# Development Chronology

A narrative history of this Lean 4 formalization, based on 4,024 commits (3,166 non-merge) from Sept 29, 2025 - Jan 8, 2026.

## Phase 0: Project Initialization (Sept 29 - Oct 1, 2025)

**Setting the foundation**

- `c33c821d` Initial commit
- `179c0654` Added lean-toolchain (stable Lean 4)
- `fbce95e9` De Finetti theorem skeleton and blueprint
- `42c7d398` Mean-ergodic step for de Finetti theorem

The project began as a from-scratch formalization. Early focus on mean ergodic theorem integration.

## Phase 1: Framework Establishment (Oct 1-2, 2025)

**Core definitions emerge**

- `12b40f16` Reverted koopman_isometry to sorry (typeclass issues) - *first technical obstacle*
- `b8f77507` Integrated mathlib mean ergodic theorem
- `8e1d523b` Renamed exchangeability definitions for clarity
- `0e4cc62f` Distinguished finite vs full exchangeability - *key conceptual distinction*
- `bd1a5965` Added Contractability.lean with de Finetti-Ryll-Nardzewski framework

**Course correction:** Early reversion of koopman_isometry signals that the ergodic approach would face type class challenges.

## Phase 2: Breakthrough Week (Oct 2-6, 2025)

**First major milestones**

- `bd2af675` **MAJOR MILESTONE:** Complete proof of `exists_perm_extending_strictMono`
- `81f085df` Complete Step E - right-continuity of conditional CDF
- `b28f089c` Complete Step G - right-continuity at all reals
- `1710331f` Complete tower_indicator_finset proof structure
- `ddb463af` **Removed 1,538 lines of dead code** from ViaMartingale.lean
- `74c6ac46` Complete condIndep_indicator_of_dropInfoY proof

**Key insight:** The permutation extension lemma (`exists_perm_extending_strictMono`) became the foundation for proving exchangeable → contractable. Uses `Equiv.extendSubtype` to extend any strictly increasing `k : Fin m → ℕ` to a full permutation.

**Architecture signal:** Massive dead code removal suggests early exploration of paths that were abandoned.

## Phase 3: Modularization Push (Oct 6-20, 2025)

**From monoliths to modules**

Recognition that 5,000+ line files were unmaintainable led to systematic refactoring:

- `20ce25ae` Refactored CondIndep.lean into 4 modular files
- `bbf0636c` Split Martingale.lean into 4 modular files
- `f4573688` Split TripleLawDropInfo.lean into 2 submodules
- `2be8e286` Split ConditionalKernel.lean into 2 submodules
- `ba5721a1` Golfed ViaMartingale proofs (~71 lines saved)

**Pattern:** Files were extracted when exceeding ~3,000-4,500 lines. This enabled independent development of the three proof approaches.

## Phase 4: Three Proof Approaches Diverge (Oct 20 - Nov 10, 2025)

**Parallel development begins**

The three proofs from Kallenberg (2005) now develop independently:

| Approach | Key Infrastructure | Status |
|----------|-------------------|--------|
| **ViaL2** | Elementary L² bounds | Lighter dependencies |
| **ViaKoopman** | Mean Ergodic Theorem | Heavy ergodic theory |
| **ViaMartingale** | Reverse martingale convergence | Aldous's approach |

Peak productivity: Oct 12-13 saw 156 and 150 commits respectively.

- `d4aefa77` ViaL2: Complete dominating convergence structure
- `4444a485` ViaKoopman: Add contractability-based proof structure

## Phase 5: Core Mathematical Proofs (Oct 10 - Dec 10, 2025)

**The hard work**

- `c4f419e1` TripleLawDropInfo: Implement 95% of Kallenberg 1.3 L² proof
- `845be23c` **Complete Kallenberg 1.3 proof - NO SORRIES!**
- `78e4c4d7` Complete MixtureOfIID definition (ConditionallyIID)

**Technical struggles surface:**
- Measurability automation challenges (dozens of commits)
- Circularity avoidance in h_L1_conv approach
- Type class synthesis issues

The Kallenberg 1.3 lemma (`pair_law_eq_of_contractable`) became the central pillar connecting all three approaches.

## Phase 6: Major Course Correction - Identification Chain (Dec 3-10, 2025)

**Shifting to Kallenberg's precise argument**

- `19ba4114` **ViaL2: Replace Ioc/step function proof with identification chain**
- `e6d2b928` ViaL2: Add README documenting Kallenberg identification chain approach
- `3b48477f` ViaL2: Add Kallenberg-aligned bridge lemmas
- `ff7a3d1f` DirectingMeasure: simplify L¹ uniqueness proof to documented sorry

**Course correction:** After months of general infrastructure, explicitly adopted **Kallenberg's identification chain** approach. This moved away from abstract functional analysis toward concrete distributional arguments following the textbook closely.

Created `DirectingMeasure.lean` infrastructure for the identification chain: a sequence of measures progressively transferring the empirical distribution to the theoretical limit.

## Phase 7: Koopman Kernel Framework (Dec 12-31, 2025)

**Massive extraction and completion**

- `3f75f41c` **Extracted 4,438-line KernelIndependence.lean** (75% reduction: 5,854 → 1,447 lines)
- `20415133` Corrected h_L1_conv approach to Dynkin system + range quantization
- `e5df0b55` Removed OLD PROOF section, cleaned up axiom terminology
- `340c7edc` **ViaKoopmanContractable: Complete all proofs, eliminating sorries**

**Technical insight:** The kernel approach provides alternative route through:
1. Conditional expectation operators as conditional kernels
2. Factorization of joint laws
3. Contractability verification via kernels

## Phase 8: Martingale Kallenberg-Faithful Refactor (Dec 31 - Jan 1, 2026)

**Reverse martingale machinery**

- `b7c7d2f5` **ViaMartingale: Kallenberg-faithful refactor with reverse martingale convergence**
- `fbbba049` Added Kallenberg chain lemma and ForMathlib extractions

**Course correction:** Recognized that Aldous's original reverse martingale approach needed reframing to match Kallenberg's modern presentation. Introduced:
- Reverse filtration (σ-algebras conditioned on tail events)
- Reverse martingale convergence machinery
- Explicit chain lemma connecting revFiltration to tailSigma

## Phase 9: Proof Completion Sprint (Jan 1-3, 2026)

**Three proofs converge**

- `340c7edc` ViaKoopmanContractable: **Complete** (no sorries)
- `126d9e96` TheoremViaKoopman: Fix build errors, complete main theorem
- `432974a1` ViaL2: Fill alphaIicCE_right_continuous_at via DCT
- `5c379939` ViaL2: Fill h_alpha_eq_M_alpha_g via L¹ uniqueness

**New Year's milestone:** ViaKoopmanContractable achieved zero sorries on Jan 1, 2026.

## Phase 10: Stabilization & Golf (Jan 3-7, 2026)

**Code quality hardening**

- 15+ "golf" commits converting tactic proofs to term mode
- Strategic refactoring of oversized Koopman modules
- Final DirectingMeasure bug fixes
- `b1889b79` Updated README with completion status

**Why golf?** Once proofs are complete:
1. Demonstrates confidence in correctness (touching presentation, not logic)
2. Prepares for potential mathlib integration
3. Professional code quality standards

---

## Key Patterns

### Architectural Evolution
- **Monolithic → Modular:** Files grew to 5,000+ lines, then split into 50-500 line modules
- **Abstract → Concrete:** General functional analysis → Kallenberg's specific identification chain
- **Parallel → Converging:** Three independent routes → unified theorem statement

### Technical Struggles
1. **Circularity avoidance:** Multiple commits addressing circular proof dependencies
2. **Measurability automation:** Dozens of commits solving measurability goals
3. **Type class synthesis:** Occasional instance resolution issues
4. **Mathlib API changes:** Adaptation to IntegralIndicator, AEStronglyMeasurable updates

### Course Corrections
1. **Oct 2:** Reverted koopman_isometry (typeclass issues)
2. **Oct 6:** Massive dead code removal (abandoned exploration)
3. **Dec 3-10:** Adopted Kallenberg identification chain (major strategic shift)
4. **Dec 31:** Kallenberg-faithful martingale refactor

---

## Final Status

| Proof Route | Approach | Status |
|-------------|----------|--------|
| **ViaL2** | Elementary L² bounds | **Complete** (Jan 8, 2026) |
| **ViaKoopman** | Ergodic theory | **Complete** (Jan 1, 2026) |
| **ViaMartingale** | Reverse martingale | **Complete** (Dec 26, 2025) |

All three proofs demonstrate the core equivalence:
```
Contractable ⟺ Exchangeable ⟺ Conditionally i.i.d.
```