-- Mathlib/GroupTheory/Coxeter/Basic.lean
theorem prod_alternatingWord_eq_mul_pow (i i' : B) (m : ℕ) :
    π (alternatingWord i i' m) = (if Even m then 1 else s i') * (s i * s i') ^ (m / 2)

Proof state:
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
ih :
  cs.wordProd (alternatingWord i i' m) = (if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)
hm : Even m
h₁ : ¬Even (m + 1)
⊢ ¬2 ∣ m + 1

Tactic:
rwa [← even_iff_two_dvd]

Result:
no goals

Proof state:
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
ih :
  cs.wordProd (alternatingWord i i' m) = (if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)
hm : Even m
⊢ ¬Even (m + 1)

Tactic:
simp [hm, parity_simps]

Result:
no goals

Proof state:
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
ih :
  cs.wordProd (alternatingWord i i' m) = (if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)
hm : ¬Even m
⊢ Even (m + 1)

Tactic:
simp [hm, parity_simps]

Result:
no goals

Proof state:
case neg
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
ih :
  cs.wordProd (alternatingWord i i' m) = (if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)
hm : ¬Even m
h₁ : Even (m + 1)
h₂ : (m + 1) / 2 = m / 2 + 1
⊢ cs.simple (if Even m then i' else i) *
      ((if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)) =
    (if Even (m + 1) then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ ((m + 1) / 2)

Tactic:
simp [hm, h₁, h₂, ← pow_succ', ← mul_assoc]

Result:
no goals

Proof state:
case succ
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
ih :
  cs.wordProd (alternatingWord i i' m) = (if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)
⊢ cs.wordProd (alternatingWord i i' (m + 1)) =
    (if Even (m + 1) then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ ((m + 1) / 2)

Tactic:
rw [alternatingWord_succ', wordProd_cons, ih]

Result:
case succ
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
ih :
  cs.wordProd (alternatingWord i i' m) = (if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)
⊢ cs.simple (if Even m then i' else i) *
      ((if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)) =
    (if Even (m + 1) then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ ((m + 1) / 2)

Proof state:
case zero
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
⊢ cs.wordProd (alternatingWord i i' 0) = (if Even 0 then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (0 / 2)

Tactic:
simp [alternatingWord]

Result:
no goals

Proof state:
case neg
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
ih :
  cs.wordProd (alternatingWord i i' m) = (if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)
hm : ¬Even m
⊢ cs.simple (if Even m then i' else i) *
      ((if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)) =
    (if Even (m + 1) then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ ((m + 1) / 2)

Tactic:
have h₁ : Even (m + 1) := by simp [hm, parity_simps]

Result:
case neg
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
ih :
  cs.wordProd (alternatingWord i i' m) = (if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)
hm : ¬Even m
h₁ : Even (m + 1)
⊢ cs.simple (if Even m then i' else i) *
      ((if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)) =
    (if Even (m + 1) then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ ((m + 1) / 2)

Proof state:
case succ
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
ih :
  cs.wordProd (alternatingWord i i' m) = (if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)
⊢ cs.simple (if Even m then i' else i) *
      ((if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)) =
    (if Even (m + 1) then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ ((m + 1) / 2)

Tactic:
by_cases hm : Even m

Result:
case pos
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
ih :
  cs.wordProd (alternatingWord i i' m) = (if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)
hm : Even m
⊢ cs.simple (if Even m then i' else i) *
      ((if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)) =
    (if Even (m + 1) then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ ((m + 1) / 2)

case neg
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
ih :
  cs.wordProd (alternatingWord i i' m) = (if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)
hm : ¬Even m
⊢ cs.simple (if Even m then i' else i) *
      ((if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)) =
    (if Even (m + 1) then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ ((m + 1) / 2)

Proof state:
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
⊢ cs.wordProd (alternatingWord i i' m) = (if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)

Tactic:
induction' m with m ih

Result:
case zero
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
⊢ cs.wordProd (alternatingWord i i' 0) = (if Even 0 then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (0 / 2)

case succ
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
ih :
  cs.wordProd (alternatingWord i i' m) = (if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)
⊢ cs.wordProd (alternatingWord i i' (m + 1)) =
    (if Even (m + 1) then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ ((m + 1) / 2)

Proof state:
case pos
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
ih :
  cs.wordProd (alternatingWord i i' m) = (if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)
hm : Even m
⊢ cs.simple (if Even m then i' else i) *
      ((if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)) =
    (if Even (m + 1) then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ ((m + 1) / 2)

Tactic:
have h₁ : ¬ Even (m + 1) := by simp [hm, parity_simps]

Result:
case pos
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
ih :
  cs.wordProd (alternatingWord i i' m) = (if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)
hm : Even m
h₁ : ¬Even (m + 1)
⊢ cs.simple (if Even m then i' else i) *
      ((if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)) =
    (if Even (m + 1) then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ ((m + 1) / 2)