-- Mathlib/GroupTheory/Coxeter/Inversion.lean
theorem getD_leftInvSeq_mul_self (ω : List B) (j : ℕ) :
    ((lis ω).getD j 1) * ((lis ω).getD j 1) = 1

Proof state:
case inl
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
j : ℕ
hj : j < ω.length
⊢ cs.wordProd (take j ω) *
      ((Option.map cs.simple (some ω[j])).getD 1 *
        ((cs.wordProd (take j ω))⁻¹ *
          (cs.wordProd (take j ω) * ((Option.map cs.simple (some ω[j])).getD 1 * (cs.wordProd (take j ω))⁻¹)))) =
    1

Tactic:
simp [← mul_assoc]

Result:
no goals

Proof state:
case inr
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
⊢ cs.wordProd (take j ω) *
      ((Option.map cs.simple ω[j]?).getD 1 *
        ((cs.wordProd (take j ω))⁻¹ *
          (cs.wordProd (take j ω) * ((Option.map cs.simple ω[j]?).getD 1 * (cs.wordProd (take j ω))⁻¹)))) =
    1

Tactic:
rw [getElem?_eq_none_iff.mpr (by omega)]

Result:
case inr
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
⊢ cs.wordProd (take j ω) *
      ((Option.map cs.simple none).getD 1 *
        ((cs.wordProd (take j ω))⁻¹ *
          (cs.wordProd (take j ω) * ((Option.map cs.simple none).getD 1 * (cs.wordProd (take j ω))⁻¹)))) =
    1

Proof state:
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 (take j ω) *
      ((Option.map cs.simple ω[j]?).getD 1 *
        ((cs.wordProd (take j ω))⁻¹ *
          (cs.wordProd (take j ω) * ((Option.map cs.simple ω[j]?).getD 1 * (cs.wordProd (take j ω))⁻¹)))) =
    1

Tactic:
rcases em (j < ω.length) with hj | nhj

Result:
case inl
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
j : ℕ
hj : j < ω.length
⊢ cs.wordProd (take j ω) *
      ((Option.map cs.simple ω[j]?).getD 1 *
        ((cs.wordProd (take j ω))⁻¹ *
          (cs.wordProd (take j ω) * ((Option.map cs.simple ω[j]?).getD 1 * (cs.wordProd (take j ω))⁻¹)))) =
    1

case inr
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
⊢ cs.wordProd (take j ω) *
      ((Option.map cs.simple ω[j]?).getD 1 *
        ((cs.wordProd (take j ω))⁻¹ *
          (cs.wordProd (take j ω) * ((Option.map cs.simple ω[j]?).getD 1 * (cs.wordProd (take j ω))⁻¹)))) =
    1

Proof state:
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:
omega

Result:
no goals

Proof state:
case inr
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
⊢ cs.wordProd (take j ω) *
      ((Option.map cs.simple none).getD 1 *
        ((cs.wordProd (take j ω))⁻¹ *
          (cs.wordProd (take j ω) * ((Option.map cs.simple none).getD 1 * (cs.wordProd (take j ω))⁻¹)))) =
    1

Tactic:
simp

Result:
no goals

Proof state:
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:
simp_rw [getD_leftInvSeq, mul_assoc]

Result:
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 (take j ω) *
      ((Option.map cs.simple ω[j]?).getD 1 *
        ((cs.wordProd (take j ω))⁻¹ *
          (cs.wordProd (take j ω) * ((Option.map cs.simple ω[j]?).getD 1 * (cs.wordProd (take j ω))⁻¹)))) =
    1

Proof state:
case inl
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
j : ℕ
hj : j < ω.length
⊢ cs.wordProd (take j ω) *
      ((Option.map cs.simple ω[j]?).getD 1 *
        ((cs.wordProd (take j ω))⁻¹ *
          (cs.wordProd (take j ω) * ((Option.map cs.simple ω[j]?).getD 1 * (cs.wordProd (take j ω))⁻¹)))) =
    1

Tactic:
rw [getElem?_eq_getElem hj]

Result:
case inl
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
j : ℕ
hj : j < ω.length
⊢ cs.wordProd (take j ω) *
      ((Option.map cs.simple (some ω[j])).getD 1 *
        ((cs.wordProd (take j ω))⁻¹ *
          (cs.wordProd (take j ω) * ((Option.map cs.simple (some ω[j])).getD 1 * (cs.wordProd (take j ω))⁻¹)))) =
    1