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{-# LANGUAGE BlockArguments #-}
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{-# LANGUAGE LambdaCase #-}
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{-# LANGUAGE OverloadedStrings #-}
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{-# LANGUAGE DerivingStrategies #-}
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{-# LANGUAGE DeriveAnyClass #-}
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{-# LANGUAGE ViewPatterns #-}
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module Elab.WiredIn where
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import Control.Exception
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import qualified Data.Map.Strict as Map
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import qualified Data.Sequence as Seq
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import qualified Data.Text as T
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import Data.Map.Strict (Map)
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import Data.Text (Text)
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import Data.Typeable
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import Elab.Eval
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import qualified Presyntax.Presyntax as P
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import Syntax
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import System.IO.Unsafe
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import Syntax.Pretty (prettyTm)
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import GHC.Stack (HasCallStack)
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import Debug.Trace
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wiType :: WiredIn -> NFType
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wiType WiType = VType
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wiType WiPretype = VTypeω
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wiType WiInterval = VTypeω
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wiType WiI0 = VI
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wiType WiI1 = VI
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wiType WiIAnd = VI ~> VI ~> VI
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wiType WiIOr = VI ~> VI ~> VI
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wiType WiINot = VI ~> VI
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wiType WiPathP = dprod (VI ~> VType) \a -> a @@ VI0 ~> a @@ VI1 ~> VType
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wiType WiIsOne = VI ~> VTypeω
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wiType WiItIsOne = VIsOne VI1
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wiType WiIsOne1 = forAll VI \i -> forAll VI \j -> VIsOne i ~> VIsOne (ior i j)
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wiType WiIsOne2 = forAll VI \i -> forAll VI \j -> VIsOne j ~> VIsOne (ior i j)
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wiType WiPartial = VI ~> VType ~> VTypeω
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wiType WiPartialP = dprod VI \x -> VPartial x VType ~> VTypeω
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wiType WiSub = dprod VType \a -> dprod VI \phi -> VPartial phi a ~> VTypeω
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wiType WiInS = forAll VType \a -> forAll VI \phi -> dprod a \u -> VSub a phi (fun (const u))
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wiType WiOutS = forAll VType \a -> forAll VI \phi -> forAll (VPartial phi a) \u -> VSub a phi u ~> a
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wiType WiComp = dprod' "A" (VI ~> VType) \a -> forAll VI \phi -> dprod (dprod VI \i -> VPartial phi (a @@ i)) \u -> VSub (a @@ VI0) phi (u @@ VI0) ~> a @@ VI1
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-- (A : Type) {phi : I} (T : Partial phi Type) (e : PartialP phi (\o -> Equiv (T o) A)) -> Type
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wiType WiGlue = dprod' "A" VType \a -> forAll' "phi" VI \phi -> dprod' "T" (VPartial phi VType) \t -> VPartialP phi (fun \o -> equiv (t @@ o) a) ~> VType
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-- {A : Type} {phi : I} {T : Partial phi Type} {e : PartialP phi (\o -> Equiv (T o) A)} -> (t : PartialP phi T) -> Sub A phi (\o -> e o (t o)) -> Glue A phi T e
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wiType WiGlueElem = forAll' "A" VType \a -> forAll' "phi" VI \phi -> forAll' "T" (VPartial phi VType) \ty -> forAll' "e" (VPartialP phi (fun \o -> equiv (ty @@ o) a)) \eqv ->
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dprod' "t" (VPartialP phi ty) \t -> VSub a phi (fun \o -> vProj1 (eqv @@ o) @@ (t @@ o)) ~> VGlueTy a phi ty eqv
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-- {A : Type} {phi : I} {T : Partial phi Type} {e : PartialP phi (\o -> Equiv (T o) A)} -> Glue A phi T e -> A
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wiType WiUnglue = forAll' "A" VType \a -> forAll' "phi" VI \phi -> forAll' "T" (VPartial phi VType) \ty -> forAll' "e" (VPartialP phi (fun \o -> equiv (ty @@ o) a)) \e -> VGlueTy a phi ty e ~> a
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wiType WiBool = VType
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wiType WiTrue = VBool
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wiType WiFalse = VBool
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wiType WiIf = dprod' "A" (VBool ~> VType) \a -> a @@ VTt ~> a @@ VFf ~> dprod' "b" VBool \b -> a @@ b
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wiValue :: WiredIn -> Value
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wiValue WiType = VType
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wiValue WiPretype = VTypeω
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wiValue WiInterval = VI
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wiValue WiI0 = VI0
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wiValue WiI1 = VI1
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wiValue WiIAnd = fun \x -> fun \y -> iand x y
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wiValue WiIOr = fun \x -> fun \y -> ior x y
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wiValue WiINot = fun inot
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wiValue WiPathP = fun \a -> fun \x -> fun \y -> VPath a x y
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wiValue WiIsOne = fun VIsOne
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wiValue WiItIsOne = VItIsOne
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wiValue WiIsOne1 = forallI \_ -> forallI \_ -> fun VIsOne1
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wiValue WiIsOne2 = forallI \_ -> forallI \_ -> fun VIsOne2
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wiValue WiPartial = fun \phi -> fun \r -> VPartial phi r
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wiValue WiPartialP = fun \phi -> fun \r -> VPartialP phi r
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wiValue WiSub = fun \a -> fun \phi -> fun \u -> VSub a phi u
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wiValue WiInS = forallI \a -> forallI \phi -> fun \u -> VInc a phi u
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wiValue WiOutS = forallI \a -> forallI \phi -> forallI \u -> fun \x -> outS a phi u x
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wiValue WiComp = fun \a -> forallI \phi -> fun \u -> fun \x -> comp a phi u x
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wiValue WiGlue = fun \a -> forallI \phi -> fun \t -> fun \e -> glueType a phi t e
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wiValue WiGlueElem = forallI \a -> forallI \phi -> forallI \ty -> forallI \eqv -> fun \x -> fun \y -> glueElem a phi ty eqv x y
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wiValue WiUnglue = forallI \a -> forallI \phi -> forallI \ty -> forallI \eqv -> fun \x -> unglue a phi ty eqv x
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wiValue WiBool = VBool
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wiValue WiTrue = VTt
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wiValue WiFalse = VFf
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wiValue WiIf = fun \a -> fun \b -> fun \c -> fun \d -> elimBool a b c d
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(~>) :: Value -> Value -> Value
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a ~> b = VPi P.Ex a (Closure (Bound "_" 0) (const b))
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infixr 7 ~>
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fun, line :: (Value -> Value) -> Value
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fun k = VLam P.Ex $ Closure (Bound "x" 0) (k . force)
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line k = VLam P.Ex $ Closure (Bound "i" 0) (k . force)
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forallI :: (Value -> Value) -> Value
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forallI k = VLam P.Im $ Closure (Bound "x" 0) (k . force)
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dprod' :: String -> Value -> (Value -> Value) -> Value
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dprod' t a b = VPi P.Ex a (Closure (Bound (T.pack t) 0) b)
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dprod :: Value -> (Value -> Value) -> Value
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dprod = dprod' "x"
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exists' :: String -> Value -> (Value -> Value) -> Value
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exists' s a b = VSigma a (Closure (Bound (T.pack s) 0) b)
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exists :: Value -> (Value -> Value) -> Value
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exists = exists' "x"
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forAll' :: String -> Value -> (Value -> Value) -> Value
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forAll' n a b = VPi P.Im a (Closure (Bound (T.pack n) 0) b)
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forAll :: Value -> (Value -> Value) -> Value
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forAll = forAll' "x"
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wiredInNames :: Map Text WiredIn
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wiredInNames = Map.fromList
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[ ("Pretype", WiPretype)
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, ("Type", WiType)
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, ("Interval", WiInterval)
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, ("i0", WiI0)
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, ("i1", WiI1)
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, ("iand", WiIAnd)
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, ("ior", WiIOr)
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, ("inot", WiINot)
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, ("PathP", WiPathP)
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, ("IsOne", WiIsOne)
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, ("itIs1", WiItIsOne)
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, ("isOneL", WiIsOne1)
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, ("isOneR", WiIsOne2)
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, ("Partial", WiPartial)
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, ("PartialP", WiPartialP)
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, ("Sub", WiSub)
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, ("inS", WiInS)
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, ("outS", WiOutS)
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, ("comp", WiComp)
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, ("Glue", WiGlue)
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, ("glue", WiGlueElem)
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, ("unglue", WiUnglue)
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, ("Bool", WiBool)
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, ("true", WiTrue)
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, ("false", WiFalse)
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, ("if", WiIf)
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]
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newtype NoSuchPrimitive = NoSuchPrimitive { getUnknownPrim :: Text }
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deriving (Show, Typeable)
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deriving anyclass (Exception)
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-- Interval operations
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iand, ior :: Value -> Value -> Value
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iand x = case force x of
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VI1 -> id
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VI0 -> const VI0
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VIAnd x y -> \z -> case force z of
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VI0 -> VI0
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VI1 -> VI1
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z -> iand x (iand y z)
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x -> \y -> case force y of
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VI0 -> VI0
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VI1 -> x
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y -> VIAnd x y
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ior x = case force x of
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VI0 -> id
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VI1 -> const VI1
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VIOr x y -> \z -> case force z of
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VI1 -> VI1
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VI0 -> VIOr x y
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_ -> ior x (ior y z)
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x -> \y -> case force y of
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VI1 -> VI1
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VI0 -> x
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y -> VIOr x y
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inot :: Value -> Value
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inot x = case force x of
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VI0 -> VI1
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VI1 -> VI0
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VIOr x y -> VIAnd (inot x) (inot y)
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VIAnd x y -> VIOr (inot x) (inot y)
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VINot x -> x
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x -> VINot x
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ielim :: Value -> Value -> Value -> Value -> NFEndp -> Value
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ielim line left right (GluedVl h sp vl) i =
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GluedVl h (sp Seq.:|> PIElim line left right i) (ielim line left right vl i)
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ielim line left right fn i =
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case force fn of
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VLine _ _ _ fun -> fun @@ i
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x -> case force i of
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VI1 -> right
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VI0 -> left
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_ -> case x of
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VNe n sp -> VNe n (sp Seq.:|> PIElim line left right i)
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VSystem map -> VSystem (fmap (flip (ielim line left right) i) map)
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VInc (VPath _ _ _) _ u -> ielim line left right u i
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_ -> error $ "can't ielim " ++ show (prettyTm (quote fn))
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outS :: NFSort -> NFEndp -> Value -> Value -> Value
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outS _ (force -> VI1) u _ = u @@ VItIsOne
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outS _ _phi _ (VInc _ _ x) = x
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outS _ VI0 _ x = x
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outS a phi u (GluedVl x sp vl) = GluedVl x (sp Seq.:|> POuc a phi u) (outS a phi u vl)
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outS a phi u (VNe x sp) = VNe x (sp Seq.:|> POuc a phi u)
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outS _ _ _ v = error $ "can't outS " ++ show (prettyTm (quote v))
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-- Composition
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comp :: HasCallStack => NFLine -> NFEndp -> Value -> Value -> Value
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comp _ VI1 u _ = u @@ VI1 @@ VItIsOne
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comp a psi@phi u (compOutS (a @@ VI1) phi (u @@ VI1 @@ VItIsOne) -> a0) =
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case force $ a @@ VVar (Bound (T.pack "i") 0) of
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VPi{} ->
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let
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plic i = let VPi p _ _ = force (a @@ i) in p
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dom i = let VPi _ d _ = force (a @@ i) in d
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rng i = let VPi _ _ (Closure _ r) = force (a @@ i) in r
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y' i y = fill (fun (dom . inot)) VI0 (fun \_ -> fun \_ -> VSystem mempty) (VInc (dom VI0) phi y) i
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ybar i y = y' (inot i) y
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in VLam (plic VI1) . Closure (Bound "x" 0) $ \arg ->
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comp (line \i -> rng i (ybar i arg))
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phi
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(system \i isone -> vApp (plic i) (u @@ i @@ isone) (ybar i arg))
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(VInc (rng VI0 (ybar VI0 arg)) phi (vApp (plic VI0) a0 (ybar VI0 arg)))
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VSigma{} ->
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let
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dom i = let VSigma d _ = force (a @@ i) in d
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rng i = let VSigma _ (Closure _ r) = force (a @@ i) in r
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w i = fill (fun dom) phi (system \i isone -> vProj1 (u @@ i @@ isone)) (VInc (dom VI0) phi (vProj1 a0)) i
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c1 = comp (fun dom) phi (system \i isone -> vProj1 (u @@ i @@ isone)) (VInc (dom VI0) phi (vProj1 a0))
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c2 = comp (fun \x -> rng x (w x)) phi (system \i isone -> vProj2 (u @@ i @@ isone)) (VInc (rng VI0 (w VI0)) phi (vProj2 a0))
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in
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VPair c1 c2
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VPath{} ->
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let
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a' i = let VPath thea _ _ = force (a @@ i) in thea
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u' i = let VPath _ theu _ = force (a @@ i) in theu
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v' i = let VPath _ _ thev = force (a @@ i) in thev
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in
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VLine (a' VI1 @@ VI1) (u' VI1) (v' VI1) $ fun \j ->
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comp (fun \x -> a' x @@ x)
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(phi `ior` j `ior` inot j)
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(system \i isone -> mkVSystem (Map.fromList [ (phi, ielim (a' VI0) (u' VI0) (v' VI0) (u @@ i @@ isone) j)
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, (j, v' i)
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, (inot j, u' i)]))
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(VInc (a' VI0 @@ VI0 @@ j) phi (ielim (a' VI0 @@ VI0) (u' VI0) (v' VI0) a0 j))
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VGlueTy{} ->
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let
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b = u
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b0 = a0
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fam = a
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in
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let
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base i = let VGlueTy base _ _ _ = force (fam @@ i) in base
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phi i = let VGlueTy _ phi _ _ = force (fam @@ i) in phi
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types i = let VGlueTy _ _ types _ = force (fam @@ i) in types
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equivs i = let VGlueTy _ _ _ equivs = force (fam @@ i) in equivs
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a i = fun \u -> unglue (base i) (phi i) (types i @@ u) (equivs i @@ u) (b @@ i @@ u)
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a0 = unglue (base VI0) (phi VI0) (types VI0) (equivs VI0) b0
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del = faceForall phi
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a1' = comp (line base) psi (line a) (VInc undefined undefined a0)
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t1' = comp (line types) psi (line (b @@)) (VInc undefined undefined b0)
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(omega_st, omega_t, omega_rep) = pres types base equivs psi (b @@) b0
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omega = outS omega_t psi omega_rep omega_st
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(t1alpha_st, t1a_t, t1a_rep) = opEquiv (base VI1) (types VI1 @@ VItIsOne) (equivs VI1 @@ VItIsOne) (del `ior` psi) (fun ts) (fun ps) a1'
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t1alpha = outS t1a_t (del `ior` psi) t1a_rep t1alpha_st
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(t1, alpha) = (vProj1 t1alpha, vProj2 t1alpha)
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ts isone = mkVSystem . Map.fromList $ [(del, t1'), (psi, (b @@ VI1 @@ isone))]
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ps _isone = mkVSystem . Map.fromList $ [(del, omega), (psi, VLine (line (const (base VI1))) a1' a1' (fun (const a1')))]
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a1 = comp
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(fun (const (base VI1 @@ VItIsOne)))
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(phi VI1 `ior` psi)
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(system \j _u -> mkVSystem (Map.fromList [ (phi VI1, ielim (base VI1) a1' (vProj1 (equivs VI1 @@ VItIsOne)) alpha j)
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, (psi, a VI1)]))
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a1'
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b1 = glueElem (base VI1) (phi VI1) (types VI1) (equivs VI1) (fun (const t1)) a1
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in b1
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VType -> VGlueTy a0 phi (system \_ _ -> mkVSystem (Map.fromList [(phi, u @@ VI1 @@ VItIsOne)]))
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(system \_ _ -> mkVSystem (Map.fromList [(phi, makeEquiv (\j -> (u @@ inot j)))]))
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-- fibrancy structure of the booleans is trivial
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VBool{} -> a0
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_ -> VComp a phi u (VInc (a @@ VI0) phi a0)
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compOutS :: NFSort -> NFEndp -> Value -> Value -> Value
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compOutS a b c d = compOutS a b c (force d) where
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compOutS _ _hi _0 vl@VComp{} = vl
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compOutS _ _hi _0 (VInc _ _ x) = x
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compOutS _ _ _ v = v
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system :: (Value -> Value -> Value) -> Value
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system k = fun \i -> fun \isone -> k i isone
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fill :: NFLine -> NFEndp -> Value -> Value -> NFEndp -> Value
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fill a phi u a0 j =
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comp (line \i -> a @@ (i `iand` j))
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(phi `ior` inot j)
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(fun \i -> fun \isone -> mkVSystem (Map.fromList [ (phi, u @@ (i `iand` j) @@ isone)
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, (inot j, a0)]))
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a0
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glueType :: NFSort -> NFEndp -> NFPartial -> NFPartial -> Value
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glueType a phi tys eqvs = VGlueTy a phi tys eqvs
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glueElem :: NFSort -> NFEndp -> NFPartial -> NFPartial -> NFPartial -> Value -> Value
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glueElem _a (force -> VI1) _tys _eqvs t _vl = t @@ VItIsOne
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glueElem a phi tys eqvs t vl = VGlue a phi tys eqvs t vl
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unglue :: NFSort -> NFEndp -> NFPartial -> NFPartial -> Value -> Value
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unglue _a (force -> VI1) _tys eqvs x = vProj1 (eqvs @@ VItIsOne) @@ x
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unglue _a _phi _tys _eqvs (force -> VGlue _ _ _ _ _ vl) = vl
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unglue a phi tys eqvs (force -> VSystem fs) = VSystem (fmap (unglue a phi tys eqvs) fs)
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unglue a phi tys eqvs vl = VUnglue a phi tys eqvs vl
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-- Definition of equivalences
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faceForall :: (NFEndp -> NFEndp) -> Value
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faceForall phi = T.length (getNameText name) `seq` go (phi (VVar name)) where
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{-# NOINLINE name #-}
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name = unsafePerformIO newName
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go x@(VVar n)
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| n == name = VI0
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| otherwise = x
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go x@(VINot (VVar n))
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| n == name = VI0
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| otherwise = x
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go (VIAnd x y) = iand (go x) (go y)
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go (VIOr x y) = ior (go x) (go y)
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go (VINot x) = inot (go x)
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go vl = vl
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isContr :: Value -> Value
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isContr a = exists' "x" a \x -> dprod' "y" a \y -> VPath (line (const a)) x y
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fiber :: NFSort -> NFSort -> Value -> Value -> Value
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fiber a b f y = exists' "x" a \x -> VPath (line (const b)) (f @@ x) y
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isEquiv :: NFSort -> NFSort -> Value -> Value
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isEquiv a b f = dprod' "y" b \y -> isContr (fiber a b f y)
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equiv :: NFSort -> NFSort -> Value
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equiv a b = exists' "f" (a ~> b) \f -> isEquiv a b f
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pres :: (NFEndp -> NFSort) -> (NFEndp -> NFSort) -> (NFEndp -> Value) -> NFEndp -> (NFEndp -> Value) -> Value -> (Value, NFSort, Value)
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pres tyT tyA f phi t t0 = (VInc pathT phi (VLine (tyA VI1) c1 c2 (line path)), pathT, fun $ \u -> VLine (fun (const (tyA VI1))) c1 c2 (fun (const (f VI1 @@ (t VI1 @@ u))))) where
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pathT = VPath (fun (const (tyA VI1))) c1 c2
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c1 = comp (line tyA) phi (system \i u -> f i @@ (t i @@ u)) (VInc (tyA VI0) phi (f VI0 @@ t0))
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c2 = f VI1 @@ comp (line tyT) phi (system \i u -> t i @@ u) t0
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a0 = f VI0 @@ t0
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v = fill (fun tyT) phi (system \i u -> t i @@ u) t0
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path j = comp (fun tyA) (phi `ior` j) (system \i _ -> f i @@ (v i)) a0
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opEquiv :: HasCallStack => Value -> Value -> Value -> NFEndp -> Value -> Value -> Value -> (Value, NFSort, Value)
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opEquiv aT tT f phi t p a = (VInc ty phi v, ty, fun \u -> VPair (t @@ u) (p @@ u)) where
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fn = vProj1 f
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ty = exists' "f" tT \x -> VPath (line (const aT)) a (fn @@ x)
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v = contr ty (vProj2 f @@ a) phi (\u -> VPair (t @@ u) (p @@ u))
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contr :: HasCallStack => Value -> Value -> NFEndp -> (Value -> Value) -> Value
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contr a aC phi u =
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comp (line (const a))
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phi
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(system \i is1 -> ielim (line (const a)) (vProj1 aC) (u is1) (vProj2 aC @@ u is1) i)
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(vProj1 aC)
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makeEquiv :: (NFEndp -> Value) -> Value
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makeEquiv line = comp (fun \i -> equiv a (line i)) VI0 (system \_ _ -> VSystem mempty) (VPair idfun idisequiv) where
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a = line VI0
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idfun = fun id
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-- idEquiv y = ((y, \i -> y), \u i -> (u.2 (inot i), \j -> u.2 (ior (inot i) j)))
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u_ty = exists' "y" a \x -> VPath (fun (const a)) x x
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idisequiv = fun \y -> VPair (id_fiber y) $ fun \u ->
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VLine u_ty (id_fiber y) u $ fun \i -> VPair (ielim (fun (const a)) y y (vProj2 u) i) $
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VLine (fun (const a)) y (vProj1 u) $ fun \j ->
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ielim (fun (const a)) y y (vProj2 u) (iand i j)
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id_fiber y = VPair y (VLine a y y (fun (const y)))
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elimBool :: NFSort -> Value -> Value -> Value -> Value
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elimBool prop x y bool =
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case force bool of
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VTt -> x
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VFf -> y
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_ -> VIf prop x y bool
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