mathlib-initiative/mathlib-tactics huggingface dataset:
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
x : M
h : IsSubmersionAtOfComplement F I J n f x
⊢ Prod.fst ∘ ⇑h.equiv '' (h.domChart.extend I).target ⊆ (h.codChart.extend J).target

Tactic:
rw [← h.writtenInCharts.image_eq, Set.image_comp, Set.image_comp,
    PartialEquiv.symm_image_target_eq_source, OpenPartialHomeomorph.extend_source,
    ← PartialEquiv.image_source_eq_target]

Goal state after:
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
x : M
h : IsSubmersionAtOfComplement F I J n f x
⊢ ↑(h.codChart.extend J) '' f '' h.domChart.source ⊆ ↑(h.codChart.extend J) '' (h.codChart.extend J).source
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
t t' : (x : M) → V x
hs : IsLocalFrameOn I F n s u
htt' : t x = t' x
i : ι
⊢ (hs.coeff i x) (t x) = (hs.coeff i x) (t' x)

Tactic:
by_cases hxe : x ∈ u <;> simp [coeff, hxe, htt']

Goal state after:
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
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
ι : Type u_7
e : Trivialization F TotalSpace.proj
inst✝¹ : MemTrivializationAtlas e
b : Basis ι 𝕜 F
inst✝ : ContMDiffVectorBundle 1 F V I
x : M
hx : x ∈ e.baseSet
s : (x : M) → V x
i : ι
he : IsLocalFrameOn I F 1 (e.localFrame b) e.baseSet
⊢ e.basisAt b hx = he.toBasisAt hx

Tactic:
ext j

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
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
ι : Type u_7
e : Trivialization F TotalSpace.proj
inst✝¹ : MemTrivializationAtlas e
b : Basis ι 𝕜 F
inst✝ : ContMDiffVectorBundle 1 F V I
x : M
hx : x ∈ e.baseSet
s : (x : M) → V x
i : ι
he : IsLocalFrameOn I F 1 (e.localFrame b) e.baseSet
j : ι
⊢ (e.basisAt b hx) j = (he.toBasisAt hx) j
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
⊢ δ * C * ((g x) (G.symm w)) (G.symm w) + ((g' y) w) w = δ * C * ((g' x) w) w + ((g' y) w) w

Tactic:
← hgx

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
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
⊢ δ * C * ((g x) (G.symm w)) (G.symm w) + ((g' y) w) w =
    δ * C * ((g x) ((trivializationAt F E x).symm x w)) ((trivializationAt F E x).symm x w) + ((g' y) w) w
Module: Mathlib.GroupTheory.CommutingProbability

Goal state before:
pp: n : ℕ
h0 : n ≠ 0
h2 : Even n
h1 : n ≠ 1
⊢ (if Even n then 3 :: reciprocalFactors (n / 2) else n % 4 * n :: reciprocalFactors (n / 4 + 1)) =
    3 :: reciprocalFactors (n / 2)

Tactic:
if_pos h2

Goal state after:
n : ℕ
h0 : n ≠ 0
h2 : Even n
h1 : n ≠ 1
⊢ 3 :: reciprocalFactors (n / 2) = 3 :: reciprocalFactors (n / 2)
Module: Mathlib.GroupTheory.CommutingProbability

Goal state before:
pp: case pos
n : ℕ
h0 : ¬n = 0
h1 : n = 1
⊢ commProb (Product (reciprocalFactors n)) = 1 / ↑n

Tactic:
h1,

Goal state after:
case pos
n : ℕ
h0 : ¬n = 0
h1 : n = 1
⊢ commProb (Product (reciprocalFactors 1)) = 1 / ↑1
Module: Mathlib.Topology.VectorBundle.Riemannian

Goal state before:
pp: case ha
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 := ⋯
C : ℝ := ⋯
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 := ⋯
hg' : ContinuousAt g' x
this✝² : ∀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✝¹ : ‖g' x - g' y‖ ≤ δ
v : E y
w : F := ⋯
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
this✝ : ((g' x) w) w ≤ δ * C * ((g' x) w) w + ((g' y) w) w
this : ((g' x) w) w ≤ (1 - δ * C)⁻¹ * ((g' y) w) w
⊢ 0 ≤ ((g' y) w) w

Tactic:
rw [← hgy, ← hg, real_inner_self_eq_norm_sq]
    positivity

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

Goal state before:
pp: case mp
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
w t : W
h : cs.IsReflection (w * t * w⁻¹)
⊢ cs.IsReflection t

Tactic:
simpa [← mul_assoc] using h.conj w⁻¹

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
w : W
i : B
⊢ cs.length (cs.simple i * w) < cs.length w → cs.length (cs.simple i * w) + 1 = cs.length w

Tactic:
intro _
    exact (cs.length_simple_mul w i).resolve_left (by lia)

Goal state after:
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: case inr
ι : Type u_1
G : Type u_4
inst✝³ : Group G
H : ι → Type u_5
inst✝² : (i : ι) → Group (H i)
f : (i : ι) → H i →* G
α : Type u_6
inst✝¹ : MulAction G α
X : ι → Set α
hXnonempty : ∀ (i : ι), (X i).Nonempty
hXdisj : Pairwise (Disjoint on X)
hpp : Pairwise fun i j ↦ ∀ (h : H i), h ≠ 1 → (f i) h • X j ⊆ X i
inst✝ : Nontrivial ι
i j : ι
w : NeWord H i j
k : ι
hcard : 3 ≤ #(H k)
⊢ (lift f) w.prod ≠ 1

Tactic:
by_cases hh : i = k <;> by_cases hl : j = k

Goal state after:
case pos
ι : Type u_1
G : Type u_4
inst✝³ : Group G
H : ι → Type u_5
inst✝² : (i : ι) → Group (H i)
f : (i : ι) → H i →* G
α : Type u_6
inst✝¹ : MulAction G α
X : ι → Set α
hXnonempty : ∀ (i : ι), (X i).Nonempty
hXdisj : Pairwise (Disjoint on X)
hpp : Pairwise fun i j ↦ ∀ (h : H i), h ≠ 1 → (f i) h • X j ⊆ X i
inst✝ : Nontrivial ι
i j : ι
w : NeWord H i j
k : ι
hcard : 3 ≤ #(H k)
hh : i = k
hl : j = k
⊢ (lift f) w.prod ≠ 1

case neg
ι : Type u_1
G : Type u_4
inst✝³ : Group G
H : ι → Type u_5
inst✝² : (i : ι) → Group (H i)
f : (i : ι) → H i →* G
α : Type u_6
inst✝¹ : MulAction G α
X : ι → Set α
hXnonempty : ∀ (i : ι), (X i).Nonempty
hXdisj : Pairwise (Disjoint on X)
hpp : Pairwise fun i j ↦ ∀ (h : H i), h ≠ 1 → (f i) h • X j ⊆ X i
inst✝ : Nontrivial ι
i j : ι
w : NeWord H i j
k : ι
hcard : 3 ≤ #(H k)
hh : i = k
hl : ¬j = k
⊢ (lift f) w.prod ≠ 1

case pos
ι : Type u_1
G : Type u_4
inst✝³ : Group G
H : ι → Type u_5
inst✝² : (i : ι) → Group (H i)
f : (i : ι) → H i →* G
α : Type u_6
inst✝¹ : MulAction G α
X : ι → Set α
hXnonempty : ∀ (i : ι), (X i).Nonempty
hXdisj : Pairwise (Disjoint on X)
hpp : Pairwise fun i j ↦ ∀ (h : H i), h ≠ 1 → (f i) h • X j ⊆ X i
inst✝ : Nontrivial ι
i j : ι
w : NeWord H i j
k : ι
hcard : 3 ≤ #(H k)
hh : ¬i = k
hl : j = k
⊢ (lift f) w.prod ≠ 1

case neg
ι : Type u_1
G : Type u_4
inst✝³ : Group G
H : ι → Type u_5
inst✝² : (i : ι) → Group (H i)
f : (i : ι) → H i →* G
α : Type u_6
inst✝¹ : MulAction G α
X : ι → Set α
hXnonempty : ∀ (i : ι), (X i).Nonempty
hXdisj : Pairwise (Disjoint on X)
hpp : Pairwise fun i j ↦ ∀ (h : H i), h ≠ 1 → (f i) h • X j ⊆ X i
inst✝ : Nontrivial ι
i j : ι
w : NeWord H i j
k : ι
hcard : 3 ≤ #(H k)
hh : ¬i = k
hl : ¬j = k
⊢ (lift f) w.prod ≠ 1
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: case mk.mk
M✝ : Type u_1
inst✝³ : AddCommMonoid M✝
M : Type u_2
inst✝² : AddCommMonoid M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
ma : M
sa : ℕ+
mb : M
sb : ℕ+
h : ↑sb • ma ≤ ↑sa • mb
h' : ↑sa • mb ≤ ↑sb • ma
⊢ ↑sb • ma = ↑sa • mb

Tactic:
exact le_antisymm h h'

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)
N : Type u_3
inst✝¹ : Monoid N
ι : Type u_4
X : ι → Type u_5
G : ι → Type u_6
inst✝ : (i : ι) → Group (G i)
B : (i : ι) → FreeGroupBasis (X i) (G i)
⊢ (lift fun i ↦ (B i).lift fun x ↦ FreeGroup.of ⟨i, x⟩).comp (FreeGroup.lift fun x ↦ of ((B x.fst) x.snd)) =
    MonoidHom.id (FreeGroup ((i : ι) × X i))

Tactic:
by ext; simp

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
x : G
y : Quotient ↑H ↑K
hy : y ∈ t
q : G
hq : q ∈ doubleCoset (out y) ↑H ↑K
hx : Quot.mk (⇑(rightRel H)) q = Quot.mk (⇑(rightRel H)) x
⊢ ∃ a, x = ↑a * q

Tactic:
obtain ⟨a, ha⟩ : ∃ a : H, a * x = q := Quotient.eq.mp hx

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
x : G
y : Quotient ↑H ↑K
hy : y ∈ t
q : G
hq : q ∈ doubleCoset (out y) ↑H ↑K
hx : Quot.mk (⇑(rightRel H)) q = Quot.mk (⇑(rightRel H)) x
a : ↥H
ha : ↑a * x = q
⊢ ∃ a, x = ↑a * q
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)
this : HasEnoughRootsOfUnity M (Monoid.exponent (G →* Mˣ))
x✝¹ x✝ : G
⊢ (Equiv.ofBijective (fun g ↦ { toFun := fun φ ↦ φ g, map_one' := ⋯, map_mul' := ⋯ }) ⋯) (x✝¹ * x✝) =
    (Equiv.ofBijective (fun g ↦ { toFun := fun φ ↦ φ g, map_one' := ⋯, map_mul' := ⋯ }) ⋯) x✝¹ *
      (Equiv.ofBijective (fun g ↦ { toFun := fun φ ↦ φ g, map_one' := ⋯, map_mul' := ⋯ }) ⋯) x✝

Tactic:
by ext; simp

Goal state after:
Module: Mathlib.GroupTheory.Transfer

Goal state before:
pp: G : Type u_1
inst✝³ : Group G
H : Subgroup G
A : Type u_2
inst✝² : CommGroup A
ϕ : ↥H →* A
inst✝¹ : H.FiniteIndex
g : G
inst✝ : Fintype (Quotient (orbitRel (↥(zpowers g)) (G ⧸ H)))
⊢ ϕ.transfer g =
    ∏ q, ϕ ⟨(Quotient.out q.out)⁻¹ * g ^ Function.minimalPeriod (fun x ↦ g • x) q.out * Quotient.out q.out, ⋯⟩

Tactic:
by
  let := H.fintypeQuotientOfFiniteIndex
  calc
    transfer ϕ g = ∏ q : G ⧸ H, _ := transfer_def ϕ (transferTransversal H g) g
    _ = _ := ((quotientEquivSigmaZMod H g).symm.prod_comp _).symm
    _ = _ := Finset.prod_sigma _ _ _
    _ = _ := by
      refine Fintype.prod_congr _ _ (fun q => ?_)
      simp only [quotientEquivSigmaZMod_symm_apply, transferTransversal_apply',
        transferTransversal_apply'']
      rw [Fintype.prod_eq_single (0 : ZMod (Function.minimalPeriod (g • ·) q.out)) _]
      · simp only [if_pos, ZMod.cast_zero, zpow_zero, one_mul, mul_assoc]
      · intro k hk
        simp only [if_neg hk, inv_mul_cancel]
        exact map_one ϕ

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

Goal state before:
pp: case isTrue
α : Type u
L : List (α × Bool)
inst✝ : DecidableEq α
a : α × Bool
l : List (α × Bool)
b : α × Bool
ih : IsReduced l → IsCyclicallyReduced (reduceCyclically l)
h : IsReduced (a :: (l ++ [b]))
h✝ : b.1 = a.1 ∧ (!b.2) = a.2
⊢ IsCyclicallyReduced (reduceCyclically l)

pp: case isFalse
α : Type u
L : List (α × Bool)
inst✝ : DecidableEq α
a : α × Bool
l : List (α × Bool)
b : α × Bool
ih : IsReduced l → IsCyclicallyReduced (reduceCyclically l)
h : IsReduced (a :: (l ++ [b]))
h✝ : ¬(b.1 = a.1 ∧ (!b.2) = a.2)
⊢ IsCyclicallyReduced (a :: l ++ [b])

Tactic:
case isTrue => exact ih (h.infix ⟨[a], [b], rfl⟩)

Goal state after:
case isFalse
α : Type u
L : List (α × Bool)
inst✝ : DecidableEq α
a : α × Bool
l : List (α × Bool)
b : α × Bool
ih : IsReduced l → IsCyclicallyReduced (reduceCyclically l)
h : IsReduced (a :: (l ++ [b]))
h✝ : ¬(b.1 = a.1 ∧ (!b.2) = a.2)
⊢ IsCyclicallyReduced (a :: l ++ [b])
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: case right
G : Type u_1
inst✝ : Group G
x : G
n : ℕ
⊢ x ∈ ⊤.lowerCentralSeries n → ∀ (g : G), ⁅x, g⁆ ∈ ⊤.lowerCentralSeries (n + 1)

Tactic:
hxn

Goal state after:
case right
G : Type u_1
inst✝ : Group G
x : G
n : ℕ
hxn : x ∈ ⊤.lowerCentralSeries n
⊢ ∀ (g : G), ⁅x, g⁆ ∈ ⊤.lowerCentralSeries (n + 1)
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: case refine_1
G : Type u_1
inst✝ : Group G
S : Subgroup G
n : ℕ
H : ℕ → Subgroup ↥S
hH : IsDescendingCentralSeries H
hn : H n = ⊥
h1 : ⊤.lowerCentralSeries n = ⊥
⊢ map S.subtype (⊤.lowerCentralSeries n) = ⊥

Tactic:
h1,

Goal state after:
case refine_1
G : Type u_1
inst✝ : Group G
S : Subgroup G
n : ℕ
H : ℕ → Subgroup ↥S
hH : IsDescendingCentralSeries H
hn : H n = ⊥
h1 : ⊤.lowerCentralSeries n = ⊥
⊢ map S.subtype ⊥ = ⊥
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: case neg
G : Type u_1
inst✝ : Group G
hH : ¬IsNilpotent G
⊢ nilpotencyClass (G ⧸ center G) = 0

Tactic:
nilpotencyClass_of_not_nilpotent

Goal state after:
case neg
G : Type u_1
inst✝ : Group G
hH : ¬IsNilpotent G
⊢ 0 = 0

case neg
G : Type u_1
inst✝ : Group G
hH : ¬IsNilpotent G
⊢ ¬IsNilpotent (G ⧸ center G)
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: case succ
P : (G : Type u_2) → [inst : Group G] → [IsNilpotent G] → Prop
hbase : ∀ (G : Type u_2) [inst : Group G] [inst_1 : Subsingleton G], P G
hstep : ∀ (G : Type u_2) [inst : Group G] [inst_1 : IsNilpotent G], P (G ⧸ center G) → P G
n : ℕ
ih : ∀ (G : Type u_2) [inst : Group G] [inst_1 : IsNilpotent G], nilpotencyClass G = n → P G
G : Type u_2
inst✝¹ : Group G
inst✝ : IsNilpotent G
h : nilpotencyClass G = n + 1
⊢ P G

Tactic:
have hn : Group.nilpotencyClass (G ⧸ center G) = n := by
      simp [nilpotencyClass_quotient_center, h]

Goal state after:
case succ
P : (G : Type u_2) → [inst : Group G] → [IsNilpotent G] → Prop
hbase : ∀ (G : Type u_2) [inst : Group G] [inst_1 : Subsingleton G], P G
hstep : ∀ (G : Type u_2) [inst : Group G] [inst_1 : IsNilpotent G], P (G ⧸ center G) → P G
n : ℕ
ih : ∀ (G : Type u_2) [inst : Group G] [inst_1 : IsNilpotent G], nilpotencyClass G = n → P G
G : Type u_2
inst✝¹ : Group G
inst✝ : IsNilpotent G
h : nilpotencyClass G = n + 1
hn : nilpotencyClass (G ⧸ center G) = n
⊢ P G
Module: Mathlib.GroupTheory.GroupAction.Transitive

Goal state before:
pp: case mp
M : Type u_3
N : Type u_4
α : Type u_5
β : Type u_6
inst✝³ : Monoid M
inst✝² : Monoid N
inst✝¹ : MulAction M α
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
⊢ IsPretransitive M α → IsPretransitive N β

Tactic:
apply IsPretransitive.of_surjective_map hf.surjective

Goal state after:
Module: Mathlib.GroupTheory.IndexNormal

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : H.index = 1
⊢ H.Normal

Tactic:
rw [index_eq_one] at hH

Goal state after:
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : H = ⊤
⊢ H.Normal
Module: Mathlib.GroupTheory.GroupAction.Primitive

Goal state before:
pp: case inl
G : Type u_1
X : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ ⦃B : Set X⦄, a ∈ B → IsBlock G B → IsTrivialBlock B
H : (orbit G a).Subsingleton
⊢ False

Tactic:
apply ha

Goal state after:
case inl
G : Type u_1
X : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ ⦃B : Set X⦄, a ∈ B → IsBlock G B → IsTrivialBlock B
H : (orbit G a).Subsingleton
⊢ a ∈ fixedPoints G X
Module: Mathlib.GroupTheory.SpecificGroups.Alternating

Goal state before:
pp: α : Type u_1
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Nontrivial α
⊢ Nat.card ↥⊤ = 2

Tactic:
mem_top,

Goal state after:
α : Type u_1
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Nontrivial α
⊢ Nat.card { x // True } = 2
Module: Mathlib.GroupTheory.GroupAction.Primitive

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝⁴ : Group G
inst✝³ : MulAction G X
M : Type u_3
inst✝² : Group M
α : Type u_4
inst✝¹ : MulAction M α
inst✝ : IsPreprimitive M α
N : Subgroup M
x✝ : N.Normal
a : α
ha : a ∉ fixedPoints (↥N) α
h : (orbit (↥N) a).Subsingleton
⊢ False

Tactic:
apply ha

Goal state after:
G : Type u_1
X : Type u_2
inst✝⁴ : Group G
inst✝³ : MulAction G X
M : Type u_3
inst✝² : Group M
α : Type u_4
inst✝¹ : MulAction M α
inst✝ : IsPreprimitive M α
N : Subgroup M
x✝ : N.Normal
a : α
ha : a ∉ fixedPoints (↥N) α
h : (orbit (↥N) a).Subsingleton
⊢ a ∈ fixedPoints (↥N) α
Module: Mathlib.GroupTheory.GroupAction.MultipleTransitivity

Goal state before:
pp: case succ
k : ℕ
hrec :
  ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],
    IsMultiplyPretransitive G α k →
      ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
Hk : IsMultiplyPretransitive G α (k + 1)
s : Set α
hs : s.ncard = k + 1
hGX : IsPretransitive G α
a : α
has : a ∈ s
t : Set ↥(ofStabilizer G a) := Subtype.val ⁻¹' s
⊢ (fixingSubgroup G s).index * (Nat.card α - (k + 1))! = (Nat.card α)!

Tactic:
have hat : Subtype.val '' t = s \ {a} := by
      rw [Set.image_preimage_eq_inter_range]
      simp only [Subtype.range_coe_subtype]
      rw [Set.sdiff_eq_compl_inter, Set.inter_comm]
      congr

Goal state after:
case succ
k : ℕ
hrec :
  ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],
    IsMultiplyPretransitive G α k →
      ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
Hk : IsMultiplyPretransitive G α (k + 1)
s : Set α
hs : s.ncard = k + 1
hGX : IsPretransitive G α
a : α
has : a ∈ s
t : Set ↥(ofStabilizer G a) := Subtype.val ⁻¹' s
hat : Subtype.val '' t = s \ {a}
⊢ (fixingSubgroup G s).index * (Nat.card α - (k + 1))! = (Nat.card α)!