-- Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/Add.lean
theorem go_denote_eq (aig : AIG α) (curr : Nat) (hcurr : curr ≤ w) (cin : Ref aig)
    (s : AIG.RefVec aig curr) (lhs rhs : AIG.RefVec aig w) (assign : α → Bool)
    (lhsExpr rhsExpr : BitVec w)
    (hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, lhs.get idx hidx, assign⟧ = lhsExpr.getLsbD idx)
    (hright : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, rhs.get idx hidx, assign⟧ = rhsExpr.getLsbD idx)
    (hcin : ⟦aig, cin, assign⟧ = BitVec.carry curr lhsExpr rhsExpr false) :
    ∀ (idx : Nat) (hidx1 : idx < w),
        curr ≤ idx
          →
        ⟦
          (go aig lhs rhs curr hcurr cin s).aig,
          (go aig lhs rhs curr hcurr cin s).vec.get idx hidx1,
          assign
        ⟧
          =
        ⟦aig, lhs.get idx hidx1, assign⟧.xor
          (⟦aig, rhs.get idx hidx1, assign⟧.xor
            (BitVec.carry idx lhsExpr rhsExpr false))

Proof state:
case inl
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
curr : Nat
hcurr : curr ≤ w
cin : aig.Ref
s : aig.RefVec curr
lhs rhs : aig.RefVec w
assign : α → Bool
lhsExpr rhsExpr : BitVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := lhs.get idx hidx }⟧ = lhsExpr.getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := rhs.get idx hidx }⟧ = rhsExpr.getLsbD idx
hcin : ⟦assign, { aig := aig, ref := cin }⟧ = BitVec.carry curr lhsExpr rhsExpr false
idx : Nat
hidx1 : idx < w
hidx2 : curr ≤ idx
res : RefVecEntry α w
hlt : curr < w
hgo :
  go (mkFullAdder aig { lhs := lhs.get curr hlt, rhs := rhs.get curr hlt, cin := cin }).aig (lhs.cast ⋯) (rhs.cast ⋯)
      (curr + 1) ⋯ (mkFullAdder aig { lhs := lhs.get curr hlt, rhs := rhs.get curr hlt, cin := cin }).cout
      ((s.cast ⋯).push (mkFullAdder aig { lhs := lhs.get curr hlt, rhs := rhs.get curr hlt, cin := cin }).out) =
    res
heq : curr = idx
⊢ ⟦assign,
      {
        aig :=
          (let res := mkFullAdderOut aig { lhs := lhs.get curr hlt, rhs := rhs.get curr hlt, cin := cin };
            let aig_1 := res.aig;
            let outRef := res.ref;
            let_fun haig1 := ⋯;
            let input := { lhs := lhs.get curr hlt, rhs := rhs.get curr hlt, cin := cin }.cast haig1;
            let res_1 := mkFullAdderCarry aig_1 input;
            let aig_2 := res_1.aig;
            let carryRef := res_1.ref;
            let_fun haig2 := ⋯;
            let outRef := outRef.cast haig2;
            let_fun hle := ⋯;
            { aig := aig_2, out := outRef, cout := carryRef, hle := hle }).aig,
        ref :=
          {
            gate :=
              ((let res := mkFullAdderOut aig { lhs := lhs.get curr hlt, rhs := rhs.get curr hlt, cin := cin };
                      let aig_1 := res.aig;
                      let outRef := res.ref;
                      let_fun haig1 := ⋯;
                      let input := { lhs := lhs.get curr hlt, rhs := rhs.get curr hlt, cin := cin }.cast haig1;
                      let res_1 := mkFullAdderCarry aig_1 input;
                      let aig_2 := res_1.aig;
                      let carryRef := res_1.ref;
                      let_fun haig2 := ⋯;
                      let outRef := outRef.cast haig2;
                      let_fun hle := ⋯;
                      { aig := aig_2, out := outRef, cout := carryRef, hle := hle }).out.cast
                  ⋯).gate,
            hgate := ?inl.hstart } }⟧ =
    (⟦assign, { aig := aig, ref := lhs.get curr ⋯ }⟧ ^^
      (⟦assign, { aig := aig, ref := rhs.get curr ⋯ }⟧ ^^ BitVec.carry curr lhsExpr rhsExpr false))

Tactic:
simp [hcin]

Result:
no goals

Proof state:
case inl
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
curr : Nat
hcurr : curr ≤ w
cin : aig.Ref
s : aig.RefVec curr
lhs rhs : aig.RefVec w
assign : α → Bool
lhsExpr rhsExpr : BitVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := lhs.get idx hidx }⟧ = lhsExpr.getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := rhs.get idx hidx }⟧ = rhsExpr.getLsbD idx
hcin : ⟦assign, { aig := aig, ref := cin }⟧ = BitVec.carry curr lhsExpr rhsExpr false
idx : Nat
hidx1 : idx < w
hidx2 : curr ≤ idx
res : RefVecEntry α w
hlt : curr < w
hgo :
  go (mkFullAdder aig { lhs := lhs.get curr hlt, rhs := rhs.get curr hlt, cin := cin }).aig (lhs.cast ⋯) (rhs.cast ⋯)
      (curr + 1) ⋯ (mkFullAdder aig { lhs := lhs.get curr hlt, rhs := rhs.get curr hlt, cin := cin }).cout
      ((s.cast ⋯).push (mkFullAdder aig { lhs := lhs.get curr hlt, rhs := rhs.get curr hlt, cin := cin }).out) =
    res
heq : curr = idx
⊢ ⟦assign,
      {
        aig :=
          (go (mkFullAdder aig { lhs := lhs.get curr hlt, rhs := rhs.get curr hlt, cin := cin }).aig (lhs.cast ⋯)
              (rhs.cast ⋯) (curr + 1) ⋯
              (mkFullAdder aig { lhs := lhs.get curr hlt, rhs := rhs.get curr hlt, cin := cin }).cout
              ((s.cast ⋯).push
                (mkFullAdder aig { lhs := lhs.get curr hlt, rhs := rhs.get curr hlt, cin := cin }).out)).aig,
        ref :=
          (((s.cast ⋯).push (mkFullAdder aig { lhs := lhs.get curr hlt, rhs := rhs.get curr hlt, cin := cin }).out).get
                idx ⋯).cast
            ⋯ }⟧ =
    (⟦assign, { aig := aig, ref := lhs.get idx hidx1 }⟧ ^^
      (⟦assign, { aig := aig, ref := rhs.get idx hidx1 }⟧ ^^ BitVec.carry idx lhsExpr rhsExpr false))

Tactic:
rw [AIG.RefVec.get_push_ref_eq' (hidx := by rw [heq])]

Result:
case inl
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
curr : Nat
hcurr : curr ≤ w
cin : aig.Ref
s : aig.RefVec curr
lhs rhs : aig.RefVec w
assign : α → Bool
lhsExpr rhsExpr : BitVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := lhs.get idx hidx }⟧ = lhsExpr.getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := rhs.get idx hidx }⟧ = rhsExpr.getLsbD idx
hcin : ⟦assign, { aig := aig, ref := cin }⟧ = BitVec.carry curr lhsExpr rhsExpr false
idx : Nat
hidx1 : idx < w
hidx2 : curr ≤ idx
res : RefVecEntry α w
hlt : curr < w
hgo :
  go (mkFullAdder aig { lhs := lhs.get curr hlt, rhs := rhs.get curr hlt, cin := cin }).aig (lhs.cast ⋯) (rhs.cast ⋯)
      (curr + 1) ⋯ (mkFullAdder aig { lhs := lhs.get curr hlt, rhs := rhs.get curr hlt, cin := cin }).cout
      ((s.cast ⋯).push (mkFullAdder aig { lhs := lhs.get curr hlt, rhs := rhs.get curr hlt, cin := cin }).out) =
    res
heq : curr = idx
⊢ ⟦assign,
      {
        aig :=
          (go (mkFullAdder aig { lhs := lhs.get curr hlt, rhs := rhs.get curr hlt, cin := cin }).aig (lhs.cast ⋯)
              (rhs.cast ⋯) (curr + 1) ⋯
              (mkFullAdder aig { lhs := lhs.get curr hlt, rhs := rhs.get curr hlt, cin := cin }).cout
              ((s.cast ⋯).push
                (mkFullAdder aig { lhs := lhs.get curr hlt, rhs := rhs.get curr hlt, cin := cin }).out)).aig,
        ref := (mkFullAdder aig { lhs := lhs.get curr hlt, rhs := rhs.get curr hlt, cin := cin }).out.cast ⋯ }⟧ =
    (⟦assign, { aig := aig, ref := lhs.get idx hidx1 }⟧ ^^
      (⟦assign, { aig := aig, ref := rhs.get idx hidx1 }⟧ ^^ BitVec.carry idx lhsExpr rhsExpr false))