fix(proofs/idris2): redesign auditTrailCompleteness as a per-insertion theorem (closes #131)

Closes the third non-theorem from the #119 Category A inventory.
Mirrors the #60 / #61 / #119A precedent: rather than discharging a
provably-false claim with `believe_me`, restate it as a structurally
honest one.

## auditTrailCompleteness (RMO.idr)

Was:

    auditTrailCompleteness :
      (entries : List AuditEntry) ->
      (p : Path) ->
      (obliterated : ObliterationProof p) ->
      Elem p (map AuditEntry.path entries)

Refutable by `entries := []`: an `ObliterationProof` for any `p`
exists (just `MkObliterationProof p Clear 0 "unlink"`), yet the empty
log contains no paths. The signature said "any obliteration is in
any log" — plainly false.

Now:

    auditTrailCompleteness :
      (log : List AuditEntry) ->
      (entry : AuditEntry) ->
      (p : Path) ->
      (insertedPath : AuditEntry.path entry = p) ->
      Elem p (map AuditEntry.path (appendAuditEntry log entry))
    auditTrailCompleteness log entry p insertedPath =
      rewrite sym insertedPath in
        elemMap AuditEntry.path (elemAppRightSelf log entry)

The corrected shape encodes the actual invariant: the path is
present in the post-append log *because* the entry recording it was
appended via the official `appendAuditEntry` constructor. The
`insertedPath` premise witnesses the caller's commitment that the
new entry indeed names the path we claim is logged.

## Closure path (zero new axioms)

1. `appendAuditEntry log entry` reduces to `log ++ [entry]` by
   definition (already proved as `appendOnlyAuditLog : Refl`).
2. `Data.List.Elem.elemMap` lifts list membership through `map`,
   giving `Elem (AuditEntry.path entry) (map AuditEntry.path
   (log ++ [entry]))`.
3. A short local helper `elemAppRightSelf` proves
   `Elem x (xs ++ [x])` by induction on the prefix — the lemma is
   not in Idris2 0.8.0 base.
4. The `insertedPath` equality rewrites `AuditEntry.path entry` to
   `p` in the goal.

## Verification

Local Idris2 0.8.0:

    $ cd proofs/idris2 && rm -rf build && idris2 --build valence-shell.ipkg
    1/4: Building Filesystem.Model        OK
    2/4: Building Filesystem.RMO          OK
    3/4: Building Filesystem.Operations   OK
    4/4: Building Filesystem.Composition  OK

    $ grep -c '?auditTrailCompleteness' proofs/idris2/src/Filesystem/RMO.idr
    0

    $ grep -c 'believe_me' proofs/idris2/src/Filesystem/RMO.idr
    2   # both inside ||| doc comments; no invocations

No regressions. `--total` honoured. One hole eliminated; tree drops
from 16 to 15. PROOF-NEEDS.md inventory + closure-history table
updated to reflect the new state.

Closes #131.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

Author: hyperpolymath

PROOF-NEEDS.md

@@ -38,9 +38,9 @@
 | `Operations.idr` | 7 (`mkdirRmdirReversibleProof`, `rmdirMkdirReversibleProof`, `touchRmReversibleProof`, `rmTouchReversibleProof`, `writeFileReversibleProof`, `operationIndependenceProof`, `cnoWriteSameContentProof`) | 4 (`?rmdirPrfAfterMkdir`, `?mkdirPrfAfterRmdir`, `?rmPrfAfterTouch`, `?touchPrfAfterRm`) |
 | `Composition.idr` | 4 (`sequenceReversibleProof`, `compositionReversibleProof`, `undoRedoIdentityProof`, `undoRedoCompositionProof`) | 0 |
 | `Model.idr` | 2 (`equivReflProof`, `equivTransProof`; `equivSymProof` is closed via `andCommutative`) | 0 |
-| `RMO.idr` | 3 (`overwriteIrreversibleProof`, `hardwareEraseIrreversibleProof`, `auditTrailCompletenessProof`; `appendOnlyAuditLogProof` is closed via `Refl`) | 0 |
+| `RMO.idr` | 2 (`overwriteIrreversibleProof`, `hardwareEraseIrreversibleProof`; `appendOnlyAuditLogProof` is closed via `Refl`; `auditTrailCompletenessProof` is closed via redesign + `elemMap` + `elemAppRightSelf` — see #131) | 0 |
 
-Drift from previous PROOF-NEEDS.md tally (22 holes) to current (16 holes) is mechanical: `equivSymProof` and `appendOnlyAuditLogProof` closed silently during the 2026-06-02 morning sweep (visible by grep but the inventory text was not updated). No body changes — this paragraph reconciles the count.
+Drift from previous PROOF-NEEDS.md tally (22 holes) to current (15 holes) is mechanical: `equivSymProof` and `appendOnlyAuditLogProof` closed silently during the 2026-06-02 morning sweep (visible by grep but the inventory text was not updated), and `auditTrailCompletenessProof` closed via signature redesign on 2026-06-03 (#131). No structural changes to the live source beyond #131 — this paragraph reconciles the count.
 
 All `partial` markers in `proofs/idris2/src/Filesystem/*.idr` were cleared 2026-06-02 (PRs #108 + #109, closing #89). The total `partial` count is zero.
 
@@ -61,6 +61,7 @@ All `partial` markers in `proofs/idris2/src/Filesystem/*.idr` were cleared 2026-
 | 2026-06-02 | Idris2 build oracle | `idris-verification.yml` workflow + Justfile recipes shipped (PR #106, closes #70) |
 | 2026-06-02 | Idris2 0.8.0 parse fixes | `AuditEntry.proof` keyword-clash rename (PR #112); `hardwareEraseIrreversible` multi-line signature fix (PR #113); `reverseConcat` closed via `Data.List.revAppend` (PR #115) |
 | 2026-06-02 PM | Coq admit triumvirate | `mkdir_two_dirs_reversible` restated to LIFO and closed (#56); `overwrite_pass_equalizes_storage` strengthened with `block_overwritten` constraint, closed with zero new axioms (#57); `obliterate_not_injective` threaded through the strengthened lemma + `multi_pass_same_start_same_result`, closed with only standard funext (#58). Coq layer now has **zero `Admitted` markers** (only the justified `Axiom is_empty_dir_dec` remains). |
+| 2026-06-03 | Idris2 RMO `auditTrailCompleteness` redesign | Signature redesigned away from the `entries = []`-refutable shape into a per-insertion claim threading `(log, entry, p, insertedPath : path entry = p)`; closed via `elemMap` (stdlib) + a local `elemAppRightSelf` induction + `sym insertedPath` rewrite (#131, mirrors the #60 / #61 / #119A precedent). Zero new axioms; zero `believe_me`. |
 
 ### What Needs Proving (current, prioritised by tractability × value)
 
@@ -94,13 +95,17 @@ about `(q == p)` on opaque `Path` values inside `elem`, which Idris2
   `HardwareEraseProof -> (Unit -> Filesystem) -> Void` is refuted by
   any non-empty `recovery` (the function exists trivially). Correct
   shape needs the recovery to take the post-erase state as input.
-- `auditTrailCompletenessProof` (`RMO.idr:270`): conclusion
-  `Elem p (map AuditEntry.path entries)` is refuted by `entries = []`.
-  Correct shape needs an "entry was appended" precondition naming the
-  insertion event in the log.
-
-These three should be filed as **`#119` sub-issues** with the
-non-theorem refutations, in line with the #60 / #61 precedent.
+- ~~`auditTrailCompletenessProof` (`RMO.idr:270`)~~ **CLOSED via #131**:
+  the previous shape `Elem p (map AuditEntry.path entries)` was refuted
+  by `entries = []`. Redesigned to thread a single `entry`, an
+  `insertedPath : AuditEntry.path entry = p` premise, and reason about
+  `appendAuditEntry log entry = log ++ [entry]`. Closure via
+  `elemMap` (stdlib) + `elemAppRightSelf` (local helper) + a
+  `sym insertedPath` rewrite. Zero new axioms, zero `believe_me`.
+
+The remaining two should be filed/handled as **`#119` sub-issues** with
+the non-theorem refutations, in line with the #60 / #61 / #131
+precedent.
 
 **4 Operations.idr sub-holes** (`?rmdirPrfAfterMkdir`,
 `?mkdirPrfAfterRmdir`, `?rmPrfAfterTouch`, `?touchPrfAfterRm`) — same

proofs/idris2/src/Filesystem/RMO.idr

@@ -259,12 +259,60 @@ appendOnlyAuditLog :
   appendAuditEntry log entry = log ++ [entry]
 appendOnlyAuditLog log entry = Refl
 
-||| Audit log provides complete history of obliterations
+||| `Elem` is preserved by right-appending the element itself.
+|||
+||| Pure list lemma — the appended item always lives at the end of
+||| the new list. Proof by straightforward induction on the prefix.
+|||
+||| Idris2 0.8.0 base does not ship this lemma directly; the closest
+||| stdlib helper is `Data.List.Elem.elemMap` (membership lifted
+||| through `map`), which we use as the second step of the audit
+||| completeness proof below.
+elemAppRightSelf : (xs : List a) -> (x : a) -> Elem x (xs ++ [x])
+elemAppRightSelf []        _ = Here
+elemAppRightSelf (_ :: ys) x = There (elemAppRightSelf ys x)
+
+||| Audit log completeness — every path inserted via the official
+||| append-only constructor appears in the resulting log.
+|||
+||| Replaces the previous non-theorem signature
+|||
+|||     auditTrailCompleteness :
+|||       (entries : List AuditEntry) ->
+|||       (p : Path) ->
+|||       (obliterated : ObliterationProof p) ->
+|||       Elem p (map AuditEntry.path entries)
+|||
+||| which was provably false (refuted by `entries = []`: an
+||| `ObliterationProof` for any `p` exists by `MkObliterationProof`,
+||| yet the empty log contains no paths). See issue #131 for the
+||| design rationale (mirrors the #60 / #61 / #119A precedent:
+||| redesign non-theorem signatures rather than close them with
+||| `believe_me`).
+|||
+||| The corrected shape encodes the actual invariant: the path is
+||| present in the post-append log *because* it was appended via
+||| `appendAuditEntry`. The premise `insertedPath` witnesses the
+||| caller's commitment that the new entry indeed records the path
+||| we claim is logged.
+|||
+||| Closure path (zero new axioms):
+||| 1. `appendAuditEntry log entry` reduces to `log ++ [entry]` by
+|||    the definition of `appendAuditEntry`.
+||| 2. `Data.List.Elem.elemMap` lifts list membership through `map`,
+|||    giving `Elem (AuditEntry.path entry) (map AuditEntry.path
+|||    (log ++ [entry]))`.
+||| 3. `elemAppRightSelf` discharges `Elem entry (log ++ [entry])`
+|||    by induction.
+||| 4. The `insertedPath` equality rewrites `AuditEntry.path entry`
+|||    to `p` in the goal.
 export
 auditTrailCompleteness :
-  (entries : List AuditEntry) ->
+  (log : List AuditEntry) ->
+  (entry : AuditEntry) ->
   (p : Path) ->
-  -- If p was obliterated, it's in the audit log
-  (obliterated : ObliterationProof p) ->
-  Elem p (map AuditEntry.path entries)  -- p appears in the log
-auditTrailCompleteness entries p oblitProof = ?auditTrailCompletenessProof
+  (insertedPath : AuditEntry.path entry = p) ->
+  Elem p (map AuditEntry.path (appendAuditEntry log entry))
+auditTrailCompleteness log entry p insertedPath =
+  rewrite sym insertedPath in
+    elemMap AuditEntry.path (elemAppRightSelf log entry)