-- Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/CachedLemmas.lean
theorem mkGateCached.go_le_size (aig : AIG α) (input : GateInput aig) :
    aig.decls.size ≤ (go aig input).aig.decls.size

Proof state:
case h_2
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
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.get? (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv) = none
⊢ aig.decls.size ≤
    (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, invariant := ⋯ }.mkConstCached false
          | Decl.const false, x, false, x_1 =>
            { decls := aig.decls, cache := aig.cache, invariant := ⋯ }.mkConstCached false
          | x, Decl.const true, x_1, true =>
            { decls := aig.decls, cache := aig.cache, invariant := ⋯ }.mkConstCached false
          | x, Decl.const false, x_1, false =>
            { decls := aig.decls, cache := aig.cache, invariant := ⋯ }.mkConstCached false
          | Decl.const true, x, false, false =>
            { aig := { decls := aig.decls, cache := aig.cache, invariant := ⋯ },
              ref := { gate := input.rhs.ref.gate, hgate := ⋯ } }
          | Decl.const false, x, true, false =>
            { aig := { decls := aig.decls, cache := aig.cache, invariant := ⋯ },
              ref := { gate := input.rhs.ref.gate, hgate := ⋯ } }
          | x, Decl.const true, false, false =>
            { aig := { decls := aig.decls, cache := aig.cache, invariant := ⋯ },
              ref := { gate := input.lhs.ref.gate, hgate := ⋯ } }
          | x, Decl.const false, false, true =>
            { aig := { decls := aig.decls, cache := aig.cache, invariant := ⋯ },
              ref := { gate := input.lhs.ref.gate, hgate := ⋯ } }
          | x, x_1, x_2, x_3 =>
            if
                (input.lhs.ref.gate == input.rhs.ref.gate && input.lhs.inv == false && input.rhs.inv == false) =
                  true then
              { aig := { decls := aig.decls, cache := aig.cache, invariant := ⋯ },
                ref := { gate := input.lhs.ref.gate, hgate := ⋯ } }
            else
              if (input.lhs.ref.gate == input.rhs.ref.gate && input.lhs.inv == !input.rhs.inv) = true then
                { decls := aig.decls, cache := aig.cache, invariant := ⋯ }.mkConstCached false
              else
                {
                  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),
                      invariant := ⋯ },
                  ref := { gate := aig.decls.size, hgate := ⋯ } }).aig.decls.size

Tactic:
split

Result:
case h_2.h_1
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.GateInput
x✝² : Option (CacheHit aig.decls (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv))
heq✝² : aig.cache.get? (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv) = none
x✝¹ x✝ : Decl α
linv✝ rinv✝ : Bool
heq✝¹ : aig.decls[input.lhs.ref.gate] = Decl.const true
heq✝ : input.lhs.inv = true
⊢ aig.decls.size ≤ ({ decls := aig.decls, cache := aig.cache, invariant := ⋯ }.mkConstCached false).aig.decls.size

case h_2.h_2
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.GateInput
x✝² : Option (CacheHit aig.decls (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv))
heq✝² : aig.cache.get? (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv) = none
x✝¹ x✝ : Decl α
linv✝ rinv✝ : Bool
heq✝¹ : aig.decls[input.lhs.ref.gate] = Decl.const false
heq✝ : input.lhs.inv = false
⊢ aig.decls.size ≤ ({ decls := aig.decls, cache := aig.cache, invariant := ⋯ }.mkConstCached false).aig.decls.size

case h_2.h_3
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.GateInput
x✝⁴ : Option (CacheHit aig.decls (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv))
heq✝² : aig.cache.get? (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv) = none
x✝³ x✝² : Decl α
linv✝ rinv✝ : Bool
heq✝¹ : aig.decls[input.rhs.ref.gate] = Decl.const true
heq✝ : input.rhs.inv = true
x✝¹ : aig.decls[input.lhs.ref.gate] = Decl.const true → input.lhs.inv = true → False
x✝ : aig.decls[input.lhs.ref.gate] = Decl.const false → input.lhs.inv = false → False
⊢ aig.decls.size ≤ ({ decls := aig.decls, cache := aig.cache, invariant := ⋯ }.mkConstCached false).aig.decls.size

case h_2.h_4
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.GateInput
x✝⁴ : Option (CacheHit aig.decls (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv))
heq✝² : aig.cache.get? (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv) = none
x✝³ x✝² : Decl α
linv✝ rinv✝ : Bool
heq✝¹ : aig.decls[input.rhs.ref.gate] = Decl.const false
heq✝ : input.rhs.inv = false
x✝¹ : aig.decls[input.lhs.ref.gate] = Decl.const true → input.lhs.inv = true → False
x✝ : aig.decls[input.lhs.ref.gate] = Decl.const false → input.lhs.inv = false → False
⊢ aig.decls.size ≤ ({ decls := aig.decls, cache := aig.cache, invariant := ⋯ }.mkConstCached false).aig.decls.size

case h_2.h_5
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.GateInput
x✝³ : Option (CacheHit aig.decls (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv))
heq✝³ : aig.cache.get? (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv) = none
x✝² x✝¹ : Decl α
linv✝ rinv✝ : Bool
heq✝² : aig.decls[input.lhs.ref.gate] = Decl.const true
heq✝¹ : input.lhs.inv = false
heq✝ : input.rhs.inv = false
x✝ : aig.decls[input.rhs.ref.gate] = Decl.const false → False
⊢ aig.decls.size ≤
    { aig := { decls := aig.decls, cache := aig.cache, invariant := ⋯ },
            ref := { gate := input.rhs.ref.gate, hgate := ⋯ } }.aig.decls.size

case h_2.h_6
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.GateInput
x✝³ : Option (CacheHit aig.decls (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv))
heq✝³ : aig.cache.get? (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv) = none
x✝² x✝¹ : Decl α
linv✝ rinv✝ : Bool
heq✝² : aig.decls[input.lhs.ref.gate] = Decl.const false
heq✝¹ : input.lhs.inv = true
heq✝ : input.rhs.inv = false
x✝ : aig.decls[input.rhs.ref.gate] = Decl.const false → False
⊢ aig.decls.size ≤
    { aig := { decls := aig.decls, cache := aig.cache, invariant := ⋯ },
            ref := { gate := input.rhs.ref.gate, hgate := ⋯ } }.aig.decls.size

case h_2.h_7
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.GateInput
x✝⁴ : Option (CacheHit aig.decls (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv))
heq✝³ : aig.cache.get? (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv) = none
x✝³ x✝² : Decl α
linv✝ rinv✝ : Bool
heq✝² : aig.decls[input.rhs.ref.gate] = Decl.const true
heq✝¹ : input.lhs.inv = false
heq✝ : input.rhs.inv = false
x✝¹ : aig.decls[input.lhs.ref.gate] = Decl.const false → False
x✝ : aig.decls[input.lhs.ref.gate] = Decl.const true → False
⊢ aig.decls.size ≤
    { aig := { decls := aig.decls, cache := aig.cache, invariant := ⋯ },
            ref := { gate := input.lhs.ref.gate, hgate := ⋯ } }.aig.decls.size

case h_2.h_8
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.GateInput
x✝³ : Option (CacheHit aig.decls (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv))
heq✝³ : aig.cache.get? (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv) = none
x✝² x✝¹ : Decl α
linv✝ rinv✝ : Bool
heq✝² : aig.decls[input.rhs.ref.gate] = Decl.const false
heq✝¹ : input.lhs.inv = false
heq✝ : input.rhs.inv = true
x✝ : aig.decls[input.lhs.ref.gate] = Decl.const false → False
⊢ aig.decls.size ≤
    { aig := { decls := aig.decls, cache := aig.cache, invariant := ⋯ },
            ref := { gate := input.lhs.ref.gate, hgate := ⋯ } }.aig.decls.size

case h_2.h_9
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.GateInput
x✝¹⁰ : Option (CacheHit aig.decls (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv))
heq✝ : aig.cache.get? (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv) = none
x✝⁹ x✝⁸ : Decl α
linv✝ rinv✝ : Bool
x✝⁷ : aig.decls[input.lhs.ref.gate] = Decl.const true → input.lhs.inv = true → False
x✝⁶ : aig.decls[input.lhs.ref.gate] = Decl.const false → input.lhs.inv = false → False
x✝⁵ : aig.decls[input.rhs.ref.gate] = Decl.const true → input.rhs.inv = true → False
x✝⁴ : aig.decls[input.rhs.ref.gate] = Decl.const false → input.rhs.inv = false → False
x✝³ : aig.decls[input.lhs.ref.gate] = Decl.const true → input.lhs.inv = false → input.rhs.inv = false → False
x✝² : aig.decls[input.lhs.ref.gate] = Decl.const false → input.lhs.inv = true → input.rhs.inv = false → False
x✝¹ : aig.decls[input.rhs.ref.gate] = Decl.const true → input.lhs.inv = false → input.rhs.inv = false → False
x✝ : aig.decls[input.rhs.ref.gate] = Decl.const false → input.lhs.inv = false → input.rhs.inv = true → False
⊢ aig.decls.size ≤
    (if (input.lhs.ref.gate == input.rhs.ref.gate && input.lhs.inv == false && input.rhs.inv == false) = true then
            { aig := { decls := aig.decls, cache := aig.cache, invariant := ⋯ },
              ref := { gate := input.lhs.ref.gate, hgate := ⋯ } }
          else
            if (input.lhs.ref.gate == input.rhs.ref.gate && input.lhs.inv == !input.rhs.inv) = true then
              { decls := aig.decls, cache := aig.cache, invariant := ⋯ }.mkConstCached false
            else
              {
                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),
                    invariant := ⋯ },
                ref := { gate := aig.decls.size, hgate := ⋯ } }).aig.decls.size