_one]
      | mul x y ihx ihy =>
        rw [map_mul, MulOpposite.unop_mul, mul_assoc, ← mul_assoc _ x y, ihx, one_mul,
          ihy]

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
A : ∀ᶠ (δ : ℝ) in 𝓝[>] 0, 0 < δ
B : Tendsto (fun δ ↦ 1 + δ * C) (𝓝[>] 0) (𝓝 (1 + 0 * C))
B' : ∀ᶠ (δ : ℝ) in 𝓝[>] 0, 1 + δ * C < r' ^ 2
⊢ ∃ δ, 0 < δ ∧ 1 + δ * C < r' ^ 2

Tactic:
exact (A.and B').exists

Goal state after:
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: ι : Type u_1
M : ι → Type u_2
inst✝² : (i : ι) → Monoid (M i)
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (M i)
i j : ι
w : Word M
m : M i
⊢ ⟨i, m⟩ ∈
      (↑(consRecOn w ⟨{ head := 1, tail := empty, fstIdx_ne := ⋯ }, ⋯⟩ fun j_1 m w h1 h2 x ↦
              if ij : j = j_1 then ⟨{ head := ⋯ ▸ m, tail := w, fstIdx_ne := ⋯ }, ⋯⟩
              else ⟨{ head := 1, tail := cons m w h1 h2, fstIdx_ne := ⋯ }, ⋯⟩)).tail.toList ↔
    ⟨i, m⟩ ∈ w.toList.tail ∨ ¬i = j ∧ ∃ (h : ¬w.toList = []), w.toList.head h = ⟨i, m⟩

Tactic:
induction w using consRecOn with
  | empty => simp
  | cons k g tail h1 h2 ih =>
    simp only [consRecOn_cons]
    split_ifs with h
    · subst k
      by_cases hij : j = i <;> simp_all
    · by_cases hik : i = k
      · subst i; simp_all [@eq_comm _ m g, @eq_comm _ k j, or_comm]
      · simp [hik, Ne.symm hik]

Goal state after:
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: case pos.mpr
ι : Type u_1
M : ι → Type u_2
inst✝² : (i : ι) → Monoid (M i)
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (M i)
w : Word M
hnil : ¬w.toList = []
m₂ m₁ : M (w.toList.head hnil).fst
hw : ⟨(w.toList.head hnil).fst, m₁⟩ ∉ w.toList.tail
hm1 : ¬m₁ = 1
⊢ (∃ m', w.toList.head hnil = ⟨(w.toList.head hnil).fst, m'⟩ ∧ m₁ = m₂ * m') ∨
      ¬w.fstIdx = some (w.toList.head hnil).fst ∧ m₁ = m₂ →
    m₁ = m₂ * (w.toList.head ⋯).snd

Tactic:
rintro (⟨_, h, rfl⟩ | hm')

Goal state after:
case pos.mpr.inl
ι : Type u_1
M : ι → Type u_2
inst✝² : (i : ι) → Monoid (M i)
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (M i)
w : Word M
hnil : ¬w.toList = []
m₂ w✝ : M (w.toList.head hnil).fst
h : w.toList.head hnil = ⟨(w.toList.head hnil).fst, w✝⟩
hw : ⟨(w.toList.head hnil).fst, m₂ * w✝⟩ ∉ w.toList.tail
hm1 : ¬m₂ * w✝ = 1
⊢ m₂ * w✝ = m₂ * (w.toList.head ⋯).snd

case pos.mpr.inr
ι : Type u_1
M : ι → Type u_2
inst✝² : (i : ι) → Monoid (M i)
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (M i)
w : Word M
hnil : ¬w.toList = []
m₂ m₁ : M (w.toList.head hnil).fst
hw : ⟨(w.toList.head hnil).fst, m₁⟩ ∉ w.toList.tail
hm1 : ¬m₁ = 1
hm' : ¬w.fstIdx = some (w.toList.head hnil).fst ∧ m₁ = m₂
⊢ m₁ = m₂ * (w.toList.head ⋯).snd
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
w t : W
ht : cs.IsReflection t
⊢ cs.length (t⁻¹ * w) < cs.length w⁻¹ ↔ cs.length (t * w) < cs.length w

Tactic:
ht.inv,

Goal state after:
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
w t : W
ht : cs.IsReflection t
⊢ cs.length (t * w) < cs.length w⁻¹ ↔ cs.length (t * w) < cs.length w
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
t : W
ht : cs.IsReflection t
w : W
⊢ cs.IsRightInversion (w⁻¹ * t⁻¹) t ↔ ¬cs.IsRightInversion w⁻¹ t

Tactic:
ht.inv,

Goal state after:
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
t : W
ht : cs.IsReflection t
w : W
⊢ cs.IsRightInversion (w⁻¹ * t) t ↔ ¬cs.IsRightInversion w⁻¹ t
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: case one
ι : Type u_1
M : ι → Type u_2
inst✝² : (i : ι) → Monoid (M i)
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (M i)
⊢ ∀ (w : Word M), (1 • w).prod = 1 * w.prod

Tactic:
intro

Goal state after:
case one
ι : Type u_1
M : ι → Type u_2
inst✝² : (i : ι) → Monoid (M i)
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (M i)
w✝ : Word M
⊢ (1 • w✝).prod = 1 * w✝.prod