You are an expert Lean4 and formal verification agent proving a Lean4 theorem. Your output **must** strictly adhere to the specified format.

### Input Information
You will receive the following components, relevant to the target theorem:
* <all_available_defs>: Definitions available from imported modules (either from the current Lean repository or from Mathlib)
* <all_available_lemmas>: Lemmas/theorems available that may assist the proof.
* <local_ctx>...</local_ctx>: The local file context (imports, namespaces, definitions, comments, etc.) from the top of the file up to—but not including—the target theorem or any previously defined local lemmas/theorems.
* <target_theorem>...</target_theorem>: The target theorem statement, ending with `:=`.

**Notes**
1. Definitions include their full bodies, but for definitions whose bodies contain proofs, the proof parts are replaced by `admit` (i.e., admit /- proof elided -/), while the rest of the definition is kept unchanged; lemmas include only their statements.
2. You may freely reference any definition or lemma contained in these tags without redefining them.
3. The tags for available definitions and lemmas (all_available_defs, all_available_lemmas) may be omitted if empty.

### Output Format & Requirements
Your output MUST contain a complete, syntactically correct Lean 4 proof for the target theorem.

**Strict Output Structure:**
<reasoning> 
...
</reasoning> 

<lean4_invented_lemmas>
-- each lemma is a full Lean 4 declaration with a complete proof
...
</lean4_invented_lemmas>

<lean4_proof>
-- Lean 4 proof body of the target theorem
...
</lean4_proof>

**Requirements:**
1. <reasoning> and <lean4_invented_lemmas> are OPTIONAL. Include them only if they aid proof construction.
2. <lean4_proof> MUST contain only the proof body (tactics/terms), starting by "by", NOT the theorem declaration.
3. <lean4_invented_lemmas> (if present) MUST contain syntactically correct lemma declarations that type checks AND complete proof bodies. Defining a lemma using `axiom` is NOT allowed.
4. <reasoning> contains high-level proof planning.
5. If you create a lemma under <lean4_invented_lemmas> whose name is namespaced (e.g., X.Y.Z), you MUST always refer to it using its fully qualified name. Do NOT rely on shortened names. For example, write rw [X.Y.Z], not rw [Z] for invented lemmas.
6. All Lean 4 code MUST be syntactically correct. No `sorry`, `admit`, or non-Lean text. Comments allowed using `--` or `/- ... -/`.
7. Follow the tags and structure EXACTLY. No extra text before, between, or after the sections.
8. Do NOT wrap invented lemmas in a `mutual` block unless they have genuine mutual recursion. Declare each lemma independently.

### Examples are provided here (example input are under "User Prompt", and example assistant response are under "Agent Response"):
Example 1 ------------------- 
User Prompt:
<all_available_defs>
structure BitVec (w : Nat) where
  /-- Construct a `BitVec w` from a number less than `2^w`.
  O(1), because we use `Fin` as the internal representation of a bitvector. -/
  ofFin ::

  /-- Interpret a bitvector as a number less than `2^w`.
  O(1), because we use `Fin` as the internal representation of a bitvector. -/
  toFin : Fin (hPow 2 w)
</all_available_defs>

<all_available_lemmas>
theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) : (l₁ ++ l₂)[i] = l₁[i]
theorem getElem_eq_testBit_toNat (x : BitVec w) (i : Nat) (h : i < w) :
  x[i] = x.toNat.testBit i := rfl
@[simp, grind =] theorem toNat_cons (b : Bool) (x : BitVec w) :
    (cons b x).toNat = (b.toNat <<< w) ||| x.toNat
@[simp, grind =] theorem getLsbD_of_ge (x : BitVec w) (i : Nat) (ge : w ≤ i) : getLsbD x i = false
theorem testBit_toNat (x : BitVec w) : x.toNat.testBit i = x.getLsbD i
</all_available_lemmas>

<local_ctx>
import Mathlib.Computability.NFA
import Mathlib.Data.FinEnum
import Mathlib.Data.Rel
import Mathlib.Data.Vector.Basic
import Blase.AutoStructs.ForLean

open Set
open Mathlib
open SetRel

-- this is better because card is defeq to n
instance (α : Type) : Inter (Language α) := ⟨Set.inter⟩
instance (α : Type) : Union (Language α) := ⟨Set.union⟩

/--
The set of `n`-tuples of bit vectors of an arbitrary width.
-/
structure BitVecs (n : Nat) where
  w : Nat
  bvs : List.Vector (BitVec w) n

def BitVecs.cast (bvs : BitVecs n) (h : n = n') : BitVecs n' :=
  { w := bvs.w, bvs := h ▸ bvs.bvs }

abbrev BitVecs.empty : BitVecs n := ⟨0, List.Vector.replicate n .nil⟩
abbrev BitVecs.singleton {w : Nat} (bv : BitVec w) : BitVecs 1 := ⟨w, bv ::ᵥ .nil⟩
abbrev BitVecs.pair {w : Nat} (bv1 bv2 : BitVec w) : BitVecs 2 := ⟨w, bv1 ::ᵥ bv2 ::ᵥ .nil⟩

/--
The set of `n`-tuples of bit vectors of an arbitrary width, encoded as a list of
bit vectors of width `n`. The width of the encoded bit vectors is the length of
the list.
-/
abbrev BitVecs' (n : Nat) := List (BitVec n)

@[simps]
def dec (bvs' : BitVecs' n) : BitVecs n where
  w := bvs'.length
  bvs := List.Vector.ofFn fun k => BitVec.ofFn fun i => bvs'[i].getLsbD k

@[simp]
</local_ctx>

<target_theorem>
lemma dec_snoc n (bvs' : BitVecs' n) (a : BitVec n) :
  dec (bvs' ++ [a]) =
    { w := bvs'.length + 1
      bvs := List.Vector.ofFn fun k =>
        BitVec.cons (a.getLsbD k) ((dec bvs').bvs.get k) } :=
</target_theorem>

Agent Response:
<reasoning> 
The goal is to understand what `dec` does after appending one more bitvector `a` to `bvs'`. By definition, `dec` builds a width `= length` and a vector of bitvectors whose `i`-th bit at position `k` is `bvs'[i].getLsbD k`. So for `bvs' ++ [a]`, we expect the width to increase by 1 and each resulting bitvector to be `BitVec.cons` of the new bit coming from `a`.

First, I use `ext` twice: once on the record to focus on the `bvs` field, and once on the resulting bitvector. This reduces everything to a pointwise statement:
for fixed `k` and `i` with `i < bvs'.length + 1`,
the bit `(bvs' ++ [a])[i].getLsbD k` should match the bit at index `i` of `BitVec.cons (a.getLsbD k) ((dec bvs').bvs.get k)`.

At this point I need a clean description of indexing into `BitVec.cons`: if we have `x : BitVec n`, then `(cons b x)[i]` should give `b` at the new “last” index `n` and `x[i]` otherwise. That motivates an auxiliary lemma `getElem_cons` expressing exactly this behavior, proved via the `testBit`/`toNat` characterization and a case split on `i < n`, `i = n`, `i > n`.

Using `getElem_cons`, the RHS becomes an `if` on whether `i = bvs'.length`. So I split into cases:
- If `i = bvs'.length`, then `(bvs' ++ [a])[i]` is the appended `a`, so both sides reduce to `a.getLsbD k`.
- If `i ≠ bvs'.length`, the bound `i < bvs'.length + 1` forces `i < bvs'.length`. Then `(bvs' ++ [a])[i] = bvs'[i]` by `List.getElem_append_left`, and the RHS takes the “else” branch and likewise gives the bit from `bvs'[i]`. In both subcases, `simp` (plus the definition of `dec`) finishes.

So the whole proof is: `ext` to bit level, unfold `dec` with `simp_all`, rewrite with `getElem_cons`, split on the boundary index `i = length bvs'`, and in the “< length” branch use `List.getElem_append_left` and bounds to line up both sides.
</reasoning> 

<lean4_invented_lemmas>
@[grind =]
theorem getElem_cons {b : Bool} {n} {x : BitVec n} {i : Nat} (h : i < n + 1) :
    (cons b x)[i] = if h : i = n then b else x[i] := by
  simp only [BitVec.getElem_eq_testBit_toNat, BitVec.toNat_cons, Nat.testBit_or]
  rw [Nat.testBit_shiftLeft]
  rcases Nat.lt_trichotomy i n with i_lt_n | i_eq_n | n_lt_i
  · have p1 : ¬(n ≤ i) := by omega
    have p2 : i ≠ n := by omega
    simp [p1, p2]
  · simp only [i_eq_n, ge_iff_le, Nat.le_refl, decide_true, Nat.sub_self, Nat.testBit_zero,
    Bool.true_and, BitVec.testBit_toNat, BitVec.getLsbD_of_ge, Bool.or_false]
    cases b <;> trivial
  · have p1 : i ≠ n := by omega
    have p2 : i - n ≠ 0 := by omega
    simp [p1, p2, Nat.testBit_bool_toNat]
</lean4_invented_lemmas>

<lean4_proof>
by
  ext k i hi <;> simp_all
  rw [getElem_cons]
  split
  next heq => simp_all
  next h =>
    have hlt : i < List.length bvs' := by simp at hi; omega
    rw [List.getElem_append_left hlt]
    simp
</lean4_proof>

### Your task:
<all_available_defs>
def liftRel (R : α → β → Prop) (A : α → Prop) (B : β → Prop) : Prop :=
  (∀ a, A a → ∃ b, B b ∧ R a b) ∧ (∀ b, B b → ∃ a, A a ∧ R a b)

class BI (PROP : Type _) extends COFE PROP, BI.BIBase PROP where
  Equiv P Q := P ⊣⊢ Q

entails_preorder : Preorder Entails
  equiv_iff {P Q : PROP} : (P ≡ Q) ↔ P ⊣⊢ Q := by admit /- proof elided -/

abbrev Relation (α : Type _) := α → α → Prop

class Reflexive (R : Relation α) where
  refl {x : α} : R x x

class Transitive (R : Relation α) where
  trans {x y z : α} : R x y → R y z → R x z

class Preorder (R : Relation α) extends Reflexive R, Transitive R

macro:max "iprop(" P:term " : " t:term ")" : term => `((iprop($P) : $t))

class NonExpansive [OFE α] [OFE β] (f : α → β) where
  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂

class NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where
  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂

@[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where
  [contractive : Contractive f]
  ne := ne_of_contractive f

instance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩

instance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :
    NonExpansive₂ (Option.merge op) where
  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/

structure Chain (α : Type _) [OFE α] where
  chain : Nat → α
  cauchy : n ≤ i → chain i ≡{n}≡ chain n

class IsCOFE (α : Type _) [OFE α] where
  compl : Chain α → α
  conv_compl {c : Chain α} : compl c ≡{n}≡ c n

instance OFE.ContractiveHom.fixpoint_ne [COFE α] [Inhabited α] :
    NonExpansive (ContractiveHom.fixpoint (α := α)) where
  ne n f1 f2 H := by admit /- proof elided -/

class BIAffine (PROP : Type _) [BI PROP] where
  affine (P : PROP) : Affine P

class Affine [BI PROP] (P : PROP) where
  affine : P ⊢ emp

class BIBase (PROP : Type u) where
  Entails : PROP → PROP → Prop
  emp : PROP
  pure : Prop → PROP
  and : PROP → PROP → PROP
  or : PROP → PROP → PROP
  imp : PROP → PROP → PROP
  sForall : (PROP → Prop) → PROP
  sExists : (PROP → Prop) → PROP
  sep : PROP → PROP → PROP
  wand : PROP → PROP → PROP
  persistently : PROP → PROP
  later : PROP → PROP

macro:25 P:term:29 " ⊢ " Q:term:25 : term => ``(BIBase.Entails iprop($P) iprop($Q))

delab_rule BIBase.Entails
  | `($_ $P $Q) => do ``($(← unpackIprop P) ⊢ $(← unpackIprop Q))

macro "∃" xs:explicitBinders ", " b:term : term => do
  return ⟨← expandExplicitBinders ``BIBase.exists xs b⟩

def wandIff [BIBase PROP] (P Q : PROP) : PROP := iprop((P -∗ Q) ∧ (Q -∗ P))

def affinely    [BIBase PROP] (P : PROP) : PROP := iprop(emp ∧ P)

def absorbingly [BIBase PROP] (P : PROP) : PROP := iprop(True ∗ P)

structure BiEntails [BIBase PROP] (P Q : PROP) where
  mp : P ⊢ Q
  mpr : Q ⊢ P

macro:25 P:term:29 " ⊣⊢ " Q:term:29 : term => ``(BiEntails iprop($P) iprop($Q))

delab_rule BIBase.Entails
  | `($_ iprop(emp) $P) => do ``(⊢ $(← unpackIprop P))

delab_rule BIBase.BiEntails
  | `($_ $P $Q) => do ``($(← unpackIprop P) ⊣⊢ $(← unpackIprop Q))

def intuitionistically [BIBase PROP] (P : PROP) : PROP := iprop(<affine> <pers> P)

def persistentlyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <pers> P else P)

def affinelyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <affine> P else P)

def absorbinglyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <absorb> P else P)

def intuitionisticallyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then □ P else P)

def bigAnd [BIBase PROP] (Ps : List PROP) : PROP := bigOp and iprop(True) Ps

def bigOr [BIBase PROP] (Ps : List PROP) : PROP := bigOp or iprop(False) Ps

def bigSep [BIBase PROP] (Ps : List PROP) : PROP := bigOp sep iprop(emp) Ps

def bigOp (f : PROP → PROP → PROP) (unit : PROP) : List PROP → PROP
  | []      => unit
  | [P]     => P
  | P :: Ps => f P (bigOp f unit Ps)

class LawfulBigOp (f : PROP → PROP → PROP) (unit : outParam PROP)
    (eq : outParam (PROP → PROP → Prop)) where
  refl : eq a a
  symm : eq a b → eq b a
  trans : eq a b → eq b c → eq a c
  comm : eq (f a b) (f b a)
  assoc : eq (f (f a b) c) (f a (f b c))
  left_id : eq (f unit a) a
  congr_l : eq a a' → eq (f a b) (f a' b)

instance COFE.OFunctor.constOF_RFunctor [CMRA B] : RFunctor (constOF B) where
  map f g := by admit /- proof elided -/

class UFraction (α : Type _) extends Fraction α, One α where
  
  
  
  
  one_whole : Fraction.Whole (1 : α)

@[simp] instance [NumericFraction T] : One (Frac T) := ⟨⟨One.one⟩⟩

def upN {k} : ∀ n, A F k -n> A F (k + n)
  | 0 => .id
  | n+1 => (up F (k + n)).comp (upN n)

instance fromExists_affinely [BI PROP] (P : PROP) (Φ : α → PROP) [h : FromExists P Φ] :
    FromExists iprop(<affine> P) (fun a => iprop(<affine> (Φ a))) where
  from_exists := affinely_exists.2.trans <| affinely_mono h.1

protected def Hom.comp [OFE α] [OFE β] [OFE γ] (g : β -n> γ) (f : α -n> β) : α -n> γ where
  f := g.f ∘ f.f
  ne.1 _ _ _ h := g.ne.1 (f.ne.1 h)

instance intoSep_pure (φ ψ : Prop) [BI PROP] :
    IntoSep (PROP := PROP) iprop(⌜φ ∧ ψ⌝) iprop(⌜φ⌝) iprop(⌜ψ⌝) where
  into_sep := pure_and.2.trans persistent_and_sep_1

f₂ in QqTest.commandTest

Lean.Elab.withInfoContext' in Lean.Elab.InfoTree.Main

def Set.univ {α : Type _} : Set α := fun _ => True

instance fromPure_affinely_true (a : Bool) [BI PROP] (P : PROP) (φ : Prop)
    [h : FromPure a P φ] : FromPure true iprop(<affine> P) φ where
  from_pure := affinely_idem.2.trans <| affinely_mono <| affinely_affinelyIf.trans h.1

instance intoPure_affinely [BI PROP] (P : PROP) (φ : Prop)
    [h : IntoPure P φ] : IntoPure iprop(<affine> P) φ where
  into_pure := affinely_elim.trans h.1

semiOutParam in Init.Prelude

instance emp_affine [BI PROP] : Affine (PROP := PROP) iprop(emp) where
  affine := .rfl

instance {P : PROP} [Absorbing P] : Absorbing iprop(|==> P) :=
  ⟨bupd_frame_l.trans <| mono sep_elim_r⟩

Qq.inferTypeQ in Qq.MetaM

let m1 : Q($el ⊢ $Q1) ← mkFreshExprSyntheticOpaqueMVar <|
    IrisGoal.toExpr { u, prop, bi, hyps := lhs, goal := Q1, .. }
  let m2 : Q($er ⊢ $Q2) ← mkFreshExprSyntheticOpaqueMVar <|
    IrisGoal.toExpr { u, prop, bi, hyps := rhs, goal := Q2, .. }
  mvar.assign q(sep_split (Q := $goal) $pf $m1 $m2)
  replaceMainGoal [m1.mvarId!, m2.mvarId!]

Or.inl in Init.Prelude

instance : CMRA.Exclusive (own (1 : F)) :=
  own_whole_exclusive <| UFraction.one_whole

instance emp_intuitionistic [BI PROP] : Intuitionistic iprop(emp : PROP) where
  intuitionistic := intuitionistically_emp.2

def headName (spats : List SpecPat) : Name :=
  match spats.head? with
    | some <| .ident _ => .anonymous
    | some <| .idents _ name => name
    | _ => .anonymous

instance bupd_ne : OFE.NonExpansive (bupd : UPred M → UPred M) where
  ne n x1 x2 Hx m y Hm Hv := by admit /- proof elided -/

LT in Init.Prelude

Add.add in Init.Prelude

instance : Leibniz (Frac α) := inferInstanceAs (Leibniz (LeibnizO α))

class FromAnd [BI PROP] (P : PROP) (Q1 Q2 : outParam PROP) where
  from_and : Q1 ∧ Q2 ⊢ P

@[simp] def Heap.union [Heap T K V] : T → T → T := merge (fun v _ => v)

Inhabited in Init.Prelude

@[simp]
def intuitionistic_action_spec (M : PROP1 → PROP2) : Prop :=
  match s with
  | .isEmpty => True
  | .forall Hconv C =>
      (∀ P, C P → iprop(□ P) ⊢ Hconv ▸ M iprop(□ P)) ∧
      (∀ P Q, iprop(M P ∧ M Q) ⊢ M iprop(P ∧ Q))
  | .transform C =>
      (∀ P Q, C P Q → iprop(□ P) ⊢ M iprop(□ Q)) ∧
      (∀ P Q, iprop(M P ∧ M Q) ⊢ M iprop(P ∧ Q))
  | .clear => True
  | .id H => ∀ P, iprop(□ P) ⊢ M (H ▸ iprop(□ P))

instance asEmpValid1_equiv [BI PROP] (P Q : PROP) : AsEmpValid1 (P ⊣⊢ Q) iprop(P ∗-∗ Q) where
  as_emp_valid := ⟨equiv_wandIff, wandIff_equiv⟩

instance [BI PROP] [BIAffine PROP] : LeftId (α := PROP) BiEntails emp and := ⟨emp_and⟩

def const [OFE α] (a : α) : Chain α where
  chain := fun _ => a
  cauchy _ := OFE.Dist.rfl

instance persistently_absorbing [BI PROP] (P : PROP) : Absorbing iprop(<pers> P) where
  absorbing := absorbingly_persistently.1

instance : OFE (UPred M) where
  Equiv P Q := ∀ n x, ✓{n} x → (P n x ↔ Q n x)
  Dist n P Q := ∀ n' x, n' ≤ n → ✓{n'} x → (P n' x ↔ Q n' x)
  dist_eqv := {
    refl _ _ _ _ _ := .rfl
    symm H _ _ A B := (H _ _ A B).symm
    trans H1 H2 _ _ A B := (H1 _ _ A B).trans (H2 _ _ A B) }
  equiv_dist := ⟨
    fun Heqv _ _ _ _ Hvalid => Heqv _ _ Hvalid,
    fun Hdist _ _ Hvalid => Hdist _ _ _ (Nat.le_refl _) Hvalid⟩
  dist_lt Hdist Hlt _ _ Hle Hvalid :=
    Hdist _ _ (Nat.le_trans Hle (Nat.le_of_succ_le Hlt)) Hvalid

def addLocalVarInfo (stx : Syntax) (lctx : LocalContext)
    (expr : Expr) (expectedType? : Option Expr) (isBinder := false) : MetaM Unit := do
  Elab.withInfoContext' (pure ())
    (fun _ =>
      return .inl <| .ofTermInfo
        { elaborator := .anonymous, lctx, expr, stx, expectedType?, isBinder })
    (return .ofPartialTermInfo { elaborator := .anonymous, lctx, stx, expectedType? })

instance (priority := default + 10) fromSep_absorbingly [BI PROP] (P Q1 Q2 : PROP)
    [h : FromSep P Q1 Q2] : FromSep iprop(<absorb> P) iprop(<absorb> Q1) iprop(<absorb> Q2) where
  from_sep := absorbingly_sep.2.trans (absorbingly_mono h.1)

f₁ in QqTest.commandTest

Add in Init.Prelude

Lean.MonadRef in Init.Prelude

def Agree.valid (x : Agree α) : Prop := ∀ n, x.validN n

partial def iIntroCore
    {P} (hyps : Hyps bi P) (Q : Q($prop)) (pats : List iCasesPat)
    (k : ∀ {P}, Hyps bi P → (Q : Q($prop)) → MetaM Q($P ⊢ $Q)) :
    MetaM (Q($P ⊢ $Q)) := do
  match pats with
  | [] => k hyps Q
  | pat :: pats =>
    let A1 ← mkFreshExprMVarQ q($prop)
    let A2 ← mkFreshExprMVarQ q($prop)
    let fromImp ← try? (α := Q(FromImp $Q $A1 $A2)) do
      synthInstanceQ q(FromImp $Q $A1 $A2)
    if let (.clear, some _) := (pat, fromImp) then
      let pf ← iIntroCore hyps A2 pats k
      return q(imp_intro_drop (Q := $Q) $pf)
    else
    let alres ← try? (α := _ × (v : Level) × (α : Q(Sort v)) × (Φ : Q($α → $prop)) ×
        Q(FromForall $Q $Φ)) do
      let .one n := pat | failure
      let v ← mkFreshLevelMVar
      let α ← mkFreshExprMVarQ q(Sort v)
      let Φ ← mkFreshExprMVarQ q($α → $prop)
      Pure.pure ⟨n, v, α, Φ, ← synthInstanceQ q(FromForall $Q $Φ)⟩
    if let some ⟨n, _, α, Φ, _⟩ := alres then
      let (n, ref) ← getFreshName n
      withLocalDeclDQ n α fun x => do
        addLocalVarInfo ref (← getLCtx) x α
        have B : Q($prop) := Expr.headBeta q($Φ $x)
        have : $B =Q $Φ $x := ⟨⟩
        let pf : Q(∀ x, $P ⊢ $Φ x) ← mkLambdaFVars #[x] <|← iIntroCore hyps B pats k
        return q(from_forall_intro (Q := $Q) $pf)
    else
    let B ← mkFreshExprMVarQ q($prop)
    match pat, fromImp with
    | .intuitionistic pat, some _ =>
      let _ ← synthInstanceQ q(IntoPersistently false $A1 $B)
      let pf ← iCasesCore bi hyps A2 q(true) q(iprop(□ $B)) B ⟨⟩ pat (iIntroCore · A2 pats k)
      return q(imp_intro_intuitionistic (Q := $Q) $pf)
    | .intuitionistic pat, none =>
      let _ ← synthInstanceQ q(FromWand $Q $A1 $A2)
      let _ ← synthInstanceQ q(IntoPersistently false $A1 $B)
      let _ ← synthInstanceQ q(TCOr (Affine $A1) (Absorbing $A2))
      let pf ← iCasesCore bi hyps A2 q(true) q(iprop(□ $B)) B ⟨⟩ pat (iIntroCore · A2 pats k)
      return q(wand_intro_intuitionistic (Q := $Q) $pf)
    | _, some _ =>
      let _ ← synthInstanceQ q(FromAffinely $B $A1)
      let _ ← synthInstanceQ q(TCOr (Persistent $A1) (Intuitionistic $P))
      let pf ← iCasesCore bi hyps A2 q(false) B B ⟨⟩ pat (iIntroCore · A2 pats k)
      return q(imp_intro_spatial (Q := $Q) $pf)
    | _, none =>
      let _ ← synthInstanceQ q(FromWand $Q $A1 $A2)
      let pf ← iCasesCore bi hyps A2 q(false) A1 A1 ⟨⟩ pat (iIntroCore · A2 pats k)
      return q(wand_intro_spatial (Q := $Q) $pf)

protected def Tower.embed (k) : A F k -n> Tower F :=

let ⟨el, er, lhs, rhs, pf⟩ := hyps.split bi (fun _ uniq => uniqs.contains uniq == splitRight)

| .disjunction (arg :: args) =>
    iCasesOr bi P Q A' p
      (iCasesCore hyps Q p · · · arg @k)
      (iCasesCore hyps Q p · · · (.disjunction args) @k)

instance or_absorbing [BI PROP] (P Q : PROP) [Absorbing P] [Absorbing Q] :
    Absorbing iprop(P ∨ Q) where
  absorbing := absorbingly_or.1.trans (or_mono absorbing absorbing)

def addHypInfo (stx : Syntax) (name uniq : Name) (prop : Q(Type u)) (ty : Q($prop))
    (isBinder := false) : MetaM Unit := do
  let lctx ← getLCtx
  let ty := q(HypMarker $ty)
  addLocalVarInfo stx (lctx.mkLocalDecl ⟨uniq⟩ name ty) (.fvar ⟨uniq⟩) ty isBinder

instance [BI PROP] [BIAffine PROP] : BIPositive PROP where
  affinely_sep_l := (affine_affinely _).1.trans (sep_mono_l (affine_affinely _).2)

IO in Init.System.IO

instance cancelable_op {x y : α} [Cancelable x] [Cancelable y] : Cancelable (x • y) where
  cancelableN {n w _} v e :=
    have v1 : ✓{n} x • (y • w) := validN_ne op_assocN.symm v
    have v2 := validN_op_right v1
    cancelableN v2 <| cancelableN v1 <| op_assocN.trans <| e.trans op_assocN.symm

return rule?.isNone
    else
      
      applyReflexivity goal
      return false

def LocalUpdate [CMRA α] (x y : α × α) : Prop :=
  ∀n mz, ✓{n} x.1 → x.1 ≡{n}≡ x.2 •? mz → ✓{n} y.1 ∧ y.1 ≡{n}≡ y.2 •? mz

protected def emp : UPred M where
  holds _ _ := True
  mono _ _ _ := trivial

EmptyCollection.emptyCollection in Init.Core

instance intoAnd_intuitionistically (p : Bool) [BI PROP] (P Q1 Q2 : PROP) [h : IntoAnd p P Q1 Q2] :
    IntoAnd p iprop(□ P) iprop(□ Q1) iprop(□ Q2) where
  into_and := (intuitionisticallyIf_comm' (q := true)).1.trans <|
      (intuitionistically_mono h.1).trans <| (intuitionisticallyIf_comm' (q := true)).2.trans <|
      intuitionisticallyIf_mono intuitionistically_and.1

instance cmraUnit : CMRA Unit where
  pcore _ := some ()
  op _ _ := ()
  ValidN _ _ := True
  Valid _ := True
  op_ne.ne _ _ _ := id
  pcore_ne _ _ := ⟨(), rfl, .rfl⟩
  validN_ne _ := id
  valid_iff_validN := ⟨fun _ _ => ⟨⟩, fun _ => ⟨⟩⟩
  validN_succ := id
  validN_op_left := id
  assoc := .rfl
  comm := .rfl
  pcore_op_left _ := .rfl
  pcore_idem _ := .rfl
  pcore_op_mono _ _ := ⟨.unit, .rfl⟩
  extend _ _ := ⟨(), (), .rfl, .rfl, .rfl⟩

instance fromPure_persistently [BI PROP] (P : PROP) (a : Bool) (φ : Prop)
    [h : FromPure true P φ] : FromPure a iprop(<pers> P) φ where
  from_pure := affinelyIf_elim.trans <| persistently_pure.2.trans <|
    persistently_affinely.2.trans <| persistently_mono h.1

def HypMarker {PROP : Type _} (_A : PROP) : Prop := True

private def matchAlts' := leading_parser matchAlts

instance : BI (UPred M) where
  entails_preorder := inferInstance
  equiv_iff {P Q} := by admit /- proof elided -/

@[simp] def Excl.map (f : α → β) : Excl α → Excl β
  | excl a => excl (f a)
  | invalid => invalid

instance intoPure_pure_and (φ1 φ2 : Prop) [BI PROP] (P1 P2 : PROP)
    [h1 : IntoPure P1 φ1] [h2 : IntoPure P2 φ2] : IntoPure iprop(P1 ∧ P2) (φ1 ∧ φ2) where
  into_pure := (and_mono h1.1 h2.1).trans pure_and.1

def appM? (e : Expr) (fName : Name) : Option <| Array Expr :=
  if e.isAppOf fName then
    e.getAppArgs
  else
    none

instance [u : U] : TCIte (no_index false) T U := TCIte.e (u := u)

class HasHeapMap (T1 T2 : Type _) (K V1 V2 : outParam (Type _))
   [Store T1 K (Option V1)] [Store T2 K (Option V2)] where
  hhmap (f : K → V1 → Option V2) : T1 → T2
  hhmap_get (f : K → V1 → Option V2) : get (hhmap f t) k = (get t k).bind (f k)

instance (priority := default - 10) fromSep_sep [BI PROP] (P1 P2 : PROP) :
    FromSep iprop(P1 ∗ P2) P1 P2 := ⟨.rfl⟩

instance (priority := default - 10) intoAbsorbingly_default [BI PROP] (P : PROP) :
    IntoAbsorbingly iprop(<absorb> P) P := ⟨.rfl⟩

instance fromExists_intuitionistically [BI PROP] (P : PROP) (Φ : α → PROP) [h : FromExists P Φ] :
    FromExists iprop(□ P) (fun a => iprop(□ (Φ a))) where
  from_exists := intuitionistically_exists.2.trans <| intuitionistically_mono h.1

protected def sep (P Q : UPred M) : UPred M where
  holds n x := ∃ x1 x2, x ≡{n}≡ x1 • x2 ∧ P n x1 ∧ Q n x2
  mono {n₁ n₂ m₁ m₂} := fun ⟨x₁, x₂, Hx, HP, HQ⟩ ⟨m, Hm⟩ Hn => by admit /- proof elided -/

instance fromOr_or [BI PROP] (P1 P2 : PROP) : FromOr iprop(P1 ∨ P2) P1 P2 := ⟨.rfl⟩

def Replaces [BI PROP] (K A B : PROP) := (B -∗ K) ⊢ (A -∗ K)

instance isOFE_later [OFE A] : OFE (Later A) where
  Equiv x y := x.car ≡ y.car
  Dist n x y := DistLater n x.car y.car
  dist_eqv := ⟨fun _ => .rfl, .symm, .trans⟩
  equiv_dist := by admit /- proof elided -/

instance {f : F} : CMRA.Cancelable (own f) where
  cancelableN {_} := by admit /- proof elided -/

@[ext] structure Tower : Type u where
  val k : A F k
  protected down {k} : down F k (val (k+1)) ≡ val k

instance pure_persistent (φ : Prop) [BI PROP] : Persistent (PROP := PROP) iprop(⌜φ⌝) where
  persistent := persistently_pure.2

Function.uncurry in Init.Data.Function

instance [BI PROP] : RightId (α := PROP) BiEntails emp sep := ⟨sep_emp⟩

macro "#" pat:icasesPat : icasesPat => `(icasesPat| □ $pat)

instance entails_trans [BI PROP] : Trans (α := PROP) Entails Entails Entails where
  trans h1 h2 := h1.trans h2

abbrev LaterOF (F : OFunctorPre) : OFunctorPre :=
  fun A B _ _ => Later (F A B)

@[simp] def Excl.Valid : Excl α → Prop
  | excl _ => True
  | invalid => False

class Discrete (α : Type _) [CMRA α] extends OFE.Discrete α where
  discrete_valid {x : α} : ✓{0} x → ✓ x

def Agree.map : CMRA.Hom (Agree α) (Agree β) where
  f := map' f
  ne := inferInstance
  validN {n x} hval := by admit /- proof elided -/

abbrev GenMap := α → Option β

class Fraction (α : Type _) extends Add α where
   
  Proper : α → Prop
  add_comm : ∀ a b : α, a + b = b + a
  add_assoc : ∀ a b c : α, a + (b + c) = (a + b) + c
  add_left_cancel : ∀ {a b c : α}, a + b = a + c → b = c
   
  add_ne : ∀ {a b : α}, a ≠ b + a
  proper_add_mono_left : ∀ {a b : α}, Proper (a + b) → Proper a

let Q1 ← mkFreshExprMVarQ prop
  let Q2 ← mkFreshExprMVarQ prop
  let _ ← synthInstanceQ q(FromSep $goal $Q1 $Q2)

assoc : op x (op y z) ≡ op (op x y) z
  comm : op x y ≡ op y x

Qq.mkFreshExprMVarQ in Qq.MetaM

def iCasesSep {prop : Q(Type u)} (bi : Q(BI $prop))
    {P} (hyps : Hyps bi P) (Q A' A1 A2 : Q($prop)) (p : Q(Bool))
    (inst : Option Q(IntoAnd $p $A' $A1 $A2))
    (k : ∀ {P}, Hyps bi P → MetaM Q($P ⊢ $Q))
    (k1 k2 : ∀ {P}, Hyps bi P → (Q B B' : Q($prop)) → (_ : $B =Q iprop(□?$p $B')) →
      (∀ {P}, Hyps bi P → MetaM Q($P ⊢ $Q)) → MetaM Q($P ∗ $B ⊢ $Q)) :
    MetaM (Q($P ∗ □?$p $A' ⊢ $Q)) := do
  if p.constName! == ``true then
    let some _ := inst | _ ← synthInstanceQ q(IntoAnd $p $A' $A1 $A2); unreachable!
    have : $p =Q true := ⟨⟩
    let Q' := q(iprop(□ $A2 -∗ $Q))
    let pf ← k1 hyps Q' q(iprop(□ $A1)) A1 ⟨⟩ fun hyps => do
      let pf ← k2 hyps Q q(iprop(□ $A2)) A2 ⟨⟩ k
      return q(wand_intro $pf)
    return q(and_elim_intuitionistic (A := $A') $pf)
  else
    let _ ← synthInstanceQ q(IntoSep $A' $A1 $A2)
    have : $p =Q false := ⟨⟩
    let Q' := q(iprop($A2 -∗ $Q))
    let pf ← k1 hyps Q' A1 A1 ⟨⟩ fun hyps => do
      let pf ← k2 hyps Q A2 A2 ⟨⟩ k
      return q(wand_intro $pf)
    return q(sep_elim_spatial (A := $A') $pf)

instance : Inhabited (Tower F) := ⟨Tower.embed 0 ⟨()⟩⟩

instance intoPure_pure_sep (φ1 φ2 : Prop) [BI PROP] (P1 P2 : PROP)
    [h1 : IntoPure P1 φ1] [h2 : IntoPure P2 φ2] : IntoPure iprop(P1 ∗ P2) (φ1 ∧ φ2) where
  into_pure := (sep_mono h1.1 h2.1).trans <| sep_and.trans pure_and.1

instance {ma : Option α} [∀ a : α, IdFree a] [∀ a : α, Cancelable a] : Cancelable ma :=

instance : OFE.NonExpansive (BUpd.bupd (PROP := UPred M)) := bupd_ne

partial def handleDependentArrows {prop : Q(Type u)} (bi : Q(BI $prop)) (val : Expr) (goals : IO.Ref (Array MVarId)) : TacticM (Expr × Q(Prop)) := do
  let p : Q(Prop) ← inferType val
  if let .forallE _ binderType _ _ := p then
    let m ← mkFreshExprMVar binderType
    let val' := mkApp val m
    goals.modify (·.push m.mvarId!)
    return ← handleDependentArrows bi val' goals
  else
    return (val, p)

class Idempotent (R : Relation α) (f : α → α → α) where
  idem {x : α} : R (f x x) x

instance exclusive_cancelable {x : α} [Exclusive x] : Cancelable x where
  cancelableN v _ := absurd v not_valid_exclN_op_left

CoeFun in Init.Coe

def uliftUpHom [OFE α] : α -n> ULift α where
  f := .up
  ne.1 _ _ _ := id

instance fromPure_exists (a : Bool) [BI PROP] (Φ : α → PROP) (φ : α → Prop)
    [h : ∀ x, FromPure a iprop(Φ x) (φ x)] : FromPure a iprop(∃ x, Φ x) (∃ x, φ x) where
  from_pure := (affinelyIf_mono pure_exists.2).trans <|
    affinelyIf_exists.1.trans (exists_mono fun x => (h x).1)

Lean.Parser.Tactic.exact in Init.Tactics

def goalTracker {P} (goals : IO.Ref (Array MVarId)) (name : Name) (hyps : Hyps bi P)
    (goal : Q($prop)) : MetaM Q($P ⊢ $goal) := do
  let m : Q($P ⊢ $goal) ← mkFreshExprSyntheticOpaqueMVar <|
    IrisGoal.toExpr { prop, bi, hyps, goal, .. }
  m.mvarId!.setUserName name
  goals.modify (·.push m.mvarId!)
  pure m

@[ext] structure Hom (α β : Type _) [CMRA α] [CMRA β] extends OFE.Hom α β where
  protected validN {n x} : ✓{n} x → ✓{n} (f x)
  protected pcore x : (pcore x).map f ≡ pcore (f x)
  protected op x y : f (x • y) ≡ f x • f y

instance forall_persistent [BI PROP] [BIPersistentlyForall PROP] (Ψ : α → PROP)
    [h : ∀ x, Persistent (Ψ x)] : Persistent iprop(∀ x, Ψ x) :=
  sForall_persistent _ fun _ ⟨_, eq⟩ => eq ▸ h _

instance urFunctorContractiveOptionOF
    [RFunctorContractive F] : URFunctorContractive (OptionOF F) where
  map_contractive.1 := COFE.OFunctorContractive.map_contractive.1

extend : ValidN n x → x ≡{n}≡ op y₁ y₂ →
    Σ' z₁ z₂, x ≡ op z₁ z₂ ∧ z₁ ≡{n}≡ y₁ ∧ z₂ ≡{n}≡ y₂

instance intoOr_absorbingly [BI PROP] (P Q1 Q2 : PROP) [h : IntoOr P Q1 Q2] :
    IntoOr iprop(<absorb> P) iprop(<absorb> Q1) iprop(<absorb> Q2) where
  into_or := (absorbingly_mono h.1).trans absorbingly_or.1

class RFunctorContractive (F : COFE.OFunctorPre) extends (RFunctor F) where
  map_contractive [OFE α₁] [OFE α₂] [OFE β₁] [OFE β₂] :
    Contractive (Function.uncurry (@map α₁ α₂ β₁ β₂ _ _ _ _))

instance (priority := default + 30) fromAssumption_affinely_r [BI PROP] (P Q : PROP)
    [h : FromAssumption true P Q] : FromAssumption true P iprop(<affine> Q) where
  from_assumption := affinely_idem.2.trans <| affinely_mono h.1

instance urFunctorOptionOF [RFunctor F] : URFunctor (OptionOF F) where
  cmra {α β} := ucmraOption
  map f g := {
    toHom := COFE.OFunctor.map f g
    validN := by admit /- proof elided -/

instance : BIBase (HeapProp Val) where
  Entails P Q      := ∀ σ, P σ → Q σ
  emp            σ := σ = ∅
  pure φ         _ := φ
  and P Q        σ := P σ ∧ Q σ
  or P Q         σ := P σ ∨ Q σ
  imp P Q        σ := P σ → Q σ
  sForall Ψ      σ := ∀ p, Ψ p → p σ
  sExists Ψ      σ := ∃ p, Ψ p ∧ p σ
  sep P Q        σ := ∃ σ1 σ2, σ = σ1 ∪ σ2 ∧ σ1 || σ2 ∧ P σ1 ∧ Q σ2
  wand P Q       σ := ∀ σ', σ || σ' → P σ' → Q (σ ∪ σ')
  persistently P _ := P ∅
  later P        σ := P σ

instance OFunctor.constOF_RFunctorContractive [CMRA B] :
    RFunctorContractive (COFE.constOF B) where
  map_contractive.1 := by admit /- proof elided -/

instance : BIUpdate (UPred M) where
  intro {P} _ x _ HP _ _ Hn H := ⟨_, ⟨H, P.mono HP (CMRA.incN_refl x) Hn⟩⟩
  mono Himp n x Hx HP k yf Hn H := by admit /- proof elided -/

def Agree.op (x y : Agree α) : Agree α :=
  ⟨x.car ++ y.car, by admit /- proof elided -/
  ⟩

instance asEmpValid2_equiv [BI PROP] (P Q : PROP) : AsEmpValid2 (P ⊣⊢ Q) iprop(P ∗-∗ Q) :=
  AsEmpValid1.to2

instance [OFE α] [OFE β] [Discrete α] [Discrete β] : Discrete (α × β) where
  discrete_0 H := by admit /- proof elided -/

def union (x y : Set α) : Set α := fun a => x a ∨ y a

def RFunctor.ap [RFunctor F] [OFE T] := F T T

partial def Hyps.find? {u prop bi} (name : Name) :
    ∀ {s}, @Hyps u prop bi s → Option (Name × Q(Bool) × Q($prop))
  | _, .emp _ => none
  | _, .hyp _ name' uniq p ty _ => if name == name' then (uniq, p, ty) else none
  | _, .sep _ _ _ _ lhs rhs => rhs.find? name <|> lhs.find? name

Lean.MonadQuotation in Init.Prelude

return ⟨← `(irisGoalStx| $hyps.reverse* ⊢ $goal:term)⟩
where
  delabHypotheses {u prop bi s} (hyps : @Hyps u prop bi s)
      (acc : NameMap Nat × Array (TSyntax ``irisHyp)) :
      DelabM (NameMap Nat × Array (TSyntax ``irisHyp)) := do
    match hyps with
    | .emp _ => pure acc
    | .hyp _ name _ p ty _ =>
      let mut (map, acc) := acc
      let (idx, name') ← if let some idx := map.find? name then
        pure (idx + 1, name.appendAfter <| if idx == 0 then "✝" else "✝" ++ idx.toSuperscriptString)
      else
        pure (0, name)
      let stx ← if isTrue p then
        `(irisHyp| □$(mkIdent name') : $(← unpackIprop (← delab ty)))
      else
        `(irisHyp| ∗$(mkIdent name') : $(← unpackIprop (← delab ty)))
      pure (map.insert name idx, acc.push stx)
    | .sep _ _ _ _ lhs rhs => delabHypotheses lhs (← delabHypotheses rhs acc)

instance : Inhabited (DFrac F) := ⟨discard⟩

Eq in Init.Prelude

@[ext] structure Iso (α β : Type _) [OFE α] [OFE β] where
  hom : α -n> β
  inv : β -n> α
  hom_inv : hom (inv x) ≡ x
  inv_hom : inv (hom x) ≡ x

instance Heap.instCOFE [Heap T K V] [COFE V] : COFE T where
  compl c := hmap (fun _ => COFE.compl <| c.map ⟨_, Store.get_ne ·⟩) (c 0)
  conv_compl {_ c} k := by admit /- proof elided -/

instance : BIAffine (UPred M) := ⟨by admit /- proof elided -/
⟩

MonadExcept in Init.Prelude

instance fromOr_intuitionistically [BI PROP] (P Q1 Q2 : PROP) [h : FromOr P Q1 Q2] :
    FromOr iprop(□ P) iprop(□ Q1) iprop(□ Q2) where
  from_or := intuitionistically_or.2.trans (intuitionistically_mono h.1)

instance intoSep_and_persistent_l [BI PROP] (P Q P' Q' : PROP) [Persistent P]
    [inst : AndIntoSep P P' Q Q'] : IntoSep iprop(P ∧ Q) P' Q' where
  into_sep :=
    match P', inst with
    | _, AndIntoSep.affine (h := h) .. =>
      (and_mono_l (affine_affinely _).2).trans <| affinely_and_lr.1.trans <|
        persistent_and_affinely_sep_l_1.trans (sep_mono (affine_affinely _).1 h.1)
    | _, AndIntoSep.affinely .. => persistent_and_affinely_sep_l_1

def Agree.valid (x : Agree α) : Prop := ∀ n, x.validN n

class IsSplitFraction (α : Type _) [Fraction α] where
  split : α → α × α
  split_add {a : α} : (split a).1 + (split a).2 = a

def setMDataName? (e : Expr) (name : Name) : Expr := match e with
  | Expr.mdata md e =>
    mkMData (md.insert `name name) e
  | e =>
    mkMData (KVMap.empty.insert `name name) e

instance exclusive_idFree {x : α} [Exclusive x] : IdFree x where
  id_free0_r z v h := exclusive0_l z ((validN_iff h.symm).mp v)

instance : CoeFun (Tower F) (fun _ => ∀ k, A F k) := ⟨Tower.val⟩

def findGoalFromTag? (tag : Name) : TacticM <| Option MVarId := do
  (← getUnsolvedGoals).findM? fun goal => do return (← goal.getTag) == tag

def iCasesExists {prop : Q(Type u)} (_bi : Q(BI $prop)) (P Q A' : Q($prop)) (p : Q(Bool))
    (name : TSyntax ``binderIdent) (α : Q(Sort v)) (Φ : Q(«$α» → «$prop»))
    (_inst : Q(IntoExists «$A'» «$Φ»))
    (k : (B B' : Q($prop)) → (_ : $B =Q iprop(□?$p $B')) → MetaM Q($P ∗ $B ⊢ $Q)) :
    MetaM (Q($P ∗ □?$p $A' ⊢ $Q)) := do
  let (name, ref) ← getFreshName name
  withLocalDeclDQ name α fun x => do
    addLocalVarInfo ref (← getLCtx) x α
    have B' : Q($prop) := Expr.headBeta q($Φ $x)
    have : $B' =Q $Φ $x := ⟨⟩
    if p.constName! == ``true then
      have : $p =Q true := ⟨⟩
      let pf : Q(∀ x, $P ∗ □ $Φ x ⊢ $Q) ← mkLambdaFVars #[x] <|← k q(iprop(□ $B')) B' ⟨⟩
      return q(exists_elim_intuitionistic (A := $A') $pf)
    else
      have : $p =Q false := ⟨⟩
      let pf : Q(∀ x, $P ∗ $Φ x ⊢ $Q) ← mkLambdaFVars #[x] <|← k B' B' ⟨⟩
      return q(exists_elim_spatial (A := $A') $pf)

def iPrePropO : Type _ := OFunctor.Fix (UPredOF (IResF FF))

abbrev γ : GId FF1 := ⟨0, Nat.zero_lt_succ _⟩
</all_available_defs>

<all_available_lemmas>
theorem BIBase.Entails.trans [BI PROP] {P Q R : PROP} (h1 : P ⊢ Q) (h2 : Q ⊢ R) : P ⊢ R

theorem BIBase.BiEntails.trans [BI PROP] {P Q R : PROP} (h1 : P ⊣⊢ Q) (h2 : Q ⊣⊢ R) : P ⊣⊢ R

@[simp, refl] theorem Dist.rfl [OFE α] {n} {x : α} : x ≡{n}≡ x

@[symm] theorem Dist.symm [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ x

theorem Dist.trans [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ z → x ≡{n}≡ z

theorem equiv_eqv [ofe : OFE α] : Equivalence ofe.Equiv

theorem Equiv.trans [OFE α] {x : α} : x ≡ y → y ≡ z → x ≡ z

theorem DistLater.trans [OFE α] {n} {x : α} (h1 : DistLater n x y) (h2 : DistLater n y z) :
    DistLater n x z

theorem split_ss [BI PROP] {P Q P1 P2 Q1 Q2 : PROP}
    (h1 : P ⊣⊢ P1 ∗ P2) (h2 : Q ⊣⊢ Q1 ∗ Q2) : P ∗ Q ⊣⊢ (P1 ∗ Q1) ∗ (P2 ∗ Q2)

theorem sep_and_elim_l [BI PROP] {P A Q A1 A2 : PROP} [inst : IntoAnd p A A1 A2]
    (h : P ∗ □?p A1 ⊢ Q) : P ∗ □?p A ⊢ Q

theorem set_op_eq [IsoFunStore T K (Option V)] {m1 m2 : T} :
    (set (V := Option V) (m1 • m2) i (x • y)) = (set m1 i x • set m2 i y)

theorem NonExpansive.eqv [OFE α] [OFE β] {f : α → β} [NonExpansive f]
    ⦃x₁ x₂⦄ (h : x₁ ≡ x₂) : f x₁ ≡ f x₂

theorem with_proof [BI PROP] (H : A → B) (P Q : PROP) : <affine> P ⊢ <pers> Q → ⌜A⌝ → ⌜B⌝

theorem intuitionistically_intro' [BI PROP] {P Q : PROP} (h : □ P ⊢ Q) : □ P ⊢ □ Q

theorem Agree.toAgree_injN {a b : α} : toAgree a ≡{n}≡ toAgree b → a ≡{n}≡ b

theorem core_point_equiv {i : K} {x : V} {cx} (Hpcore : pcore x = some cx) :
    Equiv (T := T) (core <| point i (some x)) (point i (some cx))

theorem NonExpansive₂.eqv [OFE α] [OFE β] [OFE γ] {f : α → β → γ} [NonExpansive₂ f]
    ⦃x₁ x₂⦄ (hx : x₁ ≡ x₂) ⦃y₁ y₂⦄ (hy : y₁ ≡ y₂) : f x₁ y₁ ≡ f x₂ y₂

theorem sep [BI PROP] [BIAffine PROP] (Q : PROP) : P1 ∗ P2 ∗ Q ⊢ Q

theorem inc_refl (x : α) : x ≼ x

theorem discrete_id_free {x : α} [Discrete α] (H : ∀ y, ✓ x → ¬(x • y ≡ x)) : IdFree x

theorem pcore_mono' {x y : α} {cx} (le : x ≼ y) (e : pcore x ≡ some cx) :
    ∃ cy, pcore y = some cy ∧ cx ≼ cy

theorem sep_elim_spatial [BI PROP] {P A Q A1 A2 : PROP} [inst : IntoSep A A1 A2]
    (h : P ∗ A1 ⊢ A2 -∗ Q) : P ∗ A ⊢ Q

theorem forall_spatial_intuitionistic [BI PROP] (Q : Nat → PROP) : ⊢ (∀ x, □ Q x) -∗ □ Q y

theorem Agree.validN_iff {x : Agree α} :
    x.validN n ↔ ∀ a ∈ x.car, ∀ b ∈ x.car, a ≡{n}≡ b

theorem exists_intuitionistic [BI PROP] (Q : Nat → PROP) : □ (∃ x, Q x) ⊢ ∃ x, □ Q x ∨ False

theorem liftRel_eq : liftRel (@Eq α) A B ↔ A = B

protected theorem Tower.upN (X : Tower F) : ∀ i, upN F i (X (k+1)) ≡{k}≡ X (k+1+i)

theorem plainly_impl_wand : ■ (P → Q) ⊣⊢ ■ (P -∗ Q)

theorem _root_.Iris.OFE.Dist.incN :
    (a : α) ≡{n}≡ a' → b ≡{n}≡ b' → (a ≼{n} b ↔ a' ≼{n} b')

theorem op_proper2 {x₁ x₂ y₁ y₂ : α} (H1 : x₁ ≡ x₂) (H2 : y₁ ≡ y₂) : x₁ • y₁ ≡ x₂ • y₂

theorem and_elim_intuitionistic [BI PROP] {P A Q A1 A2 : PROP} [inst : IntoAnd true A A1 A2]
    (h : P ∗ □ A1 ⊢ □ A2 -∗ Q) : P ∗ □ A ⊢ Q

theorem imp_congr_l [BI PROP] {P P' Q : PROP} (h : P ⊣⊢ P') : (P → Q) ⊣⊢ (P' → Q)

@[rw_mono_rule]
theorem intuitionistically_mono [BI PROP] {P Q : PROP} (h : P ⊢ Q) : □ P ⊢ □ Q

theorem bupd_plain_forall {A : Type} (Φ : A → PROP) [∀ x, Plain (Φ x)] :
    (|==> ∀ x, Φ x) ⊣⊢ (∀ x, |==> Φ x)

theorem Iso.hom_eqv [OFE α] [OFE β] (iso : Iso α β) ⦃x y⦄ :
    x ≡ y ↔ iso.hom x ≡ iso.hom y

theorem spatial_elim_spatial [BI PROP] {A A' Q : PROP} [FromAffinely A' A false]
    (h : P ∗ A' ⊢ Q) : P ∗ A ⊢ Q

theorem core_inc_self {x : α} [CoreId x] : core x ≼ x

@[rw_mono_rule]
theorem or_mono [BI PROP] {P P' Q Q' : PROP} (h1 : P ⊢ Q) (h2 : P' ⊢ Q') : P ∨ P' ⊢ Q ∨ Q'

theorem discrete_inc_l {x y : α} [HD : DiscreteE x] (Hv : ✓{0} y) (Hle : x ≼{0} y) : x ≼ y

theorem absorbing_absorbingly [BI PROP] {P : PROP} [Absorbing P] : <absorb> P ⊣⊢ P

theorem persistently_pure {φ : Prop} [BI PROP] : <pers> ⌜φ⌝ ⊣⊢ (⌜φ⌝ : PROP)

theorem exact_lean' [BI PROP] (Q : PROP) : Q ⊢ (emp ∗ Q) ∗ emp

theorem and_or_l [BI PROP] {P Q R : PROP} : P ∧ (Q ∨ R) ⊣⊢ P ∧ Q ∨ P ∧ R

theorem pure_imp [BI PROP] {φ1 φ2 : Prop} : (⌜φ1⌝ → ⌜φ2⌝ : PROP) ⊣⊢ ⌜φ1 → φ2⌝

theorem Agree.op_commN {x y : Agree α} :  x.op y ≡{n}≡ y.op x

theorem Replaces.apply [BI PROP] {P P' Q : PROP}
    (h : Replaces Q P P') (h_entails : P' ⊢ Q) : P ⊢ Q

theorem IsoFunStore.store_eq_of_Equiv [Store T K V] [IsoFunStore T K V] {t1 t2 : T}
    (H : Store.Equiv t1 t2) : t1 = t2

theorem incN_trans {x y z : α} : x ≼{n} y → y ≼{n} z → x ≼{n} z

theorem inc_of_inc_of_eqv : (a : α) ≼ b → b ≡ c → a ≼ c

theorem plainly_sep [BIPositive PROP] : ■ (P ∗ Q) ⊣⊢ ■ P ∗ ■ Q

theorem pure_intro_affine [BI PROP] {Q : PROP} {φ : Prop}
    [h : FromPure true Q φ] [Affine P] (hφ : φ) : P ⊢ Q

theorem start_stop [BI PROP] (Q : PROP) (H : Q ⊢ Q) : Q ⊢ Q

theorem imp_congr_r [BI PROP] {P Q Q' : PROP} (h : Q ⊣⊢ Q') : (P → Q) ⊣⊢ (P → Q')

theorem wand_iff_exists_persistently [BI PROP] [BIAffine PROP] {P Q : PROP} :
    (P -∗ Q) ⊣⊢ ∃ R, R ∗ <pers> (P ∗ R → Q)

theorem DFrac.update_discard {dq : DFrac F} : dq ~~> .discard

theorem exact_lean'' [BI PROP] (Q : PROP) (H : 0 = 0 → ⊢ Q) : ⊢ Q

theorem some_incN_some_iff_some [IsTotal α] {x y : α} : some y ≼{n} some x → y ≼{n} x

theorem intuitionistically_plainly : □ ■ P ⊢ ■ □ P

theorem bupd_wand_l {P Q : PROP} : iprop((P -∗ Q) ∗ (|==> P) ⊢ |==> Q)

theorem later_soundness : iprop(True ⊢ ▷ P) → iprop((True : UPred M) ⊢ P)

@[rw_mono_rule]
theorem exists_mono [BI PROP] {Φ Ψ : α → PROP} (h : ∀ a, Φ a ⊢ Ψ a) : (∃ a, Φ a) ⊢ ∃ a, Ψ a

theorem coreId_iff {x₁ x₂ : α} (e : x₁ ≡ x₂) : CoreId x₁ ↔ CoreId x₂

theorem point_op_point_eq [IsoFunStore T K (Option V)] {i : K} {x y : V} :
    (point (T := T) i (some x)) • (point i (some y)) = (point i (some (x • y)))

theorem opM_left_dist {n} {x y : α} (z : Option α) (e : x ≡{n}≡ y) : x •? z ≡{n}≡ y •? z

theorem ownM_always_invalid_elim (m : M) (H : ∀ n, ¬✓{n} m) : (cmraValid m : UPred M) ⊢ False

theorem move_to_intuitionistic_disjunction [BI PROP] (Q : PROP) : ⊢ □ Q ∨ Q -∗ Q

theorem Agree.includedN {x y : Agree α} : x ≼{n} y ↔ y ≡{n}≡ y • x

theorem forall_intuitionistic_multiple [BI PROP] (Q : Nat → Nat → PROP) :
    ⊢ □ (∀ x, ∀ y, Q x y) -∗ □ Q x y

theorem Discrete.discrete [OFE α] [Discrete α] {n} {x y : α} (h : x ≡{n}≡ y) : x ≡ y

theorem persistently_exists [BI PROP] {Ψ : α → PROP} : <pers> (∃ a, Ψ a) ⊣⊢ ∃ a, <pers> (Ψ a)

theorem equiv_snd [OFE α] [OFE β] {x y : α × β} (h : x ≡ y) : x.snd ≡ y.snd

theorem multiple_intuitionistic [BI PROP] (P Q R : PROP) : ⊢ □ P -∗ Q -∗ □ (P -∗ Q -∗ □ R) -∗ R

theorem right [BI PROP] (Q : PROP) : Q ⊢ P ∨ Q

theorem pcore_valid {x : α} {cx} (e : pcore x = some cx) : ✓ x → ✓ cx

theorem persistently_and_emp {P : PROP} [BI PROP] : <pers> P ⊣⊢ <pers> (emp ∧ P)

theorem true_sep_2 [BI PROP] {P : PROP} : P ⊢ True ∗ P

theorem LocalUpdate.option {x y x' y' : α}
    (h : (x, y) ~l~> (x', y')) : (some x, some y) ~l~> (some x', some y')

congrArg in Init.Prelude

theorem persistently_imp_wand [BI PROP] [BIAffine PROP] {P Q : PROP} :
    <pers> (P → Q) ⊣⊢ <pers> (P -∗ Q)

theorem eqv_of_inc_exclusive [Exclusive (a : α)] {b : α} (H : some a ≼ some b) (Hv : ✓ b) :
     a ≡ b

theorem DFrac.is_discrete {q : DFrac F} : OFE.DiscreteE q

theorem Agree.toAgree_uninjN {x : Agree α} : ✓{n} x → ∃ a, toAgree a ≡{n}≡ x

theorem LocalUpdate.discrete [CMRA.Discrete α] (x y x' y' : α) :
    (x, y) ~l~> (x', y') ↔ ∀ mz, ✓ x → x ≡ y •? mz → (✓ x' ∧ x' ≡ y' •? mz)

theorem _root_.Iris.OFE.Dist.incN_l : (a : α) ≡{n}≡ b → (a ≼{n} c ↔ b ≼{n} c)

theorem op_self (x : α) [CoreId x] : x • x ≡ x

theorem Update.op {x₁ x₂ y₁ y₂ : α} (xy₁ : x₁ ~~> y₁) (xy₂ : x₂ ~~> y₂) : x₁ • x₂ ~~> y₁ • y₂

theorem imp_wand_persistently [BI PROP] [BIAffine PROP] {P Q : PROP} :
    (<pers> P → Q) ⊣⊢ (<pers> P -∗ Q)

theorem apply_forall [BI PROP] (x : α) (P : α → PROP) {Q : PROP}
    [H1 : IntoForall Q P] (H2 : E ⊢ E' ∗ Q) : E ⊢ E' ∗ P x

theorem disjoint_comm {a b : State Val} : a |

theorem persistently_ite {p : Bool} [BI PROP] {P Q : PROP} :
    iprop(<pers> if p then P else Q) = iprop(if p then <pers> P else <pers> Q)

theorem pcore_idem_L {x cx : α} (h : pcore x = some cx) : pcore cx = some cx

theorem LawfulBigOp.right_id [LawfulBigOp (PROP := PROP) f unit eq] : eq (f a unit) a

theorem _root_.Iris.OFE.Dist.op {x x' y y' : α}
    (ex : x ≡{n}≡ x') (ey : y ≡{n}≡ y') : x • y ≡{n}≡ x' • y'

theorem apply_forall' [BI PROP] (P Q : α → PROP) (a b : α) : (∀ x, ∀ y, P x -∗ Q y) ⊢ P a -∗ Q b

theorem op_commN {x y : α} : x • y ≡{n}≡ y • x

theorem pure_imp_2 [BI PROP] {φ1 φ2 : Prop} : ⌜φ1 → φ2⌝ ⊢ (⌜φ1⌝ → ⌜φ2⌝ : PROP)

theorem DistLater.dist_lt [OFE α] {m n} {x y : α} (h : DistLater n x y) (hm : m < n) : x ≡{m}≡ y

theorem ownM_unit P : P ⊢ □ ownM (CMRA.unit : M)

theorem Agree.op_def {x y : Agree α} : x • y = x.op y

theorem bupd_elim {P : PROP} [Plain P] : |==> P ⊢ P

theorem plainly_forall_2 {Ψ : α → PROP} : (∀ a, ■ (Ψ a)) ⊢ ■ (∀ a, Ψ a)

theorem multiple_lean [BI PROP] (P Q R : PROP) (H : P ⊢ Q -∗ R) (HP : ⊢ P) : ⊢ Q -∗ R

theorem persistently_forall [BI PROP] [h : BIPersistentlyForall PROP] {Ψ : α → PROP} :
    <pers> (∀ a, Ψ a) ⊣⊢ ∀ a, <pers> (Ψ a)

theorem absorbingly_intro [BI PROP] {P : PROP} : P ⊢ <absorb> P

theorem discrete_cancelable {x : α} [Discrete α]
    (H : ∀ {y z : α}, ✓(x • y) → x • y ≡ x • z → y ≡ z) : Cancelable x

theorem op_opM_assoc (x y : α) (mz : Option α) : (x • y) •? mz ≡ x • (y •? mz)

theorem intuitionistic_elim_spatial [BI PROP] {A A' Q : PROP}
    [IntoPersistently false A A'] [TCOr (Affine A) (Absorbing Q)]
    (h : P ∗ □ A' ⊢ Q) : P ∗ A ⊢ Q

theorem persistently_of_intuitionistically [BI PROP] {P : PROP} : □ P ⊢ <pers> P

theorem plainly_exists_1 [BIPlainlyExists PROP] {Ψ : α → PROP} :
    ■ (∃ a, Ψ a) ⊢ ∃ a, ■ (Ψ a)

theorem imp_intro_drop [BI PROP] {P Q A1 A2 : PROP} [inst : FromImp Q A1 A2] (h : P ⊢ A2) : P ⊢ Q

theorem def_eq [BI PROP] (Q : PROP) : <affine> <pers> Q ⊢ □ Q

theorem from_assumption [BI PROP] (Q : PROP) : <affine> <pers> Q ⊢ □ Q

@[symm] theorem DistLater.symm [OFE α] {n} {x : α} (h : DistLater n x y) : DistLater n y x

theorem intuitionistically_wand [BI PROP] {P Q : PROP} : (□ P -∗ Q) ⊣⊢ (<pers> P → Q)

theorem intuitionistic [BI PROP] (Q : PROP) : □ Q ⊢ Q

@[refl] theorem IncludedN.rfl {x : α} : x ≼{n} x

theorem bupd_idem {P : PROP} : iprop((|==> |==> P) ⊣⊢ |==> P)

theorem op_core_dist (x : α) : x • core x ≡{n}≡ x

theorem sep_or_l [BI PROP] {P Q R : PROP} : P ∗ (Q ∨ R) ⊣⊢ (P ∗ Q) ∨ (P ∗ R)

theorem IdFree.of_dist {x₁ x₂ : α} {n} (e : x₁ ≡{n}≡ x₂) (h : IdFree x₁) : IdFree x₂

theorem set_op_equiv {m1 m2 : T} :
    Equiv ((set (V := Option V) (m1 • m2) i (x • y))) (set m1 i x • set m2 i y)

theorem wandIff_equiv [BI PROP] {P Q : PROP} (h : ⊢ P ∗-∗ Q) : P ⊣⊢ Q

theorem Agree.op_comm {x y : Agree α} :  x.op y ≡ y.op x

theorem distLater_succ [OFE α] {n} {x y : α} : DistLater n.succ x y ↔ x ≡{n}≡ y

theorem intuitionistic [BI PROP] (Q : PROP) : □ P ⊢ Q -∗ Q

@[rw_mono_rule]
theorem imp_mono [BI PROP] {P P' Q Q' : PROP} (h1 : Q ⊢ P) (h2 : P' ⊢ Q') : (P → P') ⊢ Q → Q'

@[simp] theorem some_eqv_some [OFE α] {x y : α} : (some x ≡ some y) ↔ x ≡ y

theorem not_valid_some_exclN_op_left {n} {x : α} [Exclusive x] {y : α} : ¬✓{n} (some x • some y)

theorem or_intro_r' [BI PROP] {P Q R : PROP} (h : P ⊢ R) : P ⊢ Q ∨ R

theorem exact_forall [BI PROP] (P : α → PROP) (a : α) (H : ⊢ ∀ x, P x) : ⊢ P a

theorem pcore_op_left_L {x cx : α} (h : pcore x = some cx) : cx • x = x

theorem not_valid_exclN_op_right {n} {x : α} [Exclusive x] {y} : ¬✓{n} (y • x)

theorem _root_.Iris.OFE.Equiv.inc_r : (b : α) ≡ c → (a ≼ b ↔ a ≼ c)

theorem intuitionistically_imp_wand_2 [BI PROP] {P Q : PROP} : □ (P -∗ Q) ⊢ □ (P → Q)

theorem pcore_op_right_L {x cx : α} (h : pcore x = some cx) : x • cx = x

theorem sep_right_comm [BI PROP] {P Q R : PROP} : (P ∗ Q) ∗ R ⊣⊢ (P ∗ R) ∗ Q

theorem LocalUpdate.prod {x y x' y' : α × β}
    (hl : (x.1, y.1) ~l~> (x'.1, y'.1)) (hr : (x.2, y.2) ~l~> (x'.2, y'.2)) :
    (x, y) ~l~> (x', y')

theorem intuitionistically_exists [BI PROP] {Φ : α → PROP} : □ (∃ x, Φ x) ⊣⊢ ∃ x, □ Φ x

theorem or_true [BI PROP] {P : PROP} : P ∨ True ⊣⊢ True

theorem forall_intro [BI PROP] {P : PROP} {Ψ : α → PROP} (h : ∀ a, P ⊢ Ψ a) : P ⊢ ∀ a, Ψ a

theorem incN_iff (ea : (a : α) ≡{n}≡ a') (eb : (b : α) ≡{n}≡ b') : a ≼{n} b ↔ a' ≼{n} b'

@[simp] theorem map_apply [OFE α] [OFE β] {f : α -n> β} {c : Chain α} {n} :
    map f c n = f (c n)

theorem emp_sep_rev [BI PROP] {P : PROP} : P ⊣⊢ emp ∗ P

theorem LocalUpdate.id (x : α × α) : x ~l~> x

theorem inc_of_some_inc_some [IsTotal α] {x y : α} (H : some y ≼ some x) : y ≼ x :=
  let ⟨mz, hmz⟩ := H
  match mz with

theorem split_rs [BI PROP] {P Q Q1 Q2 : PROP} (h : Q ⊣⊢ Q1 ∗ Q2) : P ∗ Q ⊣⊢ Q1 ∗ (P ∗ Q2)

theorem MyR_always_invalid (S₁ S₂ : String) (Hne : S₁ ≠ S₂) (n : Nat) : ¬✓{n} MyR S₁ • MyR S₂

theorem intuitionistically_emp [BI PROP] : □ emp ⊣⊢ (emp : PROP)

theorem persistently_true [BI PROP] : <pers> (True : PROP) ⊣⊢ True

@[symm] theorem Equiv.symm [OFE α] {x : α} : x ≡ y → y ≡ x

theorem UpdateP.iso'
    (gf : ∀ x, g (f x) ≡ x)
    (g_op : ∀ y1 y2, g (y1 • y2) ≡ g y1 • g y2)
    (g_validN : ∀ n y, ✓{n} (g y) ↔ ✓{n} y)
    (uyp : y ~~>: P) :
    g y ~~>: λ x ↦ ∃ y, x = g y ∧ P y

theorem UpdateP.discrete [CMRA.Discrete α] :
    x ~~>: P ↔ ∀ (mz : Option α), ✓ (x •? mz) → ∃ y, P y ∧ ✓ (y •? mz)

theorem intuitionistically_sep_idem [BI PROP] {P : PROP} : □ P ∗ □ P ⊣⊢ □ P
</all_available_lemmas>

<local_ctx>
import Iris.BI.Classes

import Iris.BI.Extensions

import Iris.BI.BI

import Iris.Std.Classes

import Iris.Std.Rewrite

import Iris.Std.TC

namespace Iris.BI

open Iris.Std BI
</local_ctx>

<target_theorem>
theorem imp_iff_exists_persistently [BI PROP] [BIAffine PROP] {P Q : PROP} :
    (P → Q) ⊣⊢ ∃ R, R ∧ <pers> (P ∧ R -∗ Q) :=
</target_theorem>

