-- Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/RefVecOperator/Map.lean
theorem denote_go {aig : AIG α} (curr : Nat) (hcurr : curr ≤ len) (s : RefVec aig curr)
    (input : RefVec aig len) (f : (aig : AIG α) → Ref aig → Entrypoint α)
    [LawfulOperator α Ref f] [LawfulMapOperator α f] :
    ∀ (idx : Nat) (hidx1 : idx < len),
      curr ≤ idx
        →
      ⟦(go aig curr hcurr s input f).aig, (go aig curr hcurr s input f).vec.get idx hidx1, assign⟧
        =
      ⟦f aig (input.get idx hidx1), assign⟧

Proof state:
case isTrue.inl
α : Type
inst✝³ : Hashable α
inst✝² : DecidableEq α
len : Nat
assign : α → Bool
aig : AIG α
curr : Nat
hcurr : curr ≤ len
s : aig.RefVec curr
input : aig.RefVec len
f : (aig : AIG α) → aig.Ref → Entrypoint α
inst✝¹ : LawfulOperator α Ref f
inst✝ : LawfulMapOperator α f
idx : Nat
hidx1 : idx < len
hidx2 : curr ≤ idx
res : RefVecEntry α len
h✝ : curr < len
hgo :
  go (f aig (input.get curr h✝)).aig (curr + 1) ⋯ ((s.cast ⋯).push (f aig (input.get curr h✝)).ref) (input.cast ⋯) f =
    res
heq : curr = idx
⊢ ⟦assign,
      { aig := (f aig (input.get curr h✝)).aig,
        ref := { gate := ((f aig (input.get curr h✝)).ref.cast ⋯).gate, hgate := ?isTrue.inl.hstart } }⟧ =
    ⟦assign, f aig (input.get curr ⋯)⟧

Tactic:
simp

Result:
no goals

Proof state:
case isTrue.inr
α : Type
inst✝³ : Hashable α
inst✝² : DecidableEq α
len : Nat
assign : α → Bool
aig : AIG α
curr : Nat
hcurr : curr ≤ len
s : aig.RefVec curr
input : aig.RefVec len
f : (aig : AIG α) → aig.Ref → Entrypoint α
inst✝¹ : LawfulOperator α Ref f
inst✝ : LawfulMapOperator α f
idx : Nat
hidx1 : idx < len
hidx2 : curr ≤ idx
res : RefVecEntry α len
h✝ : curr < len
hgo :
  go (f aig (input.get curr h✝)).aig (curr + 1) ⋯ ((s.cast ⋯).push (f aig (input.get curr h✝)).ref) (input.cast ⋯) f =
    res
hlt : curr < idx
⊢ ⟦assign, { aig := res.aig, ref := res.vec.get idx hidx1 }⟧ = ⟦assign, f aig (input.get idx hidx1)⟧

Tactic:
rw [← hgo]

Result:
case isTrue.inr
α : Type
inst✝³ : Hashable α
inst✝² : DecidableEq α
len : Nat
assign : α → Bool
aig : AIG α
curr : Nat
hcurr : curr ≤ len
s : aig.RefVec curr
input : aig.RefVec len
f : (aig : AIG α) → aig.Ref → Entrypoint α
inst✝¹ : LawfulOperator α Ref f
inst✝ : LawfulMapOperator α f
idx : Nat
hidx1 : idx < len
hidx2 : curr ≤ idx
res : RefVecEntry α len
h✝ : curr < len
hgo :
  go (f aig (input.get curr h✝)).aig (curr + 1) ⋯ ((s.cast ⋯).push (f aig (input.get curr h✝)).ref) (input.cast ⋯) f =
    res
hlt : curr < idx
⊢ ⟦assign,
      {
        aig :=
          (go (f aig (input.get curr h✝)).aig (curr + 1) ⋯ ((s.cast ⋯).push (f aig (input.get curr h✝)).ref)
              (input.cast ⋯) f).aig,
        ref :=
          (go (f aig (input.get curr h✝)).aig (curr + 1) ⋯ ((s.cast ⋯).push (f aig (input.get curr h✝)).ref)
                  (input.cast ⋯) f).vec.get
            idx hidx1 }⟧ =
    ⟦assign, f aig (input.get idx hidx1)⟧

Proof state:
α : Type
inst✝³ : Hashable α
inst✝² : DecidableEq α
len : Nat
assign : α → Bool
aig : AIG α
curr : Nat
hcurr : curr ≤ len
s : aig.RefVec curr
input : aig.RefVec len
f : (aig : AIG α) → aig.Ref → Entrypoint α
inst✝¹ : LawfulOperator α Ref f
inst✝ : LawfulMapOperator α f
idx : Nat
hidx1 : idx < len
hidx2 : curr ≤ idx
⊢ ⟦assign, { aig := (go aig curr hcurr s input f).aig, ref := (go aig curr hcurr s input f).vec.get idx hidx1 }⟧ =
    ⟦assign, f aig (input.get idx hidx1)⟧

Tactic:
generalize hgo : go aig curr hcurr s input f = res

Result:
α : Type
inst✝³ : Hashable α
inst✝² : DecidableEq α
len : Nat
assign : α → Bool
aig : AIG α
curr : Nat
hcurr : curr ≤ len
s : aig.RefVec curr
input : aig.RefVec len
f : (aig : AIG α) → aig.Ref → Entrypoint α
inst✝¹ : LawfulOperator α Ref f
inst✝ : LawfulMapOperator α f
idx : Nat
hidx1 : idx < len
hidx2 : curr ≤ idx
res : RefVecEntry α len
hgo : go aig curr hcurr s input f = res
⊢ ⟦assign, { aig := res.aig, ref := res.vec.get idx hidx1 }⟧ = ⟦assign, f aig (input.get idx hidx1)⟧

Proof state:
α : Type
inst✝³ : Hashable α
inst✝² : DecidableEq α
len : Nat
assign : α → Bool
aig : AIG α
curr : Nat
hcurr : curr ≤ len
s : aig.RefVec curr
input : aig.RefVec len
f : (aig : AIG α) → aig.Ref → Entrypoint α
inst✝¹ : LawfulOperator α Ref f
inst✝ : LawfulMapOperator α f
idx : Nat
hidx1 : idx < len
hidx2 : curr ≤ idx
res : RefVecEntry α len
hgo :
  (if hidx : curr < len then
      let res := f aig (input.get curr hidx);
      let aig_1 := res.aig;
      let newRef := res.ref;
      let_fun this := ⋯;
      let input_1 := input.cast this;
      let s := s.cast this;
      let s := s.push newRef;
      go aig_1 (curr + 1) ⋯ s input_1 f
    else
      let_fun this := ⋯;
      { aig := aig, vec := this ▸ s }) =
    res
⊢ ⟦assign, { aig := res.aig, ref := res.vec.get idx hidx1 }⟧ = ⟦assign, f aig (input.get idx hidx1)⟧

Tactic:
split at hgo

Result:
case isTrue
α : Type
inst✝³ : Hashable α
inst✝² : DecidableEq α
len : Nat
assign : α → Bool
aig : AIG α
curr : Nat
hcurr : curr ≤ len
s : aig.RefVec curr
input : aig.RefVec len
f : (aig : AIG α) → aig.Ref → Entrypoint α
inst✝¹ : LawfulOperator α Ref f
inst✝ : LawfulMapOperator α f
idx : Nat
hidx1 : idx < len
hidx2 : curr ≤ idx
res : RefVecEntry α len
h✝ : curr < len
hgo :
  (let res := f aig (input.get curr h✝);
    let aig_1 := res.aig;
    let newRef := res.ref;
    let_fun this := ⋯;
    let input_1 := input.cast this;
    let s := s.cast this;
    let s := s.push newRef;
    go aig_1 (curr + 1) ⋯ s input_1 f) =
    res
⊢ ⟦assign, { aig := res.aig, ref := res.vec.get idx hidx1 }⟧ = ⟦assign, f aig (input.get idx hidx1)⟧

case isFalse
α : Type
inst✝³ : Hashable α
inst✝² : DecidableEq α
len : Nat
assign : α → Bool
aig : AIG α
curr : Nat
hcurr : curr ≤ len
s : aig.RefVec curr
input : aig.RefVec len
f : (aig : AIG α) → aig.Ref → Entrypoint α
inst✝¹ : LawfulOperator α Ref f
inst✝ : LawfulMapOperator α f
idx : Nat
hidx1 : idx < len
hidx2 : curr ≤ idx
res : RefVecEntry α len
h✝ : ¬curr < len
hgo :
  (let_fun this := ⋯;
    { aig := aig, vec := this ▸ s }) =
    res
⊢ ⟦assign, { aig := res.aig, ref := res.vec.get idx hidx1 }⟧ = ⟦assign, f aig (input.get idx hidx1)⟧