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

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹² : TopologicalSpace H
E : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
H' : Type u_5
inst✝⁷ : TopologicalSpace H'
E' : Type u_6
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁴ : TopologicalSpace M'
inst✝³ : ChartedSpace H' M'
inst✝² : IsManifold I 2 M
inst✝¹ : IsManifold I' 2 M'
inst✝ : CompleteSpace E
f : M → M'
V W : (x : M') → TangentSpace I' x
x₀ : M
s : Set M
t : Set M'
hV : MDiffAt[t] (T% V) (f x₀)
hW : MDiffAt[t] (T% W) (f x₀)
hu : UniqueMDiff[s]
hf : ContMDiffWithinAt I I' 2 f s x₀
hx₀ : x₀ ∈ s
hst : f ⁻¹' t ∈ 𝓝[s] x₀
hsymm :
  IsSymmSndFDerivWithinAt 𝕜 (↑(extChartAt I' (f x₀)) ∘ f ∘ ↑(extChartAt I x₀).symm)
    (↑(extChartAt I x₀).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x₀) x₀)
A : ↑(extChartAt I x₀).symm (↑(extChartAt I x₀) x₀) = x₀
hfi : (mfderiv[s] f x₀).IsInvertible
this : CompleteSpace E'
u : Set M
u_open : IsOpen[inst✝⁹] u
x₀u : x₀ ∈ u
u_mem : u ∈ 𝓝 x₀
s' : Set M := s ∩ u
ut : s' ⊆ f ⁻¹' t
maps_u : s' ⊆ f ⁻¹' (extChartAt I' (f x₀)).source
u_smooth : ContMDiffOn I I' 2 f s'
hs' : s' = s ∩ u
s'_eq : s' =ᶠ[𝓝 x₀] s
⊢ mpullbackWithin I I' f (mlieBracketWithin I' V W t) s x₀ =
    mlieBracketWithin I (mpullbackWithin I I' f V s) (mpullbackWithin I I' f W s) s x₀

Tactic:
set t' := t ∩ (extChartAt I' (f x₀)).source with ht'

Goal state after:
case pos
𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹² : TopologicalSpace H
E : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
H' : Type u_5
inst✝⁷ : TopologicalSpace H'
E' : Type u_6
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁴ : TopologicalSpace M'
inst✝³ : ChartedSpace H' M'
inst✝² : IsManifold I 2 M
inst✝¹ : IsManifold I' 2 M'
inst✝ : CompleteSpace E
f : M → M'
V W : (x : M') → TangentSpace I' x
x₀ : M
s : Set M
t : Set M'
hV : MDiffAt[t] (T% V) (f x₀)
hW : MDiffAt[t] (T% W) (f x₀)
hu : UniqueMDiff[s]
hf : ContMDiffWithinAt I I' 2 f s x₀
hx₀ : x₀ ∈ s
hst : f ⁻¹' t ∈ 𝓝[s] x₀
hsymm :
  IsSymmSndFDerivWithinAt 𝕜 (↑(extChartAt I' (f x₀)) ∘ f ∘ ↑(extChartAt I x₀).symm)
    (↑(extChartAt I x₀).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x₀) x₀)
A : ↑(extChartAt I x₀).symm (↑(extChartAt I x₀) x₀) = x₀
hfi : (mfderiv[s] f x₀).IsInvertible
this : CompleteSpace E'
u : Set M
u_open : IsOpen[inst✝⁹] u
x₀u : x₀ ∈ u
u_mem : u ∈ 𝓝 x₀
s' : Set M := s ∩ u
ut : s' ⊆ f ⁻¹' t
maps_u : s' ⊆ f ⁻¹' (extChartAt I' (f x₀)).source
u_smooth : ContMDiffOn I I' 2 f s'
hs' : s' = s ∩ u
s'_eq : s' =ᶠ[𝓝 x₀] s
t' : Set M' := t ∩ (extChartAt I' (f x₀)).source
ht' : t' = t ∩ (extChartAt I' (f x₀)).source
⊢ mpullbackWithin I I' f (mlieBracketWithin I' V W t) s x₀ =
    mlieBracketWithin I (mpullbackWithin I I' f V s) (mpullbackWithin I I' f W s) s x₀
Module: Mathlib.Geometry.Manifold.Riemannian.Basic

Goal state before:
pp: E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type u_2
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : RiemannianBundle fun x ↦ TangentSpace I x
inst✝² : IsManifold I 1 M
inst✝¹ : IsContinuousRiemannianBundle E fun x ↦ TangentSpace I x
inst✝ : RegularSpace M
x : M
s : Set M
hs : s ∈ 𝓝 x
C : ℝ≥0
C_pos : C > 0
hC : ∀ᶠ (y : M) in 𝓝 x, ‖mfderiv% ↑(extChartAt I x) y‖ₑ < ↑C
u : Set M
u_mem : u ∈ 𝓝 x
u_closed : IsClosed[inst✝⁵] u
us : u ⊆ s
hu : u ⊆ {y | ‖mfderiv% ↑(extChartAt I x) y‖ₑ < ↑C}
uc : u ⊆ (extChartAt I x).source
uc' : u ⊆ (chartAt H x).source
v : Set M
hv : v ⊆ u
v_mem : v ∈ 𝓝 x
v_open : IsOpen[inst✝⁵] v
r : ℝ≥0
hr : ball (↑(extChartAt I x) x) ↑r ⊆ ↑(extChartAt I x).symm ⁻¹' v
r_pos : 0 < r
y : M
hy : y ∈ {y | riemannianEDist I x y < ↑(r / C)}
γ : ℝ → M
hγx : γ 0 = x
hγy : γ 1 = y
γ_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ
hγ : pathELength I γ 0 1 < ↑(r / C)
A : Set ℝ := γ ⁻¹' u
zero_mem : 0 ∈ A
A_closed : IsClosed (A ∩ Icc 0 1)
this✝ : Icc 0 1 ⊆ A
this : 1 ∈ A
⊢ y ∈ u

Tactic:
simpa [A, hγy, us]

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
⊢ IsSubmersionAt I J n g x

Tactic:
use hf.complement, by infer_instance, by infer_instance

Goal state after:
case h
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
⊢ IsSubmersionAtOfComplement hf.complement I J n g x
Module: Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Torsion

Goal state before:
pp: case mp
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
inst✝³ : CompleteSpace 𝕜
inst✝² : CompleteSpace E
inst✝¹ : FiniteDimensional 𝕜 E
inst✝ : IsManifold I 2 M
cov : CovariantDerivative I E (TangentSpace I)
⊢ cov.torsion = 0 →
    ∀ {X Y : (x : M) → TangentSpace I x} {x : M},
      MDiffAt (T% X) x → MDiffAt (T% Y) x → (↑cov Y x) (X x) - (↑cov X x) (Y x) = mlieBracket I X Y x

Tactic:
intro h

Goal state after:
case mp
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
inst✝³ : CompleteSpace 𝕜
inst✝² : CompleteSpace E
inst✝¹ : FiniteDimensional 𝕜 E
inst✝ : IsManifold I 2 M
cov : CovariantDerivative I E (TangentSpace I)
h : cov.torsion = 0
⊢ ∀ {X Y : (x : M) → TangentSpace I x} {x : M},
    MDiffAt (T% X) x → MDiffAt (T% Y) x → (↑cov Y x) (X x) - (↑cov X x) (Y x) = mlieBracket I X Y x
Module: Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Torsion

Goal state before:
pp: case mpr
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
inst✝³ : CompleteSpace 𝕜
inst✝² : CompleteSpace E
inst✝¹ : FiniteDimensional 𝕜 E
inst✝ : IsManifold I 2 M
cov : CovariantDerivative I E (TangentSpace I)
⊢ (∀ {X Y : (x : M) → TangentSpace I x} {x : M},
      MDiffAt (T% X) x → MDiffAt (T% Y) x → (↑cov Y x) (X x) - (↑cov X x) (Y x) = mlieBracket I X Y x) →
    cov.torsion = 0

Tactic:
intro h

Goal state after:
case mpr
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
inst✝³ : CompleteSpace 𝕜
inst✝² : CompleteSpace E
inst✝¹ : FiniteDimensional 𝕜 E
inst✝ : IsManifold I 2 M
cov : CovariantDerivative I E (TangentSpace I)
h :
  ∀ {X Y : (x : M) → TangentSpace I x} {x : M},
    MDiffAt (T% X) x → MDiffAt (T% Y) x → (↑cov Y x) (X x) - (↑cov X x) (Y x) = mlieBracket I X Y x
⊢ cov.torsion = 0
Module: Mathlib.Geometry.Polygon.Basic

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : Ring R
inst✝⁴ : AddCommGroup V
inst✝³ : Module R V
inst✝² : AddTorsor V P
inst✝¹ : Nontrivial R
m : ℕ
inst✝ : NeZero m.succ
poly : Polygon P m.succ
h : HasNondegenerateVertices R poly
i : Fin m.succ
⊢ 0 ≠ 1

Tactic:
by decide

Goal state after:
Module: Mathlib.Geometry.Polygon.Basic

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_3
n : ℕ
inst✝⁵ : Ring R
inst✝⁴ : AddCommGroup V
inst✝³ : Module R V
inst✝² : AddTorsor V P
inst✝¹ : Nontrivial R
inst✝ : NeZero 2
poly : Polygon P 2
h : HasNondegenerateVertices R poly
this : 2 ≤ 2
hlt : 2 < 3
⊢ 0 ≠ 2

Tactic:
by decide

Goal state after:
Module: Mathlib.GroupTheory.Commutator.Finite

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : Finite ↑(commutatorSet G)
inst✝ : Group.FG G
⊢ (center G).index ≤ Nat.card ↑(commutatorSet G) ^ Group.rank G

Tactic:
obtain ⟨S, hS1, hS2⟩ := Group.rank_spec G

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : Finite ↑(commutatorSet G)
inst✝ : Group.FG G
S : Finset G
hS1 : S.card = Group.rank G
hS2 : closure ↑S = ⊤
⊢ (center G).index ≤ Nat.card ↑(commutatorSet G) ^ Group.rank G
Module: Mathlib.GroupTheory.SpecificGroups.Dihedral

Goal state before:
pp: case inr
n : ℕ
hn : NeZero n
⊢ ¬orderOf (r 1) < n

Tactic:
intro h

Goal state after:
case inr
n : ℕ
hn : NeZero n
h : orderOf (r 1) < n
⊢ False
Module: Mathlib.GroupTheory.SpecificGroups.Dihedral

Goal state before:
pp: ⊢ ∀ (a b : DihedralGroup 1), a * b = b * a

Tactic:
by decide

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

Goal state before:
pp: ⊢ ∀ (a b : DihedralGroup 2), a * b = b * a

Tactic:
by decide

Goal state after:
Module: Mathlib.GroupTheory.CommutingProbability

Goal state before:
pp: n : ℕ
h0 : ¬n = 0
h1 : ¬n = 1
h2 : Even n
⊢ Odd 3

Tactic:
by decide

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'
M' : CoxeterMatrix B' := CoxeterMatrix.reindex e M
⊢ Set.range (uncurry M'.relation) = Set.range (uncurry M'.relation ∘ Prod.map ⇑e ⇑e)

Tactic:
by simp [Set.range_comp]

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
p : W → Prop
w : W
one : p 1
mul_simple_left : ∀ (w : W) (i : B), p w → p (cs.simple i * w)
⊢ p w

Tactic:
let p' : (w : W) → w ∈ Submonoid.closure (Set.range cs.simple) → Prop :=
    fun w _ ↦ p w
  have := cs.submonoid_closure_range_simple.symm ▸ Submonoid.mem_top w
  induction this using Submonoid.closure_induction_left with
  | one => exact one
  | mul_left i mi y my ih =>
    rw [Set.mem_range] at mi
    exact mi.choose_spec ▸ mul_simple_left _ _ ih

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
G : ι → Type u_4
inst✝ : (i : ι) → Group (G i)
⊢ ∀ (a : CoprodI G), a⁻¹ * a = 1

Tactic:
by
      intro m
      rw [inv_def]
      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
⊢ ∃ δ, 0 < δ ∧ 1 + δ * C < r' ^ 2

Tactic:
have B : Tendsto (fun δ ↦ 1 + δ * C) (𝓝[>] 0) (𝓝 (1 + 0 * C)) := by
      apply tendsto_inf_left
      exact tendsto_const_nhds.add (tendsto_id.mul tendsto_const_nhds)

Goal state after:
B✝ : Type u_1
inst✝⁷ : TopologicalSpace B✝
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
E : B✝ → Type u_3
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (x : B✝) → NormedAddCommGroup (E x)
inst✝² : (x : B✝) → InnerProductSpace ℝ (E x)
inst✝¹ : FiberBundle F E
inst✝ : VectorBundle ℝ F E
h : IsContinuousRiemannianBundle F E
x : B✝
r : ℝ
hr : 1 < r
r' : ℝ
hr' : 1 < r'
r'r : r' < r
h'x : x ∈ (trivializationAt F E x).baseSet
G : E x ≃L[ℝ] F := Trivialization.continuousLinearEquivAt ℝ (trivializationAt F E x) x h'x
C : ℝ := ‖↑G‖ ^ 2
A : ∀ᶠ (δ : ℝ) in 𝓝[>] 0, 0 < δ
B : Tendsto (fun δ ↦ 1 + δ * C) (𝓝[>] 0) (𝓝 (1 + 0 * C))
⊢ ∃ δ, 0 < δ ∧ 1 + δ * C < r' ^ 2
Module: Mathlib.GroupTheory.Coxeter.Basic

Goal state before:
pp: case h
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
w : W
i : B
ω : List B
⊢ cs.wordProd (i :: ω) = cs.simple i * cs.wordProd ω

Tactic:
rw [wordProd_cons]

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
⊢ cs.length w = 0 → w = 1

Tactic:
intro h

Goal state after:
case mp
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
w : W
h : cs.length w = 0
⊢ w = 1
Module: Mathlib.GroupTheory.Coxeter.Length

Goal state before:
pp: B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
⊢ 1 + 1 = 0

Tactic:
by decide

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
⊢ cs.IsReflection (w * t * w⁻¹) → cs.IsReflection t

Tactic:
intro h

Goal state after:
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
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
⊢ cs.length w = 1 → ∃ i, w = cs.simple i

Tactic:
intro h

Goal state after:
case mp
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
w : W
h : cs.length w = 1
⊢ ∃ i, w = cs.simple i
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
rω : cs.IsReduced ω
j j' : ℕ
j_lt_j' : j < j'
j'_lt_length : j' < (cs.rightInvSeq ω).length
dup : (cs.rightInvSeq ω)[j]? = (cs.rightInvSeq ω)[j']?
⊢ False

Tactic:
replace j'_lt_length : j' < List.length ω := by simpa using j'_lt_length

Goal state after:
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
rω : cs.IsReduced ω
j j' : ℕ
j_lt_j' : j < j'
dup : (cs.rightInvSeq ω)[j]? = (cs.rightInvSeq ω)[j']?
j'_lt_length : j' < ω.length
⊢ False
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
a b : Quotient ↑H ↑K
⊢ ¬Disjoint (doubleCoset (Quotient.out a) ↑H ↑K) (doubleCoset (Quotient.out b) ↑H ↑K) → a = b

Tactic:
intro h

Goal state after:
G : Type u_1
inst✝ : Group G
H K : Subgroup G
a b : Quotient ↑H ↑K
h : ¬Disjoint (doubleCoset (Quotient.out a) ↑H ↑K) (doubleCoset (Quotient.out b) ↑H ↑K)
⊢ a = b
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
c : DivisibleHull M
ma : M
sa : ℕ+
mb : M
sb : ℕ+
hbc : mk mb sb ≤ c
hab : mk ma sa ≤ mk mb sb
⊢ mk ma sa ≤ c

Tactic:
| mk mb sb
     => 

Goal state after:
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
c : DivisibleHull M
ma : M
sa : ℕ+
mb : M
sb : ℕ+
hbc : mk mb sb ≤ c
hab : mk ma sa ≤ mk mb sb
⊢ mk ma sa ≤ c
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 : mk ma sa ≤ mk mb sb
h' : mk mb sb ≤ mk ma sa
⊢ mk ma sa = mk mb sb

Tactic:
| mk mb sb
     => 

Goal state after:
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 : mk ma sa ≤ mk mb sb
h' : mk mb sb ≤ mk ma sa
⊢ mk ma sa = mk mb sb
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+
⊢ mk ma sa ≤ mk mb sb ∨ mk mb sb ≤ mk ma sa

Tactic:
| mk mb sb
     => 

Goal state after:
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+
⊢ mk ma sa ≤ mk mb sb ∨ mk mb sb ≤ mk ma sa
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
c : DivisibleHull M
ma : M
sa : ℕ+
mb : M
sb : ℕ+
h : mk mb sb < c
⊢ mk ma sa + mk mb sb < mk ma sa + c

Tactic:
| mk mb sb
     => 

Goal state after:
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
c : DivisibleHull M
ma : M
sa : ℕ+
mb : M
sb : ℕ+
h : mk mb sb < c
⊢ mk ma sa + mk mb sb < mk ma sa + c
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: case mk
M✝ : Type u_1
inst✝³ : AddCommMonoid M✝
M : Type u_2
inst✝² : AddCommMonoid M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
a : ℚ≥0
ha : 0 < a
c : DivisibleHull M
mb : M
sb : ℕ+
h : mk mb sb < c
⊢ a • mk mb sb < a • c

Tactic:
| mk mb sb
     => 

Goal state after:
case mk
M✝ : Type u_1
inst✝³ : AddCommMonoid M✝
M : Type u_2
inst✝² : AddCommMonoid M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
a : ℚ≥0
ha : 0 < a
c : DivisibleHull M
mb : M
sb : ℕ+
h : mk mb sb < c
⊢ a • mk mb sb < a • c
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: 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
hcard : 3 ≤ #ι ∨ ∃ i, 3 ≤ #(H i)
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
⊢ Function.Injective ⇑(lift f)

Tactic:
apply (injective_iff_map_eq_one (lift f)).mpr
    rw [(CoprodI.Word.equiv).forall_congr_left]
    intro w Heq
    dsimp [Word.equiv] at *
    rw [empty_of_word_prod_eq_one f hcard X hXnonempty hXdisj hpp Heq, Word.prod_empty]

Goal state after:
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: M✝ : Type u_1
inst✝³ : AddCommMonoid M✝
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedAddMonoid M
a : ℚ
ha : 0 < a
b c : DivisibleHull M
h : b < c
⊢ 0 < ⟨a, ⋯⟩

Tactic:
by simpa using! ha

Goal state after:
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
H : ∀ (n : ℕ), n ∣ Monoid.exponent G → ∀ (a : ZMod n), a ≠ 0 → ∃ φ, φ (Multiplicative.ofAdd a) ≠ 1
a : G
ι : Type
w✝ : Fintype ι
n : ι → ℕ
left✝ : ∀ (i : ι), 1 < n i
h : Nonempty (G ≃* ((i : ι) → Multiplicative (ZMod (n i))))
e : G ≃* ((i : ι) → Multiplicative (ZMod (n i))) := h.some
ha : ∀ (i : ι), e a i = 1
⊢ a = 1

Tactic:
exact (MulEquiv.map_eq_one_iff e).mp <| funext ha

Goal state after:
Module: Mathlib.GroupTheory.Schreier

Goal state before:
pp: G : Type u_1
inst✝¹ : CommGroup G
inst✝ : Group.FG G
⊢ Nat.card G ∣ Monoid.exponent G ^ Group.rank G

Tactic:
obtain ⟨S, hS1, hS2⟩ := Group.rank_spec G

Goal state after:
G : Type u_1
inst✝¹ : CommGroup G
inst✝ : Group.FG G
S : Finset G
hS1 : #S = Group.rank G
hS2 : closure ↑S = ⊤
⊢ Nat.card G ∣ Monoid.exponent G ^ Group.rank G
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: ι : Type u_1
inst✝² : Nontrivial ι
G : Type u_1
inst✝¹ : Group G
a : ι → G
α : Type u_4
inst✝ : MulAction G α
X Y : ι → Set α
hXnonempty : ∀ (i : ι), (X i).Nonempty
hXdisj : Pairwise (Disjoint on X)
hYdisj : Pairwise (Disjoint on Y)
hXYdisj : ∀ (i j : ι), Disjoint (X i) (Y j)
hX : ∀ (i : ι), a i • (Y i)ᶜ ⊆ X i
hY : ∀ (i : ι), a⁻¹ i • (X i)ᶜ ⊆ Y i
H : ι → Type := fun _i ↦ FreeGroup Unit
f : (i : ι) → H i →* G := fun i ↦ FreeGroup.lift fun x ↦ a i
X' : ι → Set α := fun i ↦ X i ∪ Y i
i j : ι
hij : i ≠ j
n : ℤ
h1n : n ≤ -1
⊢ a i ^ n • (X i)ᶜ ⊆ Y i

Tactic:
induction n, h1n using Int.leInductionDown with

Goal state after:
Module: Mathlib.GroupTheory.Transfer

Goal state before:
pp: G : Type u_1
inst✝³ : Group G
p : ℕ
P : Sylow p G
hP : normalizer ↑P ≤ centralizer ↑P
inst✝² : Fact (Nat.Prime p)
inst✝¹ : Finite (Sylow p G)
inst✝ : (↑P).FiniteIndex
hf : Function.Injective ⇑((transferSylow P hP).domRestrict ↑P) ∧ map (transferSylow P hP) ↑P = ⊤
this : Function.Surjective ⇑(transferSylow P hP)
⊢ (transferSylow P hP).ker.IsComplement' ↑P

Tactic:
rw [← (comap_injective this).eq_iff, comap_top, comap_map_eq, sup_comm, SetLike.ext'_iff,
    normal_mul, ← ker_eq_bot_iff, ← map_subtype_inj,
    ker_domRestrict, subgroupOf_map_subtype, Subgroup.map_bot, coe_top] at hf

Goal state after:
G : Type u_1
inst✝³ : Group G
p : ℕ
P : Sylow p G
hP : normalizer ↑P ≤ centralizer ↑P
inst✝² : Fact (Nat.Prime p)
inst✝¹ : Finite (Sylow p G)
inst✝ : (↑P).FiniteIndex
hf : (transferSylow P hP).ker ⊓ ↑P = ⊥ ∧ ↑(transferSylow P hP).ker * ↑↑P = Set.univ
this : Function.Surjective ⇑(transferSylow P hP)
⊢ (transferSylow P hP).ker.IsComplement' ↑P
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: case mp
G : Type u_1
inst✝ : Group G
hG : Group.IsNilpotent G
n : ℕ
⊢ upperCentralSeries G n = ⊤ → nilpotencyClass G ≤ n

Tactic:
intro h

Goal state after:
case mp
G : Type u_1
inst✝ : Group G
hG : Group.IsNilpotent G
n : ℕ
h : upperCentralSeries G n = ⊤
⊢ nilpotencyClass G ≤ n
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: case mpr
G : Type u_1
inst✝ : Group G
hG : Group.IsNilpotent G
n : ℕ
⊢ nilpotencyClass G ≤ n → upperCentralSeries G n = ⊤

Tactic:
intro h

Goal state after:
case mpr
G : Type u_1
inst✝ : Group G
hG : Group.IsNilpotent G
n : ℕ
h : nilpotencyClass G ≤ n
⊢ upperCentralSeries G n = ⊤
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: case mp
G : Type u_1
inst✝ : Group G
hG : Group.IsNilpotent G
n : ℕ
⊢ ⊤.lowerCentralSeries n = ⊥ → nilpotencyClass G ≤ n

Tactic:
intro h

Goal state after:
case mp
G : Type u_1
inst✝ : Group G
hG : Group.IsNilpotent G
n : ℕ
h : ⊤.lowerCentralSeries n = ⊥
⊢ nilpotencyClass G ≤ n
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: case mpr
G : Type u_1
inst✝ : Group G
hG : Group.IsNilpotent G
n : ℕ
⊢ nilpotencyClass G ≤ n → ⊤.lowerCentralSeries n = ⊥

Tactic:
intro h

Goal state after:
case mpr
G : Type u_1
inst✝ : Group G
hG : Group.IsNilpotent G
n : ℕ
h : nilpotencyClass G ≤ n
⊢ ⊤.lowerCentralSeries n = ⊥