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ABI Layer 3: type-lowering injectivity + composition (two deeper codegen theorems) (#37)
* abi(nimiser): prove C-binding signature preservation (Layer 2) Add Nimiser.ABI.Semantics, a machine-checked semantic proof that the Nim->C code generator preserves source signatures: same name, same arity, and every argument/return type lowered by the canonical injective nimToC mapping. An arity or type mismatch is unrepresentable as a SigPreserved witness. Contents: - Faithful model: NimT / CT type universes, NimSig / CSig signatures, total nimToC lowering, genBinding code generator. - Headline property SigPreserved with no constructor for any mismatch. - genBindingPreserves: the generator ALWAYS preserves (soundness). - nimToCInjective: the lowering is injective (non-vacuity engine). - decSigPreserved: sound + complete Dec decision procedure. - certifyBinding + certifyBindingSound tied to the project Result type; generatedAlwaysCertifies corollary. - Positive control samplePreserved (explicit witness) and three negative controls (wrong return type, wrong arity, wrong arg type), all machine-checked. Build: idris2 --build nimiser-abi.ipkg exits 0, zero warnings. Adversarial false proof (claiming a type-mismatched binding is preserved) is rejected by the type-checker, confirming non-vacuity. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01A6PSzJWpRxtzGDjUCEh7Mx * Add Layer-3 ABI invariants: lowering composition + generator injectivity Adds Nimiser.ABI.Invariants over the existing Semantics codegen model (NimT/CT/nimToC/NimSig/CSig/genBinding/ArgsLower), proving two facts deeper than and distinct from the Layer-2 signature-preservation theorem: 1. Composition/distributivity: lowering is a homomorphism over argument- list concatenation (lowerAppendDistrib), lifted relationally (argsLowerAppend) and to signature extension (genBindingExtendComposes, extendedArgsLower) — codegen of a grown proc is the grown codegen. 2. Whole-signature injectivity: genBindingInjective shows distinct Nim signatures lower to distinct C bindings (no symbol/ABI aliasing), strictly stronger than the Layer-2 per-type nimToCInjective lemma. Includes a sound+complete DecEq NimSig, a positive control (composed ArgsLower witness on concrete data) and negative/non-vacuity controls (distinctSigsDistinctBindings, extensionChangesBinding, decEqDistinctIsNo). Builds clean (exit 0, zero warnings); adversarial false-equality proof via genBindingInjective is rejected. No believe_me/postulate/assert_total/etc. 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/Nimiser/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 invariants for Nimiser, built over the SAME codegen model as the
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||| Layer-2 flagship (`Nimiser.ABI.Semantics`).
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|||
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||| The Layer-2 theorem was *signature preservation*: the generator's output
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||| preserves each source signature (arity + per-type lowering), with per-type
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||| injectivity (`nimToCInjective`) as a supporting lemma.
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|||
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||| This module proves two genuinely DEEPER and DISTINCT facts:
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|||
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||| 1. COMPOSITION / DISTRIBUTIVITY. The argument lowering is a homomorphism
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||| over list concatenation: appending arguments to a Nim proc corresponds
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||| exactly to appending the lowered C arguments. We prove the algebraic law
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||| `map nimToC (xs ++ ys) = map nimToC xs ++ map nimToC ys`, lift it to the
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||| relational `ArgsLower` (witnesses compose under `++`), and conclude an
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||| ARITY-AND-TYPE COMPOSITION theorem: the binding of an extended signature
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||| is the binding of the original with the extra lowered args appended.
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|||
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||| 2. WHOLE-SIGNATURE INJECTIVITY. The code generator is injective on entire
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||| signatures: distinct Nim signatures lower to distinct C bindings. This is
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||| strictly stronger than per-type injectivity and orthogonal to
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||| preservation — it says no two distinct source procs can collapse to a
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||| single C binding (no symbol/ABI aliasing introduced by codegen).
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|||
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||| Both come with a sound+complete decision procedure for Nim-signature
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||| equality, a positive control (a composed witness / concrete instance), and
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||| negative / non-vacuity controls.
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module Nimiser.ABI.Invariants
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import Nimiser.ABI.Types
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import Nimiser.ABI.Semantics
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import Decidable.Equality
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import Data.List
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%default total
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--------------------------------------------------------------------------------
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-- 1. Distributivity of the lowering over list concatenation (algebraic law)
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--------------------------------------------------------------------------------
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||| THEOREM (distributivity): lowering distributes over concatenation of Nim
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||| argument lists. Lowering `xs ++ ys` in one shot equals lowering each side and
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||| concatenating. This is the algebraic engine behind argument composition: it
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||| says the generator treats the argument list as a monoid homomorphism.
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public export
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lowerAppendDistrib :
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(xs, ys : List NimT)
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-> map Semantics.nimToC (xs ++ ys)
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= map Semantics.nimToC xs ++ map Semantics.nimToC ys
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lowerAppendDistrib [] ys = Refl
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lowerAppendDistrib (x :: xs) ys =
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rewrite lowerAppendDistrib xs ys in Refl
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--------------------------------------------------------------------------------
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-- 2. ArgsLower composes under concatenation (relational lift of the law)
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--------------------------------------------------------------------------------
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||| THEOREM (relational composition): `ArgsLower` witnesses compose under `++`.
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||| If `cs` is the lowering of `ts` and `ds` is the lowering of `us`, then
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||| `cs ++ ds` is the lowering of `ts ++ us`. Concatenating two proven-correct
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||| argument blocks yields a proven-correct combined block.
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public export
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argsLowerAppend :
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{0 ts, us : List NimT} -> {0 cs, ds : List CT}
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-> ArgsLower ts cs
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-> ArgsLower us ds
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-> ArgsLower (ts ++ us) (cs ++ ds)
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argsLowerAppend LowerNil right = right
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argsLowerAppend (LowerCons hd tlOk) right =
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LowerCons hd (argsLowerAppend tlOk right)
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--------------------------------------------------------------------------------
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-- 3. Arity-and-type composition under signature extension
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--------------------------------------------------------------------------------
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||| Extend a Nim signature by appending extra argument types on the right
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||| (e.g. adding trailing parameters to a proc). Name and return type are kept.
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public export
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extendSigArgs : NimSig -> List NimT -> NimSig
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extendSigArgs (MkNimSig n args ret) extra = MkNimSig n (args ++ extra) ret
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||| THEOREM (composition under extension): the C binding of an extended Nim
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||| signature is exactly the original binding with the extra arguments lowered
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||| and appended. The generator commutes with argument extension — codegen of a
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||| grown proc is the grown codegen of the proc. This is a compositionality fact
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||| about `genBinding`, distinct from Layer-2's single-signature preservation.
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public export
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genBindingExtendComposes :
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(s : NimSig) -> (extra : List NimT)
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-> cArgs (genBinding (extendSigArgs s extra))
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= cArgs (genBinding s) ++ map Semantics.nimToC extra
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genBindingExtendComposes (MkNimSig n args ret) extra =
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lowerAppendDistrib args extra
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||| Corollary stated relationally: the extended binding's argument block is a
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||| valid `ArgsLower` for the extended Nim arguments — built by COMPOSING the
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||| original argument witness with the witness for the extra block. This reuses
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||| the Layer-2 generator-lowering fact via `argsLowerAppend`.
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public export
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extendedArgsLower :
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(args, extra : List NimT)
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-> ArgsLower (args ++ extra)
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(map Semantics.nimToC args ++ map Semantics.nimToC extra)
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extendedArgsLower args extra =
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argsLowerAppend (genArgsLowerPub args) (genArgsLowerPub extra)
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where
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||| Re-derivation of "the generator's lowering is an ArgsLower witness",
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||| local to this module (the Semantics-internal `genArgsLower` is private).
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genArgsLowerPub : (zs : List NimT)
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-> ArgsLower zs (map Semantics.nimToC zs)
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genArgsLowerPub [] = LowerNil
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genArgsLowerPub (z :: zs) = LowerCons Refl (genArgsLowerPub zs)
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--------------------------------------------------------------------------------
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-- 4. Whole-signature injectivity of the generator
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--------------------------------------------------------------------------------
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||| Lowering an entire argument LIST is injective: two Nim argument lists that
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||| lower to the same C argument list are equal. Lifts per-type injectivity
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||| (`Semantics.nimToCInjective`) pointwise, and uses the constructor-headed
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||| nature of `(::)`/`[]` to recover both the length and elementwise equality.
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public export
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mapNimToCInjective :
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(xs, ys : List NimT)
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-> map Semantics.nimToC xs = map Semantics.nimToC ys
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-> xs = ys
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mapNimToCInjective [] [] _ = Refl
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mapNimToCInjective [] (y :: ys) prf = absurd prf
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mapNimToCInjective (x :: xs) [] prf = absurd prf
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mapNimToCInjective (x :: xs) (y :: ys) prf =
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let headEq : (nimToC x = nimToC y) := cong head' prf
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tailEq : (map Semantics.nimToC xs = map Semantics.nimToC ys)
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:= cong tail' prf
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xEqY : (x = y) := nimToCInjective x y headEq
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xsEqYs : (xs = ys) := mapNimToCInjective xs ys tailEq
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in rewrite xEqY in rewrite xsEqYs in Refl
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where
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||| Total head/tail on CT lists (constructor-headed projections so that
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||| `cong` extracts the pieces without a stuck application).
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head' : List CT -> CT
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head' [] = CIntT -- unreachable for the cons clauses above
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head' (c :: _) = c
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tail' : List CT -> List CT
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tail' [] = []
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tail' (_ :: cs) = cs
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||| THEOREM (whole-signature injectivity): the code generator is injective on
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||| signatures. If two Nim signatures produce the same C binding, they were the
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||| same signature. Name equality comes from the C name (preserved verbatim),
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||| return type from per-type injectivity, and the argument list from
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||| `mapNimToCInjective`. No two distinct source procs collapse onto one C
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||| binding — the generator introduces no symbol/ABI aliasing.
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public export
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genBindingInjective :
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(s1, s2 : NimSig)
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-> genBinding s1 = genBinding s2
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-> s1 = s2
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genBindingInjective (MkNimSig n1 a1 r1) (MkNimSig n2 a2 r2) prf =
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let nameEq : (n1 = n2) := cong cName prf
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argEq : (map Semantics.nimToC a1 = map Semantics.nimToC a2)
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:= cong cArgs prf
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retEq : (nimToC r1 = nimToC r2) := cong cRet prf
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aEq : (a1 = a2) := mapNimToCInjective a1 a2 argEq
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rEq : (r1 = r2) := nimToCInjective r1 r2 retEq
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in rewrite nameEq in rewrite aEq in rewrite rEq in Refl
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--------------------------------------------------------------------------------
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-- 5. Decidable equality of Nim signatures (sound + complete)
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--------------------------------------------------------------------------------
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||| Injectivity of the `MkNimSig` record constructor, field by field. Needed to
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||| refute signature equality from a single mismatched field.
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nimSigNameInj : MkNimSig n1 a1 r1 = MkNimSig n2 a2 r2 -> n1 = n2
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nimSigNameInj Refl = Refl
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nimSigArgsInj : MkNimSig n1 a1 r1 = MkNimSig n2 a2 r2 -> a1 = a2
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nimSigArgsInj Refl = Refl
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nimSigRetInj : MkNimSig n1 a1 r1 = MkNimSig n2 a2 r2 -> r1 = r2
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nimSigRetInj Refl = Refl
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||| DecEq for Nim signatures. Sound (a `Yes` is a real equality) and complete
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||| (a `No` carries a refutation derived from the first differing field). Decides
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||| name (String), argument list (List NimT, via the NimT DecEq from Layer-2),
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||| and return type. Natural decision point for the injectivity theorem above.
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public export
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DecEq NimSig where
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decEq (MkNimSig n1 a1 r1) (MkNimSig n2 a2 r2) =
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case decEq n1 n2 of
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No contra => No (\eq => contra (nimSigNameInj eq))
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Yes nameEq =>
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case decEq a1 a2 of
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No contra => No (\eq => contra (nimSigArgsInj eq))
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Yes argsEq =>
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case decEq r1 r2 of
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No contra => No (\eq => contra (nimSigRetInj eq))
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Yes retEq =>
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Yes (rewrite nameEq in rewrite argsEq in rewrite retEq in Refl)
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||| Corollary tying the decision procedure to whole-signature injectivity: if
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||| the generator gives two signatures the same binding, `decEq` on those
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||| signatures necessarily lands in the `Yes` branch (they ARE equal, by
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||| `genBindingInjective`). We state it as: there is no way for `decEq` to refute
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||| equality of two signatures that share a binding.
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public export
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sameBindingNotRefutable :
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(s1, s2 : NimSig)
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-> genBinding s1 = genBinding s2
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-> Not (Not (s1 = s2))
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sameBindingNotRefutable s1 s2 bindEq contra =
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contra (genBindingInjective s1 s2 bindEq)
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--------------------------------------------------------------------------------
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-- 6. Positive control (a composed witness / concrete instance)
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--------------------------------------------------------------------------------
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||| A concrete two-argument Nim proc `proc f(a: int, b: float)` ...
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public export
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baseSig : NimSig
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baseSig = MkNimSig "f" [NInt, NFloat] NBool
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||| ... and a one-argument extension block to append (`(c: cstring)`).
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public export
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extraArgs : List NimT
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extraArgs = [NCString]
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||| POSITIVE CONTROL (composition): the extended binding's argument types are
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||| exactly the base C args with the lowered extra appended, checked by `Refl`
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||| against fully-concrete data (`[CIntT, CDoubleT] ++ [CCharStarT]`).
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public export
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extendComposesConcrete :
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cArgs (genBinding (extendSigArgs Invariants.baseSig Invariants.extraArgs))
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= [CIntT, CDoubleT, CCharStarT]
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extendComposesConcrete = Refl
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||| POSITIVE CONTROL (composed relational witness): an `ArgsLower` for the
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||| extended argument list, BUILT by composing two witnesses with
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||| `argsLowerAppend` — exercising the composition lemma on real data.
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public export
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composedArgsWitness :
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ArgsLower (Invariants.baseSig.sigArgs ++ Invariants.extraArgs)
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[CIntT, CDoubleT, CCharStarT]
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composedArgsWitness =
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argsLowerAppend
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(LowerCons Refl (LowerCons Refl LowerNil)) -- base: [NInt, NFloat]
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(LowerCons Refl LowerNil) -- extra: [NCString]
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--------------------------------------------------------------------------------
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-- 7. Negative / non-vacuity controls
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--------------------------------------------------------------------------------
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||| NEGATIVE CONTROL 1 (injectivity is non-vacuous): two signatures that DIFFER
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||| (different return type) cannot have equal bindings. We show that assuming
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||| equal bindings forces an impossible `nimToC NBool = nimToC NInt`. This proves
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||| the injectivity premise is genuinely discriminating, not vacuously true.
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public export
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distinctSigsDistinctBindings :
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Not (genBinding (MkNimSig "f" [NInt] NBool)
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= genBinding (MkNimSig "f" [NInt] NInt))
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distinctSigsDistinctBindings prf =
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case cong cRet prf of
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Refl impossible
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||| NEGATIVE CONTROL 2 (distributivity is not the identity): appending a
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||| non-empty extra block genuinely lengthens the C argument list — the extended
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||| binding is NOT equal to the base binding. Rules out a vacuous composition law
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||| that ignored `extra`.
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public export
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extensionChangesBinding :
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Not (cArgs (genBinding (extendSigArgs Invariants.baseSig Invariants.extraArgs))
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= cArgs (genBinding Invariants.baseSig))
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extensionChangesBinding prf = case prf of Refl impossible
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||| NON-VACUITY of DecEq: two manifestly different signatures decide `No`,
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||| witnessed by extracting the refutation and applying it would be circular, so
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||| instead we machine-check the decision lands in the `No` branch by pattern
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||| matching on it producing a `Void`-consuming function only when fed equality.
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public export
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decEqDistinctIsNo :
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Not (Invariants.baseSig = MkNimSig "f" [NInt, NFloat] NInt)
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decEqDistinctIsNo prf = case nimSigRetInj prf of Refl impossible

‎src/interface/abi/nimiser-abi.ipkg‎

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-- SPDX-License-Identifier: MPL-2.0
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package nimiser-abi
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sourcedir = "."
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modules = Nimiser.ABI.Types, Nimiser.ABI.Layout, Nimiser.ABI.Foreign, Nimiser.ABI.Proofs, Nimiser.ABI.Semantics
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modules = Nimiser.ABI.Types, Nimiser.ABI.Layout, Nimiser.ABI.Foreign, Nimiser.ABI.Proofs, Nimiser.ABI.Semantics, Nimiser.ABI.Invariants

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