mathlib-initiative/mathlib-tactics huggingface dataset:
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
i i' : B
m : ℕ
hM : M.M i i' ≠ 0
hm : m > M.M i i'
⊢ ¬cs.IsReduced (alternatingWord i i' m)

Tactic:
by

Goal state after:
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
i : B
⊢ ¬cs.IsLeftDescent 1 i

Tactic:
by

Goal state after:
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
i : B
⊢ ¬cs.IsRightDescent 1 i

Tactic:
by

Goal state after:
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
w : W
i : B
⊢ cs.IsLeftDescent w⁻¹ i ↔ cs.IsRightDescent w i

Tactic:
by

Goal state after:
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
w : W
i : B
⊢ cs.IsRightDescent w⁻¹ i ↔ cs.IsLeftDescent w i

Tactic:
by

Goal state after:
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
ω : List B
h : cs.IsReduced ω
hw : cs.wordProd ω ≠ 1
⊢ ω ≠ []

Tactic:
by

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

Goal state before:
pp: case h
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i : B
ω' : List B
h : cs.IsReduced (i :: ω')
hw : cs.wordProd (i :: ω') ≠ 1
h₁ : i :: ω' ≠ []
⊢ cs.length (cs.simple i * cs.wordProd (i :: ω')) < (i :: ω').length

Tactic:
wordProd_cons,

Goal state after:
case h
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i : B
ω' : List B
h : cs.IsReduced (i :: ω')
hw : cs.wordProd (i :: ω') ≠ 1
h₁ : i :: ω' ≠ []
⊢ cs.length (cs.simple i * (cs.simple i * cs.wordProd ω')) < (i :: ω').length
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
i : B
ω' : List B
h : cs.IsReduced (i :: ω')
hw : cs.wordProd (i :: ω') ≠ 1
h₁ : i :: ω' ≠ []
⊢ ω'.length < (i :: ω').length

Tactic:
by

Goal state after:
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
w : W
hw : w ≠ 1
⊢ ∃ i, cs.IsLeftDescent w i

Tactic:
by

Goal state after:
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
w : W
hw : w ≠ 1
⊢ ∃ i, cs.IsRightDescent w i

Tactic:
by

Goal state after:
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
w : W
i : B
a✝ : cs.length (cs.simple i * w) < cs.length w
⊢ ¬cs.length (cs.simple i * w) = cs.length w + 1

Tactic:
by

Goal state after:
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
w : W
i : B
⊢ cs.IsLeftDescent w i ↔ cs.length (cs.simple i * w) + 1 = cs.length w

Tactic:
by

Goal state after:
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
j : ℕ
⊢ (cs.rightInvSeq ω).getD j 1 =
    (cs.wordProd (drop (j + 1) ω))⁻¹ * (Option.map cs.simple ω[j]?).getD 1 * cs.wordProd (drop (j + 1) ω)

Tactic:
by

Goal state after:
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
j : ℕ
h : j < ω.length
⊢ j < (cs.rightInvSeq ω).length

Tactic:
by

Goal state after:
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
w : W
i : B
a✝ : ¬cs.length (cs.simple i * w) < cs.length w
⊢ ¬cs.length (cs.simple i * w) + 1 = cs.length w

Tactic:
by

Goal state after:
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
j : ℕ
h : j < ω.length
⊢ (cs.rightInvSeq ω)[j] =
    (cs.wordProd (drop (j + 1) ω))⁻¹ * (Option.map cs.simple ω[j]?).getD 1 * cs.wordProd (drop (j + 1) ω)

Tactic:
by

Goal state after:
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
w : W
i : B
⊢ ¬cs.IsLeftDescent w i ↔ cs.length (cs.simple i * w) = cs.length w + 1

Tactic:
by

Goal state after:
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
w : W
i : B
a✝ : cs.length (w * cs.simple i) < cs.length w
⊢ ¬cs.length (w * cs.simple i) = cs.length w + 1

Tactic:
by

Goal state after:
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
w : W
i : B
⊢ cs.IsRightDescent w i ↔ cs.length (w * cs.simple i) + 1 = cs.length w

Tactic:
by

Goal state after:
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
w : W
i : B
a✝ : ¬cs.length (w * cs.simple i) < cs.length w
⊢ ¬cs.length (w * cs.simple i) + 1 = cs.length w

Tactic:
by

Goal state after:
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
w : W
i : B
⊢ ¬cs.IsRightDescent w i ↔ cs.length (w * cs.simple i) = cs.length w + 1

Tactic:
by

Goal state after:
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
i : B
ω : List B
ih :
  ∀ (j : ℕ),
    (cs.leftInvSeq ω).getD j 1 =
      cs.wordProd (take j ω) * (Option.map cs.simple ω[j]?).getD 1 * (cs.wordProd (take j ω))⁻¹
j' : ℕ
⊢ 1 = (MulAut.conj (cs.simple i)) 1

Tactic:
by

Goal state after:
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
w : W
i : B
⊢ cs.IsLeftDescent w i ↔ ¬cs.IsLeftDescent (cs.simple i * w) i

Tactic:
by

Goal state after:
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
w : W
i : B
⊢ cs.IsRightDescent w i ↔ ¬cs.IsRightDescent (w * cs.simple i) i

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)
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (M i)
i j : ι
m₁ : M i
m₂ : M j
w : Word M
⊢ ⟨i, m₁⟩ ∈ (of m₂ • w).toList ↔
    ¬i = j ∧ ⟨i, m₁⟩ ∈ w.toList ∨
      m₁ ≠ 1 ∧
        ∃ (hij : i = j),
          ⟨i, m₁⟩ ∈ w.toList.tail ∨
            (∃ m', ⟨j, m'⟩ ∈ w.toList.head? ∧ m₁ = ⋯ ▸ (m₂ * m')) ∨ w.fstIdx ≠ some j ∧ m₁ = ⋯ ▸ m₂

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)
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (M i)
i j : ι
hij : i ≠ j
m₁ : M i
m₂ : M j
w : Word M
⊢ ⟨i, m₁⟩ ∈ (of m₂ • w).toList ↔ ⟨i, m₁⟩ ∈ w.toList

Tactic:
by

Goal state after:
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
j : ℕ
⊢ (cs.leftInvSeq ω).getD j 1 = cs.wordProd (take j ω) * (Option.map cs.simple ω[j]?).getD 1 * (cs.wordProd (take j ω))⁻¹

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)
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (M i)
i : ι
m : M i
ls : Word M
h1 : ls.fstIdx ≠ some i
h2 : m ≠ 1
⊢ cons m ls h1 h2 = of m • ls

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)
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (M i)
i : ι
p : Pair M i
⊢ rcons p = of p.head • p.tail

Tactic:
by

Goal state after:
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
j : ℕ
h : j < ω.length
⊢ j < (cs.leftInvSeq ω).length

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)
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (M i)
i : ι
w : Word M
⊢ of ((equivPair i) w).head • ((equivPair i) w).tail = w

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)
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (M i)
motive : Word M → Prop
empty : motive Word.empty
smul : ∀ (i : ι) (m : M i) (w : Word M), motive w → motive (of m • w)
w : Word M
⊢ motive w

Tactic:
by

Goal state after:
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
j : ℕ
h : j < ω.length
⊢ (cs.leftInvSeq ω)[j] = cs.wordProd (take j ω) * cs.simple ω[j] * (cs.wordProd (take j ω))⁻¹

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)
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (M i)
m : CoprodI M
⊢ ∀ (w : Word M), (m • w).prod = m * w.prod

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: ι : Type u_1
M : ι → Type u_2
inst✝³ : (i : ι) → Monoid (M i)
N : Type u_3
inst✝² : Monoid N
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (M i)
m : CoprodI M
⊢ (fun w ↦ w.prod) ((fun m ↦ m • empty) m) = m

Tactic:
by

Goal state after:
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
j : ℕ
nhj : ¬j < ω.length
⊢ ω.length ≤ j

Tactic:
by

Goal state after:
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
j : ℕ
⊢ (cs.rightInvSeq ω).getD j 1 * (cs.rightInvSeq ω).getD j 1 = 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: ι : Type u_1
M : ι → Type u_2
inst✝³ : (i : ι) → Monoid (M i)
N : Type u_3
inst✝² : Monoid N
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (M i)
⊢ Function.RightInverse (fun w ↦ w.prod) fun m ↦ m • empty

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: ι : Type u_1
M : ι → Type u_2
inst✝ : (i : ι) → Monoid (M i)
i j : ι
w : NeWord M i j
⊢ w.toList ≠ []

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Descent

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
f : G →* G
hf : ∀ (U : Subgroup G), map f U ≤ U
s : Set G
h : G → ℝ
a b c : ℝ
ha : 0 ≤ a
H₀ : a < b
hs : s.Finite
H₁ : s * ↑f.range = Set.univ
H₂ : ∀ g ∈ s, ∀ (x : G), h x ≤ a * h (g * x) + c
H₃ : ∀ (x : G), b * h x - c ≤ h (f x)
inst✝ : Northcott h
q : G → G ⧸ map f ⊤ := QuotientGroup.mk
S : Set G := s ∪ {x | h x ≤ 2 * c / (b - a)}
U : Subgroup G := closure S
H : (↑U)ᶜ.Nonempty
g : G
hg : g ∈ s
y : G
hx₁ : g * f y ∈ (↑U)ᶜ
hx₂ : ∀ a' ∈ (↑U)ᶜ, h (g * f y) ≤ h a'
this : a * h (g * f y) + 2 * c < b * h (g * f y)
⊢ h y < h (g * f y)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: ι : Type u_1
M : ι → Type u_2
inst✝ : (i : ι) → Monoid (M i)
i j : ι
w : NeWord M i j
⊢ w.toList.head? = some ⟨i, w.head⟩

Tactic:
fun_induction toList with

Goal state after:
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: ι : Type u_1
M : ι → Type u_2
inst✝ : (i : ι) → Monoid (M i)
i j : ι
w : NeWord M i j
⊢ w.toList.head? = some ⟨i, w.head⟩

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: ι : Type u_1
M : ι → Type u_2
inst✝ : (i : ι) → Monoid (M i)
i j i✝ j✝ k✝ l✝ : ι
_w₁✝ : NeWord M i✝ j✝
_hne✝ : j✝ ≠ k✝
_w₂✝ : NeWord M k✝ l✝
_w₁_ih✝ : ⟨j✝, _w₁✝.last⟩ ∈ _w₁✝.toList.getLast?
_w₂_ih✝ : ⟨l✝, _w₂✝.last⟩ ∈ _w₂✝.toList.getLast?
⊢ ⟨l✝, (_w₁✝.append _hne✝ _w₂✝).last⟩ ∈ _w₂✝.toList.getLast?

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: ι : Type u_1
M : ι → Type u_2
inst✝ : (i : ι) → Monoid (M i)
i j : ι
w : NeWord M i j
⊢ w.toList.getLast? = some ⟨j, w.last⟩

Tactic:
by

Goal state after:
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
j : ℕ
⊢ (cs.leftInvSeq ω).getD j 1 * (cs.leftInvSeq ω).getD j 1 = 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Descent

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
f : G →* G
hf : ∀ (U : Subgroup G), map f U ≤ U
s : Set G
h : G → ℝ
a b c : ℝ
ha : 0 ≤ a
H₀ : a < b
hs : s.Finite
H₁ : s * ↑f.range = Set.univ
H₂ : ∀ g ∈ s, ∀ (x : G), h x ≤ a * h (g * x) + c
H₃ : ∀ (x : G), b * h x - c ≤ h (f x)
inst✝ : Northcott h
q : G → G ⧸ map f ⊤ := QuotientGroup.mk
S : Set G := s ∪ {x | h x ≤ 2 * c / (b - a)}
U : Subgroup G := closure S
H : (↑U)ᶜ.Nonempty
g : G
hg : g ∈ s
y : G
hx₁ : g * f y ∈ (↑U)ᶜ
hx₂ : ∀ a' ∈ (↑U)ᶜ, h (g * f y) ≤ h a'
this : 2 * c / (b - a) < h (g * f y)
⊢ a * h (g * f y) + 2 * c < b * h (g * f y)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Descent

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
f : G →* G
hf : ∀ (U : Subgroup G), map f U ≤ U
s : Set G
h : G → ℝ
a b c : ℝ
ha : 0 ≤ a
H₀ : a < b
hs : s.Finite
H₁ : s * ↑f.range = Set.univ
H₂ : ∀ g ∈ s, ∀ (x : G), h x ≤ a * h (g * x) + c
H₃ : ∀ (x : G), b * h x - c ≤ h (f x)
inst✝ : Northcott h
q : G → G ⧸ map f ⊤ := QuotientGroup.mk
S : Set G := s ∪ {x | h x ≤ 2 * c / (b - a)}
U : Subgroup G := closure S
H : (↑U)ᶜ.Nonempty
g : G
hg : g ∈ s
y : G
hx₁ : g * f y ∈ (↑U)ᶜ
hx₂ : ∀ a' ∈ (↑U)ᶜ, h (g * f y) ≤ h a'
this : g * f y ∉ S
⊢ 2 * c / (b - a) < h (g * f y)

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
i j : ι
w : NeWord M i j
⊢ ∀ l ∈ w.toList, l.snd ≠ 1

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
i j i✝ j✝ k✝ l✝ : ι
_w₁✝ : NeWord M i✝ j✝
_hne✝ : j✝ ≠ k✝
_w₂✝ : NeWord M k✝ l✝
_w₁_ih✝ : List.IsChain (fun l l' ↦ l.fst ≠ l'.fst) _w₁✝.toList
_w₂_ih✝ : List.IsChain (fun l l' ↦ l.fst ≠ l'.fst) _w₂✝.toList
⊢ List.IsChain (fun l l' ↦ l.fst ≠ l'.fst) _w₁✝.toList

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Descent

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
f : G →* G
hf : ∀ (U : Subgroup G), map f U ≤ U
s : Set G
h : G → ℝ
a b c : ℝ
ha : 0 ≤ a
H₀ : a < b
hs : s.Finite
H₁ : s * ↑f.range = Set.univ
H₂ : ∀ g ∈ s, ∀ (x : G), h x ≤ a * h (g * x) + c
H₃ : ∀ (x : G), b * h x - c ≤ h (f x)
inst✝ : Northcott h
q : G → G ⧸ map f ⊤ := QuotientGroup.mk
S : Set G := s ∪ {x | h x ≤ 2 * c / (b - a)}
U : Subgroup G := closure S
H : (↑U)ᶜ.Nonempty
g : G
hg : g ∈ s
y : G
hx₁ : g * f y ∈ (↑U)ᶜ
hx₂ : ∀ a' ∈ (↑U)ᶜ, h (g * f y) ≤ h a'
⊢ h y < h (g * f y)

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
i j i✝ j✝ k✝ l✝ : ι
_w₁✝ : NeWord M i✝ j✝
_hne✝ : j✝ ≠ k✝
_w₂✝ : NeWord M k✝ l✝
_w₁_ih✝ : List.IsChain (fun l l' ↦ l.fst ≠ l'.fst) _w₁✝.toList
_w₂_ih✝ : List.IsChain (fun l l' ↦ l.fst ≠ l'.fst) _w₂✝.toList
⊢ List.IsChain (fun l l' ↦ l.fst ≠ l'.fst) _w₂✝.toList

Tactic:
by

Goal state after:
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
j : ℕ
⊢ cs.rightInvSeq (drop j ω) = drop j (cs.rightInvSeq ω)

Tactic:
by

Goal state after:
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
j : ℕ
⊢ cs.leftInvSeq (take j ω) = take j (cs.leftInvSeq ω)

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
i j : ι
w : NeWord M i j
⊢ List.IsChain (fun l l' ↦ l.fst ≠ l'.fst) w.toList

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Descent

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
f : G →* G
hf : ∀ (U : Subgroup G), map f U ≤ U
s : Set G
h : G → ℝ
a b c : ℝ
ha : 0 ≤ a
H₀ : a < b
hs : s.Finite
H₁ : s * ↑f.range = Set.univ
H₂ : ∀ g ∈ s, ∀ (x : G), h x ≤ a * h (g * x) + c
H₃ : ∀ (x : G), b * h x - c ≤ h (f x)
inst✝ : Northcott h
⊢ FG G

Tactic:
by

Goal state after:
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
t : W
ht : t ∈ cs.rightInvSeq ω
⊢ cs.IsReflection t

Tactic:
by

Goal state after:
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
t : W
ht : t ∈ cs.leftInvSeq ω
⊢ cs.IsReflection t

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: ι : Type u_1
M : ι → Type u_2
inst✝ : (i : ι) → Monoid (M i)
w : Word M
h : w ≠ empty
this : ∃ i j w', w'.toWord.toList = w.toList
⊢ ∃ i j w', w'.toWord = w

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Descent

Goal state before:
pp: G : Type u_1
inst✝¹ : CommGroup G
n : ℕ
h : G → ℝ
a b c₀ : ℝ
c : G → ℝ
ha : 0 ≤ a
H₀ : a < b
H₁ : (powMonoidHom n).range.FiniteIndex
H₂ : ∀ (g x : G), h x ≤ a * h (g * x) + c g
H₃ : ∀ (x : G), b * h x - c₀ ≤ h (x ^ n)
inst✝ : Northcott h
f : G →* G := powMonoidHom n
q : G → G ⧸ f.range := QuotientGroup.mk
qi : G ⧸ f.range → G := Function.surjInv ⋯
s : Set G := Set.range qi
g : G
hg₁ : g ∈ s
hg₂ : ∀ b ∈ s, c b ≤ c g
x : G
⊢ qi (q x) ∈ s

Tactic:
by

Goal state after:
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
j : ℕ
⊢ cs.wordProd ω * (cs.rightInvSeq ω).getD j 1 = cs.wordProd (ω.eraseIdx j)

Tactic:
by

Goal state after:
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
j : ℕ
⊢ (cs.leftInvSeq ω).getD j 1 * cs.wordProd ω = cs.wordProd (ω.eraseIdx j)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Descent

Goal state before:
pp: G : Type u_1
inst✝¹ : CommGroup G
n : ℕ
h : G → ℝ
a b c₀ : ℝ
c : G → ℝ
ha : 0 ≤ a
H₀ : a < b
H₁ : (powMonoidHom n).range.FiniteIndex
H₂ : ∀ (g x : G), h x ≤ a * h (g * x) + c g
H₃ : ∀ (x : G), b * h x - c₀ ≤ h (x ^ n)
inst✝ : Northcott h
f : G →* G := powMonoidHom n
q : G → G ⧸ f.range := QuotientGroup.mk
qi : G ⧸ f.range → G := Function.surjInv ⋯
s : Set G := Set.range qi
g : G
hg₁ : g ∈ s
hg₂ : ∀ b ∈ s, c b ≤ c g
⊢ s * ↑f.range = Set.univ

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Descent

Goal state before:
pp: G : Type u_1
inst✝¹ : CommGroup G
n : ℕ
h : G → ℝ
a b c₀ : ℝ
c : G → ℝ
ha : 0 ≤ a
H₀ : a < b
H₁ : (powMonoidHom n).range.FiniteIndex
H₂ : ∀ (g x : G), h x ≤ a * h (g * x) + c g
H₃ : ∀ (x : G), b * h x - c₀ ≤ h (x ^ n)
inst✝ : Northcott h
f : G →* G := powMonoidHom n
q : G → G ⧸ f.range := QuotientGroup.mk
qi : G ⧸ f.range → G := Function.surjInv ⋯
s : Set G := Set.range qi
g : G
hg₁ : g ∈ s
hg₂ : ∀ b ∈ s, c b ≤ c g
H₁' : s * ↑f.range = Set.univ
c' : ℝ := max c₀ (c g)
x : G
⊢ b * h x - c' ≤ h (f x)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: ι : Type u_1
M : ι → Type u_2
inst✝ : (i : ι) → Monoid (M i)
x y : (i : ι) × M i
l : List ((i : ι) × M i)
hi :
  ∀ (hnot1 : ∀ l_1 ∈ y :: l, l_1.snd ≠ 1) (hchain : List.IsChain (fun l l' ↦ l.fst ≠ l'.fst) (y :: l)),
    { toList := y :: l, ne_one := hnot1, chain_ne := hchain } ≠ empty →
      ∃ i j w', w'.toWord.toList = { toList := y :: l, ne_one := hnot1, chain_ne := hchain }.toList
hnot1✝ : ∀ l_1 ∈ x :: y :: l, l_1.snd ≠ 1
hnot1 : x.snd ≠ 1 ∧ ∀ x ∈ y :: l, x.snd ≠ 1
hchain✝ : List.IsChain (fun l l' ↦ l.fst ≠ l'.fst) (x :: y :: l)
hchain : x.fst ≠ y.fst ∧ List.IsChain (fun l l' ↦ l.fst ≠ l'.fst) (y :: l)
h : { toList := x :: y :: l, ne_one := hnot1✝, chain_ne := hchain✝ } ≠ empty
⊢ { toList := y :: l, ne_one := ⋯, chain_ne := ⋯ } ≠ empty

Tactic:
by

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

Goal state before:
pp: B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
hω : cs.IsReduced ω
j : ℕ
hj✝ : j < (cs.rightInvSeq ω).length
hj : j < ω.length
ht : (cs.rightInvSeq ω)[j] ∈ cs.rightInvSeq ω
⊢ (ω.eraseIdx j).length < ω.length

Tactic:
by

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

Goal state before:
pp: B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
hω : cs.IsReduced ω
t : W
ht : t ∈ cs.rightInvSeq ω
⊢ cs.IsRightInversion (cs.wordProd ω) t

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: ι : Type u_1
M : ι → Type u_2
inst✝ : (i : ι) → Monoid (M i)
x y : (i : ι) × M i
l : List ((i : ι) × M i)
hnot1✝ : ∀ l_1 ∈ x :: y :: l, l_1.snd ≠ 1
hnot1 : x.snd ≠ 1 ∧ ∀ x ∈ y :: l, x.snd ≠ 1
hchain✝ : List.IsChain (fun l l' ↦ l.fst ≠ l'.fst) (x :: y :: l)
hchain : x.fst ≠ y.fst ∧ List.IsChain (fun l l' ↦ l.fst ≠ l'.fst) (y :: l)
h : { toList := x :: y :: l, ne_one := hnot1✝, chain_ne := hchain✝ } ≠ empty
i j : ι
w' : NeWord M i j
hw' : w'.toList = y :: l
⊢ y = ⟨i, w'.head⟩

Tactic:
by

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

Goal state before:
pp: B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
hω : cs.IsReduced ω
j : ℕ
hj✝ : j < (cs.leftInvSeq ω).length
hj : j < ω.length
ht : (cs.leftInvSeq ω)[j] ∈ cs.leftInvSeq ω
⊢ (ω.eraseIdx j).length < ω.length

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Descent

Goal state before:
pp: G : Type u_1
inst✝¹ : CommGroup G
n : ℕ
h : G → ℝ
a b c₀ : ℝ
c : G → ℝ
ha : 0 ≤ a
H₀ : a < b
H₁ : (powMonoidHom n).range.FiniteIndex
H₂ : ∀ (g x : G), h x ≤ a * h (g * x) + c g
H₃ : ∀ (x : G), b * h x - c₀ ≤ h (x ^ n)
inst✝ : Northcott h
⊢ Group.FG G

Tactic:
by

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

Goal state before:
pp: B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
hω : cs.IsReduced ω
t : W
ht : t ∈ cs.leftInvSeq ω
⊢ cs.IsLeftInversion (cs.wordProd ω) t

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Descent

Goal state before:
pp: G : Type u_1
inst✝¹ : CommGroup G
h : G → ℝ
C : ℝ
H₁ : (powMonoidHom 2).range.FiniteIndex
H₂ : ∀ (x : G), 0 ≤ h x
H₃ : ∀ (x y : G), |h (x * y) + h (x / y) - 2 * (h x + h y)| ≤ C
inst✝ : Northcott h
x : G
⊢ 4 * h x - (h 1 + C) ≤ h (x ^ 2)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Descent

Goal state before:
pp: G : Type u_1
inst✝¹ : CommGroup G
h : G → ℝ
C : ℝ
H₁ : (powMonoidHom 2).range.FiniteIndex
H₂ : ∀ (x : G), 0 ≤ h x
H₃ : ∀ (x y : G), |h (x * y) + h (x / y) - 2 * (h x + h y)| ≤ C
inst✝ : Northcott h
H₃' : ∀ (x : G), 4 * h x - (h 1 + C) ≤ h (x ^ 2)
g x : G
⊢ h x ≤ 2 * h (g * x) + (2 * h g⁻¹ + C)

Tactic:
by

Goal state after: