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>
structure Context where
  ref         : Syntax
   
  m           : Syntax
   
  returnType  : Syntax
  mutableVars : VarSet := {}
  insideFor   : Bool := false

private def _root_.Lean.EnvExtension.get [Inhabited σ] (ext : EnvExtension σ)
  [Monad m] [MonadEnv m] : m σ := do
  return ext.getState (<- getEnv)

private def _root_.Lean.SimpleScopedEnvExtension.get [Inhabited σ] (ext : SimpleScopedEnvExtension α σ)
  [Monad m] [MonadEnv m] : m σ := do
  return ext.getState (<- getEnv)

private def _root_.Lean.SimplePersistentEnvExtension.get [Inhabited σ] (ext : SimplePersistentEnvExtension α σ)
  [Monad m] [MonadEnv m] : m σ := do
  return ext.getState (<- getEnv)

def findNat (p : Nat -> Prop) [DecidablePred p] : Option Nat :=
  let rec aux i :=
    if p i then
      some i
    else
      aux (i + 1)
  partial_fixpoint
  aux 0

def find [Encodable α] (p : α -> Prop) [DecidablePred p] : Option α :=
  findNat (fun x => (Encodable.decode x).any (p ·)) |>.bind Encodable.decode

class Findable {α : Type u} (p : α -> Prop) where
  find : Unit -> Option α
  find_none : (find ()).isNone -> ∀ x, ¬ p x
  find_some_p : find () = some x -> p x

def findNat (p : Nat -> Prop) [DecidablePred p] : Option Nat :=
  let rec aux i :=
    if p i then
      some i
    else
      aux (i + 1)
  partial_fixpoint
  aux 0

def find [Encodable α] (p : α -> Prop) [DecidablePred p] : Option α :=
  findNat (fun x => (Encodable.decode x).any (p ·)) |>.bind Encodable.decode

class Findable {α : Type u} (p : α -> Prop) where
  find : Unit -> Option α
  find_none : (find ()).isNone -> ∀ x, ¬ p x
  find_some_p : find () = some x -> p x

def spec (pre : l) (post : α -> l) : Cont l α :=
  fun p => pre ⊓ ⌜post ≤ p⌝

class CCPOBotLawful (m : Type u -> Type v) [∀ α, Lean.Order.CCPO (m α)] [CCPOBot m] where
  prop {α} : CCPOBot.compBot (m := m) (α := α) = Lean.Order.bot

private def _root_.Lean.SimpleScopedEnvExtension.get [Inhabited σ] (ext : SimpleScopedEnvExtension α σ)
  [Monad m] [MonadEnv m] : m σ := do
  return ext.getState (<- getEnv)

private def _root_.Lean.SimplePersistentEnvExtension.get [Inhabited σ] (ext : SimplePersistentEnvExtension α σ)
  [Monad m] [MonadEnv m] : m σ := do
  return ext.getState (<- getEnv)

private def _root_.Lean.SimpleScopedEnvExtension.get [Inhabited σ] (ext : SimpleScopedEnvExtension α σ)
  [Monad m] [MonadEnv m] : m σ := do
  return ext.getState (<- getEnv)

private def _root_.Lean.SimplePersistentEnvExtension.get [Inhabited σ] (ext : SimplePersistentEnvExtension α σ)
  [Monad m] [MonadEnv m] : m σ := do
  return ext.getState (<- getEnv)

structure WPGen (x : m α) where
  get : Cont l α
  
  prop : ∀ post, get post <= wp x post

def WPGen.bind {x : m α} {f : α -> m β} (wpg : WPGen x) (wpgf : ∀ a, WPGen (f a)) :
  WPGen (x >>= f) where
  get := fun post => wpg.get (fun a => (wpgf a).get post)
  prop := by admit /- proof elided -/

macro "⌜" p:term "⌝" : term => `(LE.pure $p)

class MAlg [Monad m] (l : outParam (Type v)) where
  μ : m l -> l
  pure : ∀ l, μ (pure l) = l
  bind : ∀ {α : Type v} (x : m α) (f g : α -> m l),
    μ ∘ f = μ ∘ g ->
    μ (x >>= f) = μ (x >>= g)

abbrev MAlg.lift {m : Type u -> Type v} {l : Type u} [Monad m] [MAlg m l] :
  {α : Type u} -> m α -> Cont l α := fun x f => μ $ f <$> x

class MAlgOrdered (l : outParam (Type v)) [Monad m] [CompleteLattice l] where
  μ : m l -> l
  μ_ord_pure : ∀ l, μ (pure l) = l
  μ_ord_bind {α : Type v} :
    ∀ (f g : α -> m l), μ ∘ f ≤ μ ∘ g ->
      ∀ x : m α, μ (x >>= f) ≤ μ (x >>= g)

private def _root_.Lean.EnvExtension.get [Inhabited σ] (ext : EnvExtension σ)
  [Monad m] [MonadEnv m] : m σ := do
  return ext.getState (<- getEnv)

private def _root_.Lean.SimpleScopedEnvExtension.get [Inhabited σ] (ext : SimpleScopedEnvExtension α σ)
  [Monad m] [MonadEnv m] : m σ := do
  return ext.getState (<- getEnv)

private def _root_.Lean.SimplePersistentEnvExtension.get [Inhabited σ] (ext : SimplePersistentEnvExtension α σ)
  [Monad m] [MonadEnv m] : m σ := do
  return ext.getState (<- getEnv)

Lean.Parser.Term.termFor in Lean.Parser.Do

Lean.Parser.Tactic.tacticSeq in Lean.Parser.Term.Basic

private def elabSimpArg (indexConfig : Meta.ConfigWithKey) (eraseLocal : Bool) (kind : SimpKind)
    (arg : Syntax) : TermElabM ElabSimpArgResult := withRef arg do
  try
     
    if arg.getKind == ``Lean.Parser.Tactic.simpErase then
      let fvar? ← if eraseLocal then Term.isLocalIdent? arg[1] else pure none
      if let some fvar := fvar? then
        
        return .erase (.fvar fvar.fvarId!)
      else
        let id := arg[1]
        if let .ok declName ← observing (realizeGlobalConstNoOverloadWithInfo id) then
          if (← Simp.isSimproc declName) then
            return .eraseSimproc declName
          else
            return .erase (.decl declName)
        else
          
          
          let name := id.getId.eraseMacroScopes
          if (← Simp.isBuiltinSimproc name) then
            return .eraseSimproc name
          else
            throwUnknownConstantAt id name
    else if arg.getKind == ``Lean.Parser.Tactic.simpLemma then
      let post :=
        if arg[0].isNone then
          true
        else
          arg[0][0].getKind == ``Parser.Tactic.simpPost
      let inv  := !arg[1].isNone
      let term := arg[2]
      match (← resolveSimpIdTheorem? ⟨term⟩) with
      | .expr e  =>
        let name ← mkFreshId
        elabDeclToUnfoldOrTheorem indexConfig (.stx name arg) e post inv kind
      | .simproc declName =>
        return .addSimproc declName post
      | .ext ext₁? ext₂? h =>
        return .ext ext₁? ext₂? h
      | .none    =>
        let name ← mkFreshId
        elabSimpTheorem indexConfig (.stx name arg) term post inv
    else if arg.getKind == ``Lean.Parser.Tactic.simpStar then
      return .star
    else
      throwUnsupportedSyntax
  catch ex =>
    throw ex

def continueToTerm : M Syntax := do
  let ctx ← read
  let u ← mkUVarTuple
  match ctx.kind with
  | .regular         => unreachable!
  | .forIn           => ``(Pure.pure (ForInStep.yield $u))
  | .forInWithReturn => ``(Pure.pure (ForInStep.yield (MProdWithNames.mk none $u)))
  | .nestedBC        => ``(Pure.pure (DoResultBC.«continue» $u))
  | .nestedPR        => unreachable!
  | .nestedSBC       => ``(Pure.pure (DoResultSBC.«continue» $u))
  | .nestedPRBC      => ``(Pure.pure (DoResultPRBC.«continue» $u))

deriving Inhabited

instMonadLiftTOfMonadLift in Init.Prelude

Array.foldlM in Init.Data.Array.Basic

Lean.mkConst in Lean.Expr

Lean.Parser.Term.doLetArrow in Lean.Parser.Do

Lean.Meta.getZetaDeltaFVarIds in Lean.Meta.Basic

Encodable.decode in Mathlib.Logic.Encodable.Basic

Lean.Order.admissible in Init.Internal.Order.Basic

partial def Handle.readLineSkip (h : IO.FS.Handle) (skipLine : String := "\n") : IO String := do
  let rec loop := do
    let line ← h.getLine
    if line.isEmpty then
      return line
    if line == skipLine then
      loop
    else
      return line
  loop

scoped instance : MAlgOrdered DivM Prop where
  μ := fun x => match x with
    | .res x => ⌜x⌝
    | .div => ⊥
  μ_ord_pure := by admit /- proof elided -/

def addAssertion (s : LoomAssertionsMap) (t : Term) (n : Name) (coreName: Name): LoomAssertionsMap :=
  let maxId := s.maxId + 1
  let maxIdLocal := s.nameCounter.get! coreName + 1
  { s with
    maxId := maxId
    syntaxStore := s.syntaxStore.insert maxId t
    nameStore := s.nameStore.insert n maxId
    nameCounter := s.nameCounter.insert coreName maxIdLocal }

Array.map in Init.Data.Array.Basic

method SubstringSearch (s: Array Char) (p: Char -> Bool) return (res: SubstringResult)

Bind.bind in Init.Prelude

def Lake.unzip (file : FilePath) (dir : FilePath) : LogIO PUnit := do
  IO.FS.createDirAll dir
  proc (quiet := true) {
    cmd := "unzip"
    args := #["-d", dir.toString, file.toString]
  }

private def _root_.Lean.EnvExtension.modify (ext : EnvExtension σ) (s : σ -> σ)
  [Monad m] [MonadEnv m] : m Unit := do
  Lean.modifyEnv (ext.modifyState · s)

def mkTerminalAction (action : Syntax) : CodeBlock :=
  { code := .action action }

Lean.Loop in Init.While

inDigitSubstring := false
        currentLen := 0

instance : ToString SmtResult where
  toString
    | SmtResult.Sat none => "sat"
    | SmtResult.Sat (some m) => s!"sat\n{m}"
    | SmtResult.Unsat => s!"unsat"
    | SmtResult.Unknown none => s!"unknown"
    | SmtResult.Unknown (some r) => s!"unknown ({r})"
    | SmtResult.Failure none => "failure"
    | SmtResult.Failure (some r) => s!"failure ({r})"

method calc_fact (n: Nat) return (res: Nat)
  ensures fact n = res
  do
    if n = 0 then
      return 0
    else
      let mut i := 0
      let mut ans := 0
      let pre_res_n ← calc_fact (n - 1)
      while i < n
        invariant i <= n
        invariant i * pre_res_n = ans
        decreasing n - i
        do
          let pre_res ← calc_fact (n - 1)
          ans := ans + pre_res
          i := i + 1
      return ans

instance : MonadLift m (NonDetT m) where
  monadLift x := NonDetT.vis x pure

def cvc5Path := buildDir / "cvc5"

def cvc5.target := s!"{os}-{arch}-static"

List.length in Init.Prelude

def getMutTypes (stx : Array (TSyntax `leafny_binder)) : MetaM (Array (TSyntax `term)) := do
  let mut types := #[]
  for b in stx do
    match b with
    | `(leafny_binder| (mut $_:ident : $tp:term)) => do
      types := types.push tp
    | _ => pure ()
  return types

Lean.Order.bot in Init.Internal.Order.Basic

inline in Init.Core

def withNewMutableVars {α} (newVars : Array Var) (mutable : Bool) (x : M α) : M α :=
  withReader (fun ctx => if mutable then { ctx with mutableVars := insertVars ctx.mutableVars newVars } else ctx) x

instance {l σ : Type u} [CompleteLattice l] : LogicLift l (σ -> l) where
  lift_top := by admit /- proof elided -/

Lean.choiceKind in Init.Prelude

Auto.LamReif.reifFacts in Auto.Translation.LamReif

structure AltExpr (σ : Type) where
  ref     : Syntax
  var?    : Option Var
  funName : Syntax
  pvars   : Array Syntax
  rhs     : σ
  deriving Inhabited

le in Test.SmtTranslation.Names

Lean.mkNode in Init.Prelude

Auto.LamReif.reifInhabitations in Auto.Translation.LamReif

def NonDetT.wp {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : {α : Type u} -> NonDetT m α -> Cont l α
  | _, .pure ret => pure ret
  | _, .vis x f => fun post => _root_.wp x fun a => wp (f a) post
  | _, @NonDetT.pickCont _ _ τ p _ f => fun post => ⨆ a, ⌜p a⌝ ⊓ wp (f a) post
  | _, .repeatCont init f cont => fun post => ⨆ (inv : ForInStep _ -> l),
    ⌜ ∀ b, (inv (ForInStep.yield b)) <= wp (f b) inv⌝ ⊓
    spec (inv (.yield init)) (fun b => inv (.done b)) (fun b => wp (cont b) post)

private def elabSimpTheorem (config : Meta.ConfigWithKey) (id : Origin) (stx : Syntax)
    (post : Bool) (inv : Bool) : TermElabM ElabSimpArgResult := do
  let thm? ← Term.withoutModifyingElabMetaStateWithInfo <| withRef stx do
    let e ← Term.elabTerm stx .none
    Term.synthesizeSyntheticMVars (postpone := .no) (ignoreStuckTC := true)
    let e ← instantiateMVars e
    if e.hasSyntheticSorry then
      return .none
    let e := e.eta
    if e.hasMVar then
      let r ← abstractMVars e
      return some (r.paramNames, r.expr)
    else
      return some (#[], e)
  if let some (levelParams, proof) := thm? then
    let thms ← mkSimpTheoremFromExpr id levelParams proof (post := post) (inv := inv) (config := config)
    return .addEntries <| thms.map (SimpEntry.thm ·)
  else
    return .none

instance ExtractNonDet.pickSuchThat_weak {τ : Type u} (p : τ -> Prop) [WeakFindable p] :
  ExtractNonDet WeakFindable (MonadNonDet.pickSuchThat (m := NonDetT m) τ p) :=

structure VelvetObligation where
  binderIdents : TSyntaxArray `Lean.Parser.Term.bracketedBinder
  modIds : Array Ident
  ids : Array Ident
  retId : Ident
  ret : Term
  pre : Term
  post : Term
deriving Inhabited

parse in Auto.Parser.TPTP

register_option loom.solver : LoomSolver := {
  defValue := .grind
  descr :=
  "Loom solver options. Right now we support
   - `cvc5` and `z3` to use the SMT solvers via `auto` tactic
   - `grind` to use the grind tactic
   - `custom` to use a user-provided solver tactic

instance (t : Type v) : Monad (Cont t) where
  pure x := fun f => f x
  bind x f := fun g => x (fun a => f a g)

LawfulMonadLift in Init.Control.Lawful.MonadLift.Basic

Lake.Job.collectArray in Lake.Build.Job.Monad

def withFor {α} (x : M α) : M α :=
  withReader (fun ctx => { ctx with insideFor := true }) x

partial def extendUpdatedVarsAux (c : Code) (ws : VarSet) : TermElabM Code :=
  let rec update (c : Code) : TermElabM Code := do
    match c with
    | .joinpoint j ps b k    => return .joinpoint j ps (← update b) (← update k)
    | .seq e k               => return .seq e (← update k)
    | .match ref g ds t alts =>
      if alts.any fun alt => alt.vars.any fun x => ws.contains x.getId then
        
        pullExitPoints c
      else
        return .match ref g ds t (← alts.mapM fun alt => do pure { alt with rhs := (← update alt.rhs) })
    | .matchExpr ref «meta» d alts e =>
      if alts.any fun alt => alt.vars.any fun x => ws.contains x.getId then
        
        pullExitPoints c
      else
        let alts ← alts.mapM fun alt => do pure { alt with rhs := (← update alt.rhs) }
        let e ← update e
        return .matchExpr ref «meta» d alts e
    | .ite ref none o c t e => return .ite ref none o c (← update t) (← update e)
    | .ite ref (some h) o cond t e =>
      if ws.contains h.getId then
        
        pullExitPoints c
      else
        return Code.ite ref (some h) o cond (← update t) (← update e)
    | .reassign xs stx k => return .reassign xs stx (← update k)
    | .decl xs stx k => do
      if xs.any fun x => ws.contains x.getId then
        
        pullExitPoints c
      else
        return .decl xs stx (← update k)
    | c => return  c
  update c

Lean.Parser.Term.letDecl in Lean.Parser.Term

Lean.Parser.withForbidden in Lean.Parser.Basic

Auto.LamReif.ReifM in Auto.Translation.LamReif

instance [Monad m] [LawfulMonad m] [_root_.CompleteLattice l] [inst: MAlgOrdered m l] :
  MAlgLift m l (ReaderT σ m) (σ -> l) where
    μ_lift := by admit /- proof elided -/

Plausible.Gen in Plausible.Gen

class MAlgLiftT
  (m : (Type u -> Type v)) (l : (Type u)) [Monad m] [CompleteLattice l] [MAlgOrdered m l]
  (n : (Type u -> Type w)) (k : outParam (Type u)) [Monad n] [CompleteLattice k] [MAlgOrdered n k]
  [MonadLiftT m n]
  where
    [cl : LogicLiftT l k]
    μ_lift (x : m α) : MAlg.lift (liftM (n := n) x) f = liftM (n := Cont k) (MAlg.lift x) f

Lean.Parser.Term.letrec in Lean.Parser.Term

Nat.decreasingInduction in Mathlib.Data.Nat.Init

MonadExceptOf in Init.Prelude

lamEVarTy in Auto.Parser.TPTP

LawfulMonadLiftT in Init.Control.Lawful.MonadLift.Basic

instance (hd : ε -> _) [IsHandler hd] [_root_.CompleteLattice l] [Monad m] [LawfulMonad m] [inst: MAlgOrdered m l]
  [∀ α, CCPO (m α)] [MonoBind m]
  [MAlgPartial m] : MAlgPartial (ExceptT ε m) where
  csup_lift {α} chain := by admit /- proof elided -/

instance : Inhabited LoomSemanticsTermination := ⟨.unspecified⟩

Set in Mathlib.Data.Set.Defs

List.find? in Init.Data.List.Basic

abbrev DiscrTreeKey := Array DiscrTree.Key

method SpVSpV
  (mut out: Array Int)
  (spv1: SpV Int)
  (spv2: SpV Int) return (u: Unit)
  ensures out.size = 1
  ensures out[0]! = spv_dot spv1 spv2 0 0
  do
    out := Array.replicate 1 0
    let mut pnt1 := 0
    let mut pnt2 := 0
    while pnt1 ≠ spv1.size ∧ pnt2 ≠ spv2.size
    invariant out.size = 1
    invariant pnt1 ≤ spv1.size ∧ pnt2 ≤ spv2.size
    invariant out[0]! + spv_dot spv1 spv2 pnt1 pnt2 = spv_dot spv1 spv2 0 0
    done_with pnt1 = spv1.size ∨ pnt2 = spv2.size
    do
      if (spv1.ind)[pnt1]! = (spv2.ind)[pnt2]! then
        out[0] += (spv1.val)[pnt1]! * (spv2.val)[pnt2]!
        pnt1 := pnt1 + 1
        pnt2 := pnt2 + 1
      else
        if (spv1.ind)[pnt1]! < (spv2.ind)[pnt2]! then
          pnt1 := pnt1 + 1
        else
          pnt2 := pnt2 + 1
    return

def defineWPGen (name : Name) : TermElabM Unit := do
  unless ← isMatcher name do
    throwError s!"{name} is not a matcher"
  let targetName := name ++ `WPGen
  if (← getEnv).find? targetName matches none then
    let some res ← constructWPGen name
      | throwError s!"cannot generate WPGen for matcher {name} due to error"
    addDecl <|
      Declaration.defnDecl <|
        mkDefinitionValEx targetName res.lvls.toList (← inferType res.wpgenExpr) res.wpgenExpr
        (Lean.ReducibilityHints.regular 0)
        (DefinitionSafety.safe) []
    enableRealizationsForConst targetName
    applyAttributes targetName (← do
      let attr1 ← `(Parser.Term.attrInstance| spec )
      let attr2 ← `(Parser.Term.attrInstance| loomWpSimp)
      elabAttrs #[attr1, attr2])

Lean.BinderInfo in Lean.Expr

instance : MAlgDet Id Prop where
  demonic := by admit /- proof elided -/

elab "#reduce!" e:term : command => do
  liftTermElabM do
    let e ← ultimateReduce <| <- Term.elabTerm e none
    logInfoAt (<- getRef) e

Lean.Elab.Term.elabTermEnsuringType in Lean.Elab.Term.TermElabM

Lean.quoteNameMk in Init.Meta.Defs

instance : Inhabited (SpV Int) where
  default :=
  { ind := #[],
    val := #[],
    size := 0,
    size_eq := (by admit /- proof elided -/
    ),
    inc := (fun i j h_i h_j i_lt_j => by admit /- proof elided -/
    ) }

def dependencyFile := "dependencies.toml"

def getLetIdDeclVars (letIdDecl : Syntax) : Array Var :=
  assert! letIdDecl.isOfKind ``Parser.Term.letIdDecl
  
  
  getLetIdVars letIdDecl[0]

instance ExtractNonDet.assume' {p : Prop} [Decidable p] : ExtractNonDet findable (MonadNonDet.assume (m :=  NonDetT m) p) :=

private def expandDoLetExpr? (stx : Syntax) (doElems : List Syntax) : MacroM (Option Syntax) := match stx with
  | `(doElem| let_expr $pat:matchExprPat := $discr:term | $elseBranch:doSeq) =>
    return some (← `(doElem| match_expr (meta := false) $discr:term with
                             | $pat:matchExprPat => $(mkDoSeq doElems.toArray)
                             | _ => $elseBranch))
  | `(doElem| let_expr $pat:matchExprPat ← $discr:term | $elseBranch:doSeq) =>
    return some (← `(doElem| match_expr $discr:term with
                             | $pat:matchExprPat => $(mkDoSeq doElems.toArray)
                             | _ => $elseBranch))
  | _ => return none

def attrIgnoreAuxDef (name : Name) (default : AttrM α) (x : AttrM α) : AttrM α := do
  
  if let .str _ "_mutual" := name then

Lean.Macro.throwErrorAt in Init.Prelude

partial def toTerm (c : Code) : M Syntax := do
  let term ← go c
  if let some ref := c.getRef? then
    annotate ref term
  else
    return term
where
  go (c : Code) : M Syntax := do
    match c with
    | .return ref val     => withRef ref <| returnToTerm val
    | .continue ref       => withRef ref continueToTerm
    | .break ref          => withRef ref breakToTerm
    | .action e           => actionTerminalToTerm e
    | .joinpoint j ps b k => mkJoinPoint j ps (← toTerm b) (← toTerm k)
    | .jmp ref j args     => return mkJmp ref j args
    | .decl _ stx k       => declToTerm stx (← toTerm k)
    | .reassign _ stx k   => reassignToTerm stx (← toTerm k)
    | .seq stx k          => seqToTerm stx (← toTerm k)
    | .ite ref _ o c t e  => withRef ref <| do mkIte o c (← toTerm t) (← toTerm e)
    | .match ref genParam discrs optMotive alts =>
      let mut termAlts := #[]
      for alt in alts do
        let rhs ← toTerm alt.rhs
        let termAlt := mkNode ``Parser.Term.matchAlt #[mkAtomFrom alt.ref "|", mkNullNode #[alt.patterns], mkAtomFrom alt.ref "=>", rhs]
        termAlts := termAlts.push termAlt
      let termMatchAlts := mkNode ``Parser.Term.matchAlts #[mkNullNode termAlts]
      return mkNode ``Parser.Term.«match» #[mkAtomFrom ref "match", genParam, optMotive, discrs, mkAtomFrom ref "with", termMatchAlts]
    | .matchExpr ref «meta» d alts elseBranch => withFreshMacroScope do
      let d' ← `(discr)
      let mut termAlts := #[]
      for alt in alts do
        let rhs ← `(($(← toTerm alt.rhs) : $((← read).m) _))
        let optVar := if let some var := alt.var? then mkNullNode #[var, mkAtomFrom var "@"] else mkNullNode #[]
        let pat := mkNode ``Parser.Term.matchExprPat #[optVar, alt.funName, mkNullNode alt.pvars]
        let termAlt := mkNode ``Parser.Term.matchExprAlt #[mkAtomFrom alt.ref "|", pat, mkAtomFrom alt.ref "=>", rhs]
        termAlts := termAlts.push termAlt
      let elseBranch := mkNode ``Parser.Term.matchExprElseAlt #[mkAtomFrom ref "|", mkHole ref, mkAtomFrom ref "=>", (← toTerm elseBranch)]
      let termMatchExprAlts := mkNode ``Parser.Term.matchExprAlts #[mkNullNode termAlts, elseBranch]
      let body := mkNode ``Parser.Term.matchExpr #[mkAtomFrom ref "match_expr", d', mkAtomFrom ref "with", termMatchExprAlts]
      if «meta» then
        `(Bind.bind (instantiateMVarsIfMVarApp $d) fun discr => $body)
      else
        `(let discr := $d; $body)

def WPGen.recProd {l : Type u} {m : Type u -> Type v} [Monad m] [LawfulMonad m] [CompleteBooleanAlgebra l] [MAlgOrdered m l]
  {pr: Prod α₁ α₂}
  {mkp : α₁ → α₂ → m β}
  (wpgf : ∀ (a: α₁) (b: α₂), WPGen (mkp a b)):
  WPGen (Prod.recOn pr mkp : m β) where
  get := fun post =>
    (⨅ a, ⨅ b, (wpgf a b).get post)
  prop := by admit /- proof elided -/

def Extractable.forIn (xs : List α) (init : β) (f : α -> β -> NonDetT m (ForInStep β))
  (ex: ∀ a b, Extractable (f a b)):
  Extractable (forIn xs init f) :=

Lean.indentExpr in Lean.Message

Preorder in Mathlib.Order.Defs.PartialOrder

abbrev M := ReaderT Context TermElabM

def NonDetT.μ {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : NonDetT m l -> l := fun x => NonDetT.wp x id

private def getDoSeq (doStx : Syntax) : Syntax :=
  doStx[1]

inductive SmtResult
   
  | Sat (modelString : Option String)
  | Unsat
  | Unknown (reason : Option String)
  | Failure (reason : Option String)
deriving Inhabited

Lean.DefinitionSafety.safe in Lean.Declaration

| .atom (.symb "unsat") =>
    trace[loom.smt.result] "{solverName} says Unsat"
    let (_, solver) ← solver.takeStdin
    let stdout ← solver.stdout.readToEnd
    let stderr ← solver.stderr.readToEnd
    trace[loom.smt.debug] "stdout: {stdout}"
    trace[loom.smt.debug] "stderr: {stderr}"
    solver.kill
    return (SmtResult.Unsat, solverName)

def NonDetT.wp {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : {α : Type u} -> NonDetT m α -> Cont l α
  | _, .pure ret => pure ret
  | _, .vis x f => fun post => _root_.wp x fun a => wp (f a) post
  | _, @NonDetT.pickCont _ _ τ p _ f =>
  fun post =>
    let p : Set τ := p;
    ⨅ a ∈ (p : Set τ), wp (f a) post
  | _, @NonDetT.repeatCont _ _ β init f cont => fun post => ⨆ (inv : ForInStep β -> l),
      ⌜ ∀ b, (inv (ForInStep.yield b)) <= wp (f b) inv⌝ ⊓
      spec (inv (.yield init)) (fun b => inv (.done b)) (fun b => wp (cont b) post)

instance : CCPOBotLawful DivM where
  prop := by admit /- proof elided -/

def mkUVarTuple : M Syntax := do
  let ctx ← read
  mkTuple ctx.uvars

Lean.Parser.Term.let_delayed in Lean.Parser.Term

instance [inst: (α : Type u) -> Preorder (w α)] : PreOrderFunctor w := ⟨inst⟩

def List.nonEmpty (q : List α) : Prop :=
  ¬q.isEmpty

Lean.Meta.Simp.getSimprocs in Lean.Meta.Tactic.Simp.Simproc

Auto.computeMaxLevel in Auto.Translation.Assumptions

def mkUnless (cond : Syntax) (c : CodeBlock) : MacroM CodeBlock := do
  let thenBranch ← mkPureUnitAction
  return { c with code := .ite (← getRef) none mkNullNode cond thenBranch.code c.code }

Auto.Embedding.Lam.REntry.valid in Auto.Embedding.LamChecker

Lean.Parser.Term.doIdDecl in Lean.Parser.Do

def mkReassignCore (xs : Array Var) (stx : Syntax) (c : CodeBlock) : TermElabM CodeBlock := do
  let us := c.uvars
  let ws := insertVars us xs
  
  
  let code ← if xs.any fun x => !us.contains x.getId then extendUpdatedVarsAux c.code ws else pure c.code
  pure { code := .reassign xs stx code, uvars := ws }

Top in Mathlib.Order.Notation

instance [Monad m] [inst : ∀ α, Lean.Order.CCPO (m α)] [CCPOBot m] [CCPOBotLawful m] : CCPOBotLawful (ReaderT σ m) where
  prop := by admit /- proof elided -/

Lean.Expr.bindingDomain! in Lean.Expr

def WPGen.recOption {l : Type u} {m : Type u -> Type v} [Monad m] [LawfulMonad m] [CompleteBooleanAlgebra l] [MAlgOrdered m l]
   {x : m β} {y : α → m β} {val: Option α}
  (wpgx : WPGen x) (wpgy : ∀ a, WPGen (y a)) :
  WPGen (Option.recOn val x y : m β) where
  get := fun post =>
    (wpgx.get post) ⊓ (⨅ a, (wpgy a).get post)
  prop := by admit /- proof elided -/

let goal ← mkFreshExprMVar body
        let goals ← goal.mvarId!.casesRec fun localDecl => do
          return eqs.contains <| Expr.fvar localDecl.fvarId
        let goal := .mvar goals[0]!
        
        let goal' ← withinBranch i numBranches lExpr goal
        
        let dependencies := backup[i]!.toArray
        let (_, goal'') ← dependencies.foldlM (init := (0, goal')) fun (i, mainGoal) dep => do
          unless dep do
            let x := xs[i]!
            let xty ← inferType x
            let tmp ← mkAppOptM ``Loom.Matcher.simp1 #[.some lExpr, .none, .some xty, .some x]
            let goals ← mainGoal.mvarId!.apply tmp
            return (i.succ, .mvar goals[0]!)
          let (x, xty) ← match xs[i]? with
            | some x => pure (x, ← inferType x)
            | none =>
              let aa ← mkFreshExprMVar none
              let eq ← mkEq aa aa
              let rfl ← mkEqRefl aa
              pure (rfl, eq)
          let f ← mkFreshExprMVar (← mkArrow xty lExpr)
          let a ← mkFreshExprMVar lExpr
          let tmp ← mkAppOptM ``iInf_le_of_le #[.some lExpr, .some xty, .none,
            .some f   ,
            .some a, .some x]
          let goals ← mainGoal.mvarId!.apply tmp
          pure (i.succ, .mvar goals[0]!)
        
        let goal''' ←
          try
            let r ← goal''.mvarId!.rewrite (← goal''.mvarId!.getType) (← mkConstWithFreshMVarLevels matchEqnNames[i]!)
            let goal''' ← Expr.mvar <$> goal''.mvarId!.replaceTargetEq r.eNew r.eqProof
            for g in r.mvarIds do
              g.assumption
            pure goal'''
          catch _ => pure goal''
        
        let _ ← goal'''.mvarId!.applyConst ``WPGen.prop
        let res ← instantiateMVars goal
        mkLambdaFVars (xs ++ eqs) res
      trace[Loom.debug] "sub-proof for branch {i}: {br} ==> {res}"
      pure res
    let discrsRfls ← discrs.mapM (mkEqRefl ·)
    mkLambdaFVars #[postExpr] <| mkAppN partialSplitter (wpgenPostSubProofs ++ discrsRfls)

instance [Monad m] [LawfulMonad m] [_root_.CompleteLattice l]
  [inst: MAlgOrdered m l] :
  MAlgLift m l (NonDetT m) l where
    cl := by admit /- proof elided -/

StdGen in Init.Data.Random

instance : MAlgOrdered Id Prop where
  μ := id
  μ_ord_pure := by admit /- proof elided -/

LT in Init.Prelude

Function.comp in Init.Prelude

Lean.Parser.Term.doHave in Lean.Parser.Do

instance [MonadNonDet m] : ForIn m Lean.Loop Unit where
  forIn {β} _ _ init f := @MonadNonDet.rep m _ β init (f ())

def getMutTypes (stx : Array (TSyntax `leafny_binder)) : MetaM (Array (TSyntax `term)) := do
  let mut types := #[]
  for b in stx do
    match b with
    | `(leafny_binder| (mut $_:ident : $tp:term)) => do
      types := types.push tp
    | _ => pure ()
  return types

def Array.sumUpTo [Inhabited α] [AddCommMonoid β] (a : Array α) (f : ℕ -> α -> β) (bound : ℕ) : β :=
  ∑ i ∈ Finset.range bound, f i a[i]!

partial def expandLeafnyDoSeqItem (modIds : Array Ident) (stx : doSeqItem) : TermElabM (Array doSeqItem) := do
  match stx with
  
  | `(Term.doSeqItem| $stx ;) => expandLeafnyDoSeqItem modIds $ <- `(Term.doSeqItem| $stx:doElem)
  | `(Term.doSeqItem| return) => expandLeafnyDoSeqItem modIds $ <- `(Term.doSeqItem| return ())
  | `(Term.doSeqItem| return $t) =>
    let ret <-
      if modIds.size = 0 then
      `(term| $t)
     else
       `(term| ($t, $[$modIds:term],*))
    return #[<-`(Term.doSeqItem| return $ret)]
  | `(Term.doSeqItem| pure $t) =>
    let ret <-
      if modIds.size = 0 then
      `(term| $t)
     else
       `(term| ($t, $[$modIds:term],*))
    return #[<-`(Term.doSeqItem| pure $ret)]
  | `(Term.doSeqItem| if $h:ident : $t:term then $thn:doSeq else $els:doSeq) =>
    let thn <- expandLeafnyDoSeq modIds thn
    let els <- expandLeafnyDoSeq modIds els
    let ret ← `(Term.doSeqItem| if $h:ident : $t then $thn* else $els*)
    return #[ret]
  | `(Term.doSeqItem| if $h:ident : $t:term then $thn) =>
    expandLeafnyDoSeqItem modIds $ <- `(Term.doSeqItem| if $h:ident : $t:term then $thn else pure ())
  | `(Term.doSeqItem| if $t:term then $thn:doSeq) =>
    expandLeafnyDoSeqItem modIds $ <- `(Term.doSeqItem| if $t then $thn:doSeq else pure ())
  | `(Term.doSeqItem| match $discrs:matchDiscr,* with $[$alts:matchAlt]*) =>
    let alts' ← alts.mapM fun (alt : TSyntax _) => do
      match alt with
      | `(doMatchAltExpr| | $a,* => $rhs:doSeq) =>
        let rhs' ← expandLeafnyDoSeq modIds rhs
        let alt' ← `(doMatchAltExpr| | $a,* => $rhs'*)
        pure ⟨alt'.raw⟩
      | _ =>
        pure alt
    let ret ← `(Term.doSeqItem| match $discrs:matchDiscr,* with $[$alts':matchAlt]*)
    return #[ret]
  | `(Term.doSeqItem| if $t:term then $thn:doSeq else $els:doSeq) =>
    let thn <- expandLeafnyDoSeq modIds thn
    let els <- expandLeafnyDoSeq modIds els
    let ret ← `(Term.doSeqItem| if $t then $thn* else $els*)
    return #[ret]
  | `(Term.doSeqItem| let $id <- $trm:term) | `(Term.doSeqItem| let $id : $_ <- $trm:term) =>
    let `($t:ident $args:term*) := trm | pure #[stx]
    
    let ctx <- globalMutVarsCtx.get
    let .some ids := ctx[t.getId]? | pure #[stx]
    let mut mods := #[]
    let mut modArgs := #[]
    for (pos, a) in ids do
      let .some arg := args[pos]? | throwErrorAt stx s!"{t} is not fully applied"
      
      let argName := arg.raw.getId
      if argName.isAnonymous then throwErrorAt arg s!"mutable argument {arg} is not a name"
      if argName ∈ modArgs then throwErrorAt arg s!"mutable arguments cannot alias"
      modArgs := modArgs.push argName
      mods := mods.push $ <- withRef arg `(Term.doSeqItem| $(mkIdent argName):ident := $a:ident)
    return #[<-`(Term.doSeqItem| let $id <- $trm:term)] ++ mods
  | `(Term.doSeqItem| $trm:term) =>
    let id := mkIdent <| <- mkFreshUserName `ret
    let `($t:ident $args:term*) := trm | pure #[stx]
    
    let ctx <- globalMutVarsCtx.get
    let .some ids := ctx[t.getId]? | pure #[stx]
    let mut mods := #[]
    let mut modArgs := #[]
    for (pos, a) in ids do
      let .some arg := args[pos]? | throwErrorAt stx s!"{t} is not fully applied"
      
      let argName := arg.raw.getId
      if argName.isAnonymous then throwErrorAt arg s!"mutable argument {arg} is not a name"
      if argName ∈ modArgs then throwErrorAt arg s!"mutable arguments cannot alias"
      modArgs := modArgs.push argName
      mods := mods.push $ <- withRef arg `(Term.doSeqItem| $(mkIdent argName):ident := $a:ident)
    let ret <- withRef trm `(Term.doSeqItem| pure $id)
    return #[<- withRef trm `(Term.doSeqItem| let ⟨$id, _⟩ <- $trm:term)] ++ mods.push ret
  | `(Term.doSeqItem| while_some $id :| $p:term $invs* $done do $doSeq:doSeq) =>
    let doSeq <- expandLeafnyDoSeq modIds doSeq
    return #[<-`(Term.doSeqItem| while_some $id :| $p:term $invs* $done do $doSeq*)]
  | stx => pure #[stx]

Top.top in Mathlib.Order.Notation

List.nil in Init.Prelude

Lean.Syntax.mkApp in Init.Meta.Defs

def satGoalStr (solver : Option SmtSolver := none) : String := s!"{solverStr solver}the goal is false"

partial def doMatchToCode (doMatch : Syntax) (doElems: List Syntax) : M CodeBlock := do
    let ref       := doMatch
    let genParam  := doMatch[1]
    let optMotive := doMatch[2]
    let discrs    := doMatch[3]
    let matchAlts := doMatch[5][0].getArgs 
    let matchAlts ← matchAlts.foldlM (init := #[]) fun result matchAlt => return result ++ (← liftMacroM <| expandMatchAlt matchAlt)
    let alts ←  matchAlts.mapM fun matchAlt => do
      let patterns := matchAlt[1][0]
      let vars ← getPatternsVarsEx patterns.getSepArgs
      withRef patterns <| checkNotShadowingMutable vars
      let rhs  := matchAlt[3]
      let rhs ← doSeqToCode (getDoSeqElems rhs)
      pure { ref := matchAlt, vars := vars, patterns := patterns, rhs := rhs : Alt CodeBlock }
    let matchCode ← mkMatch ref genParam discrs optMotive alts
    concatWith matchCode doElems

instance : MonadLift m (NonDetT m) where
  monadLift x := NonDetT.vis x pure

def mkMatch (ref : Syntax) (genParam : Syntax) (discrs : Syntax) (optMotive : Syntax) (alts : Array (Alt CodeBlock)) : TermElabM CodeBlock := do
  
  let ws := alts.foldl (union · ·.rhs.uvars) {}
  let alts ← alts.mapM fun alt => do
    let rhs ← extendUpdatedVars alt.rhs ws
    return { ref := alt.ref, vars := alt.vars, patterns := alt.patterns, rhs := rhs.code : Alt Code }
  return { code := .match ref genParam discrs optMotive alts, uvars := ws }

default
  else
    
    x

Auto.lamFOL2SMT in Auto.Translation.LamFOL2SMT

structure JPDecl where
  name : Name
  params : Array (Var × Bool)
  body : Code

instance ExtractNonDet.pickSuchThat' {τ : Type u} (p : τ -> Prop) [Findable p] :
  ExtractNonDet Findable (MonadNonDet.pickSuchThat (m := NonDetT m) τ p) :=

inductive MatcherArgCase where | dropFirst | normal
deriving BEq, Repr, ToExpr, Inhabited

Lean.Elab.Tactic.getMainTarget in Lean.Elab.Tactic.Basic

Lean.mkIdentFrom in Init.Meta.Defs

instance [Monad m] [LawfulMonad m] [_root_.CompleteLattice l]
  [inst: MAlgOrdered m l] :
  MAlgLift m l (NonDetT m) l where
    cl := by admit /- proof elided -/

def mkForInBody  (_ : Syntax) (forInBody : CodeBlock) : M ToForInTermResult := do
  let ctx ← read
  let uvars := forInBody.uvars
  let uvars := varSetToArray uvars
  let term ← liftMacroM <| ToTerm.run forInBody.code ctx.m ctx.returnType uvars (if hasReturn forInBody.code then ToTerm.Kind.forInWithReturn else ToTerm.Kind.forIn)
  return ⟨uvars, term⟩

def Lean.Expr.getName (e : Expr) : MetaM Name := do
  match_expr e with
  | Name.mkStr _ n =>
    match n with
    | .lit (.strVal n) => return n.toName
    | _                => throwError "expected a string literal1"
  | _ => throwError "expected a string literal3"

Lean.Elab.Tactic.withMainContext in Lean.Elab.Tactic.Basic

ForIn in Init.Core

List.map in Init.Prelude

MonadLiftT in Init.Prelude

List.take in Init.Data.List.Basic

instance [LawfulMonad m] : LawfulMonad (NonDetT m) :=

Lean.PrettyPrinter.Unexpander in Init.Prelude

def chain_le (a : Array Int) (h_adjacent : ∀ k, 0 ≤ k ∧ k < a.size - 1 → a[k]! ≤ a[k+1]!)
    (i j : Nat) (h_i_ge : 0 ≤ i) (h_i_lt_j : i < j) (h_j_lt_size : j < a.size) : a[i]! ≤ a[j]! :=
  if h : j = i + 1 then by
    
    rw [h]
    exact h_adjacent i ⟨h_i_ge, by admit /- proof elided -/
    ⟩
  else by
    
    have h_next : i + 1 < j := by admit /- proof elided -/

private def letDeclArgHasBinders (letDeclArg : Syntax) : Bool :=
  let k := letDeclArg.getKind
  if k == ``Parser.Term.letPatDecl then
    false
  else if k == ``Parser.Term.letEqnsDecl then
    true
  else if k == ``Parser.Term.letIdDecl then
    
    let binders := letDeclArg[1]
    binders.getNumArgs > 0
  else
    false

Min in Init.Prelude

def CaseStudiesRoot : Array Glob :=
  #[`CaseStudies.Extension,
    `CaseStudies.Macro,
    `CaseStudies.Tactic,
    `CaseStudies.TestingUtil,
    `CaseStudies.Theory]

Lean.Meta.mkSimpEntryOfDeclToUnfold in Lean.Meta.Tactic.Simp.SimpTheorems

abbrev Except.getD {ε α} (default : ε -> α)  : Except ε α -> α
  | Except.ok p => p
  | Except.error e => default e

Ord in Init.Data.Ord.Basic

instance [Monad m] [LawfulMonad m] [_root_.CompleteLattice l]
  [inst: MAlgOrdered m l] :
  MAlgLift m l (NonDetT m) l where
    cl := by admit /- proof elided -/

def Lean.Expr.runUnfold (e : Expr) (defs : List Name) : TermElabM Expr := do
  let mut eu := e
  for name in defs do eu := (<- unfold eu name).expr
  return eu

Lean.Parser.Term.doSeqIndent in Lean.Parser.Do

abbrev WithName (α : Sort u) (name : Lean.Name := default) := α

instance [Monad m] [LawfulMonad m] [_root_.CompleteLattice l]
  [inst: MAlgOrdered m l] :
  MAlgLift m l (NonDetT m) l where
    cl := by admit /- proof elided -/

DoResultBC in Init.Core

MonadExcept.tryCatch in Init.Prelude

private def Array.andList (ts : Array (TSyntax `term)) : TermElabM (TSyntax `term) := do
  if ts.size = 0 then `(term| True) else
    let mut t := ts[0]!
    for t' in ts[1:] do
      t <- `(term| $t' ∧ $t)
    return t

def cvc5.baseUrl := "https://github.com/cvc5/cvc5/releases/download"

def LogicLift.refl [CompleteLattice l] : LogicLift l l where
  lift := by admit /- proof elided -/

def decodeStrLeanList (encoded_str: List Encoding) : List Char :=
  match encoded_str with
  | List.nil => List.nil
  | List.cons hd tl => (List.replicate hd.cnt hd.c) ++ (decodeStrLeanList tl)

def concat (terminal : CodeBlock) (kRef : Syntax) (y? : Option Var) (k : CodeBlock) : TermElabM CodeBlock := do
  unless hasTerminalAction terminal.code do
    throwErrorAt kRef "`do` element is unreachable"
  let (terminal, k) ← homogenize terminal k
  let xs := varSetToArray k.uvars
  let y ← match y? with | some y => pure y | none => `(y)
  let ps := xs.map fun x => (x, true)
  let ps := ps.push (y, false)
  let jpDecl ← mkFreshJP ps k.code
  let jp := jpDecl.name
  let terminal ← liftMacroM <| convertTerminalActionIntoJmp terminal.code jp xs
  return { code  := attachJP jpDecl terminal, uvars := k.uvars }

private def _root_.Lean.SimpleScopedEnvExtension.modify
  (ext : SimpleScopedEnvExtension α σ) (s : σ -> σ)
  [Monad m] [MonadEnv m] : m Unit := do
  Lean.modifyEnv (ext.modifyState · s)

def anyGoalsWithTag (tactic : MVarId → TacticM (Option (Name × α))) : TacticM (Array α) := do
  let mvarIds ← getGoals
  let mut res : Array α := #[]
  let mut mvarIdsNew : Array MVarId := #[]
  for mvarId in mvarIds do
    unless (← mvarId.isAssigned) do
      setGoals [mvarId]
      try
        let r ← tactic mvarId
        let goals <- getUnsolvedGoals
        if let some ⟨n, r⟩ := r then
          for goal in goals do
            goal.setTag n
          res := res.push r
          mvarIdsNew := mvarIdsNew ++ goals
      catch _ =>
        mvarIdsNew := mvarIdsNew.push mvarId
  setGoals mvarIdsNew.toList
  pure res

Lean.Environment in Lean.Environment

Array.flatten in Init.Data.Array.Basic

structure VelvetTestingCtx extends VelvetObligation where
   
  modBinders : Array (TSyntax `Lean.Parser.Term.bracketedBinder)
  retType : Term
deriving Inhabited

noncomputable def LE.pure {l : Type u} [inst: LE l] [OrderTop l] [OrderBot l] : Prop -> l := fun p =>
  if p then ⊤ else ⊥

Unit in Init.Prelude

partial def doForToCode (doFor : Syntax) (doElems : List Syntax) : M CodeBlock := do
    let doForDecls := doFor[1].getSepArgs
    if h : doForDecls.size > 1 then
       
      
      let doForDecl := doForDecls[1]!
      unless doForDecl[0].isNone do
        throwErrorAt doForDecl[0] "the proof annotation here has not been implemented yet"
      let y  := doForDecl[1]
      let ys := doForDecl[3]
      let doForDecls := doForDecls.eraseIdx 1
      let body := doFor[3]
      withFreshMacroScope do
         
        let toStreamApp ← withRef ys `(@toStream _ _ _ $ys)
        let auxDo ←
          `(do let mut s := $toStreamApp:term
               for $doForDecls:doForDecl,* do
                 match @Stream.next? _ _ _ s with
                 | none => break
                 | some ($y, s') =>
                   s := s'
                   do $body)
        doSeqToCode (getDoSeqElems (getDoSeq auxDo) ++ doElems)
    else withRef doFor do
      let h?        := if doForDecls[0]![0].isNone then none else some doForDecls[0]![0][0]
      let x         := doForDecls[0]![1]
      withRef x <| checkNotShadowingMutable (← getPatternVarsEx x)
      let xs        := doForDecls[0]![3]
      let forElems  := getDoSeqElems doFor[3]
      let forInBodyCodeBlock ← withFor (doSeqToCode forElems)
      let ⟨uvars, forInBody⟩ ← mkForInBody x forInBodyCodeBlock
      let ctx ← read
      
      
      
      let uvars := uvars.map fun v => ctx.mutableVars.getD v.getId v
      let uvarsTuple ← liftMacroM do mkTuple uvars
      let uvarsTuplePat ← liftMacroM do mkTuplePat uvars

def getSpecKey (ty : Expr) : MetaM (Expr × Loom.SpecType) := do
  let (_xs, _bis, body) ← forallMetaTelescope ty
  let x <- match_expr body with
  | triple _m _mInst _α _l _lInst _mPropInst _pre x _post => pure (x, .triple)
  | WPGen _m _mInst _α _l _lInst _mPropInst x => pure (x, .wpgen)
  | _ => throwError s!"not a triple: {body}"
  
  
  
  
  return x

instance MAlgLiftTRefl [Monad m] [CompleteLattice l] [LawfulMonad m] [MAlgOrdered m l] :
  MAlgLiftT m l m l where
    μ_lift := by admit /- proof elided -/

def   NonDetT.wp {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : {α : Type u} -> NonDetT m α -> Cont l α
  | _, .pure ret => pure ret
  | _, .vis x f => fun post => _root_.wp x fun a => wp (f a) post
  | _, @NonDetT.pickCont _ _ _ p _ f => fun post => ⨆ a, ⌜p a⌝ ⊓ wp (f a) post
  | _, .repeatCont init f cont => fun post => ⨆ (inv : ForInStep _ -> l) (measure : _ -> Nat),
    ⌜ ∀ b, (inv (ForInStep.yield b)) <= wp (f b) (fun | .yield b' => inv (.yield b') ⊓ ⌜ measure b' < measure b ⌝ | .done b' => inv (.done b'))⌝ ⊓
    spec (inv (.yield init)) (fun b => inv (.done b)) (fun b => wp (cont b) post)

Lean.Meta.unfoldDefinition? in Lean.Meta.WHNF

noncomputable
</all_available_defs>

<all_available_lemmas>
lemma MAlgOrdered.bind {α : Type u} {m} {l : Type u} [Monad m] [CompleteLattice l] [MAlgOrdered m l] :
    ∀ (x : m α) (f g : α -> m l), μ ∘ f = μ ∘ g ->
     μ (x >>= f) = μ (x >>= g)

omit [LawfulMonad m] in
lemma wp_wlp (c : m α) (post : α -> l) :
  wp c post <= wlp c post

lemma p_findNat_some (p : Nat -> Prop) [DecidablePred p] (i : Nat) :
  p i -> ∃ j, p j ∧ j <= i ∧ findNat p = some j

lemma ExtractNonDet.extract_refines (pre : l) (s : NonDetT m α) :
  triple pre s post ->
  triple pre s.run post

@[local simp]
private lemma compl_fun_true {α} :
  (fun (_ : α) => ⊤)ᶜ = fun _ => (⊥ : l)

@[local simp]
lemma DivM.total_decompose (α : Type) (x : DivM α) (post₁ post₂ : α -> Prop) :
  ([totl| wp x post₁] ∧ [part| wp x post₂]) = [totl| wp x (post₁ ⊓ post₂)]

omit [MAlgDet m l] in
lemma ExtractNonDet.extract_refines_wp_weak (s : NonDetT m α) (inst : ExtractNonDet WeakFindable s) :
  wp s.extractWeak post <= wp s post

lemma MonadNonDet.wp_pickSuchThat {τ : Type u} (p : τ → Prop) post :
  _root_.wp (MonadNonDet.pickSuchThat (m := NonDetT m) τ p) post = ⨆ a, ⌜p a⌝ ⊓ post a

lemma triple_bind {β} (pre : l) (x : m α) (cut : α -> l)
  (f : α -> m β) (post : β -> l) :
  triple pre x cut ->
  (∀ y, triple (cut y) (f y) post) ->
  triple pre (x >>= f) post

@[aesop unsafe]
theorem non_zero_length (q: List Nat) (hnemp: q.nonEmpty): 0 < q.length

@[ext]
lemma W_ext (t : Type v) (α : Type u) [Preorder t] (w w' : W t α) :
  w.wp = w'.wp → w = w'

@[simp]
lemma lift_cont_eq {l σ : Type u} [CompleteLattice l] [CompleteLattice σ] (c : Cont l α) :
  liftM (m := Cont l) (n := Cont (σ -> l)) c = fun post s => c (post · s)

omit [MAlgDet m l] in
lemma ExtractNonDet.extract_refines_triple_weak [∀ α, CCPO (m α)] [MAlgPartial m] [CCPOBotLawful m] (pre : l) (s : NonDetT m α) (inst : ExtractNonDet WeakFindable s) :
  triple pre s post ->
  triple pre s.extractWeak post

lemma StateT.wp_set {res: σ} (post : PUnit -> σ -> l) :
  wp (set (m := StateT σ m) res) post = fun _ => post PUnit.unit res

theorem spv_dot_pure (spv1 spv2: SpV Int) (n: ℕ)
  (sz1: ∀ i < spv1.size, spv1.ind[i]! < n) (sz2: ∀ i < spv2.size, spv2.ind[i]! < n):
    spv_dot spv1 spv2 0 0 = ∑ i ∈ Finset.range n, spv1[i] * spv2[i]

lemma ExceptT.wlp_eq (c : ExceptT ε m α) (post : α -> l) :
  [totl| wlp c post] = [part| wlp c post]

@[solverHint]
theorem spv_dot_exh
  (spv1 spv2: SpV Int) (pnt1 pnt2: ℕ):
  pnt1 = spv1.size ∨ pnt2 = spv2.size → spv_dot spv1 spv2 pnt1 pnt2 = 0

theorem triple_forIn_range {β}
  (xs : Std.Range) (init : β) (f : ℕ → β → m (ForInStep β))
  (inv : ℕ → β → l)
  (hstep : ∀ i b,
    triple
      (inv i b)
      (f i b)
      (fun | .yield b' => inv (i + xs.step) b' | .done b' => inv (xs.start + ((xs.stop - xs.start + xs.step - 1) / xs.step) * xs.step) b')) :
  triple (inv xs.start init) (forIn xs init f) (inv (xs.start + ((xs.stop - xs.start + xs.step - 1) / xs.step) * xs.step))

omit [LawfulMonad m] in
lemma WPGen.intro (x : m α) (wpg : WPGen x) :
  pre <= wpg.get post ->
  -- pre <= wpg.sideCond ->
  triple pre x post

lemma MAlg.lift_ExceptT ε (hd : ε -> Prop) [IsHandler hd] [CompleteLattice l] [inst: MAlgOrdered m l]
   (c : ExceptT ε m α) post :
  MAlg.lift c post = MAlg.lift (m := m) c (fun | .ok x => post x | .error e => ⌜hd e⌝)

@[simp]
lemma NonDetT.wp_pickCont {τ : Type u} p (f : τ → NonDetT m α) post :
  _root_.wp (NonDetT.pickCont τ p f) post = ⨅ a, ⌜p a⌝ ⇨ _root_.wp (f a) post

omit [CCPOBot m] [MAlgDet m l] in
lemma NonDetT.prop_mono (x : NonDetT m α) post post' :
  post <= post' -> x.prop post <= x.prop post'

lemma SumDigits_zero : SumDigits 0 = 0

lemma findNat_none (p : Nat -> Prop) [DecidablePred p] :
  (findNat p).isNone -> ∀ i, ¬ p i

omit [MAlgDet m l] in
lemma StateT.wlp_eq (c : StateT σ m α) (post : α -> σ -> l) :
  wlp c post = fun s => wlp (m := m) (c s) (fun xs => post xs.1 xs.2)

theorem SpMSpV_correct_triple
  (out: Array Int)
  (spm: Array (SpV Int))
  (spv: SpV Int)
  (n: ℕ):
  triple
    (∀ i < spm.size, (∀ j < spm[i]!.size, spm[i]!.ind[j]! < n) ∧ (∀ j < spv.size, spv.ind[j]! < n))
    (SpMSpV out spm spv)
    fun ⟨_, outNew⟩ =>
      outNew.size = spm.size ∧ ∀ i < spm.size, outNew[i]! = ∑ idx ∈ Finset.range n, spm[i]![idx]! * spv[idx]!

lemma wp_part_eq ε (c : ExceptT ε m α) post :
  wp c post = wp (m := m) c (fun | .ok x => post x | .error _ => ⊤)

lemma wp_compl'  (c : m α) post (wp_bot : ∀ α (c : m α), wp c ⊥ = ⊥) :
  wp c post <= (wp c postᶜ)ᶜ

lemma exists_findNat (p : Nat -> Prop) [DecidablePred p] :
  (∃ x, p x) ↔ (findNat p).isSome

@[local simp]
lemma wp_mono_part (x : NonDetT DivM α) (post₁ post₂ : α -> Prop) :
  (post₁ ≤ post₂) → ([totl|wp x post₁]) ≤ ([part| wp x post₂])

lemma Cont.monotone_lift {l : Type u} {m : Type u -> Type v} [Monad m] [LawfulMonad m] [CompleteLattice l] [MAlgOrdered m l] :
  ∀ {α : Type u} (x : m α), MAlg.lift x |>.monotone

@[grind, solverHint]
lemma sumUpToSucc
  (spv : SpV Int)
  (v : Array Int) (bound : ℕ) :
  sumUpTo spv v (bound + 1) = (sumUpTo spv v bound) +
    ((spv.val)[bound]! * v[(spv.ind)[bound]!]!)

omit [MAlgDet m l] in
lemma ExtractNonDet.extract_refines_wp_weak [∀ α, CCPO (m α)] [MAlgPartial m] [CCPOBotLawful m] (s : NonDetT m α) (inst : ExtractNonDet WeakFindable s) :
  wp s post <= wp s.extractWeak post

omit [MAlgDet m l] in
lemma wlp_cons (c : m α) (post post' : α -> l) :
  post <= post' ->
  wlp c post <= wlp c post'

lemma triple_spec (pre : l) (c : m α) (post : α -> l) :
  spec pre post <= wp c <->
  triple pre c post

@[simp]
lemma pure_intro_l {l : Type u} [CompleteLattice l] (x y : l) :
  (x ⊓ ⌜ p ⌝ <= y) = (p -> x <= y)

omit [MonoBind m] [LawfulMonad m] in
lemma wp_csup (xc : Set (m α)) (post : α -> l) :
  Lean.Order.chain xc ->
  ⨅ c ∈ xc, wp c post ≤ wp (Lean.Order.CCPO.csup xc) post

theorem VSpV_correct_triple (out: Array Int) (arr: Array Int) (spv: SpV Int):
  triple
    (∀ i < spv.size, spv.ind[i]! < arr.size)
    (VSpV out arr spv)
    fun ⟨_, outNew⟩ =>
      outNew[0]! = ∑ i ∈ Finset.range (arr.size), spv[i] * arr[i]!

lemma wlp_bind {β} (x : m α) (f : α -> m β) (post : β -> l) :
  wlp (x >>= f) post = wlp x (fun x => wlp (f x) post)
</all_available_lemmas>

<local_ctx>
import Mathlib.Logic.Function.Basic

import Mathlib.Order.CompleteBooleanAlgebra

import Mathlib.Order.Lattice

import Mathlib.Order.Basic

import Loom.MonadAlgebras.WP.Basic

import Loom.MonadAlgebras.WP.Tactic

import Loom.MonadAlgebras.WP.Gen

import Loom.MonadAlgebras.NonDetT.Findable

section NonDeterministicTransformer

inductive NonDetT (m : Type u -> Type v) : (α : Type u) -> Type _ where
  | pure {α} (ret : α) : NonDetT m α
  | vis {α} {β} (x : m β) (f : β → NonDetT m α) : NonDetT m α
  | pickCont {α} (τ : Type u) (p : τ -> Prop) [Findable p] (f : τ → NonDetT m α) : NonDetT m α
  | repeatCont {α} {β} (init : β) (f : β -> NonDetT m (ForInStep β)) (cont : β -> NonDetT m α) : NonDetT m α

variable {m : Type u -> Type v} {α β : Type u} [Monad m]

def NonDetT.bind (x : NonDetT m α) (f : α → NonDetT m β) : NonDetT m β :=
  match x with
  | pure ret => f ret
  | vis x f' => vis x fun y => bind (f' y) f
  | pickCont τ p f' => pickCont τ p fun t => bind (f' t) f
  | repeatCont init f' cont => repeatCont init f' fun t => bind (cont t) f

variable [CompleteBooleanAlgebra l] [MAlgOrdered m l]

namespace PartialCorrectness

namespace DemonicChoice

/- WP for NonDetT -/
noncomputable

def NonDetT.wp {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : {α : Type u} -> NonDetT m α -> Cont l α
  | _, .pure ret => pure ret
  | _, .vis x f => fun post => _root_.wp x fun a => wp (f a) post
  | _, @NonDetT.pickCont _ _ τ p _ f =>
  fun post =>
    let p : Set τ := p;
    ⨅ a ∈ (p : Set τ), wp (f a) post
  | _, @NonDetT.repeatCont _ _ β init f cont => fun post => ⨆ (inv : ForInStep β -> l),
      ⌜ ∀ b, (inv (ForInStep.yield b)) <= wp (f b) inv⌝ ⊓
      spec (inv (.yield init)) (fun b => inv (.done b)) (fun b => wp (cont b) post)

def NonDetT.μ  {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : NonDetT m l -> l := fun x => NonDetT.wp x id

variable [LawfulMonad m]

/- Ordered Monad Algebra instance for NonDetT -/
noncomputable
scoped

end DemonicChoice

namespace AngelicChoice

/- WP for NonDetT -/
noncomputable

def NonDetT.wp {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : {α : Type u} -> NonDetT m α -> Cont l α
  | _, .pure ret => pure ret
  | _, .vis x f => fun post => _root_.wp x fun a => wp (f a) post
  | _, @NonDetT.pickCont _ _ τ p _ f => fun post => ⨆ a, ⌜p a⌝ ⊓ wp (f a) post
  | _, .repeatCont init f cont => fun post => ⨆ (inv : ForInStep _ -> l),
    ⌜ ∀ b, (inv (ForInStep.yield b)) <= wp (f b) inv⌝ ⊓
    spec (inv (.yield init)) (fun b => inv (.done b)) (fun b => wp (cont b) post)

def NonDetT.μ {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : NonDetT m l -> l := fun x => NonDetT.wp x id

variable [LawfulMonad m]

/- Ordered Monad Algebra instance for NonDetT -/
noncomputable
scoped

end AngelicChoice

end PartialCorrectness

namespace TotalCorrectness

namespace DemonicChoice

/- WP for NonDetT -/
noncomputable

def NonDetT.wp {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : {α : Type u} -> NonDetT m α -> Cont l α
  | _, .pure ret => pure ret
  | _, .vis x f => fun post => _root_.wp x fun a => wp (f a) post
  | _, @NonDetT.pickCont _ _ τ p _ f => fun post => let p : Set τ := p; ⌜∃ h, p h⌝ ⊓  ⨅ a ∈ (p : Set τ), wp (f a) post
  | _, @NonDetT.repeatCont _ _ β init f cont => fun post => ⨆ (inv : ForInStep β -> l) (measure : β -> Nat),
      ⌜ ∀ b, (inv (ForInStep.yield b)) <= wp (f b) (fun | .yield b' => inv (.yield b') ⊓ ⌜ measure b' < measure b ⌝ | .done b' => inv (.done b'))⌝ ⊓
      spec (inv (.yield init)) (fun b => inv (.done b)) (fun b => wp (cont b) post)

def NonDetT.μ {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : NonDetT m l -> l := fun x => NonDetT.wp x id

variable [LawfulMonad m]

/- Ordered Monad Algebra instance for NonDetT -/
noncomputable
scoped
</local_ctx>

<target_theorem>
@[simp]
lemma NonDetT.wp_pickCont {τ : Type u} p (f : τ → NonDetT m α) [Findable p] post :
  _root_.wp (NonDetT.pickCont τ p f) post = ⌜∃ h, p h⌝ ⊓ ⨅ a, ⌜p a⌝ ⇨ _root_.wp (f a) post :=
</target_theorem>

