quox/lib/Quox/Syntax/Term/Base.idr

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module Quox.Syntax.Term.Base
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import public Quox.Var
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import public Quox.Scoped
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import public Quox.Syntax.Shift
import public Quox.Syntax.Subst
import public Quox.Syntax.Qty
import public Quox.Syntax.Dim
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import public Quox.Syntax.Term.TyConKind
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import public Quox.Name
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import public Quox.Loc
import public Quox.Context
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import Quox.Pretty
import public Data.DPair
import Data.List
import Data.Maybe
import Data.Nat
import public Data.So
import Data.String
import public Data.SortedMap
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import public Data.SortedMap.Dependent
import public Data.SortedSet
import Derive.Prelude
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%default total
%language ElabReflection
%hide TT.Name
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public export
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TermLike : Type
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TermLike = Nat -> Nat -> Type
public export
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TSubstLike : Type
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TSubstLike = Nat -> Nat -> Nat -> Type
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public export
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Universe : Type
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Universe = Nat
public export
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TagVal : Type
TagVal = String
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infixl 8 :#
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infixl 9 :@, :%
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mutual
public export
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TSubst : TSubstLike
TSubst d = Subst $ \n => Elim d n
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||| first argument `d` is dimension scope size;
||| second `n` is term scope size
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public export
data Term : (d, n : Nat) -> Type where
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||| type of types
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TYPE : (l : Universe) -> (loc : Loc) -> Term d n
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||| IO state token. this is a builtin because otherwise #[main] being a
||| builtin makes no sense
IOState : (loc : Loc) -> Term d n
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||| function type
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Pi : (qty : Qty) -> (arg : Term d n) ->
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(res : ScopeTerm d n) -> (loc : Loc) -> Term d n
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||| function term
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Lam : (body : ScopeTerm d n) -> (loc : Loc) -> Term d n
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||| pair type
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Sig : (fst : Term d n) -> (snd : ScopeTerm d n) -> (loc : Loc) -> Term d n
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||| pair value
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Pair : (fst, snd : Term d n) -> (loc : Loc) -> Term d n
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||| enumeration type
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Enum : (cases : SortedSet TagVal) -> (loc : Loc) -> Term d n
||| enumeration value
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Tag : (tag : TagVal) -> (loc : Loc) -> Term d n
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||| equality type
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Eq : (ty : DScopeTerm d n) -> (l, r : Term d n) -> (loc : Loc) -> Term d n
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||| equality term
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DLam : (body : DScopeTerm d n) -> (loc : Loc) -> Term d n
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||| natural numbers (temporary until 𝐖 gets added)
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NAT : (loc : Loc) -> Term d n
Nat : (val : Nat) -> (loc : Loc) -> Term d n
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Succ : (p : Term d n) -> (loc : Loc) -> Term d n
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||| strings
STRING : (loc : Loc) -> Term d n
Str : (str : String) -> (loc : Loc) -> Term d n
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||| "box" (package a value up with a certain quantity)
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BOX : (qty : Qty) -> (ty : Term d n) -> (loc : Loc) -> Term d n
Box : (val : Term d n) -> (loc : Loc) -> Term d n
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Let : (qty : Qty) -> (rhs : Elim d n) ->
(body : ScopeTerm d n) -> (loc : Loc) -> Term d n
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||| elimination
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E : (e : Elim d n) -> Term d n
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||| term closure/suspended substitution
CloT : WithSubst (Term d) (Elim d) n -> Term d n
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||| dimension closure/suspended substitution
DCloT : WithSubst (\d => Term d n) Dim d -> Term d n
%name Term s, t, r
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||| first argument `d` is dimension scope size, second `n` is term scope size
public export
data Elim : (d, n : Nat) -> Type where
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||| 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
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||| bound variable
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B : (i : Var n) -> (loc : Loc) -> Elim d n
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||| term application
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App : (fun : Elim d n) -> (arg : Term d n) -> (loc : Loc) -> Elim d n
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||| pair destruction
|||
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||| `CasePair 𝜋 𝑒 ([𝑟], 𝐴) ([𝑥, 𝑦], 𝑡)` is
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||| `𝐜𝐚𝐬𝐞 𝜋 · 𝑒 𝐫𝐞𝐭𝐮𝐫𝐧 𝑟𝐴 𝐨𝐟 { (𝑥, 𝑦) ⇒ 𝑡 }`
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CasePair : (qty : Qty) -> (pair : Elim d n) ->
(ret : ScopeTerm d n) ->
(body : ScopeTermN 2 d n) ->
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(loc : Loc) ->
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Elim d n
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||| first element of a pair. only works in non-linear contexts.
Fst : (pair : Elim d n) -> (loc : Loc) -> Elim d n
||| second element of a pair. only works in non-linear contexts.
Snd : (pair : Elim d n) -> (loc : Loc) -> Elim d n
||| enum matching
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CaseEnum : (qty : Qty) -> (tag : Elim d n) ->
(ret : ScopeTerm d n) ->
(arms : CaseEnumArms d n) ->
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(loc : Loc) ->
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Elim d n
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||| nat matching
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CaseNat : (qty, qtyIH : Qty) -> (nat : Elim d n) ->
(ret : ScopeTerm d n) ->
(zero : Term d n) ->
(succ : ScopeTermN 2 d n) ->
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(loc : Loc) ->
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Elim d n
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||| unboxing
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CaseBox : (qty : Qty) -> (box : Elim d n) ->
(ret : ScopeTerm d n) ->
(body : ScopeTerm d n) ->
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(loc : Loc) ->
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Elim d n
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||| dim application
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DApp : (fun : Elim d n) -> (arg : Dim d) -> (loc : Loc) -> Elim d n
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||| type-annotated term
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Ann : (tm, ty : Term d n) -> (loc : Loc) -> Elim d n
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||| coerce a value along a type equality, or show its coherence
||| [@xtt; §2.1.1]
Coe : (ty : DScopeTerm d n) -> (p, q : Dim d) ->
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(val : Term d n) -> (loc : Loc) -> Elim d n
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||| "generalised composition" [@xtt; §2.1.2]
Comp : (ty : Term d n) -> (p, q : Dim d) ->
(val : Term d n) -> (r : Dim d) ->
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(zero, one : DScopeTerm d n) -> (loc : Loc) -> Elim d n
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||| 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) ->
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(loc : Loc) ->
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Elim d n
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||| term closure/suspended substitution
CloE : WithSubst (Elim d) (Elim d) n -> Elim d n
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||| dimension closure/suspended substitution
DCloE : WithSubst (\d => Elim d n) Dim d -> Elim d n
%name Elim e, f
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public export
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CaseEnumArms : TermLike
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CaseEnumArms d n = SortedMap TagVal (Term d n)
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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)
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public export
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ScopeTermN, DScopeTermN : Nat -> TermLike
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ScopeTermN s d n = Scoped s (Term d) n
DScopeTermN s d n = Scoped s (\d => Term d n) d
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public export
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ScopeTerm, DScopeTerm : TermLike
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ScopeTerm = ScopeTermN 1
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DScopeTerm = DScopeTermN 1
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mutual
export %hint
EqTerm : Eq (Term d n)
EqTerm = assert_total {a = Eq (Term d n)} deriveEq
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export %hint
EqElim : Eq (Elim d n)
EqElim = assert_total {a = Eq (Elim d n)} 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
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export
Located (Elim d n) where
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(F _ _ loc).loc = loc
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(B _ loc).loc = loc
(App _ _ loc).loc = loc
(CasePair _ _ _ _ loc).loc = loc
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(Fst _ loc).loc = loc
(Snd _ loc).loc = loc
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(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
(IOState loc).loc = loc
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(Pi _ _ _ loc).loc = loc
(Lam _ loc).loc = loc
(Sig _ _ loc).loc = loc
(Pair _ _ loc).loc = loc
(Enum _ loc).loc = loc
(Tag _ loc).loc = loc
(Eq _ _ _ loc).loc = loc
(DLam _ loc).loc = loc
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(NAT loc).loc = loc
(Nat _ loc).loc = loc
(STRING loc).loc = loc
(Str _ loc).loc = loc
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(Succ _ loc).loc = loc
(BOX _ _ loc).loc = loc
(Box _ loc).loc = loc
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(Let _ _ _ loc).loc = loc
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(E e).loc = e.loc
(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
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setLoc loc (F x u _) = F x u loc
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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
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setLoc loc (Fst pair _) = Fst pair loc
setLoc loc (Snd pair _) = Fst pair loc
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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
export
Relocatable (Term d n) where
setLoc loc (TYPE l _) = TYPE l loc
setLoc loc (IOState _) = IOState loc
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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
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setLoc loc (NAT _) = NAT loc
setLoc loc (Nat n _) = Nat n loc
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setLoc loc (Succ p _) = Succ p loc
setLoc loc (STRING _) = STRING loc
setLoc loc (Str s _) = Str s loc
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setLoc loc (BOX qty ty _) = BOX qty ty loc
setLoc loc (Box val _) = Box val loc
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setLoc loc (Let qty rhs body _) = Let qty rhs body loc
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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)
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||| 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}
||| same as `F` but as a term
public export %inline
FT : Name -> Universe -> Loc -> Term d n
FT x u loc = E $ F x u loc
||| same as `B` but as a term
public export %inline
BT : Var n -> (loc : Loc) -> Term d n
BT i loc = E $ B i 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
||| 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
public export %inline
Zero : Loc -> Term d n
Zero = Nat 0
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