Establish a refinement function mapping sparse feature maps to the Verus sequence model

[coderabbitai]

Background

The Verus proof file verus/decomposition_vector_algebra.rs models vector operations on positional Seq<nat>, but the runtime implementation in common/src/decomposition_advice/vector.rs operates on sparse BTreeMap<String, u64> with key-based feature matching. There is no explicit mapping between feature keys and sequence indices, and no refinement lemma connecting the two representations.

Without establishing how the sparse, key-indexed map semantics correspond to the positional sequence semantics, the lemmas in the proof file do not justify the shipped implementation.

Raised by @leynos in PR #170: #170

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Goal

Define and use an abstraction function to lift the BTreeMap<String, u64> state into a canonical Seq<nat> representation over a shared feature universe, then prove that dot_product and norm_squared preserve their meaning under that abstraction.

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Task: Establish a refinement bridge between the runtime sparse map and the Verus sequence model

Context

• Verus proof file: verus/decomposition_vector_algebra.rs
• Runtime implementation: common/src/decomposition_advice/vector.rs
  • dot_product (line ~241) iterates &BTreeMap<String, u64>, multiplying weights for matching keys only.
  • norm_squared (line ~102) sums squared weights from the same map.
• The Verus model uses positional Seq<nat> — there is no connection to the runtime's sparse, key-indexed map.

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Steps

1. Define a feature-universe type and an abstraction function.
   In verus/decomposition_vector_algebra.rs, inside the verus! { } block, add:

   // A sorted sequence of all feature keys present in either vector.
   // The abstraction function maps a BTreeMap to a Seq<nat> aligned to this universe.
   spec fn to_seq(map: Map<nat, nat>, universe: Seq<nat>) -> Seq<nat>
       recommends universe.no_duplicates()
       decreases universe.len()
   {
       if universe.len() == 0 {
           seq![]
       } else {
           let key = universe[0];
           let weight = if map.contains_key(key) { map[key] } else { 0nat };
           seq![weight] + to_seq(map, universe.subrange(1, universe.len() as int))
       }
   }

   Adjust the key type to match whatever Verus supports for the Map abstraction (use nat indices if string keys cannot be modelled directly, and document this choice).

2. Prove that to_seq preserves length.
   Add:

   proof fn lemma_to_seq_len(map: Map<nat, nat>, universe: Seq<nat>)
       ensures to_seq(map, universe).len() == universe.len()
       decreases universe.len()
   { ... }

3. Prove that the runtime dot-product refines the sequence dot-product.
   Add a refinement lemma:

   proof fn lemma_dot_product_refinement(
       left: Map<nat, nat>,
       right: Map<nat, nat>,
       universe: Seq<nat>,
   )
       requires universe.no_duplicates(),
               forall|k: nat| left.contains_key(k) ==> universe.contains(k),
               forall|k: nat| right.contains_key(k) ==> universe.contains(k),
       ensures
           dot_product(to_seq(left, universe), to_seq(right, universe))
               == sum_matching_keys(left, right),  // spec version of the runtime loop
       decreases universe.len()
   { ... }

   where sum_matching_keys is a spec fn mirroring the runtime's key-intersection loop.

4. Update the module doc comment to explicitly state that to_seq is the abstraction function and that the proofs hold for any feature universe containing all keys from both maps.

5. Verify by running:

   make verus 2>&1 | tee /tmp/verus-algebra.log
   grep -E 'error|verified|0 errors' /tmp/verus-algebra.log
   

   Expected: all lemmas verified, 0 errors.

---

Constraints

• Do not add vstd or Verus as a dependency of any common/ crate.
• Keep the proof file under 400 lines; decompose into a separate verus/decomposition_vector_algebra_refinement.rs if needed and add it to the deterministic list in scripts/run-verus.sh.
• All comments and doc strings must use en-GB-oxendict spelling.

---

Note from the engineer:

I did not land the requested refinement bridge.

I tried several proof shapes in verus/decomposition_vector_algebra.rs: Map<nat, nat>, total spec_fn(nat) -> nat, and a sequence-of-pairs sparse-map model. In each case, the new refinement lemma either drove rust_verify into very long translation/verification runs or required lower-level sequence facts that would need a more substantial rework than the task described. I reverted those experiments so the repository stays in a healthy state.

Current status is unchanged and clean. I reran:

sh

make verus 2>&1 | tee /tmp/verus-algebra.log grep -E 'error|verified|0 errors' /tmp/verus-algebra.log

Result: existing Verus sidecars still verify with 0 errors.

If you want, I can take this on as a narrower follow-up focused just on making the refinement lemma verify, likely by first introducing small helper lemmas for sequence concatenation/subrange behaviour and then rebuilding the sparse-map abstraction on top of those.

[leynos]

@CodeRabbit plan

[coderabbitai]

Coding Plan

Summary

• Introduce a separate refinement file to isolate the new abstraction work and respect the 400-line constraint.
• Build foundation-first: small, self-contained sequence lemmas that the SMT solver can discharge, avoiding the verification timeouts previously encountered.
• Model the sparse map as Map<nat, nat> with an implicit bijection from runtime strings, then prove that lifting both maps to positional sequences preserves dot-product semantics.
• Register the new file in scripts/run-verus.sh so the refinement proofs are verified alongside the existing positional algebra proofs.

Design Choices

Design Choice 1: Key abstraction strategy

Options Considered:

1. Use Map<nat, nat> with nat keys abstracting string keys via an implicit bijection.
2. Use a spec fn(nat) -> nat total function representing the vector as a mapping from indices to weights.
3. Use a sequence-of-pairs Seq<(nat, nat)> as a sparse-map model.

Chosen Option: 1

Rationale: Verus lacks native String support in spec functions, so the runtime's BTreeMap<String, u64> cannot be modelled directly. Map<nat, nat> aligns with Verus's vstd primitives, enables direct key lookups, and the engineer's note confirms the other approaches hit verification timeouts.

Design Choice 2: File decomposition

Options Considered:

1. Add all new content to verus/decomposition_vector_algebra.rs.
2. Create a new verus/decomposition_vector_algebra_refinement.rs that imports from the existing file.

Chosen Option: 2

Rationale: The existing proof file is moderate in size and adding the full refinement bridge risks exceeding the 400-line constraint. A separate refinement file keeps each file focused, stays under the line limit, and isolates the new helper lemmas from the established proofs.

💡 User Tips

Regenerate the plan with different choices with @coderabbitai <feedback>.

Implementation Steps

Phase 1: Establish Foundation Lemmas for Sequence Operations

This phase addresses the root cause of previous verification failures by introducing small, provable helper lemmas for sequence concatenation and subrange behaviour. These form the inductive building blocks that the abstraction function will rely upon.

Task 1: Create the refinement proof file scaffold

Introduce a new file for the sparse-map abstraction so that the existing positional proofs remain undisturbed and the 400-line constraint is satisfied.

• Create verus/decomposition_vector_algebra_refinement.rs with standard preamble (use vstd::prelude::*, verus! { } macro, stub fn main() {}).
• Add a module-level doc comment explaining that this file establishes the refinement bridge between sparse Map<nat, nat> and positional Seq<nat>.
• Note in the doc comment that nat keys abstractly represent feature strings; the bijection is assumed sound but not modelled.

Task 2: Prove sequence concatenation and subrange helper lemmas

Provide foundational facts that Verus's SMT backend does not discharge automatically, enabling the recursive to_seq and refinement proofs to close.

• Add lemma_seq_concat_len: for sequences a, b, ensure (a + b).len() == a.len() + b.len().
• Add lemma_subrange_len: for a sequence s with valid bounds, ensure s.subrange(lo, hi).len() == hi - lo.
• Add lemma_singleton_concat: for a singleton seq![x] and sequence rest, ensure (seq![x] + rest)[0] == x and (seq![x] + rest).subrange(1, ...) == rest.
• Use decreases clauses for any recursive structure; keep bodies minimal (empty if the SMT solver closes the goal).

🤖 Prompt for AI agents

Create the new file `verus/decomposition_vector_algebra_refinement.rs` and
implement three foundational helper lemmas. The goal of this phase is to lay the
inductive groundwork that prevents the SMT verification timeouts encountered
previously.

- Create `verus/decomposition_vector_algebra_refinement.rs` with the standard
preamble:
  - `use vstd::prelude::*`
  - `verus! { }` macro wrapping all spec/proof content
  - A stub `fn main() {}`
- Add a module-level doc comment stating that this file establishes the
refinement bridge between sparse `Map<nat, nat>` and positional `Seq<nat>`, and
noting that `nat` keys abstractly represent feature strings with a bijection
assumed sound but not modelled
- Inside the `verus! { }` block, add the following proof lemmas:
- `lemma_seq_concat_len`: takes sequences `a` and `b`, ensures `(a + b).len() ==
a.len() + b.len()`
- `lemma_subrange_len`: takes a sequence `s` and valid bounds `lo`, `hi`,
ensures `s.subrange(lo, hi).len() == hi - lo`
- `lemma_singleton_concat`: takes a value `x` and sequence `rest`, ensures
`(seq![x] + rest)[0] == x` and `(seq![x] + rest).subrange(1, (seq![x] +
rest).len() as int) == rest`
- Add `decreases` clauses to any lemma with recursive structure; keep bodies
minimal—leave them empty if the SMT solver can close the goal unaided

Phase 2: Define the Sparse-Map Abstraction

With the foundation lemmas in place, define the abstraction function and prove its essential structural property (length preservation).

Task 1: Define `to_seq` abstraction function

Introduce the spec function that lifts a sparse map into a dense, positional sequence aligned to a feature universe.

• Define spec fn to_seq(map: Map<nat, nat>, universe: Seq<nat>) -> Seq<nat> with recommends universe.no_duplicates() and decreases universe.len().
• Base case: return seq![] when universe.len() == 0.
• Recursive case: prepend map[key] (or 0nat if absent) to the result of recursing on universe.subrange(1, universe.len() as int).

Task 2: Prove length preservation

Ensure the abstraction function produces a sequence of the expected length, which is necessary for the positional dot_product precondition.

• Add proof fn lemma_to_seq_len(map: Map<nat, nat>, universe: Seq<nat>) with ensures to_seq(map, universe).len() == universe.len() and decreases universe.len().
• Call the concatenation helper lemmas as needed; keep the body as small as possible.

Task 3: Define sparse dot-product spec

Mirror the runtime's key-intersection loop as a ghost function so that the refinement lemma has a target to equate.

• Add spec fn sum_matching_keys(left: Map<nat, nat>, right: Map<nat, nat>, universe: Seq<nat>) -> nat with decreases universe.len().
• Base case: return 0nat when universe.len() == 0.
• Recursive case: if both maps contain the current key, add the product; recurse on the tail universe.

🤖 Prompt for AI agents

In `verus/decomposition_vector_algebra_refinement.rs`, define the `to_seq`
abstraction function, the `lemma_to_seq_len` proof, and the `sum_matching_keys`
spec function. The goal is to bridge sparse `Map<nat, nat>` representations to
positional `Seq<nat>`, satisfying the length precondition required by the
positional `dot_product`.

- Define `spec fn to_seq(map: Map<nat, nat>, universe: Seq<nat>) -> Seq<nat>`:
  - Add `recommends universe.no_duplicates()` and `decreases universe.len()`
  - Base case: return `seq![]` when `universe.len() == 0`
- Recursive case: prepend `map[key]` (or `0nat` if the key is absent) to the
result of recursing on `universe.subrange(1, universe.len() as int)`
- Add `proof fn lemma_to_seq_len(map: Map<nat, nat>, universe: Seq<nat>)`:
  - `ensures to_seq(map, universe).len() == universe.len()`
  - `decreases universe.len()`
- Invoke the concatenation helper lemmas from Phase 1 as needed; keep the body
as small as possible
- Add `spec fn sum_matching_keys(left: Map<nat, nat>, right: Map<nat, nat>,
universe: Seq<nat>) -> nat`:
  - `decreases universe.len()`
  - Base case: return `0nat` when `universe.len() == 0`
- Recursive case: if both maps contain the current key, add the product of their
values; recurse on the tail universe

Phase 3: Prove the Refinement Bridge and Integrate

Complete the refinement lemma connecting the sparse and positional models, update documentation, and integrate the new file into the verification workflow.

Task 1: Prove dot-product refinement lemma

Show that the positional dot_product over abstracted sequences equals the sparse sum_matching_keys.

• Add proof fn lemma_dot_product_refinement(left: Map<nat, nat>, right: Map<nat, nat>, universe: Seq<nat>).
• Preconditions: universe.no_duplicates(), forall|k: nat| left.contains_key(k) ==> universe.contains(k), likewise for right.
• Ensures: dot_product(to_seq(left, universe), to_seq(right, universe)) == sum_matching_keys(left, right, universe).
• Use structural induction on universe.len(), invoking the length lemma and the existing dot_product spec's recursive definition.
• Apply #[verifier::nonlinear] if multiplication reasoning is required.

Task 2: Add concrete regression witnesses

Provide explicit examples that exercise the abstraction, acting as regression tests.

• Add a lemma_refinement_example that constructs small Map values and a universe, then asserts the refinement equality using by (compute).
• Ensure the example mirrors at least one scenario from the runtime unit tests in tests/vector_algebra.rs.

Task 3: Update module documentation

Make the abstraction relationship explicit for future maintainers.

• Expand the doc comment in verus/decomposition_vector_algebra_refinement.rs to explain:
  • to_seq is the abstraction function.
  • Proofs hold for any feature universe containing all keys from both maps.
  • The bijection between String and nat is an assumed modelling correspondence, not proven.
• Add a brief cross-reference comment at the top of verus/decomposition_vector_algebra.rs pointing to the refinement file.

Task 4: Register the new proof file for verification

Ensure continuous verification includes the refinement proofs.

• Open scripts/run-verus.sh and add verus/decomposition_vector_algebra_refinement.rs to the decomposition group list, immediately after decomposition_vector_algebra.rs.
• Maintain deterministic ordering (alphabetical or logical grouping).

Task 5: Verify all proofs

Run the full verification suite and confirm zero errors.

• Execute make verus 2>&1 | tee /tmp/verus-algebra.log.
• Grep for error, verified, and 0 errors to confirm success.
• If verification times out on the refinement lemma, introduce intermediate assertions or split the induction step into smaller helper lemmas (apply the same foundation-first strategy recursively).

🤖 Prompt for AI agents

Complete the refinement bridge in
`verus/decomposition_vector_algebra_refinement.rs`, then wire up the file in the
verification script and update all documentation. The goal is to prove that the
positional `dot_product` over abstracted sequences equals `sum_matching_keys`,
add regression witnesses, and ensure the new proofs run in CI.

- Add `proof fn lemma_dot_product_refinement(left: Map<nat, nat>, right:
Map<nat, nat>, universe: Seq<nat>)`:
- Preconditions: `universe.no_duplicates()`, `forall|k: nat|
left.contains_key(k) ==> universe.contains(k)`, likewise for `right`
- Ensures: `dot_product(to_seq(left, universe), to_seq(right, universe)) ==
sum_matching_keys(left, right, universe)`
- Use structural induction on `universe.len()`, invoking `lemma_to_seq_len` and
the existing `dot_product` spec's recursive definition
  - Apply `#[verifier::nonlinear]` if multiplication reasoning is required
- Add `lemma_refinement_example`:
  - Construct small `Map` values and a universe
  - Assert the refinement equality using `by (compute)`
  - Mirror at least one scenario from `tests/vector_algebra.rs`
- Expand the module-level doc comment in
`verus/decomposition_vector_algebra_refinement.rs` to state:
  - `to_seq` is the abstraction function
  - Proofs hold for any feature universe containing all keys from both maps
- The bijection between `String` and `nat` is an assumed modelling
correspondence, not proven
- Add a cross-reference comment at the top of
`verus/decomposition_vector_algebra.rs` pointing to the refinement file
- Open `scripts/run-verus.sh` and insert
`verus/decomposition_vector_algebra_refinement.rs` into the `decomposition`
group immediately after `decomposition_vector_algebra.rs`, preserving
deterministic ordering
- Run `make verus 2>&1 | tee /tmp/verus-algebra.log` and grep for `error`,
`verified`, and `0 errors` to confirm success; if the refinement lemma times
out, introduce intermediate assertions or split the induction step into smaller
helper lemmas

Research

The project maintains formal proofs in verus/ alongside runtime code in common/src/. The existing verus/decomposition_vector_algebra.rs proves algebraic properties (commutativity, non-negativity, disjoint-features-yield-zero-dot) over positional Seq<nat> using structural induction with decreases clauses. The runtime common/src/decomposition_advice/vector.rs implements dot_product and norm_squared on sparse BTreeMap<String, u64> via key-intersection iteration. No formal bridge currently connects the two representations. All Verus files follow a common pattern: vstd::prelude::*, verus! { } macro, spec functions for models, proof functions with #[verifier::nonlinear] for multiplication, and by (compute) for concrete assertions. Verification runs via scripts/run-verus.sh.

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🚀 Next Steps

🤖 All AI agent prompts combined

Task: 1

Create the new file `verus/decomposition_vector_algebra_refinement.rs` and
implement three foundational helper lemmas. The goal of this phase is to lay the
inductive groundwork that prevents the SMT verification timeouts encountered
previously.

- Create `verus/decomposition_vector_algebra_refinement.rs` with the standard
preamble:
  - `use vstd::prelude::*`
  - `verus! { }` macro wrapping all spec/proof content
  - A stub `fn main() {}`
- Add a module-level doc comment stating that this file establishes the
refinement bridge between sparse `Map<nat, nat>` and positional `Seq<nat>`, and
noting that `nat` keys abstractly represent feature strings with a bijection
assumed sound but not modelled
- Inside the `verus! { }` block, add the following proof lemmas:
- `lemma_seq_concat_len`: takes sequences `a` and `b`, ensures `(a + b).len() ==
a.len() + b.len()`
- `lemma_subrange_len`: takes a sequence `s` and valid bounds `lo`, `hi`,
ensures `s.subrange(lo, hi).len() == hi - lo`
- `lemma_singleton_concat`: takes a value `x` and sequence `rest`, ensures
`(seq![x] + rest)[0] == x` and `(seq![x] + rest).subrange(1, (seq![x] +
rest).len() as int) == rest`
- Add `decreases` clauses to any lemma with recursive structure; keep bodies
minimal—leave them empty if the SMT solver can close the goal unaided
===============================================================================

Task: 2

In `verus/decomposition_vector_algebra_refinement.rs`, define the `to_seq`
abstraction function, the `lemma_to_seq_len` proof, and the `sum_matching_keys`
spec function. The goal is to bridge sparse `Map<nat, nat>` representations to
positional `Seq<nat>`, satisfying the length precondition required by the
positional `dot_product`.

- Define `spec fn to_seq(map: Map<nat, nat>, universe: Seq<nat>) -> Seq<nat>`:
  - Add `recommends universe.no_duplicates()` and `decreases universe.len()`
  - Base case: return `seq![]` when `universe.len() == 0`
- Recursive case: prepend `map[key]` (or `0nat` if the key is absent) to the
result of recursing on `universe.subrange(1, universe.len() as int)`
- Add `proof fn lemma_to_seq_len(map: Map<nat, nat>, universe: Seq<nat>)`:
  - `ensures to_seq(map, universe).len() == universe.len()`
  - `decreases universe.len()`
- Invoke the concatenation helper lemmas from Phase 1 as needed; keep the body
as small as possible
- Add `spec fn sum_matching_keys(left: Map<nat, nat>, right: Map<nat, nat>,
universe: Seq<nat>) -> nat`:
  - `decreases universe.len()`
  - Base case: return `0nat` when `universe.len() == 0`
- Recursive case: if both maps contain the current key, add the product of their
values; recurse on the tail universe
===============================================================================

Task: 3

Complete the refinement bridge in
`verus/decomposition_vector_algebra_refinement.rs`, then wire up the file in the
verification script and update all documentation. The goal is to prove that the
positional `dot_product` over abstracted sequences equals `sum_matching_keys`,
add regression witnesses, and ensure the new proofs run in CI.

- Add `proof fn lemma_dot_product_refinement(left: Map<nat, nat>, right:
Map<nat, nat>, universe: Seq<nat>)`:
- Preconditions: `universe.no_duplicates()`, `forall|k: nat|
left.contains_key(k) ==> universe.contains(k)`, likewise for `right`
- Ensures: `dot_product(to_seq(left, universe), to_seq(right, universe)) ==
sum_matching_keys(left, right, universe)`
- Use structural induction on `universe.len()`, invoking `lemma_to_seq_len` and
the existing `dot_product` spec's recursive definition
  - Apply `#[verifier::nonlinear]` if multiplication reasoning is required
- Add `lemma_refinement_example`:
  - Construct small `Map` values and a universe
  - Assert the refinement equality using `by (compute)`
  - Mirror at least one scenario from `tests/vector_algebra.rs`
- Expand the module-level doc comment in
`verus/decomposition_vector_algebra_refinement.rs` to state:
  - `to_seq` is the abstraction function
  - Proofs hold for any feature universe containing all keys from both maps
- The bijection between `String` and `nat` is an assumed modelling
correspondence, not proven
- Add a cross-reference comment at the top of
`verus/decomposition_vector_algebra.rs` pointing to the refinement file
- Open `scripts/run-verus.sh` and insert
`verus/decomposition_vector_algebra_refinement.rs` into the `decomposition`
group immediately after `decomposition_vector_algebra.rs`, preserving
deterministic ordering
- Run `make verus 2>&1 | tee /tmp/verus-algebra.log` and grep for `error`,
`verified`, and `0 errors` to confirm success; if the refinement lemma times
out, introduce intermediate assertions or split the induction step into smaller
helper lemmas

💡 Iterate on the plan with: @coderabbitai <feedback>

Example Feedback
- `@coderabbitai` You can skip phase 3. Add a simple unit test case for phase 2.
- `@coderabbitai` For design choice 1 go ahead with option 3 and replan.

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