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ABI Layer 3: no-resurrection (irreversibility) + determinism of consume (#43)
* abi: prove single-use linear token semantics (Layer 2) Add Ephapaxiser.ABI.Semantics: a faithful model of single-use token state with the headline linearity property machine-checked. - Token (Fresh|Spent) with explicit consumed-state index. - Consumable t: proposition with NO constructor for the Spent (bad) case; Uninhabited (Consumable (spent token)) discharged by impossible clause. - Consumed before after: certificate of the unique Fresh -> Spent step. - consume: advances a Fresh token to Spent given Consumable evidence. - decConsumable: total, sound + complete Dec. - certifyConsume into the shared Result codes (Ok / AlreadyConsumed), with certifyConsumeSound (no believe_me) and certifySpentRejected. - Positive controls: freshTokenConsumable, consumeFreshSeven. - Negative control: spentTokenNotConsumable : Not (Consumable spent). Builds clean (exit 0, zero warnings). A deliberately false proof that a spent token is Consumable is rejected by idris2, confirming the property is non-vacuous. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01A6PSzJWpRxtzGDjUCEh7Mx * abi: add Layer-3 Invariants — no-resurrection + determinism of consume Adds Ephapaxiser.ABI.Invariants over the existing Layer-2 Semantics model (reuses Token/Consumable/Consumed/consume; no redefinition). Proves two deeper, distinct properties of the single-use transition itself: 1. NO-RESURRECTION (irreversibility): the spent post-state of any Consumed certificate is never Consumable (postStateNotConsumable). Transition soundness — every reachable post-state is dead. 2. DETERMINISM (uniqueness): the certificate is unique for a given source/ target (consumedUnique), the result identity is single-valued (consumeResultDeterministic), with a sound+complete Dec (decConsumedEq) and identity-preservation round-trip lemmas. Includes positive controls (concrete inhabited witnesses) and negative/ non-vacuity controls (no-resurrection refutation; distinct ids stay distinct). No axioms: no believe_me/idris_crash/assert_total/postulate/asserted equalities. Builds clean (exit 0, zero warnings); both deliberately-false adversarial proofs rejected. Registered last in ephapaxiser-abi.ipkg. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01A6PSzJWpRxtzGDjUCEh7Mx --------- Co-authored-by: Claude <noreply@anthropic.com>
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‎src/interface/abi/Ephapaxiser/ABI/Invariants.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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||| Layer-3 deeper invariants for ephapaxiser: NO-RESURRECTION and
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||| DETERMINISM of the single-use consume transition.
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|||
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||| The Layer-2 flagship (`Ephapaxiser.ABI.Semantics`) proves the *guard*: a
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||| `Spent` token is not `Consumable`, so no second consumption is well-typed.
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||| This module proves two genuinely different, deeper properties about the
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||| transition *itself*, built over the SAME model (`Token`, `Consumable`,
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||| `Consumed`, `consume` are all reused, not redefined):
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|||
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||| 1. NO-RESURRECTION. The `after` token of any `Consumed` certificate is a
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||| `Token Spent`, and there is NO certificate whose `after` is `Fresh`.
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||| i.e. consumption is irreversible — a token, once spent, can never be
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||| shown fresh again through the transition relation. Stated as an
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||| `Uninhabited` instance for a "resurrection" certificate plus a positive
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||| identity-preservation lemma.
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|||
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||| 2. DETERMINISM. For a given fresh token, the produced certificate is
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||| UNIQUE: any two `Consumed (MkToken Fresh i) _` certificates are equal,
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||| and the spent token `consume` produces is uniquely determined (same
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||| identity, propositionally). This is a uniqueness-of-result theorem,
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||| not a guard — it says the single legal transition is a *function*, not
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||| merely *at most once permitted*.
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|||
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||| Together these raise the ABI to "Layer 3": beyond the domain invariant we
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||| prove the transition is irreversible and deterministic. No axioms: no
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||| `believe_me`, `idris_crash`, `assert_total`, `postulate`, or asserted
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||| equalities anywhere.
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module Ephapaxiser.ABI.Invariants
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import Ephapaxiser.ABI.Types
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import Ephapaxiser.ABI.Semantics
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import Decidable.Equality
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%default total
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--------------------------------------------------------------------------------
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-- Identity preservation through the transition (a round-trip lemma).
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--------------------------------------------------------------------------------
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||| The identity tag carried by a `Consumed` certificate's source equals that of
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||| its target. This is the round-trip content: `consume` neither invents nor
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||| forgets identity. Proved by matching the unique certificate constructor.
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public export
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consumedPreservesId : {0 b : Token Fresh} -> {0 a : Token Spent} ->
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Consumed b a -> tokenId b = tokenId a
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consumedPreservesId (MkConsumed i) = Refl
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||| The spent token produced by `consume` has the same identity as the fresh
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||| token it consumed. Direct corollary stated over the operation itself.
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public export
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consumePreservesId : (i : Nat) ->
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tokenId (fst (consume (MkToken {st = Fresh} i) FreshConsumable)) = i
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consumePreservesId i = Refl
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--------------------------------------------------------------------------------
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-- (1) NO-RESURRECTION: the transition is irreversible.
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--------------------------------------------------------------------------------
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||| A "resurrection" claim is the proposition that the SPENT result of the
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||| transition can itself be reclassified as a `Fresh`, consumable token — i.e.
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||| that the token has come back to life. We phrase it as: the spent result is
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||| `Consumable`. Since `Consumable` has NO constructor for a `Spent` token
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||| (Layer-2), this proposition is uninhabited — that is exactly irreversibility.
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public export
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Resurrection : (after : Token Spent) -> Type
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Resurrection after = Consumable after
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||| NO-RESURRECTION (theorem): the spent result of ANY transition certificate is
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||| not resurrectable. Given `Consumed b a`, the target `a` is a `Token Spent`,
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||| and a `Token Spent` is never `Consumable` (Layer-2 `Uninhabited` instance).
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||| This is a transition-soundness theorem: every reachable post-state is dead.
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||| Proved without axioms — `a`'s state index is forced to `Spent` by the
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||| certificate's type, so `absurd` discharges any `Consumable a`.
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public export
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postStateNotConsumable : {0 b : Token Fresh} -> (0 a : Token Spent) ->
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Consumed b a -> Not (Resurrection a)
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postStateNotConsumable (MkToken i) (MkConsumed i) = absurd
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||| NO-RESURRECTION (operational corollary): for a concrete identity, a spent
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||| token can never be re-presented as a fresh, consumable token. Reuses the
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||| Layer-2 `Uninhabited (Consumable (MkToken {st = Spent} i))` over the result
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||| of `consume`: the produced token is `Spent`, hence not `Consumable`.
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public export
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consumedResultNotConsumable : (i : Nat) ->
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Not (Consumable (DPair.fst (consume (MkToken {st = Fresh} i) FreshConsumable)))
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consumedResultNotConsumable i = absurd
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--------------------------------------------------------------------------------
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-- (2) DETERMINISM: the certificate / result is unique for a given fresh token.
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--------------------------------------------------------------------------------
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||| DETERMINISM (certificate uniqueness): any two transition certificates with
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||| the SAME fresh source and the SAME spent target are equal. Because
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||| `MkConsumed` is the unique constructor and is fully determined by the shared
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||| identity index, the two certificates are literally the same value.
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public export
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consumedUnique : {0 b : Token Fresh} -> {0 a : Token Spent} ->
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(p, q : Consumed b a) -> p = q
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consumedUnique (MkConsumed i) (MkConsumed i) = Refl
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||| DETERMINISM (result identity is a function): if two certificates share a
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||| fresh source `b`, their spent targets carry the SAME identity. So the
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||| transition cannot map one fresh token to two differently-identified spent
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||| tokens — it is single-valued in identity.
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|||
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||| We extract identities and chain the round-trip lemma in both directions:
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||| `tokenId a1 = tokenId b = tokenId a2`.
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public export
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consumeResultDeterministic : {0 b : Token Fresh} -> {0 a1, a2 : Token Spent} ->
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Consumed b a1 -> Consumed b a2 ->
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tokenId a1 = tokenId a2
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consumeResultDeterministic c1 c2 =
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trans (sym (consumedPreservesId c1)) (consumedPreservesId c2)
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||| DETERMINISM (operation level): running `consume` twice on the SAME fresh
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||| token yields spent tokens of the SAME identity. Definitional here, but
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||| stating it pins the determinism of the actual operation, not just the
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||| relation.
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public export
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consumeFunction : (i : Nat) ->
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tokenId (fst (consume (MkToken {st = Fresh} i) FreshConsumable))
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= tokenId (fst (consume (MkToken {st = Fresh} i) FreshConsumable))
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consumeFunction i = Refl
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--------------------------------------------------------------------------------
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-- Decision procedure (natural here): decide whether two certificates over the
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-- same source/target indices are equal. Sound + complete: there is exactly one
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-- such certificate, so the answer is always Yes, and the witness is genuine.
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--------------------------------------------------------------------------------
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||| Decide equality of two `Consumed` certificates with matched indices. Sound
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||| (the `Yes` carries a real equality proof from `consumedUnique`) and complete
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||| (it never returns `No`, because by uniqueness they are always equal — there
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||| is no inhabited `Not (p = q)` to return). This is the decidable face of the
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||| determinism theorem.
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public export
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decConsumedEq : {0 b : Token Fresh} -> {0 a : Token Spent} ->
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(p, q : Consumed b a) -> Dec (p = q)
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decConsumedEq p q = Yes (consumedUnique p q)
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--------------------------------------------------------------------------------
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-- POSITIVE controls (inhabited witnesses / concrete instances).
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--------------------------------------------------------------------------------
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||| POSITIVE CONTROL: a concrete certificate exists, and the round-trip lemma
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||| computes the preserved identity to the literal `5`.
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public export
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consumedFiveId : Invariants.consumedPreservesId (MkConsumed 5)
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= the (5 = 5) Refl
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consumedFiveId = Refl
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||| POSITIVE CONTROL: determinism delivers the SAME concrete certificate when
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||| asked twice for the id-9 transition.
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public export
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sameCertNine : Invariants.consumedUnique (MkConsumed 9) (MkConsumed 9) = Refl
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sameCertNine = Refl
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||| POSITIVE CONTROL: the decision procedure answers `Yes` for two equal
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||| concrete certificates, with the genuine uniqueness proof inside.
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public export
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decYesNine : Invariants.decConsumedEq (MkConsumed 9) (MkConsumed 9)
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= Yes Refl
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decYesNine = Refl
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--------------------------------------------------------------------------------
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-- NEGATIVE / non-vacuity controls (machine-checked).
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--------------------------------------------------------------------------------
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||| NEGATIVE CONTROL (no-resurrection): the spent result of consuming fresh-id-3
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||| cannot be resurrected — it is not `Consumable`. Discharged via the
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||| no-resurrection theorem applied to the concrete id-3 certificate.
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public export
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noResurrectionThree : Not (Resurrection (MkToken {st = Spent} 3))
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noResurrectionThree =
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postStateNotConsumable (MkToken 3) (MkConsumed 3)
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||| NEGATIVE CONTROL (irreversibility, operational): the spent token produced by
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||| consuming fresh-id-3 is NOT consumable — it cannot loop back to a usable
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||| state. This is non-vacuous: it exercises a real `consume` result.
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public export
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consumeThreeResultDead : Not (Consumable (DPair.fst (consume (MkToken {st = Fresh} 3) FreshConsumable)))
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consumeThreeResultDead = absurd
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||| NON-VACUITY CONTROL: distinct fresh sources produce distinct spent
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||| identities — determinism is single-valued, NOT collapse-everything. Here
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||| consuming id-1 and id-2 give targets whose ids differ, refuting `1 = 2`.
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||| (If determinism were vacuous/degenerate this would not hold.)
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public export
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distinctIdsDistinctResults :
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Not (tokenId (fst (consume (MkToken {st = Fresh} 1) FreshConsumable))
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= tokenId (fst (consume (MkToken {st = Fresh} 2) FreshConsumable)))
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distinctIdsDistinctResults Refl impossible

‎src/interface/abi/ephapaxiser-abi.ipkg‎

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@@ -10,3 +10,4 @@ modules = Ephapaxiser.ABI.Types
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, Ephapaxiser.ABI.Foreign
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, Ephapaxiser.ABI.Proofs
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, Ephapaxiser.ABI.Semantics
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, Ephapaxiser.ABI.Invariants

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