mathlib-initiative/mathlib-tactics huggingface dataset:
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
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
hcard : 3 ≤ #(H i)
hheadtail : i ≠ j
⊢ (lift f) w.prod ≠ 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
a b : G
⊢ mk H K a = mk H K b ↔ ∃ h ∈ H, ∃ k ∈ K, b = h * a * k

Tactic:
rw [eq'']

Goal state after:
G : Type u_1
inst✝ : Group G
H K : Subgroup G
a b : G
⊢ (setoid ↑H ↑K) a b ↔ ∃ h ∈ H, ∃ k ∈ K, b = h * a * k
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: M✝ : Type u_1
inst✝¹ : AddCommMonoid M✝
M : Type u_2
inst✝ : AddCommGroup M
a b : ℚ
x : DivisibleHull M
⊢ (a + b) • x = a • x + b • x

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
a b : G
⊢ mk H K a = mk H K b ↔ ∃ h ∈ H, ∃ k ∈ K, b = h * a * k

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: M : Type u_2
inst✝ : AddCommGroup M
a : ℤ
m : M
s : ℕ+
⊢ a • mk m s = mk (a • m) s

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
g : G
⊢ ∃ h k, h ∈ H ∧ k ∈ K ∧ Quotient.out (mk H K g) = h * g * k

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
a b : G
h : doubleCoset a ↑H ↑K = doubleCoset b ↑H ↑K
⊢ mk H K a = mk H K b

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
i : Quotient ↑H ↑K
a : G
hg : a ∈ quotToDoubleCoset H K i
⊢ mk H K a = i

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
i : Quotient ↑H ↑K
a : G
⊢ a ∈ quotToDoubleCoset H K i ↔ mk H K a = i

Tactic:
by

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
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
⊢ 0 ≤ r'

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.DoubleCoset

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

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: M : Type u_2
inst✝² : AddCommMonoid M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
m n m' n' : M
s t s' t' : ℕ+
h : mk m s = mk m' s'
h' : mk n t = mk n' t'
⊢ (↑t • m ≤ ↑s • n) = (↑t' • m' ≤ ↑s' • n')

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
⊢ ⋃ q, quotToDoubleCoset H K q = Set.univ

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: M : Type u_2
inst✝² : AddCommMonoid M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
m m' : M
s s' : ℕ+
⊢ mk m s ≤ mk m' s' ↔ ↑s' • m ≤ ↑s • m'

Tactic:
by

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 : AddCommMonoid M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
a : DivisibleHull M
⊢ a ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
a : G
⊢ ⋃ k, op (a * ↑k) • ↑H = doubleCoset a ↑H ↑K

Tactic:
by

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 : AddCommMonoid M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
a b c : DivisibleHull M
hab : a ≤ b
hbc : b ≤ c
⊢ a ≤ c

Tactic:
by

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 : AddCommMonoid M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
a b : DivisibleHull M
h : a ≤ b
h' : b ≤ a
⊢ a = b

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
a : G
⊢ ⋃ h, (↑h * a) • ↑K = doubleCoset a ↑H ↑K

Tactic:
by

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 : AddCommMonoid M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
a b : DivisibleHull M
⊢ a ≤ b ∨ b ≤ a

Tactic:
by

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 : AddCommMonoid M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
⊢ DecidableLE (DivisibleHull M)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: M : Type u_2
inst✝² : AddCommMonoid M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
m m' : M
s s' : ℕ+
⊢ mk m s < mk m' s' ↔ ↑s' • m < ↑s • m'

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
⊢ Quotient ↑⊥ ↑H = (G ⧸ H)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
⊢ Quotient ↑H ↑⊥ = _root_.Quotient (rightRel H)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.DoubleCoset

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

Tactic:
by

Goal state after:
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:
by

Goal state after:
Module: Mathlib.GroupTheory.DoubleCoset

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

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
x : Quot ⇑(leftRel K)
y : G
hy : ⟦y⟧ = x
cover : ¬∃ i, y ∈ doubleCoset (out i) ↑H ↑K
⊢ y ∉ ⋃ q, quotToDoubleCoset H K q

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
cover : ⋃ q, quotToDoubleCoset H K q = Set.univ
x : Quot ⇑(leftRel K)
y : G
hy : ⟦y⟧ = x
⊢ ∃ i, y ∈ doubleCoset (out i) ↑H ↑K

Tactic:
by

Goal state after:
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
i j : ι
w : NeWord H i j
⊢ (lift f) w.prod ≠ 1

Tactic:
by

Goal state after:
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
w : Word H
h : (lift f) w.prod = 1
⊢ w = Word.empty

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: case mk.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+
mc : M
sc : ℕ+
h : ↑sc • mb < ↑sb • mc
this : (↑sa * ↑sb * ↑sc) • ma + (↑sa * ↑sa * ↑sc) • mb < (↑sa * ↑sb * ↑sc) • ma + (↑sa * ↑sa * ↑sb) • mc
⊢ (↑sa * (↑sc * ↑sb)) • ma + (↑sa * (↑sc * ↑sa)) • mb < (↑sa * (↑sb * ↑sc)) • ma + (↑sa * (↑sb * ↑sa)) • mc

Tactic:
convert! this using 3 <;> ring

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 : AddCommMonoid M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
a b c : DivisibleHull M
h : b < c
⊢ a + b < a + c

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
⊢ ⋃ q, Quot.mk ⇑(leftRel K) '' doubleCoset (out q) ↑H ↑K = Set.univ

Tactic:
by

Goal state after:
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:
by

Goal state after:
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
x : Quot ⇑(rightRel H)
y : G
hy : ⟦y⟧ = x
cover : ¬∃ i, y ∈ doubleCoset (out i) ↑H ↑K
⊢ y ∉ ⋃ q, quotToDoubleCoset H K q

Tactic:
by

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

Goal state after:
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
cover : ⋃ q, quotToDoubleCoset H K q = Set.univ
x : Quot ⇑(rightRel H)
y : G
hy : ⟦y⟧ = x
⊢ ∃ i, y ∈ doubleCoset (out i) ↑H ↑K

Tactic:
by

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 : AddCommMonoid M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
a : ℚ≥0
ha : 0 < a
mb : M
sb : ℕ+
mc : M
sc : ℕ+
h : ↑sc • mb < ↑sb • mc
⊢ ↑⟨a.den, ⋯⟩ * a.num ≠ 0

Tactic:
by

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)
⊢ (FreeGroup.lift fun x ↦ of ((B x.fst) x.snd)).comp (lift fun i ↦ (B i).lift fun x ↦ FreeGroup.of ⟨i, x⟩) =
    MonoidHom.id (CoprodI G)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
⊢ ⋃ q, Quot.mk ⇑(rightRel H) '' doubleCoset (out q) ↑H ↑K = Set.univ

Tactic:
by

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 : AddCommMonoid M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
a : ℚ≥0
ha : 0 < a
b c : DivisibleHull M
h : b < c
⊢ a • b < a • c

Tactic:
by

Goal state after:
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
ι : Type u_1
⊢ FreeGroup ι ≃* CoprodI fun x ↦ FreeGroup Unit

Tactic:
by

Goal state after:
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
⊢ FreeGroup.lift a = (lift fun i ↦ FreeGroup.lift fun x ↦ a i).comp freeGroupEquivCoprodI.toMonoidHom

Tactic:
by

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
hx : x ∉ ⋃ q ∈ t, doubleCoset (out q) ↑H ↑K
y : Quotient ↑H ↑K
hy : y ∈ t
q : G
hq : q ∈ doubleCoset (out y) ↑H ↑K
⊢ ¬Quot.mk (⇑(leftRel K)) q = Quot.mk (⇑(leftRel K)) x

Tactic:
contrapose 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 (⇑(leftRel K)) q = Quot.mk (⇑(leftRel K)) x
⊢ x ∈ ⋃ q ∈ t, doubleCoset (out q) ↑H ↑K
Module: Mathlib.GroupTheory.DivisibleHull

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

Tactic:
by

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

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
⊢ a • b < a • c

Tactic:
by

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 : DivisibleHull M
ha : 0 < a
b c : ℚ
h : b < c
⊢ 0 < ⟨c - b, ⋯⟩

Tactic:
by

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 : DivisibleHull M
ha : 0 < a
b c : ℚ
h : b < c
⊢ b • a < c • a

Tactic:
by

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 (⇑(leftRel K)) q = Quot.mk (⇑(leftRel K)) x
a' : ↥K.op
ha' : (fun m ↦ m • x) a' = q
⊢ (↑a')⁻¹ ∈ K.op

Tactic:
by

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 b : M
h : a ≤ b
⊢ (↑(coeAddMonoidHom M)).toFun a ≤ (↑(coeAddMonoidHom M)).toFun b

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedAddMonoid M
m : M
s s' : ℕ+
this : ArchimedeanClass.mk (↑↑s • mk m s) = ArchimedeanClass.mk (↑↑s' • mk m s')
⊢ ↑↑s ≠ 0

Tactic:
by

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 (⇑(leftRel K)) q = Quot.mk (⇑(leftRel K)) x
a' : ↥K.op
ha' : (fun m ↦ m • x) a' = q
⊢ x = 1 * q * unop (↑a')⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedAddMonoid M
m : M
s s' : ℕ+
this : ArchimedeanClass.mk (mk m s) = ArchimedeanClass.mk (↑↑s' • mk m s')
⊢ ↑↑s' ≠ 0

Tactic:
by

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 : Set.univ ⊆ ⋃ i ∈ t, Quot.mk ⇑(leftRel K) '' doubleCoset (out i) ↑H ↑K
⊢ ⋃ q ∈ t, doubleCoset (out q) ↑H ↑K = Set.univ

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedAddMonoid M
m : M
s s' : ℕ+
this : ↑↑s • mk m s = ↑↑s' • mk m s'
⊢ ArchimedeanClass.mk (mk m s) = ArchimedeanClass.mk (mk m s')

Tactic:
by

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
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✝¹ : ∀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
this : ‖(Trivialization.symmL ℝ (trivializationAt F E x) y) w‖ ^ 2 ≤ r' ^ 2 * ‖v‖ ^ 2
⊢ ‖(Trivialization.symmL ℝ (trivializationAt F E x) y ∘SL
            Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) x)
          v‖ ^
      2 ≤
    (r' * ‖v‖) ^ 2

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedAddMonoid M
m : M
s s' : ℕ+
⊢ ArchimedeanClass.mk (mk m s) = ArchimedeanClass.mk (mk m s')

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedAddMonoid M
m : M
s : ℕ+
⊢ ArchimedeanClass.mk (mk m s) = (archimedeanClassOrderHom M) (ArchimedeanClass.mk m)

Tactic:
by

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

Tactic:
contrapose 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
⊢ x ∈ ⋃ q ∈ t, doubleCoset (out q) ↑H ↑K
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
x : DivisibleHull M
x✝³ x✝² : M
x✝¹ x✝ : ℕ+
h : mk x✝³ x✝¹ = mk x✝² x✝
⊢ ArchimedeanClass.mk x✝³ = ArchimedeanClass.mk x✝²

Tactic:
by

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 : ↥H
ha : ↑a * x = q
⊢ (↑a)⁻¹ ∈ H

Tactic:
by

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:
by

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 : ↥H
ha : x = ↑a * q
⊢ x = ↑a * q * 1

Tactic:
by

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 : Set.univ ⊆ ⋃ i ∈ t, Quot.mk ⇑(rightRel H) '' doubleCoset (out i) ↑H ↑K
⊢ ⋃ q ∈ t, doubleCoset (out q) ↑H ↑K = Set.univ

Tactic:
by

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 b : DivisibleHull M
h : ArchimedeanClass.mk a ≤ ArchimedeanClass.mk b
⊢ a.liftOn (fun m s ↦ ArchimedeanClass.mk m) ⋯ ≤ b.liftOn (fun m s ↦ ArchimedeanClass.mk m) ⋯

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FiniteAbelian.Duality

Goal state before:
pp: ι : Type u_1
G : Type u_2
inst✝ : Monoid G
n : ι → ℕ
e : G ≃* ((i : ι) → Multiplicative (ZMod (n i)))
i : ι
⊢ n i = orderOf (e.symm (Pi.mulSingle i (Multiplicative.ofAdd 1)))

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FiniteAbelian.Duality

Goal state before:
pp: ι : Type u_1
G : Type u_2
inst✝ : Monoid G
n : ι → ℕ
e : G ≃* ((i : ι) → Multiplicative (ZMod (n i)))
i : ι
⊢ n i ∣ Monoid.exponent G

Tactic:
by

Goal state after: