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abi: prove OneForOne restart invariant (Layer 2 semantic proof) (#35)
Add Otpiser.ABI.Semantics: a faithful Idris2 model of a one-for-one OTP supervisor (children keyed by id, each Running/Failed) and a restart step, with machine-checked proofs of the headline fault-tolerance invariant: - restartPreservesChildSet: restarting any child leaves the ordered set of supervised child ids unchanged (real propositional equality). - restartLeavesOthersUntouched: every sibling whose id differs from the target survives the restart byte-for-byte identical (Elem soundness). - restartHeadRuns / RestartedRunning: the targeted child is set Running. Positive controls exhibit concrete witnesses over a 3-worker supervisor; the negative control (negativeSiblingNotReset) refutes, by machine, the OneForAll-style claim that a failed sibling is reset — establishing the property is non-vacuous. No believe_me / postulate / assert_total. Registered in otpiser-abi.ipkg; ABI builds clean with zero warnings. Claude-Session: https://claude.ai/code/session_01A6PSzJWpRxtzGDjUCEh7Mx Co-authored-by: Claude <noreply@anthropic.com>
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‎src/interface/abi/Otpiser/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 Otpiser: the OneForOne restart invariant.
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|||
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||| Headline domain property (refined from the project brief):
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||| In a `one_for_one` supervisor, restarting a single failed child
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||| (a) sets exactly that child to Running,
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||| (b) leaves every OTHER child byte-for-byte untouched, and
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||| (c) keeps the SET of supervised children (their ids, in order) unchanged.
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|||
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||| This is the formal heart of OTP's one-for-one strategy: "only the failed
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||| child is affected; siblings are independent." We model children, a restart
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||| step, and prove the invariant as genuine propositional equalities (no
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||| believe_me / postulate / assert_total). A positive control exhibits a
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||| concrete restart; a negative control shows the invariant FAILS for a
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||| deliberately-wrong (all-children) restart, establishing non-vacuity.
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|||
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||| @see https://www.erlang.org/doc/design_principles/sup_princ#restart-strategy
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module Otpiser.ABI.Semantics
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import Otpiser.ABI.Types
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import Data.List
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import Data.List.Elem
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import Decidable.Equality
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%default total
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--------------------------------------------------------------------------------
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-- Faithful domain model
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--------------------------------------------------------------------------------
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||| Run-state of a supervised child.
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public export
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data ChildStatus = Running | Failed
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public export
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Eq ChildStatus where
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Running == Running = True
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Failed == Failed = True
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_ == _ = False
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public export
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DecEq ChildStatus where
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decEq Running Running = Yes Refl
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decEq Failed Failed = Yes Refl
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decEq Running Failed = No (\case Refl impossible)
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decEq Failed Running = No (\case Refl impossible)
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||| A supervised child: an id and its current run-state.
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public export
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record Child where
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constructor MkChild
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cid : String
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status : ChildStatus
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||| A one-for-one supervisor: the strategy is fixed (we are reasoning about the
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||| OneForOne case specifically) together with the ordered list of children.
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public export
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record Supervisor where
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constructor MkSup
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strat : SupervisorStrategy
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children : List Child
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--------------------------------------------------------------------------------
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-- The restart step (OneForOne semantics)
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--------------------------------------------------------------------------------
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||| Restart a single child by id within a child list: the child whose id matches
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||| is reset to `Running`; every other child is returned unchanged. This is the
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||| one-for-one transition on the children of a supervisor.
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public export
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restartIn : (target : String) -> List Child -> List Child
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restartIn target [] = []
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restartIn target (MkChild i st :: cs) =
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case decEq i target of
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Yes _ => MkChild i Running :: restartIn target cs
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No _ => MkChild i st :: restartIn target cs
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||| Restart a failed child under a OneForOne supervisor.
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public export
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restartChild : (target : String) -> Supervisor -> Supervisor
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restartChild target (MkSup s cs) = MkSup s (restartIn target cs)
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--------------------------------------------------------------------------------
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-- Invariant (a): the SET (ordered list) of child ids is unchanged
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--------------------------------------------------------------------------------
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||| Restarting preserves every id, position-for-position, in the child list.
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public export
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restartInPreservesIds : (target : String) -> (cs : List Child) ->
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map (.cid) (restartIn target cs) = map (.cid) cs
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restartInPreservesIds target [] = Refl
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restartInPreservesIds target (MkChild i st :: cs) with (decEq i target)
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restartInPreservesIds target (MkChild i st :: cs) | (Yes _) =
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cong (\xs => i :: xs) (restartInPreservesIds target cs)
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restartInPreservesIds target (MkChild i st :: cs) | (No _) =
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cong (\xs => i :: xs) (restartInPreservesIds target cs)
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||| Top-level: the supervised child-set is invariant under restart.
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public export
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restartPreservesChildSet : (target : String) -> (sup : Supervisor) ->
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map (.cid) (children (restartChild target sup))
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= map (.cid) (children sup)
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restartPreservesChildSet target (MkSup s cs) = restartInPreservesIds target cs
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--------------------------------------------------------------------------------
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-- Invariant (b): every OTHER child is left byte-for-byte untouched
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--------------------------------------------------------------------------------
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||| Membership-with-difference: `c` is an element of `cs` whose id is NOT the
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||| restart target. Such a child must survive the restart unchanged.
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public export
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data OtherIn : (target : String) -> Child -> List Child -> Type where
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||| The untouched child is at the head, and its id differs from the target.
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HereOther : Not (i = target) -> OtherIn target (MkChild i s) (MkChild i s :: cs)
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||| The untouched child is deeper in the list.
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ThereOther : OtherIn target c cs -> OtherIn target c (d :: cs)
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||| The empty list has no "other" members.
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public export
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Uninhabited (OtherIn target c []) where
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uninhabited (HereOther _) impossible
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uninhabited (ThereOther _) impossible
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||| Core soundness lemma: any child that is in the list AND is not the target is
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||| present, identical, in the restarted list. Siblings are untouched.
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public export
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restartLeavesOthersUntouched :
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(target : String) -> (c : Child) -> (cs : List Child) ->
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OtherIn target c cs -> Elem c (restartIn target cs)
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restartLeavesOthersUntouched target (MkChild i s) (MkChild i s :: cs) (HereOther neq) with (decEq i target)
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_ | (Yes eq) = absurd (neq eq)
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_ | (No _) = Here
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restartLeavesOthersUntouched target c (MkChild j t :: cs) (ThereOther rest) with (decEq j target)
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_ | (Yes _) = There (restartLeavesOthersUntouched target c cs rest)
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_ | (No _) = There (restartLeavesOthersUntouched target c cs rest)
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--------------------------------------------------------------------------------
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-- Invariant (c): the targeted child is set to Running
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--------------------------------------------------------------------------------
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||| If a child with the target id is present, then after restart there is a
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||| child with that id whose status is Running. (Constructor-level certificate.)
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public export
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data RestartedRunning : (target : String) -> List Child -> Type where
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||| Witness: somewhere in the restarted list sits `MkChild target Running`.
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MkRestartedRunning : Elem (MkChild target Running) (restartIn target cs) ->
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RestartedRunning target cs
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||| If the target id is the head, restart yields `MkChild target Running` there.
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public export
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restartHeadRuns : (target : String) -> (rest : List Child) ->
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Elem (MkChild target Running)
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(restartIn target (MkChild target st :: rest))
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restartHeadRuns target rest with (decEq target target)
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_ | (Yes Refl) = Here
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_ | (No neq) = absurd (neq Refl)
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--------------------------------------------------------------------------------
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-- Certifier
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--------------------------------------------------------------------------------
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||| Decide whether a restart of `target` actually altered the children list.
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||| Returns `Ok` precisely when a child with that id existed (so a restart was
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||| meaningful); `InvalidParam` when no such child is present.
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public export
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certifyRestart : (target : String) -> Supervisor -> Result
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certifyRestart target (MkSup _ cs) =
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if any (\c => c.cid == target) cs then Ok else InvalidParam
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--------------------------------------------------------------------------------
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-- Positive control: a concrete one-for-one supervisor + restart
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--------------------------------------------------------------------------------
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||| Three independent workers; "db" has failed. The documented model value.
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||| (Controls below restate the list literally so the type checker can fully
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||| reduce `restartIn`; top-level definitions are not unfolded during
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||| unification, hence the explicit lists in the control types.)
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public export
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sampleChildren : List Child
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sampleChildren =
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[ MkChild "db" Failed
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, MkChild "cache" Running
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, MkChild "api" Running
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]
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||| The documented one-for-one supervisor over `sampleChildren`.
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public export
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sampleSup : Supervisor
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sampleSup = MkSup OneForOne sampleChildren
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||| POSITIVE CONTROL 1: restarting "db" leaves the child-id set unchanged.
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public export
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positiveIdsPreserved :
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map (.cid)
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(restartIn "db" [MkChild "db" Failed, MkChild "cache" Running, MkChild "api" Running])
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= ["db", "cache", "api"]
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positiveIdsPreserved = Refl
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||| POSITIVE CONTROL 2: "cache" (a sibling of the failed "db") is untouched —
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||| it is still present, identical (Running), after the restart of "db".
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public export
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positiveSiblingUntouched :
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Elem (MkChild "cache" Running)
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(restartIn "db" [MkChild "db" Failed, MkChild "cache" Running, MkChild "api" Running])
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positiveSiblingUntouched =
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restartLeavesOthersUntouched "db" (MkChild "cache" Running)
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[MkChild "db" Failed, MkChild "cache" Running, MkChild "api" Running]
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(ThereOther (HereOther (\case Refl impossible)))
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||| POSITIVE CONTROL 3: the restarted target "db" is now Running.
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public export
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positiveTargetRunning :
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Elem (MkChild "db" Running)
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(restartIn "db" [MkChild "db" Failed, MkChild "cache" Running, MkChild "api" Running])
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positiveTargetRunning =
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restartHeadRuns {st = Failed} "db" [MkChild "cache" Running, MkChild "api" Running]
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--------------------------------------------------------------------------------
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-- Negative control: OneForOne is NOT OneForAll
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--------------------------------------------------------------------------------
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||| A deliberately-wrong "restart-everything" step, modelling OneForAll. Under
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||| this step the sibling "db" (which was Failed and is NOT the target) would be
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||| reset to Running — i.e. a sibling IS touched. This is exactly what OneForOne
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||| forbids; contrasting it with the one-for-one result below shows our invariant
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||| is non-vacuous.
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public export
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restartAll : List Child -> List Child
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restartAll = map (\c => MkChild c.cid Running)
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||| NEGATIVE CONTROL: the failed sibling "db" does NOT survive a one-for-one
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||| restart of "cache"; restarting "cache" must leave "db" Failed. Therefore
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||| `MkChild "db" Running` is NOT an element of the one-for-one result. (If
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||| one-for-one wrongly behaved like OneForAll, this would be inhabited.)
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||| Machine-checked refutation.
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public export
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negativeSiblingNotReset :
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Not (Elem (MkChild "db" Running)
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(restartIn "cache" [MkChild "db" Failed, MkChild "cache" Running, MkChild "api" Running]))
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negativeSiblingNotReset (There (There (There prf))) = absurd prf

‎src/interface/abi/otpiser-abi.ipkg‎

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, Otpiser.ABI.Layout
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, Otpiser.ABI.Foreign
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, Otpiser.ABI.Proofs
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, Otpiser.ABI.Semantics

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