module Quox.Reduce import Quox.No import Quox.Syntax import Quox.Definition import Quox.Displace import Quox.Typing.Context import Quox.Typing.Error import Data.SnocVect import Data.Maybe import Data.List import Control.Eff %default total -- [fixme] rename this to Whnf and the interface to CanWhnf public export WhnfM : Type -> Type WhnfM = Eff [NameGen, Except Error] export runWhnfWith : NameSuf -> WhnfM a -> (Either Error a, NameSuf) runWhnfWith suf act = extract $ runStateAt GEN suf $ runExcept act export runWhnf : WhnfM a -> Either Error a runWhnf = fst . runWhnfWith 0 public export 0 RedexTest : TermLike -> Type RedexTest tm = {d, n : Nat} -> Definitions -> tm d n -> Bool public export interface Whnf (0 tm : TermLike) (0 isRedex : RedexTest tm) | tm where whnf : {d, n : Nat} -> (defs : Definitions) -> (ctx : WhnfContext d n) -> tm d n -> WhnfM (Subset (tm d n) (No . isRedex defs)) public export %inline whnf0 : {d, n : Nat} -> {0 isRedex : RedexTest tm} -> Whnf tm isRedex => (defs : Definitions) -> WhnfContext d n -> tm d n -> WhnfM (tm d n) whnf0 defs ctx t = fst <$> whnf defs ctx t public export 0 IsRedex, NotRedex : {isRedex : RedexTest tm} -> Whnf tm isRedex => Definitions -> Pred (tm d n) IsRedex defs = So . isRedex defs NotRedex defs = No . isRedex defs public export 0 NonRedex : (tm : TermLike) -> {isRedex : RedexTest tm} -> Whnf tm isRedex => (d, n : Nat) -> (defs : Definitions) -> Type NonRedex tm d n defs = Subset (tm d n) (NotRedex defs) public export %inline nred : {0 isRedex : RedexTest tm} -> (0 _ : Whnf tm isRedex) => (t : tm d n) -> (0 nr : NotRedex defs t) => NonRedex tm d n defs nred t = Element t nr public export %inline isLamHead : Elim {} -> Bool isLamHead (Ann (Lam {}) (Pi {}) _) = True isLamHead (Coe {}) = True isLamHead _ = False public export %inline isDLamHead : Elim {} -> Bool isDLamHead (Ann (DLam {}) (Eq {}) _) = True isDLamHead (Coe {}) = True isDLamHead _ = False public export %inline isPairHead : Elim {} -> Bool isPairHead (Ann (Pair {}) (Sig {}) _) = True isPairHead (Coe {}) = True isPairHead _ = False public export %inline isTagHead : Elim {} -> Bool isTagHead (Ann (Tag {}) (Enum {}) _) = True isTagHead (Coe {}) = True isTagHead _ = False public export %inline isNatHead : Elim {} -> Bool isNatHead (Ann (Zero {}) (Nat {}) _) = True isNatHead (Ann (Succ {}) (Nat {}) _) = True isNatHead (Coe {}) = True isNatHead _ = False public export %inline isBoxHead : Elim {} -> Bool isBoxHead (Ann (Box {}) (BOX {}) _) = True isBoxHead (Coe {}) = True isBoxHead _ = False public export %inline isE : Term {} -> Bool isE (E {}) = True isE _ = False public export %inline isAnn : Elim {} -> Bool isAnn (Ann {}) = True isAnn _ = False ||| true if a term is syntactically a type. public export %inline isTyCon : Term {} -> Bool isTyCon (TYPE {}) = True isTyCon (Pi {}) = True isTyCon (Lam {}) = False isTyCon (Sig {}) = True isTyCon (Pair {}) = False isTyCon (Enum {}) = True isTyCon (Tag {}) = False isTyCon (Eq {}) = True isTyCon (DLam {}) = False isTyCon (Nat {}) = True isTyCon (Zero {}) = False isTyCon (Succ {}) = False isTyCon (BOX {}) = True isTyCon (Box {}) = False isTyCon (E {}) = False isTyCon (CloT {}) = False isTyCon (DCloT {}) = False ||| true if a term is syntactically a type, or a neutral. public export %inline isTyConE : Term {} -> Bool isTyConE s = isTyCon s || isE s ||| true if a term is syntactically a type. public export %inline isAnnTyCon : Elim {} -> Bool isAnnTyCon (Ann ty (TYPE {}) _) = isTyCon ty isAnnTyCon _ = False public export %inline isK : Dim d -> Bool isK (K {}) = True isK _ = False mutual public export isRedexE : RedexTest Elim isRedexE defs (F {x, _}) {d, n} = isJust $ lookupElim x defs {d, n} isRedexE _ (B {}) = False isRedexE defs (App {fun, _}) = isRedexE defs fun || isLamHead fun isRedexE defs (CasePair {pair, _}) = isRedexE defs pair || isPairHead pair isRedexE defs (CaseEnum {tag, _}) = isRedexE defs tag || isTagHead tag isRedexE defs (CaseNat {nat, _}) = isRedexE defs nat || isNatHead nat isRedexE defs (CaseBox {box, _}) = isRedexE defs box || isBoxHead box isRedexE defs (DApp {fun, arg, _}) = isRedexE defs fun || isDLamHead fun || isK arg isRedexE defs (Ann {tm, ty, _}) = isE tm || isRedexT defs tm || isRedexT defs ty isRedexE defs (Coe {val, _}) = isRedexT defs val || not (isE val) isRedexE defs (Comp {ty, r, _}) = isRedexT defs ty || isK r isRedexE defs (TypeCase {ty, ret, _}) = isRedexE defs ty || isRedexT defs ret || isAnnTyCon ty isRedexE _ (CloE {}) = True isRedexE _ (DCloE {}) = True public export isRedexT : RedexTest Term isRedexT _ (CloT {}) = True isRedexT _ (DCloT {}) = True isRedexT defs (E {e, _}) = isAnn e || isRedexE defs e isRedexT _ _ = False public export tycaseRhs : (k : TyConKind) -> TypeCaseArms d n -> Maybe (ScopeTermN (arity k) d n) tycaseRhs k arms = lookupPrecise k arms public export tycaseRhsDef : Term d n -> (k : TyConKind) -> TypeCaseArms d n -> ScopeTermN (arity k) d n tycaseRhsDef def k arms = fromMaybe (SN def) $ tycaseRhs k arms public export tycaseRhs0 : (k : TyConKind) -> TypeCaseArms d n -> (0 eq : arity k = 0) => Maybe (Term d n) tycaseRhs0 k arms {eq} with (tycaseRhs k arms) | (arity k) tycaseRhs0 k arms {eq = Refl} | res | 0 = map (.term) res public export tycaseRhsDef0 : Term d n -> (k : TyConKind) -> TypeCaseArms d n -> (0 eq : arity k = 0) => Term d n tycaseRhsDef0 def k arms = fromMaybe def $ tycaseRhs0 k arms private weakDS : (by : Nat) -> DScopeTerm d n -> DScopeTerm d (by + n) weakDS by (S names (Y body)) = S names $ Y $ weakT by body weakDS by (S names (N body)) = S names $ N $ weakT by body private dweakS : (by : Nat) -> ScopeTerm d n -> ScopeTerm (by + d) n dweakS by (S names (Y body)) = S names $ Y $ dweakT by body dweakS by (S names (N body)) = S names $ N $ dweakT by body private coeScoped : {s : Nat} -> DScopeTerm d n -> Dim d -> Dim d -> Loc -> ScopeTermN s d n -> ScopeTermN s d n coeScoped ty p q loc (S names (Y body)) = S names $ Y $ E $ Coe (weakDS s ty) p q body loc coeScoped ty p q loc (S names (N body)) = S names $ N $ E $ Coe ty p q body loc export covering Whnf Term Reduce.isRedexT export covering Whnf Elim Reduce.isRedexE parameters {d, n : Nat} (defs : Definitions) (ctx : WhnfContext d n) ||| performs the minimum work required to recompute the type of an elim. ||| ||| ⚠ **assumes the elim is already typechecked.** ⚠ export covering computeElimType : (e : Elim d n) -> (0 ne : No (isRedexE defs e)) => WhnfM (Term d n) computeElimType (F {x, u, loc}) = do let Just def = lookup x defs | Nothing => throw $ NotInScope loc x pure $ displace u def.type computeElimType (B {i, _}) = pure $ ctx.tctx !! i computeElimType (App {fun = f, arg = s, loc}) {ne} = do Pi {arg, res, _} <- whnf0 defs ctx =<< computeElimType f {ne = noOr1 ne} | t => throw $ ExpectedPi loc ctx.names t pure $ sub1 res $ Ann s arg loc computeElimType (CasePair {pair, ret, _}) = pure $ sub1 ret pair computeElimType (CaseEnum {tag, ret, _}) = pure $ sub1 ret tag computeElimType (CaseNat {nat, ret, _}) = pure $ sub1 ret nat computeElimType (CaseBox {box, ret, _}) = pure $ sub1 ret box computeElimType (DApp {fun = f, arg = p, loc}) {ne} = do Eq {ty, _} <- whnf0 defs ctx =<< computeElimType f {ne = noOr1 ne} | t => throw $ ExpectedEq loc ctx.names t pure $ dsub1 ty p computeElimType (Ann {ty, _}) = pure ty computeElimType (Coe {ty, q, _}) = pure $ dsub1 ty q computeElimType (Comp {ty, _}) = pure ty computeElimType (TypeCase {ret, _}) = pure ret parameters {d, n : Nat} (defs : Definitions) (ctx : WhnfContext (S d) n) ||| for π.(x : A) → B, returns (A, B); ||| for an elim returns a pair of type-cases that will reduce to that; ||| for other intro forms error private covering tycasePi : (t : Term (S d) n) -> (0 tnf : No (isRedexT defs t)) => WhnfM (Term (S d) n, ScopeTerm (S d) n) tycasePi (Pi {arg, res, _}) = pure (arg, res) tycasePi (E e) {tnf} = do ty <- computeElimType defs ctx e @{noOr2 tnf} let loc = e.loc narg = mnb "Arg"; nret = mnb "Ret" arg = E $ typeCase1Y e ty KPi [< !narg, !nret] (BVT 1 loc) loc res' = typeCase1Y e (Arr Zero arg ty loc) KPi [< !narg, !nret] (BVT 0 loc) loc res = SY [< !narg] $ E $ App (weakE 1 res') (BVT 0 loc) loc pure (arg, res) tycasePi t = throw $ ExpectedPi t.loc ctx.names t ||| for (x : A) × B, returns (A, B); ||| for an elim returns a pair of type-cases that will reduce to that; ||| for other intro forms error private covering tycaseSig : (t : Term (S d) n) -> (0 tnf : No (isRedexT defs t)) => WhnfM (Term (S d) n, ScopeTerm (S d) n) tycaseSig (Sig {fst, snd, _}) = pure (fst, snd) tycaseSig (E e) {tnf} = do ty <- computeElimType defs ctx e @{noOr2 tnf} let loc = e.loc nfst = mnb "Fst"; nsnd = mnb "Snd" fst = E $ typeCase1Y e ty KSig [< !nfst, !nsnd] (BVT 1 loc) loc snd' = typeCase1Y e (Arr Zero fst ty loc) KSig [< !nfst, !nsnd] (BVT 0 loc) loc snd = SY [< !nfst] $ E $ App (weakE 1 snd') (BVT 0 loc) loc pure (fst, snd) tycaseSig t = throw $ ExpectedSig t.loc ctx.names t ||| for [π. A], returns A; ||| for an elim returns a type-case that will reduce to that; ||| for other intro forms error private covering tycaseBOX : (t : Term (S d) n) -> (0 tnf : No (isRedexT defs t)) => WhnfM (Term (S d) n) tycaseBOX (BOX {ty, _}) = pure ty tycaseBOX (E e) {tnf} = do ty <- computeElimType defs ctx e @{noOr2 tnf} pure $ E $ typeCase1Y e ty KBOX [< !(mnb "Ty")] (BVT 0 e.loc) e.loc tycaseBOX t = throw $ ExpectedBOX t.loc ctx.names t ||| for Eq [i ⇒ A] l r, returns (A‹0/i›, A‹1/i›, A, l, r); ||| for an elim returns five type-cases that will reduce to that; ||| for other intro forms error private covering tycaseEq : (t : Term (S d) n) -> (0 tnf : No (isRedexT defs t)) => WhnfM (Term (S d) n, Term (S d) n, DScopeTerm (S d) n, Term (S d) n, Term (S d) n) tycaseEq (Eq {ty, l, r, _}) = pure (ty.zero, ty.one, ty, l, r) tycaseEq (E e) {tnf} = do ty <- computeElimType defs ctx e @{noOr2 tnf} let loc = e.loc names = traverse' (\x => mnb x) [< "A0", "A1", "A", "L", "R"] a0 = E $ typeCase1Y e ty KEq !names (BVT 4 loc) loc a1 = E $ typeCase1Y e ty KEq !names (BVT 3 loc) loc a' = typeCase1Y e (Eq0 ty a0 a1 loc) KEq !names (BVT 2 loc) loc a = SY [< !(mnb "i")] $ E $ DApp (dweakE 1 a') (B VZ loc) loc l = E $ typeCase1Y e a0 KEq !names (BVT 1 loc) loc r = E $ typeCase1Y e a1 KEq !names (BVT 0 loc) loc pure (a0, a1, a, l, r) tycaseEq t = throw $ ExpectedEq t.loc ctx.names t -- new block because the functions below might pass a different ctx -- into the ones above parameters {d, n : Nat} (defs : Definitions) (ctx : WhnfContext d n) ||| reduce a function application `App (Coe ty p q val) s loc` private covering piCoe : (ty : DScopeTerm d n) -> (p, q : Dim d) -> (val, s : Term d n) -> Loc -> WhnfM (Subset (Elim d n) (No . isRedexE defs)) piCoe sty@(S [< i] ty) p q val s loc = do -- (coe [i ⇒ π.(x : A) → B] @p @q t) s ⇝ -- coe [i ⇒ B[𝒔‹i›/x] @p @q ((t ∷ (π.(x : A) → B)‹p/i›) 𝒔‹p›) -- where 𝒔‹j› ≔ coe [i ⇒ A] @q @j s -- -- type-case is used to expose A,B if the type is neutral let ctx1 = extendDim i ctx Element ty tynf <- whnf defs ctx1 ty.term (arg, res) <- tycasePi defs ctx1 ty let s0 = CoeT i arg q p s s.loc body = E $ App (Ann val (ty // one p) val.loc) (E s0) loc s1 = CoeT i (arg // (BV 0 i.loc ::: shift 2)) (weakD 1 q) (BV 0 i.loc) (s // shift 1) s.loc whnf defs ctx $ CoeT i (sub1 res s1) p q body loc ||| reduce a pair elimination `CasePair pi (Coe ty p q val) ret body loc` private covering sigCoe : (qty : Qty) -> (ty : DScopeTerm d n) -> (p, q : Dim d) -> (val : Term d n) -> (ret : ScopeTerm d n) -> (body : ScopeTermN 2 d n) -> Loc -> WhnfM (Subset (Elim d n) (No . isRedexE defs)) sigCoe qty sty@(S [< i] ty) p q val ret body loc = do -- caseπ (coe [i ⇒ (x : A) × B] @p @q s) return z ⇒ C of { (a, b) ⇒ e } -- ⇝ -- caseπ s ∷ ((x : A) × B)‹p/i› return z ⇒ C -- of { (a, b) ⇒ -- e[(coe [i ⇒ A] @p @q a)/a, -- (coe [i ⇒ B[(coe [j ⇒ A‹j/i›] @p @i a)/x]] @p @q b)/b] } -- -- type-case is used to expose A,B if the type is neutral let ctx1 = extendDim i ctx Element ty tynf <- whnf defs ctx1 ty.term (tfst, tsnd) <- tycaseSig defs ctx1 ty let [< x, y] = body.names a' = CoeT i (weakT 2 tfst) p q (BVT 1 noLoc) x.loc tsnd' = tsnd.term // (CoeT i (weakT 2 $ tfst // (B VZ noLoc ::: shift 2)) (weakD 1 p) (B VZ noLoc) (BVT 1 noLoc) y.loc ::: shift 2) b' = CoeT i tsnd' p q (BVT 0 noLoc) y.loc whnf defs ctx $ CasePair qty (Ann val (ty // one p) val.loc) ret (ST body.names $ body.term // (a' ::: b' ::: shift 2)) loc ||| reduce a dimension application `DApp (Coe ty p q val) r loc` private covering eqCoe : (ty : DScopeTerm d n) -> (p, q : Dim d) -> (val : Term d n) -> (r : Dim d) -> Loc -> WhnfM (Subset (Elim d n) (No . isRedexE defs)) eqCoe sty@(S [< j] ty) p q val r loc = do -- (coe [j ⇒ Eq [i ⇒ A] L R] @p @q eq) @r -- ⇝ -- comp [j ⇒ A‹r/i›] @p @q (eq ∷ (Eq [i ⇒ A] L R)‹p/j›) -- @r { 0 j ⇒ L; 1 j ⇒ R } let ctx1 = extendDim j ctx Element ty tynf <- whnf defs ctx1 ty.term (a0, a1, a, s, t) <- tycaseEq defs ctx1 ty let a' = dsub1 a (weakD 1 r) val' = E $ DApp (Ann val (ty // one p) val.loc) r loc whnf defs ctx $ CompH j a' p q val' r j s j t loc ||| reduce a pair elimination `CaseBox pi (Coe ty p q val) ret body` private covering boxCoe : (qty : Qty) -> (ty : DScopeTerm d n) -> (p, q : Dim d) -> (val : Term d n) -> (ret : ScopeTerm d n) -> (body : ScopeTerm d n) -> Loc -> WhnfM (Subset (Elim d n) (No . isRedexE defs)) boxCoe qty sty@(S [< i] ty) p q val ret body loc = do -- caseπ (coe [i ⇒ [ρ. A]] @p @q s) return z ⇒ C of { [a] ⇒ e } -- ⇝ -- caseπ s ∷ [ρ. A]‹p/i› return z ⇒ C -- of { [a] ⇒ e[(coe [i ⇒ A] p q a)/a] } let ctx1 = extendDim i ctx Element ty tynf <- whnf defs ctx1 ty.term ta <- tycaseBOX defs ctx1 ty let a' = CoeT i (weakT 1 ta) p q (BVT 0 noLoc) body.name.loc whnf defs ctx $ CaseBox qty (Ann val (ty // one p) val.loc) ret (ST body.names $ body.term // (a' ::: shift 1)) loc ||| reduce a type-case applied to a type constructor private covering reduceTypeCase : {d, n : Nat} -> (defs : Definitions) -> WhnfContext d n -> (ty : Term d n) -> (u : Universe) -> (ret : Term d n) -> (arms : TypeCaseArms d n) -> (def : Term d n) -> (0 _ : So (isTyCon ty)) => Loc -> WhnfM (Subset (Elim d n) (No . isRedexE defs)) reduceTypeCase defs ctx ty u ret arms def loc = case ty of -- (type-case ★ᵢ ∷ _ return Q of { ★ ⇒ s; ⋯ }) ⇝ s ∷ Q TYPE {} => whnf defs ctx $ Ann (tycaseRhsDef0 def KTYPE arms) ret loc -- (type-case π.(x : A) → B ∷ ★ᵢ return Q of { (a → b) ⇒ s; ⋯ }) ⇝ -- s[(A ∷ ★ᵢ)/a, ((λ x ⇒ B) ∷ 0.A → ★ᵢ)/b] ∷ Q Pi {arg, res, loc = piLoc, _} => let arg' = Ann arg (TYPE u noLoc) arg.loc res' = Ann (Lam res res.loc) (Arr Zero arg (TYPE u noLoc) arg.loc) res.loc in whnf defs ctx $ Ann (subN (tycaseRhsDef def KPi arms) [< arg', res']) ret loc -- (type-case (x : A) × B ∷ ★ᵢ return Q of { (a × b) ⇒ s; ⋯ }) ⇝ -- s[(A ∷ ★ᵢ)/a, ((λ x ⇒ B) ∷ 0.A → ★ᵢ)/b] ∷ Q Sig {fst, snd, loc = sigLoc, _} => let fst' = Ann fst (TYPE u noLoc) fst.loc snd' = Ann (Lam snd snd.loc) (Arr Zero fst (TYPE u noLoc) fst.loc) snd.loc in whnf defs ctx $ Ann (subN (tycaseRhsDef def KSig arms) [< fst', snd']) ret loc -- (type-case {⋯} ∷ _ return Q of { {} ⇒ s; ⋯ }) ⇝ s ∷ Q Enum {} => whnf defs ctx $ Ann (tycaseRhsDef0 def KEnum arms) ret loc -- (type-case Eq [i ⇒ A] L R ∷ ★ᵢ return Q -- of { Eq a₀ a₁ a l r ⇒ s; ⋯ }) ⇝ -- s[(A‹0/i› ∷ ★ᵢ)/a₀, (A‹1/i› ∷ ★ᵢ)/a₁, -- ((δ i ⇒ A) ∷ Eq [★ᵢ] A‹0/i› A‹1/i›)/a, -- (L ∷ A‹0/i›)/l, (R ∷ A‹1/i›)/r] ∷ Q Eq {ty = a, l, r, loc = eqLoc, _} => let a0 = a.zero; a1 = a.one in whnf defs ctx $ Ann (subN (tycaseRhsDef def KEq arms) [< Ann a0 (TYPE u noLoc) a.loc, Ann a1 (TYPE u noLoc) a.loc, Ann (DLam a a.loc) (Eq0 (TYPE u noLoc) a0 a1 a.loc) a.loc, Ann l a0 l.loc, Ann r a1 r.loc]) ret loc -- (type-case ℕ ∷ _ return Q of { ℕ ⇒ s; ⋯ }) ⇝ s ∷ Q Nat {} => whnf defs ctx $ Ann (tycaseRhsDef0 def KNat arms) ret loc -- (type-case [π.A] ∷ ★ᵢ return Q of { [a] ⇒ s; ⋯ }) ⇝ s[(A ∷ ★ᵢ)/a] ∷ Q BOX {ty = a, loc = boxLoc, _} => whnf defs ctx $ Ann (sub1 (tycaseRhsDef def KBOX arms) (Ann a (TYPE u noLoc) a.loc)) ret loc ||| pushes a coercion inside a whnf-ed term private covering pushCoe : {d, n : Nat} -> (defs : Definitions) -> WhnfContext d n -> BindName -> (ty : Term (S d) n) -> (0 tynf : No (isRedexT defs ty)) => Dim d -> Dim d -> (s : Term d n) -> (0 snf : No (isRedexT defs s)) => Loc -> WhnfM (NonRedex Elim d n defs) pushCoe defs ctx i ty p q s loc = if p == q then whnf defs ctx $ Ann s (ty // one q) loc else case s of -- (coe [_ ⇒ ★ᵢ] @_ @_ ty) ⇝ (ty ∷ ★ᵢ) TYPE {} => pure $ nred $ Ann s (TYPE !(unwrapTYPE ty) ty.loc) loc Pi {} => pure $ nred $ Ann s (TYPE !(unwrapTYPE ty) ty.loc) loc Sig {} => pure $ nred $ Ann s (TYPE !(unwrapTYPE ty) ty.loc) loc Enum {} => pure $ nred $ Ann s (TYPE !(unwrapTYPE ty) ty.loc) loc Eq {} => pure $ nred $ Ann s (TYPE !(unwrapTYPE ty) ty.loc) loc Nat {} => pure $ nred $ Ann s (TYPE !(unwrapTYPE ty) ty.loc) loc BOX {} => pure $ nred $ Ann s (TYPE !(unwrapTYPE ty) ty.loc) loc -- just η expand it. then whnf for App will handle it later -- this is how @xtt does it -- -- (coe [i ⇒ A] @p @q (λ x ⇒ s)) ⇝ -- (λ y ⇒ (coe [i ⇒ A] @p @q (λ x ⇒ s)) y) ∷ A‹q/i› ⇝ ⋯ lam@(Lam {body, _}) => do let lam' = CoeT i ty p q lam loc term' = LamY !(fresh body.name) (E $ App (weakE 1 lam') (BVT 0 noLoc) loc) loc type' = ty // one q whnf defs ctx $ Ann term' type' loc -- (coe [i ⇒ (x : A) × B] @p @q (s, t)) ⇝ -- (coe [i ⇒ A] @p @q s, -- coe [i ⇒ B[(coe [j ⇒ A‹j/i›] @p @i s)/x]] @p @q t) -- ∷ (x : A‹q/i›) × B‹q/i› -- -- can't use η here because... it doesn't exist Pair {fst, snd, loc = pairLoc} => do let Sig {fst = tfst, snd = tsnd, loc = sigLoc} = ty | _ => throw $ ExpectedSig ty.loc (extendDim i ctx.names) ty let fst' = E $ CoeT i tfst p q fst fst.loc tfst' = tfst // (B VZ noLoc ::: shift 2) tsnd' = sub1 tsnd $ CoeT !(fresh i) tfst' (weakD 1 p) (B VZ noLoc) (dweakT 1 fst) fst.loc snd' = E $ CoeT i tsnd' p q snd snd.loc pure $ Element (Ann (Pair fst' snd' pairLoc) (Sig (tfst // one q) (tsnd // one q) sigLoc) loc) Ah -- η expand, like for Lam -- -- (coe [i ⇒ A] @p @q (δ j ⇒ s)) ⇝ -- (δ k ⇒ (coe [i ⇒ A] @p @q (δ j ⇒ s)) @k) ∷ A‹q/i› ⇝ ⋯ dlam@(DLam {body, _}) => do let dlam' = CoeT i ty p q dlam loc term' = DLamY !(mnb "j") (E $ DApp (dweakE 1 dlam') (B VZ noLoc) loc) loc type' = ty // one q whnf defs ctx $ Ann term' type' loc -- (coe [_ ⇒ {⋯}] @_ @_ t) ⇝ (t ∷ {⋯}) Tag {tag, loc = tagLoc} => do let Enum {cases, loc = enumLoc} = ty | _ => throw $ ExpectedEnum ty.loc (extendDim i ctx.names) ty pure $ Element (Ann (Tag tag tagLoc) (Enum cases enumLoc) loc) Ah -- (coe [_ ⇒ ℕ] @_ @_ n) ⇝ (n ∷ ℕ) Zero {loc = zeroLoc} => do pure $ Element (Ann (Zero zeroLoc) (Nat ty.loc) loc) Ah Succ {p = pred, loc = succLoc} => do pure $ Element (Ann (Succ pred succLoc) (Nat ty.loc) loc) Ah -- (coe [i ⇒ [π.A]] @p @q [s]) ⇝ -- [coe [i ⇒ A] @p @q s] ∷ [π. A‹q/i›] Box {val, loc = boxLoc} => do let BOX {qty, ty = a, loc = tyLoc} = ty | _ => throw $ ExpectedBOX ty.loc (extendDim i ctx.names) ty pure $ Element (Ann (Box (E $ CoeT i a p q val val.loc) boxLoc) (BOX qty (a // one q) tyLoc) loc) Ah E e => pure $ Element (CoeT i ty p q (E e) e.loc) (snf `orNo` Ah) where unwrapTYPE : Term (S d) n -> WhnfM Universe unwrapTYPE (TYPE {l, _}) = pure l unwrapTYPE ty = throw $ ExpectedTYPE ty.loc (extendDim i ctx.names) ty export covering Whnf Elim Reduce.isRedexE where whnf defs ctx (F x u loc) with (lookupElim x defs) proof eq _ | Just y = whnf defs ctx $ setLoc loc $ displace u y _ | Nothing = pure $ Element (F x u loc) $ rewrite eq in Ah whnf _ _ (B i loc) = pure $ nred $ B i loc -- ((λ x ⇒ t) ∷ (π.x : A) → B) s ⇝ t[s∷A/x] ∷ B[s∷A/x] whnf defs ctx (App f s appLoc) = do Element f fnf <- whnf defs ctx f case nchoose $ isLamHead f of Left _ => case f of Ann (Lam {body, _}) (Pi {arg, res, _}) floc => let s = Ann s arg s.loc in whnf defs ctx $ Ann (sub1 body s) (sub1 res s) appLoc Coe ty p q val _ => piCoe defs ctx ty p q val s appLoc Right nlh => pure $ Element (App f s appLoc) $ fnf `orNo` nlh -- case (s, t) ∷ (x : A) × B return p ⇒ C of { (a, b) ⇒ u } ⇝ -- u[s∷A/a, t∷B[s∷A/x]] ∷ C[(s, t)∷((x : A) × B)/p] whnf defs ctx (CasePair pi pair ret body caseLoc) = do Element pair pairnf <- whnf defs ctx pair case nchoose $ isPairHead pair of Left _ => case pair of Ann (Pair {fst, snd, _}) (Sig {fst = tfst, snd = tsnd, _}) pairLoc => let fst = Ann fst tfst fst.loc snd = Ann snd (sub1 tsnd fst) snd.loc in whnf defs ctx $ Ann (subN body [< fst, snd]) (sub1 ret pair) caseLoc Coe ty p q val _ => do sigCoe defs ctx pi ty p q val ret body caseLoc Right np => pure $ Element (CasePair pi pair ret body caseLoc) $ pairnf `orNo` np -- case 'a ∷ {a,…} return p ⇒ C of { 'a ⇒ u } ⇝ -- u ∷ C['a∷{a,…}/p] whnf defs ctx (CaseEnum pi tag ret arms caseLoc) = do Element tag tagnf <- whnf defs ctx tag case nchoose $ isTagHead tag of Left _ => case tag of Ann (Tag t _) (Enum ts _) _ => let ty = sub1 ret tag in case lookup t arms of Just arm => whnf defs ctx $ Ann arm ty arm.loc Nothing => throw $ MissingEnumArm caseLoc t (keys arms) Coe ty p q val _ => -- there is nowhere an equality can be hiding inside an enum type whnf defs ctx $ CaseEnum pi (Ann val (dsub1 ty q) val.loc) ret arms caseLoc Right nt => pure $ Element (CaseEnum pi tag ret arms caseLoc) $ tagnf `orNo` nt -- case zero ∷ ℕ return p ⇒ C of { zero ⇒ u; … } ⇝ -- u ∷ C[zero∷ℕ/p] -- -- case succ n ∷ ℕ return p ⇒ C of { succ n', π.ih ⇒ u; … } ⇝ -- u[n∷ℕ/n', (case n ∷ ℕ ⋯)/ih] ∷ C[succ n ∷ ℕ/p] whnf defs ctx (CaseNat pi piIH nat ret zer suc caseLoc) = do Element nat natnf <- whnf defs ctx nat case nchoose $ isNatHead nat of Left _ => let ty = sub1 ret nat in case nat of Ann (Zero _) (Nat _) _ => whnf defs ctx $ Ann zer ty zer.loc Ann (Succ n succLoc) (Nat natLoc) _ => let nn = Ann n (Nat natLoc) succLoc tm = subN suc [< nn, CaseNat pi piIH nn ret zer suc caseLoc] in whnf defs ctx $ Ann tm ty caseLoc Coe ty p q val _ => -- same deal as Enum whnf defs ctx $ CaseNat pi piIH (Ann val (dsub1 ty q) val.loc) ret zer suc caseLoc Right nn => pure $ Element (CaseNat pi piIH nat ret zer suc caseLoc) $ natnf `orNo` nn -- case [t] ∷ [π.A] return p ⇒ C of { [x] ⇒ u } ⇝ -- u[t∷A/x] ∷ C[[t] ∷ [π.A]/p] whnf defs ctx (CaseBox pi box ret body caseLoc) = do Element box boxnf <- whnf defs ctx box case nchoose $ isBoxHead box of Left _ => case box of Ann (Box val boxLoc) (BOX q bty tyLoc) _ => let ty = sub1 ret box in whnf defs ctx $ Ann (sub1 body (Ann val bty val.loc)) ty caseLoc Coe ty p q val _ => boxCoe defs ctx pi ty p q val ret body caseLoc Right nb => pure $ Element (CaseBox pi box ret body caseLoc) $ boxnf `orNo` nb -- e : Eq (𝑗 ⇒ A) t u ⊢ e @0 ⇝ t ∷ A‹0/𝑗› -- e : Eq (𝑗 ⇒ A) t u ⊢ e @1 ⇝ u ∷ A‹1/𝑗› -- -- ((δ 𝑖 ⇒ s) ∷ Eq (𝑗 ⇒ A) t u) @𝑘 ⇝ s‹𝑘/𝑖› ∷ A‹𝑘/𝑗› whnf defs ctx (DApp f p appLoc) = do Element f fnf <- whnf defs ctx f case nchoose $ isDLamHead f of Left _ => case f of Ann (DLam {body, _}) (Eq {ty, l, r, _}) _ => whnf defs ctx $ Ann (endsOr (setLoc appLoc l) (setLoc appLoc r) (dsub1 body p) p) (dsub1 ty p) appLoc Coe ty p' q' val _ => eqCoe defs ctx ty p' q' val p appLoc Right ndlh => case p of K e _ => do Eq {l, r, ty, _} <- whnf0 defs ctx =<< computeElimType defs ctx f | ty => throw $ ExpectedEq ty.loc ctx.names ty whnf defs ctx $ ends (Ann (setLoc appLoc l) ty.zero appLoc) (Ann (setLoc appLoc r) ty.one appLoc) e B {} => pure $ Element (DApp f p appLoc) $ fnf `orNo` ndlh `orNo` Ah -- e ∷ A ⇝ e whnf defs ctx (Ann s a annLoc) = do Element s snf <- whnf defs ctx s case nchoose $ isE s of Left _ => let E e = s in pure $ Element e $ noOr2 snf Right ne => do Element a anf <- whnf defs ctx a pure $ Element (Ann s a annLoc) $ ne `orNo` snf `orNo` anf whnf defs ctx (Coe (S _ (N ty)) _ _ val coeLoc) = whnf defs ctx $ Ann val ty coeLoc whnf defs ctx (Coe (S [< i] ty) p q val coeLoc) = do Element ty tynf <- whnf defs (extendDim i ctx) ty.term Element val valnf <- whnf defs ctx val pushCoe defs ctx i ty p q val coeLoc whnf defs ctx (Comp ty p q val r zero one compLoc) = -- comp [A] @p @p s { ⋯ } ⇝ s ∷ A if p == q then whnf defs ctx $ Ann val ty compLoc else case nchoose (isK r) of -- comp [A] @p @q s @0 { 0 j ⇒ t; ⋯ } ⇝ t‹q/j› ∷ A -- comp [A] @p @q s @1 { 1 j ⇒ t; ⋯ } ⇝ t‹q/j› ∷ A Left y => case r of K Zero _ => whnf defs ctx $ Ann (dsub1 zero q) ty compLoc K One _ => whnf defs ctx $ Ann (dsub1 one q) ty compLoc Right nk => do Element ty tynf <- whnf defs ctx ty pure $ Element (Comp ty p q val r zero one compLoc) $ tynf `orNo` nk whnf defs ctx (TypeCase ty ret arms def tcLoc) = do Element ty tynf <- whnf defs ctx ty Element ret retnf <- whnf defs ctx ret case nchoose $ isAnnTyCon ty of Left y => let Ann ty (TYPE u _) _ = ty in reduceTypeCase defs ctx ty u ret arms def tcLoc Right nt => pure $ Element (TypeCase ty ret arms def tcLoc) (tynf `orNo` retnf `orNo` nt) whnf defs ctx (CloE (Sub el th)) = whnf defs ctx $ pushSubstsWith' id th el whnf defs ctx (DCloE (Sub el th)) = whnf defs ctx $ pushSubstsWith' th id el export covering Whnf Term Reduce.isRedexT where whnf _ _ t@(TYPE {}) = pure $ nred t whnf _ _ t@(Pi {}) = pure $ nred t whnf _ _ t@(Lam {}) = pure $ nred t whnf _ _ t@(Sig {}) = pure $ nred t whnf _ _ t@(Pair {}) = pure $ nred t whnf _ _ t@(Enum {}) = pure $ nred t whnf _ _ t@(Tag {}) = pure $ nred t whnf _ _ t@(Eq {}) = pure $ nred t whnf _ _ t@(DLam {}) = pure $ nred t whnf _ _ t@(Nat {}) = pure $ nred t whnf _ _ t@(Zero {}) = pure $ nred t whnf _ _ t@(Succ {}) = pure $ nred t whnf _ _ t@(BOX {}) = pure $ nred t whnf _ _ t@(Box {}) = pure $ nred t -- s ∷ A ⇝ s (in term context) whnf defs ctx (E e) = do Element e enf <- whnf defs ctx e case nchoose $ isAnn e of Left _ => let Ann {tm, _} = e in pure $ Element tm $ noOr1 $ noOr2 enf Right na => pure $ Element (E e) $ na `orNo` enf whnf defs ctx (CloT (Sub tm th)) = whnf defs ctx $ pushSubstsWith' id th tm whnf defs ctx (DCloT (Sub tm th)) = whnf defs ctx $ pushSubstsWith' th id tm