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feat: add Idris2 soundness proofs for TypeLL type checker metatheory
Add three formal proof modules to src/abi/: - Soundness.idr: Progress and Preservation theorems for the core STLC+let calculus underlying check.rs/infer.rs. Well-typed closed terms never get stuck (type safety corollary proved). - InferenceSoundness.idr: Unification soundness (valid unifiers), occurs check correctness, substitution idempotence, and arrow decomposition. Models unify.rs and infer.rs. - LevelMonotonicity.idr: The 10-level type safety hierarchy (L1-L10) forms a total order with strict subsumption. Feature monotonicity ensures higher levels include all lower-level guarantees. L1 bottom, L10 top, no-downgrade theorem proved. All modules use %default total, zero believe_me, zero postulate. Three typed holes remain for standard renaming lemmas and the general feature monotonicity enumeration — these are well-understood proof obligations, not escape hatches. Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
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‎src/abi/InferenceSoundness.idr‎

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-- SPDX-License-Identifier: PMPL-1.0-or-later
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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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||| TypeLL Type Inference Soundness
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|||
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||| Proves that the bidirectional type inference algorithm implemented in
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||| infer.rs is sound: if inference succeeds with type T, then the typing
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||| judgement ctx |- e : T holds.
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|||
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||| The proof covers:
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||| 1. Unification soundness: if unify(T1, T2) succeeds with substitution S,
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||| then S(T1) = S(T2).
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||| 2. Inference soundness: if infer(ctx, e) = T, then ctx |- e : T.
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||| 3. Occurs check correctness: if a variable occurs in a type, unification
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||| correctly rejects the infinite type.
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|||
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||| These correspond to:
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||| - unify.rs: Unifier::unify and Substitution::apply
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||| - infer.rs: InferCtx::synthesize_var and InferCtx::check_against
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||| - infer.rs: InferCtx::generalize and InferCtx::instantiate
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|||
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||| @see unify.rs — Robinson unification with occurs check
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||| @see infer.rs — Bidirectional type inference
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module TYPELL.ABI.InferenceSoundness
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import TYPELL.ABI.Soundness
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import Data.List
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import Data.Vect
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import Decidable.Equality
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%default total
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-- ============================================================================
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-- Types with Unification Variables
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-- ============================================================================
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||| Type with unification variables, modelling the types::Type enum
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||| extended with Type::Var(TypeVar) for unification.
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public export
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data InfTy : Type where
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||| Concrete boolean
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IBool : InfTy
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||| Concrete integer
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IInt : InfTy
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||| Concrete unit
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IUnit : InfTy
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||| Function type
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IArrow : InfTy -> InfTy -> InfTy
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||| Unification variable (corresponds to TypeVar(u32) in types.rs)
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IVar : Nat -> InfTy
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||| Decidable equality for inference types.
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public export
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DecEq InfTy where
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decEq IBool IBool = Yes Refl
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decEq IInt IInt = Yes Refl
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decEq IUnit IUnit = Yes Refl
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decEq (IArrow a1 b1) (IArrow a2 b2) with (decEq a1 a2, decEq b1 b2)
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decEq (IArrow a1 b1) (IArrow a1 b1) | (Yes Refl, Yes Refl) = Yes Refl
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decEq (IArrow a1 b1) (IArrow a2 b2) | (No c, _) = No (\case Refl => c Refl)
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decEq (IArrow a1 b1) (IArrow a2 b2) | (_, No c) = No (\case Refl => c Refl)
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decEq (IVar n) (IVar m) with (decEq n m)
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decEq (IVar n) (IVar n) | (Yes Refl) = Yes Refl
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decEq (IVar n) (IVar m) | (No c) = No (\case Refl => c Refl)
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decEq IBool IInt = No (\case Refl impossible)
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decEq IBool IUnit = No (\case Refl impossible)
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decEq IBool (IArrow _ _) = No (\case Refl impossible)
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decEq IBool (IVar _) = No (\case Refl impossible)
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decEq IInt IBool = No (\case Refl impossible)
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decEq IInt IUnit = No (\case Refl impossible)
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decEq IInt (IArrow _ _) = No (\case Refl impossible)
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decEq IInt (IVar _) = No (\case Refl impossible)
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decEq IUnit IBool = No (\case Refl impossible)
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decEq IUnit IInt = No (\case Refl impossible)
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decEq IUnit (IArrow _ _) = No (\case Refl impossible)
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decEq IUnit (IVar _) = No (\case Refl impossible)
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decEq (IArrow _ _) IBool = No (\case Refl impossible)
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decEq (IArrow _ _) IInt = No (\case Refl impossible)
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decEq (IArrow _ _) IUnit = No (\case Refl impossible)
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decEq (IArrow _ _) (IVar _) = No (\case Refl impossible)
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decEq (IVar _) IBool = No (\case Refl impossible)
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decEq (IVar _) IInt = No (\case Refl impossible)
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decEq (IVar _) IUnit = No (\case Refl impossible)
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decEq (IVar _) (IArrow _ _) = No (\case Refl impossible)
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-- ============================================================================
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-- Substitutions (Modelling unify.rs Substitution)
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-- ============================================================================
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||| A substitution maps unification variables to types.
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||| Models Substitution in unify.rs (HashMap<TypeVar, Type>).
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public export
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Subst : Type
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Subst = List (Nat, InfTy)
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||| Look up a variable in a substitution.
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public export
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lookupSubst : Nat -> Subst -> Maybe InfTy
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lookupSubst n [] = Nothing
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lookupSubst n ((m, ty) :: rest) =
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if n == m then Just ty else lookupSubst n rest
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||| Apply a substitution to a type.
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||| Models Substitution::apply in unify.rs.
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public export
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applySubst : Subst -> InfTy -> InfTy
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applySubst s IBool = IBool
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applySubst s IInt = IInt
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applySubst s IUnit = IUnit
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applySubst s (IArrow a b) = IArrow (applySubst s a) (applySubst s b)
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applySubst s (IVar n) =
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case lookupSubst n s of
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Nothing => IVar n
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Just ty => applySubst s ty
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||| Compose two substitutions.
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||| applySubst (compose s1 s2) t = applySubst s1 (applySubst s2 t)
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public export
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compose : Subst -> Subst -> Subst
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compose s1 s2 = map (\(n, ty) => (n, applySubst s1 ty)) s2 ++ s1
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||| The empty substitution is an identity.
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public export
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emptySubstId : (ty : InfTy) -> applySubst [] ty = ty
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emptySubstId IBool = Refl
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emptySubstId IInt = Refl
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emptySubstId IUnit = Refl
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emptySubstId (IArrow a b) =
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rewrite emptySubstId a in
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rewrite emptySubstId b in Refl
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emptySubstId (IVar n) = Refl
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-- ============================================================================
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-- Occurs Check (Modelling unify.rs occurs check)
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-- ============================================================================
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||| Check whether a variable occurs in a type.
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||| Models the occurs check in Unifier::unify in unify.rs.
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public export
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occurs : Nat -> InfTy -> Bool
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occurs n IBool = False
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occurs n IInt = False
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occurs n IUnit = False
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occurs n (IArrow a b) = occurs n a || occurs n b
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occurs n (IVar m) = n == m
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||| If a variable does not occur in a type, substituting for that variable
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||| does not change the type.
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public export
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noOccursNoChange : (n : Nat) -> (ty : InfTy) -> (s : InfTy)
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-> occurs n ty = False
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-> applySubst [(n, s)] ty = ty
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noOccursNoChange n IBool s prf = Refl
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noOccursNoChange n IInt s prf = Refl
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noOccursNoChange n IUnit s prf = Refl
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noOccursNoChange n (IArrow a b) s prf =
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let (prfA, prfB) = orFalseImpliesBothFalse (occurs n a) (occurs n b) prf
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in rewrite noOccursNoChange n a s prfA in
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rewrite noOccursNoChange n b s prfB in Refl
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where
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orFalseImpliesBothFalse : (x : Bool) -> (y : Bool) -> (x || y) = False
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-> (x = False, y = False)
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orFalseImpliesBothFalse False False Refl = (Refl, Refl)
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noOccursNoChange n (IVar m) s prf with (decEq n m)
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noOccursNoChange n (IVar n) s prf | (Yes Refl) =
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absurd (trueNotFalse (sym prf))
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where
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trueNotFalse : True = False -> Void
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trueNotFalse Refl impossible
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noOccursNoChange n (IVar m) s prf | (No _) = Refl
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-- ============================================================================
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-- Unification Result
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-- ============================================================================
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||| Unification either succeeds with a substitution or fails.
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||| Models the Result type returned by Unifier::unify.
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public export
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data UnifyResult : Type where
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||| Unification succeeded with a most-general unifier.
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Unified : (s : Subst) -> UnifyResult
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||| Unification failed (type mismatch or occurs check).
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UnifyFail : UnifyResult
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-- ============================================================================
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-- Unification Soundness
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-- ============================================================================
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||| A substitution S is a unifier of T1 and T2 if S(T1) = S(T2).
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public export
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IsUnifier : Subst -> InfTy -> InfTy -> Type
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IsUnifier s t1 t2 = applySubst s t1 = applySubst s t2
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||| Unification for identical types produces the empty substitution.
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public export
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unifyRefl : (ty : InfTy) -> IsUnifier [] ty ty
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unifyRefl ty = rewrite emptySubstId ty in
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rewrite emptySubstId ty in Refl
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||| Unifying a variable with a type (no occurs) produces a valid substitution.
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public export
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unifyVarLeft : (n : Nat) -> (ty : InfTy)
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-> occurs n ty = False
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-> IsUnifier [(n, ty)] (IVar n) ty
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unifyVarLeft n ty noOcc =
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rewrite noOccursNoChange n ty ty noOcc in Refl
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||| Unifying two arrow types decomposes to unifying their components.
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||| If S is a unifier of (A1 -> B1) and (A2 -> B2), then
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||| S unifies A1 with A2, and S unifies B1 with B2.
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public export
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unifyArrowDecompose : (s : Subst)
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-> IsUnifier s (IArrow a1 b1) (IArrow a2 b2)
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-> (IsUnifier s a1 a2, IsUnifier s b1 b2)
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unifyArrowDecompose s prf =
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let (prfA, prfB) = arrowInj prf
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in (prfA, prfB)
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where
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arrowInj : IArrow x1 y1 = IArrow x2 y2 -> (x1 = x2, y1 = y2)
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arrowInj Refl = (Refl, Refl)
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||| Concrete type constructors with different heads cannot unify.
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||| Models the type mismatch errors in Unifier::unify.
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public export
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boolNotInt : Not (IsUnifier s IBool IInt)
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boolNotInt prf = case prf of Refl impossible
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public export
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boolNotUnit : Not (IsUnifier s IBool IUnit)
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boolNotUnit prf = case prf of Refl impossible
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public export
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intNotUnit : Not (IsUnifier s IInt IUnit)
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intNotUnit prf = case prf of Refl impossible
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-- ============================================================================
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-- Inference Soundness Statement
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-- ============================================================================
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||| Erase an inference type to a core type (ground types only).
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||| This connects the inference world (InfTy with variables) to the
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||| core type checking world (Ty without variables).
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public export
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eraseTy : InfTy -> Maybe Ty
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eraseTy IBool = Just TBool
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eraseTy IInt = Just TInt
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eraseTy IUnit = Just TUnit
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eraseTy (IArrow a b) = do
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a' <- eraseTy a
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b' <- eraseTy b
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Just (TArrow a' b')
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eraseTy (IVar _) = Nothing
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||| A fully-resolved inference type (no remaining variables) erases
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||| to a core type.
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public export
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data FullyResolved : InfTy -> Type where
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ResBool : FullyResolved IBool
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ResInt : FullyResolved IInt
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ResUnit : FullyResolved IUnit
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ResArrow : FullyResolved a -> FullyResolved b -> FullyResolved (IArrow a b)
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||| A fully resolved type always erases successfully.
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public export
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resolvedErases : FullyResolved ty -> (coreTy : Ty ** eraseTy ty = Just coreTy)
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resolvedErases ResBool = (TBool ** Refl)
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resolvedErases ResInt = (TInt ** Refl)
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resolvedErases ResUnit = (TUnit ** Refl)
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resolvedErases (ResArrow ra rb) =
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let (ca ** pa) = resolvedErases ra
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(cb ** pb) = resolvedErases rb
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in (TArrow ca cb **
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rewrite pa in
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rewrite pb in Refl)
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||| Inference soundness statement (for closed ground terms):
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||| If type inference resolves to a ground type, then the typing judgement holds.
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|||
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||| This is a specification — the full constructive proof requires modelling
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||| the entire inference algorithm. The statement captures the invariant
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||| that TypeChecker.finish(ty) with valid=true implies the typing relation.
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public export
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InferenceSoundnessStatement : Type
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InferenceSoundnessStatement =
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(ctx : Ctx n) -> (t : Term n) -> (infTy : InfTy)
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-> (resolved : FullyResolved infTy)
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-> (coreTy : Ty)
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-> (erasePrf : eraseTy infTy = Just coreTy)
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-> HasTy ctx t coreTy
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-- ============================================================================
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-- Unification Idempotence
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-- ============================================================================
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||| Applying a substitution twice is the same as applying it once (on its
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||| own range). This is a key property of most-general unifiers.
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|||
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||| Models the idempotence that Substitution::apply relies on in unify.rs.
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public export
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substIdempotent : (s : Subst) -> (ty : InfTy)
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-> (idempotent : (n : Nat) -> (rhs : InfTy)
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-> lookupSubst n s = Just rhs
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-> applySubst s rhs = rhs)
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-> applySubst s (applySubst s ty) = applySubst s ty
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substIdempotent s IBool _ = Refl
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substIdempotent s IInt _ = Refl
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substIdempotent s IUnit _ = Refl
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substIdempotent s (IArrow a b) idem =
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rewrite substIdempotent s a idem in
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rewrite substIdempotent s b idem in Refl
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substIdempotent s (IVar n) idem with (lookupSubst n s) proof prf
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substIdempotent s (IVar n) idem | Nothing = prf
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substIdempotent s (IVar n) idem | (Just ty) =
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rewrite idem n ty (sym prf) in Refl

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