abi: prove cost-conservation (additivity) + budget compliance (Layer 2)

Flagship semantic proof: totalCost (xs ++ ys) = totalCost xs + totalCost ys
(conservation, by induction), plus a decidable WithinBudget proposition
with sound+complete Dec, certifier soundness, and positive + negative
controls (an over-budget ledger is provably rejected). Verified with
idris2 0.7.0 (build clean, zero warnings) + adversarial rejection.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01A6PSzJWpRxtzGDjUCEh7Mx

Author: claude

src/interface/abi/Eclexiaiser/ABI/Semantics.idr

@@ -0,0 +1,108 @@
+-- SPDX-License-Identifier: MPL-2.0
+-- Copyright (c) 2026 Jonathan D.A. Jewell (hyperpolymath) <j.d.a.jewell@open.ac.uk>
+--
+||| Flagship semantic proof for Eclexiaiser (Idris2 ABI Layer 2).
+|||
+||| Eclexiaiser's headline is "add energy/carbon/resource-cost awareness". The
+||| foundational guarantee of any cost-accounting layer is that accounting
+||| *conserves* cost — the cost of a composite computation is exactly the sum of
+||| its parts, so the accounting can neither lose nor invent cost. This module
+||| proves:
+|||
+|||   1. `additivity` — totalCost (xs ++ ys) = totalCost xs + totalCost ys, by
+|||      induction (the conservation law);
+|||   2. `WithinBudget` — budget compliance as a decidable proposition with a
+|||      sound+complete `Dec`, a certifier proven sound, and positive + negative
+|||      controls (an over-budget ledger is provably rejected).
+
+module Eclexiaiser.ABI.Semantics
+
+import Eclexiaiser.ABI.Types
+import Data.Nat
+import Decidable.Equality
+
+%default total
+
+--------------------------------------------------------------------------------
+-- A cost ledger and its total
+--------------------------------------------------------------------------------
+
+||| A ledger of per-step resource costs.
+public export
+Ledger : Type
+Ledger = List Nat
+
+public export
+totalCost : Ledger -> Nat
+totalCost []        = 0
+totalCost (c :: cs) = c + totalCost cs
+
+--------------------------------------------------------------------------------
+-- Conservation: cost is additive over composition
+--------------------------------------------------------------------------------
+
+||| The conservation law: the cost of concatenated work equals the sum of the
+||| costs — accounting neither loses nor invents resource cost.
+export
+additivity : (xs, ys : Ledger) -> totalCost (xs ++ ys) = totalCost xs + totalCost ys
+additivity []        ys = Refl
+additivity (x :: xs) ys =
+  trans (cong (x +) (additivity xs ys))
+        (plusAssociative x (totalCost xs) (totalCost ys))
+
+--------------------------------------------------------------------------------
+-- Budget compliance as a decidable proposition
+--------------------------------------------------------------------------------
+
+public export
+data WithinBudget : (budget : Nat) -> Ledger -> Type where
+  Within : {0 b : Nat} -> {0 l : Ledger} -> LTE (totalCost l) b -> WithinBudget b l
+
+public export
+decWithinBudget : (b : Nat) -> (l : Ledger) -> Dec (WithinBudget b l)
+decWithinBudget b l = case isLTE (totalCost l) b of
+  Yes prf => Yes (Within prf)
+  No  no  => No (\(Within p) => no p)
+
+--------------------------------------------------------------------------------
+-- Certifier into the ABI Result, proven sound
+--------------------------------------------------------------------------------
+
+public export
+certifyBudget : Nat -> Ledger -> Result
+certifyBudget b l = case decWithinBudget b l of
+  Yes _ => Ok
+  No  _ => Error
+
+export
+certifyBudgetSound : (b : Nat) -> (l : Ledger) ->
+  certifyBudget b l = Ok -> WithinBudget b l
+certifyBudgetSound b l prf with (decWithinBudget b l)
+  certifyBudgetSound b l prf  | Yes w = w
+  certifyBudgetSound b l Refl | No _ impossible
+
+--------------------------------------------------------------------------------
+-- Positive controls
+--------------------------------------------------------------------------------
+
+||| Conservation holds on a concrete split.
+export
+splitConserves : totalCost ([3, 4] ++ [5]) = totalCost [3, 4] + totalCost [5]
+splitConserves = additivity [3, 4] [5]
+
+export
+certifyWithinAccepts : certifyBudget 10 [3, 4] = Ok
+certifyWithinAccepts = Refl
+
+||| A genuine `WithinBudget` witness, derived from the sound certifier.
+export
+withinEx : WithinBudget 10 [3, 4]
+withinEx = certifyBudgetSound 10 [3, 4] certifyWithinAccepts
+
+--------------------------------------------------------------------------------
+-- Negative control: an over-budget ledger is rejected (non-vacuity)
+--------------------------------------------------------------------------------
+
+export
+certifyOverRejects : certifyBudget 5 [3, 4] = Error
+certifyOverRejects = Refl

src/interface/abi/eclexiaiser-abi.ipkg

@@ -7,3 +7,4 @@ modules = Eclexiaiser.ABI.Types
         , Eclexiaiser.ABI.Layout
         , Eclexiaiser.ABI.Foreign
         , Eclexiaiser.ABI.Proofs
+        , Eclexiaiser.ABI.Semantics