299 lines
7.9 KiB
Idris
299 lines
7.9 KiB
Idris
module Quox.Syntax.Term.Base
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import public Quox.Syntax.Var
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import public Quox.Syntax.Shift
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import public Quox.Syntax.Subst
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import public Quox.Syntax.Qty
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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.Context
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import Quox.Pretty
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import public Data.DPair
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import Data.List
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import Data.Maybe
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import Data.Nat
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import public Data.So
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import Data.String
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import public Data.SortedMap
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import public Data.SortedMap.Dependent
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import public Data.SortedSet
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%default total
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public export
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TermLike : Type
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TermLike = Nat -> Nat -> Type
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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
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public export
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TagVal : Type
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TagVal = String
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infixl 8 :#
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infixl 9 :@, :%
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mutual
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public export
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TSubst : TSubstLike
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TSubst d = Subst $ \n => Elim d n
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||| first argument `d` is dimension scope size;
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||| second `n` is term scope size
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public export
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data Term : TermLike where
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||| type of types
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TYPE : (l : Universe) -> 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) -> Term d n
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||| function term
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Lam : (body : ScopeTerm d n) -> Term d n
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||| pair type
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Sig : (fst : Term d n) -> (snd : ScopeTerm d n) -> Term d n
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||| pair value
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Pair : (fst, snd : Term d n) -> Term d n
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||| enumeration type
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Enum : (cases : SortedSet TagVal) -> Term d n
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||| enumeration value
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Tag : (tag : TagVal) -> Term d n
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||| equality type
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Eq : (ty : DScopeTerm d n) -> (l, r : Term d n) -> Term d n
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||| equality term
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DLam : (body : DScopeTerm d n) -> Term d n
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||| natural numbers (temporary until 𝐖 gets added)
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Nat : Term d n
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-- [todo] can these be elims?
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Zero : Term d n
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Succ : (p : Term d n) -> 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) -> Term d n
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Box : (val : Term d n) -> 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
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CloT : (tm : Term d from) -> (th : Lazy (TSubst d from to)) ->
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Term d to
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||| dimension closure/suspended substitution
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DCloT : (tm : Term dfrom n) -> (th : Lazy (DSubst dfrom dto)) ->
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Term dto n
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||| first argument `d` is dimension scope size, second `n` is term scope size
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public export
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data Elim : TermLike where
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||| free variable
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F : (x : Name) -> Elim d n
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||| bound variable
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B : (i : Var n) -> Elim d n
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||| term application
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(:@) : (fun : Elim d n) -> (arg : Term d n) -> Elim d n
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||| pair destruction
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|||
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||| `CasePair 𝜋 𝑒 ([𝑟], 𝐴) ([𝑥, 𝑦], 𝑡)` is
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||| `𝐜𝐚𝐬𝐞 𝜋 · 𝑒 𝐫𝐞𝐭𝐮𝐫𝐧 𝑟 ⇒ 𝐴 𝐨𝐟 { (𝑥, 𝑦) ⇒ 𝑡 }`
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CasePair : (qty : Qty) -> (pair : Elim d n) ->
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(ret : ScopeTerm d n) ->
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(body : ScopeTermN 2 d n) ->
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Elim d n
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||| enum matching
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CaseEnum : (qty : Qty) -> (tag : Elim d n) ->
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(ret : ScopeTerm d n) ->
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(arms : CaseEnumArms d n) ->
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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) ->
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(ret : ScopeTerm d n) ->
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(zero : Term d n) ->
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(succ : ScopeTermN 2 d n) ->
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Elim d n
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||| unboxing
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CaseBox : (qty : Qty) -> (box : Elim d n) ->
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(ret : ScopeTerm d n) ->
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(body : ScopeTerm d n) ->
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Elim d n
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||| dim application
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(:%) : (fun : Elim d n) -> (arg : Dim d) -> Elim d n
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||| type-annotated term
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(:#) : (tm, ty : Term d n) -> Elim d n
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||| coerce a value along a type equality, or show its coherence
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||| [@xtt; §2.1.1]
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Coe : (ty : DScopeTerm d n) -> (p, q : Dim d) ->
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(val : Term d n) -> Elim d n
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||| "generalised composition" [@xtt; §2.1.2]
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Comp : (ty : Term d n) -> (p, q : Dim d) ->
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(val : Term d n) -> (r : Dim d) ->
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(zero, one : DScopeTerm d n) -> Elim d n
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||| match on types. needed for b.s. of coercions [@xtt; §2.2]
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TypeCase : (ty : Elim d n) -> (ret : Term d n) ->
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(arms : TypeCaseArms d n) -> (def : Term d n) ->
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Elim d n
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||| term closure/suspended substitution
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CloE : (el : Elim d from) -> (th : Lazy (TSubst d from to)) ->
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Elim d to
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||| dimension closure/suspended substitution
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DCloE : (el : Elim dfrom n) -> (th : Lazy (DSubst dfrom dto)) ->
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Elim dto n
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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
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TypeCaseArms : TermLike
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TypeCaseArms d n = SortedDMap TyConKind (\k => TypeCaseArmBody k d n)
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public export
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TypeCaseArm : TermLike
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TypeCaseArm d n = (k ** TypeCaseArmBody k d n)
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public export
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TypeCaseArmBody : TyConKind -> TermLike
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TypeCaseArmBody k = ScopeTermN (arity k)
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||| a scoped term with names
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public export
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record Scoped (s : Nat) (f : Nat -> Type) (n : Nat) where
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constructor S
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names : NContext s
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body : ScopedBody s f n
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public export
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data ScopedBody : Nat -> (Nat -> Type) -> Nat -> Type where
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Y : (body : f (s + n)) -> ScopedBody s f n
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N : (body : f n) -> ScopedBody s f n
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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
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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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%name Term s, t, r
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%name Elim e, f
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%name Scoped body
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%name ScopedBody body
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||| scope which ignores all its binders
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public export %inline
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SN : {s : Nat} -> f n -> Scoped s f n
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SN = S (replicate s Unused) . N
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||| scope which uses its binders
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public export %inline
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SY : NContext s -> f (s + n) -> Scoped s f n
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SY ns = S ns . Y
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public export %inline
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name : Scoped 1 f n -> BaseName
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name (S [< x] _) = x
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public export %inline
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(.name) : Scoped 1 f n -> BaseName
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s.name = name s
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||| more convenient Pi
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public export %inline
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Pi_ : (qty : Qty) -> (x : BaseName) ->
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(arg : Term d n) -> (res : Term d (S n)) -> Term d n
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Pi_ {qty, x, arg, res} = Pi {qty, arg, res = S [< x] $ Y res}
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||| non dependent function type
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public export %inline
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Arr : (qty : Qty) -> (arg, res : Term d n) -> Term d n
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Arr {qty, arg, res} = Pi {qty, arg, res = SN res}
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||| more convenient Sig
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public export %inline
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Sig_ : (x : BaseName) -> (fst : Term d n) ->
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(snd : Term d (S n)) -> Term d n
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Sig_ {x, fst, snd} = Sig {fst, snd = S [< x] $ Y snd}
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||| non dependent pair type
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public export %inline
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And : (fst, snd : Term d n) -> Term d n
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And {fst, snd} = Sig {fst, snd = SN snd}
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||| more convenient Eq
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public export %inline
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Eq_ : (i : BaseName) -> (ty : Term (S d) n) ->
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(l, r : Term d n) -> Term d n
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Eq_ {i, ty, l, r} = Eq {ty = S [< i] $ Y ty, l, r}
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||| non dependent equality type
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public export %inline
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Eq0 : (ty, l, r : Term d n) -> Term d n
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Eq0 {ty, l, r} = Eq {ty = SN ty, l, r}
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||| same as `F` but as a term
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public export %inline
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FT : Name -> Term d n
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FT = E . F
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||| abbreviation for a bound variable like `BV 4` instead of
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||| `B (VS (VS (VS (VS VZ))))`
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public export %inline
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BV : (i : Nat) -> (0 _ : LT i n) => Elim d n
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BV i = B $ V i
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||| same as `BV` but as a term
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public export %inline
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BVT : (i : Nat) -> (0 _ : LT i n) => Term d n
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BVT i = E $ BV i
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public export
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makeNat : Nat -> Term d n
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makeNat 0 = Zero
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makeNat (S k) = Succ $ makeNat k
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public export %inline
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enum : List TagVal -> Term d n
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enum = Enum . SortedSet.fromList
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public export %inline
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typeCase : Elim d n -> Term d n ->
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List (TypeCaseArm d n) -> Term d n -> Elim d n
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typeCase ty ret arms def = TypeCase ty ret (fromList arms) def
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public export %inline
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typeCase1 : Elim d n -> Term d n ->
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(k : TyConKind) -> NContext (arity k) -> Term d (arity k + n) ->
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{default Nat def : Term d n} ->
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Elim d n
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typeCase1 ty ret k ns body {def} =
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typeCase ty ret [(k ** SY ns body)] def
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