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Proof Completion Strategy

Table of Contents

1. Date: 2026-02-05

2. Current Status

Total Admitted: 27 proofs across 8 files

Manual Completion Attempt: 1 proof (StatMech.v:cno_logically_reversible) - ⚠️ More complex than expected - Requires additional lemmas about state equality and eval relation - Need: "if s =st= s' and eval p s s'', then eval p s' s''"

Insight: These proofs need careful proof engineering, not simple tactics.

3. Proof Complexity Analysis

3.1. Easy (5-10 tactics) 🟢

Characteristics: - Direct application of existing lemmas - Simple unfolding + reflexivity/assumption - No new lemmas needed

Candidates (estimated): - Some LambdaCNO.v proofs - CNOCategory.v (may have direct applications)

3.2. Medium (10-30 tactics) 🟡

Characteristics: - Require case analysis - Need intermediate assertions - Build on existing theorems

Candidates: - StatMech.v:cno_logically_reversible - FilesystemCNO.v (several) - QuantumCNO.v (some)

3.3. Hard (30+ tactics, new lemmas) 🔴

Characteristics: - Need new helper lemmas - Complex induction - Non-trivial reasoning about state

Candidates: - StatMech.v:bennett_logical_implies_thermodynamic - LandauerDerivation.v (thermodynamics) - QuantumMechanicsExact.v (quantum) - MalbolgeCore.v (C register reasoning)

4. Strategy: Systematic ECHIDNA Usage

4.1. Phase 1: Assessment (Week 1, Day 1-2)

Goal: Classify all 27 proofs by difficulty

# For each file with Admitted
for file in $(find proofs/coq -name "*.v" -exec grep -l "Admitted" {} \;); do
  echo "=== Analyzing $file ==="

  # Get ECHIDNA's difficulty assessment
  just echidna-suggest "$file" | tee "analysis/$(basename $file .v)-assessment.txt"

  # Manual review: mark as Easy/Medium/Hard
done

Deliverable: Classification matrix

4.2. Phase 2: Easy Wins (Week 1, Day 3-5)

Goal: Complete 5-10 easy proofs to build confidence

Process: 1. Select easiest proofs (ECHIDNA confidence >0.8) 2. Use ECHIDNA suggestions 3. Manually verify and integrate 4. Commit incrementally (1-2 proofs per commit)

Commands:

# For each easy proof
just echidna-complete proofs/coq/category/CNOCategory.v
# Review suggestions
# Manually integrate if valid
# Test compilation
# Commit

4.3. Phase 3: Medium Proofs (Week 2-3)

Goal: Complete 10-15 medium proofs with ECHIDNA + manual effort

Process: 1. ECHIDNA provides initial tactics 2. Manual refinement needed 3. May require writing helper lemmas 4. Cross-verify in Lean 4/Z3

Example workflow:

(* Original: *)
Theorem foo : ...
Proof.
  admit.
Admitted.

(* ECHIDNA suggests: *)
(* Tactic 1: intros; unfold ... *)
(* Tactic 2: apply existing_lemma *)
(* Tactic 3: ... *)

(* Manual integration: *)
Theorem foo : ...
Proof.
  intros x H.
  unfold is_CNO in H.
  apply cno_preserves_state.
  (* Remaining gap - needs new lemma *)
  assert (helper_fact : ...).
  { (* prove helper *) }
  apply helper_fact.
  assumption.
Qed.

4.4. Phase 4: Hard Proofs (Week 4-5)

Goal: Complete remaining hard proofs, possibly with new lemmas

Process: 1. Identify missing infrastructure 2. Write and prove helper lemmas first 3. Use ECHIDNA for helper lemmas too 4. Complete main theorems 5. Extensive cross-verification

Hard proof examples:

  1. bennett_logical_implies_thermodynamic

    • Needs: bijection preservation lemmas

    • Needs: entropy preservation under bijections

    • Strategy: Prove helpers first, then main theorem

  2. Malbolge C register

    • Needs: Malbolge operational semantics lemmas

    • Needs: Encryption/decryption proofs

    • Strategy: May need to revisit Malbolge formalization

4.5. Phase 5: Verification (Week 6)

Goal: Cross-verify all completed proofs

# Multi-prover consensus
just echidna-verify

# Check no Admitted remain
grep -r "Admitted" proofs/coq/ && echo "FAIL: Admitted found" || echo "✓ All proofs complete"

# Container verification
just container-build
just container-verify

5. Required Lemmas (Preliminary Analysis)

5.1. State Equality Lemmas

(** If s =st= s' and eval p s s'', then eval p s' s'' *)
Lemma eval_state_eq_left :
  forall p s s' s'',
    s =st= s' ->
    eval p s s'' ->
    eval p s' s''.

(** If s =st= s' and eval p s'' s, then eval p s'' s' *)
Lemma eval_state_eq_right :
  forall p s s' s'',
    s =st= s' ->
    eval p s'' s ->
    eval p s'' s'.

5.2. CNO Infrastructure

(** CNOs always evaluate to the same state *)
Lemma cno_eval_self :
  forall p s,
    is_CNO p ->
    eval p s s.

(** CNO composition preserves CNO property *)
Lemma cno_compose :
  forall p1 p2,
    is_CNO p1 ->
    is_CNO p2 ->
    is_CNO (p1 ++ p2).

6. Metrics & Goals

6.1. Week 1

  • ❏ Classify all 27 proofs (Easy/Medium/Hard)

  • ❏ Complete 5 easy proofs

  • ❏ Identify required helper lemmas

6.2. Week 2-3

  • ❏ Complete 15 medium proofs

  • ❏ Write 5-10 helper lemmas

  • ❏ Cross-verify with Lean 4

6.3. Week 4-5

  • ❏ Complete 7 hard proofs

  • ❏ Write remaining helper lemmas

  • ❏ Extensive testing

6.4. Week 6

  • ❏ 27 → 0 Admitted ✅

  • ❏ Multi-prover verification ✅

  • ❏ Container integration ✅

7. Tools & Commands

7.1. Daily Workflow

# Morning: Check status
just echidna-list

# Select target proof
TARGET="proofs/coq/physics/StatMech.v"

# Get ECHIDNA suggestions
just echidna-suggest $TARGET > suggestions.txt

# Review suggestions
cat suggestions.txt

# Attempt auto-complete
just echidna-complete $TARGET

# Review generated proof
diff $TARGET ${TARGET%.v}_completed.v

# Manual integration (use editor)
vim $TARGET

# Verify (if coqc available)
# coqc -R proofs/coq/common CNO $TARGET

# Commit
git add $TARGET
git commit -m "proof: complete $(grep -B2 'Qed' $TARGET | grep 'Theorem\|Lemma' | tail -1)"

8. Risk Mitigation

8.1. Risk 1: ECHIDNA Suggestions Invalid

Likelihood: Medium Impact: Low (manual review catches this) Mitigation: Always manually verify, never blindly apply

8.2. Risk 2: Missing Infrastructure

Likelihood: High (already observed) Impact: Medium (delays completion) Mitigation: Write helper lemmas proactively

8.3. Risk 3: Time Overrun

Likelihood: Medium Impact: High (delays v1.0) Mitigation: Start with easy wins, adjust timeline if needed

8.4. Risk 4: Multi-Prover Disagreements

Likelihood: Low Impact: High (correctness concerns) Mitigation: Investigate disagreements immediately, fix root cause

9. Success Criteria

  • ❏ 0 Admitted proofs remaining

  • ❏ All proofs compile without errors

  • ❏ Multi-prover consensus (≥3/4 agreement)

  • ❏ Container verification passes

  • ❏ Documentation updated (VERIFICATION_RESULTS.adoc)

Target date: 2026-03-19 (6 weeks from today)

Strategy document created 2026-02-05