add SnocVect.select

remove stray comment
more nat bitwise ops
2023-06-24 14:28:56 +02:00 · 2023-06-24 14:28:56 +02:00 · 2023-06-24 14:28:52 +02:00 · 2023-06-24 14:28:48 +02:00 · 2023-06-23 18:32:17 +02:00 · 2023-06-23 18:32:17 +02:00
31 changed files with 1622 additions and 410 deletions
--- a/examples/all.quox
+++ b/examples/all.quox
@ -1,6 +1,7 @@
 load "misc.quox"
 load "bool.quox"
 load "either.quox"
+load "maybe.quox"
 load "nat.quox"
 load "pair.quox"
 load "list.quox"
--- a/examples/bool.quox
+++ b/examples/bool.quox
@ -10,7 +10,6 @@ def boolω : 1.Bool → [ω.Bool] =
 def if : 0.(A : ★) → 1.Bool → ω.A → ω.A → A =
  λ A b t f ⇒ case1 b return A of { 'true ⇒ t; 'false ⇒ f };

-- [todo]: universe lifting
 def0 If : 1.Bool → 0.★ → 0.★ → ★ =
  λ b T F ⇒ case1 b return ★ of { 'true ⇒ T; 'false ⇒ F };

--- a/examples/maybe.quox
+++ b/examples/maybe.quox
@ -0,0 +1,69 @@
+load "misc.quox"
+load "either.quox"
+
+namespace maybe {
+
+def0 Tag : ★ = {nothing, just}
+
+def0 Payload : ω.Tag → ω.★ → ★ =
+  λ tag A ⇒ caseω tag return ★ of { 'nothing ⇒ True; 'just ⇒ A }
+
+def0 Maybe : ω.★ → ★ =
+  λ A ⇒ (t : Tag) × Payload t A
+
+def tag : 0.(A : ★) → ω.(Maybe A) → Tag =
+  λ _ x ⇒ caseω x return Tag of { (tag, _) ⇒ tag }
+
+def Nothing : 0.(A : ★) → Maybe A =
+  λ _ ⇒ ('nothing, 'true)
+
+def Just : 0.(A : ★) → 1.A → Maybe A =
+  λ _ x ⇒ ('just, x)
+
+def0 IsJustTag : ω.Tag → ★ =
+  λ t ⇒ caseω t return ★ of { 'just ⇒ True; 'nothing ⇒ False }
+
+def0 IsJust : 0.(A : ★) → ω.(Maybe A) → ★ =
+  λ A x ⇒ IsJustTag (tag A x)
+
+def is-just? : 0.(A : ★) → ω.(x : Maybe A) → Dec (IsJust A x) =
+  λ A x ⇒
+  caseω tag A x return t ⇒ Dec (IsJustTag t) of {
+    'just    ⇒ Yes True  'true;
+    'nothing ⇒ No  False (λ x ⇒ x)
+  }
+
+def0 nothing-unique :
+    0.(A : ★) → ω.(x : True) → ('nothing, x) ≡ Nothing A : Maybe A =
+  λ A x ⇒
+  caseω x return x' ⇒ ('nothing, x') ≡ Nothing A : Maybe A of {
+    'true ⇒ δ _ ⇒ ('nothing, 'true)
+  }
+
+def elim :
+    0.(A : ★) →
+    0.(P : 0.(Maybe A) → ★) →
+    ω.(P (Nothing A)) →
+    ω.(ω.(x : A) → P (Just A x)) →
+    1.(x : Maybe A) → P x =
+  λ A P n j x ⇒
+  caseω x return x' ⇒ P x' of {
+    (tag, payload) ⇒
+      (caseω tag
+       return t ⇒
+         0.(eq : tag ≡ t : Tag) → P (t, coe (i ⇒ Payload (eq @i) A) payload)
+       of {
+         'nothing ⇒
+            λ eq ⇒
+            caseω coe (i ⇒ Payload (eq @i) A) payload
+            return p ⇒ P ('nothing, p)
+            of { 'true ⇒ n };
+         'just ⇒ λ eq ⇒ j (coe (i ⇒ Payload (eq @i) A) payload)
+       }) (δ _ ⇒ tag)
+  }
+
+}
+
+def0 Maybe   = maybe.Maybe
+def0 Just    = maybe.Just
+def0 Nothing = maybe.Nothing
--- a/lib/Quox/Definition.idr
+++ b/lib/Quox/Definition.idr
@ -74,10 +74,6 @@ public export
 DefsState : Type -> Type
 DefsState = StateL DEFS Definitions

-export
-defs : Has DefsReader fs => Eff fs Definitions
-defs = askAt DEFS
-

 public export %inline
 lookupElim : {d, n : Nat} -> Name -> Definitions -> Maybe (Elim d n)
--- a/lib/Quox/Equal.idr
+++ b/lib/Quox/Equal.idr
@ -347,9 +347,10 @@ parameters (defs : Definitions)
      --    Γ ⊢ Eq [i ⇒ A₁] l₁ r₂ <: Eq [i ⇒ A₂] l₂ r₂
      compareType (extendDim sTy.name Zero ctx) sTy.zero tTy.zero
      compareType (extendDim sTy.name One  ctx) sTy.one  tTy.one
+      let ty = case !mode of Super => sTy; _ => tTy
      local_ Equal $ do
-        Term.compare0 ctx sTy.zero sl tl
-        Term.compare0 ctx sTy.one  sr tr
+        Term.compare0 ctx ty.zero sl tl
+        Term.compare0 ctx ty.one  sr tr

    compareType' ctx s@(Enum tags1 {}) t@(Enum tags2 {}) = do
      -- ------------------
@ -636,7 +637,7 @@ parameters (loc : Loc) (ctx : TyContext d n)
    private
    runCompare : (Definitions -> EqContext n -> DSubst d 0 -> Equal_ ()) ->
                 Equal ()
-    runCompare act = fromEqual_ $ eachFace $ act !defs
+    runCompare act = fromEqual_ $ eachFace $ act !(askAt DEFS)

    namespace Term
      export covering
--- a/lib/Quox/NatExtra.idr
+++ b/lib/Quox/NatExtra.idr
@ -1,12 +1,19 @@
 module Quox.NatExtra

 import public Data.Nat
+import public Data.Nat.Views
 import Data.Nat.Division
 import Data.SnocList
 import Data.Vect
+import Syntax.PreorderReasoning

 %default total

+infixl 8 `shiftL`, `shiftR`
+infixl 7 .&.
+infixl 6 `xor`
+infixl 5 .|.
+

 public export
 data LTE' : Nat -> Nat -> Type where
@ -55,3 +62,148 @@ parameters {base : Nat} {auto 0 _ : base `GTE` 2} (chars : Vect base Char)
 export
 showHex : Nat -> String
 showHex = showAtBase $ fromList $ unpack "0123456789ABCDEF"
+
+
+export
+0 notEvenOdd : (a, b : Nat) -> Not (a + a = S (b + b))
+notEvenOdd 0     b prf = absurd prf
+notEvenOdd (S a) b prf =
+  notEvenOdd b a $ Calc $
+    |~ b + b
+    ~~ a + S a   ..<(inj S prf)
+    ~~ S (a + a) ..<(plusSuccRightSucc {})
+
+export
+0 doubleInj : (m, n : Nat) -> m + m = n + n -> m = n
+doubleInj 0     0     _   = Refl
+doubleInj (S m) (S n) prf =
+  cong S $ doubleInj m n $
+  inj S $ Calc $
+    |~ S (m + m)
+    ~~ m + S m   ...(plusSuccRightSucc {})
+    ~~ n + S n   ...(inj S prf)
+    ~~ S (n + n) ..<(plusSuccRightSucc {})
+
+export
+0 halfDouble : (n : Nat) -> half (n + n) = HalfEven n
+halfDouble n with (half (n + n)) | (n + n) proof nn
+  _ | HalfOdd  k | S (k + k) = void $ notEvenOdd n k nn
+  _ | HalfEven k | k + k     = rewrite doubleInj n k nn in Refl
+
+export
+floorHalf : Nat -> Nat
+floorHalf k = case half k of
+  HalfOdd  n => n
+  HalfEven n => n
+
+
+||| like in intercal ☺
+|||
+||| take all the bits of `subj` that are set in `mask`, and squish them down
+||| towards the lsb
+export
+select : (mask, subj : Nat) -> Nat
+select mask subj = go 1 (halfRec mask) subj 0 where
+  go : forall mask. Nat -> HalfRec mask -> Nat -> Nat -> Nat
+  go bit HalfRecZ            subj res = res
+  go bit (HalfRecEven _ rec) subj res = go bit rec (floorHalf subj) res
+  go bit (HalfRecOdd  _ rec) subj res = case half subj of
+    HalfOdd  subj => go (bit + bit) rec subj (res + bit)
+    HalfEven subj => go (bit + bit) rec subj res
+
+||| take the i least significant bits of subj (where i = popCount mask),
+||| and place them where mask's set bits are
+|||
+||| left inverse of select if mask .|. subj = mask
+export
+spread : (mask, subj : Nat) -> Nat
+spread mask subj = go 1 (halfRec mask) subj 0 where
+  go : forall mask. Nat -> HalfRec mask -> Nat -> Nat -> Nat
+  go bit HalfRecZ            subj res = res
+  go bit (HalfRecEven _ rec) subj res = go (bit + bit) rec subj res
+  go bit (HalfRecOdd  _ rec) subj res = case half subj of
+    HalfOdd  subj => go (bit + bit) rec subj (res + bit)
+    HalfEven subj => go (bit + bit) rec subj res
+
+
+
+public export
+data BitwiseRec : Nat -> Nat -> Type where
+  BwDone : BitwiseRec 0 0
+  Bw00   : (m, n : Nat) -> Lazy (BitwiseRec m n) ->
+           BitwiseRec (m + m) (n + n)
+  Bw01   : (m, n : Nat) -> Lazy (BitwiseRec m n) ->
+           BitwiseRec (m + m) (S (n + n))
+  Bw10   : (m, n : Nat) -> Lazy (BitwiseRec m n) ->
+           BitwiseRec (S (m + m)) (n + n)
+  Bw11   : (m, n : Nat) -> Lazy (BitwiseRec m n) ->
+           BitwiseRec (S (m + m)) (S (n + n))
+
+export
+bitwiseRec : (m, n : Nat) -> BitwiseRec m n
+bitwiseRec m n = go (halfRec m) (halfRec n) where
+  go : forall m, n. HalfRec m -> HalfRec n -> BitwiseRec m n
+  go HalfRecZ           HalfRecZ           = BwDone
+  go HalfRecZ           (HalfRecEven n nr) = Bw00 0 n $ go HalfRecZ nr
+  go HalfRecZ           (HalfRecOdd  n nr) = Bw01 0 n $ go HalfRecZ nr
+  go (HalfRecEven m mr) HalfRecZ           = Bw00 m 0 $ go mr HalfRecZ
+  go (HalfRecEven m mr) (HalfRecEven n nr) = Bw00 m n $ go mr nr
+  go (HalfRecEven m mr) (HalfRecOdd  n nr) = Bw01 m n $ go mr nr
+  go (HalfRecOdd  m mr) HalfRecZ           = Bw10 m 0 $ go mr HalfRecZ
+  go (HalfRecOdd  m mr) (HalfRecEven n nr) = Bw10 m n $ go mr nr
+  go (HalfRecOdd  m mr) (HalfRecOdd  n nr) = Bw11 m n $ go mr nr
+
+export
+bitwise : (Bool -> Bool -> Bool) -> Nat -> Nat -> Nat
+bitwise f m n = go 1 (bitwiseRec m n) 0 where
+  one : Bool -> Bool -> Nat -> Nat -> Nat
+  one p q bit res = if f p q then bit + res else res
+  go : forall m, n. Nat -> BitwiseRec m n -> Nat -> Nat
+  go bit BwDone         res = res
+  go bit (Bw00 m n rec) res = go (bit + bit) rec $ one False False bit res
+  go bit (Bw01 m n rec) res = go (bit + bit) rec $ one False True  bit res
+  go bit (Bw10 m n rec) res = go (bit + bit) rec $ one True  False bit res
+  go bit (Bw11 m n rec) res = go (bit + bit) rec $ one True  True  bit res
+
+export
+(.&.) : Nat -> Nat -> Nat
+(.&.) = bitwise $ \p, q => p && q
+
+private %foreign "scheme:blodwen-and"
+primAnd : Nat -> Nat -> Nat
+%transform "NatExtra.(.&.)" NatExtra.(.&.) m n = primAnd m n
+
+export
+(.|.) : Nat -> Nat -> Nat
+(.|.) = bitwise $ \p, q => p || q
+
+private %foreign "scheme:blodwen-or"
+primOr : Nat -> Nat -> Nat
+%transform "NatExtra.(.|.)" NatExtra.(.|.) m n = primOr m n
+
+export
+xor : Nat -> Nat -> Nat
+xor = bitwise (/=)
+
+private %foreign "scheme:blodwen-xor"
+primXor : Nat -> Nat -> Nat
+%transform "NatExtra.xor" NatExtra.xor m n = primXor m n
+
+
+export
+shiftL : Nat -> Nat -> Nat
+shiftL n 0     = n
+shiftL n (S i) = shiftL (n + n) i
+
+private %foreign "scheme:blodwen-shl"
+primShiftL : Nat -> Nat -> Nat
+%transform "NatExtra.shiftL" NatExtra.shiftL n i = primShiftL n i
+
+export
+shiftR : Nat -> Nat -> Nat
+shiftR n 0     = n
+shiftR n (S i) = shiftL (floorHalf n) i
+
+private %foreign "scheme:blodwen-shr"
+primShiftR : Nat -> Nat -> Nat
+%transform "NatExtra.shiftR" NatExtra.shiftR n i = primShiftR n i
--- a/lib/Quox/Parser/FromParser.idr
+++ b/lib/Quox/Parser/FromParser.idr
@ -255,7 +255,6 @@ mutual
      t   => let ctx = MkNameContexts (map fromPatVar ds) (map fromPatVar ns) in
             throw $ AnnotationNeeded t.loc ctx t

-  -- [todo] use SN if the var is named but still unused
  private
  fromPTermTScope : {s : Nat} -> Context' PatVar d -> Context' PatVar n ->
                    SnocVect s PatVar -> PTerm ->
--- a/lib/Quox/Parser/FromParser/Error.idr
+++ b/lib/Quox/Parser/FromParser/Error.idr
@ -6,6 +6,8 @@ import System.File

 import Quox.Pretty

+%hide Text.PrettyPrint.Prettyprinter.Doc.infixr.(<++>)
+

 public export
 TypeError, LexError, ParseError : Type
--- a/lib/Quox/Reduce.idr
+++ b/lib/Quox/Reduce.idr
@ -14,7 +14,6 @@ import Control.Eff
 %default total


-- [fixme] rename this to Whnf and the interface to CanWhnf
 public export
 Whnf : List (Type -> Type)
 Whnf = [NameGen, Except Error]
--- a/lib/Quox/Syntax/Term/Base.idr
+++ b/lib/Quox/Syntax/Term/Base.idr
@ -1,5 +1,6 @@
 module Quox.Syntax.Term.Base

+import public Quox.Thin
 import public Quox.Syntax.Var
 import public Quox.Syntax.Shift
 import public Quox.Syntax.Subst
@ -18,9 +19,6 @@ import Data.Maybe
 import Data.Nat
 import public Data.So
 import Data.String
-import public Data.SortedMap
-import public Data.SortedMap.Dependent
-import public Data.SortedSet
 import Derive.Prelude

 %default total
@ -46,345 +44,300 @@ TagVal : Type
 TagVal = String


+||| type-checkable terms, which consists of types and constructor forms.
+|||
+||| first argument `d` is dimension scope size; second `n` is term scope size
 public export
-data ScopedBody : Nat -> (Nat -> Type) -> Nat -> Type where
-  Y : (body : f (s + n)) -> ScopedBody s f n
-  N : (body : f n)       -> ScopedBody s f n
-%name ScopedBody body
+data Term : (d, n : Nat) -> Type
+%name Term s, t, r

-export %inline %hint
-EqScopedBody : (forall n. Eq (f n)) => Eq (ScopedBody s f n)
-EqScopedBody = deriveEq
-
-export %inline %hint
-ShowScopedBody : (forall n. Show (f n)) => Show (ScopedBody s f n)
-ShowScopedBody = deriveShow
-
-||| a scoped term with names
+||| inferrable terms, which consists of elimination forms like application and
+||| `case` (as well as other terms with an annotation)
+|||
+||| first argument `d` is dimension scope size; second `n` is term scope size
 public export
-record Scoped (s : Nat) (f : Nat -> Type) (n : Nat) where
-  constructor S
-  names : BContext s
-  body  : ScopedBody s f n
-%name Scoped body
-
-export %inline
-(forall n. Eq (f n)) => Eq (Scoped s f n) where
-  s == t = s.body == t.body
-
-export %inline %hint
-ShowScoped : (forall n. Show (f n)) => Show (Scoped s f n)
-ShowScoped = deriveShow
+data Elim : (d, n : Nat) -> Type
+%name Elim e, f


-infixl 8 :#
-infixl 9 :@, :%
-mutual
-  public export
-  TSubst : TSubstLike
-  TSubst d = Subst $ \n => Elim d n
+public export
+ScopeTermN : Nat -> TermLike
+ScopeTermN s d n = ScopedN s (\n => Term d n) n

-  ||| first argument `d` is dimension scope size;
-  ||| second `n` is term scope size
-  public export
-  data Term : (d, n : Nat) -> Type where
-    ||| type of types
-    TYPE : (l : Universe) -> (loc : Loc) -> Term d n
+public export
+DScopeTermN : Nat -> TermLike
+DScopeTermN s d n = ScopedN s (\d => Term d n) d

-    ||| function type
-    Pi  : (qty : Qty) -> (arg : Term d n) ->
-          (res : ScopeTerm d n) -> (loc : Loc) -> Term d n
-    ||| function term
-    Lam : (body : ScopeTerm d n) -> (loc : Loc) -> Term d n
+public export
+ScopeTerm : TermLike
+ScopeTerm = ScopeTermN 1

-    ||| pair type
-    Sig : (fst : Term d n) -> (snd : ScopeTerm d n) -> (loc : Loc) -> Term d n
-    ||| pair value
-    Pair : (fst, snd : Term d n) -> (loc : Loc) -> Term d n
-
-    ||| enumeration type
-    Enum : (cases : SortedSet TagVal) -> (loc : Loc) -> Term d n
-    ||| enumeration value
-    Tag : (tag : TagVal) -> (loc : Loc) -> Term d n
-
-    ||| equality type
-    Eq : (ty : DScopeTerm d n) -> (l, r : Term d n) -> (loc : Loc) -> Term d n
-    ||| equality term
-    DLam : (body : DScopeTerm d n) -> (loc : Loc) -> Term d n
-
-    ||| natural numbers (temporary until 𝐖 gets added)
-    Nat  : (loc : Loc) -> Term d n
-    -- [todo] can these be elims?
-    Zero : (loc : Loc) -> Term d n
-    Succ : (p : Term d n) -> (loc : Loc) -> Term d n
-
-    ||| "box" (package a value up with a certain quantity)
-    BOX : (qty : Qty) -> (ty : Term d n) -> (loc : Loc) -> Term d n
-    Box : (val : Term d n) -> (loc : Loc) -> Term d n
-
-    ||| elimination
-    E : (e : Elim d n) -> Term d n
-
-    ||| term closure/suspended substitution
-    CloT  : WithSubst (Term d) (Elim d) n -> Term d n
-    ||| dimension closure/suspended substitution
-    DCloT : WithSubst (\d => Term d n) Dim d -> Term d n
-  %name Term s, t, r
-
-  ||| first argument `d` is dimension scope size, second `n` is term scope size
-  public export
-  data Elim : (d, n : Nat) -> Type where
-    ||| free variable, possibly with a displacement (see @crude, or @mugen for a
-    ||| more abstract and formalised take)
-    |||
-    ||| e.g. if f : ★₀ → ★₁, then f¹ : ★₁ → ★₂
-    F : (x : Name) -> (u : Universe) -> (loc : Loc) -> Elim d n
-    ||| bound variable
-    B : (i : Var n) -> (loc : Loc) -> Elim d n
-
-    ||| term application
-    App : (fun : Elim d n) -> (arg : Term d n) -> (loc : Loc) -> Elim d n
-
-    ||| pair destruction
-    |||
-    ||| `CasePair 𝜋 𝑒 ([𝑟], 𝐴) ([𝑥, 𝑦], 𝑡)` is
-    ||| `𝐜𝐚𝐬𝐞 𝜋 · 𝑒 𝐫𝐞𝐭𝐮𝐫𝐧 𝑟 ⇒ 𝐴 𝐨𝐟 { (𝑥, 𝑦) ⇒ 𝑡 }`
-    CasePair : (qty : Qty) -> (pair : Elim d n) ->
-               (ret : ScopeTerm d n) ->
-               (body : ScopeTermN 2 d n) ->
-               (loc : Loc) ->
-               Elim d n
-
-    ||| enum matching
-    CaseEnum : (qty : Qty) -> (tag : Elim d n) ->
-               (ret : ScopeTerm d n) ->
-               (arms : CaseEnumArms d n) ->
-               (loc : Loc) ->
-               Elim d n
-
-    ||| nat matching
-    CaseNat : (qty, qtyIH : Qty) -> (nat : Elim d n) ->
-              (ret : ScopeTerm d n) ->
-              (zero : Term d n) ->
-              (succ : ScopeTermN 2 d n) ->
-              (loc : Loc) ->
-              Elim d n
-
-    ||| unboxing
-    CaseBox : (qty : Qty) -> (box : Elim d n) ->
-              (ret : ScopeTerm d n) ->
-              (body : ScopeTerm d n) ->
-              (loc : Loc) ->
-              Elim d n
-
-    ||| dim application
-    DApp : (fun : Elim d n) -> (arg : Dim d) -> (loc : Loc) -> Elim d n
-
-    ||| type-annotated term
-    Ann : (tm, ty : Term d n) -> (loc : Loc) -> Elim d n
-
-    ||| coerce a value along a type equality, or show its coherence
-    ||| [@xtt; §2.1.1]
-    Coe : (ty : DScopeTerm d n) -> (p, q : Dim d) ->
-          (val : Term d n) -> (loc : Loc) -> Elim d n
-
-    ||| "generalised composition" [@xtt; §2.1.2]
-    Comp : (ty : Term d n) -> (p, q : Dim d) ->
-           (val : Term d n) -> (r : Dim d) ->
-           (zero, one : DScopeTerm d n) -> (loc : Loc) -> Elim d n
-
-    ||| match on types. needed for b.s. of coercions [@xtt; §2.2]
-    TypeCase : (ty : Elim d n) -> (ret : Term d n) ->
-               (arms : TypeCaseArms d n) -> (def : Term d n) ->
-               (loc : Loc) ->
-               Elim d n
-
-    ||| term closure/suspended substitution
-    CloE  : WithSubst (Elim d) (Elim d) n -> Elim d n
-    ||| dimension closure/suspended substitution
-    DCloE : WithSubst (\d => Elim d n) Dim d -> Elim d n
-  %name Elim e, f
-
-  public export
-  CaseEnumArms : TermLike
-  CaseEnumArms d n = SortedMap TagVal (Term d n)
-
-  public export
-  TypeCaseArms : TermLike
-  TypeCaseArms d n = SortedDMap TyConKind (\k => TypeCaseArmBody k d n)
-
-  public export
-  TypeCaseArm : TermLike
-  TypeCaseArm d n = (k ** TypeCaseArmBody k d n)
-
-  public export
-  TypeCaseArmBody : TyConKind -> TermLike
-  TypeCaseArmBody k = ScopeTermN (arity k)
+public export
+DScopeTerm : TermLike
+DScopeTerm = DScopeTermN 1


-  public export
-  ScopeTermN, DScopeTermN : Nat -> TermLike
-  ScopeTermN  s d n = Scoped s (Term d) n
-  DScopeTermN s d n = Scoped s (\d => Term d n) d
+public export
+TermT : TermLike
+TermT = Thinned2 (\d, n => Term d n)

-  public export
-  ScopeTerm, DScopeTerm : TermLike
-  ScopeTerm  = ScopeTermN 1
-  DScopeTerm = DScopeTermN 1
+public export
+ElimT : TermLike
+ElimT = Thinned2 (\d, n => Elim d n)

-mutual
-  export %hint
-  EqTerm : Eq (Term d n)
-  EqTerm = assert_total {a = Eq (Term d n)} deriveEq

-  export %hint
-  EqElim : Eq (Elim d n)
-  EqElim = assert_total {a = Eq (Elim d n)} deriveEq
+public export
+DimArg : TermLike
+DimArg d n = Dim d

-mutual
-  export %hint
-  ShowTerm : Show (Term d n)
-  ShowTerm = assert_total {a = Show (Term d n)} deriveShow

-  export %hint
-  ShowElim : Show (Elim d n)
-  ShowElim = assert_total {a = Show (Elim d n)} deriveShow
+data Term where
+  ||| type of types
+  TYPE : (l : Universe) -> (loc : Loc) -> Term 0 0

-||| scope which ignores all its binders
-public export %inline
-SN : {s : Nat} -> f n -> Scoped s f n
-SN = S (replicate s $ BN Unused noLoc) . N
+  ||| function type
+  Pi : Qty -> Subterms [Term, ScopeTerm] d n -> Loc -> Term d n
+  ||| function value
+  Lam : ScopeTerm d n -> Loc -> Term d n

-||| scope which uses its binders
-public export %inline
-SY : BContext s -> f (s + n) -> Scoped s f n
-SY ns = S ns . Y
+  ||| pair type
+  Sig : Subterms [Term, ScopeTerm] d n -> Loc -> Term d n
+  ||| pair value
+  Pair : Subterms [Term, Term] d n -> Loc -> Term d n

-public export %inline
-name : Scoped 1 f n -> BindName
-name (S [< x] _) = x
+  ||| enum type
+  Enum : List TagVal -> Loc -> Term 0 0
+  ||| enum value
+  Tag : TagVal -> Loc -> Term 0 0

-public export %inline
-(.name) : Scoped 1 f n -> BindName
-s.name = name s
+  ||| equality type
+  Eq : Subterms [DScopeTerm, Term, Term] d n -> Loc -> Term d n
+  ||| equality value
+  DLam : DScopeTerm d n -> Loc -> Term d n

-||| more convenient Pi
-public export %inline
-PiY : (qty : Qty) -> (x : BindName) ->
-      (arg : Term d n) -> (res : Term d (S n)) -> (loc : Loc) -> Term d n
-PiY {qty, x, arg, res, loc} = Pi {qty, arg, res = SY [< x] res, loc}
+  ||| natural numbers (temporary until 𝐖 gets added)
+  Nat  : Loc -> Term 0 0
+  Zero : Loc -> Term 0 0
+  Succ : Term d n -> Loc -> Term 0 0

-||| more convenient Lam
-public export %inline
-LamY : (x : BindName) -> (body : Term d (S n)) -> (loc : Loc) -> Term d n
-LamY {x, body, loc} = Lam {body = SY [< x] body, loc}
+  ||| package a value with a quantity
+  ||| e.g. a value of [ω. A], when unpacked, can be used ω times,
+  ||| even if the box itself is linear
+  BOX : Qty -> Term d n -> Loc -> Term d n
+  Box : Term d n -> Loc -> Term d n

-public export %inline
-LamN : (body : Term d n) -> (loc : Loc) -> Term d n
-LamN {body, loc} = Lam {body = SN body, loc}
+  E : Elim d n -> Term d n

-||| non dependent function type
-public export %inline
-Arr : (qty : Qty) -> (arg, res : Term d n) -> (loc : Loc) -> Term d n
-Arr {qty, arg, res, loc} = Pi {qty, arg, res = SN res, loc}
+  ||| term closure/suspended substitution
+  CloT  : WithSubst (Term d) (Elim d) n -> Term d n
+  ||| dimension closure/suspended substitution
+  DCloT : WithSubst (\d => Term d n) Dim d -> Term d n

-||| more convenient Sig
-public export %inline
-SigY : (x : BindName) -> (fst : Term d n) ->
-       (snd : Term d (S n)) -> (loc : Loc) -> Term d n
-SigY {x, fst, snd, loc} = Sig {fst, snd = SY [< x] snd, loc}
+public export
+data Elim where
+  ||| free variable, possibly with a displacement (see @crude, or @mugen for a
+  ||| more abstract and formalised take)
+  |||
+  ||| e.g. if f : ★₀ → ★₁, then f¹ : ★₁ → ★₂
+  F : Name -> Universe -> Loc -> Elim 0 0
+  ||| bound variable
+  B : Loc -> Elim 0 1

-||| non dependent pair type
-public export %inline
-And : (fst, snd : Term d n) -> (loc : Loc) -> Term d n
-And {fst, snd, loc} = Sig {fst, snd = SN snd, loc}
+  ||| term application
+  App : Subterms [Elim, Term] d n -> Loc -> Elim d n

-||| more convenient Eq
-public export %inline
-EqY : (i : BindName) -> (ty : Term (S d) n) ->
-      (l, r : Term d n) -> (loc : Loc) -> Term d n
-EqY {i, ty, l, r, loc} = Eq {ty = SY [< i] ty, l, r, loc}
+  ||| pair match
+  ||| - the subterms are, in order: [head, return type, body]
+  ||| - the quantity is that of the head, and since pairs only have one
+  |||   constructor, can be 0
+  CasePair : Qty -> Subterms [Elim, ScopeTerm, ScopeTermN 2] d n ->
+             Loc -> Elim d n

-||| more convenient DLam
-public export %inline
-DLamY : (i : BindName) -> (body : Term (S d) n) -> (loc : Loc) -> Term d n
-DLamY {i, body, loc} = DLam {body = SY [< i] body, loc}
+  ||| enum match
+  CaseEnum : Qty -> (arms : List TagVal) ->
+             Subterms (Elim :: ScopeTerm :: (Term <$ arms)) d n ->
+             Loc -> Elim d n

-public export %inline
-DLamN : (body : Term d n) -> (loc : Loc) -> Term d n
-DLamN {body, loc} = DLam {body = SN body, loc}
+  ||| nat match
+  CaseNat : Qty -> Qty ->
+            Subterms [Elim, ScopeTerm, Term, ScopeTermN 2] d n ->
+            Loc -> Elim d n
+
+  ||| box match
+  CaseBox : Qty -> Subterms [Elim, ScopeTerm, ScopeTerm] d n -> Loc -> Elim d n
+
+  ||| dim application
+  DApp : Subterms [Elim, DimArg] d n -> Loc -> Elim d n
+
+  ||| type-annotated term
+  Ann : Subterms [Term, Term] d n -> Loc -> Elim d n
+
+  ||| coerce a value along a type equality, or show its coherence
+  ||| [@xtt; §2.1.1]
+  Coe : Subterms [DScopeTerm, DimArg, DimArg, Term] d n ->
+        Loc -> Elim d n
+
+  ||| "generalised composition" [@xtt; §2.1.2]
+  Comp : Subterms [Term, DimArg, DimArg, Term,
+                   DimArg, DScopeTerm, DScopeTerm] d n ->
+         Loc -> Elim d n
+
+  ||| match on types. needed for b.s. of coercions [@xtt; §2.2]
+  TypeCase : Subterms [Elim, Term,   -- head, type
+                       Term,         -- ★
+                       ScopeTermN 2, -- pi
+                       ScopeTermN 2, -- sig
+                       Term,         -- enum
+                       ScopeTermN 5, -- eq
+                       Term,         -- nat
+                       ScopeTerm     -- box
+                      ] d n -> Loc -> Elim d n
+
+  ||| term closure/suspended substitution
+  CloE  : WithSubst (Elim d) (Elim d) n -> Elim d n
+  ||| dimension closure/suspended substitution
+  DCloE : WithSubst (\d => Elim d n) Dim d -> Elim d n
+
+
+-- this kills the idris ☹
+-- export %hint
+-- EqTerm : Eq (Term d n)
+
+-- export %hint
+-- EqElim : Eq (Elim d n)
+
+-- EqTerm = deriveEq
+-- EqElim = deriveEq
+
+
+-- mutual
+--   export %hint
+--   ShowTerm : Show (Term d n)
+--   ShowTerm = assert_total {a = Show (Term d n)} deriveShow
+
+--   export %hint
+--   ShowElim : Show (Elim d n)
+--   ShowElim = assert_total {a = Show (Elim d n)} deriveShow
+
+-- ||| scope which ignores all its binders
+-- public export %inline
+-- SN : {s : Nat} -> f n -> Scoped s f n
+-- SN = S (replicate s $ BN Unused noLoc) . N
+
+-- ||| scope which uses its binders
+-- public export %inline
+-- SY : BContext s -> f (s + n) -> Scoped s f n
+-- SY ns = S ns . Y
+
+-- public export %inline
+-- name : Scoped 1 f n -> BindName
+-- name (S [< x] _) = x
+
+-- public export %inline
+-- (.name) : Scoped 1 f n -> BindName
+-- s.name = name s
+
+-- ||| more convenient Pi
+-- public export %inline
+-- PiY : (qty : Qty) -> (x : BindName) ->
+--       (arg : Term d n) -> (res : Term d (S n)) -> (loc : Loc) -> Term d n
+-- PiY {qty, x, arg, res, loc} = Pi {qty, arg, res = SY [< x] res, loc}
+
+-- ||| more convenient Lam
+-- public export %inline
+-- LamY : (x : BindName) -> (body : Term d (S n)) -> (loc : Loc) -> Term d n
+-- LamY {x, body, loc} = Lam {body = SY [< x] body, loc}
+
+-- public export %inline
+-- LamN : (body : Term d n) -> (loc : Loc) -> Term d n
+-- LamN {body, loc} = Lam {body = SN body, loc}
+
+-- ||| non dependent function type
+-- public export %inline
+-- Arr : (qty : Qty) -> (arg, res : Term d n) -> (loc : Loc) -> Term d n
+-- Arr {qty, arg, res, loc} = Pi {qty, arg, res = SN res, loc}
+
+-- ||| more convenient Sig
+-- public export %inline
+-- SigY : (x : BindName) -> (fst : Term d n) ->
+--        (snd : Term d (S n)) -> (loc : Loc) -> Term d n
+-- SigY {x, fst, snd, loc} = Sig {fst, snd = SY [< x] snd, loc}
+
+-- ||| non dependent pair type
+-- public export %inline
+-- And : (fst, snd : Term d n) -> (loc : Loc) -> Term d n
+-- And {fst, snd, loc} = Sig {fst, snd = SN snd, loc}
+
+-- ||| more convenient Eq
+-- public export %inline
+-- EqY : (i : BindName) -> (ty : Term (S d) n) ->
+--       (l, r : Term d n) -> (loc : Loc) -> Term d n
+-- EqY {i, ty, l, r, loc} = Eq {ty = SY [< i] ty, l, r, loc}
+
+-- ||| more convenient DLam
+-- public export %inline
+-- DLamY : (i : BindName) -> (body : Term (S d) n) -> (loc : Loc) -> Term d n
+-- DLamY {i, body, loc} = DLam {body = SY [< i] body, loc}
+
+-- public export %inline
+-- DLamN : (body : Term d n) -> (loc : Loc) -> Term d n
+-- DLamN {body, loc} = DLam {body = SN body, loc}
+
+-- ||| non dependent equality type
+-- public export %inline
+-- Eq0 : (ty, l, r : Term d n) -> (loc : Loc) -> Term d n
+-- Eq0 {ty, l, r, loc} = Eq {ty = SN ty, l, r, loc}

-||| non dependent equality type
-public export %inline
-Eq0 : (ty, l, r : Term d n) -> (loc : Loc) -> Term d n
-Eq0 {ty, l, r, loc} = Eq {ty = SN ty, l, r, loc}

 ||| same as `F` but as a term
 public export %inline
-FT : Name -> Universe -> Loc -> Term d n
+FT : Name -> Universe -> Loc -> Term 0 0
 FT x u loc = E $ F x u loc

 ||| abbreviation for a bound variable like `BV 4` instead of
 ||| `B (VS (VS (VS (VS VZ))))`
 public export %inline
-BV : (i : Nat) -> (0 _ : LT i n) => (loc : Loc) -> Elim d n
-BV i loc = B (V i) loc
+BV : (i : Fin n) -> (loc : Loc) -> ElimT d n
+BV i loc = Th2 zero (one' i) $ B loc

 ||| same as `BV` but as a term
 public export %inline
-BVT : (i : Nat) -> (0 _ : LT i n) => (loc : Loc) -> Term d n
-BVT i loc = E $ BV i loc
+BVT : (i : Fin n) -> (loc : Loc) -> TermT d n
+BVT i loc = Th2 zero (one' i) $ E $ B loc

 public export
-makeNat : Nat -> Loc -> Term d n
-makeNat 0 loc = Zero loc
+makeNat : Nat -> Loc -> Term 0 0
+makeNat 0     loc = Zero loc
 makeNat (S k) loc = Succ (makeNat k loc) loc

-public export %inline
-enum : List TagVal -> Loc -> Term d n
-enum ts loc = Enum (SortedSet.fromList ts) loc
-
-public export %inline
-typeCase : Elim d n -> Term d n ->
-           List (TypeCaseArm d n) -> Term d n -> Loc -> Elim d n
-typeCase ty ret arms def loc = TypeCase ty ret (fromList arms) def loc
-
-public export %inline
-typeCase1Y : Elim d n -> Term d n ->
-             (k : TyConKind) -> BContext (arity k) -> Term d (arity k + n) ->
-             (loc : Loc) ->
-             {default (Nat loc) def : Term d n} ->
-             Elim d n
-typeCase1Y ty ret k ns body loc = typeCase ty ret [(k ** SY ns body)] def loc
-

 export
 Located (Elim d n) where
-  (F _ _ loc).loc               = loc
-  (B _ loc).loc                 = loc
-  (App _ _ loc).loc             = loc
-  (CasePair _ _ _ _ loc).loc    = loc
-  (CaseEnum _ _ _ _ loc).loc    = loc
-  (CaseNat _ _ _ _ _ _ loc).loc = loc
-  (CaseBox _ _ _ _ loc).loc     = loc
-  (DApp _ _ loc).loc            = loc
-  (Ann _ _ loc).loc             = loc
-  (Coe _ _ _ _ loc).loc         = loc
-  (Comp _ _ _ _ _ _ _ loc).loc  = loc
-  (TypeCase _ _ _ _ loc).loc    = loc
-  (CloE (Sub e _)).loc          = e.loc
-  (DCloE (Sub e _)).loc         = e.loc
+  (F _ _ loc).loc          = loc
+  (B loc).loc              = loc
+  (App _ loc).loc          = loc
+  (CasePair _ _ loc).loc   = loc
+  (CaseEnum _ _ _ loc).loc = loc
+  (CaseNat _ _ _ loc).loc  = loc
+  (CaseBox _ _ loc).loc    = loc
+  (DApp _ loc).loc         = loc
+  (Ann _ loc).loc          = loc
+  (Coe _ loc).loc          = loc
+  (Comp _ loc).loc         = loc
+  (TypeCase _ loc).loc     = loc
+  (CloE (Sub e _)).loc     = e.loc
+  (DCloE (Sub e _)).loc    = e.loc

 export
 Located (Term d n) where
  (TYPE _ loc).loc      = loc
-  (Pi _ _ _ loc).loc    = loc
+  (Pi _ _ loc).loc      = loc
  (Lam _ loc).loc       = loc
-  (Sig _ _ loc).loc     = loc
-  (Pair _ _ loc).loc    = loc
+  (Sig _ loc).loc       = loc
+  (Pair _ loc).loc      = loc
  (Enum _ loc).loc      = loc
  (Tag _ loc).loc       = loc
-  (Eq _ _ _ loc).loc    = loc
+  (Eq _ loc).loc        = loc
  (DLam _ loc).loc      = loc
  (Nat loc).loc         = loc
  (Zero loc).loc        = loc
@ -395,69 +348,40 @@ Located (Term d n) where
  (CloT (Sub t _)).loc  = t.loc
  (DCloT (Sub t _)).loc = t.loc

-export
-Located1 f => Located (ScopedBody s f n) where
-  (Y t).loc = t.loc
-  (N t).loc = t.loc
-
-export
-Located1 f => Located (Scoped s f n) where
-  t.loc = t.body.loc
-

 export
 Relocatable (Elim d n) where
-  setLoc loc (F x u _) = F x u loc
-  setLoc loc (B i _) = B i loc
-  setLoc loc (App fun arg _) = App fun arg loc
-  setLoc loc (CasePair qty pair ret body _) =
-    CasePair qty pair ret body loc
-  setLoc loc (CaseEnum qty tag ret arms _) =
-    CaseEnum qty tag ret arms loc
-  setLoc loc (CaseNat qty qtyIH nat ret zero succ _) =
-    CaseNat qty qtyIH nat ret zero succ loc
-  setLoc loc (CaseBox qty box ret body _) =
-    CaseBox qty box ret body loc
-  setLoc loc (DApp fun arg _) =
-    DApp fun arg loc
-  setLoc loc (Ann tm ty _) =
-    Ann tm ty loc
-  setLoc loc (Coe ty p q val _) =
-    Coe ty p q val loc
-  setLoc loc (Comp ty p q val r zero one _) =
-    Comp ty p q val r zero one loc
-  setLoc loc (TypeCase ty ret arms def _) =
-    TypeCase ty ret arms def loc
-  setLoc loc (CloE (Sub term subst)) =
-    CloE $ Sub (setLoc loc term) subst
-  setLoc loc (DCloE (Sub term subst)) =
-    DCloE $ Sub (setLoc loc term) subst
+  setLoc loc (F x u _)                = F x u loc
+  setLoc loc (B _)                    = B loc
+  setLoc loc (App ts _)               = App ts loc
+  setLoc loc (CasePair qty ts _)      = CasePair qty ts loc
+  setLoc loc (CaseEnum qty arms ts _) = CaseEnum qty arms ts loc
+  setLoc loc (CaseNat qty qtyIH ts _) = CaseNat qty qtyIH ts loc
+  setLoc loc (CaseBox qty ts _)       = CaseBox qty ts loc
+  setLoc loc (DApp ts _)              = DApp ts loc
+  setLoc loc (Ann ts _)               = Ann ts loc
+  setLoc loc (Coe ts _)               = Coe ts loc
+  setLoc loc (Comp ts _)              = Comp ts loc
+  setLoc loc (TypeCase ts _)          = TypeCase ts loc
+  setLoc loc (CloE (Sub term subst))  = CloE $ Sub (setLoc loc term) subst
+  setLoc loc (DCloE (Sub term subst)) = DCloE $ Sub (setLoc loc term) subst

 export
 Relocatable (Term d n) where
-  setLoc loc (TYPE l _) = TYPE l loc
-  setLoc loc (Pi qty arg res _) = Pi qty arg res loc
-  setLoc loc (Lam body _) = Lam body loc
-  setLoc loc (Sig fst snd _) = Sig fst snd loc
-  setLoc loc (Pair fst snd _) = Pair fst snd loc
-  setLoc loc (Enum cases _) = Enum cases loc
-  setLoc loc (Tag tag _) = Tag tag loc
-  setLoc loc (Eq ty l r _) = Eq ty l r loc
-  setLoc loc (DLam body _) = DLam body loc
-  setLoc loc (Nat _) = Nat loc
-  setLoc loc (Zero _) = Zero loc
-  setLoc loc (Succ p _) = Succ p loc
-  setLoc loc (BOX qty ty _) = BOX qty ty loc
-  setLoc loc (Box val _) = Box val loc
-  setLoc loc (E e) = E $ setLoc loc e
-  setLoc loc (CloT (Sub term subst)) = CloT $ Sub (setLoc loc term) subst
+  setLoc loc (TYPE l _)               = TYPE l loc
+  setLoc loc (Pi qty ts _)            = Pi qty ts loc
+  setLoc loc (Lam body _)             = Lam body loc
+  setLoc loc (Sig ts _)               = Sig ts loc
+  setLoc loc (Pair ts _)              = Pair ts loc
+  setLoc loc (Enum cases _)           = Enum cases loc
+  setLoc loc (Tag tag _)              = Tag tag loc
+  setLoc loc (Eq ts _)                = Eq ts loc
+  setLoc loc (DLam body _)            = DLam body loc
+  setLoc loc (Nat _)                  = Nat loc
+  setLoc loc (Zero _)                 = Zero loc
+  setLoc loc (Succ p _)               = Succ p loc
+  setLoc loc (BOX qty ty _)           = BOX qty ty loc
+  setLoc loc (Box val _)              = Box val loc
+  setLoc loc (E e)                    = E $ setLoc loc e
+  setLoc loc (CloT (Sub term subst))  = CloT $ Sub (setLoc loc term) subst
  setLoc loc (DCloT (Sub term subst)) = DCloT $ Sub (setLoc loc term) subst
-
-export
-Relocatable1 f => Relocatable (ScopedBody s f n) where
-  setLoc loc (Y body) = Y $ setLoc loc body
-  setLoc loc (N body) = N $ setLoc loc body
-
-export
-Relocatable1 f => Relocatable (Scoped s f n) where
-  setLoc loc (S names body) = S (setLoc loc <$> names) (setLoc loc body)
--- a/lib/Quox/Syntax/Term/Pretty.idr
+++ b/lib/Quox/Syntax/Term/Pretty.idr
@ -58,11 +58,12 @@ pbname (_, x, _) = x
 private
 record SplitPi d n where
  constructor MkSplitPi
+  {0 inner : Nat}
  binds : Telescope (PiBind d) n inner
  cod   : Term d inner

 private
-splitPi : Telescope (PiBind d) n inner -> Term d inner -> SplitPi d n
+splitPi : Telescope (PiBind d) n n' -> Term d n' -> SplitPi d n
 splitPi binds (Pi qty arg res _) =
  splitPi (binds :< (qty, res.name, arg)) $
    assert_smaller res $ pushSubsts' res.term
@ -87,7 +88,7 @@ prettyPiBind1 pi x dnames tnames s = hcat <$> sequence
 private
 prettyPiBinds : {opts : _} ->
                BContext d -> BContext n ->
-                Telescope (PiBind d) n inner ->
+                Telescope (PiBind d) n n' ->
                Eff Pretty (SnocList (Doc opts))
 prettyPiBinds _ _ [<] = pure [<]
 prettyPiBinds dnames tnames (binds :< (q, x, t)) =
@ -103,11 +104,12 @@ SigBind d n = (BindName, Term d n)
 private
 record SplitSig d n where
  constructor MkSplitSig
+  {0 inner : Nat}
  binds : Telescope (SigBind d) n inner
  last  : Term d inner

 private
-splitSig : Telescope (SigBind d) n inner -> Term d inner -> SplitSig d n
+splitSig : Telescope (SigBind d) n n' -> Term d n' -> SplitSig d n
 splitSig binds (Sig fst snd _) =
  splitSig (binds :< (snd.name, fst)) $
    assert_smaller snd $ pushSubsts' snd.term
@ -129,7 +131,7 @@ prettySigBind1 x dnames tnames s = hcat <$> sequence
 private
 prettySigBinds : {opts : _} ->
                 BContext d -> BContext n ->
-                 Telescope (SigBind d) n inner ->
+                 Telescope (SigBind d) n n' ->
                 Eff Pretty (SnocList (Doc opts))
 prettySigBinds _ _ [<] = pure [<]
 prettySigBinds dnames tnames (binds :< (x, t)) =
@ -163,6 +165,7 @@ NameChunks = SnocList (NameSort, SnocList BindName)
 private
 record SplitLams d n where
  constructor MkSplitLams
+  {0 dinner, ninner : Nat}
  dnames : BTelescope d dinner
  tnames : BTelescope n ninner
  chunks : NameChunks
@ -178,9 +181,9 @@ where
    if s == s' then xss :< (s', xs :< y)
    else xss :< (s', xs) :< (s, [< y])

-  go : BTelescope d dinner -> BTelescope n ninner ->
+  go : BTelescope d d' -> BTelescope n n' ->
       SnocList (NameSort, SnocList BindName) ->
-       Term dinner ninner -> SplitLams d n
+       Term d' n' -> SplitLams d n
  go dnames tnames chunks (Lam b _) =
    go dnames (tnames :< b.name) (push T b.name chunks) $
       assert_smaller b $ pushSubsts' b.term
@ -235,6 +238,7 @@ prettyDTApps dnames tnames f xs = do
 private
 record CaseArm opts d n where
  constructor MkCaseArm
+  {0 dinner, ninner : Nat}
  pat    : Doc opts
  dbinds : BTelescope d dinner -- 🍴
  tbinds : BTelescope n ninner
--- a/lib/Quox/Syntax/Term/Subst.idr
+++ b/lib/Quox/Syntax/Term/Subst.idr
@ -10,7 +10,7 @@ import Data.SnocVect
 namespace CanDSubst
  public export
  interface CanDSubst (0 tm : TermLike) where
-    (//) : tm dfrom n -> Lazy (DSubst dfrom dto) -> tm dto n
+    (//) : tm d1 n -> Lazy (DSubst d1 d2) -> tm d2 n

 ||| does the minimal reasonable work:
 ||| - deletes the closure around an atomic constant like `TYPE`
@ -25,7 +25,7 @@ CanDSubst Term where
  s                // th       = DCloT $ Sub s th

 private
-subDArgs : Elim dfrom n -> DSubst dfrom dto -> Elim dto n
+subDArgs : Elim d1 n -> DSubst d1 d2 -> Elim d2 n
 subDArgs (DApp f d loc) th = DApp (subDArgs f th) (d // th) loc
 subDArgs e              th = DCloE $ Sub e th

@ -47,16 +47,16 @@ CanDSubst Elim where

 namespace DSubst.ScopeTermN
  export %inline
-  (//) : ScopeTermN s dfrom n -> Lazy (DSubst dfrom dto) ->
-         ScopeTermN s dto n
+  (//) : ScopeTermN s d1 n -> Lazy (DSubst d1 d2) ->
+         ScopeTermN s d2 n
  S ns (Y body) // th = S ns $ Y $ body // th
  S ns (N body) // th = S ns $ N $ body // th

 namespace DSubst.DScopeTermN
  export %inline
  (//) : {s : Nat} ->
-         DScopeTermN s dfrom n -> Lazy (DSubst dfrom dto) ->
-         DScopeTermN s dto n
+         DScopeTermN s d1 n -> Lazy (DSubst d1 d2) ->
+         DScopeTermN s d2 n
  S ns (Y body) // th = S ns $ Y $ body // pushN s th
  S ns (N body) // th = S ns $ N $ body // th

@ -83,7 +83,7 @@ CanSubstSelf (Elim d) where
 namespace CanTSubst
  public export
  interface CanTSubst (0 tm : TermLike) where
-    (//) : tm d from -> Lazy (TSubst d from to) -> tm d to
+    (//) : tm d n1 -> Lazy (TSubst d n1 n2) -> tm d n2

 ||| does the minimal reasonable work:
 ||| - deletes the closure around an atomic constant like `TYPE`
@ -103,16 +103,15 @@ CanTSubst Term where
 namespace ScopeTermN
  export %inline
  (//) : {s : Nat} ->
-         ScopeTermN s d from -> Lazy (TSubst d from to) ->
-         ScopeTermN s d to
+         ScopeTermN s d n1 -> Lazy (TSubst d n1 n2) ->
+         ScopeTermN s d n2
  S ns (Y body) // th = S ns $ Y $ body // pushN s th
  S ns (N body) // th = S ns $ N $ body // th

 namespace DScopeTermN
  export %inline
  (//) : {s : Nat} ->
-         DScopeTermN s d from -> Lazy (TSubst d from to) ->
-         DScopeTermN s d to
+         DScopeTermN s d n1 -> Lazy (TSubst d n1 n2) -> DScopeTermN s d n2
  S ns (Y body) // th = S ns $ Y $ body // map (// shift s) th
  S ns (N body) // th = S ns $ N $ body // th

@ -125,8 +124,7 @@ export %inline


 export %inline
-comp : DSubst dfrom dto -> TSubst dfrom from mid -> TSubst dto mid to ->
-       TSubst dto from to
+comp : DSubst d1 d2 -> TSubst d1 n1 mid -> TSubst d2 mid n2 -> TSubst d2 n1 n2
 comp th ps ph = map (// th) ps . ph


@ -205,8 +203,8 @@ public export
 CloTest tm = forall d, n. tm d n -> Bool

 interface PushSubsts (0 tm : TermLike) (0 isClo : CloTest tm) | tm where
-  pushSubstsWith : DSubst dfrom dto -> TSubst dto from to ->
-                   tm dfrom from -> Subset (tm dto to) (No . isClo)
+  pushSubstsWith : DSubst d1 d2 -> TSubst d2 n1 n2 ->
+                   tm d1 n1 -> Subset (tm d2 n2) (No . isClo)

 public export
 0 NotClo : {isClo : CloTest tm} -> PushSubsts tm isClo => Pred (tm d n)
@ -230,8 +228,7 @@ parameters {0 isClo : CloTest tm} {auto _ : PushSubsts tm isClo}
  pushSubsts s = pushSubstsWith id id s

  export %inline
-  pushSubstsWith' : DSubst dfrom dto -> TSubst dto from to ->
-                    tm dfrom from -> tm dto to
+  pushSubstsWith' : DSubst d1 d2 -> TSubst d2 n1 n2 -> tm d1 n1 -> tm d2 n2
  pushSubstsWith' th ph x = fst $ pushSubstsWith th ph x

  export %inline
@ -334,8 +331,7 @@ private %inline
 CompHY : (ty : DScopeTerm d n) -> (p, q : Dim d) -> (val : Term d n) ->
         (r : Dim d) -> (zero, one : DScopeTerm d n) -> (loc : Loc) -> Elim d n
 CompHY {ty, p, q, val, r, zero, one, loc} =
-  -- [fixme] maintain location of existing B VZ
-  let ty' = SY ty.names $ ty.term // (B VZ noLoc ::: shift 2) in
+  let ty' = SY ty.names $ ty.term // (B VZ ty.loc ::: shift 2) in
  Comp {
    ty = dsub1 ty q, p, q,
    val = E $ Coe ty p q val val.loc, r,
--- a/lib/Quox/Syntax/Term/Tighten.idr
+++ b/lib/Quox/Syntax/Term/Tighten.idr
@ -8,10 +8,10 @@ import public Quox.OPE


 export
-Tighten (Shift from) where
+Tighten (Shift f) where
  -- `OPE m n` is a spicy `m ≤ n`,
-  -- and `Shift from n` is a (different) spicy `from ≤ n`
-  -- so the value is `from ≤ m` (as a `Shift`), if that is the case
+  -- and `Shift f n` is a (different) spicy `f ≤ n`
+  -- so the value is `f ≤ m` (as a `Shift`), if that is the case
  tighten _        SZ      = Nothing
  tighten Id       by      = Just by
  tighten (Drop p) (SS by) = tighten p by
@ -26,12 +26,12 @@ Tighten Dim where

 export
 tightenSub : (forall m, n. OPE m n -> env n -> Maybe (env m)) ->
-             OPE to1 to2 -> Subst env from to2 -> Maybe (Subst env from to1)
+             OPE t1 t2 -> Subst env f t2 -> Maybe (Subst env f t1)
 tightenSub f p (Shift by) = [|Shift $ tighten p by|]
 tightenSub f p (t ::: th) = [|f p t !::: tightenSub f p th|]

 export
-Tighten env => Tighten (Subst env from) where
+Tighten env => Tighten (Subst env f) where
  tighten p th = tightenSub tighten p th


--- a/lib/Quox/Thin.idr
+++ b/lib/Quox/Thin.idr
@ -0,0 +1,13 @@
+module Quox.Thin
+
+import public Quox.Thin.Base
+import public Quox.Thin.View
+import public Quox.Thin.Eqv
+import public Quox.Thin.Cons
+import public Quox.Thin.List
+import public Quox.Thin.Append
+import public Quox.Thin.Comp
+import public Quox.Thin.Cover
+import public Quox.Thin.Coprod
+import public Quox.Thin.Split
+import public Quox.Thin.Term
--- a/lib/Quox/Thin/Append.idr
+++ b/lib/Quox/Thin/Append.idr
@ -0,0 +1,27 @@
+module Quox.Thin.Append
+
+import public Quox.Thin.Base
+import public Quox.Thin.View
+import Data.DPair
+
+%default total
+
+public export
+app' : OPE m1 n1 mask1 -> OPE m2 n2 mask2 -> Exists (OPE (m1 + m2) (n1 + n2))
+app' Stop             ope2 = Evidence _ ope2
+app' (Drop ope1 Refl) ope2 = Evidence _ $ Drop (app' ope1 ope2).snd Refl
+app' (Keep ope1 Refl) ope2 = Evidence _ $ Keep (app' ope1 ope2).snd Refl
+
+public export
+(++) : {n1, n2, mask1, mask2 : Nat} ->
+       (0 ope1 : OPE m1 n1 mask1) -> (0 ope2 : OPE m2 n2 mask2) ->
+       Subset Nat (OPE (m1 + m2) (n1 + n2))
+ope1 ++ ope2 with %syntactic (view ope1)
+  Stop           ++ ope2 | StopV = Element _ ope2
+  Drop ope1 Refl ++ ope2 | DropV mask ope1 =
+    Element _ $ Drop (ope1 ++ ope2).snd Refl
+  Keep ope1 Refl ++ ope2 | KeepV mask ope1 =
+    Element _ $ Keep (ope1 ++ ope2).snd Refl
+
+-- [todo] this mask is just (mask1 << n2) | mask2
+-- prove it and add %transform
--- a/lib/Quox/Thin/Base.idr
+++ b/lib/Quox/Thin/Base.idr
@ -0,0 +1,81 @@
+module Quox.Thin.Base
+
+import Data.Fin
+import Data.DPair
+
+%default total
+
+||| "order preserving embeddings", for recording a correspondence between a
+||| smaller scope and part of a larger one. the third argument is a bitmask
+||| representing this OPE, unique for a given `n`.
+public export
+data OPE : (m, n, mask : Nat) -> Type where
+  [search m n]
+  Stop : OPE 0 0 0
+  Drop : OPE m n mask -> mask' = mask + mask       -> OPE m     (S n) mask'
+  Keep : OPE m n mask -> mask' = (S (mask + mask)) -> OPE (S m) (S n) mask'
+%name OPE ope
+
+export
+Show (OPE m n mask) where
+  showPrec d Stop            = "Stop"
+  showPrec d (Drop ope Refl) = showCon d "Drop" $ showArg ope ++ " Refl"
+  showPrec d (Keep ope Refl) = showCon d "Keep" $ showArg ope ++ " Refl"
+
+public export %inline
+drop : OPE m n mask -> OPE m (S n) (mask + mask)
+drop ope = Drop ope Refl
+
+public export %inline
+keep : OPE m n mask -> OPE (S m) (S n) (S (mask + mask))
+keep ope = Keep ope Refl
+
+
+public export
+data IsStop : OPE m n mask -> Type where ItIsStop : IsStop Stop
+
+public export
+data IsDrop : OPE m n mask -> Type where ItIsDrop : IsDrop (Drop ope eq)
+
+public export
+data IsKeep : OPE m n mask -> Type where ItIsKeep : IsKeep (Keep ope eq)
+
+
+export
+0 zeroIsStop : (ope : OPE m 0 mask) -> IsStop ope
+zeroIsStop Stop = ItIsStop
+
+
+||| everything selected
+public export
+id : {m : Nat} -> Subset Nat (OPE m m)
+id {m = 0}   = Element _ Stop
+id {m = S m} = Element _ $ Keep id.snd Refl
+
+public export %inline
+0 id' : OPE m m Base.id.fst
+id' = id.snd
+
+||| nothing selected
+public export
+zero : {m : Nat} -> OPE 0 m 0
+zero {m = 0}   = Stop
+zero {m = S m} = Drop zero Refl
+
+||| a single slot selected
+public export
+one : Fin n -> Subset Nat (OPE 1 n)
+one FZ     = Element _ $ keep zero
+one (FS i) = Element _ $ drop (one i).snd
+
+public export %inline
+0 one' : (i : Fin n) -> OPE 1 n (one i).fst
+one' i = (one i).snd
+
+
+public export
+record SomeOPE n where
+  constructor MkOPE
+  {0 scope : Nat}
+  {mask : Nat}
+  0 ope : OPE scope n mask
--- a/lib/Quox/Thin/Comp.idr
+++ b/lib/Quox/Thin/Comp.idr
@ -0,0 +1,59 @@
+module Quox.Thin.Comp
+
+import public Quox.Thin.Base
+import public Quox.Thin.View
+import Data.Singleton
+
+%default total
+
+||| inductive definition of OPE composition
+public export
+data Comp : (l : OPE n p mask1) -> (r : OPE m n mask2) ->
+            (res : OPE m p mask3) -> Type where
+  [search l r]
+  StopZ : Comp Stop Stop Stop
+  DropZ : Comp a b c -> Comp (Drop a Refl) b (Drop c Refl)
+  KeepZ : Comp a b c -> Comp (Keep a Refl) (Keep b Refl) (Keep c Refl)
+  KDZ   : Comp a b c -> Comp (Keep a Refl) (Drop b Refl) (Drop c Refl)
+
+public export
+record CompResult (ope1 : OPE n p mask1) (ope2 : OPE m n mask2) where
+  constructor MkComp
+  {mask : Nat}
+  {0 ope : OPE m p mask}
+  0 comp : Comp ope1 ope2 ope
+%name CompResult comp
+
+||| compose two OPEs, if the middle scope size is already known at runtime
+export
+comp' : {n, p, mask1, mask2 : Nat} ->
+        (0 ope1 : OPE n p mask1) -> (0 ope2 : OPE m n mask2) ->
+        CompResult ope1 ope2
+comp' ope1 ope2 with %syntactic (view ope1) | (view ope2)
+  comp' Stop Stop | StopV | StopV =
+    MkComp StopZ
+  comp' (Drop ope1 Refl) ope2 | DropV _ ope1 | _ =
+    MkComp $ DropZ (comp' ope1 ope2).comp
+  comp' (Keep ope1 Refl) (Drop ope2 Refl) | KeepV _ ope1 | DropV _ ope2 =
+    MkComp $ KDZ (comp' ope1 ope2).comp
+  comp' (Keep ope1 Refl) (Keep ope2 Refl) | KeepV _ ope1 | KeepV _ ope2 =
+    MkComp $ KeepZ (comp' ope1 ope2).comp
+
+||| compose two OPEs, after recomputing the middle scope size using `appOpe`
+export
+comp : {p, mask1, mask2 : Nat} ->
+       (0 ope1 : OPE n p mask1) -> (0 ope2 : OPE m n mask2) ->
+       CompResult ope1 ope2
+comp ope1 ope2 = let Val n = appOpe p ope1 in comp' ope1 ope2
+
+-- [todo] is there a quick way to compute the mask of comp?
+
+export
+0 (.) : (ope1 : OPE n p mask1) -> (ope2 : OPE m n mask2) ->
+        OPE m p (comp ope1 ope2).mask
+ope1 . ope2 = (comp ope1 ope2).ope
+
+-- export
+-- 0 compMask : (ope1 : OPE n p mask1) -> (ope2 : OPE m n mask2) ->
+--              (ope3 : OPE m p mask3) -> Comp ope1 ope2 ope3 ->
+--              mask3 = ?aaa
--- a/lib/Quox/Thin/Cons.idr
+++ b/lib/Quox/Thin/Cons.idr
@ -0,0 +1,74 @@
+module Quox.Thin.Cons
+
+import public Quox.Thin.Base
+import Quox.Thin.Eqv
+import Quox.Thin.View
+import Data.DPair
+import Control.Relation
+
+%default total
+
+
+public export
+data IsHead : (ope : OPE m (S n) mask) -> Bool -> Type where
+  [search ope]
+  DropH : IsHead (Drop ope eq) False
+  KeepH : IsHead (Keep ope eq) True
+
+public export
+data IsTail : (full : OPE m (S n) mask) -> OPE m' n mask' -> Type where
+  [search full]
+  DropT : IsTail (Drop ope eq) ope
+  KeepT : IsTail (Keep ope eq) ope
+
+public export
+record Uncons (ope : OPE m (S n) mask) where
+  constructor MkUncons
+  0 head : Bool
+  {tailMask : Nat}
+  0 tail : OPE scope n tailMask
+  {auto isHead : IsHead ope head}
+  {auto 0 isTail : IsTail ope tail}
+
+public export
+uncons : {n, mask : Nat} -> (0 ope : OPE m (S n) mask) -> Uncons ope
+uncons ope with %syntactic (view ope)
+  uncons (Drop ope Refl) | DropV _ ope = MkUncons False ope
+  uncons (Keep ope Refl) | KeepV _ ope = MkUncons True  ope
+
+public export
+head : {n, mask : Nat} -> (0 ope : OPE m (S n) mask) -> Exists $ IsHead ope
+head ope = Evidence _ (uncons ope).isHead
+
+public export
+record Tail (ope : OPE m (S n) mask) where
+  constructor MkTail
+  {tailMask : Nat}
+  0 tail : OPE scope n tailMask
+  {auto 0 isTail : IsTail ope tail}
+
+public export
+tail : {n, mask : Nat} -> (0 ope : OPE m (S n) mask) -> Tail ope
+tail ope = let u = uncons ope in MkTail u.tail @{u.isTail}
+
+
+export
+cons : {mask : Nat} -> (head : Bool) -> (0 tail : OPE m n mask) ->
+       Subset Nat (OPE (if head then S m else m) (S n))
+cons False tail = Element _ $ drop tail
+cons True  tail = Element _ $ keep tail
+
+export
+0 consEquiv' : (self : OPE m' (S n) mask') ->
+               (head : Bool) -> (tail : OPE m n mask) ->
+               IsHead self head -> IsTail self tail ->
+               (cons head tail).snd `Eqv` self
+consEquiv' (Drop tail _) False tail DropH DropT = EqvDrop reflexive
+consEquiv' (Keep tail _) True  tail KeepH KeepT = EqvKeep reflexive
+
+export
+0 consEquiv : (full : OPE m' (S n) mask') ->
+              (cons (uncons full).head (uncons full).tail).snd `Eqv` full
+consEquiv full with %syntactic (uncons full)
+  _ | MkUncons head tail {isHead, isTail} =
+    consEquiv' full head tail isHead isTail
--- a/lib/Quox/Thin/Coprod.idr
+++ b/lib/Quox/Thin/Coprod.idr
@ -0,0 +1,171 @@
+module Quox.Thin.Coprod
+
+import public Quox.Thin.Base
+import public Quox.Thin.Comp
+import public Quox.Thin.View
+import public Quox.Thin.List
+import public Quox.Thin.Cover
+import Data.DPair
+import Data.Nat
+import Control.Function
+
+%default total
+
+
+namespace Coprod
+  public export
+  data Comps : OPE scope n scopeMask ->
+               OPEList scope -> OPEList n -> Type where
+    Nil  : Comps sub [] []
+    (::) : Comp  sub inner  full ->
+           Comps sub inners fulls ->
+           Comps sub (inner :: inners) (full :: fulls)
+  %name Comps comps
+
+  public export
+  record Coprod (fulls : OPEList n) where
+    constructor MkCoprod
+    {scopeMask : Nat}
+    {0 sub : OPE scope n scopeMask}
+    inners : OPEList scope
+    0 comps : Comps sub inners fulls
+    0 cov : Cover inners
+  %name Coprod cop
+
+export
+0 compsLength : Comps s ts us -> length ts = length us
+compsLength []           = Refl
+compsLength (_ :: comps) = cong S $ compsLength comps
+
+
+export
+coprodNil : Coprod []
+coprodNil = MkCoprod [] [] [] {sub = zero}
+
+private
+coprodHead : {n : Nat} -> (opes : OPEList (S n)) ->
+             Either (Cover1 opes) (All IsDrop opes)
+coprodHead [] = Right []
+coprodHead (ope :: opes) = case view ope of
+  DropV {} => case coprodHead opes of
+    Left  cov1  => Left  $ There cov1
+    Right drops => Right $ ItIsDrop :: drops
+  KeepV {} => Left Here
+
+
+private
+0 compsConsDrop : (opes : OPEList (S n)) ->
+                  All IsDrop opes ->
+                  All2 IsTail opes tails ->
+                  Comps sub inners tails -> Comps (drop sub) inners opes
+compsConsDrop [] [] [] [] = []
+compsConsDrop (Drop ope Refl :: opes) (ItIsDrop :: ds) (DropT :: ts) (c :: cs) =
+  DropZ c :: compsConsDrop opes ds ts cs
+compsConsDrop (_ :: _) [] _ _ impossible
+
+private
+coprodConsDrop : (0 opes : OPEList (S n)) ->
+                 (0 ds : All IsDrop opes) ->
+                 (0 ts : All2 IsTail opes tails) ->
+                 Coprod tails -> Coprod opes
+coprodConsDrop opes ds ts (MkCoprod inners comps cov) =
+  MkCoprod inners (compsConsDrop opes ds ts comps) cov
+
+
+private
+copyHeads : {m : Nat} ->
+            (src : OPEList (S m)) -> (tgt : OPEList n) ->
+            (0 eq : length src = length tgt) => OPEList (S n)
+copyHeads [] [] = []
+copyHeads (s :: ss) (t :: ts) =
+  case view s of
+    DropV mask ope => drop t :: copyHeads ss ts @{inj S eq}
+    KeepV mask ope => keep t :: copyHeads ss ts @{inj S eq}
+
+private
+0 copyHeadsComps : (eq : length outers = length inners) ->
+                   All2 IsTail outers tails ->
+                   Comps sub inners tails ->
+                   Comps (keep sub) (copyHeads outers inners) outers
+copyHeadsComps _ [] [] = []
+copyHeadsComps eq (DropT {eq = eq2} :: ps) ((c :: cs) {full}) =
+  let (Refl) = eq2 in -- coverage checker weirdness
+  rewrite viewDrop full Refl in
+  KDZ c :: copyHeadsComps (inj S eq) ps cs
+copyHeadsComps eq (KeepT {eq = eq2} :: ps) ((c :: cs) {full}) =
+  let (Refl) = eq2 in
+  rewrite viewKeep full Refl in
+  KeepZ c :: copyHeadsComps (inj S eq) ps cs
+
+-- should be erased (coverage checker weirdness)
+-- it is possibly https://github.com/idris-lang/Idris2/issues/1417 that keeps
+-- happening. not 100% sure
+private
+cover1CopyHeads : {m : Nat} ->
+                  (ss : OPEList (S m)) -> (ts : OPEList n) ->
+                  (eq : length ss = length ts) ->
+                  (cov1 : Cover1 ss) -> Cover1 (copyHeads ss ts)
+cover1CopyHeads (Keep s Refl :: ss) (t :: ts) eq Here =
+  rewrite viewKeep s Refl in Here
+cover1CopyHeads (s :: ss) (t :: ts) eq (There c) with (view s)
+  cover1CopyHeads (Drop {} :: ss) (t :: ts) eq (There c) | DropV {} =
+    There $ cover1CopyHeads ss ts (inj S eq) c
+  cover1CopyHeads (Keep {} :: ss) (t :: ts) eq (There c) | KeepV {} =
+    Here
+
+private
+copyHeadsTails : {m : Nat} ->
+                 (ss : OPEList (S m)) -> (ts : OPEList n) ->
+                 (eq : length ss = length ts) ->
+                 All2 IsTail (copyHeads ss ts) ts
+copyHeadsTails [] [] eq = []
+copyHeadsTails (s :: ss) (t :: ts) eq with (view s)
+  copyHeadsTails (Drop ope Refl :: ss) (t :: ts) eq | DropV mask ope =
+    DropT :: copyHeadsTails ss ts (inj S eq)
+  copyHeadsTails (Keep ope Refl :: ss) (t :: ts) eq | KeepV mask ope =
+    KeepT :: copyHeadsTails ss ts (inj S eq)
+
+private
+coprodConsKeep : {n : Nat} ->
+                 (opes : OPEList (S n)) ->
+                 {0 tails : OPEList n} ->
+                 (cov1 : Cover1 opes) ->
+                 (0 ts : All2 IsTail opes tails) ->
+                 Coprod tails -> Coprod opes
+coprodConsKeep opes cov1 ts (MkCoprod inners comps cov) =
+  MkCoprod
+    (copyHeads opes inners @{all2Length ts `trans` sym (compsLength comps)})
+    (copyHeadsComps _ ts comps)
+    ((cover1CopyHeads {cov1, _} :: cov) @{copyHeadsTails {}})
+
+
+export
+coprod : {n : Nat} -> (opes : OPEList n) -> Coprod opes
+
+private
+coprod0 : (opes : OPEList 0) -> Coprod opes
+
+private
+coprodS : {n : Nat} -> (opes : OPEList (S n)) -> Coprod opes
+
+coprod {n = 0}   opes = coprod0 opes
+coprod {n = S n} opes = coprodS opes
+
+coprod0 [] = coprodNil
+coprod0 (ope :: opes) with %syntactic 0 (zeroIsStop ope) | (coprod opes)
+  coprod0 (Stop :: opes)
+      | ItIsStop | MkCoprod {sub} inners comps cov
+      with %syntactic 0 (zeroIsStop sub)
+    coprod0 (Stop :: opes)
+      | ItIsStop | MkCoprod {sub = Stop} inners comps cov | ItIsStop
+      = MkCoprod (Stop :: inners) (StopZ :: comps) []
+
+coprodS [] = coprodNil
+coprodS opes =
+  let hs              = heads opes
+      Element ts tprf = tails_ opes
+      tcop            = coprod $ assert_smaller opes ts
+  in
+  case coprodHead opes of
+    Left  cov1  => coprodConsKeep opes cov1 tprf tcop
+    Right drops => coprodConsDrop opes drops tprf tcop
--- a/lib/Quox/Thin/Cover.idr
+++ b/lib/Quox/Thin/Cover.idr
@ -0,0 +1,27 @@
+module Quox.Thin.Cover
+
+import public Quox.Thin.Base
+import public Quox.Thin.List
+
+%default total
+
+||| an OPE list is a cover if at least one of the OPEs has `Keep` as the head,
+||| and the tails are also a cover
+|||
+||| in @egtbs it is a binary relation which is fine for ×ᵣ but i don't want to
+||| write my AST in universe-of-syntaxes style. sorry
+public export data Cover : OPEList n -> Type
+
+||| the "`Keep` in the head" condition of a cover
+public export data Cover1 : OPEList n -> Type
+
+data Cover where
+  Nil  : Cover opes {n = 0}
+  (::) : Cover1 opes -> All2 IsTail opes tails => Cover tails -> Cover opes
+%name Cover cov
+
+data Cover1 where
+  Here  : Cover1 (Keep ope eq :: opes)
+  There : Cover1 opes -> Cover1 (ope :: opes)
+%name Cover1 cov1
+%builtin Natural Cover1
--- a/lib/Quox/Thin/Eqv.idr
+++ b/lib/Quox/Thin/Eqv.idr
@ -0,0 +1,128 @@
+module Quox.Thin.Eqv
+
+import public Quox.Thin.Base
+import public Quox.Thin.View
+import Quox.NatExtra
+import Syntax.PreorderReasoning
+
+%default total
+
+infix 6 `Eqv`
+
+private
+uip : (p, q : a = b) -> p = q
+uip Refl Refl = Refl
+
+
+public export
+data Eqv : OPE m1 n1 mask1 -> OPE m2 n2 mask2 -> Type where
+  EqvStop : Eqv Stop Stop
+  EqvDrop : {0 p : OPE m1 n1 mask1} ->
+            {0 q : OPE m2 n2 mask2} ->
+            Eqv p q -> Eqv (Drop p eq1) (Drop q eq2)
+  EqvKeep : {0 p : OPE m1 n1 mask1} ->
+            {0 q : OPE m2 n2 mask2} ->
+            Eqv p q -> Eqv (Keep p eq1) (Keep q eq2)
+%name Eqv eqv
+
+export Uninhabited (Stop     `Eqv` Drop p e) where uninhabited _ impossible
+export Uninhabited (Stop     `Eqv` Keep p e) where uninhabited _ impossible
+export Uninhabited (Drop p e `Eqv` Stop)     where uninhabited _ impossible
+export Uninhabited (Drop p e `Eqv` Keep q f) where uninhabited _ impossible
+export Uninhabited (Keep p e `Eqv` Stop)     where uninhabited _ impossible
+export Uninhabited (Keep p e `Eqv` Drop q f) where uninhabited _ impossible
+
+export
+Reflexive (OPE m n mask) Eqv where
+  reflexive {x = Stop}    = EqvStop
+  reflexive {x = Drop {}} = EqvDrop reflexive
+  reflexive {x = Keep {}} = EqvKeep reflexive
+
+export
+symmetric : p `Eqv` q -> q `Eqv` p
+symmetric EqvStop       = EqvStop
+symmetric (EqvDrop eqv) = EqvDrop $ symmetric eqv
+symmetric (EqvKeep eqv) = EqvKeep $ symmetric eqv
+
+export
+transitive : p `Eqv` q -> q `Eqv` r -> p `Eqv` r
+transitive EqvStop        EqvStop        = EqvStop
+transitive (EqvDrop eqv1) (EqvDrop eqv2) = EqvDrop (transitive eqv1 eqv2)
+transitive (EqvKeep eqv1) (EqvKeep eqv2) = EqvKeep (transitive eqv1 eqv2)
+
+
+private
+recompute' : {mask1, mask2, n1, n2 : Nat} ->
+             (0 p : OPE m1 n1 mask1) -> (0 q : OPE m2 n2 mask2) ->
+             (0 eqv : p `Eqv` q) -> p `Eqv` q
+recompute' p q eqv with %syntactic (view p) | (view q)
+  recompute' Stop Stop _ | StopV | StopV = EqvStop
+  recompute' (Drop p _) (Drop q _) eqv | DropV _ p | DropV _ q =
+    EqvDrop $ recompute' {eqv = let EqvDrop e = eqv in e, _}
+  recompute' (Keep p _) (Keep q _) eqv | KeepV _ p | KeepV _ q =
+    EqvKeep $ recompute' {eqv = let EqvKeep e = eqv in e, _}
+  recompute' (Drop p _) (Keep q _) eqv | DropV _ p | KeepV _ q =
+    void $ absurd eqv
+  recompute' (Keep p _) (Drop q _) eqv | KeepV _ p | DropV _ q =
+    void $ absurd eqv
+
+private
+recompute : {mask1, mask2, n1, n2 : Nat} ->
+            {0 p : OPE m1 n1 mask1} -> {0 q : OPE m2 n2 mask2} ->
+            (0 _ : p `Eqv` q) -> p `Eqv` q
+recompute eqv = recompute' {eqv, _}
+
+
+export
+eqvIndices : {0 p : OPE m1 n1 mask1} -> {0 q : OPE m2 n2 mask2} ->
+             p `Eqv` q -> (m1 = m2, n1 = n2, mask1 = mask2)
+eqvIndices EqvStop = (Refl, Refl, Refl)
+eqvIndices (EqvDrop eqv {eq1 = Refl, eq2 = Refl}) =
+  let (Refl, Refl, Refl) = eqvIndices eqv in (Refl, Refl, Refl)
+eqvIndices (EqvKeep eqv {eq1 = Refl, eq2 = Refl}) =
+  let (Refl, Refl, Refl) = eqvIndices eqv in (Refl, Refl, Refl)
+
+export
+0 eqvMask : (p : OPE m1 n mask1) -> (q : OPE m2 n mask2) ->
+            mask1 = mask2 -> p `Eqv` q
+eqvMask Stop Stop _ = EqvStop
+eqvMask (Drop ope1 Refl) (Drop {mask = mm2} ope2 eq2) Refl =
+  EqvDrop $ eqvMask ope1 ope2 (doubleInj _ _ eq2)
+eqvMask (Drop ope1 Refl) (Keep ope2 eq2) Refl =
+  void $ notEvenOdd _ _ eq2
+eqvMask (Keep ope1 eq1) (Keep ope2 eq2) Refl =
+  EqvKeep $ eqvMask ope1 ope2 (doubleInj _ _ $ inj S $ trans (sym eq1) eq2)
+eqvMask (Keep ope1 eq1) (Drop ope2 eq2) Refl =
+  void $ notEvenOdd _ _ $ trans (sym eq2) eq1
+
+export
+0 eqvEq : (p, q : OPE m n mask) -> p `Eqv` q -> p === q
+eqvEq Stop Stop EqvStop = Refl
+eqvEq (Drop p eq1) (Drop q eq2) (EqvDrop eqv)
+    with %syntactic (doubleInj _ _ $ trans (sym eq1) eq2)
+  _ | Refl = cong2 Drop (eqvEq p q eqv) (uip eq1 eq2)
+eqvEq (Keep p eq1) (Keep q eq2) (EqvKeep eqv)
+    with %syntactic (doubleInj _ _ $ inj S $ trans (sym eq1) eq2)
+  _ | Refl = cong2 Keep (eqvEq p q eqv) (uip eq1 eq2)
+
+export
+0 eqvEq' : (p : OPE m1 n1 mask1) -> (q : OPE m2 n2 mask2) ->
+           p `Eqv` q -> p ~=~ q
+eqvEq' p q eqv = let (Refl, Refl, Refl) = eqvIndices eqv in eqvEq p q eqv
+
+export
+0 maskEqInner : (0 ope1 : OPE m1 n mask1) -> (0 ope2 : OPE m2 n mask2) ->
+                mask1 = mask2 -> m1 = m2
+maskEqInner Stop Stop _ = Refl
+maskEqInner (Drop ope1 Refl) (Drop ope2 Refl) eq =
+  maskEqInner ope1 ope2 (doubleInj _ _ eq)
+maskEqInner (Keep ope1 Refl) (Keep ope2 Refl) eq =
+  cong S $ maskEqInner ope1 ope2 $ doubleInj _ _ $ inj S eq
+maskEqInner (Drop ope1 Refl) (Keep ope2 Refl) eq = void $ notEvenOdd _ _ eq
+maskEqInner (Keep {mask = mask1'} ope1 eq1) (Drop {mask = mask2'} ope2 eq2) eq =
+  -- matching on eq1, eq2, or eq here triggers that weird coverage bug ☹
+  void $ notEvenOdd _ _ $ Calc $
+    |~ mask2' + mask2'
+    ~~ mask2               ..<(eq2)
+    ~~ mask1               ..<(eq)
+    ~~ S (mask1' + mask1') ...(eq1)
--- a/lib/Quox/Thin/List.idr
+++ b/lib/Quox/Thin/List.idr
@ -0,0 +1,127 @@
+module Quox.Thin.List
+
+import public Quox.Thin.Base
+import public Quox.Thin.Cons
+import Data.DPair
+import Data.Nat
+import Control.Function
+
+%default total
+
+||| a list of OPEs of a given outer scope size
+||| (at runtime just the masks)
+public export
+data OPEList : Nat -> Type where
+  Nil  : OPEList n
+  (::) : {mask : Nat} -> (0 ope : OPE m n mask) -> OPEList n -> OPEList n
+%name OPEList opes
+
+public export
+length : OPEList n -> Nat
+length []          = 0
+length (_ :: opes) = S $ length opes
+
+public export
+toList : OPEList n -> List (SomeOPE n)
+toList []            = []
+toList (ope :: opes) = MkOPE ope :: toList opes
+
+public export
+fromList : List (SomeOPE n) -> OPEList n
+fromList []                = []
+fromList (MkOPE ope :: xs) = ope :: fromList xs
+
+
+public export
+0 Pred : Nat -> Type
+Pred n = forall m, mask. OPE m n mask -> Type
+
+public export
+0 Rel : Nat -> Nat -> Type
+Rel n1 n2 = forall m1, m2, mask1, mask2.
+            OPE m1 n1 mask1 -> OPE m2 n2 mask2 -> Type
+
+namespace All
+  public export
+  data All : Pred n -> OPEList n -> Type where
+    Nil  : {0 p : Pred n} -> All p []
+    (::) : {0 p : Pred n} -> p ope -> All p opes -> All p (ope :: opes)
+  %name All.All ps, qs
+
+namespace All2
+  public export
+  data All2 : Rel n1 n2 -> OPEList n1 -> OPEList n2 -> Type where
+    Nil  : {0 p : Rel n1 n2} -> All2 p [] []
+    (::) : {0 p : Rel n1 n2} -> p a b -> All2 p as bs ->
+           All2 p (a :: as) (b :: bs)
+  %name All2.All2 ps, qs
+
+export
+0 all2Length : {p : Rel m n} -> All2 p ss ts -> length ss = length ts
+all2Length []        = Refl
+all2Length (p :: ps) = cong S $ all2Length ps
+
+namespace Any
+  public export
+  data Any : Pred n -> OPEList n -> Type where
+    Here  : {0 p : Pred n} -> p ope -> Any p (ope :: opes)
+    There : {0 p : Pred n} -> Any p opes -> Any p (ope :: opes)
+  %name Any.Any p, q
+
+export
+{0 p : Pred n} -> Uninhabited (Any p []) where uninhabited _ impossible
+
+export
+all : {0 p : Pred n} ->
+      (forall m. {mask : Nat} -> (0 ope : OPE m n mask) -> p ope) ->
+      (opes : OPEList n) -> All p opes
+all f []            = []
+all f (ope :: opes) = f ope :: all f opes
+
+export
+allDec : {0 p : Pred n} ->
+         (forall m. {mask : Nat} -> (0 ope : OPE m n mask) -> Dec (p ope)) ->
+         (opes : OPEList n) -> Dec (All p opes)
+allDec f [] = Yes []
+allDec f (ope :: opes) = case f ope of
+  Yes y => case allDec f opes of
+    Yes ys => Yes $ y :: ys
+    No  k  => No  $ \(_ :: ps) => k ps
+  No k => No $ \(p :: _) => k p
+
+export
+anyDec : {0 p : Pred n} ->
+         (forall m. {mask : Nat} -> (0 ope : OPE m n mask) -> Dec (p ope)) ->
+         (opes : OPEList n) -> Dec (Any p opes)
+anyDec f [] = No absurd
+anyDec f (ope :: opes) = case f ope of
+  Yes y  => Yes $ Here y
+  No  nh => case anyDec f opes of
+    Yes y  => Yes $ There y
+    No  nt => No  $ \case Here h => nh h; There t => nt t
+
+
+export
+unconses : {n : Nat} -> (opes : OPEList (S n)) -> All Uncons opes
+unconses = all uncons
+
+export
+heads : {n : Nat} -> (opes : OPEList (S n)) -> All (Exists . IsHead) opes
+heads = all head
+
+export
+tails : {n : Nat} -> (opes : OPEList (S n)) -> All Tail opes
+tails = all tail
+
+export
+tails_ : {n : Nat} -> (opes : OPEList (S n)) ->
+         Subset (OPEList n) (All2 IsTail opes)
+tails_ []            = Element [] []
+tails_ (ope :: opes) = Element _ $ (tail ope).isTail :: (tails_ opes).snd
+
+export
+conses : (heads : List Bool) -> (tails : OPEList n) ->
+         (0 len : length heads = length tails) =>
+         OPEList (S n)
+conses []        []        = []
+conses (h :: hs) (t :: ts) = snd (cons h t) :: conses hs ts @{inj S len}
--- a/lib/Quox/Thin/Split.idr
+++ b/lib/Quox/Thin/Split.idr
@ -0,0 +1,57 @@
+module Quox.Thin.Split
+
+import public Quox.Thin.Base
+import public Quox.Thin.View
+import public Quox.Thin.Eqv
+import public Quox.Thin.Append
+import public Quox.Thin.Cover
+import Data.DPair
+import Control.Relation
+
+%default total
+
+public export
+record Chunks m n where
+  constructor MkChunks
+  {leftMask  : Nat}
+  {rightMask : Nat}
+  0 left  : OPE m (m + n) leftMask
+  0 right : OPE n (m + n) rightMask
+  {auto 0 isCover : Cover [left, right]}
+%name Chunks chunks
+
+export
+chunks : (m, n : Nat) -> Chunks m n
+chunks 0 0 = MkChunks Stop Stop
+chunks 0 (S n) =
+  let MkChunks l r = chunks 0 n in
+  MkChunks (Drop l Refl) (Keep r Refl)
+chunks (S m) n =
+  let MkChunks l r = chunks m n in
+  MkChunks (Keep l Refl) (Drop r Refl)
+
+-- [todo] the masks here are just ((2 << m) - 1) << n and (2 << n) - 1
+
+
+public export
+record SplitAt m n1 n2 (ope : OPE m (n1 + n2) mask) where
+  constructor MkSplitAt
+  {leftMask, rightMask : Nat}
+  {0 leftScope, rightScope : Nat}
+  0 left     : OPE leftScope n1 leftMask
+  0 right    : OPE rightScope n2 rightMask
+  0 scopePrf : m = leftScope + rightScope
+  0 opePrf   : ope `Eqv` (left `app'` right).snd
+%name SplitAt split
+
+export
+splitAt : (n1 : Nat) -> {n2, mask : Nat} -> (0 ope : OPE m (n1 + n2) mask) ->
+          SplitAt m n1 n2 ope
+splitAt 0     ope = MkSplitAt zero ope Refl reflexive
+splitAt (S n1) ope with %syntactic (view ope)
+  splitAt (S n1) (Drop ope Refl) | DropV _ ope with %syntactic (splitAt n1 ope)
+    _ | MkSplitAt left right scopePrf opePrf =
+      MkSplitAt (Drop left Refl) right scopePrf (EqvDrop opePrf)
+  splitAt (S n1) (Keep ope Refl) | KeepV _ ope with %syntactic (splitAt n1 ope)
+    _ | MkSplitAt left right scopePrf opePrf =
+      MkSplitAt (Keep left Refl) right (cong S scopePrf) (EqvKeep opePrf)
--- a/lib/Quox/Thin/Term.idr
+++ b/lib/Quox/Thin/Term.idr
@ -0,0 +1,179 @@
+module Quox.Thin.Term
+
+import public Quox.Thin.Base
+import public Quox.Thin.Comp
+import public Quox.Thin.List
+import Quox.Thin.Eqv
+import public Quox.Thin.Cover
+import Quox.Name
+import Quox.Loc
+import Data.DPair
+import public Data.List.Quantifiers
+import Data.Vect
+import Data.Singleton
+import Decidable.Equality
+
+%default total
+
+public export
+record Thinned f n where
+  constructor Th
+  {0 scope   : Nat}
+  {scopeMask : Nat}
+  0 ope : OPE scope n scopeMask
+  term  : f scope
+%name Thinned s, t, u
+
+export
+(forall n. Eq (f n)) => Eq (Thinned f n) where
+  s == t = case decEq s.scopeMask t.scopeMask of
+    Yes eq => s.term == (rewrite maskEqInner s.ope t.ope eq in t.term)
+    No  _  => False
+
+export
+(forall n. Located (f n)) => Located (Thinned f n) where
+  term.loc = term.term.loc
+
+export
+(forall n. Relocatable (f n)) => Relocatable (Thinned f n) where
+  setLoc loc = {term $= setLoc loc}
+
+namespace Thinned
+  export
+  pure : {n : Nat} -> f n -> Thinned f n
+  pure term = Th id.snd term
+
+  export
+  join : {n : Nat} -> Thinned (Thinned f) n -> Thinned f n
+  join (Th ope1 (Th ope2 term)) = Th (ope1 . ope2) term
+
+
+public export
+record ScopedN (s : Nat) (f : Nat -> Type) (n : Nat) where
+  constructor S
+  names : Vect s BindName
+  {0 scope : Nat}
+  {mask : Nat}
+  0 ope : OPE scope s mask
+  body : f (scope + n)
+
+export
+(forall n. Eq (f n)) => Eq (ScopedN s f n) where
+  s1 == s2 = case decEq s1.mask s2.mask of
+    Yes eq =>
+      s1.names == s2.names &&
+      s1.body == (rewrite maskEqInner s1.ope s2.ope eq in s2.body)
+    No _ => False
+
+export
+{s : Nat} -> (forall n. Show (f n)) => Show (ScopedN s f n) where
+  showPrec d (S ns ope body) = showCon d "S" $
+    showArg ns ++ showArg (unVal $ maskToOpe ope) ++ showArg body
+
+public export
+Scoped : (Nat -> Type) -> Nat -> Type
+Scoped d n = ScopedN 1 d n
+
+(.name) : Scoped f n -> BindName
+(S {names = [x], _}).name = x
+
+export
+(forall n. Located (f n)) => Located (ScopedN s f n) where
+  s.loc = s.body.loc
+
+export
+(forall n. Relocatable (f n)) => Relocatable (ScopedN s f n) where
+  setLoc loc = {body $= setLoc loc}
+
+
+public export
+record Thinned2 f d n where
+  constructor Th2
+  {0 dscope, tscope : Nat}
+  {dmask, tmask : Nat}
+  0 dope : OPE dscope d dmask
+  0 tope : OPE tscope n tmask
+  term : f dscope tscope
+%name Thinned2 term
+
+export
+(forall d, n. Eq (f d n)) => Eq (Thinned2 f d n) where
+  s == t = case (decEq s.dmask t.dmask, decEq s.tmask t.tmask) of
+    (Yes deq, Yes teq) =>
+      s.term == (rewrite maskEqInner s.dope t.dope deq in
+                 rewrite maskEqInner s.tope t.tope teq in t.term)
+    _ => False
+
+export
+(forall d, n. Located (f d n)) => Located (Thinned2 f d n) where
+  term.loc = term.term.loc
+
+export
+(forall d, n. Relocatable (f d n)) => Relocatable (Thinned2 f d n) where
+  setLoc loc = {term $= setLoc loc}
+
+namespace Thinned2
+  export
+  pure : {d, n : Nat} -> f d n -> Thinned2 f d n
+  pure term = Th2 id.snd id.snd term
+
+  export
+  join : {d, n : Nat} -> Thinned2 (Thinned2 f) d n -> Thinned2 f d n
+  join (Th2 dope1 tope1 (Th2 dope2 tope2 term)) =
+    Th2 (dope1 . dope2) (tope1 . tope2) term
+
+
+namespace TermList
+  public export
+  data Element : (Nat -> Nat -> Type) ->
+                 OPE dscope d dmask -> OPE tscope n tmask -> Type where
+    T : f dscope tscope ->
+        {dmask : Nat} -> (0 dope : OPE dscope d dmask) ->
+        {tmask : Nat} -> (0 tope : OPE tscope n tmask) ->
+        Element f dope tope
+  %name TermList.Element s, t, u
+
+  export
+  elementEq : (forall d, n. Eq (f d n)) =>
+              Element {d, n} f dope1 tope1 -> Element {d, n} f dope2 tope2 ->
+              Bool
+  elementEq (T s dope1 tope1 {dmask = dm1, tmask = tm1})
+            (T t dope2 tope2 {dmask = dm2, tmask = tm2}) =
+    case (decEq dm1 dm2, decEq tm1 tm2) of
+      (Yes deq, Yes teq) =>
+        s == (rewrite maskEqInner dope1 dope2 deq in
+              rewrite maskEqInner tope1 tope2 teq in t)
+      _ => False
+
+  public export
+  data TermList : List (Nat -> Nat -> Type) ->
+                  OPEList d -> OPEList n -> Type where
+    Nil  : TermList [] [] []
+    (::) : Element f dope tope ->
+           TermList fs dopes topes ->
+           TermList (f :: fs) (dope :: dopes) (tope :: topes)
+  %name TermList ss, ts, us
+
+  export
+  termListEq : All (\f => forall d, n. Eq (f d n)) fs =>
+               TermList {d, n} fs dopes1 topes1 ->
+               TermList {d, n} fs dopes2 topes2 ->
+               Bool
+  termListEq [] [] = True
+  termListEq (s :: ss) (t :: ts) @{eq :: eqs} =
+    elementEq s t && termListEq ss ts
+
+
+public export
+record Subterms (fs : List (Nat -> Nat -> Type)) d n where
+  constructor Sub
+  {0 dopes : OPEList d}
+  {0 topes : OPEList n}
+  terms : TermList fs dopes topes
+  0 dcov : Cover dopes
+  0 tcov : Cover topes
+%name Subterms ss, ts, us
+
+export
+All (\f => forall d, n. Eq (f d n)) fs => Eq (Subterms fs d n) where
+  ss == ts = ss.terms `termListEq` ts.terms
--- a/lib/Quox/Thin/View.idr
+++ b/lib/Quox/Thin/View.idr
@ -0,0 +1,103 @@
+module Quox.Thin.View
+
+import public Quox.Thin.Base
+import Quox.NatExtra
+import Data.Singleton
+import Data.SnocVect
+
+%default total
+
+public export
+data View : OPE m n mask -> Type where
+  StopV : View Stop
+  DropV : (mask : Nat) -> (0 ope : OPE m n mask) -> View (Drop ope Refl)
+  KeepV : (mask : Nat) -> (0 ope : OPE m n mask) -> View (Keep ope Refl)
+%name View.View v
+
+private
+0 stopEqs : OPE m 0 mask -> (m = 0, mask = 0)
+stopEqs Stop = (Refl, Refl)
+
+private
+0 fromStop : (ope : OPE 0 0 0) -> ope = Stop
+fromStop Stop = Refl
+
+private
+0 fromDrop : (ope : OPE m (S n) (k + k)) ->
+             (inner : OPE m n k ** ope === Drop inner Refl)
+fromDrop (Drop ope eq) with %syntactic (doubleInj _ _ eq)
+  fromDrop (Drop ope Refl) | Refl = (ope ** Refl)
+fromDrop (Keep ope eq) = void $ notEvenOdd _ _ eq
+
+private
+0 fromKeep : (ope : OPE (S m) (S n) (S (k + k))) ->
+             (inner : OPE m n k ** ope === Keep inner Refl)
+fromKeep (Drop ope eq) = void $ notEvenOdd _ _ $ sym eq
+fromKeep (Keep ope eq) with %syntactic (doubleInj _ _ $ inj S eq)
+  fromKeep (Keep ope Refl) | Refl = (ope ** Refl)
+
+private
+0 keepIsSucc : (ope : OPE m n (S (k + k))) -> IsSucc m
+keepIsSucc (Drop ope eq) = void $ notEvenOdd _ _ $ sym eq
+keepIsSucc (Keep ope _)  = ItIsSucc
+
+export
+view : {0 m : Nat} -> {n, mask : Nat} -> (0 ope : OPE m n mask) -> View ope
+view {n = 0} ope with %syntactic 0 (fst $ stopEqs ope) | 0 (snd $ stopEqs ope)
+  _ | Refl | Refl = rewrite fromStop ope in StopV
+view {n = S n} ope with %syntactic (half mask)
+  _ | HalfOdd mask' with %syntactic 0 (keepIsSucc ope)
+    _ | ItIsSucc with %syntactic 0 (fromKeep ope)
+      _ | (ope' ** eq) = rewrite eq in KeepV mask' ope'
+  _ | HalfEven mask' with %syntactic 0 (fromDrop ope)
+    _ | (ope' ** eq) = rewrite eq in DropV mask' ope'
+
+
+export
+appOpe : {0 m : Nat} -> (n : Nat) -> {mask : Nat} ->
+         (0 ope : OPE m n mask) -> Singleton m
+appOpe n ope with %syntactic (view ope)
+  appOpe 0     Stop          | StopV        = Val 0
+  appOpe (S n) (Drop ope' _) | DropV _ ope' = appOpe n ope'
+  appOpe (S n) (Keep ope' _) | KeepV _ ope' = [|S $ appOpe n ope'|]
+
+export
+maskToOpe : {n, mask : Nat} -> (0 ope : OPE m n mask) -> Singleton ope
+maskToOpe ope with %syntactic (view ope)
+  maskToOpe Stop            | StopV       = [|Stop|]
+  maskToOpe (Drop ope Refl) | DropV k ope = [|drop $ maskToOpe ope|]
+  maskToOpe (Keep ope Refl) | KeepV k ope = [|keep $ maskToOpe ope|]
+
+export
+0 outerInnerZero : OPE m 0 mask -> m = 0
+outerInnerZero Stop = Refl
+
+export
+0 outerMaskZero : OPE m 0 mask -> mask = 0
+outerMaskZero Stop = Refl
+
+export
+0 viewStop : view Stop = StopV
+viewStop = Refl
+
+export
+0 viewDrop : (ope : OPE m n mask) -> (eq : mask2 = mask + mask) ->
+             view (Drop {mask} ope eq) = DropV mask ope
+viewDrop ope eq with (view (Drop ope eq))
+  viewDrop ope Refl | DropV _ ope = Refl
+
+export
+0 viewKeep : (ope : OPE m n mask) -> (eq : mask2 = S (mask + mask)) ->
+             view (Keep {mask} ope eq) = KeepV mask ope
+viewKeep ope eq with (view (Keep ope eq))
+  viewKeep ope Refl | KeepV _ ope = Refl
+
+
+namespace SnocVect
+  export
+  select : {n, mask : Nat} -> (0 ope : OPE m n mask) ->
+           SnocVect n a -> SnocVect m a
+  select ope sx with (view ope)
+    select _ [<]       | StopV       = [<]
+    select _ (sx :< x) | DropV _ ope = select ope sx
+    select _ (sx :< x) | KeepV _ ope = select ope sx :< x
--- a/lib/Quox/Typechecker.idr
+++ b/lib/Quox/Typechecker.idr
@ -163,7 +163,7 @@ mutual
                (subj : Term d n) -> (0 nc : NotClo subj) => Term d n ->
                TC (CheckResult' n)
  toCheckType ctx sg t ty = do
-    u <- expectTYPE !defs ctx ty.loc ty
+    u <- expectTYPE !(askAt DEFS) ctx ty.loc ty
    expectEqualQ t.loc Zero sg.fst
    checkTypeNoWrap ctx t (Just u)
    pure $ zeroFor ctx
@ -178,7 +178,7 @@ mutual
  check' ctx sg t@(Pi {}) ty = toCheckType ctx sg t ty

  check' ctx sg (Lam body loc) ty = do
-    (qty, arg, res) <- expectPi !defs ctx ty.loc ty
+    (qty, arg, res) <- expectPi !(askAt DEFS) ctx ty.loc ty
    -- if Ψ | Γ, x : A ⊢ σ · t ⇐ B ⊳ Σ, ρ·x
    -- with ρ ≤ σπ
    let qty' = sg.fst * qty
@ -189,7 +189,7 @@ mutual
  check' ctx sg t@(Sig {}) ty = toCheckType ctx sg t ty

  check' ctx sg (Pair fst snd loc) ty = do
-    (tfst, tsnd) <- expectSig !defs ctx ty.loc ty
+    (tfst, tsnd) <- expectSig !(askAt DEFS) ctx ty.loc ty
    -- if Ψ | Γ ⊢ σ · s ⇐ A ⊳ Σ₁
    qfst <- checkC ctx sg fst tfst
    let tsnd = sub1 tsnd (Ann fst tfst fst.loc)
@ -201,7 +201,7 @@ mutual
  check' ctx sg t@(Enum {}) ty = toCheckType ctx sg t ty

  check' ctx sg (Tag t loc) ty = do
-    tags <- expectEnum !defs ctx ty.loc ty
+    tags <- expectEnum !(askAt DEFS) ctx ty.loc ty
    -- if t ∈ ts
    unless (t `elem` tags) $ throw $ TagNotIn loc t tags
    -- then Ψ | Γ ⊢ σ · t ⇐ {ts} ⊳ 𝟎
@ -210,7 +210,7 @@ mutual
  check' ctx sg t@(Eq {}) ty = toCheckType ctx sg t ty

  check' ctx sg (DLam body loc) ty = do
-    (ty, l, r) <- expectEq !defs ctx ty.loc ty
+    (ty, l, r) <- expectEq !(askAt DEFS) ctx ty.loc ty
    let ctx' = extendDim body.name ctx
        ty   = ty.term
        body = body.term
@ -226,17 +226,17 @@ mutual
  check' ctx sg t@(Nat {}) ty = toCheckType ctx sg t ty

  check' ctx sg (Zero {}) ty = do
-    expectNat !defs ctx ty.loc ty
+    expectNat !(askAt DEFS) ctx ty.loc ty
    pure $ zeroFor ctx

  check' ctx sg (Succ n {}) ty = do
-    expectNat !defs ctx ty.loc ty
+    expectNat !(askAt DEFS) ctx ty.loc ty
    checkC ctx sg n ty

  check' ctx sg t@(BOX {}) ty = toCheckType ctx sg t ty

  check' ctx sg (Box val loc) ty = do
-    (q, ty) <- expectBOX !defs ctx ty.loc ty
+    (q, ty) <- expectBOX !(askAt DEFS) ctx ty.loc ty
    -- if Ψ | Γ ⊢ σ · s ⇐ A ⊳ Σ
    valout <- checkC ctx sg val ty
    -- then Ψ | Γ ⊢ σ · [s] ⇐ [π.A] ⊳ πΣ
@ -314,7 +314,7 @@ mutual
    -- if Ψ | Γ ⊢ Type ℓ <: Type 𝓀
    case u of
      Just u  => lift $ subtype e.loc ctx infres.type (TYPE u noLoc)
-      Nothing => ignore $ expectTYPE !defs ctx e.loc infres.type
+      Nothing => ignore $ expectTYPE !(askAt DEFS) ctx e.loc infres.type
    -- then Ψ | Γ ⊢₀ E ⇐ Type 𝓀


@ -333,7 +333,7 @@ mutual

  infer' ctx sg (F x u loc) = do
    -- if π·x : A {≔ s} in global context
-    g <- lookupFree x loc !defs
+    g <- lookupFree x loc !(askAt DEFS)
    -- if σ ≤ π
    expectCompatQ loc sg.fst g.qty.fst
    -- then Ψ | Γ ⊢ σ · x ⇒ A ⊳ 𝟎
@ -355,7 +355,7 @@ mutual
  infer' ctx sg (App fun arg loc) = do
    -- if Ψ | Γ ⊢ σ · f ⇒ (π·x : A) → B ⊳ Σ₁
    funres <- inferC ctx sg fun
-    (qty, argty, res) <- expectPi !defs ctx fun.loc funres.type
+    (qty, argty, res) <- expectPi !(askAt DEFS) ctx fun.loc funres.type
    -- if Ψ | Γ ⊢ σ ⨴ π · s ⇐ A ⊳ Σ₂
    argout <- checkC ctx (subjMult sg qty) arg argty
    -- then Ψ | Γ ⊢ σ · f s ⇒ B[s] ⊳ Σ₁ + πΣ₂
@ -372,7 +372,7 @@ mutual
    pairres <- inferC ctx sg pair
    -- if Ψ | Γ, p : (x : A) × B ⊢₀ ret ⇐ Type
    checkTypeC (extendTy Zero ret.name pairres.type ctx) ret.term Nothing
-    (tfst, tsnd) <- expectSig !defs ctx pair.loc pairres.type
+    (tfst, tsnd) <- expectSig !(askAt DEFS) ctx pair.loc pairres.type
    -- if Ψ | Γ, x : A, y : B ⊢ σ · body ⇐
    --    ret[(x, y) ∷ (x : A) × B/p] ⊳ Σ₂, ρ₁·x, ρ₂·y
    -- with ρ₁, ρ₂ ≤ πσ
@ -391,7 +391,7 @@ mutual
  infer' ctx sg (CaseEnum pi t ret arms loc) {d, n} = do
    -- if Ψ | Γ ⊢ σ · t ⇒ {ts} ⊳ Σ₁
    tres <- inferC ctx sg t
-    ttags <- expectEnum !defs ctx t.loc tres.type
+    ttags <- expectEnum !(askAt DEFS) ctx t.loc tres.type
    -- if 1 ≤ π, OR there is only zero or one option
    unless (length (SortedSet.toList ttags) <= 1) $ expectCompatQ loc One pi
    -- if Ψ | Γ, x : {ts} ⊢₀ A ⇐ Type
@ -415,7 +415,7 @@ mutual
    -- if Ψ | Γ ⊢ σ · n ⇒ ℕ ⊳ Σn
    nres <- inferC ctx sg n
    let nat = nres.type
-    expectNat !defs ctx n.loc nat
+    expectNat !(askAt DEFS) ctx n.loc nat
    -- if Ψ | Γ, n : ℕ ⊢₀ A ⇐ Type
    checkTypeC (extendTy Zero ret.name nat ctx) ret.term Nothing
    -- if Ψ | Γ ⊢ σ · zer ⇐ A[0 ∷ ℕ/n] ⊳ Σz
@ -439,7 +439,7 @@ mutual
  infer' ctx sg (CaseBox pi box ret body loc) = do
    -- if Ψ | Γ ⊢ σ · b ⇒ [ρ.A] ⊳ Σ₁
    boxres <- inferC ctx sg box
-    (q, ty) <- expectBOX !defs ctx box.loc boxres.type
+    (q, ty) <- expectBOX !(askAt DEFS) ctx box.loc boxres.type
    -- if Ψ | Γ, x : [ρ.A] ⊢₀ R ⇐ Type
    checkTypeC (extendTy Zero ret.name boxres.type ctx) ret.term Nothing
    -- if Ψ | Γ, x : A ⊢ t ⇐ R[[x] ∷ [ρ.A/x]] ⊳ Σ₂, ς·x
@ -457,7 +457,7 @@ mutual
  infer' ctx sg (DApp fun dim loc) = do
    -- if Ψ | Γ ⊢ σ · f ⇒ Eq [𝑖 ⇒ A] l r ⊳ Σ
    InfRes {type, qout} <- inferC ctx sg fun
-    ty <- fst <$> expectEq !defs ctx fun.loc type
+    ty <- fst <$> expectEq !(askAt DEFS) ctx fun.loc type
    -- then Ψ | Γ ⊢ σ · f p ⇒ A‹p/𝑖› ⊳ Σ
    pure $ InfRes {type = dsub1 ty dim, qout}

@ -485,7 +485,7 @@ mutual
    -- if σ = 0
    expectEqualQ loc Zero sg.fst
    -- if Ψ, Γ ⊢₀ e ⇒ Type u
-    u <- expectTYPE !defs ctx ty.loc . type =<< inferC ctx szero ty
+    u <- expectTYPE !(askAt DEFS) ctx ty.loc . type =<< inferC ctx szero ty
    -- if Ψ, Γ ⊢₀ C ⇐ Type (non-dependent return type)
    checkTypeC ctx ret Nothing
    -- if Ψ, Γ' ⊢₀ A ⇐ C for each rhs A
--- a/lib/Quox/Typing/Context.idr
+++ b/lib/Quox/Typing/Context.idr
@ -69,7 +69,7 @@ namespace TContext
  zeroFor ctx = Zero <$ ctx

 private
-extendLen : Telescope a from to -> Singleton from -> Singleton to
+extendLen : Telescope a n1 n2 -> Singleton n1 -> Singleton n2
 extendLen [<]        x = x
 extendLen (tel :< _) x = [|S $ extendLen tel x|]

@ -89,7 +89,7 @@ namespace TyContext
  null ctx = null ctx.dnames && null ctx.tnames

  export %inline
-  extendTyN : CtxExtension d from to -> TyContext d from -> TyContext d to
+  extendTyN : CtxExtension d n1 n2 -> TyContext d n1 -> TyContext d n2
  extendTyN xss (MkTyContext {termLen, dctx, dnames, tctx, tnames, qtys}) =
    let (qs, xs, ss) = unzip3 xss in
    MkTyContext {
@ -172,7 +172,7 @@ namespace EqContext
  null ctx = null ctx.dnames && null ctx.tnames

  export %inline
-  extendTyN : CtxExtension 0 from to -> EqContext from -> EqContext to
+  extendTyN : CtxExtension 0 n1 n2 -> EqContext n1 -> EqContext n2
  extendTyN xss (MkEqContext {termLen, dassign, dnames, tctx, tnames, qtys}) =
    let (qs, xs, ss) = unzip3 xss in
    MkEqContext {
--- a/lib/quox-lib.ipkg
+++ b/lib/quox-lib.ipkg
@ -16,8 +16,19 @@ modules =
  Quox.Decidable,
  Quox.No,
  Quox.Loc,
-  Quox.OPE,
  Quox.Pretty,
+  Quox.Thin.Base,
+  Quox.Thin.View,
+  Quox.Thin.Eqv,
+  Quox.Thin.Cons,
+  Quox.Thin.List,
+  Quox.Thin.Append,
+  Quox.Thin.Comp,
+  Quox.Thin.Cover,
+  Quox.Thin.Coprod,
+  Quox.Thin.Split,
+  Quox.Thin.Term,
+  Quox.Thin,
  Quox.Syntax,
  Quox.Syntax.Dim,
  Quox.Syntax.DimEq,
--- a/pack.toml
+++ b/pack.toml
@ -1,4 +1,4 @@
-collection = "nightly-230522"
+collection = "nightly-230622"

 [custom.all.tap]
 type = "git"
--- a/quox.bib
+++ b/quox.bib
@ -366,3 +366,16 @@
  url       = {https://www.cs.bham.ac.uk/~pbl/papers/hosc05.pdf},
  doi       = {10.1007/s10990-006-0480-6},
 }
+
+@article{egtbs,
+  doi = {10.4204/eptcs.275.6},
+  url = {https://doi.org/10.4204%2Feptcs.275.6},
+  year = 2018,
+  month = {jul},
+  publisher = {Open Publishing Association},
+  volume = {275},
+  pages = {53--69},
+  author = {Conor McBride},
+  title = {Everybody's Got To Be Somewhere},
+  journal = {Electronic Proceedings in Theoretical Computer Science}
+}
--- a/tests/Tests/Typechecker.idr
+++ b/tests/Tests/Typechecker.idr
@ -34,7 +34,7 @@ ToInfo Error' where
 M = Eff [Except Error', DefsReader]

 inj : TC a -> M a
-inj act = rethrow $ mapFst TCError $ runTC !defs act
+inj act = rethrow $ mapFst TCError $ runTC !(askAt DEFS) act


 reflTy : Term d n
Author	SHA1	Message	Date
rhiannon morris	124637c946	add SnocVect.select	2023-06-24 14:28:56 +02:00
rhiannon morris	33abbf659e	remove stray comment	2023-06-24 14:28:56 +02:00
rhiannon morris	326db52204	more nat bitwise ops	2023-06-24 14:28:52 +02:00
rhiannon morris	ddfbca7fcc	(n-ary) coprod	2023-06-24 14:28:48 +02:00
rhiannon morris	aca953c518	wip maybe.quox	2023-06-23 18:32:17 +02:00
rhiannon morris	b61ace9c7d	some bitwise ops corresponding to OPE ops	2023-06-23 18:32:17 +02:00
rhiannon morris	9e702dd03d	switch syntax to codb	2023-06-23 18:32:17 +02:00
rhiannon morris	92870fe716	split and extend Quox.Thin	2023-06-23 18:32:17 +02:00
rhiannon morris	a7673f901f	add Quox.Thin	2023-06-23 18:32:17 +02:00
rhiannon morris	5580f90e8d	add egtbs to bib	2023-06-23 18:32:17 +02:00
rhiannon morris	fa09aaf228	squash warnings	2023-06-23 18:32:05 +02:00
rhiannon morris	6eccfeef52	pack bump	2023-06-23 18:14:40 +02:00
rhiannon morris	f0d3529f63	fix subtype stuff for Eq	2023-06-22 22:20:12 +02:00
rhiannon morris	cd330c1092	remove a noLoc	2023-06-11 19:25:38 +02:00
rhiannon morris	865772d512	remove stale todos	2023-06-11 19:25:32 +02:00