Theorem: BVExpr.bitblast.blastAdd.go_eq_eval_getLsb
Proof state before:
case inr
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
curr : Nat
hcurr : curr ≤ w
cin : aig.Ref
s : aig.RefStream curr
lhs rhs : aig.RefStream w
assign : α → Bool
lhsExpr rhsExpr : BitVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦{ aig := aig, ref := lhs.getRef idx hidx }, assign⟧ = lhsExpr.getLsb idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦{ aig := aig, ref := rhs.getRef idx hidx }, assign⟧ = rhsExpr.getLsb idx
hcin : ⟦{ aig := aig, ref := cin }, assign⟧ = BitVec.carry curr lhsExpr rhsExpr false
idx : Nat
hidx1 : idx < w
hidx2 : curr ≤ idx
res : RefStreamEntry α w
hlt✝ : curr < w
hgo :
  go (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).aig (curr + 1) ⋯
      (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).cout
      ((s.cast ⋯).pushRef
        (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).out)
      (lhs.cast ⋯) (rhs.cast ⋯) =
    res
hlt : curr < idx
⊢ ⟦{ aig := (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).aig,
            ref := (lhs.cast ⋯).getRef idx hidx1 },
          assign⟧.xor
      (⟦{ aig := (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).aig,
              ref := (rhs.cast ⋯).getRef idx hidx1 },
            assign⟧.xor
        (BitVec.carry idx lhsExpr rhsExpr false)) =
    ⟦{ aig := aig, ref := lhs.getRef idx hidx1 }, assign⟧.xor
      (⟦{ aig := aig, ref := rhs.getRef idx hidx1 }, assign⟧.xor (BitVec.carry idx lhsExpr rhsExpr false))

case inr.hleft
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
curr : Nat
hcurr : curr ≤ w
cin : aig.Ref
s : aig.RefStream curr
lhs rhs : aig.RefStream w
assign : α → Bool
lhsExpr rhsExpr : BitVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦{ aig := aig, ref := lhs.getRef idx hidx }, assign⟧ = lhsExpr.getLsb idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦{ aig := aig, ref := rhs.getRef idx hidx }, assign⟧ = rhsExpr.getLsb idx
hcin : ⟦{ aig := aig, ref := cin }, assign⟧ = BitVec.carry curr lhsExpr rhsExpr false
idx : Nat
hidx1 : idx < w
hidx2 : curr ≤ idx
res : RefStreamEntry α w
hlt✝ : curr < w
hgo :
  go (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).aig (curr + 1) ⋯
      (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).cout
      ((s.cast ⋯).pushRef
        (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).out)
      (lhs.cast ⋯) (rhs.cast ⋯) =
    res
hlt : curr < idx
⊢ ∀ (idx : Nat) (hidx : idx < w),
    ⟦{ aig := (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).aig,
          ref := (lhs.cast ⋯).getRef idx hidx },
        assign⟧ =
      lhsExpr.getLsb idx

case inr.hright
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
curr : Nat
hcurr : curr ≤ w
cin : aig.Ref
s : aig.RefStream curr
lhs rhs : aig.RefStream w
assign : α → Bool
lhsExpr rhsExpr : BitVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦{ aig := aig, ref := lhs.getRef idx hidx }, assign⟧ = lhsExpr.getLsb idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦{ aig := aig, ref := rhs.getRef idx hidx }, assign⟧ = rhsExpr.getLsb idx
hcin : ⟦{ aig := aig, ref := cin }, assign⟧ = BitVec.carry curr lhsExpr rhsExpr false
idx : Nat
hidx1 : idx < w
hidx2 : curr ≤ idx
res : RefStreamEntry α w
hlt✝ : curr < w
hgo :
  go (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).aig (curr + 1) ⋯
      (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).cout
      ((s.cast ⋯).pushRef
        (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).out)
      (lhs.cast ⋯) (rhs.cast ⋯) =
    res
hlt : curr < idx
⊢ ∀ (idx : Nat) (hidx : idx < w),
    ⟦{ aig := (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).aig,
          ref := (rhs.cast ⋯).getRef idx hidx },
        assign⟧ =
      rhsExpr.getLsb idx

case inr.hcin
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
curr : Nat
hcurr : curr ≤ w
cin : aig.Ref
s : aig.RefStream curr
lhs rhs : aig.RefStream w
assign : α → Bool
lhsExpr rhsExpr : BitVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦{ aig := aig, ref := lhs.getRef idx hidx }, assign⟧ = lhsExpr.getLsb idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦{ aig := aig, ref := rhs.getRef idx hidx }, assign⟧ = rhsExpr.getLsb idx
hcin : ⟦{ aig := aig, ref := cin }, assign⟧ = BitVec.carry curr lhsExpr rhsExpr false
idx : Nat
hidx1 : idx < w
hidx2 : curr ≤ idx
res : RefStreamEntry α w
hlt✝ : curr < w
hgo :
  go (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).aig (curr + 1) ⋯
      (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).cout
      ((s.cast ⋯).pushRef
        (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).out)
      (lhs.cast ⋯) (rhs.cast ⋯) =
    res
hlt : curr < idx
⊢ ⟦{ aig := (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).aig,
        ref := (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).cout },
      assign⟧ =
    BitVec.carry (curr + 1) lhsExpr rhsExpr false

case inr.a
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
curr : Nat
hcurr : curr ≤ w
cin : aig.Ref
s : aig.RefStream curr
lhs rhs : aig.RefStream w
assign : α → Bool
lhsExpr rhsExpr : BitVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦{ aig := aig, ref := lhs.getRef idx hidx }, assign⟧ = lhsExpr.getLsb idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦{ aig := aig, ref := rhs.getRef idx hidx }, assign⟧ = rhsExpr.getLsb idx
hcin : ⟦{ aig := aig, ref := cin }, assign⟧ = BitVec.carry curr lhsExpr rhsExpr false
idx : Nat
hidx1 : idx < w
hidx2 : curr ≤ idx
res : RefStreamEntry α w
hlt✝ : curr < w
hgo :
  go (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).aig (curr + 1) ⋯
      (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).cout
      ((s.cast ⋯).pushRef
        (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).out)
      (lhs.cast ⋯) (rhs.cast ⋯) =
    res
hlt : curr < idx
⊢ curr + 1 ≤ idx
Tactic:
. rw [mkFullAdder_denote_mem_prefix]
  rw [mkFullAdder_denote_mem_prefix]
  . simp
  . simp [Ref.hgate]
  . simp [Ref.hgate]
Proof state after:
case inr.hleft
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
curr : Nat
hcurr : curr ≤ w
cin : aig.Ref
s : aig.RefStream curr
lhs rhs : aig.RefStream w
assign : α → Bool
lhsExpr rhsExpr : BitVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦{ aig := aig, ref := lhs.getRef idx hidx }, assign⟧ = lhsExpr.getLsb idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦{ aig := aig, ref := rhs.getRef idx hidx }, assign⟧ = rhsExpr.getLsb idx
hcin : ⟦{ aig := aig, ref := cin }, assign⟧ = BitVec.carry curr lhsExpr rhsExpr false
idx : Nat
hidx1 : idx < w
hidx2 : curr ≤ idx
res : RefStreamEntry α w
hlt✝ : curr < w
hgo :
  go (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).aig (curr + 1) ⋯
      (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).cout
      ((s.cast ⋯).pushRef
        (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).out)
      (lhs.cast ⋯) (rhs.cast ⋯) =
    res
hlt : curr < idx
⊢ ∀ (idx : Nat) (hidx : idx < w),
    ⟦{ aig := (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).aig,
          ref := (lhs.cast ⋯).getRef idx hidx },
        assign⟧ =
      lhsExpr.getLsb idx

case inr.hright
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
curr : Nat
hcurr : curr ≤ w
cin : aig.Ref
s : aig.RefStream curr
lhs rhs : aig.RefStream w
assign : α → Bool
lhsExpr rhsExpr : BitVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦{ aig := aig, ref := lhs.getRef idx hidx }, assign⟧ = lhsExpr.getLsb idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦{ aig := aig, ref := rhs.getRef idx hidx }, assign⟧ = rhsExpr.getLsb idx
hcin : ⟦{ aig := aig, ref := cin }, assign⟧ = BitVec.carry curr lhsExpr rhsExpr false
idx : Nat
hidx1 : idx < w
hidx2 : curr ≤ idx
res : RefStreamEntry α w
hlt✝ : curr < w
hgo :
  go (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).aig (curr + 1) ⋯
      (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).cout
      ((s.cast ⋯).pushRef
        (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).out)
      (lhs.cast ⋯) (rhs.cast ⋯) =
    res
hlt : curr < idx
⊢ ∀ (idx : Nat) (hidx : idx < w),
    ⟦{ aig := (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).aig,
          ref := (rhs.cast ⋯).getRef idx hidx },
        assign⟧ =
      rhsExpr.getLsb idx

case inr.hcin
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
curr : Nat
hcurr : curr ≤ w
cin : aig.Ref
s : aig.RefStream curr
lhs rhs : aig.RefStream w
assign : α → Bool
lhsExpr rhsExpr : BitVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦{ aig := aig, ref := lhs.getRef idx hidx }, assign⟧ = lhsExpr.getLsb idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦{ aig := aig, ref := rhs.getRef idx hidx }, assign⟧ = rhsExpr.getLsb idx
hcin : ⟦{ aig := aig, ref := cin }, assign⟧ = BitVec.carry curr lhsExpr rhsExpr false
idx : Nat
hidx1 : idx < w
hidx2 : curr ≤ idx
res : RefStreamEntry α w
hlt✝ : curr < w
hgo :
  go (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).aig (curr + 1) ⋯
      (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).cout
      ((s.cast ⋯).pushRef
        (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).out)
      (lhs.cast ⋯) (rhs.cast ⋯) =
    res
hlt : curr < idx
⊢ ⟦{ aig := (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).aig,
        ref := (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).cout },
      assign⟧ =
    BitVec.carry (curr + 1) lhsExpr rhsExpr false

case inr.a
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
curr : Nat
hcurr : curr ≤ w
cin : aig.Ref
s : aig.RefStream curr
lhs rhs : aig.RefStream w
assign : α → Bool
lhsExpr rhsExpr : BitVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦{ aig := aig, ref := lhs.getRef idx hidx }, assign⟧ = lhsExpr.getLsb idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦{ aig := aig, ref := rhs.getRef idx hidx }, assign⟧ = rhsExpr.getLsb idx
hcin : ⟦{ aig := aig, ref := cin }, assign⟧ = BitVec.carry curr lhsExpr rhsExpr false
idx : Nat
hidx1 : idx < w
hidx2 : curr ≤ idx
res : RefStreamEntry α w
hlt✝ : curr < w
hgo :
  go (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).aig (curr + 1) ⋯
      (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).cout
      ((s.cast ⋯).pushRef
        (mkFullAdder aig { lhs := lhs.getRef curr hlt✝, rhs := rhs.getRef curr hlt✝, cin := cin }).out)
      (lhs.cast ⋯) (rhs.cast ⋯) =
    res
hlt : curr < idx
⊢ curr + 1 ≤ idx