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

Goal state before:
pp: case inr
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
t : (x : M) → V x
hs : IsLocalFrameOn I F n s u
inst✝¹ : VectorBundle 𝕜 F V
inst✝ : FiniteDimensional 𝕜 F
h : ∀ (i : ι), ContMDiffOn I 𝓘(𝕜, 𝕜) n ((LinearMap.piApply (hs.coeff i)) t) u
x : M
hx : x ∈ u
this✝ : Fintype ι
this : ∀ (i : ι), ContMDiffOn I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, ((LinearMap.piApply (hs.coeff i)) t • s i) x⟩) u
almost : ContMDiffOn I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, ∑ i, (LinearMap.piApply (hs.coeff i)) t x • s i x⟩) u
⊢ ∀ y ∈ u, ⟨y, t y⟩ = ⟨y, ∑ i, (LinearMap.piApply (hs.coeff i)) t y • s i y⟩

Tactic:
intro y hy

Goal state after:
case inr
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
t : (x : M) → V x
hs : IsLocalFrameOn I F n s u
inst✝¹ : VectorBundle 𝕜 F V
inst✝ : FiniteDimensional 𝕜 F
h : ∀ (i : ι), ContMDiffOn I 𝓘(𝕜, 𝕜) n ((LinearMap.piApply (hs.coeff i)) t) u
x : M
hx : x ∈ u
this✝ : Fintype ι
this : ∀ (i : ι), ContMDiffOn I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, ((LinearMap.piApply (hs.coeff i)) t • s i) x⟩) u
almost : ContMDiffOn I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, ∑ i, (LinearMap.piApply (hs.coeff i)) t x • s i x⟩) u
y : M
hy : y ∈ u
⊢ ⟨y, t y⟩ = ⟨y, ∑ i, (LinearMap.piApply (hs.coeff i)) t y • s i y⟩
Module: Mathlib.Geometry.Manifold.VectorBundle.LocalFrame

Goal state before:
pp: case inr
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
t : (x : M) → V x
inst✝¹ : VectorBundle 𝕜 F V
hs : IsLocalFrameOn I F 1 s u
inst✝ : FiniteDimensional 𝕜 F
h : ∀ (i : ι), MDiff[u] ((LinearMap.piApply (hs.coeff i)) t)
x : M
hx : x ∈ u
this✝ : Fintype ι
this : ∀ (i : ι), MDiff[u] fun x ↦ ⟨x, ((LinearMap.piApply (hs.coeff i)) t • s i) x⟩
almost : MDiff[u] fun x ↦ ⟨x, ∑ i, (hs.coeff i x) (t x) • s i x⟩
⊢ ∀ y ∈ u, ⟨y, t y⟩ = ⟨y, ∑ i, (hs.coeff i y) (t y) • s i y⟩

Tactic:
intro y hy

Goal state after:
case inr
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
t : (x : M) → V x
inst✝¹ : VectorBundle 𝕜 F V
hs : IsLocalFrameOn I F 1 s u
inst✝ : FiniteDimensional 𝕜 F
h : ∀ (i : ι), MDiff[u] ((LinearMap.piApply (hs.coeff i)) t)
x : M
hx : x ∈ u
this✝ : Fintype ι
this : ∀ (i : ι), MDiff[u] fun x ↦ ⟨x, ((LinearMap.piApply (hs.coeff i)) t • s i) x⟩
almost : MDiff[u] fun x ↦ ⟨x, ∑ i, (hs.coeff i x) (t x) • s i x⟩
y : M
hy : y ∈ u
⊢ ⟨y, t y⟩ = ⟨y, ∑ i, (hs.coeff i y) (t y) • s i y⟩
Module: Mathlib.Geometry.Manifold.VectorBundle.LocalFrame

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
F : Type u_5
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
n : ℕ∞ω
V : M → Type u_6
inst✝⁷ : TopologicalSpace (TotalSpace F V)
inst✝⁶ : (x : M) → AddCommGroup (V x)
inst✝⁵ : (x : M) → Module 𝕜 (V x)
inst✝⁴ : (x : M) → TopologicalSpace (V x)
inst✝³ : FiberBundle F V
inst✝² : VectorBundle 𝕜 F V
inst✝¹ : ContMDiffVectorBundle n F V I
ι : Type u_7
e : Trivialization F TotalSpace.proj
inst✝ : MemTrivializationAtlas e
b : Basis ι 𝕜 F
i : ι
⊢ ∀ y ∈ e.baseSet, (↑e ⟨y, e.localFrame b i y⟩).2 = b i

Tactic:
intro y hy

Goal state after:
𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
F : Type u_5
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
n : ℕ∞ω
V : M → Type u_6
inst✝⁷ : TopologicalSpace (TotalSpace F V)
inst✝⁶ : (x : M) → AddCommGroup (V x)
inst✝⁵ : (x : M) → Module 𝕜 (V x)
inst✝⁴ : (x : M) → TopologicalSpace (V x)
inst✝³ : FiberBundle F V
inst✝² : VectorBundle 𝕜 F V
inst✝¹ : ContMDiffVectorBundle n F V I
ι : Type u_7
e : Trivialization F TotalSpace.proj
inst✝ : MemTrivializationAtlas e
b : Basis ι 𝕜 F
i : ι
y : M
hy : y ∈ e.baseSet
⊢ (↑e ⟨y, e.localFrame b i y⟩).2 = b i
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 := ⋯
this : ContMDiffAt I 𝓘(𝕜, 𝕜) k aux x
⊢ Set.EqOn ((LinearMap.piApply (Trivialization.localFrameCoeff I e b i)) s) aux e.baseSet

Tactic:
intro y hy

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
this : ContMDiffAt I 𝓘(𝕜, 𝕜) k aux x
y : M
hy : y ∈ e.baseSet
⊢ (LinearMap.piApply (Trivialization.localFrameCoeff I e b i)) s y = aux y
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 := ⋯
this : MDiffAt aux x
⊢ Set.EqOn ((LinearMap.piApply (Trivialization.localFrameCoeff I e b i)) s) aux e.baseSet

Tactic:
intro y hy

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
this : MDiffAt aux x
y : M
hy : y ∈ e.baseSet
⊢ (LinearMap.piApply (Trivialization.localFrameCoeff I e b i)) s y = aux y
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
⊢ Function.RightInverse (fun f x ↦ f.comp of) fun fi ↦
    (conGen (Rel M)).lift (FreeMonoid.lift fun p ↦ (fi p.fst) p.snd) ⋯

Tactic:
intro f
    ext i x
    rfl

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

Goal state before:
pp: case mp
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
hω : cs.IsReduced ω
h : cs.length (cs.wordProd ω) = 0
⊢ cs.wordProd ω = 1

Tactic:
have : ω = [] := eq_nil_of_length_eq_zero (hω.symm.trans h)

Goal state after:
case mp
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
hω : cs.IsReduced ω
h : cs.length (cs.wordProd ω) = 0
this : ω = []
⊢ cs.wordProd ω = 1
Module: Mathlib.GroupTheory.Coxeter.Length

Goal state before:
pp: case inr
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
w : W
i : B
h : cs.length w + 1 ≤ cs.length (w * cs.simple i)
⊢ cs.length (w * cs.simple i) = cs.length w + 1 ∨ cs.length (w * cs.simple i) + 1 = cs.length w

Tactic:
refine .inl (h.antisymm' ?_)

Goal state after:
case inr
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
w : W
i : B
h : cs.length w + 1 ≤ cs.length (w * cs.simple i)
⊢ cs.length (w * cs.simple i) ≤ cs.length w + 1
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
hω : cs.IsReduced ω
t : W
ht : t ∈ cs.rightInvSeq ω
⊢ cs.IsRightInversion (cs.wordProd ω) t

Tactic:
by
  constructor
  · exact cs.isReflection_of_mem_rightInvSeq ω ht
  · obtain ⟨j, hj, rfl⟩ := List.mem_iff_getElem.mp ht
    rw [← List.getD_eq_getElem _ 1 hj, wordProd_mul_getD_rightInvSeq]
    rw [cs.length_rightInvSeq] at hj
    calc
      ℓ (π (ω.eraseIdx j))
      _ ≤ (ω.eraseIdx j).length := cs.length_wordProd_le _
      _ < ω.length := by rw [← List.length_eraseIdx_add_one hj]; exact lt_add_one _
      _ = ℓ (π ω) := hω.symm

Goal state after:
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: M : Type u_2
inst✝ : AddCommGroup M
a : ℚ
m : M
s : ℕ+
⊢ a • mk m s = mk (a.num • m) (⟨a.den, ⋯⟩ * s)

Tactic:
obtain h | h := le_total 0 a
  · rw [qsmul_of_nonneg h, nnqsmul_mk, ← natCast_zsmul]
    congr
    simpa using h
  · rw [qsmul_of_nonpos h]
    have : a.num.natAbs • m = -a.num • m := by
      rw [← natCast_zsmul]
      congr
      simpa using h
    simp [nnqsmul_mk, this, ← neg_mk]

Goal state after:
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
a h : G
hh : h ∈ ↑H
k : G
hk : k ∈ ↑K
hb : h * a * k ∈ doubleCoset a ↑H ↑K
⊢ ↑H * {h} * {a} * ↑K = ↑H * {a} * ↑K

Tactic:
Subgroup.subgroup_mul_singleton hh

Goal state after:
G : Type u_1
inst✝ : Group G
H K : Subgroup G
a h : G
hh : h ∈ ↑H
k : G
hk : k ∈ ↑K
hb : h * a * k ∈ doubleCoset a ↑H ↑K
⊢ ↑H * {a} * ↑K = ↑H * {a} * ↑K
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
I : Finset (Quotient ↑H ↑K)
hI : ⋃ i ∈ I, quotToDoubleCoset H K i = Set.univ
this : ↑I = Set.univ
⊢ Finite (Quotient ↑H ↑K)

Tactic:
simp_rw [← Set.finite_univ_iff, ← this, I.finite_toSet]

Goal state after:
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
t : Finset (Quotient ↑H ↑K)
ht : ¬⋃ q ∈ t, doubleCoset (out q) ↑H ↑K = Set.univ
⊢ ¬Set.univ ⊆ ⋃ i ∈ t, Quot.mk ⇑(leftRel K) '' doubleCoset (out i) ↑H ↑K

Tactic:
simp only [Set.univ_subset_iff, ← ne_eq] at ⊢ ht

Goal state after:
G : Type u_1
inst✝ : Group G
H K : Subgroup G
t : Finset (Quotient ↑H ↑K)
ht : ⋃ q ∈ t, doubleCoset (out q) ↑H ↑K ≠ Set.univ
⊢ ⋃ i ∈ t, Quot.mk ⇑(leftRel K) '' doubleCoset (out i) ↑H ↑K ≠ Set.univ
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
t : Finset (Quotient ↑H ↑K)
ht : ¬⋃ q ∈ t, doubleCoset (out q) ↑H ↑K = Set.univ
⊢ ¬Set.univ ⊆ ⋃ i ∈ t, Quot.mk ⇑(rightRel H) '' doubleCoset (out i) ↑H ↑K

Tactic:
simp only [Set.univ_subset_iff, ← ne_eq] at ⊢ ht

Goal state after:
G : Type u_1
inst✝ : Group G
H K : Subgroup G
t : Finset (Quotient ↑H ↑K)
ht : ⋃ q ∈ t, doubleCoset (out q) ↑H ↑K ≠ Set.univ
⊢ ⋃ i ∈ t, Quot.mk ⇑(rightRel H) '' doubleCoset (out i) ↑H ↑K ≠ Set.univ
Module: Mathlib.GroupTheory.Transfer

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
g : G
q : G ⧸ H
⊢ ↑(H.transferFunction g q) = q

Tactic:
transferFunction_apply,

Goal state after:
G : Type u_1
inst✝ : Group G
H : Subgroup G
g : G
q : G ⧸ H
⊢ ↑(g ^ ((H.quotientEquivSigmaZMod g) q).snd.cast * Quotient.out (Quotient.out ((H.quotientEquivSigmaZMod g) q).fst)) =
    q
Module: Mathlib.GroupTheory.Transfer

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
g : G
q : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)
k : ZMod (minimalPeriod (fun x ↦ g • x) (Quotient.out q))
⊢ H.transferFunction g (g ^ k.cast • Quotient.out q) = g ^ k.cast * Quotient.out (Quotient.out q)

Tactic:
transferFunction_apply,

Goal state after:
G : Type u_1
inst✝ : Group G
H : Subgroup G
g : G
q : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)
k : ZMod (minimalPeriod (fun x ↦ g • x) (Quotient.out q))
⊢ g ^ ((H.quotientEquivSigmaZMod g) (g ^ k.cast • Quotient.out q)).snd.cast *
      Quotient.out (Quotient.out ((H.quotientEquivSigmaZMod g) (g ^ k.cast • Quotient.out q)).fst) =
    g ^ k.cast * Quotient.out (Quotient.out q)
Module: Mathlib.GroupTheory.Transfer

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
g : G
q : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)
k : ZMod (minimalPeriod (fun x ↦ g • x) (Quotient.out q))
⊢ g • H.transferFunction g (g⁻¹ • g ^ k.cast • Quotient.out q) =
    if k = 0 then g ^ minimalPeriod (fun x ↦ g • x) (Quotient.out q) * Quotient.out (Quotient.out q)
    else g ^ k.cast * Quotient.out (Quotient.out q)

Tactic:
transferFunction_apply,

Goal state after:
G : Type u_1
inst✝ : Group G
H : Subgroup G
g : G
q : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)
k : ZMod (minimalPeriod (fun x ↦ g • x) (Quotient.out q))
⊢ g •
      (g ^ ((H.quotientEquivSigmaZMod g) (g⁻¹ • g ^ k.cast • Quotient.out q)).snd.cast *
        Quotient.out (Quotient.out ((H.quotientEquivSigmaZMod g) (g⁻¹ • g ^ k.cast • Quotient.out q)).fst)) =
    if k = 0 then g ^ minimalPeriod (fun x ↦ g • x) (Quotient.out q) * Quotient.out (Quotient.out q)
    else g ^ k.cast * Quotient.out (Quotient.out q)
Module: Mathlib.GroupTheory.Frattini

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
φ : G →* H
⊢ (frattini G).Characteristic

Tactic:
rw [characteristic_iff_comap_eq]
  intro φ
  apply φ.comapSubgroup.map_radical

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: G : Type u_1
inst✝ : Group G
⊢ Monotone (upperCentralSeries G)

Tactic:
refine monotone_nat_of_le_succ ?_
  intro n x hx y
  rw [commutatorElement_def, mul_assoc, mul_assoc, ← mul_assoc y x⁻¹ y⁻¹]
  exact mul_mem hx (Normal.conj_mem inferInstance x⁻¹ (inv_mem hx) y)

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: case refine_2
G : Type u_1
inst✝ : Group G
S : Subgroup G
⊢ (∃ n, S.lowerCentralSeries n = ⊥) → ∃ n H, IsDescendingCentralSeries H ∧ H n = ⊥

Tactic:
rintro ⟨n, hn⟩
    refine ⟨n, lowerCentralSeries ⊤, lowerCentralSeries_isDescendingCentralSeries, ?_⟩
    rwa [← map_subtype_inj, map_bot, top_subtype_lowerCentralSeries]

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

Goal state before:
pp: α : Type u_1
X : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction (FreeGroup α) X
x : X
w : α × Bool
g : FreeGroup α
hg : g ∈ startsWith w
l✝ : List (α × Bool) := g.toWord
h : ⟨g, hg⟩ = ⟨mk g.toWord, ⋯⟩
a b : α × Bool
l : List (α × Bool)
hl : a :: b :: l = g.toWord
ha : a = w
h1 : (a.1 = b.1 → a.2 = b.2) ∧ IsReduced (b :: l)
⊢ (mk [w])⁻¹ • ⟨mk ([w] ++ b :: l), ⋯⟩ • x ∈ MulAction.orbit (↑(startsWith b)) x

Tactic:
← mul_mk,

Goal state after:
α : Type u_1
X : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction (FreeGroup α) X
x : X
w : α × Bool
g : FreeGroup α
hg : g ∈ startsWith w
l✝ : List (α × Bool) := g.toWord
h : ⟨g, hg⟩ = ⟨mk g.toWord, ⋯⟩
a b : α × Bool
l : List (α × Bool)
hl : a :: b :: l = g.toWord
ha : a = w
h1 : (a.1 = b.1 → a.2 = b.2) ∧ IsReduced (b :: l)
⊢ (mk [w])⁻¹ • ⟨mk [w] * mk (b :: l), ⋯⟩ • x ∈ MulAction.orbit (↑(startsWith b)) x
Module: Mathlib.GroupTheory.GroupAction.Transitive

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
h : G
hy : h • a = y
⊢ (h * g⁻¹) • x = y

Tactic:
← hx,

Goal state after:
G : Type u_1
X : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
h : G
hy : h • a = y
⊢ (h * g⁻¹) • g • a = y
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
h₁ h₂ : ↥H
h : (fun x ↦ g • x) ((fun m ↦ m • a) h₁) = (fun m ↦ m • a) h₂
⊢ ↑h₂⁻¹ * g * ↑h₁ ∈ stabilizer G a

Tactic:
simpa only [mem_stabilizer_iff, InvMemClass.coe_inv, mul_smul, inv_smul_eq_iff]

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

Goal state before:
pp: M : Type u_1
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
inst✝¹ : Nontrivial M
is_perfect : commutator M = ⊤
inst✝ : IsQuasiPreprimitive M α
IwaS : IwasawaStructure M α
is_faithful : FaithfulSMul M α
⊢ ∀ (H : Subgroup M), H.Normal → H = ⊥ ∨ H = ⊤

Tactic:
intro N

Goal state after:
M : Type u_1
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
inst✝¹ : Nontrivial M
is_perfect : commutator M = ⊤
inst✝ : IsQuasiPreprimitive M α
IwaS : IwasawaStructure M α
is_faithful : FaithfulSMul M α
N : Subgroup M
⊢ N.Normal → N = ⊥ ∨ N = ⊤
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: case mp
G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
B : Set X
inst✝ : IsPretransitive G X
hB : IsBlock G B
a : X
ha : a ∈ B
k : G
k_mem : k ∈ stabilizer G B
⊢ k • a ∈ B

Tactic:
rw [← k_mem, Set.smul_mem_smul_set_iff]

Goal state after:
case mp
G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
B : Set X
inst✝ : IsPretransitive G X
hB : IsBlock G B
a : X
ha : a ∈ B
k : G
k_mem : k ∈ stabilizer G B
⊢ a ∈ B
Module: Mathlib.GroupTheory.Perm.ConjAct

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Finite α
g : Perm α
s : Finset α
⊢ ConjAct.toConjAct g • ofSubtype.range = ofSubtype.range

Tactic:
by
  have : Fintype α := Fintype.ofFinite α
  ext k
  simp_rw [Subgroup.mem_pointwise_smul_iff_inv_smul_mem, mem_range_ofSubtype_iff]
  simp [support_conj_eq_smul_support, Set.subset_smul_set_iff]

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

Goal state before:
pp: α : Type u_1
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Nontrivial α
⊢ (alternatingGroup α).index = 2

Tactic:
rw [alternatingGroup, index_ker, MonoidHom.range_eq_top.mpr (sign_surjective α)]
  simp_rw [mem_top, Nat.card_eq_fintype_card, card_subtype_true, card_units_int]

Goal state after:
Module: Mathlib.GroupTheory.GroupAction.SubMulAction.OfStabilizer

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
a : α
g : ↥(stabilizer G a)
x : α
⊢ x ∈ {a}ᶜ → g • x ∈ {a}ᶜ

Tactic:
simp only [Set.mem_compl_iff, Set.mem_singleton_iff]

Goal state after:
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
a : α
g : ↥(stabilizer G a)
x : α
⊢ ¬x = a → ¬g • x = a
Module: Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s t : Set α
g : M
hg : g • t = s
k : M
hk : ∀ y ∈ t, k • y = y
⊢ ∀ y ∈ s, (MulAut.conj g) k • y = y

Tactic:
intro y hy

Goal state after:
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s t : Set α
g : M
hg : g • t = s
k : M
hk : ∀ y ∈ t, k • y = y
y : α
hy : y ∈ s
⊢ (MulAut.conj g) k • y = y
Module: Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup

Goal state before:
pp: case w
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup M s) ↥(ofFixingSubgroup M s)
g : M
a : α
ha : a ∈ sᶜ ∩ (g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ ofFixingSubgroup M (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ ofFixingSubgroup M s
hg'a : g⁻¹ • a ∈ ofFixingSubgroup M s
k : M
hk✝ : k ∈ fixingSubgroup M s
hk : ∀ y ∈ s, k • y = y
hkax : ⟨k, hk✝⟩ • ⟨g⁻¹ • a, hg'a⟩ = ⟨g⁻¹ • x, hg'x⟩
⊢ ∀ y ∈ s ∩ g • s, (g * k * g⁻¹) • y = y

Tactic:
intro y hy

Goal state after:
case w
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup M s) ↥(ofFixingSubgroup M s)
g : M
a : α
ha : a ∈ sᶜ ∩ (g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ ofFixingSubgroup M (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ ofFixingSubgroup M s
hg'a : g⁻¹ • a ∈ ofFixingSubgroup M s
k : M
hk✝ : k ∈ fixingSubgroup M s
hk : ∀ y ∈ s, k • y = y
hkax : ⟨k, hk✝⟩ • ⟨g⁻¹ • a, hg'a⟩ = ⟨g⁻¹ • x, hg'x⟩
y : α
hy : y ∈ s ∩ g • s
⊢ (g * k * g⁻¹) • y = y