it works ! a lot has changed
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2
pi.ipkg
2
pi.ipkg
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@ -2,11 +2,11 @@ package pi
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modules = Term
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, Value
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, Ctx
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, Normalize
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, Convert
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, Check
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, Misc
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, Tests
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options = "-p contrib --warnpartial"
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@ -0,0 +1,59 @@
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module Check
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import Control.Monad.RWS
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import Control.Monad.Either
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import Control.Monad.Identity
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import Data.Vect
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import Data.Fin
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import Term
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import Value
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import Normalize
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import Misc
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import Convert
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%default total
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mutual
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public export
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-- terms types expected term
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check : Ctx n -> Ctx n -> Value -> Term n -> PI Bool
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check trs tys xpt' tr = do
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xpt <- whnf xpt'
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case tr of
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TLam sc => case xpt of
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VClos env (TPi a b) => do
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v <- VGen <$> fresh
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check (v :: trs) (VClos env a :: tys) (VClos (v :: env) b) sc
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_ => oops "expected pi"
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TPi a b => case xpt of
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VType => do
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v <- VGen <$> fresh
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(&&) <$> check trs tys VType a
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<*> delay <$> check (v :: trs) (VClos trs a :: tys) VType b
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_ => oops "expected type"
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_ => convert xpt =<< infer trs tys tr
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-- terms types term
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infer : Ctx n -> Ctx n -> Term n -> PI Value
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infer trs tys (TVar i _) = pure (index (natToFinLT i) tys)
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infer trs tys TType = pure VType
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infer trs tys (TApp f x) = infer trs tys f >>= whnf >>=
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\case
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VClos env (TPi a b) => do
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if !(check trs tys (VClos env a) x)
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then pure (VClos (VClos trs x :: env) b)
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else oops "application"
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_ => oops "expected infer pi"
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infer trs tys _ = oops "cannot infer type"
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public export
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typecheck : Term 0 -> Term 0 -> Either String (Bool, List String)
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typecheck tr ty = resolve $ (&&) <$> check [] [] VType ty
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<*> delay <$> check [] [] (VClos [] ty) tr
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@ -0,0 +1,42 @@
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module Convert
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import Term
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import Value
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import Misc
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import Normalize
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import Control.Monad.RWS
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import Control.Monad.Identity
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import Control.Monad.Either
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import Data.Nat
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import Data.Vect
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%default total
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public export
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convert : Value -> Value -> PI Bool
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convert u1 u2 = do
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u1' <- whnf u1
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u2' <- whnf u2
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logS ("checking equality of terms '" ++ show u1 ++ "' and '" ++ show u2 ++ "'.")
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logS ("with value representations '" ++ show u1' ++ "' and '" ++ show u2' ++ "'.")
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assert_total $ -- :(
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case (u1', u2') of
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(VType, VType) => pure True
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(VApp f1 x1, VApp f2 x2) => (&&) <$> convert f1 f2
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<*> delay <$> convert x1 x2
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(VGen k1, VGen k2) => pure (k1 == k2)
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(VClos env1 (TLam sc1), VClos env2 (TLam sc2)) => do
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v <- VGen <$> fresh
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convert (VClos (v :: env1) sc1)
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(VClos (v :: env2) sc2)
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(VClos env1 (TPi a1 b1), VClos env2 (TPi a2 b2)) => do
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v <- VGen <$> fresh
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(&&) <$> convert (VClos env1 a1)
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(VClos env2 a2)
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<*> delay <$> convert (VClos (v :: env1) b1)
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(VClos (v :: env2) b2)
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_ => pure False
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40
src/Ctx.idr
40
src/Ctx.idr
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@ -1,40 +0,0 @@
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module Ctx
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import Misc
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import Data.Nat
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%default total
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infixr 7 ::
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public export
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data Ctx : (Index -> Type) -> Index -> Type where
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Nil : Ctx ty 0
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(::) : {n : _} -> ty n -> Ctx ty n -> Ctx ty (S n)
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-- indexed by amount of free variables
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public export
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data Closure : (Index -> Type) -> Index -> Type where
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MkClosure : {n : _} -> Ctx ty n -> ty (m + n) -> Closure ty m
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public export
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enclose : {n : _} -> ty n -> Closure ty n
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enclose {n = n} ty = MkClosure Nil (rewrite plusZeroRightNeutral n in ty)
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public export
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lookup : {m : _} -> (n : Nat) -> Ctx ty m -> LT n m -> ty (minus m (S n))
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lookup {m = S m} Z (x :: _) _ = rewrite minusZeroRight m in x
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lookup (S n) (_ :: ctx) (LTESucc p) = lookup n ctx p
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public export
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strengthen : Ctx ty (S n) -> Ctx ty n
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strengthen (_ :: ctx) = ctx
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public export
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strengthenTo : {n,m : _} -> Ctx ty m -> LTE n m -> Ctx ty n
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strengthenTo {n = Z} _ _ = Nil
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strengthenTo (x :: ctx) (LTESucc p)
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= case lteSplit p of
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Left Refl => x :: ctx
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Right p2 => strengthenTo ctx p2
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25
src/Misc.idr
25
src/Misc.idr
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@ -1,5 +1,9 @@
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module Misc
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import Control.Monad.RWS
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import Control.Monad.Identity
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import Control.Monad.Either
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import Data.Nat
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%default total
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@ -14,11 +18,28 @@ Name = String
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public export
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PI : Type -> Type
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PI = Either String
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PI = EitherT String (RWS () (List String) Nat)
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public export
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resolve : PI a -> Either String (a, List String)
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resolve a = case runRWS (runEitherT a) () 0 of
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(Left e, _) => Left e
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(Right r, _, s) => Right (r, s)
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public export
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oops : String -> PI a
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oops = Left
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oops = left
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public export
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fresh : PI Nat
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fresh = do
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i <- get
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put (S i)
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pure i
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public export
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logS : String -> PI ()
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logS = tell . (:: [])
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public export
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lteTransp : LTE a b -> a = c -> b = d -> LTE c d
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@ -2,54 +2,38 @@ module Normalize
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import Term
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import Value
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import Ctx
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import Misc
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import Control.Monad.RWS
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import Control.Monad.Identity
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import Control.Monad.Either
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import Data.Nat
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import Data.Vect
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%default total
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-- There is some nasty assert_total here since I cannot solve the halting problem
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public export
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asTerm : NF -> Term 0
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asTerm NType = TType
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asTerm (NLam ty sc) = TLam (asTerm ty) sc
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asTerm (NPi ty sc) = TPi (asTerm ty) sc
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public export
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fromNF : NF -> Glue
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fromNF nf = MkGlue (asTerm nf) (const (pure nf))
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substStep : {n,m : _} -> Term m -> LTE m n -> Term (S n) -> Term n
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substStep {m = m} tr p tr' = case tr' of -- case somehow needed for idris not to loop over metavariable
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TType => TType
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(TLam ty sc) => TLam (substStep tr p ty) (substStep tr (lteSuccRight p) sc)
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(TPi ty sc) => TPi (substStep tr p ty) (substStep tr (lteSuccRight p) sc)
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(TApp f x) => TApp (substStep tr p f) (substStep tr p x)
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(TVar i p) => case natEqDecid m i of
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Left p => ?que1
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Right p => ?que2
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mutual
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evalClosure : {n : Nat} -> Closure Term 0 -> PI (Term n)
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evalClosure (MkClosure ctx term) = eval [] ctx term >>= pure . weaken0 . asTerm
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public export
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app : Value -> Value -> PI Value
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app (VClos env (TLam sc)) x = eval (x :: env) sc
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app f x = pure (VApp f x)
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public export
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eval : {n : Nat} -> List (Closure Term 0) -> Ctx Term n -> Term n -> PI NF
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eval stack env TType = pure NType
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-- dirty
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eval (thunk :: stack) env (TLam _ sc) = eval stack (!(assert_total (evalClosure thunk)) :: env) sc
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eval stack env (TApp f x) = eval (MkClosure env x :: stack) env x
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eval stack env (TVar m p) = assert_total $ eval stack (strengthenTo env minusLte) (lookup m env p)
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eval stack env (TPi ty sc) = ?idk
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eval [] env (TLam ty sc) = ?what
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eval : Ctx n -> Term n -> PI Value
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eval env (TVar i _) = pure (index (natToFinLT i) env)
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eval env (TApp f x) = do
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f' <- eval env f
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x' <- eval env x
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assert_total (app f' x') -- :(
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eval env TType = pure VType
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eval env tr = pure (VClos env tr)
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public export
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fromTerm : Term 0 -> Glue
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fromTerm term = MkGlue term (eval [] Nil)
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whnf : Value -> PI Value
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whnf (VApp f x) = do
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f' <- whnf f
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x' <- whnf x
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app f' x'
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whnf (VClos env tr) = eval env tr
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whnf v = pure v
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34
src/Term.idr
34
src/Term.idr
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@ -12,17 +12,25 @@ import Misc
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-}
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public export
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data Term : (_ : Index) -> Type where
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TType : Term n -- Type of types
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TLam : Term n -> Term (S n) -> Term n -- Lambda abstraction (λ Type -> Scope)
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TPi : Term n -> Term (S n) -> Term n -- Pi type (∏ A -> B a )
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TApp : Term n -> Term n -> Term n -- Appliction
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TVar : (n : Nat) -> LT n m -> Term m -- Variable
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TType : Term n -- Type of types
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TLam : Term (S n) -> Term n -- Lambda abstraction (λ _ . Scope)
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TPi : Term n -> Term (S n) -> Term n -- Pi type (∏ _ : A . B _ )
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TApp : Term n -> Term n -> Term n -- Appliction
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TVar : (n : Nat) -> LT n m -> Term m -- Variable
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public export
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Show (Term n) where
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show TType = "TType"
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show (TLam sc) = "TLam (" ++ show sc ++ ")"
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show (TPi ty sc) = "TPi (" ++ show ty ++ ") (" ++ show sc ++ ")"
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show (TApp f x) = "TApp (" ++ show f ++ ") (" ++ show x ++ ")"
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show (TVar i _) = "TVar " ++ show i
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public export
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weaken : {p, q : _} -> LTE p q -> Term p -> Term q
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weaken _ TType = TType
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weaken p (TLam ty sc) = TLam (weaken p ty) (weaken (LTESucc p) sc)
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weaken p (TPi a bx) = TLam (weaken p a) (weaken (LTESucc p) bx)
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weaken p (TLam sc) = TLam (weaken (LTESucc p) sc)
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weaken p (TPi a bx) = TPi (weaken p a) (weaken (LTESucc p) bx)
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weaken p (TApp f x) = TApp (weaken p f) (weaken p x)
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{-
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Getting new index
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@ -50,11 +58,11 @@ weaken p (TApp f x) = TApp (weaken p f) (weaken p x)
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(r + q) - p < q ∎
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-}
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weaken {p = S p} {q = S q} (LTESucc p1) (TVar r p2') =
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weaken {p = S p} {q = S q} (LTESucc p1) (TVar r p2' ) =
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case p2' of
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LTESucc p2 => TVar (minus (r + q) p)
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(LTESucc (lteTransp (minusLteMonotone (plusLteMonotoneRight q r p p2)) Refl (minusPlus p)))
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LTESucc p2 => TVar (minus (r + q) p)
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(LTESucc (lteTransp (minusLteMonotone (plusLteMonotoneRight q r p p2)) Refl (minusPlus p)))
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public export
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weaken0 : {n : _} -> Term 0 -> Term n
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weaken0 = weaken LTEZero
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weaken1 : {n : _} -> Term n -> Term (S n)
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weaken1 = weaken lteS
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23
src/Tests.idr
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23
src/Tests.idr
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@ -0,0 +1,23 @@
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module Tests
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import Term
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import Check
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import Check
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import Data.Nat
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%default total
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a : {p, q : Nat} -> lt p q = True -> LT p q
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a {p} {q} eq = ltReflectsLT p q eq
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test1 : Either String (Bool, List String)
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test1 = typecheck (TLam (TLam (TVar 0 (a Refl))))
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(TPi TType (TPi (TVar 0 (a Refl)) (TVar 1 (a Refl))))
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test2 : Either String (Bool, List String)
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test2 = typecheck (TLam (TLam (TLam (TLam (TApp (TVar 1 (a Refl)) (TVar 0 (a Refl)))))))
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(TPi TType
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(TPi (TPi (TVar 0 (a Refl)) TType)
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(TPi (TPi (TVar 1 (a Refl)) (TApp (TVar 1 (a Refl)) (TVar 0 (a Refl))))
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(TPi (TVar 2 (a Refl)) (TApp (TVar 2 (a Refl)) (TVar 0 (a Refl)))))))
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@ -2,24 +2,26 @@ module Value
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import Term
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import Misc
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import Ctx
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import Data.Vect
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%default total
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public export
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data NF : Type where
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NType : NF -- Type of types
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NLam : NF -> Term 1 -> NF -- Lambda abstraction
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NPi : NF -> Term 1 -> NF -- Pi Type
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mutual
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public export
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data Value : Type where
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VType : Value
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VGen : Nat -> Value
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VApp : Value -> Value -> Value
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VClos : Ctx n -> Term n -> Value
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public export
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Ctx : Index -> Type
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Ctx i = Vect i Value
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public export
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data Glue : Type where
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MkGlue : Term 0 -> (Term 0 -> PI NF) -> Glue
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public export
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getTerm : Glue -> Term 0
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getTerm (MkGlue term _) = term
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public export
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getNF : Glue -> PI NF
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getNF (MkGlue term f) = f term
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Show Value where
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show VType = "VType"
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show (VGen i) = "VGen " ++ show i
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show (VApp f x) = "VApp (" ++ show f ++ ") (" ++ show x ++ ")"
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show (VClos e t) = "VClos (" ++ assert_total (show e) ++ ") (" ++ show t ++ ")"
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