mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.GroupTheory.Transfer

Goal state before:
pp: G : Type u_1
inst✝² : Group G
H : Subgroup G
A : Type u_2
inst✝¹ : CommGroup A
ϕ : ↥H →* A
T : H.LeftTransversal
inst✝ : H.FiniteIndex
g : G
⊢ { toFun := fun g ↦ diff ϕ default (g • default), map_one' := ⋯, map_mul' := ⋯ } g =
    diff ϕ ?T (g • T) * diff ϕ (?g • T) (?g • ?T)

pp: case T
G : Type u_1
inst✝² : Group G
H : Subgroup G
A : Type u_2
inst✝¹ : CommGroup A
ϕ : ↥H →* A
T : H.LeftTransversal
inst✝ : H.FiniteIndex
g : G
⊢ H.LeftTransversal

pp: case g
G : Type u_1
inst✝² : Group G
H : Subgroup G
A : Type u_2
inst✝¹ : CommGroup A
ϕ : ↥H →* A
T : H.LeftTransversal
inst✝ : H.FiniteIndex
g : G
⊢ G

Tactic:
diff_mul_diff

Goal state after:
G : Type u_1
inst✝² : Group G
H : Subgroup G
A : Type u_2
inst✝¹ : CommGroup A
ϕ : ↥H →* A
T : H.LeftTransversal
inst✝ : H.FiniteIndex
g : G
⊢ { toFun := fun g ↦ diff ϕ default (g • default), map_one' := ⋯, map_mul' := ⋯ } g = diff ϕ ?R (g • ?R)

case R
G : Type u_1
inst✝² : Group G
H : Subgroup G
A : Type u_2
inst✝¹ : CommGroup A
ϕ : ↥H →* A
T : H.LeftTransversal
inst✝ : H.FiniteIndex
g : G
⊢ H.LeftTransversal
Module: Mathlib.GroupTheory.Focal

Goal state before:
pp: case mem
G : Type u_1
inst✝ : Group G
H : Subgroup G
n g : G
hg : g ∈ H
z : G
hz : z ∈ {g | g ∈ H ∧ ∃ x ∈ H, ∃ u, g = ⁅x, u⁆}
⊢ g * z * g⁻¹ ∈ H.focalSubgroup

Tactic:
| mem z hz =>

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
N : Subgroup G
inst✝ : N.Normal
x✝ y : G
⊢ ⁅x✝, y⁆ ∈ N ↔ ↑(y * x✝) = ↑(x✝ * y)

Tactic:
eq_comm,

Goal state after:
G : Type u_1
inst✝¹ : Group G
N : Subgroup G
inst✝ : N.Normal
x✝ y : G
⊢ ⁅x✝, y⁆ ∈ N ↔ ↑(x✝ * y) = ↑(y * x✝)
Module: Mathlib.GroupTheory.Generators

Goal state before:
pp: case mp
G : Type u_1
inst✝ : Group G
⊢ FG G → ∃ n, Nonempty (Generators G (Fin n))

Tactic:
rintro ⟨S, hS⟩

Goal state after:
case mp
G : Type u_1
inst✝ : Group G
S : Finset G
hS : Subgroup.closure ↑S = ⊤
⊢ ∃ n, Nonempty (Generators G (Fin n))
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
h'δ : 1 + δ * C < r' ^ 2
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' y - g' x‖ ≤ δ
v : E x
w : F := (Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) x) v
hgx : ((g x) v) v = ((g' x) w) w
hgy : ((g y) ((trivializationAt F E x).symm y w)) ((trivializationAt F E x).symm y w) = ((g' y) w) w
⊢ ‖g' y - g' x‖ * ‖w‖ * ‖w‖ + ((g' x) w) w ≤ δ * ‖w‖ ^ 2 + ((g' x) w) w

Tactic:
rw [pow_two, mul_assoc]

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
h'δ : 1 + δ * C < r' ^ 2
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' y - g' x‖ ≤ δ
v : E x
w : F := (Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) x) v
hgx : ((g x) v) v = ((g' x) w) w
hgy : ((g y) ((trivializationAt F E x).symm y w)) ((trivializationAt F E x).symm y w) = ((g' y) w) w
⊢ ‖g' y - g' x‖ * (‖w‖ * ‖w‖) + ((g' x) w) w ≤ δ * (‖w‖ * ‖w‖) + ((g' x) w) w
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : IsNilpotent G
⊢ Nat.find ⋯ = 0 ↔ Subsingleton G

Tactic:
Nat.find_eq_zero,

Goal state after:
G : Type u_1
inst✝¹ : Group G
inst✝ : IsNilpotent G
⊢ upperCentralSeries G 0 = ⊤ ↔ Subsingleton G
Module: Mathlib.GroupTheory.FreeGroup.NielsenSchreier

Goal state before:
pp: G : Type u
inst✝¹ : Groupoid G
inst✝ : IsFreeGroupoid G
a b : G
p : a ⟶ b
⊢ WeaklyConnectedComponent.mk (symgen a) = WeaklyConnectedComponent.mk (symgen b)

Tactic:
eq_comm,

Goal state after:
G : Type u
inst✝¹ : Groupoid G
inst✝ : IsFreeGroupoid G
a b : G
p : a ⟶ b
⊢ WeaklyConnectedComponent.mk (symgen b) = WeaklyConnectedComponent.mk (symgen a)
Module: Mathlib.GroupTheory.IndexNormal

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : H.index = 2
x : G
hxH : x ∈ H
g : G
⊢ g * x * g⁻¹ ∈ H

Tactic:
by simp_rw [mul_mem_iff_of_index_two hH, hxH, iff_true, inv_mem_iff]

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

Goal state before:
pp: G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
x : X
hx : x ∈ B
⊢ B.ncard = (stabilizer G x).relIndex (stabilizer G B)

Tactic:
have key : (stabilizer G x).subgroupOf (stabilizer G B) = stabilizer (stabilizer G B) x := by
    ext; rfl

Goal state after:
G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
x : X
hx : x ∈ B
key : (stabilizer G x).subgroupOf (stabilizer G B) = stabilizer (↥(stabilizer G B)) x
⊢ B.ncard = (stabilizer G x).relIndex (stabilizer G B)
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: case inr.«0»
G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
inst✝ : IsPretransitive G X
B : Set X
hX : Finite X
hB : IsBlock G B
hB_ne : B.Nonempty
hB' : 0 < 2
key : 0 = Nat.card X
⊢ B = Set.univ

Tactic:
eq_comm,

Goal state after:
case inr.«0»
G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
inst✝ : IsPretransitive G X
B : Set X
hX : Finite X
hB : IsBlock G B
hB_ne : B.Nonempty
hB' : 0 < 2
key : Nat.card X = 0
⊢ B = Set.univ
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
inst✝ : IsPretransitive G X
B : Set X
a : X
hfB : B.Finite
B' : Set X := ⋂ k, ⋂ (_ : a ∈ k • B), k • B
hfB_ne : B.Nonempty
hB'₀ : ∀ (k : G), a ∈ k • B → B' ⊆ k • B
hfB' : B'.Finite
hag : ∀ (g : G), a ∈ g • B' → B' ⊆ g • B'
g : G
hg : a ∈ g • B'
⊢ B' = g • B'

Tactic:
eq_comm,

Goal state after:
G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
inst✝ : IsPretransitive G X
B : Set X
a : X
hfB : B.Finite
B' : Set X := ⋂ k, ⋂ (_ : a ∈ k • B), k • B
hfB_ne : B.Nonempty
hB'₀ : ∀ (k : G), a ∈ k • B → B' ⊆ k • B
hfB' : B'.Finite
hag : ∀ (g : G), a ∈ g • B' → B' ⊆ g • B'
g : G
hg : a ∈ g • B'
⊢ g • B' = B'
Module: Mathlib.GroupTheory.GroupAction.Primitive

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝² : Group G
inst✝¹ : MulAction G X
inst✝ : IsPretransitive G X
a : X
H : ∀ {B : Set X}, a ∈ B → IsBlock G B → IsTrivialBlock B
B : Set X
hB : IsBlock G B
⊢ IsTrivialBlock B

Tactic:
by
    obtain rfl | ⟨b, hb⟩ := B.eq_empty_or_nonempty
    · simp [IsTrivialBlock]
    · obtain ⟨g, hg⟩ := exists_smul_eq G b a
      rw [← IsTrivialBlock.smul_iff g]
      apply H _ (hB.translate g)
      rw [← hg]
      use b

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

Goal state before:
pp: G : Type u_3
inst✝³ : Group G
X : Type u_4
inst✝² : MulAction G X
inst✝¹ : IsPretransitive G X
inst✝ : Nontrivial X
a : X
⊢ IsSimpleOrder (BlockMem G a) ↔ IsPreprimitive G X

Tactic:
by
  constructor
  · intro h; let h_bot_or_top := h.eq_bot_or_eq_top
    apply IsPreprimitive.of_isTrivialBlock_base a
    intro B haB hB
    rcases h_bot_or_top ⟨B, haB, hB⟩ with hB' | hB' <;>
      simp only [← Subtype.coe_inj] at hB'
    · left; rw [hB']; exact Set.subsingleton_singleton
    · right; rw [hB']; rfl
  · intro hGX'; apply IsSimpleOrder.mk
    rintro ⟨B, haB, hB⟩
    simp only [← Subtype.coe_inj]
    cases hGX'.isTrivialBlock_of_isBlock hB with
    | inl h =>
      simp [BlockMem.coe_bot, h.eq_singleton_of_mem haB]
    | inr h =>
      simp [BlockMem.coe_top, h]

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

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G X
hGX : IsPretransitive G X
hp : Nat.Prime (Nat.card X)
B : Set X
hB : IsBlock G B
hB' : B.Nontrivial
this : Finite X
⊢ B.ncard = Nat.card X

Tactic:
eq_comm,

Goal state after:
G : Type u_1
X : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G X
hGX : IsPretransitive G X
hp : Nat.Prime (Nat.card X)
B : Set X
hB : IsBlock G B
hB' : B.Nontrivial
this : Finite X
⊢ Nat.card X = B.ncard
Module: Mathlib.SetTheory.Cardinal.Embedding

Goal state before:
pp: α : Type u_1
m n : ℕ
inst✝ : Finite α
hn : m + n ≤ Nat.card α
⊢ Surjective fun x ↦ (castAddEmb n).trans x

Tactic:
by
  apply restrictSurjective_of_add_le_ENatCard
  rwa [← ENat.natCast_add, ENat.card_eq_coe_natCard, ENat.natCast_le_natCast]

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Finset α
⊢ (Perm.ofSubtype.comp (alternatingGroup ↥s).subtype).range =
    map (alternatingGroup α).subtype (Perm.ofSubtype.range.subgroupOf (alternatingGroup α))

Tactic:
MonoidHom.range_comp,

Goal state after:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Finset α
⊢ map Perm.ofSubtype (alternatingGroup ↥s).subtype.range =
    map (alternatingGroup α).subtype (Perm.ofSubtype.range.subgroupOf (alternatingGroup α))
Module: Mathlib.GroupTheory.GroupAction.MultipleTransitivity

Goal state before:
pp: case neg
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
h2 : ∀ {a b c d : α}, a ≠ b → c ≠ d → ∃ g, g • a = c ∧ g • b = d
this : IsPretransitive G α
B : Set α
hB : ∀ ⦃g : G⦄ ⦃a : α⦄, a ∈ B → g • a ∈ B → g • B = B
a : α
ha : a ∈ B
b : α
hb : b ∈ B
h : a ≠ b
c : α
a✝ : c ∈ ⊤
h' : ¬a = c
g : G
hga : g • a = a
hgb : g • b = c
⊢ g • b ∈ g • B

Tactic:
exact smul_mem_smul_set hb

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

Goal state before:
pp: B✝ : Type u_1
inst✝¹⁸ : TopologicalSpace B✝
F✝ : Type u_2
inst✝¹⁷ : NormedAddCommGroup F✝
inst✝¹⁶ : NormedSpace ℝ F✝
E✝ : B✝ → Type u_3
inst✝¹⁵ : TopologicalSpace (TotalSpace F✝ E✝)
inst✝¹⁴ : (x : B✝) → NormedAddCommGroup (E✝ x)
inst✝¹³ : (x : B✝) → InnerProductSpace ℝ (E✝ x)
inst✝¹² : FiberBundle F✝ E✝
inst✝¹¹ : VectorBundle ℝ F✝ E✝
B : Type u_4
inst✝¹⁰ : TopologicalSpace B
F : Type u_5
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
E : B → Type u_6
inst✝⁷ : TopologicalSpace (TotalSpace F E)
inst✝⁶ : (b : B) → TopologicalSpace (E b)
inst✝⁵ : (b : B) → AddCommGroup (E b)
inst✝⁴ : (b : B) → Module ℝ (E b)
inst✝³ : ∀ (b : B), IsTopologicalAddGroup (E b)
inst✝² : ∀ (b : B), ContinuousConstSMul ℝ (E b)
inst✝¹ : FiberBundle F E
inst✝ : VectorBundle ℝ F E
g : RiemannianMetric E
b : B
v : E b
⊢ 0 ≤ RCLike.re (((g.inner b) v) v)

Tactic:
rcases eq_or_ne v 0 with rfl | hv
    · simp
    · simpa using (g.pos b v hv).le

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

Goal state before:
pp: case right
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hsn' : 2 < ENat.card α
hG' : IsMultiplyPretransitive G α 2
a : α
this✝¹ : IsPretransitive G α
this✝ : Nontrivial α
hGa : IsCoatom (stabilizer G a)
hyp : normalClosure ↑(stabilizer G a) ≤ stabilizer G a
b c : ↥(ofStabilizer G a)
hbc : b ≠ c
this : IsPretransitive ↥(stabilizer G a) ↥(ofStabilizer G a)
g : G
hg : g ∈ stabilizer G a
hgbc : ↑(⟨g, hg⟩ • b) = ↑c
h : G
hinvab : h • ↑b = a
⊢ ↑b = ↑c

Tactic:
eq_comm,

Goal state after:
case right
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hsn' : 2 < ENat.card α
hG' : IsMultiplyPretransitive G α 2
a : α
this✝¹ : IsPretransitive G α
this✝ : Nontrivial α
hGa : IsCoatom (stabilizer G a)
hyp : normalClosure ↑(stabilizer G a) ≤ stabilizer G a
b c : ↥(ofStabilizer G a)
hbc : b ≠ c
this : IsPretransitive ↥(stabilizer G a) ↥(ofStabilizer G a)
g : G
hg : g ∈ stabilizer G a
hgbc : ↑(⟨g, hg⟩ • b) = ↑c
h : G
hinvab : a = h • ↑b
⊢ ↑b = ↑c
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 : α
hy : y ∈ s
⊢ (g * k * g⁻¹) • y = y

Tactic:
eq_comm,

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
⊢ y = (g * k * g⁻¹) • y
Module: Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup

Goal state before:
pp: case right
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M t
⊢ ∃ a, (conjMap_ofFixingSubgroup hst) a = ⟨x, hx⟩

Tactic:
eq_comm,

Goal state after:
case right
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s t : Set α
g : M
hst✝ : g • s = t
hst : t = g • s
x : α
hx : x ∈ ofFixingSubgroup M t
⊢ ∃ a, (conjMap_ofFixingSubgroup hst✝) a = ⟨x, hx⟩
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✝ s t : Set α
m : ↥(fixingSubgroup M (s ∪ t))
⊢ ⟨↑m, ⋯⟩ ∈ fixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)

Tactic:
by
      let hm := m.prop
      simp only [fixingSubgroup_union, Subgroup.mem_inf] at hm
      rintro ⟨⟨x, hx⟩, hx'⟩
      simp only [Set.mem_preimage] at hx'
      simp only [← SetLike.coe_eq_coe, SubMulAction.val_smul_of_tower]
      exact hm.right ⟨x, hx'⟩

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

Goal state before:
pp: case right
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s t : Set α
a : α
ha : a ∈ ofFixingSubgroup M s
ha' : ⟨a, ha⟩ ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)
hy : a ∈ s ∪ t
⊢ False

Tactic:
rcases (Set.mem_union a s t).mp hy with h | h

Goal state after:
case right.inl
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s t : Set α
a : α
ha : a ∈ ofFixingSubgroup M s
ha' : ⟨a, ha⟩ ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)
hy : a ∈ s ∪ t
h : a ∈ s
⊢ False

case right.inr
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s t : Set α
a : α
ha : a ∈ ofFixingSubgroup M s
ha' : ⟨a, ha⟩ ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)
hy : a ∈ s ∪ t
h : a ∈ t
⊢ False
Module: Mathlib.GroupTheory.GroupAction.MultiplePrimitivity

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝³ : Group M
inst✝² : MulAction M α
m n : ℕ
inst✝¹ : IsMultiplyPreprimitive M α n
s : Set α
inst✝ : Finite ↑s
hs : s.ncard + m = n
t : Set ↥(ofFixingSubgroup M s)
ht : t.encard + 1 = ↑m
t' : Set α := Subtype.val '' t
htt' : t = Subtype.val ⁻¹' t'
this : IsPreprimitive ↥(fixingSubgroup M (s ∪ t')) ↥(ofFixingSubgroup M (s ∪ t'))
⊢ IsPreprimitive ↥(fixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t'))
    ↥(ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t'))

Tactic:
by
      apply IsPreprimitive.of_surjective map_ofFixingSubgroupUnion_bijective.surjective

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

Goal state before:
pp: n : ℕ
hrec :
  ∀ m < n,
    ∀ {G : Type u_1} {α : Type u_2} [inst : Group G] [inst_1 : MulAction G α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          s.ncard = m + 1 →
            m + 2 < Nat.card α →
              (IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2) ∧
                (IsPreprimitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPreprimitive G α 2)
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hG : IsPreprimitive G α
s : Set α
hsn : s.ncard = n + 1
hsn' : n + 2 < Nat.card α
this✝ : Finite α
hs_ne_univ : s ≠ Set.univ
hs_nonempty : s.Nonempty
hn : n + 1 ≥ 2
hn1 : 2 * (n + 1) < Nat.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := ⋯
ht : t.Finite
htm : t.ncard = t.ncard - 1 + 1
hmn : t.ncard - 1 < n
htm' : t.ncard - 1 + 2 < Nat.card α
this : IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
hs_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s)
⊢ s ∪ g • s ≠ ⊤

Tactic:
apply Set.union_ne_univ_of_ncard_add_ncard_lt

Goal state after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {G : Type u_1} {α : Type u_2} [inst : Group G] [inst_1 : MulAction G α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          s.ncard = m + 1 →
            m + 2 < Nat.card α →
              (IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2) ∧
                (IsPreprimitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPreprimitive G α 2)
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hG : IsPreprimitive G α
s : Set α
hsn : s.ncard = n + 1
hsn' : n + 2 < Nat.card α
this✝ : Finite α
hs_ne_univ : s ≠ Set.univ
hs_nonempty : s.Nonempty
hn : n + 1 ≥ 2
hn1 : 2 * (n + 1) < Nat.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := ⋯
ht : t.Finite
htm : t.ncard = t.ncard - 1 + 1
hmn : t.ncard - 1 < n
htm' : t.ncard - 1 + 2 < Nat.card α
this : IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
hs_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s)
⊢ s.ncard + (g • s).ncard < Nat.card α
Module: Mathlib.GroupTheory.GroupAction.MultiplePrimitivity

Goal state before:
pp: case succ.inl
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
m n : ℕ
hrec : IsMultiplyPreprimitive M α n → m ≤ n → ↑n ≤ ENat.card α → IsMultiplyPreprimitive M α m
hn : IsMultiplyPreprimitive M α (n + 1)
hmn✝ : m ≤ n + 1
hα : ↑(n + 1) ≤ ENat.card α
hmn : m = n + 1
⊢ IsMultiplyPreprimitive M α m

pp: case succ.inr
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
m n : ℕ
hrec : IsMultiplyPreprimitive M α n → m ≤ n → ↑n ≤ ENat.card α → IsMultiplyPreprimitive M α m
hn : IsMultiplyPreprimitive M α (n + 1)
hmn : m ≤ n + 1
hα : ↑(n + 1) ≤ ENat.card α
hmn' : m < n + 1
⊢ IsMultiplyPreprimitive M α m

Tactic:
· rw [hmn]; exact hn

Goal state after:
case succ.inr
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
m n : ℕ
hrec : IsMultiplyPreprimitive M α n → m ≤ n → ↑n ≤ ENat.card α → IsMultiplyPreprimitive M α m
hn : IsMultiplyPreprimitive M α (n + 1)
hmn : m ≤ n + 1
hα : ↑(n + 1) ≤ ENat.card α
hmn' : m < n + 1
⊢ IsMultiplyPreprimitive M α m
Module: Mathlib.GroupTheory.GroupAction.Jordan

Goal state before:
pp: n : ℕ
hrec :
  ∀ m < n,
    ∀ {G : Type u_1} {α : Type u_2} [inst : Group G] [inst_1 : MulAction G α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          s.ncard = m + 1 →
            m + 2 < Nat.card α →
              (IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2) ∧
                (IsPreprimitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPreprimitive G α 2)
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hG : IsPreprimitive G α
s : Set α
hsn : s.ncard = n + 1
hsn' : n + 2 < Nat.card α
this : Finite α
hs_ne_univ : s ≠ Set.univ
hs_nonempty : s.Nonempty
hn : n + 1 ≥ 2
hn1 : 2 * (n + 1) < Nat.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht : t.Finite
htm : t.ncard = t.ncard - 1 + 1
hmn : t.ncard - 1 < n
htm' : t.ncard - 1 + 2 < Nat.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s)
⊢ s ∪ g • s ≠ ⊤

Tactic:
apply Set.union_ne_univ_of_ncard_add_ncard_lt

Goal state after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {G : Type u_1} {α : Type u_2} [inst : Group G] [inst_1 : MulAction G α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          s.ncard = m + 1 →
            m + 2 < Nat.card α →
              (IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2) ∧
                (IsPreprimitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPreprimitive G α 2)
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hG : IsPreprimitive G α
s : Set α
hsn : s.ncard = n + 1
hsn' : n + 2 < Nat.card α
this : Finite α
hs_ne_univ : s ≠ Set.univ
hs_nonempty : s.Nonempty
hn : n + 1 ≥ 2
hn1 : 2 * (n + 1) < Nat.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := ⋯
ht : t.Finite
htm : t.ncard = t.ncard - 1 + 1
hmn : t.ncard - 1 < n
htm' : t.ncard - 1 + 2 < Nat.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s)
⊢ s.ncard + (g • s).ncard < Nat.card α
Module: Mathlib.GroupTheory.GroupAction.Period

Goal state before:
pp: α : Type v
G : Type u
inst✝¹ : Group G
inst✝ : MulAction G α
g : G
a : α
n : ℕ
⊢ a = (g⁻¹ ^ n)⁻¹ • a ↔ g ^ n • a = a

Tactic:
eq_comm,

Goal state after:
α : Type v
G : Type u
inst✝¹ : Group G
inst✝ : MulAction G α
g : G
a : α
n : ℕ
⊢ (g⁻¹ ^ n)⁻¹ • a = a ↔ g ^ n • a = a
Module: Mathlib.GroupTheory.GroupAction.Jordan

Goal state before:
pp: n : ℕ
hrec :
  ∀ m < n,
    ∀ {G : Type u_1} {α : Type u_2} [inst : Group G] [inst_1 : MulAction G α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          s.ncard = m + 1 →
            m + 2 < Nat.card α →
              (IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2) ∧
                (IsPreprimitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPreprimitive G α 2)
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hG : IsPreprimitive G α
s : Set α
hsn : s.ncard = n + 1
hsn' : n + 2 < Nat.card α
this : Finite α
hs_ne_univ : s ≠ Set.univ
hs_nonempty : s.Nonempty
hn : n + 1 ≥ 2
hn2 : 2 * (n + 1) ≥ Nat.card α
a : α
ha✝ : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
ha : a ∉ s ∪ g • s
⊢ 1 ≤ t.ncard

Tactic:
rw [Nat.one_le_iff_ne_zero, ← Nat.pos_iff_ne_zero, Set.ncard_pos]

Goal state after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {G : Type u_1} {α : Type u_2} [inst : Group G] [inst_1 : MulAction G α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          s.ncard = m + 1 →
            m + 2 < Nat.card α →
              (IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2) ∧
                (IsPreprimitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPreprimitive G α 2)
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hG : IsPreprimitive G α
s : Set α
hsn : s.ncard = n + 1
hsn' : n + 2 < Nat.card α
this : Finite α
hs_ne_univ : s ≠ Set.univ
hs_nonempty : s.Nonempty
hn : n + 1 ≥ 2
hn2 : 2 * (n + 1) ≥ Nat.card α
a : α
ha✝ : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
ha : a ∉ s ∪ g • s
⊢ t.Nonempty