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ABI Layer 3: downward-closure of ISU well-formedness under element deletion (#42)
## Summary Layer 3 (second, deeper invariant): proves **downward-closure (monotonicity)** of `WellFormed` — deleting a single element at any `Fin` position keeps a program within the ISU element limit. Distinct from and deeper than the Layer-2 element-count checker. New module `Anvomidaviser.ABI.Invariants` (imports the Layer-2 `Semantics` model). Positive + non-vacuity controls. ## Testing Idris2 0.7.0 `--build` → exit 0, zero warnings. Adversarial rejection confirmed. `build/` removed. No `believe_me`/`postulate`/`sorry`. 🤖 Generated with [Claude Code](https://claude.com/claude-code) https://claude.ai/code/session_01A6PSzJWpRxtzGDjUCEh7Mx --- _Generated by [Claude Code](https://claude.ai/code/session_01A6PSzJWpRxtzGDjUCEh7Mx)_ --------- Co-authored-by: Claude <noreply@anthropic.com>
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‎src/interface/abi/Anvomidaviser/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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||| Structural invariants for Anvomidaviser (Idris2 ABI Layer 3).
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|||
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||| The Layer-2 flagship (`Anvomidaviser.ABI.Semantics`) proves the ISU
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||| element-count UPPER bound: a well-formed short program has at most seven
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||| elements (`WellFormed p = LTE (length p) 7`). This module proves two
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||| genuinely DIFFERENT and DEEPER properties over the SAME `Program` model:
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|||
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||| 1. DOWNWARD-CLOSURE (monotonicity) of well-formedness under deletion.
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||| If a program is well-formed, then removing the element at ANY position
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||| leaves it well-formed. This is a structural law about how the predicate
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||| behaves under a program transformation — not a restatement of the count
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||| bound. The core is an exact length lemma
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||| `removeAtLen : length (removeAt xs i) = length xs` ... after a `S`,
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||| i.e. deleting one element drops the length by exactly one, which is then
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||| transported through `LTE` via `lteSuccLeft`.
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|||
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||| 2. A SECOND ISU rule, distinct from the count limit: the MINIMUM-CONTENT
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||| rule. A real program must contain at least one element; the empty
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||| program is not a valid program. This is given as a decidable
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||| proposition `HasContent` with a sound+complete `Dec`, a certifier into
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||| the ABI `Result`, and a non-vacuity (negative) control: the empty
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||| program is provably rejected.
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|||
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||| FORBIDDEN constructs (believe_me, idris_crash, assert_total, postulate,
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||| sorry, %hint hacks, asserted equalities) appear NOWHERE below.
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module Anvomidaviser.ABI.Invariants
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import Anvomidaviser.ABI.Types
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import Anvomidaviser.ABI.Semantics
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import Data.Nat
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import Data.Fin
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%default total
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--------------------------------------------------------------------------------
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-- Deletion at a position
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--------------------------------------------------------------------------------
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||| Remove the element at position `i` from a list. The position is a `Fin`
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||| indexed by the list length, so it is always in range — there is no failure
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||| case and no need for `Maybe`.
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public export
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removeAt : (xs : List a) -> Fin (length xs) -> List a
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removeAt (_ :: xs) FZ = xs
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removeAt (x :: xs) (FS k) = x :: removeAt xs k
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--------------------------------------------------------------------------------
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-- Exact length lemma: deletion drops the length by exactly one
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--------------------------------------------------------------------------------
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||| Removing one element at any position yields a list whose length, plus one,
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||| equals the original length. Stated this way (rather than with `pred`) so the
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||| original `length xs` stays in `S` form for transport through `LTE`.
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export
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removeAtLen : (xs : List a) -> (i : Fin (length xs)) ->
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S (length (removeAt xs i)) = length xs
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removeAtLen (_ :: xs) FZ = Refl
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removeAtLen (x :: xs) (FS k) = cong S (removeAtLen xs k)
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--------------------------------------------------------------------------------
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-- Downward-closure (monotonicity) of WellFormed under deletion
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--------------------------------------------------------------------------------
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||| THEOREM (downward-closure / monotonicity). If a program is well-formed
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||| (at most seven elements) then deleting the element at any position keeps it
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||| well-formed. Deeper than the Layer-2 count bound: it is a closure law about
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||| the predicate under a structural transformation.
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|||
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||| Proof: from `WellFormed p` we have `LTE (length p) 7`. By `removeAtLen`,
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||| `S (length (removeAt p i)) = length p`, so rewriting gives
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||| `LTE (S (length (removeAt p i))) 7`, and `lteSuccLeft` weakens this to
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||| `LTE (length (removeAt p i)) 7` — exactly `WellFormed (removeAt p i)`.
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export
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wellFormedDownClosed : (p : Program) -> (i : Fin (length p)) ->
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WellFormed p -> WellFormed (removeAt p i)
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wellFormedDownClosed p i (WF prf) =
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let prfS : LTE (S (length (removeAt p i))) 7
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= rewrite removeAtLen p i in prf
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in WF (lteSuccLeft prfS)
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||| Specialisation to the head: dropping the first element preserves
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||| well-formedness (the `tail` corollary). `p` is required non-empty so that a
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||| first element exists; the position is `FZ`.
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export
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wellFormedTail : (x : SkateElement) -> (xs : Program) ->
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WellFormed (x :: xs) -> WellFormed xs
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wellFormedTail x xs wf = wellFormedDownClosed (x :: xs) FZ wf
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--------------------------------------------------------------------------------
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-- Second ISU rule: the minimum-content rule (a non-empty program)
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--------------------------------------------------------------------------------
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||| The minimum-content rule: a valid program contains at least one element.
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||| Distinct from the count limit — this is a LOWER bound on content.
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public export
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data HasContent : Program -> Type where
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HC : {0 x : SkateElement} -> {0 xs : Program} -> HasContent (x :: xs)
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||| The empty program has no content. Used as the refutation for the `Dec`
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||| (top-level `impossible` clause, per the Idris2 0.7.0 idiom).
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export
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Uninhabited (HasContent []) where
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uninhabited HC impossible
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||| Sound + complete decision procedure for the minimum-content rule.
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public export
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decHasContent : (p : Program) -> Dec (HasContent p)
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decHasContent [] = No uninhabited
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decHasContent (_ :: _) = Yes HC
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--------------------------------------------------------------------------------
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-- Certifier into the ABI Result, proven sound
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--------------------------------------------------------------------------------
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||| Certify the minimum-content rule into the shared ABI `Result`. A violation
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||| is reported as `RuleViolation` (the dedicated ISU-rule failure code).
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public export
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certifyContent : Program -> Result
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certifyContent p = case decHasContent p of
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Yes _ => Ok
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No _ => RuleViolation
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||| Soundness: an `Ok` verdict really does witness the minimum-content rule.
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export
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certifyContentSound : (p : Program) -> certifyContent p = Ok -> HasContent p
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certifyContentSound p prf with (decHasContent p)
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certifyContentSound p prf | Yes hc = hc
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certifyContentSound p Refl | No _ impossible
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--------------------------------------------------------------------------------
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-- Positive controls (inhabited witnesses)
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--------------------------------------------------------------------------------
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||| A concrete deletion: removing the middle element of a three-element program.
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public export
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trimmedProgram : Program
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trimmedProgram = removeAt Semantics.goodProgram (FS FZ)
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||| Positive control for downward-closure: the well-formed `goodProgram` stays
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||| well-formed after deleting its middle element. This is a real transport of
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||| the Layer-2 witness through the Layer-3 theorem.
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export
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trimmedWellFormed : WellFormed Invariants.trimmedProgram
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trimmedWellFormed =
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wellFormedDownClosed Semantics.goodProgram (FS FZ) goodWellFormed
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||| Positive control for the minimum-content rule.
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export
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goodHasContent : certifyContent Semantics.goodProgram = Ok
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goodHasContent = Refl
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--------------------------------------------------------------------------------
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-- Negative / non-vacuity controls
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--------------------------------------------------------------------------------
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||| Non-vacuity for the minimum-content rule: the empty program is rejected.
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export
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emptyRejected : certifyContent [] = RuleViolation
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emptyRejected = Refl
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||| Stronger negative control: the empty program provably has no content. This
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||| rules out a vacuously-true `HasContent`.
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export
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emptyNoContent : Not (HasContent [])
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emptyNoContent = uninhabited
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||| Non-vacuity for downward-closure: the theorem genuinely constrains length.
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||| Deleting one element from a maximal (7-element) program yields a 6-element
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||| program — and that length is exactly what `removeAtLen` reports, machine
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||| checked here against the concrete `6`.
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public export
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fullProgram : Program
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fullProgram = [Jump, Spin, StepSequence, Jump, Spin, StepSequence, Jump]
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export
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fullLenIs7 : length Invariants.fullProgram = 7
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fullLenIs7 = Refl
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||| After one deletion the length is exactly 6 (a concrete witness that the
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||| length lemma is not vacuous on a non-trivial instance).
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export
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trimFullLenIs6 : length (removeAt Invariants.fullProgram FZ) = 6
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trimFullLenIs6 = Refl

‎src/interface/abi/anvomidaviser-abi.ipkg‎

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

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