apply denote.eq_of_aig_eq (entry := ⟨aig, start,hstart⟩)
w : Nat
assign : BVBit → Bool
aig : AIG BVBit
idx : Nat
hidx : idx ≤ w
s : aig.RefStream idx
a start : Nat
hstart : start < aig.decls.size
⊢ ⟦{ aig := (go w aig idx s a hidx).aig, ref := { gate := start, hgate := ⋯ } }, assign⟧ =
    ⟦{ aig := aig, ref := { gate := start, hgate := hstart } }, assign⟧
case hprefix
w : Nat
assign : BVBit → Bool
aig : AIG BVBit
idx : Nat
hidx : idx ≤ w
s : aig.RefStream idx
a start : Nat
hstart : start < aig.decls.size
⊢ IsPrefix { aig := aig, ref := { gate := start, hgate := hstart } }.aig.decls (go w aig idx s a hidx).aig.decls