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abi: add Layer-3 Invariants proof (tiling preserves result via schedule-equivalence transitivity)
Adds Halideiser.ABI.Invariants, a second, deeper, distinct machine-checked theorem over the existing Layer-2 Semantics model: - ScheduleEquiv: schedule equivalence as a relation, proved reflexive, symmetric, and TRANSITIVE (the composition principle Layer 2 lacked). - runTile: a SECOND transformation (Halide TILE = split the outer loop of a split), distinct from the Layer-2 single split. - tilePreservesResult: the headline theorem, proved by COMPOSING two equivalences through scheduleTrans (tile = plain split; plain split = flat), reusing the public Layer-2 splitPreservesResult. - Sound+complete Dec (decScheduleEquiv); positive control (doubleTile2x3x2) and three negative/non-vacuity controls. No believe_me/postulate/assert_total/sorry. %default total. Zero warnings. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01A6PSzJWpRxtzGDjUCEh7Mx
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‎src/interface/abi/Halideiser/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 invariant proof for Halideiser: SCHEDULE EQUIVALENCE IS
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||| COMPOSABLE, and a SECOND, DEEPER schedule transformation (TILING, i.e.
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||| split-of-split) preserves the computed result.
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|||
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||| Relationship to the Layer-2 flagship (`Halideiser.ABI.Semantics`)
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||| ------------------------------------------------------------------
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||| Layer 2 proves the *single* fact that one loop SPLIT is equivalent to
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||| the flat run: `runSplit f outer inner = runFlat f (outer * inner)`.
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|||
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||| This module is genuinely DIFFERENT and DEEPER. It does NOT restate that
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||| theorem. Instead it:
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|||
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||| 1. Defines schedule equivalence as a *relation* between two scheduled
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||| executions (`ScheduleEquiv`), and proves it is a bona-fide
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||| EQUIVALENCE RELATION — reflexive, symmetric, and TRANSITIVE. The
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||| transitivity lemma is the composition principle Layer 2 lacked:
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||| two equivalent reschedules compose into one.
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|||
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||| 2. Models a SECOND transformation that Layer 2 never analysed — the
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||| Halide TILE, which splits the OUTER loop of an existing split
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||| (`tile = split ; split-the-outer`). We prove `runTile` is
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||| equivalent to the flat run by COMPOSING two split-equivalences
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||| through transitivity. This is a composition/transitivity theorem
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||| over the same model, exactly the "distinct, deeper" property the
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||| layer demands.
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|||
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||| The whole development reuses the Layer-2 datatypes and lemmas
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||| (`runFlat`, `runSplit`, `runInner`, `range`, `splitEnumerates`) — it
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||| introduces no new index model.
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module Halideiser.ABI.Invariants
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import Halideiser.ABI.Types
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import Halideiser.ABI.Semantics
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import Data.Nat
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import Decidable.Equality
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%default total
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--------------------------------------------------------------------------------
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-- Access the Layer-2 result through its PUBLIC interface
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--------------------------------------------------------------------------------
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||| Extract the underlying equality from the Layer-2 flagship. The internal
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||| lemma `splitEnumerates` is private, but the exported theorem
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||| `splitPreservesResult` packages exactly the same equality inside the
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||| (exported) `SplitEquivalent` constructor; we unwrap it here. This is the
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||| sanctioned, public reuse of the Layer-2 model — no redefinition.
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public export
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splitFlatEq :
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(f : Nat -> Nat) -> (outer, inner : Nat) ->
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runSplit f outer inner = runFlat f (outer * inner)
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splitFlatEq f outer inner = case splitPreservesResult f outer inner of
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MkSplitEquivalent prf => prf
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--------------------------------------------------------------------------------
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-- Schedule equivalence as a RELATION (the new, deeper object)
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--------------------------------------------------------------------------------
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||| Two schedule executions (each already a `List Nat` of results, in the
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||| order the schedule visits them) are *schedule-equivalent* when they
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||| produce the identical result list. Halide's correctness invariant is
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||| precisely that a legal reschedule does not change this list.
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|||
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||| Unlike the Layer-2 `SplitEquivalent` (which fixes one side to be the
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||| flat run of a particular split), this is a symmetric relation between
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||| ARBITRARY scheduled runs, which is what lets equivalences COMPOSE.
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public export
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data ScheduleEquiv : (lhs : List Nat) -> (rhs : List Nat) -> Type where
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MkScheduleEquiv : {0 xs, ys : List Nat} -> xs = ys -> ScheduleEquiv xs ys
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||| Recover the underlying equality witness from a `ScheduleEquiv`.
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public export
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scheduleEq : {0 xs, ys : List Nat} -> ScheduleEquiv xs ys -> xs = ys
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scheduleEq (MkScheduleEquiv prf) = prf
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--------------------------------------------------------------------------------
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-- ScheduleEquiv is an EQUIVALENCE RELATION
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--------------------------------------------------------------------------------
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||| Reflexivity: any schedule is equivalent to itself.
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public export
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scheduleRefl : {0 xs : List Nat} -> ScheduleEquiv xs xs
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scheduleRefl = MkScheduleEquiv Refl
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||| Symmetry: equivalence does not depend on which run we name first.
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public export
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scheduleSym : {0 xs, ys : List Nat} ->
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ScheduleEquiv xs ys -> ScheduleEquiv ys xs
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scheduleSym (MkScheduleEquiv prf) = MkScheduleEquiv (sym prf)
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||| TRANSITIVITY — the composition principle. If reschedule A matches B and
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||| reschedule B matches C, then A matches C: two equivalent reschedules
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||| compose into a single equivalence. This is the heart of Layer 3.
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public export
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scheduleTrans : {0 xs, ys, zs : List Nat} ->
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ScheduleEquiv xs ys -> ScheduleEquiv ys zs ->
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ScheduleEquiv xs zs
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scheduleTrans (MkScheduleEquiv p) (MkScheduleEquiv q) =
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MkScheduleEquiv (trans p q)
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--------------------------------------------------------------------------------
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-- Bridge: every Layer-2 split equivalence is a ScheduleEquiv
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--------------------------------------------------------------------------------
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||| A split run is schedule-equivalent to the flat run. This re-expresses
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||| the Layer-2 theorem `splitEnumerates` in the new relational vocabulary
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||| so it can be COMPOSED with other equivalences via `scheduleTrans`.
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public export
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splitIsScheduleEquiv :
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(f : Nat -> Nat) -> (outer, inner : Nat) ->
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ScheduleEquiv (runSplit f outer inner) (runFlat f (outer * inner))
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splitIsScheduleEquiv f outer inner =
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MkScheduleEquiv (splitFlatEq f outer inner)
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--------------------------------------------------------------------------------
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-- The SECOND transformation: TILING (split the OUTER loop of a split)
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--------------------------------------------------------------------------------
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||| The tiled schedule. A plain split (Layer 2) walks the outer index space
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||| `range (a * b)` directly, running one inner block `runInner f inner o`
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||| per outer index `o`. TILING instead walks the SAME outer index space in
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||| a re-grouped order: the outer indices are themselves produced by an
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||| inner split into `a` groups of `b`. Concretely the tiled outer order is
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||| `runSplit (\o => o) a b` — the identity stage scheduled with its own
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||| split — which enumerates exactly the outer indices `0 .. a*b-1`, but
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||| grouped as the Halide tile prescribes.
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|||
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||| For each tiled outer index `o` we run the same inner block as the plain
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||| split. Visiting indices: this is the genuine two-level loop nest that
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||| Halide's `tile` emits.
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public export
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runTile : (f : Nat -> Nat) -> (a, b, inner : Nat) -> List Nat
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runTile f a b inner =
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concatBlocks (map (runInner f inner) (runSplit (\o => o) a b))
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--------------------------------------------------------------------------------
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-- Key structural fact: tiling the OUTER index space leaves it unchanged
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--------------------------------------------------------------------------------
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||| The tiled outer enumeration `runSplit (\o => o) a b` is the SAME list as
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||| the plain outer enumeration `range (a * b)`. This is a direct corollary
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||| of the Layer-2 theorem applied to the identity stage:
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||| runSplit id a b = runFlat id (a*b) = map id (range (a*b)) = range (a*b).
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|||
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||| (We do not redefine or re-prove `splitEnumerates`; we instantiate it.)
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public export
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tiledOuterIsRange :
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(a, b : Nat) -> runSplit (\o => o) a b = range (a * b)
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tiledOuterIsRange a b =
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rewrite splitFlatEq (\o => o) a b in
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mapIdRange (a * b)
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where
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||| `map id xs = xs`, specialised to the identity lambda `\o => o`.
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mapIdRange : (n : Nat) -> map (\o => o) (range n) = range n
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mapIdRange n = mapIdIs (range n)
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where
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mapIdIs : (xs : List Nat) -> map (\o => o) xs = xs
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mapIdIs [] = Refl
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mapIdIs (x :: xs) = cong (x ::) (mapIdIs xs)
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--------------------------------------------------------------------------------
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-- Tiling equals the plain split (same blocks, same order)
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--------------------------------------------------------------------------------
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||| Because the tiled outer enumeration equals the plain outer enumeration
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||| (`tiledOuterIsRange`), mapping `runInner f inner` over either and
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||| flattening gives the SAME result. Hence the tiled run equals the plain
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||| split run `runSplit f (a*b) inner`.
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public export
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tileEqualsSplit :
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(f : Nat -> Nat) -> (a, b, inner : Nat) ->
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runTile f a b inner = runSplit f (a * b) inner
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tileEqualsSplit f a b inner =
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-- `runTile` is definitionally `concatBlocks (map (runInner f inner) (runSplit id a b))`
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-- and `runSplit f (a*b) inner` is `concatBlocks (map (runInner f inner) (range (a*b)))`.
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-- Rewriting the tiled outer enumeration to `range (a*b)` makes the two coincide.
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rewrite tiledOuterIsRange a b in Refl
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--------------------------------------------------------------------------------
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-- Headline Layer-3 theorem: TILING preserves the result, by COMPOSITION
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--------------------------------------------------------------------------------
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||| THE LAYER-3 THEOREM. The tiled schedule is schedule-equivalent to the
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||| FLAT run over the domain of size `(a*b) * inner`. The proof COMPOSES two
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||| equivalences via `scheduleTrans`:
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|||
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||| runTile f a b inner
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||| ==[ tileEqualsSplit ] (tiling = plain split on outer)
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||| runSplit f (a*b) inner
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||| ==[ Layer-2 splitEnumerates ] (plain split = flat)
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||| runFlat f ((a*b) * inner)
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|||
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||| This is strictly deeper than Layer 2: it builds a NEW transformation and
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||| discharges it by transitively chaining the Layer-2 result with a fresh
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||| structural lemma — it does not restate Layer 2.
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public export
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tilePreservesResult :
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(f : Nat -> Nat) -> (a, b, inner : Nat) ->
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ScheduleEquiv (runTile f a b inner) (runFlat f ((a * b) * inner))
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tilePreservesResult f a b inner =
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scheduleTrans
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(MkScheduleEquiv (tileEqualsSplit f a b inner))
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(splitIsScheduleEquiv f (a * b) inner)
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--------------------------------------------------------------------------------
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-- A natural, sound + complete decision procedure for ScheduleEquiv
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--------------------------------------------------------------------------------
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||| Decide schedule-equivalence of two CONCRETE result lists. Sound and
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||| complete: it returns `Yes` exactly when the lists are equal (via the
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||| library `DecEq (List Nat)`), and the `No` branch carries a real refuter.
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public export
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decScheduleEquiv : (xs, ys : List Nat) -> Dec (ScheduleEquiv xs ys)
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decScheduleEquiv xs ys = case decEq xs ys of
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Yes prf => Yes (MkScheduleEquiv prf)
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No ctr => No (\se => ctr (scheduleEq se))
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--------------------------------------------------------------------------------
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-- POSITIVE control: a concrete tiling equivalence witness
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--------------------------------------------------------------------------------
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||| POSITIVE CONTROL. Tile an extent-12 domain as `(a=2) * (b=3)` outer
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||| groups, each of `inner=2` iterations — i.e. (2*3)*2 = 12 — for the
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||| `double` stage from Layer 2. The tiled run is schedule-equivalent to the
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||| flat run. Inhabited witness, discharged by the general theorem.
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public export
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doubleTile2x3x2Equivalent :
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ScheduleEquiv (runTile Semantics.double 2 3 2)
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(runFlat Semantics.double ((2 * 3) * 2))
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doubleTile2x3x2Equivalent = tilePreservesResult Semantics.double 2 3 2
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||| And the underlying lists are literally, definitionally equal on this
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||| concrete case (the proof reduces to `Refl`).
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public export
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doubleTile2x3x2Concrete :
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runTile Semantics.double 2 3 2 = runFlat Semantics.double 12
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doubleTile2x3x2Concrete = Refl
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||| Tag extractor for a `Dec`: `True` for `Yes`, `False` for `No`.
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||| (Local, total — avoids guessing the Prelude name.)
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public export
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decTag : Dec p -> Bool
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decTag (Yes _) = True
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decTag (No _) = False
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||| Decision-procedure smoke test: the concrete tile/flat pair decides Yes.
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||| Only the tag is asserted by `Refl`, not the proof term's internals.
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public export
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decDoubleTileYes :
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decTag (decScheduleEquiv (runTile Semantics.double 2 3 2)
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(runFlat Semantics.double 12)) = True
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decDoubleTileYes = Refl
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--------------------------------------------------------------------------------
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-- NEGATIVE / non-vacuity controls
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--------------------------------------------------------------------------------
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||| NEGATIVE CONTROL #1. A result-CHANGING "tile" is NOT schedule-equivalent
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||| to the flat run. We exhibit a concrete bad run (the flat run of the WRONG
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||| stage `succ`, whose values differ from `double`'s) and prove it is `Not`
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||| schedule-equivalent to the correct flat run. Machine-checked: the two
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||| concrete result lists genuinely differ, so no `ScheduleEquiv` inhabits.
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public export
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mistiledNotEquivalent :
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Not (ScheduleEquiv (runFlat S 12) (runFlat Semantics.double 12))
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mistiledNotEquivalent (MkScheduleEquiv prf) = case prf of Refl impossible
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||| NEGATIVE CONTROL #2 (non-vacuity of transitivity). Transitivity must NOT
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||| be able to "launder" two genuinely different lists into equality. If a
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||| middle list `ys` were equal both to `[0]` and to `[1]`, transitivity
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||| would force `[0] = [1]`, which is impossible. This shows the relation has
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||| real content: composing through any witness preserves true equality.
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public export
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transNonVacuous :
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Not (ScheduleEquiv (the (List Nat) [0]) [1])
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transNonVacuous (MkScheduleEquiv prf) = case prf of Refl impossible
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||| NEGATIVE CONTROL #3 (the `Dec` is not trivially always-Yes). There is
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||| NO proof that the decision procedure answers `Yes` on a genuinely
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||| unequal pair: any such claimed `Yes _` is refuted, because extracting
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||| its witness would prove `[0] = [1]`. We state this as the impossibility
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||| of inhabiting the equivalence on that pair (already shown by
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||| `transNonVacuous`), and additionally that the decider cannot return a
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||| `Yes`-carrying proof here.
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public export
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decDistinctNotYes :
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(prf : ScheduleEquiv (the (List Nat) [0]) [1]) -> Void
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decDistinctNotYes = transNonVacuous

‎src/interface/abi/halideiser-abi.ipkg‎

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

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