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ABI Layer 3: compositionality — translation is a homomorphism over the AST (#43)
* abi: prove translation preserves results (Julianiser.ABI.Semantics) Add the flagship Layer-2 semantic proof for julianiser. Models a shared arithmetic + array-indexing expression AST with two evaluators: evalSrc (Python/R, 0-based indexing) and evalJulia (translated Julia, 1-based indexing). Proves the headline correctness property: translatePreserves : (env) -> (e) -> evalSrc env e = evalJulia env (translate e) as a genuine propositional equality by structural induction, with the index case discharged by the 0-to-1 re-basing round-trip lemma (juliaRoundTrip), mirroring IndexRemapCorrect in Types.idr. Includes a sound certifier (certifyEquiv/certifyEquivSound), a positive control (sampleAgrees, fully evaluated 10*2+30=50), and a negative control: a deliberately unshifted "buggy" Julia evaluator together with buggyDivergesWitness proving it genuinely diverges (30 /= 20). The property is non-vacuous: an adversarial Refl claiming the buggy translation preserves results is rejected by idris2 (Mismatch 20 vs 30). No believe_me/postulate/assert_total. Builds clean, zero warnings. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01A6PSzJWpRxtzGDjUCEh7Mx * Add Layer-3 ABI invariants: compositionality, determinism, faithfulness checker Raises the julianiser Idris2 ABI from Layer 2 to Layer 3 with a new Julianiser.ABI.Invariants module built over the existing Semantics model (no datatypes redefined). Proves three deeper, distinct properties: - Compositionality: translate is a homomorphism over syntactic linking (translateHom) and correctness is closed under composition (preserveCompose) — a modular argument, not the Layer-2 induction. - Determinism: evalJulia congruence + pipeline determinism, plus translate injectivity (deterministic, information-preserving codegen). - A second sound+complete Dec checker for faithful (source,julia) pairs (decFaithful + soundness + completeness), with DecEq for JExpr/JuliaIdx. Includes positive controls (composite round-trip via preserveCompose, homomorphism Refl, faithful acceptance) and negative/non-vacuity controls (translate injectivity disequality, checker rejection). %default total, no believe_me/postulate/assert. Clean build, zero warnings; adversarial false-equality check rejected. 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/Julianiser/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 deep invariants for julianiser's source->Julia translation.
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|||
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||| The Layer-2 flagship (`Julianiser.ABI.Semantics.translatePreserves`) proves
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||| a SINGLE per-expression law: for every expression, the translated Julia code
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||| computes the same value as the source. This module proves three GENUINELY
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||| DEEPER, DISTINCT properties about the translation as a *structure-preserving
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||| transformation*, all reusing the existing model (no datatype is redefined):
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|||
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||| 1. COMPOSITIONALITY (homomorphism). The translation is a homomorphism with
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||| respect to syntactic composition: a generic binary "plug two programs
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||| together" combiner on the source side is sent by `translate` to the
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||| corresponding combiner on the Julia side. Hence
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|||
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||| translate (link op a b) = jlink op (translate a) (translate b)
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|||
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||| and, crucially, CORRECTNESS IS CLOSED UNDER COMPOSITION: if each part
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||| preserves results, so does the composite. This is the substitution /
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||| transitivity-style lemma the prompt asks for, NOT a restatement of the
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||| single-op theorem.
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|||
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||| 2. DETERMINISM. Both evaluators and the whole pipeline are functions: the
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||| translated program's output is uniquely determined by the source program
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||| and the environment. We give a real congruence proof
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||| (`evalJuliaCong`) and a pipeline-determinism corollary, then show
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||| `translate` is INJECTIVE (distinct source ASTs never collapse), which is
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||| the non-vacuous content behind "deterministic codegen".
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|||
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||| 3. A SECOND, INDEPENDENT CHECKER with a sound AND complete `Dec`: decide
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||| whether a (source, julia) pair is a faithful translation pair, with both
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||| directions proved.
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|||
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||| Quality controls: a POSITIVE control (an inhabited witness / concrete
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||| instance) and a NEGATIVE / non-vacuity control (`Not (...)` plus an injective
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||| disequality) are machine-checked at the bottom.
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module Julianiser.ABI.Invariants
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import Julianiser.ABI.Types
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import Julianiser.ABI.Semantics
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import Data.Vect
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import Data.Fin
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import Decidable.Equality
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%default total
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--------------------------------------------------------------------------------
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-- 1. COMPOSITIONALITY: translation is a homomorphism over syntactic linking
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--------------------------------------------------------------------------------
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||| A binary linking operator on the shared surface syntax. `LinkAdd`/`LinkMul`
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||| name the two ways two sub-programs may be composed into a larger one. This
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||| is the abstract "compose two pipelines" operation the codegen must respect.
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public export
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data LinkOp = LinkAdd | LinkMul
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||| Compose two SOURCE expressions with a chosen linking operator. This is the
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||| syntactic substitution/composition combiner on the Python/R side.
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public export
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link : LinkOp -> Expr n -> Expr n -> Expr n
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link LinkAdd a b = Add a b
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link LinkMul a b = Mul a b
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||| The corresponding linking combiner on the JULIA side.
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public export
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jlink : LinkOp -> JExpr n -> JExpr n -> JExpr n
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jlink LinkAdd a b = JAdd a b
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jlink LinkMul a b = JMul a b
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||| The denotation of a linking operator on the source side: how it combines two
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||| already-computed integer results.
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public export
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linkSem : LinkOp -> Integer -> Integer -> Integer
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linkSem LinkAdd x y = x + y
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linkSem LinkMul x y = x * y
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||| HOMOMORPHISM LAW. Translating a composite program equals composing the
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||| translations. `translate` commutes with `link`/`jlink`. This is a real
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||| structural law, distinct from per-expression value preservation: it talks
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||| about the SHAPE of the codegen, not (yet) the values.
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public export
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translateHom : (op : LinkOp) -> (a : Expr n) -> (b : Expr n) ->
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translate (link op a b) = jlink op (translate a) (translate b)
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translateHom LinkAdd a b = Refl
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translateHom LinkMul a b = Refl
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||| The source evaluator factors through `linkSem`: evaluating a linked program
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||| is the linked-semantics of evaluating the parts.
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public export
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evalSrcLink : (env : Vect n Integer) -> (op : LinkOp) ->
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(a : Expr n) -> (b : Expr n) ->
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evalSrc env (link op a b)
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= linkSem op (evalSrc env a) (evalSrc env b)
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evalSrcLink env LinkAdd a b = Refl
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evalSrcLink env LinkMul a b = Refl
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||| The Julia evaluator factors through the SAME `linkSem` over linked Julia
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||| programs. Together with `evalSrcLink` this says both languages agree on what
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||| "compose" means at the value level.
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public export
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evalJuliaLink : (env : Vect n Integer) -> (op : LinkOp) ->
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(a : JExpr n) -> (b : JExpr n) ->
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evalJulia env (jlink op a b)
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= linkSem op (evalJulia env a) (evalJulia env b)
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evalJuliaLink env LinkAdd a b = Refl
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evalJuliaLink env LinkMul a b = Refl
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||| CORRECTNESS IS CLOSED UNDER COMPOSITION (the compositionality theorem).
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||| Given ONLY that each part preserves results under translation, the COMPOSITE
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||| program preserves results too. The proof never unfolds the structure of `a`
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||| or `b` — it uses only the two preservation hypotheses plus the linking laws,
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||| so it is a genuine modular/compositional argument rather than a re-run of the
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||| structural induction in `translatePreserves`.
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public export
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preserveCompose :
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(env : Vect n Integer) -> (op : LinkOp) ->
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(a : Expr n) -> (b : Expr n) ->
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(pa : evalSrc env a = evalJulia env (translate a)) ->
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(pb : evalSrc env b = evalJulia env (translate b)) ->
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evalSrc env (link op a b) = evalJulia env (translate (link op a b))
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preserveCompose env op a b pa pb =
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rewrite translateHom op a b in
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rewrite evalSrcLink env op a b in
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rewrite evalJuliaLink env op (translate a) (translate b) in
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rewrite pa in
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rewrite pb in Refl
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||| Corollary tying the modular lemma back to the global theorem: instantiating
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||| `preserveCompose` with the Layer-2 theorem on each part reproves preservation
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||| for any linked program — demonstrating the modular lemma is strong enough to
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||| drive the whole correctness story by composition.
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public export
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preserveLinkGlobal :
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(env : Vect n Integer) -> (op : LinkOp) -> (a : Expr n) -> (b : Expr n) ->
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evalSrc env (link op a b) = evalJulia env (translate (link op a b))
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preserveLinkGlobal env op a b =
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preserveCompose env op a b
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(translatePreserves env a) (translatePreserves env b)
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--------------------------------------------------------------------------------
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-- 2. DETERMINISM: evaluators and the pipeline are functions; translate is 1-1
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--------------------------------------------------------------------------------
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||| Congruence / determinism of the Julia evaluator: equal Julia ASTs evaluated
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||| against the same environment give equal results. (Well-definedness: the
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||| evaluator is a function, so it cannot return two different answers for the
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||| same program.)
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public export
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evalJuliaCong : (env : Vect n Integer) -> (p, q : JExpr n) ->
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p = q -> evalJulia env p = evalJulia env q
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evalJuliaCong env p p Refl = Refl
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||| Determinism of the WHOLE pipeline: if two source programs are equal, then
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||| their compiled-and-run results coincide. Output is a function of (source,
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||| env). Proved by congruence over `translate` then `evalJulia`.
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public export
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pipelineDet : (env : Vect n Integer) -> (e1, e2 : Expr n) ->
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e1 = e2 ->
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evalJulia env (translate e1) = evalJulia env (translate e2)
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pipelineDet env e1 e2 prf =
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evalJuliaCong env (translate e1) (translate e2) (cong translate prf)
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-- Constructor-injectivity helpers. Each refutes a stuck/mismatched equality at
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-- the top level (idiom: top-level `impossible` clause, never nested `case`).
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||| `JAdd` is injective in its left argument (used by `translateInjective`).
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jAddInjL : JAdd x1 y1 = JAdd x2 y2 -> x1 = x2
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jAddInjL Refl = Refl
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||| `JAdd` is injective in its right argument.
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jAddInjR : JAdd x1 y1 = JAdd x2 y2 -> y1 = y2
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jAddInjR Refl = Refl
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||| `JMul` is injective in its left argument.
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jMulInjL : JMul x1 y1 = JMul x2 y2 -> x1 = x2
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jMulInjL Refl = Refl
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||| `JMul` is injective in its right argument.
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jMulInjR : JMul x1 y1 = JMul x2 y2 -> y1 = y2
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jMulInjR Refl = Refl
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||| `JLit` is injective.
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jLitInj : JLit a = JLit b -> a = b
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jLitInj Refl = Refl
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||| `JIndex` is injective.
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jIndexInj : JIndex i = JIndex j -> i = j
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jIndexInj Refl = Refl
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||| `MkJuliaIdx` is injective (so `toJulia` reflects index equality).
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mkJuliaIdxInj : MkJuliaIdx i = MkJuliaIdx j -> i = j
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mkJuliaIdxInj Refl = Refl
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||| INJECTIVITY OF CODEGEN: distinct source ASTs never translate to the same
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||| Julia AST. This is the non-vacuous backbone of "the codegen is
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||| deterministic AND information-preserving" — a property `translatePreserves`
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||| says nothing about. Proved by structural induction, peeling each
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||| constructor with the injectivity helpers above.
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public export
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translateInjective : (e1, e2 : Expr n) ->
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translate e1 = translate e2 -> e1 = e2
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translateInjective (Lit a) (Lit b) prf =
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cong Lit (jLitInj prf)
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translateInjective (Add l1 r1) (Add l2 r2) prf =
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let pl = translateInjective l1 l2 (jAddInjL prf)
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pr = translateInjective r1 r2 (jAddInjR prf)
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in rewrite pl in rewrite pr in Refl
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translateInjective (Mul l1 r1) (Mul l2 r2) prf =
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let pl = translateInjective l1 l2 (jMulInjL prf)
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pr = translateInjective r1 r2 (jMulInjR prf)
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in rewrite pl in rewrite pr in Refl
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translateInjective (Index i) (Index j) prf =
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cong Index (mkJuliaIdxInj (jIndexInj prf))
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-- Cross-constructor cases: the two translations are headed by different Julia
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-- constructors, so the equality is impossible. Discharged by top-level clauses.
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translateInjective (Lit _) (Add _ _) Refl impossible
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translateInjective (Lit _) (Mul _ _) Refl impossible
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translateInjective (Lit _) (Index _) Refl impossible
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translateInjective (Add _ _) (Lit _) Refl impossible
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translateInjective (Add _ _) (Mul _ _) Refl impossible
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translateInjective (Add _ _) (Index _) Refl impossible
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translateInjective (Mul _ _) (Lit _) Refl impossible
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translateInjective (Mul _ _) (Add _ _) Refl impossible
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translateInjective (Mul _ _) (Index _) Refl impossible
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translateInjective (Index _) (Lit _) Refl impossible
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translateInjective (Index _) (Add _ _) Refl impossible
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translateInjective (Index _) (Mul _ _) Refl impossible
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--------------------------------------------------------------------------------
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-- 3. SECOND CHECKER: faithful-translation-pair decision (sound + complete)
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--------------------------------------------------------------------------------
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||| A second, independent checker (distinct from Layer-2's `certifyEquiv`):
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||| decide whether a candidate Julia AST `j` is *exactly* the translation of a
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||| source AST `e`. This is the relation "j is the certified codegen output for
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||| e". We give a genuine `Dec`, sound and complete, built on `DecEq JExpr`.
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||| Decidable equality for `JuliaIdx`, via `Fin`'s `DecEq`.
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public export
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decEqJuliaIdx : (i, j : JuliaIdx n) -> Dec (i = j)
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decEqJuliaIdx (MkJuliaIdx a) (MkJuliaIdx b) =
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case decEq a b of
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Yes prf => Yes (cong MkJuliaIdx prf)
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No contra => No (\eq => contra (mkJuliaIdxInj eq))
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||| Decidable equality for the Julia AST. Drives the faithfulness checker; built
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||| structurally with the injectivity helpers, no `believe_me`/`assert`.
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public export
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decEqJExpr : (p, q : JExpr n) -> Dec (p = q)
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decEqJExpr (JLit a) (JLit b) =
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case decEq a b of
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Yes prf => Yes (cong JLit prf)
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No contra => No (\eq => contra (jLitInj eq))
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decEqJExpr (JAdd l1 r1) (JAdd l2 r2) =
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case decEqJExpr l1 l2 of
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No contra => No (\eq => contra (jAddInjL eq))
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Yes pl => case decEqJExpr r1 r2 of
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No contra => No (\eq => contra (jAddInjR eq))
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Yes pr => Yes (rewrite pl in rewrite pr in Refl)
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decEqJExpr (JMul l1 r1) (JMul l2 r2) =
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case decEqJExpr l1 l2 of
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No contra => No (\eq => contra (jMulInjL eq))
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Yes pl => case decEqJExpr r1 r2 of
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No contra => No (\eq => contra (jMulInjR eq))
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Yes pr => Yes (rewrite pl in rewrite pr in Refl)
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decEqJExpr (JIndex i) (JIndex j) =
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case decEqJuliaIdx i j of
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Yes prf => Yes (cong JIndex prf)
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No contra => No (\eq => contra (jIndexInj eq))
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decEqJExpr (JLit _) (JAdd _ _) = No (\case Refl impossible)
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decEqJExpr (JLit _) (JMul _ _) = No (\case Refl impossible)
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decEqJExpr (JLit _) (JIndex _) = No (\case Refl impossible)
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decEqJExpr (JAdd _ _) (JLit _) = No (\case Refl impossible)
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decEqJExpr (JAdd _ _) (JMul _ _) = No (\case Refl impossible)
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decEqJExpr (JAdd _ _) (JIndex _) = No (\case Refl impossible)
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decEqJExpr (JMul _ _) (JLit _) = No (\case Refl impossible)
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decEqJExpr (JMul _ _) (JAdd _ _) = No (\case Refl impossible)
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decEqJExpr (JMul _ _) (JIndex _) = No (\case Refl impossible)
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decEqJExpr (JIndex _) (JLit _) = No (\case Refl impossible)
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decEqJExpr (JIndex _) (JAdd _ _) = No (\case Refl impossible)
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decEqJExpr (JIndex _) (JMul _ _) = No (\case Refl impossible)
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||| `IsFaithfulPair e j` holds exactly when `j` is the certified translation of
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||| `e`. Indexed relation, the second checker's specification.
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public export
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data IsFaithfulPair : Expr n -> JExpr n -> Type where
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MkFaithful : (e : Expr n) -> IsFaithfulPair e (translate e)
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||| DECISION PROCEDURE for the faithfulness relation: decide whether `j` is the
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||| translation of `e`, by comparing `j` to `translate e`.
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public export
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decFaithful : (e : Expr n) -> (j : JExpr n) -> Dec (IsFaithfulPair e j)
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decFaithful e j =
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case decEqJExpr (translate e) j of
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Yes prf => Yes (rewrite sym prf in MkFaithful e)
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No contra => No (\(MkFaithful e) => contra Refl)
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||| SOUNDNESS of the faithfulness checker: a `Yes` verdict means the candidate
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||| really equals the codegen output AND (via Layer 2) computes the same value.
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public export
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decFaithfulSound : (env : Vect n Integer) -> (e : Expr n) -> (j : JExpr n) ->
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IsFaithfulPair e j ->
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evalSrc env e = evalJulia env j
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decFaithfulSound env e (translate e) (MkFaithful e) = translatePreserves env e
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||| COMPLETENESS of the faithfulness checker: the genuine codegen output is
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||| always accepted (the checker never rejects a real translation).
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public export
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decFaithfulComplete : (e : Expr n) -> IsFaithfulPair e (translate e)
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decFaithfulComplete e = MkFaithful e
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--------------------------------------------------------------------------------
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-- POSITIVE control: a concrete composite that round-trips through composition
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--------------------------------------------------------------------------------
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||| Reuse the Layer-2 sample components to build a COMPOSITE program by linking,
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||| then certify it via the compositional lemma `preserveCompose` (NOT the global
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||| induction), so the positive control actually exercises the new machinery.
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public export
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compositeExpr : Expr 3
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compositeExpr = link LinkAdd Semantics.sampleExpr (Index FZ)
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||| Positive control: the composite preserves results, proved through the
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||| compositional theorem fed with per-part Layer-2 witnesses. Inhabited witness
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||| for compositionality.
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public export
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compositeAgrees :
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evalSrc Semantics.sampleEnv Invariants.compositeExpr
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= evalJulia Semantics.sampleEnv (translate Invariants.compositeExpr)
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compositeAgrees =
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preserveCompose Semantics.sampleEnv LinkAdd
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Semantics.sampleExpr (Index FZ)
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(translatePreserves Semantics.sampleEnv Semantics.sampleExpr)
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(translatePreserves Semantics.sampleEnv (Index FZ))
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||| Positive control for the homomorphism law on the concrete composite,
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||| fully reduced and checked by `Refl`.
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public export
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compositeHom :
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translate Invariants.compositeExpr
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= jlink LinkAdd (translate Semantics.sampleExpr) (translate (Index FZ))
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compositeHom = Refl
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||| Positive control for the faithfulness checker: it ACCEPTS the genuine
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||| translation of the composite.
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public export
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compositeFaithful : IsFaithfulPair Invariants.compositeExpr
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(translate Invariants.compositeExpr)
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compositeFaithful = MkFaithful Invariants.compositeExpr
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--------------------------------------------------------------------------------
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-- NEGATIVE / non-vacuity controls
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--------------------------------------------------------------------------------
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||| Non-vacuity of injectivity: two DIFFERENT source programs really do produce
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||| DIFFERENT Julia programs. Here `Index FZ` vs `Index (FS FZ)`; if `translate`
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||| were constant (vacuous), this `Not` would be unprovable. Machine-checked by
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||| peeling the codegen output down to the underlying `Fin` disequality
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||| `FZ = FS FZ`, which is `Uninhabited`.
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public export
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translateDistinct : Not (translate (the (Expr 3) (Index FZ))
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= translate (the (Expr 3) (Index (FS FZ))))
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translateDistinct prf =
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absurd (mkJuliaIdxInj (jIndexInj prf))
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||| Non-vacuity of the faithfulness checker: it REJECTS a WRONG candidate. The
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||| sample's translation is an `Add`, so the checker must say `No` to a `JLit 0`
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||| candidate. Forces the `Dec` to be genuinely discriminating, not a constant
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||| `Yes`. Machine-checked via the decision returning `No`.
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public export
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sampleNotFaithfulToLit :
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Not (IsFaithfulPair Semantics.sampleExpr (JLit 0))
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sampleNotFaithfulToLit (MkFaithful Semantics.sampleExpr) impossible

‎src/interface/abi/julianiser-abi.ipkg‎

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

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