From 17ed8145b371ce578db7fdd67aced8dd2013e623 Mon Sep 17 00:00:00 2001 From: Adam Megacz Date: Sat, 19 Mar 2011 16:23:03 -0700 Subject: [PATCH] checkpoint --- examples/tutorial.hs | 332 ++++++++++++++++++++++++++++++++++++++++++++++- src/HaskProofToStrong.v | 104 +++++++++++---- 2 files changed, 404 insertions(+), 32 deletions(-) diff --git a/examples/tutorial.hs b/examples/tutorial.hs index 29939b6..bace8d4 100644 --- a/examples/tutorial.hs +++ b/examples/tutorial.hs @@ -1,10 +1,13 @@ -{-# OPTIONS_GHC -XModalTypes -XScopedTypeVariables -XFlexibleContexts -XMultiParamTypeClasses -ddump-types -XNoMonoPatBinds #-} +{-# OPTIONS_GHC -XModalTypes -XScopedTypeVariables -XFlexibleContexts -XMultiParamTypeClasses -ddump-types -XNoMonoPatBinds -XFlexibleInstances -XGADTs #-} module GArrowsTutorial where import Data.Bits import Data.Bool (not) import GHC.HetMet.CodeTypes hiding ((-)) import GHC.HetMet.GArrow +import Control.Category +import Control.Arrow +import Prelude hiding ( id, (.) ) -- The best way to understand heterogeneous metaprogramming and -- generalized arrows is to play around with this file, poking at the @@ -205,7 +208,7 @@ increment_at_level1 = <[ \x -> x + 1 ]> -------------------------------------------------------------------------------- -- Ye Olde and Most Venerable "pow" Function - +{- pow n = if n==0 then <[ \x -> 1 ]> @@ -289,7 +292,7 @@ dotproduct'' v1 = [] -> 0 (b:bx) -> ~~(guestIntegerLiteral a) * b + ~~(dotproduct'' ax) bx ]> - +-} @@ -351,7 +354,7 @@ class Stream a where s_empty :: a -> Bool s_head :: a -> Char s_tail :: a -> a - +{- -- a continuation-passing-style matcher accept :: Stream s => Regex -> (s -> Bool) -> s -> Bool @@ -433,6 +436,140 @@ staged_accept (Const c) k = + +-------------------------------------------------------------------------------- +-- Unflattening + +-- This more or less "undoes" the flatten function. People often ask +-- me how you "translate generalized arrows back into multi-level +-- terms".. I'm not sure why you'd want to do that, but this is how: +newtype Code x y = Code { unCode :: forall a. <[ x -> y ]>@a } + +instance Category Code where + id = Code <[ \x -> x ]> + f . g = Code <[ \x -> ~~(unCode f) (~~(unCode g) x) ]> + +instance GArrow Code (,) where + ga_first f = Code <[ \(x,y) -> ((~~(unCode f) x),y) ]> + ga_second f = Code <[ \(x,y) -> (x ,(~~(unCode f) y)) ]> + ga_cancell = Code <[ \(_,x) -> x ]> + ga_cancelr = Code <[ \(x,_) -> x ]> + ga_uncancell = Code <[ \x -> (%%(),x) ]> + ga_uncancelr = Code <[ \x -> (x,%%()) ]> + ga_assoc = Code <[ \((x,y),z) -> (x,(y,z)) ]> + ga_unassoc = Code <[ \(x,(y,z)) -> ((x,y),z) ]> + + + + +-------------------------------------------------------------------------------- +-- BiGArrows + +class GArrow g (**) => BiGArrow g (**) where + -- Note that we trust the user's pair of functions actually are + -- mutually inverse; confirming this in the type system would + -- require very powerful dependent types (such as Coq's). However, + -- the consequences of failure here are much more mild than failures + -- in BiArrow.inv: if the functions below are not mutually inverse, + -- the LoggingBiGArrow will simply compute the wrong result rather + -- than fail in some manner outside the language's semantics. + biga_arr :: (x -> y) -> (y -> x) -> g x y + biga_inv :: g x y -> g y x + +-- For any GArrow instance, its mutually inverse pairs form a BiGArrow +data GArrow g (**) => GArrowInversePair g (**) x y = + GArrowInversePair { forward :: g x y , backward :: g y x } +instance GArrow g (**) => Category (GArrowInversePair g (**)) where + id = GArrowInversePair { forward = id , backward = id } + f . g = GArrowInversePair { forward = (forward f) . (forward g) , backward = (backward g) . (backward f) } +instance GArrow g (**) => GArrow (GArrowInversePair g (**)) (**) where + ga_first f = GArrowInversePair { forward = ga_first (forward f), backward = ga_first (backward f) } + ga_second f = GArrowInversePair { forward = ga_second (forward f), backward = ga_second (backward f) } + ga_cancell = GArrowInversePair { forward = ga_cancell , backward = ga_uncancell } + ga_cancelr = GArrowInversePair { forward = ga_cancelr , backward = ga_uncancelr } + ga_uncancell = GArrowInversePair { forward = ga_uncancell , backward = ga_cancell } + ga_uncancelr = GArrowInversePair { forward = ga_uncancelr , backward = ga_cancelr } + ga_assoc = GArrowInversePair { forward = ga_assoc , backward = ga_unassoc } + ga_unassoc = GArrowInversePair { forward = ga_unassoc , backward = ga_assoc } +instance GArrowSwap g (**) => GArrowSwap (GArrowInversePair g (**)) (**) where + ga_swap = GArrowInversePair { forward = ga_swap , backward = ga_swap } +instance (GArrowDrop g (**), GArrowCopy g (**)) => GArrowCopy (GArrowInversePair g (**)) (**) where + ga_copy = GArrowInversePair { forward = ga_copy , backward = ga_second ga_drop >>> ga_cancelr } +-- but notice that we can't (in general) get +-- instance GArrowDrop g => GArrowDrop (GArrowInversePair g) where ... + + +-- For that, we need PreLenses, which "log the history" where necessary. +-- I call this a "PreLens" because it consists of the data required +-- for a Lens (as in BCPierce's Lenses) but does not necessarily +-- satisfy the putget/getput laws. Specifically, the "extra stuff" we +-- store is the inversion function. +newtype PreLens x y = PreLens { preLens :: x -> (y , y->x) } + +instance Category PreLens where + id = PreLens { preLens = \x -> (x, (\x -> x)) } + f . g = PreLens { preLens = \x -> let (gx,g') = (preLens g) x in let (fgx,f') = (preLens f) gx in (fgx , \q -> g' (f' q)) } + +instance GArrow PreLens (,) where + ga_first f = PreLens { preLens = \(x,z) -> let (y,f') = (preLens f) x in ((y,z),(\(q1,q2) -> (f' q1,q2))) } + ga_second f = PreLens { preLens = \(z,x) -> let (y,f') = (preLens f) x in ((z,y),(\(q1,q2) -> (q1,f' q2))) } + ga_cancell = PreLens { preLens = \(_,x) -> (x, (\x -> ((),x))) } + ga_cancelr = PreLens { preLens = \(x,_) -> (x, (\x -> (x,()))) } + ga_uncancell = PreLens { preLens = \x -> (((),x), (\(_,x) -> x)) } + ga_uncancelr = PreLens { preLens = \x -> ((x,()), (\(x,_) -> x)) } + ga_assoc = PreLens { preLens = \((x,y),z) -> ( (x,(y,z)) , (\(x,(y,z)) -> ((x,y),z)) ) } + ga_unassoc = PreLens { preLens = \(x,(y,z)) -> ( ((x,y),z) , (\((x,y),z) -> (x,(y,z))) ) } + +instance GArrowDrop PreLens (,) where + ga_drop = PreLens { preLens = \x -> (() , (\() -> x)) } +instance GArrowCopy PreLens (,) where + ga_copy = PreLens { preLens = \x -> ((x,x) , fst) } +instance GArrowSwap PreLens (,) where + ga_swap = PreLens { preLens = \(x,y) -> ((y,x) , (\(z,q) -> (q,z))) } + + + +data Lens x y where + Lens :: forall x y c1 c2 . ((x,c1)->(y,c2)) -> ((y,c2)->(x,c1)) -> Lens x y + +-- can we make lenses out of GArrows other than (->)? +instance Category Lens where + id = Lens (\x -> x) (\x -> x) + (Lens g1 g2) . (Lens f1 f2) = Lens (\(x,(c1,c2)) -> let (y,fc) = f1 (x,c1) in let (z,gc) = g1 (y,c2) in (z,(fc,gc))) + (\(z,(c1,c2)) -> let (y,gc) = g2 (z,c2) in let (x,fc) = f2 (y,c1) in (x,(fc,gc))) + +instance GArrow Lens (,) where + ga_first (Lens f1 f2) = Lens (\((x1,x2),c) -> let (y,c') = f1 (x1,c) in ((y,x2),c')) + (\((x1,x2),c) -> let (y,c') = f2 (x1,c) in ((y,x2),c')) + ga_second (Lens f1 f2) = Lens (\((x1,x2),c) -> let (y,c') = f1 (x2,c) in ((x1,y),c')) + (\((x1,x2),c) -> let (y,c') = f2 (x2,c) in ((x1,y),c')) + ga_cancell = Lens (\(((),x),()) -> ( x ,())) + (\( x ,()) -> (((),x),())) + ga_uncancell = Lens (\( x ,()) -> (((),x),())) + (\(((),x),()) -> ( x ,())) + ga_cancelr = Lens (\((x,()),()) -> ( x ,())) + (\( x ,()) -> ((x,()),())) + ga_uncancelr = Lens (\( x ,()) -> ((x,()),())) + (\((x,()),()) -> ( x ,())) + ga_assoc = Lens (\(((x,y),z),()) -> ((x,(y,z)),())) + (\((x,(y,z)),()) -> (((x,y),z),())) + ga_unassoc = Lens (\((x,(y,z)),()) -> (((x,y),z),())) + (\(((x,y),z),()) -> ((x,(y,z)),())) + +instance GArrowDrop Lens (,) where + ga_drop = Lens (\(x,()) -> ((),x)) (\((),x) -> (x,())) +instance GArrowCopy Lens (,) where + ga_copy = Lens (\(x,()) -> ((x,x),())) (\((x,_),()) -> (x,())) +instance GArrowSwap Lens (,) where + ga_swap = Lens (\((x,y),()) -> ((y,x),())) (\((x,y),()) -> ((y,x),())) + +instance BiGArrow Lens (,) where + biga_arr f f' = Lens (\(x,()) -> ((f x),())) (\(x,()) -> ((f' x),())) + biga_inv (Lens f1 f2) = Lens f2 f1 + + + + -------------------------------------------------------------------------------- -- An example generalized arrow @@ -498,3 +635,190 @@ instance GArrowLiteral GArrowVerilog (,) where + +{- +lambda calculus interpreter + +data Val = + Num Int +| Fun <[Val -> Val]> + +This requires higher-order functions in the second level... + +eval :: Exp -> a Env Val +eval (Var s) = <[ lookup s ]> +eval (Add e1 e2) = <[ let (Num v1) = ~(eval e1) + in let (Num v2) = ~(eval e2) + in (Num (v1+v2)) ]> +eval (If e1 e2 e3) = <[ let v1 = ~(eval e1) in + in if v1 + then ~(eval e2) + else ~(eval e3) ]> +eval (Lam x e) = ??? + +eval (Var s) = proc env -> + returnA -< fromJust (lookup s env) +eval (Add e1 e2) = proc env -> + (eval e1 -< env) `bind` \ ~(Num u) -> + (eval e2 -< env) `bind` \ ~(Num v) -> + returnA -< Num (u + v) +eval (If e1 e2 e3) = proc env -> + (eval e1 -< env) `bind` \ ~(Bl b) -> + if b then eval e2 -< env + else eval e3 -< env +eval (Lam x e) = proc env -> + returnA -< Fun (proc v -> eval e -< (x,v):env) +eval (App e1 e2) = proc env -> + (eval e1 -< env) `bind` \ ~(Fun f) -> + (eval e2 -< env) `bind` \ v -> + f -< v + +eval (Var s) = <[ \env -> fromJust (lookup s env) ]> +eval (Add e1 e2) = <[ \env -> (~(eval e1) env) + (~(eval e2) env) ]> +eval (If e1 e2 e3) = <[ \env -> if ~(eval e1) env + then ~(eval e2) env + else ~(eval e2) env +eval (Lam x e) = <[ \env -> Fun (\v -> ~(eval e) ((x,v):env)) ]> +eval (App e1 e2) = <[ \env -> case ~(eval e1) env of + (Fun f) -> f (~(eval e2) env) ]> +eval (Var s) <[env]> = <[ fromJust (lookup s env) ]> +eval (Add e1 e2) <[env]> = <[ (~(eval e1) env) + (~(eval e2) env) ]> +-} + + + + + +{- +immutable heap with cycles + +-- an immutable heap; maps Int->(Int,Int) + +alloc :: A (Int,Int) Int +lookup :: A Int (Int,Int) + +onetwocycle :: A (Int,Int) (Int,Int) +onetwocycle = + proc \(x,y)-> do + x' <- alloc -< (1,y) + y' <- alloc -< (2,x) + return (x',y') +\end{verbatim} + +\begin{verbatim} +alloc :: <[ (Int,Int) -> Int ]> +lookup :: <[ Int -> (Int,Int) ]> + +onetwocycle :: <[ (Int,Int) ]> -> <[ (Int,Int) ]> +onetwocycle x y = <[ + let x' = ~alloc (1,~y) + in let y' = ~alloc (2,~x) + in (x',y') +]> + +onetwocycle' :: <[ (Int,Int) -> (Int,Int) ]> +onetwocycle' = back2 onetwocycle +\end{verbatim} +-} + + + + +{- +The example may seem a little contrived, but its purpose is to +illustrate the be- haviour when the argument of mapC refers both to +its parameter and a free vari- able (n). + +\begin{verbatim} +-- we can use mapA rather than mapC (from page 100) + +mapA f = proc xs -> case xs of +[] -> returnA -< [] x:xs’ -> do y <- f -< x +ys’ <- mapA f -< xs’ returnA -< y:ys + +example2 = + <[ \(n,xs) -> + ~(mapA <[ \x-> (~(delay 0) n, x) ]> ) + xs + ]> + +<[ example2 (n,xs) = + ~(mapA <[ \x-> (~(delay 0) n, x) ]> ) xs ]> +\end{verbatim} +-} + + + + + + +{- +delaysA = + arr listcase >>> + arr (const []) ||| + (arr id *** (delaysA >>> delay []) >>> + arr (uncurry (:))) + +nor :: SF (Bool,Bool) Bool +nor = arr (not.uncurry (||)) + +edge :: SF Bool Bool +edge = + proc a -> do + b <- delay False -< a + returnA -< a && not b + +flipflop = + proc (reset,set) -> do + rec c <- delay False -< nor + reset d d <- delay True -< nor set c + returnA -< (c,d) + +halfAdd :: Arrow arr => arr (Bool,Bool) (Bool,Bool) +halfAdd = + proc (x,y) -> returnA -< (x&&y, x/=y) + +fullAdd :: + Arrow arr => arr (Bool,Bool,Bool) (Bool,Bool) +fullAdd = + proc (x,y,c) -> do + (c1,s1) <- halfAdd -< (x,y) + (c2,s2) <- halfAdd -< (s1,c) + returnA -< (c1||c2,s2) + +Here is the appendix of Hughes04: +module Circuits where +import Control.Arrow import List +class ArrowLoop a => ArrowCircuit a where delay :: b -> a b b +nor :: Arrow a => a (Bool,Bool) Bool nor = arr (not.uncurry (||)) +flipflop :: ArrowCircuit a => a (Bool,Bool) (Bool,Bool) flipflop = loop (arr (\((a,b),~(c,d)) -> ((a,d),(b,c))) >>> +nor *** nor >>> delay (False,True) >>> arr id &&& arr id) +class Signal a where showSignal :: [a] -> String +instance Signal Bool where showSignal bs = concat top++"\n"++concat bot++"\n" +where (top,bot) = unzip (zipWith sh (False:bs) bs) sh True True = ("__"," ") sh True False = (" ","|_") sh False True = (" _","| ") +sh False False = (" ","__") +instance (Signal a,Signal b) => Signal showSignal xys = showSignal (map fst showSignal (map snd +instance Signal a => Signal [a] where showSignal = concat . map showSignal +sig = concat . map (uncurry replicate) +(a,b) where xys) ++ xys) +. transpose +flipflopInput = sig [(5,(False,False)),(2,(False,True)),(5,(False,False)), +(2,(True,False)),(5,(False,False)),(2,(True,True)), (6,(False,False))] + + + + + +-- from Hughes' "programming with Arrows" + +mapC :: ArrowChoice arr => arr (env,a) b -> arr (env,[a]) [b] mapC c = proc (env,xs) -> +case xs of [] -> returnA -< [] x:xs’ -> do y <- c -< (env,x) +ys <- mapC c -< (env,xs’) returnA -< y:ys + +example2 = proc (n,xs) -> (| mapC (\x-> do delay 0 -< n +&&& do returnA -< x) |) xs +-} + + + +-} \ No newline at end of file diff --git a/src/HaskProofToStrong.v b/src/HaskProofToStrong.v index ece5801..65c638e 100644 --- a/src/HaskProofToStrong.v +++ b/src/HaskProofToStrong.v @@ -401,17 +401,16 @@ Section HaskProofToStrong. destruct l0 as [τ l']. simpl. apply ileaf in X. simpl in X. - assert (unlev (ξ' v) = τ). - admit. - rewrite <- H. + destruct (eqd_dec (unlev (ξ' v)) τ). + rewrite <- e. apply ELR_leaf. - rewrite H. destruct (ξ' v). - rewrite <- H. simpl. - assert (h0=l). admit. - rewrite H0 in X. + destruct (eqd_dec h0 l). + rewrite <- e0. apply X. + apply (Prelude_error "level mismatch; should never happen"). + apply (Prelude_error "letrec type mismatch; should never happen"). apply ELR_nil. @@ -425,11 +424,17 @@ Section HaskProofToStrong. Defined. + Lemma leaves_unleaves {T}(t:list T) : leaves (unleaves t) = t. + induction t; auto. + simpl. + rewrite IHt; auto. + Qed. -(* - Definition case_helper tc Γ Δ lev tbranches avars ξ (Σ:Tree ??VV) tys : + Definition case_helper tc Γ Δ lev tbranches avars ξ (Σ:Tree ??VV) : forall pcb : ProofCaseBranch tc Γ Δ lev tbranches avars, - judg2exprType (pcb_judg pcb) -> FreshM + judg2exprType (pcb_judg pcb) -> + (pcb_freevars pcb) = mapOptionTree ξ Σ -> + FreshM {scb : StrongCaseBranchWithVVs VV eqdec_vv tc avars & Expr (sac_Γ scb Γ) (sac_Δ scb Γ avars (weakCK'' Δ)) (scbwv_ξ scb ξ lev) (weakLT' (tbranches @@ lev))}. @@ -437,10 +442,56 @@ Section HaskProofToStrong. simpl in X. destruct pcb. simpl in *. - refine (bind ξvars = fresh_lemma' Γ pcb_freevars Σ [] ξ _ ; _). apply FreshMon. - destruct ξvars as [vars [ξ' - Defined. -*) + set (sac_types pcb_scb _ avars) as boundvars. + refine (fresh_lemma' _ (unleaves (map (fun x => x@@(weakL' lev)) (vec2list boundvars))) Σ + (mapOptionTree weakLT' pcb_freevars) + (weakLT' ○ ξ) _ + >>>= fun ξvars => _). apply FreshMon. + rewrite H. + rewrite <- mapOptionTree_compose. + reflexivity. + destruct ξvars as [ exprvars [pf1 pf2 ]]. + set (list2vec (leaves (mapOptionTree (@fst _ _) exprvars))) as exprvars'. + assert (sac_numExprVars pcb_scb = Datatypes.length (leaves (mapOptionTree (@fst _ _) exprvars))) as H'. + rewrite <- mapOptionTree_compose in pf2. + simpl in pf2. + rewrite mapleaves. + rewrite <- map_preserves_length. + rewrite map_preserves_length with (f:=update_ξ (weakLT' ○ ξ) (leaves exprvars) ○ (@fst _ _)). + rewrite <- mapleaves. + rewrite pf2. + rewrite leaves_unleaves. + rewrite vec2list_map_list2vec. + rewrite vec2list_len. + reflexivity. + rewrite <- H' in exprvars'. + clear H'. + + set (@Build_StrongCaseBranchWithVVs VV _ tc _ avars pcb_scb exprvars') as scb. + set (scbwv_ξ scb ξ lev) as ξ'. + refine (X ξ' (Σ,,(unleaves (vec2list exprvars'))) _ >>>= fun X' => return _). apply FreshMon. + simpl. + unfold ξ'. + unfold scbwv_ξ. + simpl. + rewrite <- vec2list_map_list2vec. + rewrite <- pf1. + admit. + + apply ileaf in X'. + simpl in X'. + exists scb. + unfold weakCK''. + unfold ξ' in X'. + apply X'. + Defined. + + Fixpoint treeM {T}(t:Tree ??(FreshM T)) : FreshM (Tree ??T) := + match t with + | T_Leaf None => return [] + | T_Leaf (Some x) => bind x' = x ; return [x'] + | T_Branch b1 b2 => bind b1' = treeM b1 ; bind b2' = treeM b2 ; return (b1',,b2') + end. Lemma itree_mapOptionTree : forall T T' F (f:T->T') t, ITree _ F (mapOptionTree f t) -> @@ -685,13 +736,6 @@ Section HaskProofToStrong. apply (letrec_helper _ _ _ _ _ X1). destruct case_RCase. - apply ILeaf. -simpl. -intros. -apply (Prelude_error "FIXME"). - - -(* apply ILeaf; simpl; intros. inversion X_. clear X_. @@ -705,17 +749,21 @@ apply (Prelude_error "FIXME"). rename vars1 into varsalts. rename vars2 into varsΣ. - refine (X0 ξ varsΣ _ >>>= fun X => return ILeaf _ _); auto. apply FreshMon. - clear X0. - eapply (ECase _ _ _ _ _ _ _ (ileaf X1)). - clear X1. + refine ( _ >>>= fun Y => X0 ξ varsΣ _ >>>= fun X => return ILeaf _ (@ECase _ _ _ _ _ _ _ _ _ (ileaf X) Y)); auto. + apply FreshMon. destruct ξvars as [varstypes [pf1 pf2]]. - + + apply treeM. apply itree_mapOptionTree in X. refine (itree_to_tree (itmap _ X)). - apply case_helper. -*) + intros. + eapply case_helper. + apply X1. + instantiate (1:=varsΣ). + rewrite <- H2. + admit. + apply FreshMon. Defined. Definition closed2expr : forall c (pn:@ClosedND _ Rule c), ITree _ judg2exprType c. -- 1.7.10.4