. rw [← hres]
  intros
  simp
case h_1
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.GateInput
res : Entrypoint α
x✝ : Option (CacheHit aig.decls (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv))
hit✝ : CacheHit aig.decls (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv)
heq✝ : aig.cache.find? (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv) = some hit✝
hres : { aig := { decls := aig.decls, cache := aig.cache, inv := ⋯ }, ref := { gate := hit✝.idx, hgate := ⋯ } } = res
⊢ ∀ (idx : Nat) (h1 : idx < res.aig.decls.size) (h2 : idx < aig.decls.size), res.aig.decls[idx] = aig.decls[idx]

case h_2
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.GateInput
res : Entrypoint α
x✝ : Option (CacheHit aig.decls (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv))
heq✝ : aig.cache.find? (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv) = none
hres :
  (match aig.decls[input.lhs.ref.gate], aig.decls[input.rhs.ref.gate], input.lhs.inv, input.rhs.inv with
    | Decl.const true, x, true, x_1 => { decls := aig.decls, cache := aig.cache, inv := ⋯ }.mkConstCached false
    | Decl.const false, x, false, x_1 => { decls := aig.decls, cache := aig.cache, inv := ⋯ }.mkConstCached false
    | x, Decl.const true, x_1, true => { decls := aig.decls, cache := aig.cache, inv := ⋯ }.mkConstCached false
    | x, Decl.const false, x_1, false => { decls := aig.decls, cache := aig.cache, inv := ⋯ }.mkConstCached false
    | Decl.const true, x, false, false =>
      { aig := { decls := aig.decls, cache := aig.cache, inv := ⋯ }, ref := { gate := input.rhs.ref.gate, hgate := ⋯ } }
    | Decl.const false, x, true, false =>
      { aig := { decls := aig.decls, cache := aig.cache, inv := ⋯ }, ref := { gate := input.rhs.ref.gate, hgate := ⋯ } }
    | x, Decl.const true, false, false =>
      { aig := { decls := aig.decls, cache := aig.cache, inv := ⋯ }, ref := { gate := input.lhs.ref.gate, hgate := ⋯ } }
    | x, Decl.const false, false, true =>
      { aig := { decls := aig.decls, cache := aig.cache, inv := ⋯ }, ref := { gate := input.lhs.ref.gate, hgate := ⋯ } }
    | x, x_1, x_2, x_3 =>
      {
        aig :=
          { decls := aig.decls.push (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv),
            cache :=
              Cache.insert aig.decls aig.cache
                (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv),
            inv := ⋯ },
        ref := { gate := aig.decls.size, hgate := ⋯ } }) =
    res
⊢ ∀ (idx : Nat) (h1 : idx < res.aig.decls.size) (h2 : idx < aig.decls.size), res.aig.decls[idx] = aig.decls[idx]
case h_2
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.GateInput
res : Entrypoint α
x✝ : Option (CacheHit aig.decls (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv))
heq✝ : aig.cache.find? (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv) = none
hres :
  (match aig.decls[input.lhs.ref.gate], aig.decls[input.rhs.ref.gate], input.lhs.inv, input.rhs.inv with
    | Decl.const true, x, true, x_1 => { decls := aig.decls, cache := aig.cache, inv := ⋯ }.mkConstCached false
    | Decl.const false, x, false, x_1 => { decls := aig.decls, cache := aig.cache, inv := ⋯ }.mkConstCached false
    | x, Decl.const true, x_1, true => { decls := aig.decls, cache := aig.cache, inv := ⋯ }.mkConstCached false
    | x, Decl.const false, x_1, false => { decls := aig.decls, cache := aig.cache, inv := ⋯ }.mkConstCached false
    | Decl.const true, x, false, false =>
      { aig := { decls := aig.decls, cache := aig.cache, inv := ⋯ }, ref := { gate := input.rhs.ref.gate, hgate := ⋯ } }
    | Decl.const false, x, true, false =>
      { aig := { decls := aig.decls, cache := aig.cache, inv := ⋯ }, ref := { gate := input.rhs.ref.gate, hgate := ⋯ } }
    | x, Decl.const true, false, false =>
      { aig := { decls := aig.decls, cache := aig.cache, inv := ⋯ }, ref := { gate := input.lhs.ref.gate, hgate := ⋯ } }
    | x, Decl.const false, false, true =>
      { aig := { decls := aig.decls, cache := aig.cache, inv := ⋯ }, ref := { gate := input.lhs.ref.gate, hgate := ⋯ } }
    | x, x_1, x_2, x_3 =>
      {
        aig :=
          { decls := aig.decls.push (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv),
            cache :=
              Cache.insert aig.decls aig.cache
                (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv),
            inv := ⋯ },
        ref := { gate := aig.decls.size, hgate := ⋯ } }) =
    res
⊢ ∀ (idx : Nat) (h1 : idx < res.aig.decls.size) (h2 : idx < aig.decls.size), res.aig.decls[idx] = aig.decls[idx]