Theorem: CoxeterSystem.getD_leftInvSeq
File: Mathlib/GroupTheory/Coxeter/Inversion.lean

tactic: induction' ω with i ω ih generalizing j
state_before: 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.get? j)).getD 1 * (cs.wordProd (take j ω))⁻¹
state_after: case nil
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
j : ℕ
⊢ (cs.leftInvSeq []).getD j 1 =
    cs.wordProd (take j []) * (Option.map cs.simple ([].get? j)).getD 1 * (cs.wordProd (take j []))⁻¹

case cons
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.get? j)).getD 1 * (cs.wordProd (take j ω))⁻¹
j : ℕ
⊢ (cs.leftInvSeq (i :: ω)).getD j 1 =
    cs.wordProd (take j (i :: ω)) * (Option.map cs.simple ((i :: ω).get? j)).getD 1 * (cs.wordProd (take j (i :: ω)))⁻¹

tactic: simp
state_before: case nil
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
j : ℕ
⊢ (cs.leftInvSeq []).getD j 1 =
    cs.wordProd (take j []) * (Option.map cs.simple ([].get? j)).getD 1 * (cs.wordProd (take j []))⁻¹
state_after: no goals

tactic: dsimp [leftInvSeq]
state_before: case cons
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.get? j)).getD 1 * (cs.wordProd (take j ω))⁻¹
j : ℕ
⊢ (cs.leftInvSeq (i :: ω)).getD j 1 =
    cs.wordProd (take j (i :: ω)) * (Option.map cs.simple ((i :: ω).get? j)).getD 1 * (cs.wordProd (take j (i :: ω)))⁻¹
state_after: case cons
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.get? j)).getD 1 * (cs.wordProd (take j ω))⁻¹
j : ℕ
⊢ (cs.simple i :: List.map (⇑(MulAut.conj (cs.simple i))) (cs.leftInvSeq ω)).getD j 1 =
    cs.wordProd (take j (i :: ω)) * (Option.map cs.simple ((i :: ω).get? j)).getD 1 * (cs.wordProd (take j (i :: ω)))⁻¹

tactic: rcases j with _ | j'
state_before: case cons
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.get? j)).getD 1 * (cs.wordProd (take j ω))⁻¹
j : ℕ
⊢ (cs.simple i :: List.map (⇑(MulAut.conj (cs.simple i))) (cs.leftInvSeq ω)).getD j 1 =
    cs.wordProd (take j (i :: ω)) * (Option.map cs.simple ((i :: ω).get? j)).getD 1 * (cs.wordProd (take j (i :: ω)))⁻¹
state_after: case cons.zero
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.get? j)).getD 1 * (cs.wordProd (take j ω))⁻¹
⊢ (cs.simple i :: List.map (⇑(MulAut.conj (cs.simple i))) (cs.leftInvSeq ω)).getD 0 1 =
    cs.wordProd (take 0 (i :: ω)) * (Option.map cs.simple ((i :: ω).get? 0)).getD 1 * (cs.wordProd (take 0 (i :: ω)))⁻¹

case cons.succ
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.get? j)).getD 1 * (cs.wordProd (take j ω))⁻¹
j' : ℕ
⊢ (cs.simple i :: List.map (⇑(MulAut.conj (cs.simple i))) (cs.leftInvSeq ω)).getD (j' + 1) 1 =
    cs.wordProd (take (j' + 1) (i :: ω)) * (Option.map cs.simple ((i :: ω).get? (j' + 1))).getD 1 *
      (cs.wordProd (take (j' + 1) (i :: ω)))⁻¹

tactic: simp [getD_cons_zero]
state_before: case cons.zero
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.get? j)).getD 1 * (cs.wordProd (take j ω))⁻¹
⊢ (cs.simple i :: List.map (⇑(MulAut.conj (cs.simple i))) (cs.leftInvSeq ω)).getD 0 1 =
    cs.wordProd (take 0 (i :: ω)) * (Option.map cs.simple ((i :: ω).get? 0)).getD 1 * (cs.wordProd (take 0 (i :: ω)))⁻¹
state_after: no goals

tactic: rw [getD_cons_succ]
state_before: case cons.succ
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.get? j)).getD 1 * (cs.wordProd (take j ω))⁻¹
j' : ℕ
⊢ (cs.simple i :: List.map (⇑(MulAut.conj (cs.simple i))) (cs.leftInvSeq ω)).getD (j' + 1) 1 =
    cs.wordProd (take (j' + 1) (i :: ω)) * (Option.map cs.simple ((i :: ω).get? (j' + 1))).getD 1 *
      (cs.wordProd (take (j' + 1) (i :: ω)))⁻¹
state_after: case cons.succ
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.get? j)).getD 1 * (cs.wordProd (take j ω))⁻¹
j' : ℕ
⊢ (List.map (⇑(MulAut.conj (cs.simple i))) (cs.leftInvSeq ω)).getD j' 1 =
    cs.wordProd (take (j' + 1) (i :: ω)) * (Option.map cs.simple ((i :: ω).get? (j' + 1))).getD 1 *
      (cs.wordProd (take (j' + 1) (i :: ω)))⁻¹

tactic: rw [(by simp : 1 = ⇑(MulAut.conj (s i)) 1)]
state_before: case cons.succ
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.get? j)).getD 1 * (cs.wordProd (take j ω))⁻¹
j' : ℕ
⊢ (List.map (⇑(MulAut.conj (cs.simple i))) (cs.leftInvSeq ω)).getD j' 1 =
    cs.wordProd (take (j' + 1) (i :: ω)) * (Option.map cs.simple ((i :: ω).get? (j' + 1))).getD 1 *
      (cs.wordProd (take (j' + 1) (i :: ω)))⁻¹
state_after: case cons.succ
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.get? j)).getD 1 * (cs.wordProd (take j ω))⁻¹
j' : ℕ
⊢ (List.map (⇑(MulAut.conj (cs.simple i))) (cs.leftInvSeq ω)).getD j' ((MulAut.conj (cs.simple i)) 1) =
    cs.wordProd (take (j' + 1) (i :: ω)) *
        (Option.map cs.simple ((i :: ω).get? (j' + 1))).getD ((MulAut.conj (cs.simple i)) 1) *
      (cs.wordProd (take (j' + 1) (i :: ω)))⁻¹

tactic: rw [getD_map]
state_before: case cons.succ
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.get? j)).getD 1 * (cs.wordProd (take j ω))⁻¹
j' : ℕ
⊢ (List.map (⇑(MulAut.conj (cs.simple i))) (cs.leftInvSeq ω)).getD j' ((MulAut.conj (cs.simple i)) 1) =
    cs.wordProd (take (j' + 1) (i :: ω)) *
        (Option.map cs.simple ((i :: ω).get? (j' + 1))).getD ((MulAut.conj (cs.simple i)) 1) *
      (cs.wordProd (take (j' + 1) (i :: ω)))⁻¹
state_after: case cons.succ
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.get? j)).getD 1 * (cs.wordProd (take j ω))⁻¹
j' : ℕ
⊢ (MulAut.conj (cs.simple i)) ((cs.leftInvSeq ω).getD j' 1) =
    cs.wordProd (take (j' + 1) (i :: ω)) *
        (Option.map cs.simple ((i :: ω).get? (j' + 1))).getD ((MulAut.conj (cs.simple i)) 1) *
      (cs.wordProd (take (j' + 1) (i :: ω)))⁻¹

tactic: rw [ih j']
state_before: case cons.succ
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.get? j)).getD 1 * (cs.wordProd (take j ω))⁻¹
j' : ℕ
⊢ (MulAut.conj (cs.simple i)) ((cs.leftInvSeq ω).getD j' 1) =
    cs.wordProd (take (j' + 1) (i :: ω)) *
        (Option.map cs.simple ((i :: ω).get? (j' + 1))).getD ((MulAut.conj (cs.simple i)) 1) *
      (cs.wordProd (take (j' + 1) (i :: ω)))⁻¹
state_after: case cons.succ
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.get? j)).getD 1 * (cs.wordProd (take j ω))⁻¹
j' : ℕ
⊢ (MulAut.conj (cs.simple i))
      (cs.wordProd (take j' ω) * (Option.map cs.simple (ω.get? j')).getD 1 * (cs.wordProd (take j' ω))⁻¹) =
    cs.wordProd (take (j' + 1) (i :: ω)) *
        (Option.map cs.simple ((i :: ω).get? (j' + 1))).getD ((MulAut.conj (cs.simple i)) 1) *
      (cs.wordProd (take (j' + 1) (i :: ω)))⁻¹

tactic: simp [← mul_assoc, wordProd_cons]
state_before: case cons.succ
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.get? j)).getD 1 * (cs.wordProd (take j ω))⁻¹
j' : ℕ
⊢ (MulAut.conj (cs.simple i))
      (cs.wordProd (take j' ω) * (Option.map cs.simple (ω.get? j')).getD 1 * (cs.wordProd (take j' ω))⁻¹) =
    cs.wordProd (take (j' + 1) (i :: ω)) *
        (Option.map cs.simple ((i :: ω).get? (j' + 1))).getD ((MulAut.conj (cs.simple i)) 1) *
      (cs.wordProd (take (j' + 1) (i :: ω)))⁻¹
state_after: no goals

tactic: simp
state_before: 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.get? j)).getD 1 * (cs.wordProd (take j ω))⁻¹
j' : ℕ
⊢ 1 = (MulAut.conj (cs.simple i)) 1
state_after: no goals