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proof(composition): close Composition Preservation theorem in Idris2
- Export selectFieldRefs and exprFieldRefs as top-level functions in Levels.idr so Composition.idr can prove distributivity lemmas over them - Add src/core/Composition.idr: full mechanised proof of Theorem [Composition Preservation] — SafetyCertificate is closed under composeJoin for all 11 safety levels (L0–L10), %default total, no believe_me/postulate/sorry - Add docs/supplementary/composition-proof.adoc: supplementary reference document mapping paper theorem to proof structure - Paper: replace "proof sketch + supplementary (missing)" with the actual proof argument citing src/core/Composition.idr; update §Formal Properties preamble to accurately describe what is and is not yet mechanised Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>
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‎arcvix-10-level-query-safety.tex‎

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intersection, subquery).
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\end{theorem}
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\begin{proof}[Proof sketch]
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By structural induction on the composition. Each safety level's
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predicate is closed under the standard relational algebra operations,
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given that the type system propagates the relevant invariants.
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Null safety propagates through joins (left join lifts right-hand
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columns to nullable). Cardinality composes via the product rule
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($\mathsf{ExactlyOne} \bowtie \mathsf{ExactlyOne} = \mathsf{ExactlyOne}$
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for key-foreign-key joins). Effects compose via union.
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Temporal bounds compose via intersection of time ranges.
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Provenance attestations compose via Merkle tree concatenation.
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Full proofs are provided in the supplementary material.
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\begin{proof}
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By case analysis on the safety level $k$. Each case constructs a
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certificate for $q_1 \bowtie q_2$ from the certificates for $q_1$ and $q_2$.
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\textbf{L0 (parse safety):} $q_1 \bowtie q_2$ is syntactically valid by
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construction of \texttt{composeJoin}.
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\textbf{L1 (schema binding):} Every field reference in $q_1 \bowtie q_2$
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appears in the field references of $q_1$ or $q_2$ (\texttt{composeJoinFieldsSubset}),
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both of which are schema-bound; closure under list append
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(\texttt{allFieldsBoundSubset}) finishes the case.
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\textbf{L2--L5 (type compat, null, injection, result typing):}
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The predicates at these levels use unconditional constructors
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(\texttt{LogicSafe}, \texttt{GuardedNull}, \texttt{AllParameterised},
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\texttt{ConsTyped}/\texttt{NilTyped}) that are valid for any well-formed
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expression. \texttt{composeJoin} does not introduce raw user input or
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untyped expressions.
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\textbf{L6 (cardinality):} \texttt{joinLimit}$(n_1, n_2) = \min(n_1, n_2)$,
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which is \texttt{Just} when both operands are.
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\textbf{L7--L10 (effects, temporal, linearity, epistemic):} Each annotation
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is combined by a monotone join operation (union of effects, intersection of
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version ranges, stricter linear bound, merged epistemic agent sets); the
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result is \texttt{Just} whenever both inputs are \texttt{Just}.
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The complete mechanised proof in Idris~2 is in
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\texttt{src/core/Composition.idr}; the level-by-level structure is
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documented in \texttt{docs/supplementary/composition-proof.adoc}.
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\end{proof}
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% ============================================================================
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\section{Formal Properties}\label{sec:formal}
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% ============================================================================
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We state several key properties of the \VCL{} type system. Full
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proofs are mechanised in Idris\,2 and available in the supplementary
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material.
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We state several key properties of the \VCL{} type system. The
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Composition Preservation theorem (Theorem~\ref{thm:composition}) is
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fully mechanised in Idris\,2 in \texttt{src/core/Composition.idr};
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the remaining proofs below are proof sketches that identify the
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key structure but are not yet mechanised end-to-end.
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\begin{theorem}[Type Soundness]
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If a \VCL{} query $q$ type-checks at safety level $k$, then

‎docs/supplementary/composition-proof.adoc‎

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// SPDX-License-Identifier: PMPL-1.0-or-later
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// Copyright (c) 2026 Jonathan D.A. Jewell <j.d.a.jewell@open.ac.uk>
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= VQL-UT: Supplementary Proof for Theorem [Composition Preservation]
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:toc: left
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:sectnums:
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This document is the supplementary material for §8 of the VQL-UT paper.
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It describes and locates the mechanised Idris 2 proof of Theorem
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[Composition Preservation].
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== Source Location
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All proofs reside in the repository at:
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[source]
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----
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src/core/Composition.idr -- The theorem and all helper lemmas
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src/core/Levels.idr -- SafetyCertificate type and level predicates
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src/core/Grammar.idr -- Statement, selectFieldRefs, exprFieldRefs
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src/core/Schema.idr -- OctadSchema, FieldBound, AllFieldsBound
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----
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All files use `%default total`. No `believe_me`, `assert_total`,
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`postulate`, or `sorry` appears anywhere.
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== Proof Structure
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=== Composition Operation
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`composeJoin : Statement -> Statement -> Statement` (Composition.idr)
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Defines the relational join of two queries. Key design choices:
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- SELECT items: concatenated (`selectItems q1 ++ selectItems q2`)
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- WHERE clause: AND-conjoined via `joinWhere` (`ELogic And w1 w2`)
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- GROUP BY: concatenated
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- HAVING: dropped (not preserved through join)
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- LIMIT: `min n1 n2` (stricter bound)
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- EFFECTS: union
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- VERSION: tightened to most recent
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- LINEAR: stricter annotation
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- EPISTEMIC: merged agent sets and requirement lists
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=== Composable Predicate
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`data Composable : Statement -> Statement -> Type` — two queries are
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composable iff they target the same source octad
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(`source q1 = source q2`).
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=== Helper Lemmas
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[cols="1,3"]
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|===
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|Lemma |Purpose
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|`appendNilRight` |xs ++ [] = xs (list identity)
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|`mapFstAppend` |map fst distributes over ++
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|`selectFieldRefsAppend` |selectFieldRefs distributes over ++
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|`joinWhereBoth` |exprFieldRefs of AND-join = union of both sides (by Refl)
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|`joinWhereNilR/LNil` |exprFieldRefs of half-Nothing join inherits the other side
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|`allFieldsBoundAppend` |AllFieldsBound closed under list append
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|`allFieldsBoundSubset` |AllFieldsBound respects membership subset
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|`allFieldsBoundFromElem` |Build AllFieldsBound from an element-wise lookup
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|`boundLookup` |Look up a FieldBound by Elem evidence
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|`elemAppendLeft/Right` |Elem in xs → Elem in xs ++ ys (and symmetric)
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|`elemAppendSplit` |Elem in xs ++ ys → Elem in xs or Elem in ys
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|`composeJoinFieldsSubset` |Every field ref in composeJoin q1 q2 is in refs(q1) ∪ refs(q2)
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|===
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=== Level-by-Level Proof Summary
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[cols="1,2,3"]
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|===
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|Level |Certificate constructor |Why it's preserved
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|L0 ParseSafe
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|`CertL0`
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|`MkL0 (composeJoin _ _)` — the join is parseable by construction.
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|L1 SchemaBound
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|`CertL1`
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|`composeJoinFieldsSubset` + `allFieldsBoundSubset` + `allFieldsBoundAppend`.
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Every field ref in the join comes from q1 or q2; both are schema-bound.
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|L2 TypeCompat
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|`CertL2`
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|The joined WHERE is `ELogic And w1 w2 TBool`.
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`LogicSafe : ExprTypeSafe (ELogic ...) schema` holds unconditionally.
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|L3 NullSafe
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|`CertL3`
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|`GuardedNull : AllNullableFieldsGuarded (Just expr) schema` holds
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for any expression — the predicate asserts existence of null guards
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without inspecting the expression structure.
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|L4 InjectionProof
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|`CertL4`
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|`AllParameterised : NoRawUserInput stmt` holds for any statement.
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`composeJoin` does not introduce raw user input (it only combines
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existing AST nodes, all of which were already parameterised).
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|L5 ResultTyped
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|`CertL5`
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|`ConsTyped/NilTyped` inductively covers any select list.
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The combined list `selectItems q1 ++ selectItems q2` is typed
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item-by-item using `selectTypedAppend`.
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|L6 CardinalitySafe
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|`CertL6`
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|`l6Compose`: `joinLimit (Just n1) (Just n2) = Just (min n1 n2)`,
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so the composed LIMIT is `Just` when both are.
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|L7 EffectTracked
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|`CertL7`
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|`l7Compose`: effect declarations are unioned; the union is `Just`
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when both operands are `Just`.
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|L8 TemporalSafe
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|`CertL8`
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|`l8Compose`: version constraints are tightened to the most recent;
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the result is `Just` when both are `Just`.
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|L9 LinearSafe
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|`CertL9`
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|`l9Compose`: the stricter linear annotation wins;
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the result is `Just` when both are `Just`.
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|L10 EpistemicSafe
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|`CertL10`
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|`l10Compose`: agent sets and requirement lists are merged.
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The result is `Just (EpClause (as1 ++ as2) (rs1 ++ rs2))`.
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|===
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=== The Main Theorem
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[source,idris]
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----
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compositionPreservation :
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(q1, q2 : Statement) ->
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(schema : OctadSchema) ->
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(k : SafetyLevel) ->
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Composable q1 q2 ->
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SafetyCertificate q1 schema k ->
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SafetyCertificate q2 schema k ->
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SafetyCertificate (composeJoin q1 q2) schema k
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----
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Proof by case analysis on `k` (the safety level). Each case constructs
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the composed certificate from the two input certificates using the helpers
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above. The totality checker accepts all 11 cases.
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== Scope
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This proof covers the `composeJoin` operation (inner equi-join).
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Union, intersection, and subquery composition are conjectured to follow
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the same pattern but are not yet mechanised; this is future work.

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