-- Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/Model.lean
theorem insert_eq_insertₘ [BEq α] [Hashable α] (m : Raw₀ α β) (a : α) (b : β a) :
    m.insert a b = m.insertₘ a b

Proof state:
case isTrue
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : Hashable α
m : Raw₀ α β
a : α
b : β a
h✝ : AssocList.contains a m.1.buckets[(mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat] = true
⊢ ⟨{ size := m.1.size,
        buckets :=
          (m.1.buckets.set (mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat AssocList.nil ⋯).set
            (mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat
            (AssocList.replace a b m.1.buckets[(mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat]) ⋯ },
      ⋯⟩ =
    m.replaceₘ a b

Tactic:
simp only [replaceₘ, Subtype.mk.injEq, Raw.mk.injEq, true_and]

Result:
case isTrue
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : Hashable α
m : Raw₀ α β
a : α
b : β a
h✝ : AssocList.contains a m.1.buckets[(mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat] = true
⊢ (m.1.buckets.set (mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat AssocList.nil ⋯).set
      (mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat
      (AssocList.replace a b m.1.buckets[(mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat]) ⋯ =
    updateBucket m.val.buckets ⋯ a fun l => AssocList.replace a b l

Proof state:
case isTrue
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : Hashable α
m : Raw₀ α β
a : α
b : β a
h✝ : AssocList.contains a m.1.buckets[(mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat] = true
⊢ (m.1.buckets.set (mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat AssocList.nil ⋯).set
      (mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat
      (AssocList.replace a b m.1.buckets[(mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat]) ⋯ =
    updateBucket m.val.buckets ⋯ a fun l => AssocList.replace a b l

Tactic:
rw [Array.set_set, updateBucket]

Result:
case isTrue
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : Hashable α
m : Raw₀ α β
a : α
b : β a
h✝ : AssocList.contains a m.1.buckets[(mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat] = true
⊢ m.1.buckets.set (mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat
      (AssocList.replace a b m.1.buckets[(mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat]) ⋯ =
    match mkIdx m.val.buckets.size ⋯ (hash a) with
    | ⟨i, h⟩ => m.val.buckets.uset i (AssocList.replace a b m.val.buckets[i]) h

Proof state:
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : Hashable α
m : Raw₀ α β
a : α
b : β a
⊢ (if AssocList.contains a m.1.buckets[(mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat] = true then
      ⟨{ size := m.1.size,
          buckets :=
            (m.1.buckets.set (mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat AssocList.nil ⋯).set
              (mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat
              (AssocList.replace a b m.1.buckets[(mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat]) ⋯ },
        ⋯⟩
    else
      expandIfNecessary
        ⟨{ size := m.1.size + 1,
            buckets :=
              m.1.buckets.set (mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat
                (AssocList.cons a b m.1.buckets[(mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat]) ⋯ },
          ⋯⟩) =
    if AssocList.contains a m.val.buckets[(mkIdx m.val.buckets.size ⋯ (hash a)).val.toNat] = true then m.replaceₘ a b
    else (m.consₘ a b).expandIfNecessary

Tactic:
split

Result:
case isTrue
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : Hashable α
m : Raw₀ α β
a : α
b : β a
h✝ : AssocList.contains a m.1.buckets[(mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat] = true
⊢ ⟨{ size := m.1.size,
        buckets :=
          (m.1.buckets.set (mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat AssocList.nil ⋯).set
            (mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat
            (AssocList.replace a b m.1.buckets[(mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat]) ⋯ },
      ⋯⟩ =
    m.replaceₘ a b

case isFalse
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : Hashable α
m : Raw₀ α β
a : α
b : β a
h✝ : ¬AssocList.contains a m.1.buckets[(mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat] = true
⊢ expandIfNecessary
      ⟨{ size := m.1.size + 1,
          buckets :=
            m.1.buckets.set (mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat
              (AssocList.cons a b m.1.buckets[(mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat]) ⋯ },
        ⋯⟩ =
    (m.consₘ a b).expandIfNecessary

Proof state:
case isFalse
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : Hashable α
m : Raw₀ α β
a : α
b : β a
h✝ : ¬AssocList.contains a m.1.buckets[(mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat] = true
⊢ expandIfNecessary
      ⟨{ size := m.1.size + 1,
          buckets :=
            m.1.buckets.set (mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat
              (AssocList.cons a b m.1.buckets[(mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat]) ⋯ },
        ⋯⟩ =
    (m.consₘ a b).expandIfNecessary

Tactic:
rfl

Result:
no goals

Proof state:
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : Hashable α
m : Raw₀ α β
a : α
b : β a
⊢ (match mkIdx m.1.buckets.size ⋯ (hash a) with
    | ⟨i, h⟩ =>
      let bkt := m.1.buckets[i];
      if AssocList.contains a bkt = true then
        let buckets' := m.1.buckets.uset i AssocList.nil h;
        ⟨{ size := m.1.size, buckets := buckets'.uset i (AssocList.replace a b bkt) ⋯ }, ⋯⟩
      else
        let size' := m.1.size + 1;
        let buckets' := m.1.buckets.uset i (AssocList.cons a b bkt) h;
        expandIfNecessary ⟨{ size := size', buckets := buckets' }, ⋯⟩) =
    if
        AssocList.contains a
            (match mkIdx m.val.buckets.size ⋯ (hash a) with
            | ⟨i, h⟩ => m.val.buckets[i]) =
          true then
      m.replaceₘ a b
    else (m.consₘ a b).expandIfNecessary

Tactic:
dsimp only [Array.ugetElem_eq_getElem, Array.uset]

Result:
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : Hashable α
m : Raw₀ α β
a : α
b : β a
⊢ (if AssocList.contains a m.1.buckets[(mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat] = true then
      ⟨{ size := m.1.size,
          buckets :=
            (m.1.buckets.set (mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat AssocList.nil ⋯).set
              (mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat
              (AssocList.replace a b m.1.buckets[(mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat]) ⋯ },
        ⋯⟩
    else
      expandIfNecessary
        ⟨{ size := m.1.size + 1,
            buckets :=
              m.1.buckets.set (mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat
                (AssocList.cons a b m.1.buckets[(mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat]) ⋯ },
          ⋯⟩) =
    if AssocList.contains a m.val.buckets[(mkIdx m.val.buckets.size ⋯ (hash a)).val.toNat] = true then m.replaceₘ a b
    else (m.consₘ a b).expandIfNecessary

Proof state:
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : Hashable α
m : Raw₀ α β
a : α
b : β a
⊢ m.insert a b = m.insertₘ a b

Tactic:
rw [insert, insertₘ, containsₘ, bucket]

Result:
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : Hashable α
m : Raw₀ α β
a : α
b : β a
⊢ (match mkIdx m.1.buckets.size ⋯ (hash a) with
    | ⟨i, h⟩ =>
      let bkt := m.1.buckets[i];
      if AssocList.contains a bkt = true then
        let buckets' := m.1.buckets.uset i AssocList.nil h;
        ⟨{ size := m.1.size, buckets := buckets'.uset i (AssocList.replace a b bkt) ⋯ }, ⋯⟩
      else
        let size' := m.1.size + 1;
        let buckets' := m.1.buckets.uset i (AssocList.cons a b bkt) h;
        expandIfNecessary ⟨{ size := size', buckets := buckets' }, ⋯⟩) =
    if
        AssocList.contains a
            (match mkIdx m.val.buckets.size ⋯ (hash a) with
            | ⟨i, h⟩ => m.val.buckets[i]) =
          true then
      m.replaceₘ a b
    else (m.consₘ a b).expandIfNecessary