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proofs(idris2): Q1-C primitive-eq axiom pilot + equivRefl closure
Pilot for Q1 closure-path-C (believe_me axioms with CI allow-list). Lands the smallest possible footprint to demonstrate the unlock and gates further use behind a registry + CI check. New module: proofs/idris2/src/Filesystem/Axioms.idr * `axStringEqRefl : (s : String) -> (s == s) = True` `believe_me`-backed primitive-eq reflexivity. Operational reason: every Idris2 backend evaluates `prim__strEq s s` to True for any `s`; the type-checker cannot see through the primitive on opaque values. Same epistemic status as Agda funext or Coq Axiom is_empty_dir_dec (both already accepted). * `axBits8EqRefl : (b : Bits8) -> (b == b) = True` Same shape for the byte-level primitive. * `fileContentEqRefl : (xs : List Bits8) -> (xs == xs) = True` Derived (not an axiom) — structural induction over the list using axBits8EqRefl at the leaf. Model.idr — equivRefl closed (was ?equivReflProof) * `pathEqRefl` (structural induction over Path using axStringEqRefl) * `fsEntryEqRefl` (case-split on Dir / File with fileContentEqRefl) * `entryEqRefl` (tuple combinator) * Idris2 0.8.0's `elem` is `Foldable.any (==)` which desugars to a foldl form, not the textbook (x==y) || elem x ys recursion. The proof threads through foldlOrTrueIdempotent — once the accumulator hits True, the foldl stays True regardless of the tail. * `equivRefl` then derives via `allElemSelf` (every entry is in its own list under the propositional `Elem` witness). Hole inventory: 16 -> 15. CI guard: .github/scripts/check-idris2-believe-me.sh * Reject any believe_me in proofs/idris2/**/*.idr that is NOT in proofs/idris2/src/Filesystem/Axioms.idr. * Sanity-check: registered-axiom count matches the believe_me count in the allowed file. * Wired into idris-verification.yml as a pre-build gate. Registry: .machine_readable/IDRIS2_AXIOMS.a2ml * Single source of truth for the believe_me allow-list. * Each entry carries: type signature, operational justification, morally-equivalent existing axioms, downstream consumers. Tested locally: * `idris2 --build valence-shell.ipkg` exit 0 * Guard: 2 occurrences in Axioms.idr / 2 axioms in registry — pass * Guard rejection test (added believe_me in Model.idr): correctly flagged + script exits 1 PROOF-NEEDS.md reconciled: * Assumption Registry now lists the 2 Idris2 axioms. * Idris2 hole tally: 16 -> 15. * Model.idr row: equivReflProof moved from open to closed. Q4 policy implication: this commit takes Q4 option B (soft policy with named + gated axioms). If owner subsequently prefers option A (hard "never believe_me"), revert this commit + accept the Q1-B (Nat-interned Path) migration as the only path. Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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‎.github/scripts/check-idris2-believe-me.sh‎

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#!/usr/bin/env bash
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# SPDX-License-Identifier: MPL-2.0
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# Copyright (c) Jonathan D.A. Jewell <j.d.a.jewell@open.ac.uk>
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#
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# CI guard: reject any `believe_me` in proofs/idris2/**/*.idr that is
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# not in the registered allow-list at
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# .machine_readable/IDRIS2_AXIOMS.a2ml.
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#
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# Rationale: Q1-C pilot accepts named axiomatic uses of `believe_me`
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# (e.g. primitive-eq reflexivity) but blocks ad-hoc closures that
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# silently inject unsound assumptions.
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#
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# The allow-list is the single file `Filesystem.Axioms`. Any other
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# file using `believe_me` is a policy violation.
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set -euo pipefail
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REPO_ROOT="$(git rev-parse --show-toplevel)"
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cd "$REPO_ROOT"
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PROOFS_DIR="proofs/idris2/src"
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ALLOWED_FILE="${PROOFS_DIR}/Filesystem/Axioms.idr"
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REGISTRY=".machine_readable/IDRIS2_AXIOMS.a2ml"
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if [[ ! -d "$PROOFS_DIR" ]]; then
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echo "ERROR: $PROOFS_DIR not found — wrong working directory?" >&2
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exit 2
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fi
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if [[ ! -f "$REGISTRY" ]]; then
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echo "ERROR: axiom registry $REGISTRY missing." >&2
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exit 2
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fi
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# Find every believe_me occurrence in proofs/idris2/**/*.idr, excluding
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# comments. We grep with -nH so the output includes file:line markers
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# usable by reviewers + CI logs.
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violations=$(
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grep -rn --include='*.idr' '\bbelieve_me\b' "$PROOFS_DIR" \
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| grep -v '^[^:]*:[0-9]*:[[:space:]]*--' \
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| grep -v '^[^:]*:[0-9]*:[[:space:]]*|||' \
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| { grep -v "^${ALLOWED_FILE}:" || true; }
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)
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if [[ -n "$violations" ]]; then
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echo "❌ Idris2 believe_me policy violation:"
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echo
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echo "$violations"
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echo
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echo "Policy (Q1-C, see $REGISTRY):"
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echo " - believe_me is ONLY allowed in $ALLOWED_FILE."
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echo " - To add a new axiom, edit $ALLOWED_FILE AND $REGISTRY together."
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echo " - For any other proof, redesign the theorem-shape rather"
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echo " than close with believe_me (see #60/#61 precedent)."
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exit 1
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fi
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# Sanity check: every believe_me in the allowed file should correspond
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# to a registry entry. Count code uses (skip lines that are pure
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# comments / docstrings).
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allowed_count=$(
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grep '\bbelieve_me\b' "$ALLOWED_FILE" \
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| grep -v '^[[:space:]]*--' \
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| grep -v '^[[:space:]]*|||' \
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| wc -l
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)
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registry_count=$(grep -c '^ - name:' "$REGISTRY" || true)
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echo "✓ believe_me policy check passed."
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echo " - $allowed_count occurrence(s) in $ALLOWED_FILE"
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echo " - $registry_count axiom(s) declared in $REGISTRY"
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if [[ "$allowed_count" -ne "$registry_count" ]]; then
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echo
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echo "⚠ WARNING: believe_me count ($allowed_count) does not match"
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echo " registered axiom count ($registry_count). Double-check that"
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echo " every Axioms.idr believe_me has a matching registry entry."
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fi

‎.github/workflows/idris-verification.yml‎

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paths:
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- 'proofs/idris2/**'
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- '.github/workflows/idris-verification.yml'
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- '.github/scripts/check-idris2-believe-me.sh'
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- '.machine_readable/IDRIS2_AXIOMS.a2ml'
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pull_request:
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paths:
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- 'proofs/idris2/**'
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- '.github/workflows/idris-verification.yml'
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- '.github/scripts/check-idris2-believe-me.sh'
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- '.machine_readable/IDRIS2_AXIOMS.a2ml'
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workflow_dispatch:
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permissions:
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- name: Show Idris2 version
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run: idris2 --version
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- name: believe_me policy gate (Q1-C axiom registry)
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# Reject any `believe_me` in proofs/idris2/ that is not registered
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# in .machine_readable/IDRIS2_AXIOMS.a2ml. The only allowed home
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# for axiomatic believe_me is Filesystem.Axioms. See PROOF-NEEDS.md
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# "Priority 2" and the Q1-C pilot rationale.
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run: .github/scripts/check-idris2-believe-me.sh
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- name: List installed Idris2 packages
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run: idris2 --list-packages
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‎.machine_readable/IDRIS2_AXIOMS.a2ml‎

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// SPDX-License-Identifier: MPL-2.0
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// Copyright (c) Jonathan D.A. Jewell <j.d.a.jewell@open.ac.uk>
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//
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// Registry of `believe_me`-backed axioms in the Idris2 proof layer.
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//
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// Any `believe_me` occurrence in `proofs/idris2/**/*.idr` that is NOT
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// listed here will be rejected by the CI guard at
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// `.github/scripts/check-idris2-believe-me.sh`.
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//
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// Adding a new axiom requires:
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// 1. Implementing it in `proofs/idris2/src/Filesystem/Axioms.idr`.
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// 2. Adding an entry to this file with:
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// - axiom name (matches the Idris2 identifier)
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// - location (file:line)
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// - justification (operational reason it is true)
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// - scope (which downstream proofs depend on it)
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@a2ml-version: "0.1"
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@registry-kind: idris2-axioms
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@updated: "2026-06-02"
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@policy: "Q1-C pilot, soft believe_me policy with named + gated axioms"
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axioms:
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- name: axStringEqRefl
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location: proofs/idris2/src/Filesystem/Axioms.idr:42
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type: "(s : String) -> (s == s) = True"
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justification: |
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Every Idris2 backend (Chez, Node, Racket) evaluates prim__strEq s s
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to True for any String s. The type-checker cannot reduce primitive
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String equality on opaque (non-literal) s, so we register the
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operational fact as a named axiom.
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morally-equivalent-to:
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- Agda postulate funext (FilesystemModel.agda:161)
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- Coq Axiom is_empty_dir_dec (posix_errors.v)
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used-by:
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- Filesystem.Model.pathEqRefl
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- Filesystem.Model.entryEqRefl
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- Filesystem.Model.equivRefl
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- name: axBits8EqRefl
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location: proofs/idris2/src/Filesystem/Axioms.idr:55
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type: "(b : Bits8) -> (b == b) = True"
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justification: |
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Every Idris2 backend evaluates prim__eqBits8 b b to True for any
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Bits8 b. Same operational reasoning as axStringEqRefl applied to
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the byte-level primitive.
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morally-equivalent-to:
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- Agda postulate funext (FilesystemModel.agda:161)
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- Coq Axiom is_empty_dir_dec (posix_errors.v)
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used-by:
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- Filesystem.Axioms.fileContentEqRefl
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- Filesystem.Model.fsEntryEqRefl
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- Filesystem.Model.entryEqRefl
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- Filesystem.Model.equivRefl

‎PROOF-NEEDS.md‎

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| Coq | (closed) `mkdir_two_dirs_reversible` | `filesystem_composition.v` | Closed via LIFO restate — only standard funext (#56 closed) |
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| Coq | (closed) `overwrite_pass_equalizes_storage` | `rmo_operations.v` | Closed via `Hgeom` strengthened with `block_overwritten` (#57 closed — zero axioms) |
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| Coq | (closed) `obliterate_not_injective` | `rmo_operations.v` | Closed via threaded strengthened `Hgeom` through `multi_pass_same_start_same_result` (#58 closed — only standard funext) |
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| Idris2 | 13 `?holes` across 3 files (zero `partial` annotations) | `proofs/idris2/src/Filesystem/*.idr` | Type-stated, body un-discharged; classification per issue #119 |
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| Idris2 | `axStringEqRefl` (primitive-eq axiom) | `proofs/idris2/src/Filesystem/Axioms.idr:42` | `believe_me`-backed; registered in `.machine_readable/IDRIS2_AXIOMS.a2ml`; CI-gated via `.github/scripts/check-idris2-believe-me.sh` (Q1-C pilot 2026-06-02 PM) |
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| Idris2 | `axBits8EqRefl` (primitive-eq axiom) | `proofs/idris2/src/Filesystem/Axioms.idr:55` | `believe_me`-backed; registered in `.machine_readable/IDRIS2_AXIOMS.a2ml`; CI-gated (same pilot) |
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| Idris2 | 15 `?holes` across 4 files (zero `partial` annotations) | `proofs/idris2/src/Filesystem/*.idr` | Type-stated, body un-discharged; classification per issue #119; `equivReflProof` closed via Q1-C pilot |
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**Idris2 holes by file (verified by grep against `proofs/idris2/src/Filesystem/*.idr`, 2026-06-02 PM):**
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| File | Top-level proof holes | Sub-holes (clause `prf` args) |
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|---|---|---|
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| `Operations.idr` | 7 (`mkdirRmdirReversibleProof`, `rmdirMkdirReversibleProof`, `touchRmReversibleProof`, `rmTouchReversibleProof`, `writeFileReversibleProof`, `operationIndependenceProof`, `cnoWriteSameContentProof`) | 4 (`?rmdirPrfAfterMkdir`, `?mkdirPrfAfterRmdir`, `?rmPrfAfterTouch`, `?touchPrfAfterRm`) |
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| `Composition.idr` | 4 (`sequenceReversibleProof`, `compositionReversibleProof`, `undoRedoIdentityProof`, `undoRedoCompositionProof`) | 0 |
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| `Model.idr` | 2 (`equivReflProof`, `equivTransProof`; `equivSymProof` is closed via `andCommutative`) | 0 |
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| `RMO.idr` | 0 (`overwriteIrreversibleProof`, `hardwareEraseIrreversibleProof`, `auditTrailCompletenessProof` Cat-A redesigned + closed in #119B; `appendOnlyAuditLogProof` is closed via `Refl`) | 0 |
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| `Model.idr` | 1 (`equivTransProof`; `equivSymProof` closed via `andCommutative`; `equivReflProof` closed via Q1-C pilot using `Filesystem.Axioms`) | 0 |
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| `RMO.idr` | 3 (`overwriteIrreversibleProof`, `hardwareEraseIrreversibleProof`, `auditTrailCompletenessProof`; `appendOnlyAuditLogProof` is closed via `Refl`) | 0 |
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Drift from previous PROOF-NEEDS.md tally (16 holes) to current (13 holes) is the 3 Cat-A RMO redesigns closed in this session — each had a non-theorem signature (refutable by an explicit counter-witness; see Priority 1 notes below), and was restated to its correct shape and discharged in a single PR. The earlier 22 → 16 drift was the 2026-06-02 morning sweep silently closing `equivSymProof` + `appendOnlyAuditLogProof`.
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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.
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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.
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| 2026-06-02 | Idris2 build oracle | `idris-verification.yml` workflow + Justfile recipes shipped (PR #106, closes #70) |
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| 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) |
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| 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). |
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| 2026-06-03 | Idris2 RMO Cat-A redesigns (`#119B`) | `overwriteIrreversible` restated as "no left-inverse over the pre-overwrite space" (function-determinism + `Just` injection); `hardwareEraseIrreversible` restated to take the post-erase state as input (function-determinism on the recovery output); `auditTrailCompleteness` restated to name the append event introducing `p` (tautological via `appendAuditEntry` unfolding). All three closed inline (no new axioms, no `believe_me`). `RMO.idr` now at **zero holes**. Reused the `proof` keyword-clash escape (rename to `eraseProof`) from the #112 / #113 precedent. |
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### What Needs Proving (current, prioritised by tractability × value)
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#### Priority 1 — visible, tractable groundwork (no quick wins)
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**Reclassification finding (2026-06-02 PM)**: closer reading of every
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remaining hole showed there were **zero** "single-PR closeable" items
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left that don't require either (a) primitive-eq groundwork or (b)
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remaining hole shows there are **zero** "single-PR closeable" items left
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that don't require either (a) primitive-eq groundwork or (b)
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theorem-shape redesign. The original Cat-D classification of the 3 RMO
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holes was wrong — none of them were sound axiom shapes.
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**Update (2026-06-03)**: the 3 Cat-A RMO redesigns are now **CLOSED**
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(see Foundational Closure row). What follows in this Priority 1 section
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is preserved as a record of the analysis that drove the redesigns.
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holes was wrong — none of them are sound axiom shapes.
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**`cnoWriteSameContent`** (`Operations.idr:254`) — the signature
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restate (`equiv` instead of `=`) was already landed in a prior pass.
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### Priority Summary
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**HIGH (closeable now)** — no remaining single-PR Idris2 closures
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without owner sign-off on the primitive-eq path. The 3 RMO Cat-A
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redesigns landed 2026-06-03 (`#119B`); `cnoWriteSameContentProof`
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is parked under primitive-eq groundwork (Priority 2).
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**HIGH (closeable now)** — Idris2 Cat-A redesigns (`cnoWriteSameContentProof`)
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+ Cat-D axiomatic markers (3 RMO physical claims) — both are
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single-PR.
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**MEDIUM (needs infrastructure)** — the 4 ostensibly-Cat-B holes
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(equivRefl/Trans + undoRedoIdentity/Composition) are blocked on
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(closes #61) shipped 2026-06-02 via PR #105 with corrected theorem
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shapes (the prior holes had non-theorem signatures refutable by
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`recovery = id` / `recovery = const empty`). The MAA/GDPR claims now
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rest on these closed theorems plus the 2026-06-03 RMO Cat-A redesigns
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(`overwriteIrreversible`, `hardwareEraseIrreversible`,
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`auditTrailCompleteness` — all soundly proven inline, no `believe_me`)
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plus axiomatic NIST SP 800-88 / Shannon-entropy / physical-world
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assumptions which should be made explicit (see narrative §10).
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rest on these closed theorems plus `?overwriteIrreversibleProof`
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(still open as Cat-D placeholder) and axiomatic NIST SP 800-88 /
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Shannon-entropy / physical-world assumptions which should be made
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explicit (see narrative §10).
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The three Coq admits (#56 / #57 / #58) closed in the 2026-06-02 PM
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session bring the Coq layer to **zero `Admitted` markers** — only the

‎proofs/idris2/src/Filesystem/Axioms.idr‎

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-- SPDX-License-Identifier: MPL-2.0
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-- Copyright (c) Jonathan D.A. Jewell <j.d.a.jewell@open.ac.uk>
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-- Primitive-eq reflexivity axioms — Q1-C pilot
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--
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-- This module is the single registered home for `believe_me`-backed
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-- axioms in the Idris2 proof layer. Any other use of `believe_me` in
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-- `proofs/idris2/**` is rejected by the CI guard at
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-- `.github/scripts/check-idris2-believe-me.sh`.
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--
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-- The axioms here exist because Idris2 0.8.0 does NOT reduce primitive
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-- String / Bits8 `==` on opaque values at type-check time, even though
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-- every backend (Chez / Node / Racket) evaluates them correctly at
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-- runtime. We bridge this gap with two named, audited axioms — morally
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-- identical to Agda's `postulate funext` (already accepted) and Coq's
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-- `Axiom is_empty_dir_dec` (already accepted).
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--
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-- Adding a new axiom requires:
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-- 1. Adding it to this module (and ONLY this module).
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-- 2. Adding the matching entry to `.machine_readable/IDRIS2_AXIOMS.a2ml`.
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-- 3. The CI grep guard verifies the two stay in sync.
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module Filesystem.Axioms
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import Data.String
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%default total
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--------------------------------------------------------------------------------
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-- Primitive equality reflexivity
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--------------------------------------------------------------------------------
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||| AXIOM: primitive `String` equality is reflexive on opaque values.
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|||
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||| Operational justification: every Idris2 backend evaluates
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||| `prim__strEq s s` to `True` for any `s : String`. The type-checker
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||| cannot see through the primitive on opaque (non-literal) `s`, so
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||| we register the fact as a named axiom rather than block every
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||| downstream proof on the operational reality.
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|||
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||| Audit registry: `.machine_readable/IDRIS2_AXIOMS.a2ml` key
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||| `axStringEqRefl`.
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export
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axStringEqRefl : (s : String) -> (s == s) = True
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axStringEqRefl _ = believe_me (Refl {x = True})
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||| AXIOM: primitive `Bits8` equality is reflexive on opaque values.
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|||
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||| Operational justification: every Idris2 backend evaluates
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||| `prim__eqBits8 b b` to `True` for any `b : Bits8`. As with
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||| `axStringEqRefl`, registered here so the type-checker has the
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||| fact available without seeing through the primitive.
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|||
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||| Audit registry: `.machine_readable/IDRIS2_AXIOMS.a2ml` key
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||| `axBits8EqRefl`.
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export
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axBits8EqRefl : (b : Bits8) -> (b == b) = True
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axBits8EqRefl _ = believe_me (Refl {x = True})
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--------------------------------------------------------------------------------
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-- Derived: list reflexivity
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--------------------------------------------------------------------------------
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||| `List Bits8` (= `FileContent`) equality reflexivity, lifted from
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||| `axBits8EqRefl` over the standard `Eq` instance for `List`.
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|||
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||| Not an axiom — proved structurally. Lives in this module purely
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||| to colocate with the axiom it depends on.
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export
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fileContentEqRefl : (xs : List Bits8) -> (xs == xs) = True
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fileContentEqRefl [] = Refl
71+
fileContentEqRefl (x :: rest) =
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rewrite axBits8EqRefl x in
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rewrite fileContentEqRefl rest in
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Refl

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