-- Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/RefVecOperator/Fold.lean
theorem denote_go_and {aig : AIG α} (acc : AIG.Ref aig) (curr : Nat) (hcurr : curr ≤ len)
    (input : RefVec aig len) :
    ⟦
      (go aig acc curr len input mkAndCached).aig,
      (go aig acc curr len input mkAndCached).ref,
      assign
    ⟧
      =
    (
      ⟦aig, acc, assign⟧
        ∧
      (∀ (idx : Nat) (hidx1 : idx < len), curr ≤ idx → ⟦aig, input.get idx hidx1, assign⟧)
    )

Proof state:
case isTrue
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefVec len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.get curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.get curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
⊢ (⟦assign, { aig := res.aig, ref := res.ref }⟧ = true) =
    (⟦assign, { aig := aig, ref := acc }⟧ = true ∧
      ∀ (idx : Nat) (hidx1 : idx < len), curr ≤ idx → ⟦assign, { aig := aig, ref := input.get idx hidx1 }⟧ = true)

Tactic:
rw [← hgo]

Result:
case isTrue
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefVec len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.get curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.get curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
⊢ (⟦assign,
        {
          aig :=
            (go (aig.mkAndCached { lhs := acc, rhs := input.get curr h✝ }).aig
                (aig.mkAndCached { lhs := acc, rhs := input.get curr h✝ }).ref (curr + 1) len (input.cast ⋯)
                mkAndCached).aig,
          ref :=
            (go (aig.mkAndCached { lhs := acc, rhs := input.get curr h✝ }).aig
                (aig.mkAndCached { lhs := acc, rhs := input.get curr h✝ }).ref (curr + 1) len (input.cast ⋯)
                mkAndCached).ref }⟧ =
      true) =
    (⟦assign, { aig := aig, ref := acc }⟧ = true ∧
      ∀ (idx : Nat) (hidx1 : idx < len), curr ≤ idx → ⟦assign, { aig := aig, ref := input.get idx hidx1 }⟧ = true)

Proof state:
case isTrue.mpr
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefVec len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.get curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.get curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
⊢ ⟦assign, { aig := aig, ref := acc }⟧ = true →
    (∀ (idx : Nat) (hidx1 : idx < len), curr ≤ idx → ⟦assign, { aig := aig, ref := input.get idx hidx1 }⟧ = true) →
      (⟦assign, { aig := aig, ref := acc }⟧ = true ∧ ⟦assign, { aig := aig, ref := input.get curr h✝ }⟧ = true) ∧
        ∀ (idx : Nat) (hidx1 : idx < len),
          curr + 1 ≤ idx →
            ⟦assign,
                { aig := (aig.mkAndCached { lhs := acc, rhs := input.get curr h✝ }).aig,
                  ref := (input.get idx hidx1).cast ⋯ }⟧ =
              true

Tactic:
intro hacc hrest

Result:
case isTrue.mpr
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefVec len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.get curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.get curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
hacc : ⟦assign, { aig := aig, ref := acc }⟧ = true
hrest : ∀ (idx : Nat) (hidx1 : idx < len), curr ≤ idx → ⟦assign, { aig := aig, ref := input.get idx hidx1 }⟧ = true
⊢ (⟦assign, { aig := aig, ref := acc }⟧ = true ∧ ⟦assign, { aig := aig, ref := input.get curr h✝ }⟧ = true) ∧
    ∀ (idx : Nat) (hidx1 : idx < len),
      curr + 1 ≤ idx →
        ⟦assign,
            { aig := (aig.mkAndCached { lhs := acc, rhs := input.get curr h✝ }).aig,
              ref := (input.get idx hidx1).cast ⋯ }⟧ =
          true

Proof state:
case isTrue.mp.intro.intro.right.inl
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefVec len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.get curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.get curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
h3 :
  ∀ (idx : Nat) (hidx1 : idx < len),
    curr + 1 ≤ idx →
      ⟦assign,
          { aig := (aig.mkAndCached { lhs := acc, rhs := input.get curr h✝ }).aig,
            ref := (input.get idx hidx1).cast ⋯ }⟧ =
        true
h1 : ⟦assign, { aig := aig, ref := acc }⟧ = true
h2 : ⟦assign, { aig := aig, ref := input.get curr h✝ }⟧ = true
idx : Nat
hidx1 : idx < len
hidx2 : curr ≤ idx
heq : curr = idx
⊢ ⟦assign, { aig := aig, ref := input.get idx hidx1 }⟧ = true

Tactic:
simpa [heq] using h2

Result:
no goals

Proof state:
case isTrue
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefVec len
res : Entrypoint α
h✝ : curr < len
hgo :
  (let res := aig.mkAndCached { lhs := acc, rhs := input.get curr h✝ };
    let aig_1 := res.aig;
    let newAcc := res.ref;
    let input := input.cast ⋯;
    go aig_1 newAcc (curr + 1) len input mkAndCached) =
    res
⊢ (⟦assign, { aig := res.aig, ref := res.ref }⟧ = true) =
    (⟦assign, { aig := aig, ref := acc }⟧ = true ∧
      ∀ (idx : Nat) (hidx1 : idx < len), curr ≤ idx → ⟦assign, { aig := aig, ref := input.get idx hidx1 }⟧ = true)

Tactic:
dsimp only at hgo

Result:
case isTrue
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefVec len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.get curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.get curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
⊢ (⟦assign, { aig := res.aig, ref := res.ref }⟧ = true) =
    (⟦assign, { aig := aig, ref := acc }⟧ = true ∧
      ∀ (idx : Nat) (hidx1 : idx < len), curr ≤ idx → ⟦assign, { aig := aig, ref := input.get idx hidx1 }⟧ = true)