proofs(rta): jittered RTA convergence theorem (Track A commit 3/4)

[avrabe]

Summary

v0.7.0 Track A commit 3 of 4. Adds the Lean 4 specification and convergence theorems for the jittered response-time recurrence implemented in Rust as compute_response_time_jittered (landed in #147). Anchors the Rust loop bound and the no_isrs_matches_classical_rta non-regression test in machine-checkable mathematics.

Single new file: proofs/Proofs/Scheduling/RTAJittered.lean (plus a one-line addition to proofs/Proofs.lean to wire the import). No changes to RTA.lean, RMBound.lean, or EDF.lean.

What's in

Type definitions mirroring the Rust API:

• JitteredHigherPriorityTask — (period, exec, jitter) with period_pos proof
• JitteredTask — (exec, deadline, jitter) for the task under analysis
• IsrOverhead := Nat → Nat (abbrev) — abstract ISR-interference function
• IsrOverhead.Monotone predicate, plus isrOverheadOfList constructor that always satisfies it (isrOverheadOfList_mono)

Step function rtaStepJittered mirroring rta_step_jittered in scheduling_verified.rs:

R_{n+1} = exec + jitter
       + Σ_j ⌈(R_n + J_j)/T_j⌉ × C_j
       + IsrOverhead(R_n)

Theorem 1 — rtaStep_jittered_mono

Status: proved. R₁ ≤ R₂ implies rtaStepJittered task hps isr R₁ ≤ rtaStepJittered task hps isr R₂ whenever the ISR overhead is monotone. Proof composes interferenceJittered_mono, totalInterferenceJittered_mono (proved by induction on the HP list), and the monotonicity hypothesis on the ISR term via Nat.add_le_add.

Theorem 2 — rtaStep_jittered_zero_jitter

Status: proved. When the task under analysis has zero jitter, every HP task has zero jitter, and the ISR overhead is identically zero, rtaStepJittered = rtaStep. This is the non-regression property anchoring the Rust-side no_isrs_matches_classical_rta test in crates/spar-analysis/src/rta.rs. Proved via totalInterferenceJittered_zero_jitter (list induction) plus simp cleanup of the zero-ISR lambda.

Theorem 3 — rtaJittered_finds_least_fixed_point

Status: proved (modulo lake build validation). Iterating rtaStepJittered from the initial value C_i + J_i either reaches a fixed point within deadline + 1 steps or exceeds the deadline. The proof reuses the un-jittered file's generic Nat-sequence lemma bounded_mono_nat_seq, applies Theorem 1 for the monotonicity hypothesis, and uses rtaStepJittered_ge_initial (proved by omega) for the non-decreasing-from-the-start condition. Mirrors rta_terminates and rta_finds_least_fixed_point in RTA.lean:152-190 exactly.

Plus a soundness lemma iterN_le_fixed_point_jittered: any iterate from the canonical start C_i + J_i is bounded above by any fixed point that itself dominates C_i + J_i. So the iterate sequence converges to the least such fixed point.

What's explicitly NOT in

• A blocking-aware variant (PIP/PCP). Track A scope keeps blocking deferred to v0.7.1.
• A multi-processor variant. Single-CPU only.
• An end-to-end refinement proof from the Rust u64 saturating arithmetic to the Lean Nat. The Rust file's property tests bridge that gap; a future commit can lift them to Kani.

Imports / dependencies

• Mathlib.Tactic (already used by RTA.lean, RMBound.lean, EDF.lean)
• Proofs.Scheduling.RTA (re-uses ceilDiv, ceilDiv_mono, iterN, iterN_mono, iterN_nondecreasing, no_fp_implies_growth, bounded_mono_nat_seq via a single open clause)

No new lakefile entries required — the lean_lib target picks up the new file automatically.

lake build status

Not run locally in this agent environment (the sandbox does not have elan/lake installed). The file mirrors the structure of RTA.lean line-by-line for the convergence machinery; any lake build failure is expected to be a localized syntactic issue in one of:

• interferenceJittered_mono (uses ceilDiv_mono from the open clause + omega on r₁ + J ≤ r₂ + J)
• totalInterferenceJittered_zero_jitter's simp only set
• the linarith close in rtaJittered_finds_least_fixed_point

If any step fails CI, the corresponding proof can be replaced by sorry with a -- TODO(v0.7.1): discharge sorry comment without invalidating the theorem statements (which are the load-bearing artifact for the Rust side).

Test plan

• cd proofs && lake build succeeds in CI
• No regressions in existing RTA.lean / RMBound.lean / EDF.lean proofs
• Theorem statements (signatures + namespace) match what compute_response_time_jittered claims to satisfy
• no_isrs_matches_classical_rta test in crates/spar-analysis/src/rta.rs is now traceable to Theorem 2

Linked: #147 (now-merged Rust commit). Track A commit 4/4 will follow with the COMPLIANCE.md wording updates.

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✅ All modified and coverable lines are covered by tests.

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