https://github.com/velvetmonkey/trace-logic-lean
Lean 4 / Mathlib formalisation of Hoffman trace logic for finite Markov chains. Trace tower, semantic order transitivity, transient hidden blocks. 0 sorry.
https://github.com/velvetmonkey/trace-logic-lean
formal-verification lean4 markov-chains mathlib schur-complement stochastic-matrices trace-logic
Last synced: 6 days ago
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Lean 4 / Mathlib formalisation of Hoffman trace logic for finite Markov chains. Trace tower, semantic order transitivity, transient hidden blocks. 0 sorry.
- Host: GitHub
- URL: https://github.com/velvetmonkey/trace-logic-lean
- Owner: velvetmonkey
- License: mit
- Created: 2026-06-02T18:31:52.000Z (8 days ago)
- Default Branch: main
- Last Pushed: 2026-06-02T23:34:49.000Z (8 days ago)
- Last Synced: 2026-06-03T01:16:15.455Z (8 days ago)
- Topics: formal-verification, lean4, markov-chains, mathlib, schur-complement, stochastic-matrices, trace-logic
- Language: Lean
- Size: 18.6 KB
- Stars: 0
- Watchers: 0
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
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README
# trace-logic-lean
[](https://lean-lang.org/)
[](https://github.com/leanprover-community/mathlib4)
[](LICENSE)
[](TraceLogic)
Lean 4 / Mathlib formalisation of Phase 5 Hoffman trace logic for finite Markov chains.
## Mathematical Setting
The library works with finite row-stochastic Markov transition matrices over `ℝ`, decomposed into visible and hidden states. It defines the visible-visible, visible-hidden, hidden-visible, and hidden-hidden blocks, and the trace matrix
```lean
P_VV + P_VH * (1 - P_HH)⁻¹ * P_HV
```
as a Schur-complement-style effective transition matrix on visible states. It also includes nonnegativity predicates and a matrix operator norm wrapper via `Matrix.toEuclideanCLM`.
## Theorems Proved
| Name | Statement |
| --- | --- |
| `hidden_block_transient` | If `‖P_HH‖ < 1`, then `I - P_HH` is invertible. |
| `neumann_series` | Under `‖P_HH‖ < 1`, the inverse `(I - P_HH)⁻¹` exists as the hidden-block resolvent. |
| `row_stochastic_mulVec_ones` | A row-stochastic matrix maps the all-ones vector to the all-ones vector. |
| `blockHV_mulVec_ones` | The hidden-visible and hidden-hidden blocks satisfy the hidden-row all-ones identity. |
| `blockVV_blockVH_mulVec_ones` | The visible-visible and visible-hidden blocks satisfy the visible-row all-ones identity. |
| `IsNonneg.mul` | Products of nonnegative matrices are nonnegative. |
| `IsNonneg.add` | Sums of nonnegative matrices are nonnegative. |
| `IsNonneg.submatrix` | Submatrices of nonnegative matrices are nonnegative. |
| `trace_nonneg` | Under the stated Phase 1 hypotheses, the trace matrix is nonnegative. |
| `inv_blockHV_mulVec_ones` | The hidden-block inverse identity used in the trace row-sum proof. |
| `trace_row_sum` | Each row of the trace matrix sums to `1`. |
| `trace_row_stochastic` | The trace matrix is row-stochastic. |
| `trace_order_refl` | Trace order is reflexive in the Phase 1 skeleton. |
| `trace_order` | Entrywise dominance order on matrices: `A <= B` iff `A i j <= B i j` for all entries. |
| `trace_order_trans` | Transitivity of the entrywise dominance order, proved by `le_trans`. |
| `trace_order_Trans` | `Trans` instance for the entrywise dominance order, enabling `calc` chains. |
## Target 5 Scope
Target 5 proves transitivity of the **entrywise dominance order**:
```lean
trace_order A B := forall i j, A i j <= B i j
```
This is not the Schur-complement / trace-quotient order. The deeper trace-quotient direction is handled separately in Phase 3 through identities such as
```text
Tr_H2 (Tr_H1 P) = Tr_(H1+H2) P
```
requires a block-matrix-inversion formalisation for nested hidden-state eliminations.
## Phase 3: Trace Tower
Phase 3 proves the **Trace Tower Property**, also called the Schur-complement composition law:
```lean
trace_tower :
traceTwo P = traceMatrix P
```
This states that tracing out `h1` and then `h2` agrees with tracing out the combined hidden block `h1 ⊕ h2` in one step. The theorem assumes the three invertibility conditions required for the Schur complements to be defined:
- `IsUnit (1 - blockHH (reindexForTrace P))`
- `IsUnit (1 - blockHH (traceMatrix (reindexForTrace P)))`
- `IsUnit (1 - blockHH P)`
The proof is zero-sorry. Because Mathlib does not currently provide the full 2x2 block-matrix inversion formula needed here, the development proves roughly fifteen block-equation helper lemmas, including the reindexing block identities, the `Y1`/`Y2` elimination equations, the `Q_matrix` block identities, and the final correction identity used to assemble the tower law.
Phase 3 also introduces the trace-semantic order:
```lean
trace_sem_le A B
```
This means that `B` is obtained from `A` by extending with finitely many hidden states and applying `traceMatrix`. Reflexivity is proved by the empty-hidden witness, and `trace_sem_le_implies_trace_order` proves that, under nonnegativity hypotheses on the witness matrix and hidden-block inverse, the trace-semantic order refines the entrywise dominance order.
## Phase 4: Semantic Transitivity
Phase 4 proves `trace_sem_le_trans`. The theorem composes a witness `P1` on `n ⊕ Fin k1` and a witness `P2` on `n ⊕ Fin k2` into a combined hidden-state witness, using six private helper lemmas for hidden-type reindexing, block-diagonal inverses, and Schur-correction splitting.
The transitivity result is unconditional modulo the pre-existing invertibility hypotheses inherent to the Schur complement. No definitions were changed and no positivity or nonnegativity assumptions were added.
## Phase 5: Transient Hidden Blocks
Phase 5 discharges the Schur-complement invertibility hypotheses for transient hidden blocks. It introduces:
- `IsStrictlySubstochastic`: entrywise nonnegative square matrices whose row sums are all strictly less than `1`.
- `IsTransientHidden`: entrywise nonnegative square matrices whose some power has all row sums strictly less than `1`.
- `mulVec_one_sub_injective`: injectivity of `(1 - A).mulVec` for strictly substochastic `A`.
- `isUnit_one_sub_of_strictly_substochastic`: `1 - A` is a unit for strictly substochastic `A`.
- `isUnit_one_sub_of_transient`: `1 - A` is a unit for transient hidden blocks.
- `trace_row_stochastic_substochastic`: trace row-stochasticity without a separate invertibility hypothesis.
- `trace_tower_substochastic`: the trace tower law with invertibility discharged from strict substochasticity.
- `trace_sem_le_trans_substochastic`: trace-semantic order transitivity with invertibility discharged from strict substochasticity.
- `trace_tower_transient`: the trace tower law with invertibility discharged from transient-hidden-block hypotheses.
- `trace_sem_le_trans_transient`: trace-semantic order transitivity with invertibility discharged from transient-hidden-block hypotheses.
For all transient Markov-chain hidden blocks in this formal sense, the trace tower and trace-semantic transitivity results are unconditional: the required inverses are proved from the finite-power row-sum escape condition rather than assumed separately. The proof avoids Perron-Frobenius by reducing the transient case to the same maximum-entry argument applied to `A^N`.
## Future Work
Optional future work is to connect this finite-power row-sum definition of transience to spectral-radius and Perron-Frobenius formulations once Mathlib has more infrastructure for nonnegative matrices.
## Building
```bash
lake exe cache get
lake build
```
## Verification
The Phase 5 source compiles cleanly with Mathlib v4.28.0 and contains no `sorry` or `admit`.