Theorem: AIG.mkGateCached_eval_eq_mkGate_eval
Proof state before:
case h_1
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
assign : α → Bool
aig : AIG α
input : aig.GateInput
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✝
⊢ ⟦{ aig := { decls := aig.decls, cache := aig.cache, inv := ⋯ }, ref := { gate := hit✝.idx, hgate := ⋯ } }, assign⟧ =
    ⟦aig.mkGate input, assign⟧

case h_2
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
assign : α → Bool
aig : AIG α
input : aig.GateInput
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
⊢ ⟦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 := ⋯ } },
      assign⟧ =
    ⟦aig.mkGate input, assign⟧
Tactic:
. next heq1 =>
  rw [denote_mkGate_cached heq1]
Proof state after:
case h_2
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
assign : α → Bool
aig : AIG α
input : aig.GateInput
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
⊢ ⟦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 := ⋯ } },
      assign⟧ =
    ⟦aig.mkGate input, assign⟧