lean-array-list

name	lean-array-list
description	Use when Lean 4 proofs involve ByteArray, Array, List indexing, getElem, length lemmas, take/drop, or roundtrip proofs over byte collections.
allowed-tools	Read, Bash, Grep

Lean 4 ByteArray/Array/List Proof Patterns

ByteArray/Array/List Indexing

• data.data[pos] = data[pos] (where data : ByteArray) is rfl
• For data.data.toList[pos] = data[pos]: simp only [Array.getElem_toList] suffices (as of v4.29.0-rc2; on rc1 a trailing ; rfl was also needed)
• When List.getElem_map is involved (e.g. (arr.toList.map f)[i] = f arr[i]): simp only [List.getElem_map, Array.getElem_toList] closes the goal

Length Conversions

• Array.length_toList: arr.toList.length = arr.size
• List.size_toArray: (l.toArray).size = l.length — bridges Array.size to List.length for array literals (#[a, b, c] elaborates as List.toArray [a, b, c])
• ByteArray.size_data: ba.data.size = ba.size
• Chain them for ba.data.toList.length

Concrete array size in simp only: To reduce #[a, b, ...].size to a number, use List.size_toArray + List.length_cons + List.length_nil:

simp only [myArray, List.size_toArray, List.length_cons, List.length_nil]
-- Reduces #[a, b, c].size to 3

Note: Array.size_toArray does NOT exist — use List.size_toArray.

getElem?_pos/getElem!_pos for Array Lookups

To prove arr[i]? = some arr[i]!, use the two-step pattern:

rw [getElem!_pos arr i h]  -- rewrites arr[i]! to arr[i] (bounds-checked)
exact getElem?_pos arr i h  -- proves arr[i]? = some arr[i]

getElem?_pos needs the explicit container argument (not _) to avoid GetElem? type class synthesis failures.

getElem!_def + getElem?_eq_some_iff for Panic-Indexed Array Proofs

For arr[idx]!, unfold with getElem!_def and case-split on in-bounds:

simp only [getElem!_def]
split
· -- some case: arr[idx]? = some e
  rename_i e he
  obtain ⟨hi, heq⟩ := Array.getElem?_eq_some_iff.mp he
  -- hi : idx < arr.size, heq : arr[idx] = e
  rw [← heq]
  exact some_property ⟨idx, hi⟩
· -- none case: idx ≥ arr.size, result is `default`
  have : (default : MyType).field = 0 := by decide
  omega

Key lemma: Array.getElem?_eq_some_iff : xs[i]? = some b ↔ ∃ h, xs[i] = b

Common mistake: Trying Array.getElem?_eq_some (doesn't exist), Array.get!_pos/Array.get!_neg (don't exist), or Array.getElem?_pos (different type — proves arr[i]? = some arr[i], not the reverse direction needed here).

Fin Coercion Mismatch in omega

When a lemma over Fin n is applied as lemma ⟨k, hk⟩, omega treats arr[(⟨k, hk⟩ : Fin n).val]! and arr[k]! as different variables.

Fix by annotating the result type:

have : arr[k]! ≥ 1 := lemma ⟨k, hk⟩

Critical: have := lemma ⟨k, hk⟩ (without type annotation) does NOT work — the anonymous hypothesis retains the Fin.val form and omega still sees two distinct variables. Always use the annotated form.

Deeper mismatch — GetElem vs getInternal: After unfold f at h, the goal may use Array.getInternal while a helper lemma uses arr[i]'hi (GetElem). Even with type annotation, omega treats these as distinct. Fix: use exact with Nat.le_trans (or similar) instead of omega — exact does definitional unification:

-- BAD: omega can't unify GetElem-based and getInternal-based array access
have := helper_lemma code hlt  -- uses arr[code]'hlt via GetElem
omega  -- fails: sees arr[code].fst and (arr.getInternal code hlt).fst as distinct

-- GOOD: exact does definitional unification
exact Nat.le_trans (helper_lemma code hlt) (Nat.le_add_right _ _)

decide_cbv on Fin-Bounded Array Properties

To verify a property holds for all entries of a concrete array, use decide_cbv on a Fin-bounded universal:

private theorem all_baselines_ge_three :
    ∀ i : Fin myTable.size, (myTable[i.val]'i.isLt).1 ≥ 3 := by
  decide_cbv

Then wrap in a Nat-indexed helper to avoid Fin coercion issues in callers:

private theorem baseline_ge_three (i : Nat) (hi : i < myTable.size) :
    (myTable[i]'hi).1 ≥ 3 :=
  all_baselines_ge_three ⟨i, hi⟩

Key constraints:

• The ∀ i : Fin n, P i form is needed for decide_cbv — it must be a closed proposition (no free Nat variables)
• decide (without _cbv) has the same constraint but may be slower
• The Nat wrapper eliminates the Fin coercion so callers can use exact or Nat.le_trans without the GetElem/getInternal mismatch

List.getElem_of_eq for Extracting from List Equality

When hih : l1 = l2 and you need l1[i] = l2[i], use List.getElem_of_eq hih hbound where hbound : i < l1.length. This avoids dependent-type rewriting issues with direct rw [hih] on getElem.

n + 0 Normalization Breaks rw Patterns

As of v4.29.0-rc2, Lean normalizes n + 0 to n earlier. If a lemma's conclusion contains arr[pfx.size + k] and you instantiate k = 0, the rewrite target arr[pfx.size + 0] won't match the goal's arr[pfx.size]. Fix: add a specialized _zero variant of the lemma that states the result with arr[pfx.size] directly.

take/drop ↔ Array.extract

To bridge List.take/List.drop (from spec) with Array.extract (from native):

simp only [Array.toList_extract, List.extract, Nat.sub_zero, List.drop_zero]

Then ← List.map_drop + List.drop_take for drop-inside-map-take.

Array.toArray_toList

a.toList.toArray = a for any Array. Use Array.toArray_toList. NOT Array.toList_toArray or List.toArray_toList — those don't exist.

readBitsLSB_bound for omega

readBitsLSB n bits = some (val, rest) implies val < 2^n. Essential for bounding UInt values (e.g., hlit_v.toNat < 32) before omega can prove ≤ UInt16.size.

List Nat ↔ Array UInt8 Roundtrip

To prove l = (l.toArray.map Nat.toUInt8).toList.map UInt8.toNat when all elements are ≤ 15 (from ValidLengths):

simp only [Array.toList_map, List.map_map]; symm
rw [List.map_congr_left (fun n hn => by
  show UInt8.toNat (Nat.toUInt8 n) = n
  simp only [Nat.toUInt8, UInt8.toNat, UInt8.ofNat, BitVec.toNat_ofNat]
  exact Nat.mod_eq_of_lt (by have := hv.1 n hn; omega))]
simp  -- closes `List.map (fun n => n) l = l` (not `List.map id l`)

Note: List.map_congr_left produces fun n => n not id, so List.map_id won't match — use simp instead.

UInt8→Nat→UInt8 Roundtrip

To prove Nat.toUInt8 (UInt8.toNat u) = u:

unfold Nat.toUInt8 UInt8.ofNat UInt8.toNat
rw [BitVec.ofNat_toNat, BitVec.setWidth_eq]

Do NOT use simp [Nat.toUInt8, UInt8.toNat, ...] — it loops via UInt8.toNat.eq_1 / UInt8.toNat_toBitVec. Do NOT try congr 1 (max recursion) or UInt8.ext / UInt8.eq_of_toNat_eq (don't exist).

For lists: l.map (Nat.toUInt8 ∘ UInt8.toNat) = l via List.map_congr_left with the above per-element proof, then simp.

ByteArray Concatenation Indexing

When proving properties about concatenated ByteArrays (e.g. header ++ payload ++ trailer), use the getElem!_append_left/getElem!_append_right chain. Key lemmas (in Zip/Spec/BinaryCorrect.lean):

Two-part concatenation:

getElem!_append_left  (a b : ByteArray) (i : Nat) (h : i < a.size) :
    (a ++ b)[i]! = a[i]!
getElem!_append_right (a b : ByteArray) (i : Nat) (h : i < b.size) :
    (a ++ b)[a.size + i]! = b[i]!

Three-part concatenation (a ++ b ++ c):

getElem!_append3_left  (a b c) (i) (h : i < a.size) :
    (a ++ b ++ c)[i]! = a[i]!
getElem!_append3_mid   (a b c) (i) (h : i < b.size) :
    (a ++ b ++ c)[a.size + i]! = b[i]!
getElem!_append3_right (a b c) (i) (h : i < c.size) :
    (a ++ b ++ c)[(a ++ b).size + i]! = c[i]!

Reading integers from concatenated arrays:

readUInt32LE_append_left   (a b) (offset) (h : offset + 4 ≤ a.size) :
    readUInt32LE (a ++ b) offset = readUInt32LE a offset
readUInt32LE_append_right  (a b) (offset) (h : offset + 4 ≤ b.size) :
    readUInt32LE (a ++ b) (a.size + offset) = readUInt32LE b offset
readUInt32LE_append3_mid   (a b c) (offset) (h : offset + 4 ≤ b.size) :
    readUInt32LE (a ++ b ++ c) (a.size + offset) = readUInt32LE b offset

Pattern for three-part concat proofs (common in gzip/zlib framing):

1. Unfold the function to expose individual getElem! or readUInt32LE calls
2. Apply getElem!_append3_* or readUInt32LE_append3_* lemma for each byte/field
3. Resolve arithmetic offsets with show a.size + offset + k = a.size + (offset + k) from by omega
4. Close size bounds with by simp [ByteArray.size_append]; omega

Size of concatenation: ByteArray.size_append : (a ++ b).size = a.size + b.size

get! ↔ getD Bridge (Array.getElem!_eq_getD)

a[i]! (for types with Inhabited) is definitionally a.getD i default. For Nat, default = 0, so a[i]! = a.getD i 0 by rfl. The library provides Array.getElem!_eq_getD : xs[i]! = xs.getD i default.

When predicates use getD but goals have get! (common when loop invariants are stated with getD but do-notation desugaring produces get!):

-- Direct: get! and getD are definitionally equal for Nat
have hcount : v3[sym]! ≤ bound := h3_counts sym
-- h3_counts : ∀ sym, v3.getD sym 0 ≤ bound
-- Works because v3[sym]! = v3.getD sym 0 definitionally

When you need explicit conversion: simp only [Array.getElem!_eq_getD] rewrites a[i]! to a.getD i default in the goal.

getD After set! (Array.getElem?_setIfInBounds)

simp only [Array.getElem_setIfInBounds] often fails after unfold Array.getD; simp only [Array.size_setIfInBounds] because the bound proof gets transported and the simp lemma can't match. This is because Array.getElem_setIfInBounds is @[grind =], not @[simp].

Workaround: Prove a getD-level helper using Array.getD_eq_getD_getElem? and Array.getElem?_setIfInBounds (which IS @[simp]-compatible):

private theorem getD_set! (a : Array Nat) (i v s : Nat) :
    (a.set! i v).getD s 0 = if i = s ∧ i < a.size then v else a.getD s 0 := by
  simp only [Array.set!_eq_setIfInBounds, Array.getD_eq_getD_getElem?,
    Array.getElem?_setIfInBounds]
  split <;> split <;> simp_all <;> intro <;> omega

Then use rw [getD_set!]; split instead of manual unfolding.

Singleton Array Fin Elimination

For arr[i] with arr size 1 and i : Fin arr.size, rw/simp on the index fails (the bound proof depends on i). Use Fin.ext (by omega) + subst to eliminate i, then show matches the concrete value:

have hsz : arr.size = 1 := rfl  -- by rfl or existing theorem
have : i = ⟨0, hsz ▸ Nat.zero_lt_one⟩ := Fin.ext (by omega)
subst this  -- now arr[⟨0, _⟩] reduces
show someConcreteValue

hsz ▸ Nat.zero_lt_one builds a proof term subst can handle. Approaches that fail: rw [show i.val = 0 ...] (dependent type errors), Array.getElem_singleton (needs Nat index, not Fin), Fin.getElem_fin (doesn't exist here), show/change before subst (can't reduce arr[i] with abstract i).

Build Missing API, Don't Work Around It

If a proof is blocked by a missing lemma for a standard type (ByteArray, Array, List, UInt32, ...), add it to ZipForStd/ in the matching namespace (e.g. a missing ByteArray.foldl_toList goes in ZipForStd/ByteArray.lean). Write it as upstream-quality — these are candidates for Lean's stdlib. Don't route around the gap (e.g. via .data.data.foldl) when a proper API lemma is the right fix.