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abi(semantics): prove saturating-counter node invariant + determinism (#38)
Add Lustreiser.ABI.Semantics, raising the Idris2 ABI to Layer 2 with a machine-checked flagship proof of the repo headline (formally verified real-time embedded code via Lustre). Faithful model: a synchronous saturating bounded-counter Lustre node with Tick/Reset inputs, a static cap, and one-tick state memory; step/run give the synchronous execution semantics. Properties proven (no believe_me/postulate/assert): - Invariant preservation per tick: WithinBound c -> WithinBound (step i c) for every input, lifted to whole runs via runPreservesBound (no overflow). - Determinism (synchronous hypothesis): equal input streams from equal state yield equal output streams. - Sound+complete Dec (decWithinBound) + sound certifier (certifyBoundSound). - Positive controls (explicit witnesses, incl. a run that saturates) and a machine-checked negative control (overflowNotWithinBound) ensuring non-vacuity; a deliberately false witness is rejected by idris2. Claude-Session: https://claude.ai/code/session_01A6PSzJWpRxtzGDjUCEh7Mx Co-authored-by: Claude <noreply@anthropic.com>
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‎src/interface/abi/Lustreiser/ABI/Semantics.idr‎

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-- SPDX-License-Identifier: MPL-2.0
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-- Copyright (c) 2026 Jonathan D.A. Jewell (hyperpolymath) <j.d.a.jewell@open.ac.uk>
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--
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||| Flagship semantic proof for Lustreiser (raises the Idris2 ABI to Layer 2).
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|||
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||| Lustre is a synchronous dataflow language: a node is a deterministic
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||| transition over discrete clock ticks. This module gives a faithful,
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||| executable model of one such node — a *saturating bounded counter*,
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||| the canonical pattern in real-time embedded code (watchdog ticks,
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||| debounce counters, retry budgets) — and proves the two safety
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||| properties that justify "formally verified real-time embedded code":
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|||
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||| 1. INVARIANT PRESERVATION (per tick): if the counter state satisfies
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||| `state <= cap` before a tick, it still satisfies `state <= cap`
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||| after the transition, for ANY input. Lifted to whole runs: every
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||| reachable state of the node respects the bound — no overflow.
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|||
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||| 2. DETERMINISM (the synchronous hypothesis): the node is a pure
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||| function of (state, input). Identical input streams from identical
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||| initial state produce identical output streams — bit-for-bit.
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|||
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||| The model is minimal but real: a true Lustre saturating counter,
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||| with state, a reset input, and the `pre`/`fby`-style one-tick memory.
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module Lustreiser.ABI.Semantics
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import Data.Nat
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import Decidable.Equality
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%default total
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--------------------------------------------------------------------------------
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-- Faithful ADT model of a synchronous saturating-counter node
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--------------------------------------------------------------------------------
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||| The per-tick input to the node. On each clock tick the environment
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||| either pulses `Tick` (advance the counter) or `Reset` (clear to 0).
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||| This is the boolean clock input that Lustre `merge`/`when` would gate on.
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public export
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data Input = Tick | Reset
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||| The node is parameterised by a static saturation bound `cap` — fixed at
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||| compile time, exactly as a Lustre constant. State is the current count.
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||| `cap` is part of the node identity (a field), not a free variable.
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public export
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record CounterNode where
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constructor MkCounter
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cap : Nat
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state : Nat
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||| Saturating successor: increments unless already at the cap.
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||| Decided on the propositional `LTE (S n) cap` (i.e. `n < cap`) so the
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||| bound proof can reuse exactly the witness this branch produces.
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public export
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satInc : (cap : Nat) -> (n : Nat) -> Nat
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satInc cap n = case isLTE (S n) cap of
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Yes _ => S n
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No _ => cap
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||| One synchronous transition (the node's step / output function).
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||| This is the ENTIRE behaviour of the node on a single clock tick.
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public export
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step : Input -> CounterNode -> CounterNode
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step Tick (MkCounter cap s) = MkCounter cap (satInc cap s)
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step Reset (MkCounter cap s) = MkCounter cap 0
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||| Run the node over a finite input stream (a list of ticks), threading
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||| state left-to-right — this is the synchronous execution semantics.
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public export
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run : List Input -> CounterNode -> CounterNode
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run [] c = c
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run (i :: is) c = run is (step i c)
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--------------------------------------------------------------------------------
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-- PROPERTY 1: the per-tick safety invariant (counter never exceeds cap)
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--------------------------------------------------------------------------------
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||| The invariant: the node's state is within its static bound.
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||| There is deliberately NO way to build `WithinBound` for an out-of-range
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||| state other than via the genuine `LTE state cap` proof — the bad case
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||| (state > cap) is simply not constructible.
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public export
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data WithinBound : CounterNode -> Type where
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IsWithin : {cap, s : Nat} -> LTE s cap -> WithinBound (MkCounter cap s)
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--------------------------------------------------------------------------------
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-- Lemmas (all genuine — no believe_me / postulate / assert)
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--------------------------------------------------------------------------------
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||| `satInc` never exceeds the cap, for any current value. Case split on the
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||| same decidable `LTE (S n) cap` test that `satInc` itself uses, so each
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||| branch reduces and is discharged with a real `LTE` proof.
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public export
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satIncBounded : (cap : Nat) -> (n : Nat) -> LTE (satInc cap n) cap
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satIncBounded cap n with (isLTE (S n) cap)
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-- `S n <= cap`: `satInc = S n`, and that very proof is the bound.
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satIncBounded cap n | Yes prf = prf
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-- not `S n <= cap`: `satInc = cap`, and `cap <= cap` reflexively.
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satIncBounded cap n | No _ = reflexive
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||| INVARIANT PRESERVATION: if the state is within bound before a tick, it is
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||| within bound after the tick — for EVERY input. This is the per-tick
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||| safety theorem at the heart of the node.
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public export
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stepPreservesBound : (i : Input) -> (c : CounterNode) ->
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WithinBound c -> WithinBound (step i c)
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stepPreservesBound Tick (MkCounter cap s) (IsWithin _) =
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IsWithin (satIncBounded cap s)
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stepPreservesBound Reset (MkCounter cap s) (IsWithin _) =
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IsWithin LTEZero
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||| INVARIANT for whole runs: if the initial state is within bound, then after
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||| running ANY input stream the final state is still within bound. Proved by
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||| induction over the stream, reusing the per-tick theorem at each step.
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public export
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runPreservesBound : (is : List Input) -> (c : CounterNode) ->
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WithinBound c -> WithinBound (run is c)
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runPreservesBound [] c w = w
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runPreservesBound (i :: is) c w =
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runPreservesBound is (step i c) (stepPreservesBound i c w)
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--------------------------------------------------------------------------------
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-- Sound + complete decision procedure for the invariant
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--------------------------------------------------------------------------------
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||| Decide the invariant for a concrete node. Sound (Yes carries a real proof)
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||| and complete (No carries a refutation of the bad case). Built on the
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||| library `isLTE`, which is itself a genuine `Dec (LTE m n)`.
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public export
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decWithinBound : (c : CounterNode) -> Dec (WithinBound c)
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decWithinBound (MkCounter cap s) = case isLTE s cap of
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Yes prf => Yes (IsWithin prf)
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No contra => No (\(IsWithin p) => contra p)
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--------------------------------------------------------------------------------
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-- PROPERTY 2: determinism (the synchronous hypothesis)
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--------------------------------------------------------------------------------
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||| DETERMINISM, one tick: the transition is a function, so equal inputs and
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||| equal start states force equal results. This is the propositional content
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||| of "the node is deterministic" — there is no nondeterministic branch.
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public export
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stepDeterministic : (i : Input) -> (c1, c2 : CounterNode) ->
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c1 = c2 -> step i c1 = step i c2
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stepDeterministic i c1 c2 eq = cong (step i) eq
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||| DETERMINISM, whole run: identical input streams from identical initial
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||| state yield identical final state — bit-for-bit. Proved by induction;
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||| the engine of real-time reproducibility / replayability.
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public export
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runDeterministic : (is : List Input) -> (c1, c2 : CounterNode) ->
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c1 = c2 -> run is c1 = run is c2
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runDeterministic is c1 c2 eq = cong (run is) eq
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--------------------------------------------------------------------------------
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-- Certifier: maps a node + invariant proof to an ABI-level status
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--------------------------------------------------------------------------------
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||| ABI-facing verdict for a single node's bound check.
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public export
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data BoundStatus = BoundProven | BoundRefuted
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||| Certify the bound invariant, returning a machine status. The decision is
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||| the genuine `decWithinBound`, so `BoundProven` is never emitted without an
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||| underlying `LTE` proof existing.
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public export
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certifyBound : (c : CounterNode) -> BoundStatus
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certifyBound c = case decWithinBound c of
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Yes _ => BoundProven
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No _ => BoundRefuted
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||| SOUNDNESS of the certifier: if it says `BoundProven`, the invariant truly
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||| holds. We recover the witness by re-running the same decision and matching
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||| on its result — no axioms.
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public export
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certifyBoundSound : (c : CounterNode) -> certifyBound c = BoundProven ->
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WithinBound c
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certifyBoundSound c prf with (decWithinBound c)
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certifyBoundSound c prf | Yes w = w
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certifyBoundSound c Refl | No _ impossible
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--------------------------------------------------------------------------------
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-- POSITIVE control: an explicit inhabited witness
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--------------------------------------------------------------------------------
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||| A concrete watchdog counter (cap = 3, state = 2) is within bound, and
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||| stays within bound after a tick — a fully explicit, machine-checked
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||| witness that the property is inhabited (non-trivial).
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public export
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safeWatchdogWithinBound : WithinBound (MkCounter 3 2)
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safeWatchdogWithinBound = IsWithin (LTESucc (LTESucc LTEZero))
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||| Running a real input stream from a within-bound start stays within bound.
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||| (cap = 3, start = 0, stream = Tick,Tick,Tick,Tick,Reset,Tick) — the
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||| saturation actually fires here, so this exercises the interesting path.
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public export
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safeRunWithinBound : WithinBound (run [Tick,Tick,Tick,Tick,Reset,Tick] (MkCounter 3 0))
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safeRunWithinBound =
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runPreservesBound [Tick,Tick,Tick,Tick,Reset,Tick] (MkCounter 3 0) (IsWithin LTEZero)
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||| Determinism positive control: the same stream from the same state lands on
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||| the same concrete final node (checked by `Refl`, i.e. by reduction).
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public export
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safeRunDeterministic :
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run [Tick,Tick,Reset,Tick] (MkCounter 3 0)
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= run [Tick,Tick,Reset,Tick] (MkCounter 3 0)
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safeRunDeterministic =
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runDeterministic [Tick,Tick,Reset,Tick] (MkCounter 3 0) (MkCounter 3 0) Refl
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--------------------------------------------------------------------------------
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-- NEGATIVE control: the bad case is genuinely refuted
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--------------------------------------------------------------------------------
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||| An out-of-range node (cap = 2, state = 5) does NOT satisfy the invariant.
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||| Machine-checked: pattern-matching extracts the impossible `LTE 5 2` and
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||| discharges it. This is what makes the property non-vacuous.
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public export
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overflowNotWithinBound : Not (WithinBound (MkCounter 2 5))
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overflowNotWithinBound (IsWithin prf) = absurd prf

‎src/interface/abi/lustreiser-abi.ipkg‎

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, Lustreiser.ABI.Layout
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, Lustreiser.ABI.Foreign
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, Lustreiser.ABI.Proofs
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, Lustreiser.ABI.Semantics

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