internlm/Lean-Github huggingface dataset:
Theorem: AIG.RefStream.fold.denote_go_and
Proof state before:
case isTrue.mp
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig✝ : AIG α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefStream len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
⊢ ((⟦{ aig := aig, ref := acc }, assign⟧ = true ∧ ⟦{ aig := aig, ref := input.getRef curr h✝ }, assign⟧ = true) ∧
      ∀ (idx : Nat) (hidx1 : idx < len),
        curr + 1 ≤ idx →
          ⟦{ aig := (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig,
                ref := (input.getRef idx hidx1).cast ⋯ },
              assign⟧ =
            true) →
    ⟦{ aig := aig, ref := acc }, assign⟧ = true ∧
      ∀ (idx : Nat) (hidx1 : idx < len), curr ≤ idx → ⟦{ aig := aig, ref := input.getRef idx hidx1 }, assign⟧ = true
Tactic:
intro h
Proof state after:
case isTrue.mp
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig✝ : AIG α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefStream len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
h :
  (⟦{ aig := aig, ref := acc }, assign⟧ = true ∧ ⟦{ aig := aig, ref := input.getRef curr h✝ }, assign⟧ = true) ∧
    ∀ (idx : Nat) (hidx1 : idx < len),
      curr + 1 ≤ idx →
        ⟦{ aig := (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig,
              ref := (input.getRef idx hidx1).cast ⋯ },
            assign⟧ =
          true
⊢ ⟦{ aig := aig, ref := acc }, assign⟧ = true ∧
    ∀ (idx : Nat) (hidx1 : idx < len), curr ≤ idx → ⟦{ aig := aig, ref := input.getRef idx hidx1 }, assign⟧ = true
Theorem: AIG.RefStream.fold.denote_go_and
Proof state before:
case isTrue.mp
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig✝ : AIG α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefStream len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
h :
  (⟦{ aig := aig, ref := acc }, assign⟧ = true ∧ ⟦{ aig := aig, ref := input.getRef curr h✝ }, assign⟧ = true) ∧
    ∀ (idx : Nat) (hidx1 : idx < len),
      curr + 1 ≤ idx →
        ⟦{ aig := (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig,
              ref := (input.getRef idx hidx1).cast ⋯ },
            assign⟧ =
          true
⊢ ⟦{ aig := aig, ref := acc }, assign⟧ = true ∧
    ∀ (idx : Nat) (hidx1 : idx < len), curr ≤ idx → ⟦{ aig := aig, ref := input.getRef idx hidx1 }, assign⟧ = true
Tactic:
rcases h with ⟨⟨h1, h2⟩, h3⟩
Proof state after:
case isTrue.mp.intro.intro
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig✝ : AIG α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefStream len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
h3 :
  ∀ (idx : Nat) (hidx1 : idx < len),
    curr + 1 ≤ idx →
      ⟦{ aig := (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig,
            ref := (input.getRef idx hidx1).cast ⋯ },
          assign⟧ =
        true
h1 : ⟦{ aig := aig, ref := acc }, assign⟧ = true
h2 : ⟦{ aig := aig, ref := input.getRef curr h✝ }, assign⟧ = true
⊢ ⟦{ aig := aig, ref := acc }, assign⟧ = true ∧
    ∀ (idx : Nat) (hidx1 : idx < len), curr ≤ idx → ⟦{ aig := aig, ref := input.getRef idx hidx1 }, assign⟧ = true
Theorem: AIG.RefStream.fold.denote_go_and
Proof state before:
case isTrue.mp.intro.intro
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig✝ : AIG α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefStream len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
h3 :
  ∀ (idx : Nat) (hidx1 : idx < len),
    curr + 1 ≤ idx →
      ⟦{ aig := (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig,
            ref := (input.getRef idx hidx1).cast ⋯ },
          assign⟧ =
        true
h1 : ⟦{ aig := aig, ref := acc }, assign⟧ = true
h2 : ⟦{ aig := aig, ref := input.getRef curr h✝ }, assign⟧ = true
⊢ ⟦{ aig := aig, ref := acc }, assign⟧ = true ∧
    ∀ (idx : Nat) (hidx1 : idx < len), curr ≤ idx → ⟦{ aig := aig, ref := input.getRef idx hidx1 }, assign⟧ = true
Tactic:
constructor
Proof state after:
case isTrue.mp.intro.intro.left
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig✝ : AIG α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefStream len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
h3 :
  ∀ (idx : Nat) (hidx1 : idx < len),
    curr + 1 ≤ idx →
      ⟦{ aig := (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig,
            ref := (input.getRef idx hidx1).cast ⋯ },
          assign⟧ =
        true
h1 : ⟦{ aig := aig, ref := acc }, assign⟧ = true
h2 : ⟦{ aig := aig, ref := input.getRef curr h✝ }, assign⟧ = true
⊢ ⟦{ aig := aig, ref := acc }, assign⟧ = true

case isTrue.mp.intro.intro.right
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig✝ : AIG α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefStream len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
h3 :
  ∀ (idx : Nat) (hidx1 : idx < len),
    curr + 1 ≤ idx →
      ⟦{ aig := (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig,
            ref := (input.getRef idx hidx1).cast ⋯ },
          assign⟧ =
        true
h1 : ⟦{ aig := aig, ref := acc }, assign⟧ = true
h2 : ⟦{ aig := aig, ref := input.getRef curr h✝ }, assign⟧ = true
⊢ ∀ (idx : Nat) (hidx1 : idx < len), curr ≤ idx → ⟦{ aig := aig, ref := input.getRef idx hidx1 }, assign⟧ = true
Theorem: AIG.RefStream.fold.denote_go_and
Proof state before:
case isTrue.mp.intro.intro.left
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig✝ : AIG α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefStream len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
h3 :
  ∀ (idx : Nat) (hidx1 : idx < len),
    curr + 1 ≤ idx →
      ⟦{ aig := (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig,
            ref := (input.getRef idx hidx1).cast ⋯ },
          assign⟧ =
        true
h1 : ⟦{ aig := aig, ref := acc }, assign⟧ = true
h2 : ⟦{ aig := aig, ref := input.getRef curr h✝ }, assign⟧ = true
⊢ ⟦{ aig := aig, ref := acc }, assign⟧ = true

case isTrue.mp.intro.intro.right
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig✝ : AIG α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefStream len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
h3 :
  ∀ (idx : Nat) (hidx1 : idx < len),
    curr + 1 ≤ idx →
      ⟦{ aig := (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig,
            ref := (input.getRef idx hidx1).cast ⋯ },
          assign⟧ =
        true
h1 : ⟦{ aig := aig, ref := acc }, assign⟧ = true
h2 : ⟦{ aig := aig, ref := input.getRef curr h✝ }, assign⟧ = true
⊢ ∀ (idx : Nat) (hidx1 : idx < len), curr ≤ idx → ⟦{ aig := aig, ref := input.getRef idx hidx1 }, assign⟧ = true
Tactic:
. assumption
Proof state after:
case isTrue.mp.intro.intro.right
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig✝ : AIG α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefStream len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
h3 :
  ∀ (idx : Nat) (hidx1 : idx < len),
    curr + 1 ≤ idx →
      ⟦{ aig := (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig,
            ref := (input.getRef idx hidx1).cast ⋯ },
          assign⟧ =
        true
h1 : ⟦{ aig := aig, ref := acc }, assign⟧ = true
h2 : ⟦{ aig := aig, ref := input.getRef curr h✝ }, assign⟧ = true
⊢ ∀ (idx : Nat) (hidx1 : idx < len), curr ≤ idx → ⟦{ aig := aig, ref := input.getRef idx hidx1 }, assign⟧ = true
Theorem: AIG.RefStream.fold.denote_go_and
Proof state before:
case isTrue.mp.intro.intro.right
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig✝ : AIG α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefStream len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
h3 :
  ∀ (idx : Nat) (hidx1 : idx < len),
    curr + 1 ≤ idx →
      ⟦{ aig := (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig,
            ref := (input.getRef idx hidx1).cast ⋯ },
          assign⟧ =
        true
h1 : ⟦{ aig := aig, ref := acc }, assign⟧ = true
h2 : ⟦{ aig := aig, ref := input.getRef curr h✝ }, assign⟧ = true
⊢ ∀ (idx : Nat) (hidx1 : idx < len), curr ≤ idx → ⟦{ aig := aig, ref := input.getRef idx hidx1 }, assign⟧ = true
Tactic:
. intro idx hidx1 hidx2
  cases Nat.eq_or_lt_of_le hidx2 with
  | inl heq => simpa [heq] using h2
  | inr hlt =>
    specialize h3 idx hidx1 (by omega)
    rw [← h3]
    rw [AIG.LawfulOperator.denote_mem_prefix (f := AIG.mkAndCached)]
    . simp
    . simp [Ref.hgate]
Proof state after:
no goals
Theorem: AIG.RefStream.fold.denote_go_and
Proof state before:
case isTrue.mp.intro.intro.left
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig✝ : AIG α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefStream len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
h3 :
  ∀ (idx : Nat) (hidx1 : idx < len),
    curr + 1 ≤ idx →
      ⟦{ aig := (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig,
            ref := (input.getRef idx hidx1).cast ⋯ },
          assign⟧ =
        true
h1 : ⟦{ aig := aig, ref := acc }, assign⟧ = true
h2 : ⟦{ aig := aig, ref := input.getRef curr h✝ }, assign⟧ = true
⊢ ⟦{ aig := aig, ref := acc }, assign⟧ = true
Tactic:
assumption
Proof state after:
no goals
Theorem: AIG.RefStream.fold.denote_go_and
Proof state before:
case isTrue.mp.intro.intro.right
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig✝ : AIG α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefStream len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
h3 :
  ∀ (idx : Nat) (hidx1 : idx < len),
    curr + 1 ≤ idx →
      ⟦{ aig := (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig,
            ref := (input.getRef idx hidx1).cast ⋯ },
          assign⟧ =
        true
h1 : ⟦{ aig := aig, ref := acc }, assign⟧ = true
h2 : ⟦{ aig := aig, ref := input.getRef curr h✝ }, assign⟧ = true
⊢ ∀ (idx : Nat) (hidx1 : idx < len), curr ≤ idx → ⟦{ aig := aig, ref := input.getRef idx hidx1 }, assign⟧ = true
Tactic:
intro idx hidx1 hidx2
Proof state after:
case isTrue.mp.intro.intro.right
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig✝ : AIG α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefStream len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
h3 :
  ∀ (idx : Nat) (hidx1 : idx < len),
    curr + 1 ≤ idx →
      ⟦{ aig := (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig,
            ref := (input.getRef idx hidx1).cast ⋯ },
          assign⟧ =
        true
h1 : ⟦{ aig := aig, ref := acc }, assign⟧ = true
h2 : ⟦{ aig := aig, ref := input.getRef curr h✝ }, assign⟧ = true
idx : Nat
hidx1 : idx < len
hidx2 : curr ≤ idx
⊢ ⟦{ aig := aig, ref := input.getRef idx hidx1 }, assign⟧ = true
Theorem: AIG.RefStream.fold.denote_go_and
Proof state before:
case isTrue.mp.intro.intro.right
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig✝ : AIG α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefStream len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
h3 :
  ∀ (idx : Nat) (hidx1 : idx < len),
    curr + 1 ≤ idx →
      ⟦{ aig := (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig,
            ref := (input.getRef idx hidx1).cast ⋯ },
          assign⟧ =
        true
h1 : ⟦{ aig := aig, ref := acc }, assign⟧ = true
h2 : ⟦{ aig := aig, ref := input.getRef curr h✝ }, assign⟧ = true
idx : Nat
hidx1 : idx < len
hidx2 : curr ≤ idx
⊢ ⟦{ aig := aig, ref := input.getRef idx hidx1 }, assign⟧ = true
Tactic:
cases Nat.eq_or_lt_of_le hidx2 with
| inl heq => simpa [heq] using h2
| inr hlt =>
  specialize h3 idx hidx1 (by omega)
  rw [← h3]
  rw [AIG.LawfulOperator.denote_mem_prefix (f := AIG.mkAndCached)]
  . simp
  . simp [Ref.hgate]
Proof state after:
no goals
Theorem: AIG.RefStream.fold.denote_go_and
Proof state before:
case isTrue.mp.intro.intro.right.inl
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig✝ : AIG α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefStream len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
h3 :
  ∀ (idx : Nat) (hidx1 : idx < len),
    curr + 1 ≤ idx →
      ⟦{ aig := (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig,
            ref := (input.getRef idx hidx1).cast ⋯ },
          assign⟧ =
        true
h1 : ⟦{ aig := aig, ref := acc }, assign⟧ = true
h2 : ⟦{ aig := aig, ref := input.getRef curr h✝ }, assign⟧ = true
idx : Nat
hidx1 : idx < len
hidx2 : curr ≤ idx
heq : curr = idx
⊢ ⟦{ aig := aig, ref := input.getRef idx hidx1 }, assign⟧ = true
Tactic:
simpa [heq] using h2
Proof state after:
no goals
Theorem: AIG.RefStream.fold.denote_go_and
Proof state before:
case isTrue.mp.intro.intro.right.inr
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig✝ : AIG α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefStream len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
h3 :
  ∀ (idx : Nat) (hidx1 : idx < len),
    curr + 1 ≤ idx →
      ⟦{ aig := (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig,
            ref := (input.getRef idx hidx1).cast ⋯ },
          assign⟧ =
        true
h1 : ⟦{ aig := aig, ref := acc }, assign⟧ = true
h2 : ⟦{ aig := aig, ref := input.getRef curr h✝ }, assign⟧ = true
idx : Nat
hidx1 : idx < len
hidx2 : curr ≤ idx
hlt : curr < idx
⊢ ⟦{ aig := aig, ref := input.getRef idx hidx1 }, assign⟧ = true
Tactic:
specialize h3 idx hidx1 (by omega)
Proof state after:
case isTrue.mp.intro.intro.right.inr
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig✝ : AIG α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefStream len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
h1 : ⟦{ aig := aig, ref := acc }, assign⟧ = true
h2 : ⟦{ aig := aig, ref := input.getRef curr h✝ }, assign⟧ = true
idx : Nat
hidx1 : idx < len
hidx2 : curr ≤ idx
hlt : curr < idx
h3 :
  ⟦{ aig := (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig, ref := (input.getRef idx hidx1).cast ⋯ },
      assign⟧ =
    true
⊢ ⟦{ aig := aig, ref := input.getRef idx hidx1 }, assign⟧ = true
Theorem: AIG.RefStream.fold.denote_go_and
Proof state before:
case isTrue.mp.intro.intro.right.inr
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig✝ : AIG α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefStream len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
h1 : ⟦{ aig := aig, ref := acc }, assign⟧ = true
h2 : ⟦{ aig := aig, ref := input.getRef curr h✝ }, assign⟧ = true
idx : Nat
hidx1 : idx < len
hidx2 : curr ≤ idx
hlt : curr < idx
h3 :
  ⟦{ aig := (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig, ref := (input.getRef idx hidx1).cast ⋯ },
      assign⟧ =
    true
⊢ ⟦{ aig := aig, ref := input.getRef idx hidx1 }, assign⟧ = true
Tactic:
rw [← h3]
Proof state after:
case isTrue.mp.intro.intro.right.inr
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig✝ : AIG α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefStream len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
h1 : ⟦{ aig := aig, ref := acc }, assign⟧ = true
h2 : ⟦{ aig := aig, ref := input.getRef curr h✝ }, assign⟧ = true
idx : Nat
hidx1 : idx < len
hidx2 : curr ≤ idx
hlt : curr < idx
h3 :
  ⟦{ aig := (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig, ref := (input.getRef idx hidx1).cast ⋯ },
      assign⟧ =
    true
⊢ ⟦{ aig := aig, ref := input.getRef idx hidx1 }, assign⟧ =
    ⟦{ aig := (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig,
        ref := (input.getRef idx hidx1).cast ⋯ },
      assign⟧
Theorem: AIG.RefStream.fold.denote_go_and
Proof state before:
case isTrue.mp.intro.intro.right.inr
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig✝ : AIG α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefStream len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
h1 : ⟦{ aig := aig, ref := acc }, assign⟧ = true
h2 : ⟦{ aig := aig, ref := input.getRef curr h✝ }, assign⟧ = true
idx : Nat
hidx1 : idx < len
hidx2 : curr ≤ idx
hlt : curr < idx
h3 :
  ⟦{ aig := (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig, ref := (input.getRef idx hidx1).cast ⋯ },
      assign⟧ =
    true
⊢ ⟦{ aig := aig, ref := input.getRef idx hidx1 }, assign⟧ =
    ⟦{ aig := (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig,
        ref := (input.getRef idx hidx1).cast ⋯ },
      assign⟧
Tactic:
rw [AIG.LawfulOperator.denote_mem_prefix (f := AIG.mkAndCached)]
Proof state after:
case isTrue.mp.intro.intro.right.inr
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig✝ : AIG α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefStream len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
h1 : ⟦{ aig := aig, ref := acc }, assign⟧ = true
h2 : ⟦{ aig := aig, ref := input.getRef curr h✝ }, assign⟧ = true
idx : Nat
hidx1 : idx < len
hidx2 : curr ≤ idx
hlt : curr < idx
h3 :
  ⟦{ aig := (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig, ref := (input.getRef idx hidx1).cast ⋯ },
      assign⟧ =
    true
⊢ ⟦{ aig := aig, ref := input.getRef idx hidx1 }, assign⟧ =
    ⟦{ aig := aig,
        ref := { gate := ((input.getRef idx hidx1).cast ⋯).gate, hgate := ?isTrue.mp.intro.intro.right.inr } },
      assign⟧

case isTrue.mp.intro.intro.right.inr
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig✝ : AIG α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefStream len
res : Entrypoint α
h✝ : curr < len
hgo :
  go (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig
      (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
    res
h1 : ⟦{ aig := aig, ref := acc }, assign⟧ = true
h2 : ⟦{ aig := aig, ref := input.getRef curr h✝ }, assign⟧ = true
idx : Nat
hidx1 : idx < len
hidx2 : curr ≤ idx
hlt : curr < idx
h3 :
  ⟦{ aig := (aig.mkAndCached { lhs := acc, rhs := input.getRef curr h✝ }).aig, ref := (input.getRef idx hidx1).cast ⋯ },
      assign⟧ =
    true
⊢ ((input.getRef idx hidx1).cast ⋯).gate < aig.decls.size