aver/codegen/lean/

mod.rs

/// Lean 4 backend for the Aver transpiler.
///
/// Transpiles only pure core logic: functions without effects, type definitions,
/// verify blocks (as `example` proofs), and decision blocks (as comments).
/// Effectful functions and `main` are skipped.
mod builtins;
mod expr;
mod law_auto;
mod pattern;
pub(crate) mod recurrence;
mod shared;
mod toplevel;
mod types;

use std::collections::{HashMap, HashSet};

use crate::ast::{FnDef, Spanned, TopLevel, VerifyKind};
use crate::codegen::{CodegenContext, ProjectOutput};

/// How verify blocks should be emitted in generated Lean.
#[derive(Clone, Copy, Debug, Eq, PartialEq)]
pub enum VerifyEmitMode {
    /// `example : lhs = rhs := by native_decide`
    NativeDecide,
    /// `example : lhs = rhs := by sorry`
    Sorry,
    /// Named theorem stubs:
    /// `theorem <fn>_verify_<n> : lhs = rhs := by`
    /// `  sorry`
    TheoremSkeleton,
}

// RecursionPlan / ProofModeIssue moved to shared `crate::codegen::recursion`
// so the Dafny backend can reuse the same classifier. Re-export here so
// existing `lean::RecursionPlan` / `lean::ProofModeIssue` call sites keep
// working without churn.
pub use crate::codegen::recursion::{ProofModeIssue, RecursionPlan};

const LEAN_PRELUDE_HEADER: &str = r#"-- Generated by the Aver → Lean 4 transpiler
-- Pure core logic plus Oracle-lifted classified effects

set_option linter.unusedVariables false

-- Prelude: helper definitions for Aver builtins"#;

const LEAN_PRELUDE_FLOAT_COE: &str = r#"instance : Coe Int Float := ⟨fun n => Float.ofInt n⟩

def Float.fromInt (n : Int) : Float := Float.ofInt n

-- Aver's Float-to-Int operations match the runtime semantics
-- (`f64::floor() as i64` in VM, Rust codegen, WASM — all three use the
-- same IEEE 754 floor/round/ceil followed by Rust's saturating
-- `f64 as i64` cast):
--   * finite values within [i64::MIN, i64::MAX]: truncate toward zero
--   * finite > i64::MAX:              saturate to i64::MAX
--   * finite < i64::MIN:              saturate to i64::MIN
--   * +Inf:                           saturate to i64::MAX
--   * -Inf:                           saturate to i64::MIN
--   * NaN:                            0 (Rust 1.45+ defined behavior)
--
-- Lean's `Float.floor : Float → Float` doesn't directly satisfy Aver's
-- `Float.floor : Float → Int`, so we synthesize via the saturating
-- `Float.toUInt64` (returns 0 for NaN/negative) with sign handling and
-- explicit bounds. Per-case correctness is asserted by `native_decide`
-- examples below; total semantic agreement with `f64 as i64` would
-- need a formal IEEE spec in Lean, which is out of scope.
--
-- Asymmetry with the Dafny backend: Lean has IEEE 754 `Float` natively
-- (`double` at runtime), so we use it. Dafny only offers mathematical
-- `real` (Cauchy-style, no NaN/Inf/overflow), which is a fundamental
-- type mismatch with Aver's IEEE Float — Dafny operations stay opaque
-- (`function FloatPi(): real` etc.) rather than synthesizing IEEE on
-- top of `bv64`, which would mean implementing the entire IEEE
-- arithmetic in Dafny by hand.
namespace AverFloat
def toInt (x : Float) : Int :=
  if x.isNaN then 0
  -- 2^63 is exactly representable in f64; values ≥ that saturate up.
  else if x ≥ 9223372036854775808.0 then 9223372036854775807
  -- -2^63 is exactly representable; values strictly below saturate down.
  else if x < -9223372036854775808.0 then -9223372036854775808
  else if x ≥ 0.0 then Int.ofNat x.toUInt64.toNat
  else -(Int.ofNat (-x).toUInt64.toNat)

def floor (x : Float) : Int := toInt x.floor
def ceil  (x : Float) : Int := toInt x.ceil
def round (x : Float) : Int := toInt x.round

def pow (x y : Float) : Float := x ^ y

-- Edge-case smoke checks: each `example` is discharged by reduction,
-- so any drift from these documented values fails Lake build.
example : AverFloat.toInt 0.0 = 0                                := by native_decide
example : AverFloat.toInt 3.7 = 3                                := by native_decide
example : AverFloat.toInt (-3.7) = -3                            := by native_decide
example : AverFloat.toInt (1.0 / 0.0) = 9223372036854775807      := by native_decide
example : AverFloat.toInt (-1.0 / 0.0) = -9223372036854775808    := by native_decide
example : AverFloat.toInt (0.0 / 0.0) = 0                        := by native_decide
example : AverFloat.floor 3.7 = 3                                := by native_decide
example : AverFloat.floor (-3.7) = -4                            := by native_decide
example : AverFloat.ceil  3.2 = 4                                := by native_decide
example : AverFloat.ceil  (-3.2) = -3                            := by native_decide
-- Rounding mode (half-away-from-zero, matching Rust's `f64::round`):
example : AverFloat.round 0.5 = 1                                := by native_decide
example : AverFloat.round (-0.5) = -1                            := by native_decide
example : AverFloat.round 2.5 = 3                                := by native_decide
example : AverFloat.round (-2.5) = -3                            := by native_decide
end AverFloat"#;

const LEAN_PRELUDE_FLOAT_DEC_EQ: &str = r#"private unsafe def Float.unsafeDecEq (a b : Float) : Decidable (a = b) :=
  if a == b then isTrue (unsafeCast ()) else isFalse (unsafeCast ())
@[implemented_by Float.unsafeDecEq]
private opaque Float.compDecEq (a b : Float) : Decidable (a = b)
instance : DecidableEq Float := Float.compDecEq"#;

const LEAN_PRELUDE_EXCEPT_DEC_EQ: &str = r#"instance [DecidableEq ε] [DecidableEq α] : DecidableEq (Except ε α)
  | .ok a, .ok b =>
    if h : a = b then isTrue (h ▸ rfl) else isFalse (by intro h'; cases h'; exact h rfl)
  | .error a, .error b =>
    if h : a = b then isTrue (h ▸ rfl) else isFalse (by intro h'; cases h'; exact h rfl)
  | .ok _, .error _ => isFalse (by intro h; cases h)
  | .error _, .ok _ => isFalse (by intro h; cases h)"#;

const LEAN_PRELUDE_EXCEPT_NS: &str = r#"namespace Except
def withDefault (r : Except ε α) (d : α) : α :=
  match r with
  | .ok v => v
  | .error _ => d
end Except"#;

const LEAN_PRELUDE_OPTION_TO_EXCEPT: &str = r#"def Option.toExcept (o : Option α) (e : ε) : Except ε α :=
  match o with
  | some v => .ok v
  | none => .error e"#;

const LEAN_PRELUDE_STRING_HADD: &str = r#"instance : HAdd String String String := ⟨String.append⟩"#;

/// Oracle v1: BranchPath mirrors the Aver-source opaque builtin. The
/// dewey-decimal string under the hood is not user-observable — users
/// construct paths through `.root`, `.child`, `.parse`.
const LEAN_PRELUDE_BRANCH_PATH: &str = r#"structure BranchPath where
  dewey : String
  deriving Repr, BEq, DecidableEq

def BranchPath.Root : BranchPath := { dewey := "" }

def BranchPath.child (p : BranchPath) (idx : Int) : BranchPath :=
  if p.dewey.isEmpty then { dewey := toString idx }
  else { dewey := p.dewey ++ "." ++ toString idx }

def BranchPath.parse (s : String) : BranchPath := { dewey := s }"#;

const LEAN_PRELUDE_PROOF_FUEL: &str = r#"def averStringPosFuel (s : String) (pos : Int) (rankBudget : Nat) : Nat :=
  (((s.data.length) - pos.toNat) + 1) * rankBudget"#;

const LEAN_PRELUDE_AVER_MEASURE: &str = r#"namespace AverMeasure
def list (elemMeasure : α → Nat) : List α → Nat
  | [] => 1
  | x :: xs => elemMeasure x + list elemMeasure xs + 1
def option (elemMeasure : α → Nat) : Option α → Nat
  | none => 1
  | some x => elemMeasure x + 1
def except (errMeasure : ε → Nat) (okMeasure : α → Nat) : Except ε α → Nat
  | .error e => errMeasure e + 1
  | .ok v => okMeasure v + 1
end AverMeasure"#;

const AVER_MAP_PRELUDE_BASE: &str = r#"namespace AverMap
def empty : List (α × β) := []
def get [DecidableEq α] (m : List (α × β)) (k : α) : Option β :=
  match m with
  | [] => none
  | (k', v) :: rest => if k = k' then some v else AverMap.get rest k
def set [DecidableEq α] (m : List (α × β)) (k : α) (v : β) : List (α × β) :=
  let rec go : List (α × β) → List (α × β)
    | [] => [(k, v)]
    | (k', v') :: rest => if k = k' then (k, v) :: rest else (k', v') :: go rest
  go m
def has [DecidableEq α] (m : List (α × β)) (k : α) : Bool :=
  m.any (fun p => decide (k = p.1))
def remove [DecidableEq α] (m : List (α × β)) (k : α) : List (α × β) :=
  m.filter (fun p => !(decide (k = p.1)))
def keys (m : List (α × β)) : List α := m.map Prod.fst
def values (m : List (α × β)) : List β := m.map Prod.snd
def entries (m : List (α × β)) : List (α × β) := m
def len (m : List (α × β)) : Nat := m.length
def fromList (entries : List (α × β)) : List (α × β) := entries"#;

const AVER_MAP_PRELUDE_HAS_SET_SELF: &str = r#"private theorem any_set_go_self [DecidableEq α] (k : α) (v : β) :
    ∀ (m : List (α × β)), List.any (AverMap.set.go k v m) (fun p => decide (k = p.1)) = true := by
  intro m
  induction m with
  | nil =>
      simp [AverMap.set.go, List.any]
  | cons p tl ih =>
      cases p with
      | mk k' v' =>
          by_cases h : k = k'
          · simp [AverMap.set.go, List.any, h]
          · simp [AverMap.set.go, List.any, h, ih]

theorem has_set_self [DecidableEq α] (m : List (α × β)) (k : α) (v : β) :
    AverMap.has (AverMap.set m k v) k = true := by
  simpa [AverMap.has, AverMap.set] using any_set_go_self k v m"#;

const AVER_MAP_PRELUDE_GET_SET_SELF: &str = r#"private theorem get_set_go_self [DecidableEq α] (k : α) (v : β) :
    ∀ (m : List (α × β)), AverMap.get (AverMap.set.go k v m) k = some v := by
  intro m
  induction m with
  | nil =>
      simp [AverMap.set.go, AverMap.get]
  | cons p tl ih =>
      cases p with
      | mk k' v' =>
          by_cases h : k = k'
          · simp [AverMap.set.go, AverMap.get, h]
          · simp [AverMap.set.go, AverMap.get, h, ih]

theorem get_set_self [DecidableEq α] (m : List (α × β)) (k : α) (v : β) :
    AverMap.get (AverMap.set m k v) k = some v := by
  simpa [AverMap.set] using get_set_go_self k v m"#;

const AVER_MAP_PRELUDE_GET_SET_OTHER: &str = r#"private theorem get_set_go_other [DecidableEq α] (k key : α) (v : β) (h : key ≠ k) :
    ∀ (m : List (α × β)), AverMap.get (AverMap.set.go k v m) key = AverMap.get m key := by
  intro m
  induction m with
  | nil =>
      simp [AverMap.set.go, AverMap.get, h]
  | cons p tl ih =>
      cases p with
      | mk k' v' =>
          by_cases hk : k = k'
          · have hkey : key ≠ k' := by simpa [hk] using h
            simp [AverMap.set.go, AverMap.get, hk, hkey]
          · by_cases hkey : key = k'
            · simp [AverMap.set.go, AverMap.get, hk, hkey]
            · simp [AverMap.set.go, AverMap.get, hk, hkey, ih]

theorem get_set_other [DecidableEq α] (m : List (α × β)) (k key : α) (v : β) (h : key ≠ k) :
    AverMap.get (AverMap.set m k v) key = AverMap.get m key := by
  simpa [AverMap.set] using get_set_go_other k key v h m"#;

const AVER_MAP_PRELUDE_HAS_SET_OTHER: &str = r#"theorem has_eq_isSome_get [DecidableEq α] (m : List (α × β)) (k : α) :
    AverMap.has m k = (AverMap.get m k).isSome := by
  induction m with
  | nil =>
      simp [AverMap.has, AverMap.get]
  | cons p tl ih =>
      cases p with
      | mk k' v' =>
          by_cases h : k = k'
          · simp [AverMap.has, AverMap.get, List.any, h]
          · simpa [AverMap.has, AverMap.get, List.any, h] using ih

theorem has_set_other [DecidableEq α] (m : List (α × β)) (k key : α) (v : β) (h : key ≠ k) :
    AverMap.has (AverMap.set m k v) key = AverMap.has m key := by
  rw [AverMap.has_eq_isSome_get, AverMap.has_eq_isSome_get]
  simp [AverMap.get_set_other, h]"#;

const AVER_MAP_PRELUDE_END: &str = r#"end AverMap"#;

const LEAN_PRELUDE_AVER_LIST: &str = r#"namespace AverList
def get (xs : List α) (i : Int) : Option α :=
  if i < 0 then none else xs.get? i.toNat
private def insertSorted [Ord α] (x : α) : List α → List α
  | [] => [x]
  | y :: ys =>
    if compare x y == Ordering.lt || compare x y == Ordering.eq then
      x :: y :: ys
    else
      y :: insertSorted x ys
def sort [Ord α] (xs : List α) : List α :=
  xs.foldl (fun acc x => insertSorted x acc) []
end AverList"#;

// Built-in record types (Header, HttpResponse, HttpRequest,
// Tcp.Connection, Terminal.Size) used to live as hard-coded literals
// here. They now live in `crate::codegen::builtin_records` —
// declarative descriptions consumed by Lean, Dafny, and WASM via
// shared `needed_records()` and `render_lean()`. Drift between
// backends is no longer possible.

const LEAN_PRELUDE_STRING_HELPERS: &str = r#"def String.charAt (s : String) (i : Int) : Option String :=
  if i < 0 then none
  else (s.toList.get? i.toNat).map Char.toString
theorem String.charAt_length_none (s : String) : String.charAt s s.length = none := by
  have hs : ¬ ((s.length : Int) < 0) := by omega
  unfold String.charAt
  simp [hs]
  change s.data.length ≤ s.length
  exact Nat.le_refl _
def String.slice (s : String) (start stop : Int) : String :=
  let startN := if start < 0 then 0 else start.toNat
  let stopN := if stop < 0 then 0 else stop.toNat
  let chars := s.toList
  String.mk ((chars.drop startN).take (stopN - startN))
private def trimFloatTrailingZerosChars (chars : List Char) : List Char :=
  let noZeros := (chars.reverse.dropWhile (fun c => c == '0')).reverse
  match noZeros.reverse with
  | '.' :: rest => rest.reverse
  | _ => noZeros
private def normalizeFloatString (s : String) : String :=
  if s.toList.any (fun c => c == '.') then
    String.mk (trimFloatTrailingZerosChars s.toList)
  else s
def String.fromFloat (f : Float) : String := normalizeFloatString (toString f)
def String.chars (s : String) : List String := s.toList.map Char.toString
def String.containsSubstr (haystack needle : String) : Bool :=
  if needle.length == 0 then true
  else decide ((haystack.splitOn needle).length > 1)
private theorem char_to_string_append_mk (c : Char) (chars : List Char) :
    Char.toString c ++ String.mk chars = String.mk (c :: chars) := by
  rfl
private theorem string_intercalate_empty_char_strings_go (acc : String) (chars : List Char) :
    String.intercalate.go acc "" (chars.map Char.toString) = acc ++ String.mk chars := by
  induction chars generalizing acc with
  | nil =>
      simp [String.intercalate.go]
  | cons c rest ih =>
      calc
        String.intercalate.go acc "" ((c :: rest).map Char.toString)
            = String.intercalate.go (acc ++ Char.toString c) "" (rest.map Char.toString) := by
                simp [String.intercalate.go]
        _ = (acc ++ Char.toString c) ++ String.mk rest := by
                simpa using ih (acc ++ Char.toString c)
        _ = acc ++ String.mk (c :: rest) := by
                simp [String.append_assoc, char_to_string_append_mk]
private theorem string_intercalate_empty_char_strings (chars : List Char) :
    String.intercalate "" (chars.map Char.toString) = String.mk chars := by
  cases chars with
  | nil =>
      simp [String.intercalate]
  | cons c rest =>
      simpa [String.intercalate] using string_intercalate_empty_char_strings_go c.toString rest
theorem String.intercalate_empty_chars (s : String) :
    String.intercalate "" (String.chars s) = s := by
  cases s with
  | mk chars =>
      simpa [String.chars] using string_intercalate_empty_char_strings chars
namespace AverString
def splitOnCharGo (currentRev : List Char) (sep : Char) : List Char → List String
  | [] => [String.mk currentRev.reverse]
  | c :: cs =>
      if c == sep then
        String.mk currentRev.reverse :: splitOnCharGo [] sep cs
      else
        splitOnCharGo (c :: currentRev) sep cs
def splitOnChar (s : String) (sep : Char) : List String :=
  splitOnCharGo [] sep s.toList
def split (s delim : String) : List String :=
  match delim.toList with
  | [] => "" :: (s.toList.map Char.toString) ++ [""]
  | [c] => splitOnChar s c
  | _ => s.splitOn delim
@[simp] private theorem char_toString_data (c : Char) : c.toString.data = [c] := by
  rfl
private theorem splitOnCharGo_until_sep
    (prefixRev part tail : List Char) (sep : Char) :
    part.all (fun c => c != sep) = true ->
    splitOnCharGo prefixRev sep (part ++ sep :: tail) =
      String.mk (prefixRev.reverse ++ part) :: splitOnCharGo [] sep tail := by
  intro h_safe
  induction part generalizing prefixRev with
  | nil =>
      simp [splitOnCharGo]
  | cons c rest ih =>
      simp at h_safe
      have h_rest : (rest.all fun c => c != sep) = true := by
        simpa using h_safe.2
      simpa [splitOnCharGo, h_safe.1, List.reverse_cons, List.append_assoc] using
        (ih (c :: prefixRev) h_rest)
private theorem splitOnCharGo_no_sep
    (prefixRev chars : List Char) (sep : Char) :
    chars.all (fun c => c != sep) = true ->
    splitOnCharGo prefixRev sep chars =
      [String.mk (prefixRev.reverse ++ chars)] := by
  intro h_safe
  induction chars generalizing prefixRev with
  | nil =>
      simp [splitOnCharGo]
  | cons c rest ih =>
      simp at h_safe
      have h_rest : (rest.all fun c => c != sep) = true := by
        simpa using h_safe.2
      simpa [splitOnCharGo, h_safe.1, List.reverse_cons, List.append_assoc] using
        (ih (c :: prefixRev) h_rest)
@[simp] theorem split_single_char_append
    (head tail : String) (sep : Char) :
    head.toList.all (fun c => c != sep) = true ->
    split (head ++ Char.toString sep ++ tail) (Char.toString sep) =
      head :: split tail (Char.toString sep) := by
  intro h_safe
  simpa [split, splitOnChar] using
    (splitOnCharGo_until_sep [] head.data tail.data sep h_safe)
@[simp] theorem split_single_char_no_sep
    (s : String) (sep : Char) :
    s.toList.all (fun c => c != sep) = true ->
    split s (Char.toString sep) = [s] := by
  intro h_safe
  simpa [split, splitOnChar] using
    (splitOnCharGo_no_sep [] s.data sep h_safe)
private theorem intercalate_go_prefix
    (pref acc sep : String) (rest : List String) :
    String.intercalate.go (pref ++ sep ++ acc) sep rest =
      pref ++ sep ++ String.intercalate.go acc sep rest := by
  induction rest generalizing acc with
  | nil =>
      simp [String.intercalate.go, String.append_assoc]
  | cons x xs ih =>
      simpa [String.intercalate.go, String.append_assoc] using
        (ih (acc ++ sep ++ x))
@[simp] theorem split_intercalate_trailing_single_char
    (parts : List String) (sep : Char) :
    parts.all (fun part => part.toList.all (fun c => c != sep)) = true ->
    split (String.intercalate (Char.toString sep) parts ++ Char.toString sep) (Char.toString sep) =
      match parts with
      | [] => ["", ""]
      | _ => parts ++ [""] := by
  intro h_safe
  induction parts with
  | nil =>
      simp [split, splitOnChar, splitOnCharGo]
  | cons part rest ih =>
      simp at h_safe
      have h_part : (part.toList.all fun c => c != sep) = true := by
        simpa using h_safe.1
      cases rest with
      | nil =>
          have h_empty : ("".toList.all fun c => c != sep) = true := by simp
          calc
            split (String.intercalate.go part (Char.toString sep) [] ++ Char.toString sep) (Char.toString sep)
                = split (part ++ Char.toString sep) (Char.toString sep) := by
                    simp [String.intercalate.go]
            _ = part :: split "" (Char.toString sep) := by
                    simpa using split_single_char_append part "" sep h_part
            _ = [part, ""] := by
                    simp [split_single_char_no_sep, h_empty]
      | cons next rest' =>
          have h_rest : ((next :: rest').all fun part => part.toList.all fun c => c != sep) = true := by
            simpa using h_safe.2
          calc
            split
                  (String.intercalate.go part (Char.toString sep) (next :: rest') ++ Char.toString sep)
                  (Char.toString sep)
                =
                  split
                    (part ++ Char.toString sep ++ (String.intercalate (Char.toString sep) (next :: rest') ++ Char.toString sep))
                    (Char.toString sep) := by
                    have h_join :
                        String.intercalate.go part (Char.toString sep) (next :: rest') ++ Char.toString sep
                          = part ++ Char.toString sep ++ (String.intercalate (Char.toString sep) (next :: rest') ++ Char.toString sep) := by
                        calc
                          String.intercalate.go part (Char.toString sep) (next :: rest') ++ Char.toString sep
                              = String.intercalate.go (part ++ Char.toString sep ++ next) (Char.toString sep) rest' ++ Char.toString sep := by
                                  simp [String.intercalate.go]
                          _ = part ++ Char.toString sep ++ (String.intercalate.go next (Char.toString sep) rest' ++ Char.toString sep) := by
                                  rw [intercalate_go_prefix part next (Char.toString sep) rest']
                                  simp [String.append_assoc]
                          _ = part ++ Char.toString sep ++ (String.intercalate (Char.toString sep) (next :: rest') ++ Char.toString sep) := by
                                  simp [String.intercalate, String.intercalate.go]
                    simpa using congrArg (fun s => split s (Char.toString sep)) h_join
            _ = part :: split
                    (String.intercalate (Char.toString sep) (next :: rest') ++ Char.toString sep)
                    (Char.toString sep) := by
                    simpa using split_single_char_append
                      part
                      (String.intercalate (Char.toString sep) (next :: rest') ++ Char.toString sep)
                      sep
                      h_part
            _ = part :: (next :: rest' ++ [""]) := by
                    simpa using ih h_rest
end AverString"#;

const LEAN_PRELUDE_NUMERIC_PARSE: &str = r#"namespace AverDigits
def foldDigitsAcc (acc : Nat) : List Nat -> Nat
  | [] => acc
  | d :: ds => foldDigitsAcc (acc * 10 + d) ds

def foldDigits (digits : List Nat) : Nat :=
  foldDigitsAcc 0 digits

private theorem foldDigitsAcc_append_singleton (acc : Nat) (xs : List Nat) (d : Nat) :
    foldDigitsAcc acc (xs ++ [d]) = foldDigitsAcc acc xs * 10 + d := by
  induction xs generalizing acc with
  | nil =>
      simp [foldDigitsAcc]
  | cons x xs ih =>
      simp [foldDigitsAcc, ih, Nat.left_distrib, Nat.add_assoc, Nat.add_left_comm]

private theorem foldDigits_append_singleton (xs : List Nat) (d : Nat) :
    foldDigits (xs ++ [d]) = foldDigits xs * 10 + d := by
  simpa [foldDigits] using foldDigitsAcc_append_singleton 0 xs d

def natDigits : Nat -> List Nat
  | n =>
      if n < 10 then
        [n]
      else
        natDigits (n / 10) ++ [n % 10]
termination_by
  n => n

theorem natDigits_nonempty (n : Nat) : natDigits n ≠ [] := by
  by_cases h : n < 10
  · rw [natDigits.eq_1]
    simp [h]
  · rw [natDigits.eq_1]
    simp [h]

theorem natDigits_digits_lt_ten : ∀ n : Nat, ∀ d ∈ natDigits n, d < 10 := by
  intro n d hd
  by_cases h : n < 10
  · rw [natDigits.eq_1] at hd
    simp [h] at hd
    rcases hd with rfl
    exact h
  · rw [natDigits.eq_1] at hd
    simp [h] at hd
    rcases hd with hd | hd
    · exact natDigits_digits_lt_ten (n / 10) d hd
    · subst hd
      exact Nat.mod_lt n (by omega)

theorem foldDigits_natDigits : ∀ n : Nat, foldDigits (natDigits n) = n := by
  intro n
  by_cases h : n < 10
  · rw [natDigits.eq_1]
    simp [h, foldDigits, foldDigitsAcc]
  · rw [natDigits.eq_1]
    simp [h]
    rw [foldDigits_append_singleton]
    rw [foldDigits_natDigits (n / 10)]
    omega

def digitChar : Nat -> Char
  | 0 => '0' | 1 => '1' | 2 => '2' | 3 => '3' | 4 => '4'
  | 5 => '5' | 6 => '6' | 7 => '7' | 8 => '8' | 9 => '9'
  | _ => '0'

def charToDigit? : Char -> Option Nat
  | '0' => some 0 | '1' => some 1 | '2' => some 2 | '3' => some 3 | '4' => some 4
  | '5' => some 5 | '6' => some 6 | '7' => some 7 | '8' => some 8 | '9' => some 9
  | _ => none

theorem charToDigit_digitChar : ∀ d : Nat, d < 10 -> charToDigit? (digitChar d) = some d
  | 0, _ => by simp [digitChar, charToDigit?]
  | 1, _ => by simp [digitChar, charToDigit?]
  | 2, _ => by simp [digitChar, charToDigit?]
  | 3, _ => by simp [digitChar, charToDigit?]
  | 4, _ => by simp [digitChar, charToDigit?]
  | 5, _ => by simp [digitChar, charToDigit?]
  | 6, _ => by simp [digitChar, charToDigit?]
  | 7, _ => by simp [digitChar, charToDigit?]
  | 8, _ => by simp [digitChar, charToDigit?]
  | 9, _ => by simp [digitChar, charToDigit?]
  | Nat.succ (Nat.succ (Nat.succ (Nat.succ (Nat.succ (Nat.succ (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))))), h => by
      omega

theorem digitChar_ne_minus : ∀ d : Nat, d < 10 -> digitChar d ≠ '-'
  | 0, _ => by decide
  | 1, _ => by decide
  | 2, _ => by decide
  | 3, _ => by decide
  | 4, _ => by decide
  | 5, _ => by decide
  | 6, _ => by decide
  | 7, _ => by decide
  | 8, _ => by decide
  | 9, _ => by decide
  | Nat.succ (Nat.succ (Nat.succ (Nat.succ (Nat.succ (Nat.succ (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))))), h => by
      omega

theorem digitChar_not_ws : ∀ d : Nat, d < 10 ->
    Char.toString (digitChar d) ≠ " " ∧
    Char.toString (digitChar d) ≠ "\t" ∧
    Char.toString (digitChar d) ≠ "\n" ∧
    Char.toString (digitChar d) ≠ "\r"
  | 0, _ => by decide
  | 1, _ => by decide
  | 2, _ => by decide
  | 3, _ => by decide
  | 4, _ => by decide
  | 5, _ => by decide
  | 6, _ => by decide
  | 7, _ => by decide
  | 8, _ => by decide
  | 9, _ => by decide
  | Nat.succ (Nat.succ (Nat.succ (Nat.succ (Nat.succ (Nat.succ (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))))), h => by
      omega

theorem mapM_charToDigit_digits : ∀ ds : List Nat,
    (∀ d ∈ ds, d < 10) -> List.mapM charToDigit? (ds.map digitChar) = some ds := by
  intro ds hds
  induction ds with
  | nil =>
      simp
  | cons d ds ih =>
      have hd : d < 10 := hds d (by simp)
      have htail : ∀ x ∈ ds, x < 10 := by
        intro x hx
        exact hds x (by simp [hx])
      simp [charToDigit_digitChar d hd, ih htail]

def natDigitsChars (n : Nat) : List Char :=
  (natDigits n).map digitChar

def parseNatChars (chars : List Char) : Option Nat :=
  match chars with
  | [] => none
  | _ => do
      let digits <- List.mapM charToDigit? chars
      pure (foldDigits digits)

theorem parseNatChars_nat (n : Nat) :
    parseNatChars (natDigitsChars n) = some n := by
  unfold parseNatChars natDigitsChars
  cases h : (natDigits n).map digitChar with
  | nil =>
      exfalso
      exact natDigits_nonempty n (List.map_eq_nil_iff.mp h)
  | cons hd tl =>
      have hdigits : List.mapM charToDigit? (List.map digitChar (natDigits n)) = some (natDigits n) :=
        mapM_charToDigit_digits (natDigits n) (fun d hd => natDigits_digits_lt_ten n d hd)
      rw [h] at hdigits
      simp [h, hdigits, foldDigits_natDigits]
end AverDigits

def String.fromInt (n : Int) : String :=
  match n with
  | .ofNat m => String.mk (AverDigits.natDigitsChars m)
  | .negSucc m => String.mk ('-' :: AverDigits.natDigitsChars (m + 1))

def Int.fromString (s : String) : Except String Int :=
  match s.toList with
  | [] => .error ("Cannot parse '" ++ s ++ "' as Int")
  | '-' :: rest =>
    match AverDigits.parseNatChars rest with
    | some n => .ok (-Int.ofNat n)
    | none => .error ("Cannot parse '" ++ s ++ "' as Int")
  | chars =>
    match AverDigits.parseNatChars chars with
    | some n => .ok (Int.ofNat n)
    | none => .error ("Cannot parse '" ++ s ++ "' as Int")

theorem Int.fromString_fromInt : ∀ n : Int, Int.fromString (String.fromInt n) = .ok n
  | .ofNat m => by
      cases h : AverDigits.natDigits m with
      | nil =>
          exfalso
          exact AverDigits.natDigits_nonempty m h
      | cons d ds =>
          have hd : d < 10 := AverDigits.natDigits_digits_lt_ten m d (by simp [h])
          have hne : AverDigits.digitChar d ≠ '-' := AverDigits.digitChar_ne_minus d hd
          have hparse : AverDigits.parseNatChars (AverDigits.digitChar d :: List.map AverDigits.digitChar ds) = some m := by
            simpa [AverDigits.natDigitsChars, h] using AverDigits.parseNatChars_nat m
          simp [String.fromInt, Int.fromString, AverDigits.natDigitsChars, h, hne, hparse]
  | .negSucc m => by
      simp [String.fromInt, Int.fromString, AverDigits.parseNatChars_nat]
      rfl

private def charDigitsToNat (cs : List Char) : Nat :=
  cs.foldl (fun acc c => acc * 10 + (c.toNat - '0'.toNat)) 0

private def parseExpPart : List Char → (Bool × List Char)
  | '-' :: rest => (true, rest.takeWhile Char.isDigit)
  | '+' :: rest => (false, rest.takeWhile Char.isDigit)
  | rest => (false, rest.takeWhile Char.isDigit)

def Float.fromString (s : String) : Except String Float :=
  let chars := s.toList
  let (neg, chars) := match chars with
    | '-' :: rest => (true, rest)
    | _ => (false, chars)
  let intPart := chars.takeWhile Char.isDigit
  let rest := chars.dropWhile Char.isDigit
  let (fracPart, rest) := match rest with
    | '.' :: rest => (rest.takeWhile Char.isDigit, rest.dropWhile Char.isDigit)
    | _ => ([], rest)
  let (expNeg, expDigits) := match rest with
    | 'e' :: rest => parseExpPart rest
    | 'E' :: rest => parseExpPart rest
    | _ => (false, [])
  if intPart.isEmpty && fracPart.isEmpty then .error ("Invalid float: " ++ s)
  else
    let mantissa := charDigitsToNat (intPart ++ fracPart)
    let fracLen : Int := fracPart.length
    let expVal : Int := charDigitsToNat expDigits
    let shift : Int := (if expNeg then -expVal else expVal) - fracLen
    let f := if shift >= 0 then Float.ofScientific mantissa false shift.toNat
             else Float.ofScientific mantissa true ((-shift).toNat)
    .ok (if neg then -f else f)"#;

const LEAN_PRELUDE_CHAR_BYTE: &str = r#"def Char.toCode (s : String) : Int :=
  match s.toList.head? with
  | some c => (c.toNat : Int)
  | none => panic! "Char.toCode: string is empty"
def Char.fromCode (n : Int) : Option String :=
  if n < 0 || n > 1114111 then none
  else if n >= 55296 && n <= 57343 then none
  else some (Char.toString (Char.ofNat n.toNat))

def hexDigit (n : Int) : String :=
  match n with
  | 0 => "0" | 1 => "1" | 2 => "2" | 3 => "3"
  | 4 => "4" | 5 => "5" | 6 => "6" | 7 => "7"
  | 8 => "8" | 9 => "9" | 10 => "a" | 11 => "b"
  | 12 => "c" | 13 => "d" | 14 => "e" | 15 => "f"
  | _ => "?"

def byteToHex (code : Int) : String :=
  hexDigit (code / 16) ++ hexDigit (code % 16)

namespace AverByte
private def hexValue (c : Char) : Option Int :=
  match c with
  | '0' => some 0  | '1' => some 1  | '2' => some 2  | '3' => some 3
  | '4' => some 4  | '5' => some 5  | '6' => some 6  | '7' => some 7
  | '8' => some 8  | '9' => some 9  | 'a' => some 10 | 'b' => some 11
  | 'c' => some 12 | 'd' => some 13 | 'e' => some 14 | 'f' => some 15
  | 'A' => some 10 | 'B' => some 11 | 'C' => some 12 | 'D' => some 13
  | 'E' => some 14 | 'F' => some 15
  | _ => none
def toHex (n : Int) : Except String String :=
  if n < 0 || n > 255 then
    .error ("Byte.toHex: " ++ toString n ++ " is out of range 0-255")
  else
    .ok (byteToHex n)
def fromHex (s : String) : Except String Int :=
  match s.toList with
  | [hi, lo] =>
    match hexValue hi, hexValue lo with
    | some h, some l => .ok (h * 16 + l)
    | _, _ => .error ("Byte.fromHex: invalid hex '" ++ s ++ "'")
  | _ => .error ("Byte.fromHex: expected exactly 2 hex chars, got '" ++ s ++ "'")
end AverByte"#;

pub(crate) fn pure_fns(ctx: &CodegenContext) -> Vec<&FnDef> {
    ctx.modules
        .iter()
        .flat_map(|m| m.fn_defs.iter())
        .chain(ctx.fn_defs.iter())
        .filter(|fd| toplevel::is_pure_fn(fd))
        .collect()
}

pub(crate) fn recursive_type_names(ctx: &CodegenContext) -> HashSet<String> {
    ctx.modules
        .iter()
        .flat_map(|m| m.type_defs.iter())
        .chain(ctx.type_defs.iter())
        .filter(|td| toplevel::is_recursive_type_def(td))
        .map(|td| toplevel::type_def_name(td).to_string())
        .collect()
}

pub(crate) fn recursive_pure_fn_names(ctx: &CodegenContext) -> HashSet<String> {
    // `ctx.recursive_fns` is the single source of truth — populated
    // by `build_context` from analyze in production, by
    // `refresh_facts()` (called from each `transpile*` entry point)
    // in test stubs. After phase C it's keyed by opaque `FnId`;
    // project pure-fn ids back through the symbol table and surface
    // bare names for Lean's downstream scope-local classifiers.
    let symbols = &ctx.symbol_table;
    let pure_ids: HashSet<crate::ir::FnId> = pure_fns(ctx)
        .into_iter()
        .filter_map(|fd| crate::codegen::common::fn_id_for_decl(ctx, fd))
        .collect();
    ctx.recursive_fns
        .intersection(&pure_ids)
        .map(|id| symbols.fn_entry(*id).key.name.clone())
        .collect()
}

fn verify_counter_key(vb: &crate::ast::VerifyBlock) -> String {
    match &vb.kind {
        VerifyKind::Cases => format!("fn:{}", vb.fn_name),
        VerifyKind::Law(law) => format!("law:{}::{}", vb.fn_name, law.name),
    }
}

fn lean_project_name(ctx: &CodegenContext) -> String {
    crate::codegen::common::entry_basename(ctx)
}

pub(super) fn bound_expr_to_lean(expr: &Spanned<crate::ir::hir::ResolvedExpr>) -> String {
    use crate::ir::hir::{ResolvedCallee, ResolvedExpr};
    match &expr.node {
        ResolvedExpr::Literal(crate::ast::Literal::Int(n)) => format!("{}", n),
        ResolvedExpr::Ident(name) | ResolvedExpr::Resolved { name, .. } => {
            expr::aver_name_to_lean(name)
        }
        ResolvedExpr::Call(callee, args) => {
            // Bound expressions are linear arithmetic over the param —
            // typically `List.len(xs)` or an int constant. The resolver
            // lifts `List.len(...)` to `ResolvedCallee::Builtin`; user-fn
            // refs in a bound expression would be a malformed metric
            // anyway, so we only render builtins and fall through to
            // `"0"` (the same fallback the original AST helper used for
            // shapes it couldn't classify).
            let dotted = match callee {
                ResolvedCallee::Builtin(name) => Some(name.clone()),
                _ => None,
            };
            if let Some(dotted) = dotted {
                // List.len(xs) → xs.length in Lean
                if dotted == "List.len" && args.len() == 1 {
                    return format!("{}.length", bound_expr_to_lean(&args[0]));
                }
                let lean_args: Vec<String> = args.iter().map(bound_expr_to_lean).collect();
                format!(
                    "({} {})",
                    expr::aver_name_to_lean(&dotted),
                    lean_args.join(" ")
                )
            } else {
                "0".to_string()
            }
        }
        ResolvedExpr::Attr(obj, field) => format!(
            "{}.{}",
            bound_expr_to_lean(obj),
            expr::aver_name_to_lean(field)
        ),
        _ => "0".to_string(),
    }
}

pub(crate) use crate::codegen::recursion::detect::sizeof_measure_param_indices;

/// Proof-mode diagnostics for Lean transpilation.
///
/// Returns human-readable notices for recursive shapes that still fall back to
/// regular `partial` Lean defs instead of total proof-mode emission.
pub fn proof_mode_findings(ctx: &CodegenContext) -> Vec<ProofModeIssue> {
    // ProofIR carries `unclassified_fns` populated by the ContractLower
    // pipeline stage — same data analyze_plans used to return, just
    // read off the IR instead of re-running the classifier.
    ctx.proof_ir
        .unclassified_fns
        .iter()
        .map(|uf| ProofModeIssue {
            line: uf.line,
            message: uf.message.clone(),
        })
        .collect()
}

pub fn proof_mode_issues(ctx: &CodegenContext) -> Vec<String> {
    proof_mode_findings(ctx)
        .into_iter()
        .map(|issue| issue.message)
        .collect()
}

/// Transpile an Aver program to a Lean 4 project.
///
/// Takes `&mut ctx` so it can run `ctx.refresh_facts()` upfront — keeps
/// the derived sets (`mutual_tco_members`, `recursive_fns`) in sync with
/// the current items + modules. Idempotent: production callers go through
/// `build_context` which already populated them, so refresh recomputes
/// the same answer; test stubs that build the ctx piecewise (push items
/// in-place, bypass `build_context`) get the fresh sets they need.
pub fn transpile(ctx: &mut CodegenContext) -> ProjectOutput {
    transpile_with_verify_mode(ctx, VerifyEmitMode::NativeDecide)
}

/// Proof-mode transpilation.
///
/// Uses recursion plans validated by `proof_mode_issues` and emits supported
/// recursive functions without `partial`, adding `termination_by` scaffolding.
pub fn transpile_for_proof_mode(
    ctx: &mut CodegenContext,
    verify_mode: VerifyEmitMode,
) -> ProjectOutput {
    // No refresh_facts call here: production callers go through
    // build_codegen_context → pipeline, which populates every derived
    // fact (recursive_fns, mutual_tco_members, proof_ir) once.
    // Synthetic-AST tests that bypass the pipeline call refresh_facts
    // themselves before reaching this fn.
    transpile_unified(ctx, verify_mode, LeanEmitMode::Proof)
}

/// Transpile an Aver program to a Lean 4 project with configurable verify proof mode.
///
/// - `NativeDecide` emits `example ... := by native_decide`
/// - `Sorry` emits `example ... := by sorry`
/// - `TheoremSkeleton` emits named theorem skeletons with `sorry`
pub fn transpile_with_verify_mode(
    ctx: &mut CodegenContext,
    verify_mode: VerifyEmitMode,
) -> ProjectOutput {
    // No refresh_facts call here — same reasoning as
    // `transpile_for_proof_mode`. Synthetic-AST tests refresh
    // themselves; production paths come pre-populated.
    transpile_unified(ctx, verify_mode, LeanEmitMode::Standard)
}

/// Oracle v1: for each effectful FnDef whose effects are all classified,
/// run effect_lifting::lift_fn_def to produce a pure (lifted) FnDef and
/// emit it via the standard pure-fn path. Effectful functions that use
/// unclassified (stateful / interactive / higher-order-callback) effects
/// are still skipped entirely — matches the pre-Oracle behavior.
///
/// Mutual recursion among effectful fns is out of scope for v0: this
/// emits each lifted fn independently.
fn emit_lifted_effectful_functions(
    ctx: &CodegenContext,
    recursive_fns: &HashSet<String>,
    sections: &mut Vec<String>,
) {
    use crate::types::checker::effect_classification::is_classified;

    // Oracle v1: only emit effectful fns reachable from some verify
    // block. Non-terminating effectful fns (e.g. REPL loops that loop
    // forever on `Console.readLine`) would otherwise make Lean reject
    // the whole module — and if nobody is proving anything about them,
    // that's dead code in the proof output.
    let reachable = crate::codegen::common::verify_reachable_fn_names(&ctx.items);

    // Oracle v1: collect the effect list for every eligible
    // effectful fn *first* — call sites to these helpers in any
    // lifted body get `(path, oracle...)` injected so the arity
    // matches the helper's lifted form.
    let mut helpers: std::collections::HashMap<String, Vec<String>> =
        std::collections::HashMap::new();
    for item in &ctx.items {
        let TopLevel::FnDef(fd) = item else { continue };
        if fd.effects.is_empty() || fd.name == "main" {
            continue;
        }
        if !fd.effects.iter().all(|e| is_classified(&e.node)) {
            continue;
        }
        if !reachable.contains(&fd.name) {
            continue;
        }
        helpers.insert(
            fd.name.clone(),
            fd.effects.iter().map(|e| e.node.clone()).collect(),
        );
    }

    let mut lifted_fns: Vec<(String, crate::ast::FnDef)> = Vec::new();
    for item in &ctx.items {
        let TopLevel::FnDef(fd) = item else { continue };
        if fd.effects.is_empty() || fd.name == "main" {
            continue;
        }
        if !fd.effects.iter().all(|e| is_classified(&e.node)) {
            continue;
        }
        if !reachable.contains(&fd.name) {
            continue;
        }
        let Ok(Some(lifted)) =
            crate::types::checker::effect_lifting::lift_fn_def_with_helpers(fd, &helpers)
        else {
            continue;
        };
        lifted_fns.push((fd.name.clone(), lifted));
    }

    // Oracle v1: topologically sort so callees are emitted before
    // callers. Without this, a lifted effectful fn that calls another
    // lifted effectful helper (e.g. `handle(msg) -> printErr(msg)`)
    // can land before the helper and Lean complains about an unknown
    // identifier. The pure-fn emission goes through SCC analysis for
    // the same reason — this is a cheap approximation good enough
    // for non-mutual effectful chains.
    let eligible_names: std::collections::HashSet<String> =
        lifted_fns.iter().map(|(n, _)| n.clone()).collect();
    let mut emitted: std::collections::HashSet<String> = std::collections::HashSet::new();
    let mut order: Vec<usize> = Vec::new();
    let mut remaining: Vec<usize> = (0..lifted_fns.len()).collect();
    while !remaining.is_empty() {
        let before = remaining.len();
        remaining.retain(|&idx| {
            let body_calls = collect_called_idents_in_body(&lifted_fns[idx].1.body);
            let ready = body_calls
                .iter()
                .all(|name| !eligible_names.contains(name) || emitted.contains(name));
            if ready {
                emitted.insert(lifted_fns[idx].0.clone());
                order.push(idx);
                false
            } else {
                true
            }
        });
        if remaining.len() == before {
            // Cycle or deadlock — fall back to source order for what's
            // left rather than looping forever. Lean will complain at
            // build time, which is the right signal for users.
            order.append(&mut remaining);
        }
    }

    for idx in order {
        let (_, lifted) = &lifted_fns[idx];
        if let Some(code) = toplevel::emit_fn_def(lifted, recursive_fns, ctx) {
            sections.push(code);
            sections.push(String::new());
        }
    }
}

fn collect_called_idents_in_body(body: &crate::ast::FnBody) -> std::collections::HashSet<String> {
    use crate::ast::{Expr, Spanned, Stmt};
    let mut out = std::collections::HashSet::new();
    fn walk(expr: &Spanned<Expr>, out: &mut std::collections::HashSet<String>) {
        match &expr.node {
            Expr::FnCall(callee, args) => {
                if let Expr::Ident(name) | Expr::Resolved { name, .. } = &callee.node {
                    out.insert(name.clone());
                }
                walk(callee, out);
                for a in args {
                    walk(a, out);
                }
            }
            Expr::BinOp(_, l, r) => {
                walk(l, out);
                walk(r, out);
            }
            Expr::Match { subject, arms } => {
                walk(subject, out);
                for arm in arms {
                    walk(&arm.body, out);
                }
            }
            Expr::Attr(inner, _) | Expr::ErrorProp(inner) => walk(inner, out),
            Expr::Constructor(_, Some(inner)) => walk(inner, out),
            Expr::List(items) | Expr::Tuple(items) | Expr::IndependentProduct(items, _) => {
                for i in items {
                    walk(i, out);
                }
            }
            Expr::MapLiteral(pairs) => {
                for (k, v) in pairs {
                    walk(k, out);
                    walk(v, out);
                }
            }
            Expr::RecordCreate { fields, .. } => {
                for (_, v) in fields {
                    walk(v, out);
                }
            }
            Expr::RecordUpdate { base, updates, .. } => {
                walk(base, out);
                for (_, v) in updates {
                    walk(v, out);
                }
            }
            Expr::InterpolatedStr(parts) => {
                for part in parts {
                    if let crate::ast::StrPart::Parsed(inner) = part {
                        walk(inner, out);
                    }
                }
            }
            _ => {}
        }
    }
    for stmt in body.stmts() {
        match stmt {
            Stmt::Expr(e) => walk(e, &mut out),
            Stmt::Binding(_, _, e) => walk(e, &mut out),
        }
    }
    out
}

#[cfg(test)]
fn generate_prelude() -> String {
    generate_prelude_for_body("", true)
}

#[cfg(test)]
fn generate_prelude_for_body(body: &str, include_all_helpers: bool) -> String {
    // Oracle v1: trust-assumption header first so the emitted file opens with
    // the explicit claim block before any prelude or definitions. Skipped when
    // the body has no Oracle lifting at all — pure-math files don't depend on
    // any of the trust claims, so emitting the block would just add noise.
    let mut parts = vec![LEAN_PRELUDE_HEADER.to_string()];
    if include_all_helpers || crate::codegen::builtin_records::needs_trust_header(body) {
        // This branch is only reachable from #[cfg(test)] code that
        // calls `generate_prelude` without a real ctx; pass an empty
        // declared_effects set so the test fixture exercises the
        // "no effects" rendering. Production calls go through the
        // ctx-aware path in `transpile_unified`.
        let empty = crate::codegen::common::DeclaredEffects {
            bare_namespaces: std::collections::HashSet::new(),
            methods: std::collections::HashSet::new(),
        };
        let has_ip = body.contains("BranchPath");
        parts.push(
            crate::types::checker::proof_trust_header::generate_commented("-- ", &empty, has_ip),
        );
    }
    // Built-in record types — shared decision module decides which ones.
    for record in crate::codegen::builtin_records::needed_records(body, include_all_helpers) {
        parts.push(crate::codegen::builtin_records::render_lean(record));
    }

    // Built-in helpers — same shared decision pattern. Each key has a
    // backend-native body resolved here (Lean preludes); other backends
    // use the same shared decision against their own implementations.
    for helper in crate::codegen::builtin_helpers::needed_helpers(body, include_all_helpers) {
        match helper.key {
            "BranchPath" => parts.push(LEAN_PRELUDE_BRANCH_PATH.to_string()),
            "AverList" => parts.push(LEAN_PRELUDE_AVER_LIST.to_string()),
            "StringHelpers" => parts.push(LEAN_PRELUDE_STRING_HELPERS.to_string()),
            "NumericParse" => parts.push(LEAN_PRELUDE_NUMERIC_PARSE.to_string()),
            "CharByte" => parts.push(LEAN_PRELUDE_CHAR_BYTE.to_string()),
            "AverMeasure" => parts.push(LEAN_PRELUDE_AVER_MEASURE.to_string()),
            "AverMap" => parts.push(generate_map_prelude(body, include_all_helpers)),
            "ProofFuel" => parts.push(LEAN_PRELUDE_PROOF_FUEL.to_string()),
            "FloatInstances" => parts.extend([
                LEAN_PRELUDE_FLOAT_COE.to_string(),
                LEAN_PRELUDE_FLOAT_DEC_EQ.to_string(),
            ]),
            "ExceptInstances" => parts.extend([
                LEAN_PRELUDE_EXCEPT_DEC_EQ.to_string(),
                LEAN_PRELUDE_EXCEPT_NS.to_string(),
                LEAN_PRELUDE_OPTION_TO_EXCEPT.to_string(),
            ]),
            "StringHadd" => parts.push(LEAN_PRELUDE_STRING_HADD.to_string()),
            // Dafny-side datatype declarations — Lean has Result/Option
            // natively (`Except`/`Option`) and BranchPath ships as part
            // of the BranchPath helper key, so all four are no-ops here.
            "ResultDatatype" | "OptionDatatype" | "OptionToResult" | "BranchPathDatatype" => {}
            other => panic!(
                "Lean backend has no implementation for builtin helper key '{}'. \
                 Add a match arm in generate_prelude_for_body or remove the key \
                 from BUILTIN_HELPERS.",
                other
            ),
        }
    }

    parts.join("\n\n")
}

fn generate_map_prelude(body: &str, include_all_helpers: bool) -> String {
    let mut parts = vec![AVER_MAP_PRELUDE_BASE.to_string()];

    let needs_has_set_self = include_all_helpers || body.contains("AverMap.has_set_self");
    let needs_get_set_self = include_all_helpers || body.contains("AverMap.get_set_self");
    let needs_get_set_other = include_all_helpers
        || body.contains("AverMap.get_set_other")
        || body.contains("AverMap.has_set_other");
    let needs_has_set_other = include_all_helpers || body.contains("AverMap.has_set_other");

    if needs_has_set_self {
        parts.push(AVER_MAP_PRELUDE_HAS_SET_SELF.to_string());
    }
    if needs_get_set_self {
        parts.push(AVER_MAP_PRELUDE_GET_SET_SELF.to_string());
    }
    if needs_get_set_other {
        parts.push(AVER_MAP_PRELUDE_GET_SET_OTHER.to_string());
    }
    if needs_has_set_other {
        parts.push(AVER_MAP_PRELUDE_HAS_SET_OTHER.to_string());
    }

    parts.push(AVER_MAP_PRELUDE_END.to_string());
    parts.join("\n\n")
}

fn generate_lakefile_with_roots(project_name: &str, extra_roots: &[String]) -> String {
    let mut roots: Vec<String> = vec![format!("`{}", project_name)];
    for r in extra_roots {
        roots.push(format!("`{}", r));
    }
    let roots_str = roots.join(", ");
    format!(
        r#"import Lake
open Lake DSL

package «{}» where
  version := v!"0.1.0"

@[default_target]
lean_lib «{}» where
  srcDir := "."
  roots := #[{}]
"#,
        project_name.to_lowercase(),
        project_name,
        roots_str
    )
}

fn generate_toolchain() -> String {
    "leanprover/lean4:v4.15.0\n".to_string()
}

#[derive(Clone, Copy)]
enum LeanEmitMode {
    Standard,
    Proof,
}

/// Multi-file Lean output for multi-module Aver projects:
/// - `AverCommon.lean` carries built-in helpers + records (UNION decision
///   over every module + entry body, so a helper is included only if
///   something actually references it).
/// - `<Module>.lean` (one per `depends [...]` entry) wraps that module's
///   types and pure fns in `namespace M ... end M`. Submodules like
///   `Models.User` land at `Models/User.lean` to match Lean's path-as-
///   module-name convention.
/// - `<ProjectName>.lean` is the entry: trust header (here only),
///   top-level entry items, lifted effectful fns, decisions, verify
///   blocks. Imports `AverCommon` plus every dependent module.
fn transpile_unified(
    ctx: &CodegenContext,
    verify_mode: VerifyEmitMode,
    emit_mode: LeanEmitMode,
) -> ProjectOutput {
    // Read recursion fact from `ctx.recursive_fns` — populated upstream
    // by `refresh_facts()` (test stubs) or `build_context` (production).
    // After phase C the set is keyed by `FnId`; project back to bare
    // names for scope-local `emit_fn_def` consumers via the symbol
    // table (the DAG invariant keeps bare-name unambiguous within a
    // single scope). The proof-mode auto-prove path also needs the
    // same bare-name projection.
    let recursive_fns: HashSet<String> = ctx
        .recursive_fns
        .iter()
        .map(|id| ctx.symbol_table.fn_entry(*id).key.name.clone())
        .collect();
    let recursive_names = recursive_pure_fn_names(ctx);
    let recursive_types = recursive_type_names(ctx);

    // Pure fns are SCC-routed per scope (per dependent module + entry)
    // independently — shared `route_pure_components_per_scope` handles
    // the loop. A global SCC pass would conflate same-bare-name fns from
    // different modules (rogue's `Map.getT` vs `Fov.getT`).
    let pure_per_scope = crate::codegen::common::route_pure_components_per_scope(
        ctx,
        toplevel::is_pure_fn,
        |comp, scope| {
            let scope_opt = if scope.is_empty() { None } else { Some(scope) };
            ctx.with_module_scope(scope_opt, || {
                let mut out: Vec<String> = Vec::new();
                if comp.len() > 1 {
                    let code = match emit_mode {
                        LeanEmitMode::Proof => {
                            let all_supported = comp.iter().all(|fd| {
                                crate::codegen::common::fn_contract_exists_for_fn(ctx, fd)
                            });
                            if all_supported {
                                toplevel::emit_mutual_group_proof(comp, ctx)
                            } else {
                                toplevel::emit_mutual_group(comp, ctx)
                            }
                        }
                        LeanEmitMode::Standard => toplevel::emit_mutual_group(comp, ctx),
                    };
                    out.push(code);
                    out.push(String::new());
                } else if let Some(fd) = comp.first() {
                    let emitted = match emit_mode {
                        LeanEmitMode::Proof => {
                            let is_recursive = recursive_names.contains(&fd.name);
                            // ProofIR's `fn_contracts` holds an entry only for
                            // recursive fns the ContractLower stage could
                            // classify. Recursive fns without a contract land
                            // in `unclassified_fns` and fall through to the
                            // partial/non-recursive emit.
                            if is_recursive
                                && !crate::codegen::common::fn_contract_exists_for_fn(ctx, fd)
                            {
                                toplevel::emit_fn_def(fd, &recursive_names, ctx)
                            } else {
                                toplevel::emit_fn_def_proof(fd, ctx)
                            }
                        }
                        LeanEmitMode::Standard => toplevel::emit_fn_def(fd, &recursive_fns, ctx),
                    };
                    if let Some(code) = emitted {
                        out.push(code);
                        out.push(String::new());
                    }
                }
                out
            })
        },
    );

    // Lifted effectful fns + decisions + verifies remain entry-only.
    let mut entry_lifted_sections: Vec<String> = Vec::new();
    let lifted_recursive_names = match emit_mode {
        LeanEmitMode::Proof => &recursive_names,
        LeanEmitMode::Standard => &recursive_fns,
    };
    emit_lifted_effectful_functions(ctx, lifted_recursive_names, &mut entry_lifted_sections);

    let mut entry_decision_sections: Vec<String> = Vec::new();
    for item in &ctx.items {
        if let TopLevel::Decision(db) = item {
            entry_decision_sections.push(toplevel::emit_decision(db));
            entry_decision_sections.push(String::new());
        }
    }

    let mut entry_verify_sections: Vec<String> = Vec::new();
    let mut verify_case_counters: HashMap<String, usize> = HashMap::new();
    for item in &ctx.items {
        if let TopLevel::Verify(vb) = item {
            let key = verify_counter_key(vb);
            let start_idx = *verify_case_counters.get(&key).unwrap_or(&0);
            let (emitted, next_idx) = toplevel::emit_verify_block(vb, ctx, verify_mode, start_idx);
            verify_case_counters.insert(key, next_idx);
            entry_verify_sections.push(emitted);
            entry_verify_sections.push(String::new());
        }
    }

    // ---- Per-module file bodies ----
    let mut module_files: Vec<(String, String)> = Vec::new();
    let mut union_body = String::new();

    for module in &ctx.modules {
        let mut body_sections: Vec<String> = Vec::new();
        ctx.with_module_scope(Some(module.prefix.as_str()), || {
            for td in &module.type_defs {
                body_sections.push(toplevel::emit_type_def_in_scope(
                    td,
                    ctx,
                    Some(module.prefix.as_str()),
                ));
                if toplevel::is_recursive_type_def(td) {
                    body_sections.push(toplevel::emit_recursive_decidable_eq(
                        toplevel::type_def_name(td),
                    ));
                    if matches!(emit_mode, LeanEmitMode::Proof)
                        && let Some(measure) =
                            toplevel::emit_recursive_measure(td, &recursive_types)
                    {
                        body_sections.push(measure);
                    }
                }
                body_sections.push(String::new());
            }
        });
        if let Some(scope_sections) = pure_per_scope.by_scope.get(&module.prefix) {
            body_sections.extend(scope_sections.clone());
        }
        let body = body_sections.join("\n");
        union_body.push_str(&body);
        union_body.push('\n');

        let mut imports = vec!["import AverCommon".to_string()];
        for d in &module.depends {
            imports.push(format!("import {}", d));
        }
        // AverCommon has no surrounding namespace (top-level helpers / instances),
        // so `import` already brings them into scope. We `open` only the
        // user-defined dependent modules.
        let opens: Vec<String> = module
            .depends
            .iter()
            .map(|d| format!("open {}", d))
            .collect();

        let opens_str = if opens.is_empty() {
            String::new()
        } else {
            format!("\n{}\n", opens.join("\n"))
        };
        let content = format!(
            "{}\n\nset_option linter.unusedVariables false\n{}\nnamespace {}\n\n{}\nend {}\n",
            imports.join("\n"),
            opens_str,
            module.prefix,
            body,
            module.prefix
        );
        let path = module.prefix.replace('.', "/");
        module_files.push((format!("{}.lean", path), content));
    }

    // ---- Entry sections ----
    let mut entry_body_sections: Vec<String> = Vec::new();
    for td in &ctx.type_defs {
        entry_body_sections.push(toplevel::emit_type_def(td, ctx));
        if toplevel::is_recursive_type_def(td) {
            entry_body_sections.push(toplevel::emit_recursive_decidable_eq(
                toplevel::type_def_name(td),
            ));
            if matches!(emit_mode, LeanEmitMode::Proof)
                && let Some(measure) = toplevel::emit_recursive_measure(td, &recursive_types)
            {
                entry_body_sections.push(measure);
            }
        }
        entry_body_sections.push(String::new());
    }
    if let Some(entry_pure) = pure_per_scope.by_scope.get("") {
        entry_body_sections.extend(entry_pure.clone());
    }
    entry_body_sections.extend(entry_lifted_sections);
    entry_body_sections.extend(entry_decision_sections);
    entry_body_sections.extend(entry_verify_sections);

    let entry_body = entry_body_sections.join("\n");
    union_body.push_str(&entry_body);
    union_body.push('\n');

    let project_name = lean_project_name(ctx);
    let mut entry_imports = vec!["import AverCommon".to_string()];
    for m in &ctx.modules {
        entry_imports.push(format!("import {}", m.prefix));
    }
    let entry_opens: Vec<String> = ctx
        .modules
        .iter()
        .map(|m| format!("open {}", m.prefix))
        .collect();
    let mut entry_parts = vec![entry_imports.join("\n")];
    if !entry_opens.is_empty() {
        entry_parts.push(entry_opens.join("\n"));
    }
    // Silence `unused variable` warnings for the named-match equation
    // binders (`h_NN :`) that the wf elaborator needs but the user-
    // source body never references. Without this every ListStructural
    // recursion would surface a warning per nested match. Per-file
    // because `set_option` is local; AverCommon already has the same
    // option for its prelude defs.
    entry_parts.push("set_option linter.unusedVariables false".to_string());
    let declared = crate::codegen::common::collect_declared_effects(ctx);
    let has_ip = union_body.contains("BranchPath");
    let has_classified =
        crate::types::checker::effect_classification::classifications_for_proof_subset()
            .iter()
            .any(|c| declared.includes(c.method));
    if has_ip || has_classified {
        entry_parts.push(
            crate::types::checker::proof_trust_header::generate_commented("-- ", &declared, has_ip),
        );
    }
    let subtype_block = crate::types::checker::oracle_subtypes::lean_subtypes(&declared);
    if !subtype_block.is_empty() {
        // Fold subtype block into the union body BEFORE computing
        // `needed_helpers` — the Oracle subtype block is what
        // introduces `BranchPath` references (e.g. `abbrev
        // TimeUnixMsOracle := BranchPath → Int → Int`) for files that
        // declare classified effects but never spell `BranchPath` in
        // user code. Without this, AverCommon.lean misses the
        // `structure BranchPath` block and Main.lean fails build with
        // `unknown identifier 'BranchPath'`.
        union_body.push_str(&subtype_block);
        union_body.push('\n');
        entry_parts.push(subtype_block);
    }
    entry_parts.push(entry_body);
    let entry_content = entry_parts.join("\n\n");

    // ---- AverCommon.lean ----
    let common_content = build_common_lean(&union_body);

    // Project files
    let mut extra_roots: Vec<String> = vec!["AverCommon".to_string()];
    for m in &ctx.modules {
        extra_roots.push(m.prefix.clone());
    }
    let lakefile = generate_lakefile_with_roots(&project_name, &extra_roots);
    let toolchain = generate_toolchain();

    let mut files = module_files;
    files.push((format!("{}.lean", project_name), entry_content));
    files.push(("AverCommon.lean".to_string(), common_content));
    files.push(("lakefile.lean".to_string(), lakefile));
    files.push(("lean-toolchain".to_string(), toolchain));
    ProjectOutput { files }
}

fn build_common_lean(union_body: &str) -> String {
    let mut parts = vec![LEAN_PRELUDE_HEADER.to_string()];
    for record in crate::codegen::builtin_records::needed_records(union_body, false) {
        parts.push(crate::codegen::builtin_records::render_lean(record));
    }
    for helper in crate::codegen::builtin_helpers::needed_helpers(union_body, false) {
        match helper.key {
            "BranchPath" => parts.push(LEAN_PRELUDE_BRANCH_PATH.to_string()),
            "AverList" => parts.push(LEAN_PRELUDE_AVER_LIST.to_string()),
            "StringHelpers" => parts.push(LEAN_PRELUDE_STRING_HELPERS.to_string()),
            "NumericParse" => parts.push(LEAN_PRELUDE_NUMERIC_PARSE.to_string()),
            "CharByte" => parts.push(LEAN_PRELUDE_CHAR_BYTE.to_string()),
            "AverMeasure" => parts.push(LEAN_PRELUDE_AVER_MEASURE.to_string()),
            "AverMap" => parts.push(generate_map_prelude(union_body, false)),
            "ProofFuel" => parts.push(LEAN_PRELUDE_PROOF_FUEL.to_string()),
            "FloatInstances" => parts.extend([
                LEAN_PRELUDE_FLOAT_COE.to_string(),
                LEAN_PRELUDE_FLOAT_DEC_EQ.to_string(),
            ]),
            "ExceptInstances" => parts.extend([
                LEAN_PRELUDE_EXCEPT_DEC_EQ.to_string(),
                LEAN_PRELUDE_EXCEPT_NS.to_string(),
                LEAN_PRELUDE_OPTION_TO_EXCEPT.to_string(),
            ]),
            "StringHadd" => parts.push(LEAN_PRELUDE_STRING_HADD.to_string()),
            "ResultDatatype" | "OptionDatatype" | "OptionToResult" | "BranchPathDatatype" => {}
            other => panic!(
                "Lean backend has no implementation for builtin helper key '{}'. \
                 Add a match arm in build_common_lean or remove the key from BUILTIN_HELPERS.",
                other
            ),
        }
    }
    parts.join("\n\n")
}

#[cfg(test)]
mod tests {
    use super::{
        VerifyEmitMode, generate_prelude, proof_mode_issues, recurrence, transpile,
        transpile_for_proof_mode, transpile_with_verify_mode,
    };
    use crate::ast::{
        BinOp, Expr, FnBody, FnDef, Literal, MatchArm, Pattern, Spanned, Stmt, TailCallData,
        TopLevel, TypeDef, TypeVariant, VerifyBlock, VerifyGiven, VerifyGivenDomain, VerifyKind,
        VerifyLaw,
    };

    /// Shorthand: wrap an Expr in Spanned with line=0.
    fn sb(e: Expr) -> Spanned<Expr> {
        Spanned::bare(e)
    }
    /// Shorthand: wrap an Expr in Box<Spanned> with line=0.
    fn sbb(e: Expr) -> Box<Spanned<Expr>> {
        Box::new(Spanned::bare(e))
    }
    use crate::codegen::{CodegenContext, build_context};
    use crate::source::parse_source;

    use std::collections::{HashMap, HashSet};
    use std::sync::Arc as Rc;

    /// Populate `ctx.proof_ir` from the current items. Synthetic-AST
    /// tests bypass the pipeline (where ProofLower runs automatically);
    /// IR-pinned law strategies rely on this being populated so the
    /// backend's IR-pin lookups can fire.
    fn populate_proof_ir(ctx: &mut CodegenContext) {
        // Mirror the production order: BuildSymbols → proof_lower. The
        // populate side asserts the symbol table is present (it's a
        // hard prerequisite for FnId-keyed fn_contracts), so synthetic-
        // ctx tests have to build it the same way `refresh_facts` does.
        ctx.symbol_table = crate::ir::SymbolTable::build(&ctx.items, &ctx.modules);
        let inputs = crate::codegen::proof_lower::ProofLowerInputs::from_ctx(ctx);
        ctx.proof_ir = crate::codegen::proof_lower::lower(&inputs);
    }

    fn empty_ctx() -> CodegenContext {
        CodegenContext {
            items: vec![],
            memo_fns: HashSet::new(),
            memo_safe_types: HashSet::new(),
            type_defs: vec![],
            fn_defs: vec![],
            project_name: "verify_mode".to_string(),
            modules: vec![],
            module_prefixes: HashSet::new(),
            policy: None,
            emit_replay_runtime: false,
            runtime_policy_from_env: false,
            guest_entry: None,
            emit_self_host_support: false,
            extra_fn_defs: Vec::new(),
            mutual_tco_members: HashSet::<crate::ir::FnId>::new(),
            recursive_fns: HashSet::<crate::ir::FnId>::new(),
            buffer_build_sinks: HashMap::new(),
            buffer_fusion_sites: Vec::new(),
            synthesized_buffered_fns: Vec::new(),
            proof_ir: crate::ir::ProofIR::default(),
            symbol_table: crate::ir::SymbolTable::default(),
            resolved_fn_defs: Vec::new(),
            resolved_module_fn_defs: Vec::new(),
            current_module_scope: std::cell::RefCell::new(None),
            resolved_program: crate::codegen::program_view::ResolvedProgramView::default(),
        }
    }

    fn ctx_from_source(source: &str, project_name: &str) -> CodegenContext {
        let mut items = parse_source(source).expect("source should parse");
        // Proof-mode minimal pipeline: only the stages a proof
        // exporter actually consumes. Resolve / last_use / escape /
        // interp_lower / buffer_build all rewrite item shapes in
        // ways that break the recursion classifier's source-level
        // pattern matching (e.g. escape inlines a record into the
        // caller, dropping the recursive call's structural shape).
        // Analyze stays on — `recursive_fns` is what `proof_lower`
        // reads to decide which fns to classify.
        let pipeline_result = crate::ir::pipeline::run(
            &mut items,
            crate::ir::PipelineConfig {
                run_tco: true,
                typecheck: Some(crate::ir::TypecheckMode::Full { base_dir: None }),
                run_interp_lower: false,
                run_buffer_build: false,
                run_resolve: false,
                run_last_use: false,
                run_analyze: true,
                run_escape: false,
                run_refinement_lower: true,
                run_contract_lower: true,
                run_law_lower: true,
                // BuildSymbols is needed for fn_contracts lookup
                // (keyed by opaque FnId resolved through the symbol
                // table since the FnKey → FnId migration).
                run_build_symbols: true,
                dep_modules: &[],
                alloc_policy: None,
                call_ctx: None,
                on_after_pass: None,
            },
        );
        let tc = pipeline_result.typecheck.expect("typecheck requested");
        assert!(
            tc.errors.is_empty(),
            "source should typecheck without errors: {:?}",
            tc.errors
        );
        let proof_ir = pipeline_result.proof_ir;
        let mut ctx = build_context(
            items,
            &tc,
            pipeline_result.analysis.as_ref(),
            HashSet::new(),
            project_name.to_string(),
            vec![],
            pipeline_result.symbol_table,
            pipeline_result.resolved_items,
        );
        if let Some(ir) = proof_ir {
            ctx.proof_ir = ir;
        }
        ctx
    }

    /// Concatenate every emitted `.lean` source (entry + per-module +
    /// `AverCommon`) into a single string for content assertions. The
    /// unified emitter splits prelude (`AverCommon.lean`) and body
    /// (`<Project>.lean`) into separate files; tests originally checked
    /// for substrings against the legacy single-file output, so the
    /// helper now returns the concatenation so those substring assertions
    /// keep working regardless of which file the content lands in.
    fn generated_lean_file(out: &crate::codegen::ProjectOutput) -> String {
        out.files
            .iter()
            .filter_map(|(name, content)| {
                (name.ends_with(".lean") && name != "lakefile.lean").then_some(content.as_str())
            })
            .collect::<Vec<&str>>()
            .join("\n")
    }

    fn empty_ctx_with_verify_case() -> CodegenContext {
        let mut ctx = empty_ctx();
        ctx.items.push(TopLevel::Verify(VerifyBlock {
            fn_name: "f".to_string(),
            line: 1,
            cases: vec![(
                sb(Expr::Literal(Literal::Int(1))),
                sb(Expr::Literal(Literal::Int(1))),
            )],
            case_spans: vec![],
            case_givens: vec![],
            case_hostile_origins: vec![],
            case_hostile_profiles: vec![],
            case_reverse_order: vec![],
            kind: VerifyKind::Cases,
            trace: false,
            cases_givens: vec![],
        }));
        ctx
    }

    fn empty_ctx_with_two_verify_blocks_same_fn() -> CodegenContext {
        let mut ctx = empty_ctx();
        ctx.items.push(TopLevel::Verify(VerifyBlock {
            fn_name: "f".to_string(),
            line: 1,
            cases: vec![(
                sb(Expr::Literal(Literal::Int(1))),
                sb(Expr::Literal(Literal::Int(1))),
            )],
            case_spans: vec![],
            case_givens: vec![],
            case_hostile_origins: vec![],
            case_hostile_profiles: vec![],
            case_reverse_order: vec![],
            kind: VerifyKind::Cases,
            trace: false,
            cases_givens: vec![],
        }));
        ctx.items.push(TopLevel::Verify(VerifyBlock {
            fn_name: "f".to_string(),
            line: 2,
            cases: vec![(
                sb(Expr::Literal(Literal::Int(2))),
                sb(Expr::Literal(Literal::Int(2))),
            )],
            case_spans: vec![],
            case_givens: vec![],
            case_hostile_origins: vec![],
            case_hostile_profiles: vec![],
            case_reverse_order: vec![],
            kind: VerifyKind::Cases,
            trace: false,
            cases_givens: vec![],
        }));
        ctx
    }

    fn empty_ctx_with_verify_law() -> CodegenContext {
        let mut ctx = empty_ctx();
        let add = FnDef {
            name: "add".to_string(),
            line: 1,
            params: vec![
                ("a".to_string(), "Int".to_string()),
                ("b".to_string(), "Int".to_string()),
            ],
            return_type: "Int".to_string(),
            effects: vec![],
            desc: None,
            body: Rc::new(FnBody::from_expr(sb(Expr::BinOp(
                BinOp::Add,
                sbb(Expr::Ident("a".to_string())),
                sbb(Expr::Ident("b".to_string())),
            )))),
            resolution: None,
        };
        ctx.fn_defs.push(add.clone());
        ctx.items.push(TopLevel::FnDef(add));
        ctx.items.push(TopLevel::Verify(VerifyBlock {
            fn_name: "add".to_string(),
            line: 1,
            cases: vec![
                (
                    sb(Expr::FnCall(
                        sbb(Expr::Ident("add".to_string())),
                        vec![
                            sb(Expr::Literal(Literal::Int(1))),
                            sb(Expr::Literal(Literal::Int(2))),
                        ],
                    )),
                    sb(Expr::FnCall(
                        sbb(Expr::Ident("add".to_string())),
                        vec![
                            sb(Expr::Literal(Literal::Int(2))),
                            sb(Expr::Literal(Literal::Int(1))),
                        ],
                    )),
                ),
                (
                    sb(Expr::FnCall(
                        sbb(Expr::Ident("add".to_string())),
                        vec![
                            sb(Expr::Literal(Literal::Int(2))),
                            sb(Expr::Literal(Literal::Int(3))),
                        ],
                    )),
                    sb(Expr::FnCall(
                        sbb(Expr::Ident("add".to_string())),
                        vec![
                            sb(Expr::Literal(Literal::Int(3))),
                            sb(Expr::Literal(Literal::Int(2))),
                        ],
                    )),
                ),
            ],
            case_spans: vec![],
            case_givens: vec![],
            case_hostile_origins: vec![],
            case_hostile_profiles: vec![],
            case_reverse_order: vec![],
            kind: VerifyKind::Law(Box::new(VerifyLaw {
                name: "commutative".to_string(),
                givens: vec![
                    VerifyGiven {
                        name: "a".to_string(),
                        type_name: "Int".to_string(),
                        domain: VerifyGivenDomain::IntRange { start: 1, end: 2 },
                    },
                    VerifyGiven {
                        name: "b".to_string(),
                        type_name: "Int".to_string(),
                        domain: VerifyGivenDomain::Explicit(vec![
                            sb(Expr::Literal(Literal::Int(2))),
                            sb(Expr::Literal(Literal::Int(3))),
                        ]),
                    },
                ],
                when: None,
                lhs: sb(Expr::FnCall(
                    sbb(Expr::Ident("add".to_string())),
                    vec![
                        sb(Expr::Ident("a".to_string())),
                        sb(Expr::Ident("b".to_string())),
                    ],
                )),
                rhs: sb(Expr::FnCall(
                    sbb(Expr::Ident("add".to_string())),
                    vec![
                        sb(Expr::Ident("b".to_string())),
                        sb(Expr::Ident("a".to_string())),
                    ],
                )),
                sample_guards: vec![],
            })),
            trace: false,
            cases_givens: vec![],
        }));
        ctx
    }

    #[test]
    fn prelude_normalizes_float_string_format() {
        let prelude = generate_prelude();
        assert!(
            prelude.contains("private def normalizeFloatString (s : String) : String :="),
            "missing normalizeFloatString helper in prelude"
        );
        assert!(
            prelude.contains(
                "def String.fromFloat (f : Float) : String := normalizeFloatString (toString f)"
            ),
            "String.fromFloat should normalize Lean float formatting"
        );
    }

    #[test]
    fn prelude_validates_char_from_code_unicode_bounds() {
        let prelude = generate_prelude();
        assert!(
            prelude.contains("if n < 0 || n > 1114111 then none"),
            "Char.fromCode should reject code points above Unicode max"
        );
        assert!(
            prelude.contains("else if n >= 55296 && n <= 57343 then none"),
            "Char.fromCode should reject surrogate code points"
        );
    }

    #[test]
    fn prelude_includes_map_set_helper_lemmas() {
        let prelude = generate_prelude();
        assert!(
            prelude.contains("theorem has_set_self [DecidableEq α]"),
            "missing AverMap.has_set_self helper theorem"
        );
        assert!(
            prelude.contains("theorem get_set_self [DecidableEq α]"),
            "missing AverMap.get_set_self helper theorem"
        );
    }

    #[test]
    fn lean_output_without_map_usage_omits_map_prelude() {
        let mut ctx = ctx_from_source(
            r#"
module NoMap
    intent = "Simple pure program without maps."

fn addOne(n: Int) -> Int
    n + 1

verify addOne
    addOne(1) => 2
"#,
            "nomap",
        );
        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);

        assert!(
            !lean.contains("namespace AverMap"),
            "did not expect AverMap prelude in program without map usage:\n{}",
            lean
        );
    }

    #[test]
    fn transpile_emits_native_decide_for_verify_by_default() {
        let mut ctx = empty_ctx_with_verify_case();
        let out = transpile(&mut ctx);
        let lean = out
            .files
            .iter()
            .find_map(|(name, content)| (name == "Verify_mode.lean").then_some(content))
            .expect("expected generated Lean file");
        assert!(lean.contains("example : 1 = 1 := by native_decide"));
    }

    #[test]
    fn transpile_can_emit_sorry_for_verify_when_requested() {
        let mut ctx = empty_ctx_with_verify_case();
        let out = transpile_with_verify_mode(&mut ctx, VerifyEmitMode::Sorry);
        let lean = out
            .files
            .iter()
            .find_map(|(name, content)| (name == "Verify_mode.lean").then_some(content))
            .expect("expected generated Lean file");
        assert!(lean.contains("example : 1 = 1 := by sorry"));
    }

    #[test]
    fn transpile_can_emit_theorem_skeletons_for_verify() {
        let mut ctx = empty_ctx_with_verify_case();
        let out = transpile_with_verify_mode(&mut ctx, VerifyEmitMode::TheoremSkeleton);
        let lean = out
            .files
            .iter()
            .find_map(|(name, content)| (name == "Verify_mode.lean").then_some(content))
            .expect("expected generated Lean file");
        assert!(lean.contains("theorem f_verify_1 : 1 = 1 := by"));
        assert!(lean.contains("  sorry"));
    }

    #[test]
    fn theorem_skeleton_numbering_is_global_per_function_across_verify_blocks() {
        let mut ctx = empty_ctx_with_two_verify_blocks_same_fn();
        let out = transpile_with_verify_mode(&mut ctx, VerifyEmitMode::TheoremSkeleton);
        let lean = out
            .files
            .iter()
            .find_map(|(name, content)| (name == "Verify_mode.lean").then_some(content))
            .expect("expected generated Lean file");
        assert!(lean.contains("theorem f_verify_1 : 1 = 1 := by"));
        assert!(lean.contains("theorem f_verify_2 : 2 = 2 := by"));
    }

    #[test]
    fn transpile_emits_named_theorems_for_verify_law() {
        let mut ctx = empty_ctx_with_verify_law();
        populate_proof_ir(&mut ctx);
        let out = transpile(&mut ctx);
        let lean = out
            .files
            .iter()
            .find_map(|(name, content)| (name == "Verify_mode.lean").then_some(content))
            .expect("expected generated Lean file");
        assert!(lean.contains("-- verify law add.commutative (2 cases)"));
        assert!(lean.contains("-- given a: Int = 1..2"));
        assert!(lean.contains("-- given b: Int = [2, 3]"));
        assert!(lean.contains(
            "theorem add_law_commutative : ∀ (a : Int) (b : Int), add a b = add b a := by"
        ));
        assert!(lean.contains("  intro a b"));
        assert!(lean.contains("  simp [add, Int.add_comm]"));
        assert!(lean.contains(
            "theorem add_law_commutative_sample_1 : add 1 2 = add 2 1 := by native_decide"
        ));
        assert!(lean.contains(
            "theorem add_law_commutative_sample_2 : add 2 3 = add 3 2 := by native_decide"
        ));
    }

    #[test]
    fn generate_prelude_emits_int_roundtrip_theorem() {
        let lean = generate_prelude();
        assert!(lean.contains(
            "theorem Int.fromString_fromInt : ∀ n : Int, Int.fromString (String.fromInt n) = .ok n"
        ));
        assert!(lean.contains("theorem String.intercalate_empty_chars (s : String) :"));
        assert!(lean.contains("def splitOnCharGo"));
        assert!(lean.contains("theorem split_single_char_append"));
        assert!(lean.contains("theorem split_intercalate_trailing_single_char"));
        assert!(lean.contains("namespace AverDigits"));
        assert!(lean.contains("theorem String.charAt_length_none (s : String)"));
        assert!(lean.contains("theorem digitChar_not_ws : ∀ d : Nat, d < 10 ->"));
    }

    #[test]
    fn transpile_emits_guarded_theorems_for_verify_law_when_clause() {
        let mut ctx = ctx_from_source(
            r#"
module GuardedLaw
    intent =
        "verify law with precondition"

fn pickGreater(a: Int, b: Int) -> Int
    match a > b
        true -> a
        false -> b

verify pickGreater law ordered
    given a: Int = [1, 2]
    given b: Int = [1, 2]
    when a > b
    pickGreater(a, b) => a
"#,
            "guarded_law",
        );
        let out = transpile_with_verify_mode(&mut ctx, VerifyEmitMode::TheoremSkeleton);
        let lean = generated_lean_file(&out);

        assert!(lean.contains("-- when (a > b)"));
        assert!(lean.contains(
            "theorem pickGreater_law_ordered : ∀ (a : Int) (b : Int), a = 1 ∨ a = 2 -> b = 1 ∨ b = 2 -> (a > b) = true -> pickGreater a b = a := by"
        ));
        assert!(lean.contains(
            "theorem pickGreater_law_ordered_sample_1 : (1 > 1) = true -> pickGreater 1 1 = 1 := by"
        ));
        assert!(lean.contains(
            "theorem pickGreater_law_ordered_sample_4 : (2 > 2) = true -> pickGreater 2 2 = 2 := by"
        ));
    }

    #[test]
    fn transpile_uses_spec_theorem_names_for_declared_spec_laws() {
        let mut ctx = ctx_from_source(
            r#"
module SpecDemo
    intent =
        "spec demo"

fn absVal(x: Int) -> Int
    match x < 0
        true -> 0 - x
        false -> x

fn absValSpec(x: Int) -> Int
    match x < 0
        true -> 0 - x
        false -> x

verify absVal law absValSpec
    given x: Int = [-2, -1, 0, 1, 2]
    absVal(x) => absValSpec(x)
"#,
            "spec_demo",
        );
        let out = transpile_with_verify_mode(&mut ctx, VerifyEmitMode::TheoremSkeleton);
        let lean = generated_lean_file(&out);

        assert!(lean.contains("-- verify law absVal.spec absValSpec (5 cases)"));
        assert!(
            lean.contains(
                "theorem absVal_eq_absValSpec : ∀ (x : Int), absVal x = absValSpec x := by"
            )
        );
        assert!(lean.contains("theorem absVal_eq_absValSpec_checked_domain :"));
        assert!(lean.contains("theorem absVal_eq_absValSpec_sample_1 :"));
        assert!(!lean.contains("theorem absVal_law_absValSpec :"));
    }

    #[test]
    fn transpile_keeps_noncanonical_spec_laws_as_regular_law_names() {
        let mut ctx = ctx_from_source(
            r#"
module SpecLawShape
    intent =
        "shape probe"

fn foo(x: Int) -> Int
    x + 1

fn fooSpec(seed: Int, x: Int) -> Int
    x + seed

verify foo law fooSpec
    given x: Int = [1, 2]
    foo(x) => fooSpec(1, x)
"#,
            "spec_law_shape",
        );
        let out = transpile_with_verify_mode(&mut ctx, VerifyEmitMode::TheoremSkeleton);
        let lean = generated_lean_file(&out);

        assert!(lean.contains("-- verify law foo.fooSpec (2 cases)"));
        assert!(lean.contains("theorem foo_law_fooSpec : ∀ (x : Int), foo x = fooSpec 1 x := by"));
        assert!(!lean.contains("theorem foo_eq_fooSpec :"));
    }

    #[test]
    fn transpile_auto_proves_linear_int_canonical_spec_law_in_auto_mode() {
        let mut ctx = ctx_from_source(
            r#"
module SpecGap
    intent =
        "nontrivial canonical spec law"

fn inc(x: Int) -> Int
    x + 1

fn incSpec(x: Int) -> Int
    x + 2 - 1

verify inc law incSpec
    given x: Int = [0, 1, 2]
    inc(x) => incSpec(x)
"#,
            "spec_gap",
        );
        let out = transpile(&mut ctx);
        let lean = generated_lean_file(&out);

        assert!(lean.contains("-- verify law inc.spec incSpec (3 cases)"));
        assert!(lean.contains("theorem inc_eq_incSpec : ∀ (x : Int), inc x = incSpec x := by"));
        assert!(lean.contains("change (x + 1) = ((x + 2) - 1)"));
        assert!(lean.contains("omega"));
        assert!(!lean.contains(
            "-- universal theorem inc_eq_incSpec omitted: sampled law shape is not auto-proved yet"
        ));
        assert!(lean.contains("theorem inc_eq_incSpec_checked_domain :"));
    }

    #[test]
    fn transpile_auto_proves_guarded_canonical_spec_law_in_auto_mode() {
        let mut ctx = ctx_from_source(
            r#"
module GuardedSpecGap
    intent =
        "guarded canonical spec law"

fn clampNonNegative(x: Int) -> Int
    match x < 0
        true -> 0
        false -> x

fn clampNonNegativeSpec(x: Int) -> Int
    match x < 0
        true -> 0
        false -> x

verify clampNonNegative law clampNonNegativeSpec
    given x: Int = [-2, -1, 0, 1, 2]
    when x >= 0
    clampNonNegative(x) => clampNonNegativeSpec(x)
"#,
            "guarded_spec_gap",
        );
        let out = transpile(&mut ctx);
        let lean = generated_lean_file(&out);

        assert!(lean.contains("-- when (x >= 0)"));
        assert!(lean.contains(
            "theorem clampNonNegative_eq_clampNonNegativeSpec : ∀ (x : Int), x = (-2) ∨ x = (-1) ∨ x = 0 ∨ x = 1 ∨ x = 2 -> (x >= 0) = true -> clampNonNegative x = clampNonNegativeSpec x := by"
        ));
        assert!(lean.contains("intro x h_x h_when"));
        assert!(lean.contains("simpa [clampNonNegative, clampNonNegativeSpec]"));
        assert!(!lean.contains(
            "-- universal theorem clampNonNegative_eq_clampNonNegativeSpec omitted: sampled law shape is not auto-proved yet"
        ));
        assert!(!lean.contains("cases h_x"));
    }

    #[test]
    fn transpile_auto_proves_simp_normalized_canonical_spec_law_in_auto_mode() {
        let mut ctx = ctx_from_source(
            r#"
module SpecGapNonlinear
    intent =
        "nonlinear canonical spec law"

fn square(x: Int) -> Int
    x * x

fn squareSpec(x: Int) -> Int
    x * x + 0

verify square law squareSpec
    given x: Int = [0, 1, 2]
    square(x) => squareSpec(x)
"#,
            "spec_gap_nonlinear",
        );
        let out = transpile(&mut ctx);
        let lean = generated_lean_file(&out);

        assert!(lean.contains("-- verify law square.spec squareSpec (3 cases)"));
        assert!(
            lean.contains(
                "theorem square_eq_squareSpec : ∀ (x : Int), square x = squareSpec x := by"
            )
        );
        assert!(lean.contains("simp [square, squareSpec]"));
        assert!(!lean.contains(
            "-- universal theorem square_eq_squareSpec omitted: sampled law shape is not auto-proved yet"
        ));
        assert!(lean.contains("theorem square_eq_squareSpec_checked_domain :"));
        assert!(lean.contains("theorem square_eq_squareSpec_sample_1 :"));
    }

    #[test]
    fn transpile_auto_proves_reflexive_law_with_rfl() {
        let mut ctx = empty_ctx();
        // After the phase-E3 migration LawTheorem.fn_id is resolved
        // through the symbol table at populate time, so the law's
        // target fn must exist as a FnDef in `ctx.items`. Pre-fix
        // the verify block alone was enough (FnKey was constructed
        // from the bare name without checking).
        let id_law = FnDef {
            name: "idLaw".to_string(),
            line: 1,
            params: vec![("x".to_string(), "Int".to_string())],
            return_type: "Int".to_string(),
            effects: vec![],
            desc: None,
            body: Rc::new(FnBody::from_expr(sb(Expr::Ident("x".to_string())))),
            resolution: None,
        };
        ctx.fn_defs.push(id_law.clone());
        ctx.items.push(TopLevel::FnDef(id_law));
        ctx.items.push(TopLevel::Verify(VerifyBlock {
            fn_name: "idLaw".to_string(),
            line: 1,
            cases: vec![(
                sb(Expr::Literal(Literal::Int(1))),
                sb(Expr::Literal(Literal::Int(1))),
            )],
            case_spans: vec![],
            case_givens: vec![],
            case_hostile_origins: vec![],
            case_hostile_profiles: vec![],
            case_reverse_order: vec![],
            kind: VerifyKind::Law(Box::new(VerifyLaw {
                name: "reflexive".to_string(),
                givens: vec![VerifyGiven {
                    name: "x".to_string(),
                    type_name: "Int".to_string(),
                    domain: VerifyGivenDomain::IntRange { start: 1, end: 2 },
                }],
                when: None,
                lhs: sb(Expr::Ident("x".to_string())),
                rhs: sb(Expr::Ident("x".to_string())),
                sample_guards: vec![],
            })),
            trace: false,
            cases_givens: vec![],
        }));
        // Synthetic-AST test bypasses the parser + pipeline, so the
        // LawLower stage hasn't run. refresh_facts populates
        // ProofIR.law_theorems with Reflexive on `x => x` — backend's
        // Step-24 reader checks the IR to emit rfl.
        ctx.refresh_facts();
        let out = transpile(&mut ctx);
        let lean = out
            .files
            .iter()
            .find_map(|(name, content)| (name == "Verify_mode.lean").then_some(content))
            .expect("expected generated Lean file");
        assert!(lean.contains("theorem idLaw_law_reflexive : ∀ (x : Int), x = x := by"));
        assert!(lean.contains("  intro x"));
        assert!(lean.contains("  rfl"));
    }

    #[test]
    fn transpile_auto_proves_identity_law_for_int_add_wrapper() {
        let mut ctx = empty_ctx_with_verify_law();
        ctx.items.push(TopLevel::Verify(VerifyBlock {
            fn_name: "add".to_string(),
            line: 10,
            cases: vec![(
                sb(Expr::FnCall(
                    sbb(Expr::Ident("add".to_string())),
                    vec![
                        sb(Expr::Literal(Literal::Int(1))),
                        sb(Expr::Literal(Literal::Int(0))),
                    ],
                )),
                sb(Expr::Literal(Literal::Int(1))),
            )],
            case_spans: vec![],
            case_givens: vec![],
            case_hostile_origins: vec![],
            case_hostile_profiles: vec![],
            case_reverse_order: vec![],
            kind: VerifyKind::Law(Box::new(VerifyLaw {
                name: "identityZero".to_string(),
                givens: vec![VerifyGiven {
                    name: "a".to_string(),
                    type_name: "Int".to_string(),
                    domain: VerifyGivenDomain::Explicit(vec![
                        sb(Expr::Literal(Literal::Int(0))),
                        sb(Expr::Literal(Literal::Int(1))),
                    ]),
                }],
                when: None,
                lhs: sb(Expr::FnCall(
                    sbb(Expr::Ident("add".to_string())),
                    vec![
                        sb(Expr::Ident("a".to_string())),
                        sb(Expr::Literal(Literal::Int(0))),
                    ],
                )),
                rhs: sb(Expr::Ident("a".to_string())),
                sample_guards: vec![],
            })),
            trace: false,
            cases_givens: vec![],
        }));
        populate_proof_ir(&mut ctx);
        let out = transpile(&mut ctx);
        let lean = out
            .files
            .iter()
            .find_map(|(name, content)| (name == "Verify_mode.lean").then_some(content))
            .expect("expected generated Lean file");
        assert!(lean.contains("theorem add_law_identityZero : ∀ (a : Int), add a 0 = a := by"));
        assert!(lean.contains("  intro a"));
        assert!(lean.contains("  simp [add]"));
    }

    #[test]
    fn transpile_auto_proves_associative_law_for_int_add_wrapper() {
        let mut ctx = empty_ctx_with_verify_law();
        ctx.items.push(TopLevel::Verify(VerifyBlock {
            fn_name: "add".to_string(),
            line: 20,
            cases: vec![(
                sb(Expr::FnCall(
                    sbb(Expr::Ident("add".to_string())),
                    vec![
                        sb(Expr::FnCall(
                            sbb(Expr::Ident("add".to_string())),
                            vec![
                                sb(Expr::Literal(Literal::Int(1))),
                                sb(Expr::Literal(Literal::Int(2))),
                            ],
                        )),
                        sb(Expr::Literal(Literal::Int(3))),
                    ],
                )),
                sb(Expr::FnCall(
                    sbb(Expr::Ident("add".to_string())),
                    vec![
                        sb(Expr::Literal(Literal::Int(1))),
                        sb(Expr::FnCall(
                            sbb(Expr::Ident("add".to_string())),
                            vec![
                                sb(Expr::Literal(Literal::Int(2))),
                                sb(Expr::Literal(Literal::Int(3))),
                            ],
                        )),
                    ],
                )),
            )],
            case_spans: vec![],
            case_givens: vec![],
            case_hostile_origins: vec![],
            case_hostile_profiles: vec![],
            case_reverse_order: vec![],
            kind: VerifyKind::Law(Box::new(VerifyLaw {
                name: "associative".to_string(),
                givens: vec![
                    VerifyGiven {
                        name: "a".to_string(),
                        type_name: "Int".to_string(),
                        domain: VerifyGivenDomain::Explicit(vec![sb(Expr::Literal(Literal::Int(
                            1,
                        )))]),
                    },
                    VerifyGiven {
                        name: "b".to_string(),
                        type_name: "Int".to_string(),
                        domain: VerifyGivenDomain::Explicit(vec![sb(Expr::Literal(Literal::Int(
                            2,
                        )))]),
                    },
                    VerifyGiven {
                        name: "c".to_string(),
                        type_name: "Int".to_string(),
                        domain: VerifyGivenDomain::Explicit(vec![sb(Expr::Literal(Literal::Int(
                            3,
                        )))]),
                    },
                ],
                when: None,
                lhs: sb(Expr::FnCall(
                    sbb(Expr::Ident("add".to_string())),
                    vec![
                        sb(Expr::FnCall(
                            sbb(Expr::Ident("add".to_string())),
                            vec![
                                sb(Expr::Ident("a".to_string())),
                                sb(Expr::Ident("b".to_string())),
                            ],
                        )),
                        sb(Expr::Ident("c".to_string())),
                    ],
                )),
                rhs: sb(Expr::FnCall(
                    sbb(Expr::Ident("add".to_string())),
                    vec![
                        sb(Expr::Ident("a".to_string())),
                        sb(Expr::FnCall(
                            sbb(Expr::Ident("add".to_string())),
                            vec![
                                sb(Expr::Ident("b".to_string())),
                                sb(Expr::Ident("c".to_string())),
                            ],
                        )),
                    ],
                )),
                sample_guards: vec![],
            })),
            trace: false,
            cases_givens: vec![],
        }));
        populate_proof_ir(&mut ctx);
        let out = transpile(&mut ctx);
        let lean = out
            .files
            .iter()
            .find_map(|(name, content)| (name == "Verify_mode.lean").then_some(content))
            .expect("expected generated Lean file");
        assert!(lean.contains(
            "theorem add_law_associative : ∀ (a : Int) (b : Int) (c : Int), add (add a b) c = add a (add b c) := by"
        ));
        assert!(lean.contains("  intro a b c"));
        assert!(lean.contains("  simp [add, Int.add_assoc]"));
    }

    #[test]
    fn transpile_auto_proves_sub_laws() {
        let mut ctx = empty_ctx();
        let sub = FnDef {
            name: "sub".to_string(),
            line: 1,
            params: vec![
                ("a".to_string(), "Int".to_string()),
                ("b".to_string(), "Int".to_string()),
            ],
            return_type: "Int".to_string(),
            effects: vec![],
            desc: None,
            body: Rc::new(FnBody::from_expr(sb(Expr::BinOp(
                BinOp::Sub,
                sbb(Expr::Ident("a".to_string())),
                sbb(Expr::Ident("b".to_string())),
            )))),
            resolution: None,
        };
        ctx.fn_defs.push(sub.clone());
        ctx.items.push(TopLevel::FnDef(sub));

        ctx.items.push(TopLevel::Verify(VerifyBlock {
            fn_name: "sub".to_string(),
            line: 10,
            cases: vec![(
                sb(Expr::FnCall(
                    sbb(Expr::Ident("sub".to_string())),
                    vec![
                        sb(Expr::Literal(Literal::Int(2))),
                        sb(Expr::Literal(Literal::Int(0))),
                    ],
                )),
                sb(Expr::Literal(Literal::Int(2))),
            )],
            case_spans: vec![],
            case_givens: vec![],
            case_hostile_origins: vec![],
            case_hostile_profiles: vec![],
            case_reverse_order: vec![],
            kind: VerifyKind::Law(Box::new(VerifyLaw {
                name: "rightIdentity".to_string(),
                givens: vec![VerifyGiven {
                    name: "a".to_string(),
                    type_name: "Int".to_string(),
                    domain: VerifyGivenDomain::Explicit(vec![sb(Expr::Literal(Literal::Int(2)))]),
                }],
                when: None,
                lhs: sb(Expr::FnCall(
                    sbb(Expr::Ident("sub".to_string())),
                    vec![
                        sb(Expr::Ident("a".to_string())),
                        sb(Expr::Literal(Literal::Int(0))),
                    ],
                )),
                rhs: sb(Expr::Ident("a".to_string())),
                sample_guards: vec![],
            })),
            trace: false,
            cases_givens: vec![],
        }));
        ctx.items.push(TopLevel::Verify(VerifyBlock {
            fn_name: "sub".to_string(),
            line: 20,
            cases: vec![(
                sb(Expr::FnCall(
                    sbb(Expr::Ident("sub".to_string())),
                    vec![
                        sb(Expr::Literal(Literal::Int(2))),
                        sb(Expr::Literal(Literal::Int(1))),
                    ],
                )),
                sb(Expr::Neg(sbb(Expr::FnCall(
                    sbb(Expr::Ident("sub".to_string())),
                    vec![
                        sb(Expr::Literal(Literal::Int(1))),
                        sb(Expr::Literal(Literal::Int(2))),
                    ],
                )))),
            )],
            case_spans: vec![],
            case_givens: vec![],
            case_hostile_origins: vec![],
            case_hostile_profiles: vec![],
            case_reverse_order: vec![],
            kind: VerifyKind::Law(Box::new(VerifyLaw {
                name: "antiCommutative".to_string(),
                givens: vec![
                    VerifyGiven {
                        name: "a".to_string(),
                        type_name: "Int".to_string(),
                        domain: VerifyGivenDomain::Explicit(vec![sb(Expr::Literal(Literal::Int(
                            2,
                        )))]),
                    },
                    VerifyGiven {
                        name: "b".to_string(),
                        type_name: "Int".to_string(),
                        domain: VerifyGivenDomain::Explicit(vec![sb(Expr::Literal(Literal::Int(
                            1,
                        )))]),
                    },
                ],
                when: None,
                lhs: sb(Expr::FnCall(
                    sbb(Expr::Ident("sub".to_string())),
                    vec![
                        sb(Expr::Ident("a".to_string())),
                        sb(Expr::Ident("b".to_string())),
                    ],
                )),
                rhs: sb(Expr::Neg(sbb(Expr::FnCall(
                    sbb(Expr::Ident("sub".to_string())),
                    vec![
                        sb(Expr::Ident("b".to_string())),
                        sb(Expr::Ident("a".to_string())),
                    ],
                )))),
                sample_guards: vec![],
            })),
            trace: false,
            cases_givens: vec![],
        }));

        populate_proof_ir(&mut ctx);
        let out = transpile(&mut ctx);
        let lean = out
            .files
            .iter()
            .find_map(|(name, content)| (name == "Verify_mode.lean").then_some(content))
            .expect("expected generated Lean file");
        assert!(lean.contains("theorem sub_law_rightIdentity : ∀ (a : Int), sub a 0 = a := by"));
        assert!(lean.contains("  simp [sub]"));
        assert!(lean.contains(
            "theorem sub_law_antiCommutative : ∀ (a : Int) (b : Int), sub a b = (-sub b a) := by"
        ));
        assert!(lean.contains("  simpa [sub] using (Int.neg_sub b a).symm"));
    }

    #[test]
    fn transpile_auto_proves_unary_wrapper_equivalence_law() {
        let mut ctx = empty_ctx();
        let add = FnDef {
            name: "add".to_string(),
            line: 1,
            params: vec![
                ("a".to_string(), "Int".to_string()),
                ("b".to_string(), "Int".to_string()),
            ],
            return_type: "Int".to_string(),
            effects: vec![],
            desc: None,
            body: Rc::new(FnBody::from_expr(sb(Expr::BinOp(
                BinOp::Add,
                sbb(Expr::Ident("a".to_string())),
                sbb(Expr::Ident("b".to_string())),
            )))),
            resolution: None,
        };
        let add_one = FnDef {
            name: "addOne".to_string(),
            line: 2,
            params: vec![("n".to_string(), "Int".to_string())],
            return_type: "Int".to_string(),
            effects: vec![],
            desc: None,
            body: Rc::new(FnBody::from_expr(sb(Expr::BinOp(
                BinOp::Add,
                sbb(Expr::Ident("n".to_string())),
                sbb(Expr::Literal(Literal::Int(1))),
            )))),
            resolution: None,
        };
        ctx.fn_defs.push(add.clone());
        ctx.fn_defs.push(add_one.clone());
        ctx.items.push(TopLevel::FnDef(add));
        ctx.items.push(TopLevel::FnDef(add_one));
        ctx.items.push(TopLevel::Verify(VerifyBlock {
            fn_name: "addOne".to_string(),
            line: 3,
            cases: vec![(
                sb(Expr::FnCall(
                    sbb(Expr::Ident("addOne".to_string())),
                    vec![sb(Expr::Literal(Literal::Int(2)))],
                )),
                sb(Expr::FnCall(
                    sbb(Expr::Ident("add".to_string())),
                    vec![
                        sb(Expr::Literal(Literal::Int(2))),
                        sb(Expr::Literal(Literal::Int(1))),
                    ],
                )),
            )],
            case_spans: vec![],
            case_givens: vec![],
            case_hostile_origins: vec![],
            case_hostile_profiles: vec![],
            case_reverse_order: vec![],
            kind: VerifyKind::Law(Box::new(VerifyLaw {
                name: "identityViaAdd".to_string(),
                givens: vec![VerifyGiven {
                    name: "n".to_string(),
                    type_name: "Int".to_string(),
                    domain: VerifyGivenDomain::Explicit(vec![sb(Expr::Literal(Literal::Int(2)))]),
                }],
                when: None,
                lhs: sb(Expr::FnCall(
                    sbb(Expr::Ident("addOne".to_string())),
                    vec![sb(Expr::Ident("n".to_string()))],
                )),
                rhs: sb(Expr::FnCall(
                    sbb(Expr::Ident("add".to_string())),
                    vec![
                        sb(Expr::Ident("n".to_string())),
                        sb(Expr::Literal(Literal::Int(1))),
                    ],
                )),
                sample_guards: vec![],
            })),
            trace: false,
            cases_givens: vec![],
        }));
        populate_proof_ir(&mut ctx);
        let out = transpile(&mut ctx);
        let lean = out
            .files
            .iter()
            .find_map(|(name, content)| (name == "Verify_mode.lean").then_some(content))
            .expect("expected generated Lean file");
        assert!(
            lean.contains(
                "theorem addOne_law_identityViaAdd : ∀ (n : Int), addOne n = add n 1 := by"
            )
        );
        assert!(lean.contains("  simp [addOne, add]"));
    }

    #[test]
    fn transpile_auto_proves_direct_map_set_laws() {
        let mut ctx = empty_ctx();

        // Stub FnDef for the verify target — see analogous note in
        // `transpile_auto_proves_reflexive_law_with_rfl`.
        let map_fn = FnDef {
            name: "map".to_string(),
            line: 1,
            params: vec![],
            return_type: "Int".to_string(),
            effects: vec![],
            desc: None,
            body: Rc::new(FnBody::from_expr(sb(Expr::Literal(Literal::Int(0))))),
            resolution: None,
        };
        ctx.fn_defs.push(map_fn.clone());
        ctx.items.push(TopLevel::FnDef(map_fn));

        let map_set = |m: Spanned<Expr>, k: Spanned<Expr>, v: Spanned<Expr>| {
            sb(Expr::FnCall(
                sbb(Expr::Attr(
                    sbb(Expr::Ident("Map".to_string())),
                    "set".to_string(),
                )),
                vec![m, k, v],
            ))
        };
        let map_has = |m: Spanned<Expr>, k: Spanned<Expr>| {
            sb(Expr::FnCall(
                sbb(Expr::Attr(
                    sbb(Expr::Ident("Map".to_string())),
                    "has".to_string(),
                )),
                vec![m, k],
            ))
        };
        let map_get = |m: Spanned<Expr>, k: Spanned<Expr>| {
            sb(Expr::FnCall(
                sbb(Expr::Attr(
                    sbb(Expr::Ident("Map".to_string())),
                    "get".to_string(),
                )),
                vec![m, k],
            ))
        };
        let some = |v: Spanned<Expr>| {
            sb(Expr::FnCall(
                sbb(Expr::Attr(
                    sbb(Expr::Ident("Option".to_string())),
                    "Some".to_string(),
                )),
                vec![v],
            ))
        };
        let map_empty = || {
            sb(Expr::FnCall(
                sbb(Expr::Attr(
                    sbb(Expr::Ident("Map".to_string())),
                    "empty".to_string(),
                )),
                vec![],
            ))
        };

        ctx.items.push(TopLevel::Verify(VerifyBlock {
            fn_name: "map".to_string(),
            line: 1,
            cases: vec![(
                map_has(
                    map_set(
                        sb(Expr::Ident("m".to_string())),
                        sb(Expr::Ident("k".to_string())),
                        sb(Expr::Ident("v".to_string())),
                    ),
                    sb(Expr::Ident("k".to_string())),
                ),
                sb(Expr::Literal(Literal::Bool(true))),
            )],
            case_spans: vec![],
            case_givens: vec![],
            case_hostile_origins: vec![],
            case_hostile_profiles: vec![],
            case_reverse_order: vec![],
            kind: VerifyKind::Law(Box::new(VerifyLaw {
                name: "setHasKey".to_string(),
                givens: vec![
                    VerifyGiven {
                        name: "m".to_string(),
                        type_name: "Map<String, Int>".to_string(),
                        domain: VerifyGivenDomain::Explicit(vec![map_empty()]),
                    },
                    VerifyGiven {
                        name: "k".to_string(),
                        type_name: "String".to_string(),
                        domain: VerifyGivenDomain::Explicit(vec![sb(Expr::Literal(Literal::Str(
                            "a".to_string(),
                        )))]),
                    },
                    VerifyGiven {
                        name: "v".to_string(),
                        type_name: "Int".to_string(),
                        domain: VerifyGivenDomain::Explicit(vec![sb(Expr::Literal(Literal::Int(
                            1,
                        )))]),
                    },
                ],
                when: None,
                lhs: map_has(
                    map_set(
                        sb(Expr::Ident("m".to_string())),
                        sb(Expr::Ident("k".to_string())),
                        sb(Expr::Ident("v".to_string())),
                    ),
                    sb(Expr::Ident("k".to_string())),
                ),
                rhs: sb(Expr::Literal(Literal::Bool(true))),
                sample_guards: vec![],
            })),
            trace: false,
            cases_givens: vec![],
        }));

        ctx.items.push(TopLevel::Verify(VerifyBlock {
            fn_name: "map".to_string(),
            line: 2,
            cases: vec![(
                map_get(
                    map_set(
                        sb(Expr::Ident("m".to_string())),
                        sb(Expr::Ident("k".to_string())),
                        sb(Expr::Ident("v".to_string())),
                    ),
                    sb(Expr::Ident("k".to_string())),
                ),
                some(sb(Expr::Ident("v".to_string()))),
            )],
            case_spans: vec![],
            case_givens: vec![],
            case_hostile_origins: vec![],
            case_hostile_profiles: vec![],
            case_reverse_order: vec![],
            kind: VerifyKind::Law(Box::new(VerifyLaw {
                name: "setGetKey".to_string(),
                givens: vec![
                    VerifyGiven {
                        name: "m".to_string(),
                        type_name: "Map<String, Int>".to_string(),
                        domain: VerifyGivenDomain::Explicit(vec![map_empty()]),
                    },
                    VerifyGiven {
                        name: "k".to_string(),
                        type_name: "String".to_string(),
                        domain: VerifyGivenDomain::Explicit(vec![sb(Expr::Literal(Literal::Str(
                            "a".to_string(),
                        )))]),
                    },
                    VerifyGiven {
                        name: "v".to_string(),
                        type_name: "Int".to_string(),
                        domain: VerifyGivenDomain::Explicit(vec![sb(Expr::Literal(Literal::Int(
                            1,
                        )))]),
                    },
                ],
                when: None,
                lhs: map_get(
                    map_set(
                        sb(Expr::Ident("m".to_string())),
                        sb(Expr::Ident("k".to_string())),
                        sb(Expr::Ident("v".to_string())),
                    ),
                    sb(Expr::Ident("k".to_string())),
                ),
                rhs: some(sb(Expr::Ident("v".to_string()))),
                sample_guards: vec![],
            })),
            trace: false,
            cases_givens: vec![],
        }));

        populate_proof_ir(&mut ctx);
        let out = transpile(&mut ctx);
        let lean = out
            .files
            .iter()
            .find_map(|(name, content)| (name == "Verify_mode.lean").then_some(content))
            .expect("expected generated Lean file");
        assert!(lean.contains("simpa using AverMap.has_set_self m k v"));
        assert!(lean.contains("simpa using AverMap.get_set_self m k v"));
    }

    #[test]
    fn transpile_auto_proves_direct_recursive_sum_law_by_structural_induction() {
        let mut ctx = ctx_from_source(
            r#"
module Mirror
    intent =
        "direct recursive sum induction probe"

type Tree
    Leaf(Int)
    Node(Tree, Tree)

fn mirror(t: Tree) -> Tree
    match t
        Tree.Leaf(v) -> Tree.Leaf(v)
        Tree.Node(left, right) -> Tree.Node(mirror(right), mirror(left))

verify mirror law involutive
    given t: Tree = [Tree.Leaf(1), Tree.Node(Tree.Leaf(1), Tree.Leaf(2))]
    mirror(mirror(t)) => t
"#,
            "mirror",
        );
        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);

        assert!(
            lean.contains(
                "theorem mirror_law_involutive : ∀ (t : Tree), mirror (mirror t) = t := by"
            )
        );
        assert!(lean.contains("  induction t with"));
        assert!(lean.contains("  | leaf f0 => simp [mirror]"));
        assert!(lean.contains("  | node f0 f1 ih0 ih1 => simp_all [mirror]"));
        assert!(!lean.contains(
            "-- universal theorem mirror_law_involutive omitted: sampled law shape is not auto-proved yet"
        ));
    }

    #[test]
    fn transpile_auto_proves_map_update_laws() {
        let mut ctx = empty_ctx();

        let map_get = |m: Spanned<Expr>, k: Spanned<Expr>| {
            sb(Expr::FnCall(
                sbb(Expr::Attr(
                    sbb(Expr::Ident("Map".to_string())),
                    "get".to_string(),
                )),
                vec![m, k],
            ))
        };
        let map_set = |m: Spanned<Expr>, k: Spanned<Expr>, v: Spanned<Expr>| {
            sb(Expr::FnCall(
                sbb(Expr::Attr(
                    sbb(Expr::Ident("Map".to_string())),
                    "set".to_string(),
                )),
                vec![m, k, v],
            ))
        };
        let map_has = |m: Spanned<Expr>, k: Spanned<Expr>| {
            sb(Expr::FnCall(
                sbb(Expr::Attr(
                    sbb(Expr::Ident("Map".to_string())),
                    "has".to_string(),
                )),
                vec![m, k],
            ))
        };
        let option_some = |v: Spanned<Expr>| {
            sb(Expr::FnCall(
                sbb(Expr::Attr(
                    sbb(Expr::Ident("Option".to_string())),
                    "Some".to_string(),
                )),
                vec![v],
            ))
        };
        let option_with_default = |opt: Spanned<Expr>, def: Spanned<Expr>| {
            sb(Expr::FnCall(
                sbb(Expr::Attr(
                    sbb(Expr::Ident("Option".to_string())),
                    "withDefault".to_string(),
                )),
                vec![opt, def],
            ))
        };
        let map_empty = || {
            sb(Expr::FnCall(
                sbb(Expr::Attr(
                    sbb(Expr::Ident("Map".to_string())),
                    "empty".to_string(),
                )),
                vec![],
            ))
        };

        let add_one = FnDef {
            name: "addOne".to_string(),
            line: 1,
            params: vec![("n".to_string(), "Int".to_string())],
            return_type: "Int".to_string(),
            effects: vec![],
            desc: None,
            body: Rc::new(FnBody::from_expr(sb(Expr::BinOp(
                BinOp::Add,
                sbb(Expr::Ident("n".to_string())),
                sbb(Expr::Literal(Literal::Int(1))),
            )))),
            resolution: None,
        };
        ctx.fn_defs.push(add_one.clone());
        ctx.items.push(TopLevel::FnDef(add_one));

        let inc_count = FnDef {
            name: "incCount".to_string(),
            line: 2,
            params: vec![
                ("counts".to_string(), "Map<String, Int>".to_string()),
                ("word".to_string(), "String".to_string()),
            ],
            return_type: "Map<String, Int>".to_string(),
            effects: vec![],
            desc: None,
            body: Rc::new(FnBody::Block(vec![
                Stmt::Binding(
                    "current".to_string(),
                    None,
                    map_get(
                        sb(Expr::Ident("counts".to_string())),
                        sb(Expr::Ident("word".to_string())),
                    ),
                ),
                Stmt::Expr(sb(Expr::Match {
                    subject: sbb(Expr::Ident("current".to_string())),
                    arms: vec![
                        MatchArm {
                            pattern: Pattern::Constructor(
                                "Option.Some".to_string(),
                                vec!["n".to_string()],
                            ),
                            body: Box::new(map_set(
                                sb(Expr::Ident("counts".to_string())),
                                sb(Expr::Ident("word".to_string())),
                                sb(Expr::BinOp(
                                    BinOp::Add,
                                    sbb(Expr::Ident("n".to_string())),
                                    sbb(Expr::Literal(Literal::Int(1))),
                                )),
                            )),
                            binding_slots: std::sync::OnceLock::new(),
                        },
                        MatchArm {
                            pattern: Pattern::Constructor("Option.None".to_string(), vec![]),
                            body: Box::new(map_set(
                                sb(Expr::Ident("counts".to_string())),
                                sb(Expr::Ident("word".to_string())),
                                sb(Expr::Literal(Literal::Int(1))),
                            )),
                            binding_slots: std::sync::OnceLock::new(),
                        },
                    ],
                })),
            ])),
            resolution: None,
        };
        ctx.fn_defs.push(inc_count.clone());
        ctx.items.push(TopLevel::FnDef(inc_count));

        ctx.items.push(TopLevel::Verify(VerifyBlock {
            fn_name: "incCount".to_string(),
            line: 10,
            cases: vec![(
                map_has(
                    sb(Expr::FnCall(
                        sbb(Expr::Ident("incCount".to_string())),
                        vec![
                            sb(Expr::Ident("counts".to_string())),
                            sb(Expr::Ident("word".to_string())),
                        ],
                    )),
                    sb(Expr::Ident("word".to_string())),
                ),
                sb(Expr::Literal(Literal::Bool(true))),
            )],
            case_spans: vec![],
            case_givens: vec![],
            case_hostile_origins: vec![],
            case_hostile_profiles: vec![],
            case_reverse_order: vec![],
            kind: VerifyKind::Law(Box::new(VerifyLaw {
                name: "keyPresent".to_string(),
                givens: vec![
                    VerifyGiven {
                        name: "counts".to_string(),
                        type_name: "Map<String, Int>".to_string(),
                        domain: VerifyGivenDomain::Explicit(vec![map_empty()]),
                    },
                    VerifyGiven {
                        name: "word".to_string(),
                        type_name: "String".to_string(),
                        domain: VerifyGivenDomain::Explicit(vec![sb(Expr::Literal(Literal::Str(
                            "a".to_string(),
                        )))]),
                    },
                ],
                when: None,
                lhs: map_has(
                    sb(Expr::FnCall(
                        sbb(Expr::Ident("incCount".to_string())),
                        vec![
                            sb(Expr::Ident("counts".to_string())),
                            sb(Expr::Ident("word".to_string())),
                        ],
                    )),
                    sb(Expr::Ident("word".to_string())),
                ),
                rhs: sb(Expr::Literal(Literal::Bool(true))),
                sample_guards: vec![],
            })),
            trace: false,
            cases_givens: vec![],
        }));

        ctx.items.push(TopLevel::Verify(VerifyBlock {
            fn_name: "incCount".to_string(),
            line: 20,
            cases: vec![(
                map_get(
                    sb(Expr::FnCall(
                        sbb(Expr::Ident("incCount".to_string())),
                        vec![
                            sb(Expr::Ident("counts".to_string())),
                            sb(Expr::Literal(Literal::Str("a".to_string()))),
                        ],
                    )),
                    sb(Expr::Literal(Literal::Str("a".to_string()))),
                ),
                option_some(sb(Expr::FnCall(
                    sbb(Expr::Ident("addOne".to_string())),
                    vec![option_with_default(
                        map_get(
                            sb(Expr::Ident("counts".to_string())),
                            sb(Expr::Literal(Literal::Str("a".to_string()))),
                        ),
                        sb(Expr::Literal(Literal::Int(0))),
                    )],
                ))),
            )],
            case_spans: vec![],
            case_givens: vec![],
            case_hostile_origins: vec![],
            case_hostile_profiles: vec![],
            case_reverse_order: vec![],
            kind: VerifyKind::Law(Box::new(VerifyLaw {
                name: "existingKeyIncrements".to_string(),
                givens: vec![VerifyGiven {
                    name: "counts".to_string(),
                    type_name: "Map<String, Int>".to_string(),
                    domain: VerifyGivenDomain::Explicit(vec![map_empty()]),
                }],
                when: None,
                lhs: map_get(
                    sb(Expr::FnCall(
                        sbb(Expr::Ident("incCount".to_string())),
                        vec![
                            sb(Expr::Ident("counts".to_string())),
                            sb(Expr::Literal(Literal::Str("a".to_string()))),
                        ],
                    )),
                    sb(Expr::Literal(Literal::Str("a".to_string()))),
                ),
                rhs: option_some(sb(Expr::FnCall(
                    sbb(Expr::Ident("addOne".to_string())),
                    vec![option_with_default(
                        map_get(
                            sb(Expr::Ident("counts".to_string())),
                            sb(Expr::Literal(Literal::Str("a".to_string()))),
                        ),
                        sb(Expr::Literal(Literal::Int(0))),
                    )],
                ))),
                sample_guards: vec![],
            })),
            trace: false,
            cases_givens: vec![],
        }));

        populate_proof_ir(&mut ctx);
        let out = transpile(&mut ctx);
        let lean = out
            .files
            .iter()
            .find_map(|(name, content)| (name == "Verify_mode.lean").then_some(content))
            .expect("expected generated Lean file");
        assert!(
            lean.contains("cases h : AverMap.get counts word <;> simp [AverMap.has_set_self]"),
            "expected keyPresent auto-proof with has_set_self"
        );
        assert!(
            lean.contains("cases h : AverMap.get counts \"a\" <;> simp [AverMap.get_set_self, incCount, addOne]"),
            "expected existingKeyIncrements auto-proof with get_set_self"
        );
    }

    #[test]
    fn transpile_parenthesizes_negative_int_call_args_in_law_samples() {
        let mut ctx = empty_ctx();
        let add = FnDef {
            name: "add".to_string(),
            line: 1,
            params: vec![
                ("a".to_string(), "Int".to_string()),
                ("b".to_string(), "Int".to_string()),
            ],
            return_type: "Int".to_string(),
            effects: vec![],
            desc: None,
            body: Rc::new(FnBody::from_expr(sb(Expr::BinOp(
                BinOp::Add,
                sbb(Expr::Ident("a".to_string())),
                sbb(Expr::Ident("b".to_string())),
            )))),
            resolution: None,
        };
        ctx.fn_defs.push(add.clone());
        ctx.items.push(TopLevel::FnDef(add));
        ctx.items.push(TopLevel::Verify(VerifyBlock {
            fn_name: "add".to_string(),
            line: 1,
            cases: vec![(
                sb(Expr::FnCall(
                    sbb(Expr::Ident("add".to_string())),
                    vec![
                        sb(Expr::Literal(Literal::Int(-2))),
                        sb(Expr::Literal(Literal::Int(-1))),
                    ],
                )),
                sb(Expr::FnCall(
                    sbb(Expr::Ident("add".to_string())),
                    vec![
                        sb(Expr::Literal(Literal::Int(-1))),
                        sb(Expr::Literal(Literal::Int(-2))),
                    ],
                )),
            )],
            case_spans: vec![],
            case_givens: vec![],
            case_hostile_origins: vec![],
            case_hostile_profiles: vec![],
            case_reverse_order: vec![],
            kind: VerifyKind::Law(Box::new(VerifyLaw {
                name: "commutative".to_string(),
                givens: vec![
                    VerifyGiven {
                        name: "a".to_string(),
                        type_name: "Int".to_string(),
                        domain: VerifyGivenDomain::Explicit(vec![sb(Expr::Literal(Literal::Int(
                            -2,
                        )))]),
                    },
                    VerifyGiven {
                        name: "b".to_string(),
                        type_name: "Int".to_string(),
                        domain: VerifyGivenDomain::Explicit(vec![sb(Expr::Literal(Literal::Int(
                            -1,
                        )))]),
                    },
                ],
                when: None,
                lhs: sb(Expr::FnCall(
                    sbb(Expr::Ident("add".to_string())),
                    vec![
                        sb(Expr::Ident("a".to_string())),
                        sb(Expr::Ident("b".to_string())),
                    ],
                )),
                rhs: sb(Expr::FnCall(
                    sbb(Expr::Ident("add".to_string())),
                    vec![
                        sb(Expr::Ident("b".to_string())),
                        sb(Expr::Ident("a".to_string())),
                    ],
                )),
                sample_guards: vec![],
            })),
            trace: false,
            cases_givens: vec![],
        }));

        populate_proof_ir(&mut ctx);
        let out = transpile(&mut ctx);
        let lean = out
            .files
            .iter()
            .find_map(|(name, content)| (name == "Verify_mode.lean").then_some(content))
            .expect("expected generated Lean file");
        assert!(lean.contains(
            "theorem add_law_commutative_sample_1 : add (-2) (-1) = add (-1) (-2) := by native_decide"
        ));
    }

    #[test]
    fn verify_law_numbering_is_scoped_per_law_name() {
        let mut ctx = empty_ctx();
        let f = FnDef {
            name: "f".to_string(),
            line: 1,
            params: vec![("x".to_string(), "Int".to_string())],
            return_type: "Int".to_string(),
            effects: vec![],
            desc: None,
            body: Rc::new(FnBody::from_expr(sb(Expr::Ident("x".to_string())))),
            resolution: None,
        };
        ctx.fn_defs.push(f.clone());
        ctx.items.push(TopLevel::FnDef(f));
        ctx.items.push(TopLevel::Verify(VerifyBlock {
            fn_name: "f".to_string(),
            line: 1,
            cases: vec![(
                sb(Expr::Literal(Literal::Int(1))),
                sb(Expr::Literal(Literal::Int(1))),
            )],
            case_spans: vec![],
            case_givens: vec![],
            case_hostile_origins: vec![],
            case_hostile_profiles: vec![],
            case_reverse_order: vec![],
            kind: VerifyKind::Cases,
            trace: false,
            cases_givens: vec![],
        }));
        ctx.items.push(TopLevel::Verify(VerifyBlock {
            fn_name: "f".to_string(),
            line: 2,
            cases: vec![(
                sb(Expr::Literal(Literal::Int(2))),
                sb(Expr::Literal(Literal::Int(2))),
            )],
            case_spans: vec![],
            case_givens: vec![],
            case_hostile_origins: vec![],
            case_hostile_profiles: vec![],
            case_reverse_order: vec![],
            kind: VerifyKind::Law(Box::new(VerifyLaw {
                name: "identity".to_string(),
                givens: vec![VerifyGiven {
                    name: "x".to_string(),
                    type_name: "Int".to_string(),
                    domain: VerifyGivenDomain::Explicit(vec![sb(Expr::Literal(Literal::Int(2)))]),
                }],
                when: None,
                lhs: sb(Expr::Ident("x".to_string())),
                rhs: sb(Expr::Ident("x".to_string())),
                sample_guards: vec![],
            })),
            trace: false,
            cases_givens: vec![],
        }));
        let out = transpile_with_verify_mode(&mut ctx, VerifyEmitMode::TheoremSkeleton);
        let lean = out
            .files
            .iter()
            .find_map(|(name, content)| (name == "Verify_mode.lean").then_some(content))
            .expect("expected generated Lean file");
        assert!(lean.contains("theorem f_verify_1 : 1 = 1 := by"));
        assert!(lean.contains("theorem f_law_identity : ∀ (x : Int), x = x := by"));
        assert!(lean.contains("theorem f_law_identity_sample_1 : 2 = 2 := by"));
        assert!(!lean.contains("theorem f_law_identity_sample_2 : 2 = 2 := by"));
    }

    #[test]
    fn proof_mode_accepts_single_int_countdown_recursion() {
        let mut ctx = empty_ctx();
        let down = FnDef {
            name: "down".to_string(),
            line: 1,
            params: vec![("n".to_string(), "Int".to_string())],
            return_type: "Int".to_string(),
            effects: vec![],
            desc: None,
            body: Rc::new(FnBody::from_expr(sb(Expr::Match {
                subject: sbb(Expr::Ident("n".to_string())),
                arms: vec![
                    MatchArm {
                        pattern: Pattern::Literal(Literal::Int(0)),
                        body: sbb(Expr::Literal(Literal::Int(0))),
                        binding_slots: std::sync::OnceLock::new(),
                    },
                    MatchArm {
                        pattern: Pattern::Wildcard,
                        body: sbb(Expr::TailCall(Box::new(TailCallData::new(
                            "down".to_string(),
                            vec![sb(Expr::BinOp(
                                BinOp::Sub,
                                sbb(Expr::Ident("n".to_string())),
                                sbb(Expr::Literal(Literal::Int(1))),
                            ))],
                        )))),
                        binding_slots: std::sync::OnceLock::new(),
                    },
                ],
            }))),
            resolution: None,
        };
        ctx.items.push(TopLevel::FnDef(down.clone()));
        ctx.fn_defs.push(down);

        ctx.refresh_facts();
        let issues = proof_mode_issues(&ctx);
        assert!(
            issues.is_empty(),
            "expected Int countdown recursion to be accepted, got: {:?}",
            issues
        );

        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = out
            .files
            .iter()
            .find_map(|(name, content)| (name == "Verify_mode.lean").then_some(content))
            .expect("expected generated Lean file");
        // No `Module` declaration in `ctx.items` ⇒ closed-world by the
        // entry-script rule in `is_closed_world_pure_fn`. No external
        // caller for `down` in this synthetic ctx, so the
        // single-caller-predicate extractor returns empty and the Lean
        // emitter defaults to `(h_dom : n ≥ 0)` — same fallback the
        // legacy fibTR path used. Body has the canonical `match n { 0 ->
        // 0; _ -> down(n-1) }` shape, so we land on the native guarded
        // emit instead of fuel.
        assert!(
            lean.contains("def down__aux (n : Int) (h_dom : n ≥ 0) : Int :="),
            "expected native aux def with default precondition, got:\n{}",
            lean
        );
        assert!(
            lean.contains("else down__aux (n - 1) (by omega)"),
            "expected aux recursive call with omega proof, got:\n{}",
            lean
        );
        assert!(lean.contains("termination_by Int.natAbs n"));
        assert!(lean.contains("def down (n : Int) : Int :="));
        assert!(lean.contains("if h_dom : n ≥ 0 then down__aux n h_dom"));
        assert!(!lean.contains("def down__fuel"));
    }

    #[test]
    fn proof_mode_when_stronger_than_refinement_invariant_stays_in_theorem() {
        // Before fix: `when` was dropped unconditionally whenever a
        // given was refinement-lifted, on the assumption that `when`
        // restated the type's invariant. A user-written stronger
        // predicate (`when a >= 10` over `Natural` whose invariant is
        // `a.val >= 0`) would silently disappear from the emitted
        // theorem — the proof artifact would universally quantify
        // over ALL Naturals while the user's source claim was
        // restricted to `a >= 10`. After fix: `when_is_redundant_
        // with_refinement_lifts` compares user's predicate to the
        // type's invariant (via commutator-relaxed compare). Drop
        // only fires when they match.
        let src = "module Stronger\n\
             \x20   intent = \"t\"\n\
             \n\
             record Natural\n\
             \x20   value: Int\n\
             \n\
             fn fromInt(n: Int) -> Result<Natural, String>\n\
             \x20   match n >= 0\n\
             \x20       true  -> Result.Ok(Natural(value = n))\n\
             \x20       false -> Result.Err(\"must be >= 0\")\n\
             \n\
             fn identity(a: Natural) -> Natural\n\
             \x20   a\n\
             \n\
             verify identity law selfEq\n\
             \x20   given a: Int = [10, 20, 30]\n\
             \x20   when a >= 10\n\
             \x20   identity(Natural(value = a)) => identity(Natural(value = a))\n";
        let mut ctx = ctx_from_source(src, "stronger");
        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);
        // `when a >= 10` is STRONGER than Natural's `n >= 0` invariant
        // so the universal theorem must keep it as a premise — AND
        // project `a` to `a.val` because the quantifier is now over
        // the Subtype carrier, not the underlying Int. The bare-`a`
        // shape would fail `lake build` with `failed to synthesize LE
        // Natural / OfNat Natural 10`.
        let universal_theorem = lean
            .lines()
            .find(|l| l.contains("theorem identity_law_selfEq"))
            .unwrap_or_else(|| panic!("expected universal theorem line, got:\n{}", lean));
        assert!(
            universal_theorem.contains("a.val >= 10"),
            "expected `when a.val >= 10` (projected) in universal theorem premise, got:\n{}",
            universal_theorem
        );
        assert!(
            !universal_theorem.contains(" a >= 10"),
            "must NOT emit bare `a >= 10` — Subtype carrier has no `LE Natural` instance, got:\n{}",
            universal_theorem
        );
    }

    #[test]
    fn proof_mode_when_compound_equivalent_to_compound_invariant_drops_cleanly() {
        // Regression guard for `examples/refinement/int_range/int_range.av`:
        // the refinement predicate itself is `Bool.and(n >= 0, n <=
        // 100)`. A naive bijective match (lift `[Bool.and(a >= 0, a <=
        // 100), Bool.and(b >= 0, b <= 100)]` against flattened `when`
        // `[a >= 0, a <= 100, b >= 0, b <= 100]`) would length-
        // mismatch and keep the redundant premise — re-introducing
        // the type-mismatch shape that pre-fix already worked around.
        // The fix flattens BOTH sides; this test pins that.
        let src = "module IR\n\
             \x20   intent = \"t\"\n\
             \n\
             record IntRange\n\
             \x20   value: Int\n\
             \n\
             fn fromInt(n: Int) -> Result<IntRange, String>\n\
             \x20   match Bool.and(n >= 0, n <= 100)\n\
             \x20       true  -> Result.Ok(IntRange(value = n))\n\
             \x20       false -> Result.Err(\"oob\")\n\
             \n\
             fn identity(a: IntRange) -> IntRange\n\
             \x20   a\n\
             \n\
             verify identity law selfEq\n\
             \x20   given a: Int = [0, 50, 100]\n\
             \x20   when Bool.and(a >= 0, a <= 100)\n\
             \x20   identity(IntRange(value = a)) => identity(IntRange(value = a))\n";
        let mut ctx = ctx_from_source(src, "ir");
        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);
        let universal_theorem = lean
            .lines()
            .find(|l| l.contains("theorem identity_law_selfEq"))
            .unwrap_or_else(|| panic!("expected universal theorem line, got:\n{}", lean));
        assert!(
            !universal_theorem.contains("a >= 0"),
            "expected compound `when` to be dropped when it matches compound invariant, got:\n{}",
            universal_theorem
        );
        assert!(
            !universal_theorem.contains("a <= 100"),
            "expected compound `when` to be dropped when it matches compound invariant, got:\n{}",
            universal_theorem
        );
        assert!(
            universal_theorem.contains("∀ (a : IntRange)"),
            "expected universal to quantify over IntRange, got:\n{}",
            universal_theorem
        );
    }

    #[test]
    fn proof_mode_when_equivalent_to_refinement_invariant_drops_cleanly() {
        // Regression: the typical natural.av-style case — `when a >=
        // 0` over `Natural` (invariant `n >= 0`) — must continue to
        // drop the redundant `when` so the universal theorem is `∀ (a
        // : Natural), ...` not `∀ (a : Natural), a.val >= 0 -> ...`
        // (which Lean type-mismatches against the Subtype carrier).
        let src = "module Equiv\n\
             \x20   intent = \"t\"\n\
             \n\
             record Natural\n\
             \x20   value: Int\n\
             \n\
             fn fromInt(n: Int) -> Result<Natural, String>\n\
             \x20   match n >= 0\n\
             \x20       true  -> Result.Ok(Natural(value = n))\n\
             \x20       false -> Result.Err(\"must be >= 0\")\n\
             \n\
             fn identity(a: Natural) -> Natural\n\
             \x20   a\n\
             \n\
             verify identity law selfEq\n\
             \x20   given a: Int = [0, 1, 2]\n\
             \x20   when a >= 0\n\
             \x20   identity(Natural(value = a)) => identity(Natural(value = a))\n";
        let mut ctx = ctx_from_source(src, "equiv");
        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);
        let universal_theorem = lean
            .lines()
            .find(|l| l.contains("theorem identity_law_selfEq"))
            .unwrap_or_else(|| panic!("expected universal theorem line, got:\n{}", lean));
        // Predicate equivalent to invariant → drop.
        assert!(
            !universal_theorem.contains("a >= 0"),
            "expected redundant `when a >= 0` to be dropped from universal theorem, got:\n{}",
            universal_theorem
        );
        assert!(
            universal_theorem.contains("∀ (a : Natural)"),
            "expected universal to quantify over Natural, got:\n{}",
            universal_theorem
        );
    }

    #[test]
    fn proof_mode_non_zero_base_literal_falls_back_to_fuel() {
        // Conservative guard: native IntCountdownGuarded emit assumes
        // the aux's default `(h_dom : p ≥ 0)` precondition, under
        // which only `match p { 0 -> ... }` proves preservation
        // (wildcard arm gives `p ≠ 0`, with `p ≥ 0` that's `p ≥ 1`,
        // so `p - 1 ≥ 0`). A non-zero base literal like `match p { 5
        // -> ... }` would let `p = 0` reach the wildcard arm and
        // recurse with `p - 1 = -1`, breaking the precondition.
        // `omega` would rightly reject it at lake build. Compiler
        // must reject the shape upfront — generalising to arbitrary
        // literals needs a real preservation check that doesn't
        // exist yet (follow-up). Falls back to fuel encoding.
        let src = "module Worker\n\
             \x20   intent = \"t\"\n\
             \n\
             fn worker(n: Int) -> Int\n\
             \x20   match n\n\
             \x20       3 -> n\n\
             \x20       _ -> worker(n - 1)\n\
             \n\
             fn caller(n: Int) -> Int\n\
             \x20   match n > 2\n\
             \x20       true -> match n < 500\n\
             \x20           true -> worker(n)\n\
             \x20           false -> 0\n\
             \x20       false -> 0\n";
        let mut ctx = ctx_from_source(src, "worker");
        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);
        assert!(
            !lean.contains("worker__aux"),
            "must NOT emit native aux when base literal != 0 — preservation isn't provable without a real linear-int check; got:\n{}",
            lean
        );
        assert!(
            lean.contains("def worker__fuel"),
            "expected fuel fallback for non-zero base literal, got:\n{}",
            lean
        );
    }

    #[test]
    fn proof_mode_exposed_int_countdown_falls_back_to_fuel() {
        // Closed-world check: a fn that lives in a module with an
        // explicit `exposes [...]` listing the fn is open-world, so the
        // native-guarded path is unsafe (callers outside this artifact
        // could pass negative ints). The classifier must keep the fuel
        // encoding for these.
        let src = "module Down\n\
             \x20   intent = \"t\"\n\
             \x20   exposes [down]\n\
             \n\
             fn down(n: Int) -> Int\n\
             \x20   match n\n\
             \x20       0 -> 0\n\
             \x20       _ -> down(n - 1)\n";
        let mut ctx = ctx_from_source(src, "downmod");
        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);
        assert!(
            lean.contains("def down__fuel"),
            "expected fuel emission for exposed fn, got:\n{}",
            lean
        );
        assert!(
            !lean.contains("down__aux"),
            "should not emit native aux for exposed fn, got:\n{}",
            lean
        );
    }

    #[test]
    fn proof_mode_accepts_single_int_countdown_on_nonfirst_param() {
        let mut ctx = empty_ctx();
        let repeat_like = FnDef {
            name: "repeatLike".to_string(),
            line: 1,
            params: vec![
                ("char".to_string(), "String".to_string()),
                ("n".to_string(), "Int".to_string()),
            ],
            return_type: "List<String>".to_string(),
            effects: vec![],
            desc: None,
            body: Rc::new(FnBody::from_expr(sb(Expr::Match {
                subject: sbb(Expr::BinOp(
                    BinOp::Lte,
                    sbb(Expr::Ident("n".to_string())),
                    sbb(Expr::Literal(Literal::Int(0))),
                )),
                arms: vec![
                    MatchArm {
                        pattern: Pattern::Literal(Literal::Bool(true)),
                        body: sbb(Expr::List(vec![])),
                        binding_slots: std::sync::OnceLock::new(),
                    },
                    MatchArm {
                        pattern: Pattern::Literal(Literal::Bool(false)),
                        body: sbb(Expr::TailCall(Box::new(TailCallData::new(
                            "repeatLike".to_string(),
                            vec![
                                sb(Expr::Ident("char".to_string())),
                                sb(Expr::BinOp(
                                    BinOp::Sub,
                                    sbb(Expr::Ident("n".to_string())),
                                    sbb(Expr::Literal(Literal::Int(1))),
                                )),
                            ],
                        )))),
                        binding_slots: std::sync::OnceLock::new(),
                    },
                ],
            }))),
            resolution: None,
        };
        ctx.items.push(TopLevel::FnDef(repeat_like.clone()));
        ctx.fn_defs.push(repeat_like);

        ctx.refresh_facts();
        let issues = proof_mode_issues(&ctx);
        assert!(
            issues.is_empty(),
            "expected non-first Int countdown recursion to be accepted, got: {:?}",
            issues
        );

        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = out
            .files
            .iter()
            .find_map(|(name, content)| (name == "Verify_mode.lean").then_some(content))
            .expect("expected generated Lean file");
        assert!(lean.contains("def repeatLike__fuel"));
        assert!(lean.contains("def repeatLike (char : String) (n : Int) : List String :="));
        assert!(lean.contains("repeatLike__fuel ((Int.natAbs n) + 1) char n"));
    }

    #[test]
    fn proof_mode_accepts_negative_guarded_int_ascent() {
        let mut ctx = empty_ctx();
        let normalize = FnDef {
            name: "normalize".to_string(),
            line: 1,
            params: vec![("angle".to_string(), "Int".to_string())],
            return_type: "Int".to_string(),
            effects: vec![],
            desc: None,
            body: Rc::new(FnBody::from_expr(sb(Expr::Match {
                subject: sbb(Expr::BinOp(
                    BinOp::Lt,
                    sbb(Expr::Ident("angle".to_string())),
                    sbb(Expr::Literal(Literal::Int(0))),
                )),
                arms: vec![
                    MatchArm {
                        pattern: Pattern::Literal(Literal::Bool(true)),
                        body: sbb(Expr::TailCall(Box::new(TailCallData::new(
                            "normalize".to_string(),
                            vec![sb(Expr::BinOp(
                                BinOp::Add,
                                sbb(Expr::Ident("angle".to_string())),
                                sbb(Expr::Literal(Literal::Int(360))),
                            ))],
                        )))),
                        binding_slots: std::sync::OnceLock::new(),
                    },
                    MatchArm {
                        pattern: Pattern::Literal(Literal::Bool(false)),
                        body: sbb(Expr::Ident("angle".to_string())),
                        binding_slots: std::sync::OnceLock::new(),
                    },
                ],
            }))),
            resolution: None,
        };
        ctx.items.push(TopLevel::FnDef(normalize.clone()));
        ctx.fn_defs.push(normalize);

        ctx.refresh_facts();
        let issues = proof_mode_issues(&ctx);
        assert!(
            issues.is_empty(),
            "expected negative-guarded Int ascent recursion to be accepted, got: {:?}",
            issues
        );

        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = out
            .files
            .iter()
            .find_map(|(name, content)| (name == "Verify_mode.lean").then_some(content))
            .expect("expected generated Lean file");
        assert!(lean.contains("def normalize__fuel"));
        assert!(lean.contains("normalize__fuel ((Int.natAbs angle) + 1) angle"));
    }

    #[test]
    fn proof_mode_accepts_single_list_structural_recursion() {
        let mut ctx = empty_ctx();
        let len = FnDef {
            name: "len".to_string(),
            line: 1,
            params: vec![("xs".to_string(), "List<Int>".to_string())],
            return_type: "Int".to_string(),
            effects: vec![],
            desc: None,
            body: Rc::new(FnBody::from_expr(sb(Expr::Match {
                subject: sbb(Expr::Ident("xs".to_string())),
                arms: vec![
                    MatchArm {
                        pattern: Pattern::EmptyList,
                        body: sbb(Expr::Literal(Literal::Int(0))),
                        binding_slots: std::sync::OnceLock::new(),
                    },
                    MatchArm {
                        pattern: Pattern::Cons("h".to_string(), "t".to_string()),
                        body: sbb(Expr::TailCall(Box::new(TailCallData::new(
                            "len".to_string(),
                            vec![sb(Expr::Ident("t".to_string()))],
                        )))),
                        binding_slots: std::sync::OnceLock::new(),
                    },
                ],
            }))),
            resolution: None,
        };
        ctx.items.push(TopLevel::FnDef(len.clone()));
        ctx.fn_defs.push(len);

        ctx.refresh_facts();
        let issues = proof_mode_issues(&ctx);
        assert!(
            issues.is_empty(),
            "expected List structural recursion to be accepted, got: {:?}",
            issues
        );
    }

    #[test]
    fn proof_mode_accepts_single_list_structural_recursion_on_nonfirst_param() {
        let mut ctx = empty_ctx();
        let len_from = FnDef {
            name: "lenFrom".to_string(),
            line: 1,
            params: vec![
                ("count".to_string(), "Int".to_string()),
                ("xs".to_string(), "List<Int>".to_string()),
            ],
            return_type: "Int".to_string(),
            effects: vec![],
            desc: None,
            body: Rc::new(FnBody::from_expr(sb(Expr::Match {
                subject: sbb(Expr::Ident("xs".to_string())),
                arms: vec![
                    MatchArm {
                        pattern: Pattern::EmptyList,
                        body: sbb(Expr::Ident("count".to_string())),
                        binding_slots: std::sync::OnceLock::new(),
                    },
                    MatchArm {
                        pattern: Pattern::Cons("h".to_string(), "t".to_string()),
                        body: sbb(Expr::TailCall(Box::new(TailCallData::new(
                            "lenFrom".to_string(),
                            vec![
                                sb(Expr::BinOp(
                                    BinOp::Add,
                                    sbb(Expr::Ident("count".to_string())),
                                    sbb(Expr::Literal(Literal::Int(1))),
                                )),
                                sb(Expr::Ident("t".to_string())),
                            ],
                        )))),
                        binding_slots: std::sync::OnceLock::new(),
                    },
                ],
            }))),
            resolution: None,
        };
        ctx.items.push(TopLevel::FnDef(len_from.clone()));
        ctx.fn_defs.push(len_from);

        ctx.refresh_facts();
        let issues = proof_mode_issues(&ctx);
        assert!(
            issues.is_empty(),
            "expected non-first List structural recursion to be accepted, got: {:?}",
            issues
        );

        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);
        assert!(lean.contains("termination_by xs.length"));
        assert!(!lean.contains("partial def lenFrom"));
    }

    #[test]
    fn proof_mode_accepts_single_string_pos_advance_recursion() {
        let mut ctx = empty_ctx();
        let skip_ws = FnDef {
            name: "skipWs".to_string(),
            line: 1,
            params: vec![
                ("s".to_string(), "String".to_string()),
                ("pos".to_string(), "Int".to_string()),
            ],
            return_type: "Int".to_string(),
            effects: vec![],
            desc: None,
            body: Rc::new(FnBody::from_expr(sb(Expr::Match {
                subject: sbb(Expr::FnCall(
                    sbb(Expr::Attr(
                        sbb(Expr::Ident("String".to_string())),
                        "charAt".to_string(),
                    )),
                    vec![
                        sb(Expr::Ident("s".to_string())),
                        sb(Expr::Ident("pos".to_string())),
                    ],
                )),
                arms: vec![
                    MatchArm {
                        pattern: Pattern::Constructor("Option.None".to_string(), vec![]),
                        body: sbb(Expr::Ident("pos".to_string())),
                        binding_slots: std::sync::OnceLock::new(),
                    },
                    MatchArm {
                        pattern: Pattern::Wildcard,
                        body: sbb(Expr::TailCall(Box::new(TailCallData::new(
                            "skipWs".to_string(),
                            vec![
                                sb(Expr::Ident("s".to_string())),
                                sb(Expr::BinOp(
                                    BinOp::Add,
                                    sbb(Expr::Ident("pos".to_string())),
                                    sbb(Expr::Literal(Literal::Int(1))),
                                )),
                            ],
                        )))),
                        binding_slots: std::sync::OnceLock::new(),
                    },
                ],
            }))),
            resolution: None,
        };
        ctx.items.push(TopLevel::FnDef(skip_ws.clone()));
        ctx.fn_defs.push(skip_ws);

        ctx.refresh_facts();
        let issues = proof_mode_issues(&ctx);
        assert!(
            issues.is_empty(),
            "expected String+pos recursion to be accepted, got: {:?}",
            issues
        );

        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);
        assert!(lean.contains("def skipWs__fuel"));
        assert!(!lean.contains("partial def skipWs"));
    }

    #[test]
    fn proof_mode_accepts_mutual_int_countdown_recursion() {
        let mut ctx = empty_ctx();
        let even = FnDef {
            name: "even".to_string(),
            line: 1,
            params: vec![("n".to_string(), "Int".to_string())],
            return_type: "Bool".to_string(),
            effects: vec![],
            desc: None,
            body: Rc::new(FnBody::from_expr(sb(Expr::Match {
                subject: sbb(Expr::Ident("n".to_string())),
                arms: vec![
                    MatchArm {
                        pattern: Pattern::Literal(Literal::Int(0)),
                        body: sbb(Expr::Literal(Literal::Bool(true))),
                        binding_slots: std::sync::OnceLock::new(),
                    },
                    MatchArm {
                        pattern: Pattern::Wildcard,
                        body: sbb(Expr::TailCall(Box::new(TailCallData::new(
                            "odd".to_string(),
                            vec![sb(Expr::BinOp(
                                BinOp::Sub,
                                sbb(Expr::Ident("n".to_string())),
                                sbb(Expr::Literal(Literal::Int(1))),
                            ))],
                        )))),
                        binding_slots: std::sync::OnceLock::new(),
                    },
                ],
            }))),
            resolution: None,
        };
        let odd = FnDef {
            name: "odd".to_string(),
            line: 2,
            params: vec![("n".to_string(), "Int".to_string())],
            return_type: "Bool".to_string(),
            effects: vec![],
            desc: None,
            body: Rc::new(FnBody::from_expr(sb(Expr::Match {
                subject: sbb(Expr::Ident("n".to_string())),
                arms: vec![
                    MatchArm {
                        pattern: Pattern::Literal(Literal::Int(0)),
                        body: sbb(Expr::Literal(Literal::Bool(false))),
                        binding_slots: std::sync::OnceLock::new(),
                    },
                    MatchArm {
                        pattern: Pattern::Wildcard,
                        body: sbb(Expr::TailCall(Box::new(TailCallData::new(
                            "even".to_string(),
                            vec![sb(Expr::BinOp(
                                BinOp::Sub,
                                sbb(Expr::Ident("n".to_string())),
                                sbb(Expr::Literal(Literal::Int(1))),
                            ))],
                        )))),
                        binding_slots: std::sync::OnceLock::new(),
                    },
                ],
            }))),
            resolution: None,
        };
        ctx.items.push(TopLevel::FnDef(even.clone()));
        ctx.items.push(TopLevel::FnDef(odd.clone()));
        ctx.fn_defs.push(even);
        ctx.fn_defs.push(odd);

        ctx.refresh_facts();
        let issues = proof_mode_issues(&ctx);
        assert!(
            issues.is_empty(),
            "expected mutual Int countdown recursion to be accepted, got: {:?}",
            issues
        );

        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);
        assert!(lean.contains("def even__fuel"));
        assert!(lean.contains("def odd__fuel"));
        assert!(lean.contains("def even (n : Int) : Bool :="));
        assert!(lean.contains("even__fuel ((Int.natAbs n) + 1) n"));
    }

    #[test]
    fn proof_mode_accepts_mutual_string_pos_recursion_with_ranked_same_edges() {
        let mut ctx = empty_ctx();
        let f = FnDef {
            name: "f".to_string(),
            line: 1,
            params: vec![
                ("s".to_string(), "String".to_string()),
                ("pos".to_string(), "Int".to_string()),
            ],
            return_type: "Int".to_string(),
            effects: vec![],
            desc: None,
            body: Rc::new(FnBody::from_expr(sb(Expr::Match {
                subject: sbb(Expr::BinOp(
                    BinOp::Gte,
                    sbb(Expr::Ident("pos".to_string())),
                    sbb(Expr::Literal(Literal::Int(3))),
                )),
                arms: vec![
                    MatchArm {
                        pattern: Pattern::Literal(Literal::Bool(true)),
                        body: sbb(Expr::Ident("pos".to_string())),
                        binding_slots: std::sync::OnceLock::new(),
                    },
                    MatchArm {
                        pattern: Pattern::Wildcard,
                        body: sbb(Expr::TailCall(Box::new(TailCallData::new(
                            "g".to_string(),
                            vec![
                                sb(Expr::Ident("s".to_string())),
                                sb(Expr::Ident("pos".to_string())),
                            ],
                        )))),
                        binding_slots: std::sync::OnceLock::new(),
                    },
                ],
            }))),
            resolution: None,
        };
        let g = FnDef {
            name: "g".to_string(),
            line: 2,
            params: vec![
                ("s".to_string(), "String".to_string()),
                ("pos".to_string(), "Int".to_string()),
            ],
            return_type: "Int".to_string(),
            effects: vec![],
            desc: None,
            body: Rc::new(FnBody::from_expr(sb(Expr::Match {
                subject: sbb(Expr::BinOp(
                    BinOp::Gte,
                    sbb(Expr::Ident("pos".to_string())),
                    sbb(Expr::Literal(Literal::Int(3))),
                )),
                arms: vec![
                    MatchArm {
                        pattern: Pattern::Literal(Literal::Bool(true)),
                        body: sbb(Expr::Ident("pos".to_string())),
                        binding_slots: std::sync::OnceLock::new(),
                    },
                    MatchArm {
                        pattern: Pattern::Wildcard,
                        body: sbb(Expr::TailCall(Box::new(TailCallData::new(
                            "f".to_string(),
                            vec![
                                sb(Expr::Ident("s".to_string())),
                                sb(Expr::BinOp(
                                    BinOp::Add,
                                    sbb(Expr::Ident("pos".to_string())),
                                    sbb(Expr::Literal(Literal::Int(1))),
                                )),
                            ],
                        )))),
                        binding_slots: std::sync::OnceLock::new(),
                    },
                ],
            }))),
            resolution: None,
        };
        ctx.items.push(TopLevel::FnDef(f.clone()));
        ctx.items.push(TopLevel::FnDef(g.clone()));
        ctx.fn_defs.push(f);
        ctx.fn_defs.push(g);

        ctx.refresh_facts();
        let issues = proof_mode_issues(&ctx);
        assert!(
            issues.is_empty(),
            "expected mutual String+pos recursion to be accepted, got: {:?}",
            issues
        );

        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);
        assert!(lean.contains("def f__fuel"));
        assert!(lean.contains("def g__fuel"));
        assert!(!lean.contains("partial def f"));
    }

    #[test]
    fn proof_mode_accepts_mutual_ranked_sizeof_recursion() {
        let mut ctx = empty_ctx();
        let f = FnDef {
            name: "f".to_string(),
            line: 1,
            params: vec![("xs".to_string(), "List<Int>".to_string())],
            return_type: "Int".to_string(),
            effects: vec![],
            desc: None,
            body: Rc::new(FnBody::from_expr(sb(Expr::TailCall(Box::new(
                TailCallData::new(
                    "g".to_string(),
                    vec![
                        sb(Expr::Literal(Literal::Str("acc".to_string()))),
                        sb(Expr::Ident("xs".to_string())),
                    ],
                ),
            ))))),
            resolution: None,
        };
        let g = FnDef {
            name: "g".to_string(),
            line: 2,
            params: vec![
                ("acc".to_string(), "String".to_string()),
                ("xs".to_string(), "List<Int>".to_string()),
            ],
            return_type: "Int".to_string(),
            effects: vec![],
            desc: None,
            body: Rc::new(FnBody::from_expr(sb(Expr::Match {
                subject: sbb(Expr::Ident("xs".to_string())),
                arms: vec![
                    MatchArm {
                        pattern: Pattern::EmptyList,
                        body: sbb(Expr::Literal(Literal::Int(0))),
                        binding_slots: std::sync::OnceLock::new(),
                    },
                    MatchArm {
                        pattern: Pattern::Cons("h".to_string(), "t".to_string()),
                        body: sbb(Expr::TailCall(Box::new(TailCallData::new(
                            "f".to_string(),
                            vec![sb(Expr::Ident("t".to_string()))],
                        )))),
                        binding_slots: std::sync::OnceLock::new(),
                    },
                ],
            }))),
            resolution: None,
        };
        ctx.items.push(TopLevel::FnDef(f.clone()));
        ctx.items.push(TopLevel::FnDef(g.clone()));
        ctx.fn_defs.push(f);
        ctx.fn_defs.push(g);

        ctx.refresh_facts();
        let issues = proof_mode_issues(&ctx);
        assert!(
            issues.is_empty(),
            "expected mutual ranked-sizeOf recursion to be accepted, got: {:?}",
            issues
        );

        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);
        // After the native-decreases path landed for mutual sizeOf
        // SCCs (PR #84), this group emits as a plain `mutual ...
        // end` block with `termination_by` instead of the older
        // `__fuel` helper-and-wrapper pair. The classifier still
        // recognises the recursion (no proof-mode issues raised),
        // which is what this test pins.
        assert!(lean.contains("mutual"));
        assert!(lean.contains("def f"));
        assert!(lean.contains("def g"));
        assert!(lean.contains("termination_by"));
        assert!(!lean.contains("partial def f"));
        assert!(!lean.contains("partial def g"));
    }

    #[test]
    fn proof_mode_rejects_recursive_pure_functions() {
        let mut ctx = empty_ctx();
        let recursive_fn = FnDef {
            name: "loop".to_string(),
            line: 1,
            params: vec![("n".to_string(), "Int".to_string())],
            return_type: "Int".to_string(),
            effects: vec![],
            desc: None,
            body: Rc::new(FnBody::from_expr(sb(Expr::FnCall(
                sbb(Expr::Ident("loop".to_string())),
                vec![sb(Expr::Ident("n".to_string()))],
            )))),
            resolution: None,
        };
        ctx.items.push(TopLevel::FnDef(recursive_fn.clone()));
        ctx.fn_defs.push(recursive_fn);

        ctx.refresh_facts();
        let issues = proof_mode_issues(&ctx);
        assert!(
            issues.iter().any(|i| i.contains("outside proof subset")),
            "expected recursive function blocker, got: {:?}",
            issues
        );
    }

    #[test]
    fn proof_mode_allows_recursive_types() {
        let mut ctx = empty_ctx();
        let recursive_type = TypeDef::Sum {
            name: "Node".to_string(),
            variants: vec![TypeVariant {
                name: "Cons".to_string(),
                fields: vec!["Node".to_string()],
            }],
            line: 1,
        };
        ctx.items.push(TopLevel::TypeDef(recursive_type.clone()));
        ctx.type_defs.push(recursive_type);

        ctx.refresh_facts();
        let issues = proof_mode_issues(&ctx);
        assert!(
            issues
                .iter()
                .all(|i| !i.contains("recursive types require unsafe DecidableEq shim")),
            "did not expect recursive type blocker, got: {:?}",
            issues
        );
    }

    #[test]
    fn law_auto_example_exports_real_proof_artifacts() {
        let mut ctx = ctx_from_source(
            include_str!("../../../examples/formal/law_auto.av"),
            "law_auto",
        );
        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);

        assert!(lean.contains("theorem add_law_commutative :"));
        assert!(lean.contains("theorem id'_law_reflexive : ∀ (x : Int), x = x := by"));
        assert!(lean.contains("theorem incCount_law_keyPresent :"));
        assert!(lean.contains("AverMap.has_set_self"));
        assert!(lean.contains("theorem add_law_commutative_sample_1 :"));
        assert!(lean.contains(":= by native_decide"));
    }

    #[test]
    fn json_example_stays_inside_proof_subset() {
        let mut ctx = ctx_from_source(include_str!("../../../examples/data/json.av"), "json");
        ctx.refresh_facts();
        let issues = proof_mode_issues(&ctx);
        assert!(
            issues.is_empty(),
            "expected json example to stay inside proof subset, got: {:?}",
            issues
        );
    }

    #[test]
    fn json_example_uses_total_defs_and_domain_guarded_laws_in_proof_mode() {
        let mut ctx = ctx_from_source(include_str!("../../../examples/data/json.av"), "json");
        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);

        assert!(!lean.contains("partial def"));
        assert!(lean.contains("def skipWs__fuel"));
        assert!(lean.contains("def parseValue__fuel"));
        assert!(lean.contains("def toString' (j : Json) : String :="));
        assert!(
            lean.contains(
                "def averMeasureJsonEntries_String (items : List (String × Json)) : Nat :="
            )
        );
        assert!(lean.contains(
            "| .jsonObject x0 => (averMeasureJsonEntries_String (AverMap.entries x0)) + 1"
        ));
        assert!(lean.contains("-- when jsonRoundtripSafe j"));
        assert!(!lean.contains("-- hint: verify law '"));
        assert!(!lean.contains("private theorem toString'_law_parseRoundtrip_aux"));
        assert!(
            lean.contains(
                "theorem toString'_law_parseRoundtrip : ∀ (j : Json), j = Json.jsonNull ∨"
            )
        );
        assert!(lean.contains(
            "jsonRoundtripSafe j = true -> fromString (toString' j) = Except.ok j := by"
        ));
        assert!(
            lean.contains("theorem finishFloat_law_fromCanonicalFloat : ∀ (f : Float), f = 3.5 ∨")
        );
        assert!(lean.contains("theorem finishInt_law_fromCanonicalInt_checked_domain :"));
        assert!(lean.contains(
            "theorem toString'_law_parseValueRoundtrip : ∀ (j : Json), j = Json.jsonNull ∨"
        ));
        assert!(lean.contains("theorem toString'_law_parseRoundtrip_sample_1 :"));
        assert!(lean.contains(
            "example : fromString \"null\" = Except.ok Json.jsonNull := by native_decide"
        ));
    }

    #[test]
    fn transpile_injects_builtin_network_types_and_vector_get_support() {
        let mut ctx = ctx_from_source(
            r#"
fn firstOrMissing(xs: Vector<String>) -> Result<String, String>
    Option.toResult(Vector.get(xs, 0), "missing")

fn defaultHeaders() -> Map<String, List<String>>
    {"content-type" => ["application/json"]}

fn mkResponse(body: String) -> HttpResponse
    HttpResponse(status = 200, body = body, headers = defaultHeaders())

fn requestPath(req: HttpRequest) -> String
    req.path

fn echoConn(conn: Tcp.Connection) -> Tcp.Connection
    conn
"#,
            "network_helpers",
        );
        let out = transpile(&mut ctx);
        let lean = generated_lean_file(&out);

        assert!(lean.contains("structure HttpResponse where"));
        assert!(lean.contains("structure HttpRequest where"));
        // `Tcp.Connection` is opaque from the surface (Phase 4.7+
        // fix #11), but the Lean prelude still ships its struct
        // so functions that take/return `Tcp.Connection` typecheck.
        assert!(lean.contains("structure Tcp_Connection where"));
        assert!(lean.contains("port : Int"));
        // Headers field renders as the Map shape (Lean uses List of pairs).
        assert!(lean.contains("List (String × List String)"));
    }

    #[test]
    fn law_auto_example_has_no_sorry_in_proof_mode() {
        let mut ctx = ctx_from_source(
            include_str!("../../../examples/formal/law_auto.av"),
            "law_auto",
        );
        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);
        assert!(
            !lean.contains("sorry"),
            "expected law_auto proof export to avoid sorry, got:\n{}",
            lean
        );
    }

    #[test]
    fn map_example_has_no_sorry_in_proof_mode() {
        let mut ctx = ctx_from_source(include_str!("../../../examples/data/map.av"), "map");
        ctx.refresh_facts();
        let issues = proof_mode_issues(&ctx);
        assert!(
            issues.is_empty(),
            "expected map example to stay inside proof subset, got: {:?}",
            issues
        );

        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);
        // After codegen change: universal theorems that can't be auto-proved get sorry
        assert!(lean.contains("theorem incCount_law_trackedCountStepsByOne :"));
        assert!(lean.contains("sorry"));
        // Universal theorems that can't be auto-proved now get sorry instead of being omitted
        assert!(lean.contains("theorem countWords_law_presenceMatchesContains_sample_1 :"));
        assert!(lean.contains("theorem countWords_law_trackedWordCount_sample_1 :"));
        assert!(lean.contains("AverMap.has_set_self"));
        assert!(lean.contains("AverMap.get_set_self"));
    }

    #[test]
    fn spec_laws_example_has_no_sorry_in_proof_mode() {
        let mut ctx = ctx_from_source(
            include_str!("../../../examples/formal/spec_laws.av"),
            "spec_laws",
        );
        ctx.refresh_facts();
        let issues = proof_mode_issues(&ctx);
        assert!(
            issues.is_empty(),
            "expected spec_laws example to stay inside proof subset, got: {:?}",
            issues
        );

        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);
        assert!(
            !lean.contains("sorry"),
            "expected spec_laws proof export to avoid sorry, got:\n{}",
            lean
        );
        assert!(lean.contains("theorem absVal_eq_absValSpec :"));
        assert!(lean.contains("theorem clampNonNegative_eq_clampNonNegativeSpec :"));
    }

    #[test]
    fn rle_example_exports_sampled_roundtrip_laws_without_sorry() {
        let mut ctx = ctx_from_source(include_str!("../../../examples/data/rle.av"), "rle");
        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);

        assert!(
            lean.contains("sorry"),
            "expected rle proof export to contain sorry for unproved universal theorems"
        );
        assert!(lean.contains(
            "theorem encode_law_roundtrip_sample_1 : decode (encode []) = [] := by native_decide"
        ));
        assert!(lean.contains(
            "theorem encodeString_law_string_roundtrip_sample_1 : decodeString (encodeString \"\") = \"\" := by native_decide"
        ));
    }

    #[test]
    fn fibonacci_example_uses_native_guarded_int_countdown_in_proof_mode() {
        let mut ctx = ctx_from_source(
            include_str!("../../../examples/data/fibonacci.av"),
            "fibonacci",
        );
        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);

        // Native guarded emission: an aux fn with the precondition
        // derived from `fib`'s `match (n < 0) { false -> fibTR(n,...) }`
        // arm (flipped to positive form `n >= 0` at extract time) +
        // termination on `n.natAbs`, plus a thin wrapper preserving the
        // source signature. Replaces the historical `def fibTR__fuel`
        // path so Lean can `simp`/`decide` through recursive calls
        // instead of treating the helper as opaque.
        assert!(
            lean.contains(
                "def fibTR__aux (n : Int) (a : Int) (b : Int) (h_dom : ((n >= 0))) : Int :="
            ),
            "expected fibTR aux with caller-derived precondition, got:\n{}",
            lean
        );
        assert!(
            lean.contains("if h_zero : n = 0 then a"),
            "expected dependent-if on literal 0, got:\n{}",
            lean
        );
        assert!(
            lean.contains("else fibTR__aux (n - 1) b (a + b) (by omega)"),
            "expected recursive call carrying (by omega), got:\n{}",
            lean
        );
        assert!(lean.contains("termination_by Int.natAbs n"));
        assert!(lean.contains("def fibTR (n : Int) (a : Int) (b : Int) : Int :="));
        assert!(lean.contains("if h_dom : ((n >= 0)) then fibTR__aux n a b h_dom"));
        assert!(!lean.contains("def fibTR__fuel"));
        assert!(!lean.contains("partial def fibTR"));
    }

    #[test]
    fn fibonacci_example_stays_inside_proof_subset() {
        let mut ctx = ctx_from_source(
            include_str!("../../../examples/data/fibonacci.av"),
            "fibonacci",
        );
        ctx.refresh_facts();
        let issues = proof_mode_issues(&ctx);
        assert!(
            issues.is_empty(),
            "expected fibonacci example to stay inside proof subset, got: {:?}",
            issues
        );
    }

    #[test]
    fn fibonacci_example_matches_general_linear_recurrence_shapes() {
        let ctx = ctx_from_source(
            include_str!("../../../examples/data/fibonacci.av"),
            "fibonacci",
        );
        let fib = ctx.fn_defs.iter().find(|fd| fd.name == "fib").unwrap();
        let fib_tr = ctx.fn_defs.iter().find(|fd| fd.name == "fibTR").unwrap();
        let fib_spec = ctx.fn_defs.iter().find(|fd| fd.name == "fibSpec").unwrap();

        assert!(recurrence::detect_tailrec_int_linear_pair_wrapper(fib).is_some());
        assert!(recurrence::detect_tailrec_int_linear_pair_worker(fib_tr).is_some());
        assert!(recurrence::detect_second_order_int_linear_recurrence(fib_spec).is_some());
    }

    #[test]
    fn fibonacci_example_auto_proves_general_linear_recurrence_spec_law() {
        let mut ctx = ctx_from_source(
            include_str!("../../../examples/data/fibonacci.av"),
            "fibonacci",
        );
        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);

        assert!(lean.contains("private def fibSpec__nat : Nat -> Int"));
        assert!(!lean.contains("partial def fibSpec"));
        assert!(lean.contains("private theorem fib_eq_fibSpec__worker_nat_shift"));
        assert!(lean.contains("private theorem fib_eq_fibSpec__helper_nat"));
        assert!(lean.contains("private theorem fib_eq_fibSpec__helper_seed"));
        assert!(lean.contains("theorem fib_eq_fibSpec : ∀ (n : Int), fib n = fibSpec n := by"));
        assert!(!lean.contains(
            "-- universal theorem fib_eq_fibSpec omitted: sampled law shape is not auto-proved yet"
        ));
    }

    #[test]
    fn pell_like_example_auto_proves_same_general_shape() {
        let mut ctx = ctx_from_source(
            r#"
module Pell
    intent =
        "linear recurrence probe"

fn pellTR(n: Int, a: Int, b: Int) -> Int
    match n
        0 -> a
        _ -> pellTR(n - 1, b, a + 2 * b)

fn pell(n: Int) -> Int
    match n < 0
        true -> 0
        false -> pellTR(n, 0, 1)

fn pellSpec(n: Int) -> Int
    match n < 0
        true -> 0
        false -> match n
            0 -> 0
            1 -> 1
            _ -> pellSpec(n - 2) + 2 * pellSpec(n - 1)

verify pell law pellSpec
    given n: Int = [0, 1, 2, 3]
    pell(n) => pellSpec(n)
"#,
            "pell",
        );
        ctx.refresh_facts();
        let issues = proof_mode_issues(&ctx);
        assert!(
            issues.is_empty(),
            "expected pell example to stay inside proof subset, got: {:?}",
            issues
        );

        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);
        assert!(lean.contains("private def pellSpec__nat : Nat -> Int"));
        assert!(lean.contains("private theorem pell_eq_pellSpec__worker_nat_shift"));
        assert!(lean.contains("theorem pell_eq_pellSpec : ∀ (n : Int), pell n = pellSpec n := by"));
        assert!(!lean.contains(
            "-- universal theorem pell_eq_pellSpec omitted: sampled law shape is not auto-proved yet"
        ));
    }

    #[test]
    fn nonlinear_pair_state_recurrence_is_not_auto_proved_as_linear_shape() {
        let mut ctx = ctx_from_source(
            r#"
module WeirdRec
    intent =
        "reject nonlinear pair-state recurrence from linear recurrence prover"

fn weirdTR(n: Int, a: Int, b: Int) -> Int
    match n
        0 -> a
        _ -> weirdTR(n - 1, b, a * b)

fn weird(n: Int) -> Int
    match n < 0
        true -> 0
        false -> weirdTR(n, 0, 1)

fn weirdSpec(n: Int) -> Int
    match n < 0
        true -> 0
        false -> match n
            0 -> 0
            1 -> 1
            _ -> weirdSpec(n - 1) * weirdSpec(n - 2)

verify weird law weirdSpec
    given n: Int = [0, 1, 2, 3]
    weird(n) => weirdSpec(n)
"#,
            "weirdrec",
        );
        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);

        // After codegen change: emit sorry instead of omitting universal theorems
        assert!(lean.contains("sorry"));
        assert!(!lean.contains("private theorem weird_eq_weirdSpec__worker_nat_shift"));
        assert!(lean.contains("theorem weird_eq_weirdSpec_sample_1 :"));
    }

    #[test]
    fn date_example_stays_inside_proof_subset() {
        let mut ctx = ctx_from_source(include_str!("../../../examples/data/date.av"), "date");
        ctx.refresh_facts();
        let issues = proof_mode_issues(&ctx);
        assert!(
            issues.is_empty(),
            "expected date example to stay inside proof subset, got: {:?}",
            issues
        );

        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);
        assert!(!lean.contains("partial def"));
        assert!(lean.contains("def parseIntSlice (s : String) (from' : Int) (to : Int) : Int :="));
    }

    #[test]
    fn temperature_example_stays_inside_proof_subset() {
        let mut ctx = ctx_from_source(
            include_str!("../../../examples/core/temperature.av"),
            "temperature",
        );
        ctx.refresh_facts();
        let issues = proof_mode_issues(&ctx);
        assert!(
            issues.is_empty(),
            "expected temperature example to stay inside proof subset, got: {:?}",
            issues
        );

        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);
        assert!(!lean.contains("partial def"));
        assert!(
            lean.contains("example : celsiusToFahr 0.0 = 32.0 := by native_decide"),
            "expected verify examples to survive proof export, got:\n{}",
            lean
        );
    }

    #[test]
    fn quicksort_example_stays_inside_proof_subset() {
        let mut ctx = ctx_from_source(
            include_str!("../../../examples/data/quicksort.av"),
            "quicksort",
        );
        ctx.refresh_facts();
        let issues = proof_mode_issues(&ctx);
        assert!(
            issues.is_empty(),
            "expected quicksort example to stay inside proof subset, got: {:?}",
            issues
        );

        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);
        assert!(lean.contains("def isOrderedFrom"));
        assert!(!lean.contains("partial def isOrderedFrom"));
        assert!(lean.contains("termination_by xs.length"));
    }

    #[test]
    fn grok_s_language_example_uses_total_ranked_sizeof_mutual_recursion() {
        let mut ctx = ctx_from_source(
            include_str!("../../../examples/core/grok_s_language.av"),
            "grok_s_language",
        );
        ctx.refresh_facts();
        let issues = proof_mode_issues(&ctx);
        assert!(
            issues.is_empty(),
            "expected grok_s_language example to stay inside proof subset, got: {:?}",
            issues
        );

        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);
        assert!(lean.contains("mutual"));
        assert!(lean.contains("def eval__fuel"));
        assert!(lean.contains("def parseListItems__fuel"));
        assert!(!lean.contains("partial def eval"));
        assert!(!lean.contains("termination_by (sizeOf e,"));
        assert!(lean.contains("-- when validSymbolNames e"));
        assert!(!lean.contains("private theorem toString'_law_parseRoundtrip_aux"));
        assert!(lean.contains(
            "theorem toString'_law_parseRoundtrip : ∀ (e : Sexpr), e = Sexpr.atomNum 42 ∨"
        ));
        assert!(
            lean.contains("validSymbolNames e = true -> parse (toString' e) = Except.ok e := by")
        );
        assert!(lean.contains("theorem toString'_law_parseSexprRoundtrip :"));
        assert!(lean.contains("theorem toString'_law_parseRoundtrip_sample_1 :"));
    }

    #[test]
    fn lambda_example_keeps_only_eval_outside_proof_subset() {
        let mut ctx = ctx_from_source(include_str!("../../../examples/core/lambda.av"), "lambda");
        ctx.refresh_facts();
        let issues = proof_mode_issues(&ctx);
        assert_eq!(
            issues,
            vec!["recursive function 'eval' is outside proof subset (currently supported: Int countdown, second-order affine Int recurrences with pair-state worker, structural recursion on List/recursive ADTs, String+position, mutual Int countdown, mutual String+position, and ranked sizeOf recursion)".to_string()]
        );

        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);
        assert!(lean.contains("def termToString__fuel"));
        assert!(lean.contains("def subst__fuel"));
        assert!(lean.contains("def countS__fuel"));
        assert!(!lean.contains("partial def termToString"));
        assert!(!lean.contains("partial def subst"));
        assert!(!lean.contains("partial def countS"));
        assert!(lean.contains("partial def eval"));
    }

    #[test]
    fn mission_control_example_stays_inside_proof_subset() {
        let mut ctx = ctx_from_source(
            include_str!("../../../examples/apps/mission_control.av"),
            "mission_control",
        );
        ctx.refresh_facts();
        let issues = proof_mode_issues(&ctx);
        assert!(
            issues.is_empty(),
            "expected mission_control example to stay inside proof subset, got: {:?}",
            issues
        );

        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);
        assert!(!lean.contains("partial def normalizeAngle"));
        assert!(lean.contains("def normalizeAngle__fuel"));
    }

    #[test]
    fn notepad_store_example_stays_inside_proof_subset() {
        let mut ctx = ctx_from_source(
            include_str!("../../../examples/apps/notepad/store.av"),
            "notepad_store",
        );
        ctx.refresh_facts();
        let issues = proof_mode_issues(&ctx);
        assert!(
            issues.is_empty(),
            "expected notepad/store example to stay inside proof subset, got: {:?}",
            issues
        );

        let out = transpile_for_proof_mode(&mut ctx, VerifyEmitMode::NativeDecide);
        let lean = generated_lean_file(&out);
        assert!(lean.contains("def deserializeLine (line : String) : Except String Note :="));
        assert!(lean.contains("Except String (List Note)"));
        assert!(!lean.contains("partial def deserializeLine"));
        assert!(lean.contains("-- when noteRoundtripSafe note"));
        assert!(lean.contains("-- when notesRoundtripSafe notes"));
        assert!(lean.contains(
            "theorem serializeLine_law_lineRoundtrip : ∀ (note : Note), note = { id' := 1, title := \"Hello\", body := \"World\" : Note } ∨"
        ));
        assert!(lean.contains(
            "theorem serializeLines_law_notesRoundtrip : ∀ (notes : List Note), notes = [] ∨"
        ));
        assert!(lean.contains("notesRoundtripSafe notes = true ->"));
        assert!(lean.contains("parseNotes (s!\"{String.intercalate \"\\n\" (serializeLines notes)}\\n\") = Except.ok notes"));
        assert!(lean.contains("theorem serializeLine_law_lineRoundtrip_sample_1 :"));
        assert!(lean.contains("theorem serializeLines_law_notesRoundtrip_sample_1 :"));
    }
}