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ABI Layer 3: affine composition law over disjoint variable sets (#38)
## Summary Layer 3 (second, deeper invariant): proves the **affine composition law** — `concatAffine : Disjoint t1 t2 -> AffineOk t1 -> AffineOk t2 -> AffineOk (t1 ++ t2)` — so two affine usage traces over disjoint variable sets compose into an affine trace. Distinct from and deeper than the Layer-2 use-at-most-once checker. New module `Affinescriptiser.ABI.Invariants` (imports the Layer-2 `Semantics` model). Positive + non-vacuity controls. ## Testing Idris2 0.7.0 `--build` → exit 0, zero warnings. Adversarial rejection confirmed. `build/` removed. No `believe_me`/`postulate`/`sorry`. 🤖 Generated with [Claude Code](https://claude.com/claude-code) https://claude.ai/code/session_01A6PSzJWpRxtzGDjUCEh7Mx --- _Generated by [Claude Code](https://claude.ai/code/session_01A6PSzJWpRxtzGDjUCEh7Mx)_ --------- Signed-off-by: Jonathan D.A. Jewell <6759885+hyperpolymath@users.noreply.github.com> Co-authored-by: Claude <noreply@anthropic.com>
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‎src/interface/abi/Affinescriptiser/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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||| Structural invariants for Affinescriptiser (ABI Layer 3).
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|||
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||| This module proves a SECOND, deeper property over the SAME affine-usage
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||| model defined in `Affinescriptiser.ABI.Semantics` (the `Trace`/`AffineOk`
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||| datatypes are imported, not redefined). Where Layer 2 establishes a
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||| sound+complete *checker* for a single trace ("no variable used twice"),
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||| Layer 3 establishes an *algebraic composition law* for that checker:
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|||
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||| COMPOSITION (the headline Layer-3 theorem):
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||| If two usage traces are each affine-safe AND their variable sets are
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||| disjoint, then their concatenation is affine-safe. Formally
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||| Disjoint t1 t2 -> AffineOk t1 -> AffineOk t2 -> AffineOk (t1 ++ t2)
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||| This is the soundness justification for the codegen *sequencing* two
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||| already-checked program fragments over disjoint locals into one fragment
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||| without re-running the whole affine check.
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|||
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||| DECOMPOSITION (the converse structural law):
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||| Affine-safety is preserved under taking the *suffix* of a concatenation:
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||| AffineOk (t1 ++ t2) -> AffineOk t2
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||| i.e. dropping a safe prefix can never break affine-safety of the rest.
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|||
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||| Both directions are genuine, machine-checked, total proofs. Disjointness is
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||| given a sound+complete decision procedure (`decDisjoint`). We supply a
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||| POSITIVE control (a concrete inhabited composition witness) and TWO NEGATIVE
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||| controls: a `Not` showing overlapping traces compose to an unsafe trace, and
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||| a `Not` showing a wide-gap double-use trace is rejected.
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|||
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||| Quality bar (identical to Layer 2): no believe_me, no idris_crash, no
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||| assert_total, no postulate, no sorry, no %hint hacks, no asserted equalities.
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module Affinescriptiser.ABI.Invariants
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import Affinescriptiser.ABI.Types
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import Affinescriptiser.ABI.Semantics
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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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-- Membership-over-append lemmas (the technical core)
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--------------------------------------------------------------------------------
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||| If `x` occurs in `xs ++ ys` then it occurs in `xs` or in `ys`.
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||| Proved by induction on `xs`. `Here`/`There` are constructor-headed, so the
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||| recursive case-splits reduce cleanly with no stuck `case ... of Refl`.
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export
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elemAppSplit : {0 x : Var} -> (xs : Trace) -> Elem x (xs ++ ys) -> Either (Elem x xs) (Elem x ys)
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elemAppSplit [] el = Right el
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elemAppSplit (y :: ys) Here = Left Here
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elemAppSplit (y :: ys) (There el) =
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case elemAppSplit ys el of
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Left inXs => Left (There inXs)
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Right inYs => Right inYs
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||| Conversely, membership in `ys` injects into membership in `xs ++ ys`.
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||| Induction on `xs` again; each step is just one `There`.
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export
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elemAppRight : {0 x : Var} -> (xs : Trace) -> Elem x ys -> Elem x (xs ++ ys)
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elemAppRight [] el = el
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elemAppRight (y :: ys) el = There (elemAppRight ys el)
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||| If `x` is in neither `xs` nor `ys`, it is not in `xs ++ ys`.
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||| This is the contrapositive of `elemAppSplit`, packaged for direct use.
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export
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notElemApp : {0 x : Var}
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-> (xs : Trace)
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-> Not (Elem x xs)
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-> Not (Elem x ys)
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-> Not (Elem x (xs ++ ys))
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notElemApp xs nxs nys el =
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case elemAppSplit xs el of
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Left inXs => nxs inXs
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Right inYs => nys inYs
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--------------------------------------------------------------------------------
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-- Disjointness of variable sets
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--------------------------------------------------------------------------------
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||| Two traces are `Disjoint` when no variable appears in both. This is the
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||| precondition under which sequencing two affine fragments stays affine: each
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||| fragment may reuse names internally only if those names do not collide.
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public export
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Disjoint : Trace -> Trace -> Type
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Disjoint xs ys = (v : Var) -> Elem v xs -> Not (Elem v ys)
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||| The empty trace is disjoint from anything (vacuously: nothing is in `[]`).
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export
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disjointNilLeft : (ys : Trace) -> Disjoint [] ys
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disjointNilLeft ys v el = absurd el
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||| Drop the head of the left trace and disjointness is preserved.
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export
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disjointTail : {0 x : Var} -> Disjoint (x :: xs) ys -> Disjoint xs ys
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disjointTail dj v inXs = dj v (There inXs)
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||| The head of a disjoint left trace is not present in the right trace.
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||| `x` is bound explicitly (quantity 1) because we use it at the term level.
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export
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disjointHeadNotRight : {x : Var} -> Disjoint (x :: xs) ys -> Not (Elem x ys)
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disjointHeadNotRight dj = dj x Here
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||| Build a disjointness witness by consing onto the left trace: if the new
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||| head `x` is absent from `ys` and `xs` is already disjoint from `ys`, then
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||| `x :: xs` is disjoint from `ys`. Matching on the LEFT membership refines the
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||| variable cleanly (avoiding the Nat-literal coverage pitfalls).
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export
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consDisjoint : {0 x : Var} -> Not (Elem x ys) -> Disjoint xs ys -> Disjoint (x :: xs) ys
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consDisjoint nx _ v Here = nx
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consDisjoint _ dxs v (There inXs) = dxs v inXs
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--------------------------------------------------------------------------------
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-- Sound + complete decision procedure for disjointness
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--------------------------------------------------------------------------------
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||| Decide whether two traces are disjoint. A `Yes` carries a real witness
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||| (a function refuting any shared membership), a `No` carries a real overlap.
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||| Built only from `isElem` (Data.List.Elem) and the structural lemmas above.
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public export
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decDisjoint : (xs : Trace) -> (ys : Trace) -> Dec (Disjoint xs ys)
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decDisjoint [] ys = Yes (disjointNilLeft ys)
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decDisjoint (x :: xs) ys =
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case isElem x ys of
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-- x is shared: the pair cannot be disjoint, since x is in both sides.
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Yes xInYs => No (\dj => disjointHeadNotRight dj xInYs)
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No xNotYs =>
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case decDisjoint xs ys of
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Yes djRest =>
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Yes (\v, inHead =>
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case inHead of
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Here => xNotYs
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There inXs => djRest v inXs)
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No notRest =>
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No (\dj => notRest (disjointTail dj))
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--------------------------------------------------------------------------------
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-- HEADLINE Layer-3 theorem: composition of affine traces over disjoint vars
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--------------------------------------------------------------------------------
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||| COMPOSITION. Sequencing two affine-safe traces whose variable sets are
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||| disjoint yields an affine-safe trace. Proved by induction on the structure
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||| of the first trace via its `AffineOk` witness:
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|||
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||| * `[] ++ t2` reduces to `t2`, whose safety we already have.
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||| * `(v :: vs) ++ t2` reduces to `v :: (vs ++ t2)`. The new `notLater`
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||| obligation — `v` not in `vs ++ t2` — is discharged by `notElemApp`:
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||| `v` is absent from `vs` (head of t1's witness) and absent from `t2`
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||| (disjointness). The tail is handled by recursion.
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export
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concatAffine : (t1 : Trace)
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-> Disjoint t1 t2
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-> AffineOk t1
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-> AffineOk t2
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-> AffineOk (t1 ++ t2)
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concatAffine [] _ _ ok2 = ok2
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concatAffine (v :: vs) dj (ConsOk nv ok1) ok2 =
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let notInVs : Not (Elem v vs) := nv
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notInT2 : Not (Elem v t2) := disjointHeadNotRight dj
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notInBoth : Not (Elem v (vs ++ t2)) := notElemApp vs notInVs notInT2
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restOk : AffineOk (vs ++ t2) := concatAffine vs (disjointTail dj) ok1 ok2
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in ConsOk notInBoth restOk
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--------------------------------------------------------------------------------
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-- Converse structural law: affine-safety of a suffix
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--------------------------------------------------------------------------------
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||| DECOMPOSITION. If a concatenation is affine-safe, so is its suffix `t2`.
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||| (Dropping a safe prefix never breaks the rest.) Induction on `t1`: each
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||| `ConsOk` we strip exposes the `AffineOk (vs ++ t2)` witness for recursion;
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||| the base case `[] ++ t2 = t2` returns the witness directly.
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export
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suffixAffine : (t1 : Trace) -> AffineOk (t1 ++ t2) -> AffineOk t2
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suffixAffine [] ok = ok
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suffixAffine (v :: vs) (ConsOk _ rest) = suffixAffine vs rest
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--------------------------------------------------------------------------------
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-- Positive control: a concrete inhabited composition
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--------------------------------------------------------------------------------
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||| Left fragment uses variables 0 and 1 (each once).
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export
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fragLeftOk : AffineOk [0, 1]
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fragLeftOk =
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ConsOk (\el => case el of
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There el1 => absurd el1)
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(ConsOk absurd NilOk)
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||| Right fragment uses variables 2 and 3 (each once).
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export
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fragRightOk : AffineOk [2, 3]
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fragRightOk =
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ConsOk (\el => case el of
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There el1 => absurd el1)
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(ConsOk absurd NilOk)
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||| 0 is not an element of [2,3]: the diagonal `Here`/`There Here` shapes would
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||| require 0 = 2 / 0 = 3 (automatically pruned), so only the empty tail
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||| `There (There el)` remains, discharged by `absurd`.
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zeroNotIn23 : Not (Elem (the Var 0) [2, 3])
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zeroNotIn23 (There (There el)) = absurd el
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||| 1 is not an element of [2,3]. Same structure as `zeroNotIn23`.
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oneNotIn23 : Not (Elem (the Var 1) [2, 3])
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oneNotIn23 (There (There el)) = absurd el
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||| The two fragments use disjoint variable sets {0,1} and {2,3}. Built
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||| compositionally with `consDisjoint` from the per-element refutations
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||| `zeroNotIn23` / `oneNotIn23` and the vacuous `disjointNilLeft` — a genuine
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||| witness, no assertion.
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export
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fragsDisjoint : Disjoint [0, 1] [2, 3]
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fragsDisjoint =
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consDisjoint zeroNotIn23 (consDisjoint oneNotIn23 (disjointNilLeft [2, 3]))
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||| POSITIVE CONTROL: composing the two disjoint affine fragments produces an
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||| affine-safe trace [0,1,2,3] — built by the headline theorem, not by hand.
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export
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composedOk : AffineOk ([0, 1] ++ [2, 3])
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composedOk = concatAffine [0, 1] fragsDisjoint fragLeftOk fragRightOk
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--------------------------------------------------------------------------------
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-- Negative control 1: overlapping fragments are NOT disjoint
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--------------------------------------------------------------------------------
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||| Fragments [0,1] and [1,2] share variable 1, so they are NOT disjoint.
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||| Machine-checked refutation: any disjointness witness would have to refute
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||| `Elem 1 [1,2]`, but 1 is `Here`.
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export
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overlapNotDisjoint : Not (Disjoint [0, 1] [1, 2])
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overlapNotDisjoint dj = dj 1 (There Here) Here
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||| And their concatenation [0,1,1,2] is genuinely unsafe: variable 1 occurs
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||| twice. No `AffineOk` witness exists — the head-1 obligation would have to
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||| refute `Elem 1 [1,2]`, which is `Here`.
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export
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overlapConcatNotAffine : Not (AffineOk ([0, 1] ++ [1, 2]))
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overlapConcatNotAffine ok = headNotLater (tailOk ok) Here
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--------------------------------------------------------------------------------
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-- Negative control 2: a wide-gap double use is rejected
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--------------------------------------------------------------------------------
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||| A reuse separated by several distinct intervening uses is still caught:
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||| variable 9 appears at the head and again after 4 other variables in
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||| [9, 1, 2, 3, 4, 9]. Refuted because 9 reappears later in the tail.
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export
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wideGapDoubleUseNotAffine : Not (AffineOk [9, 1, 2, 3, 4, 9])
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wideGapDoubleUseNotAffine ok =
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headNotLater ok (There (There (There (There Here))))

‎src/interface/abi/affinescriptiser-abi.ipkg‎

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, Affinescriptiser.ABI.Foreign
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, Affinescriptiser.ABI.Proofs
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, Affinescriptiser.ABI.Semantics
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, Affinescriptiser.ABI.Invariants

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