From 3c534cfde2dea5384113b891362bbfc1ee6b98fe Mon Sep 17 00:00:00 2001 From: Adam Chlipala Date: Tue, 17 Feb 2009 14:49:28 -0500 Subject: Start of Featherweight Ur semantics --- src/coq/Syntax.v | 213 +++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 213 insertions(+) create mode 100644 src/coq/Syntax.v (limited to 'src/coq/Syntax.v') diff --git a/src/coq/Syntax.v b/src/coq/Syntax.v new file mode 100644 index 00000000..786751d0 --- /dev/null +++ b/src/coq/Syntax.v @@ -0,0 +1,213 @@ +(* Copyright (c) 2009, Adam Chlipala + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * - Redistributions of source code must retain the above copyright notice, + * this list of conditions and the following disclaimer. + * - Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * - The names of contributors may not be used to endorse or promote products + * derived from this software without specific prior written permission. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE + * LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR + * CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF + * SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS + * INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN + * CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) + * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE + * POSSIBILITY OF SUCH DAMAGE. + *) + +Set Implicit Arguments. + + +Definition name := nat. + + +(** Syntax of Featherweight Ur *) + +Inductive kind : Type := +| KType : kind +| KName : kind +| KArrow : kind -> kind -> kind +| KRecord : kind -> kind. + +Section vars. + Variable cvar : kind -> Type. + + Inductive con : kind -> Type := + | CVar : forall k, cvar k -> con k + | Arrow : con KType -> con KType -> con KType + | Poly : forall k, (cvar k -> con KType) -> con KType + | CAbs : forall k1 k2, (cvar k1 -> con k2) -> con (KArrow k1 k2) + | CApp : forall k1 k2, con (KArrow k1 k2) -> con k1 -> con k2 + | Name : name -> con KName + | TRecord : con (KRecord KType) -> con KType + | CEmpty : forall k, con (KRecord k) + | CSingle : forall k, con KName -> con k -> con (KRecord k) + | CConcat : forall k, con (KRecord k) -> con (KRecord k) -> con (KRecord k) + | CFold : forall k1 k2, con (KArrow (KArrow KName (KArrow k1 (KArrow k2 k2))) + (KArrow k2 (KArrow (KRecord k1) k2))) + | CGuarded : forall k1 k2, con (KRecord k1) -> con (KRecord k1) -> con k2 -> con k2. + + Variable dvar : forall k, con (KRecord k) -> con (KRecord k) -> Type. + + Section subs. + Variable k1 : kind. + Variable c1 : con k1. + + Inductive subs : forall k2, (cvar k1 -> con k2) -> con k2 -> Type := + | S_Unchanged : forall k2 (c2 : con k2), + subs (fun _ => c2) c2 + | S_CVar : subs (fun x => CVar x) c1 + | S_Arrow : forall c2 c3 c2' c3', + subs c2 c2' + -> subs c3 c3' + -> subs (fun x => Arrow (c2 x) (c3 x)) (Arrow c2' c3') + | S_Poly : forall k (c2 : cvar k1 -> cvar k -> _) (c2' : cvar k -> _), + (forall x', subs (fun x => c2 x x') (c2' x')) + -> subs (fun x => Poly (c2 x)) (Poly c2') + | S_CAbs : forall k2 k3 (c2 : cvar k1 -> cvar k2 -> con k3) (c2' : cvar k2 -> _), + (forall x', subs (fun x => c2 x x') (c2' x')) + -> subs (fun x => CAbs (c2 x)) (CAbs c2') + | S_CApp : forall k1 k2 (c2 : _ -> con (KArrow k1 k2)) c3 c2' c3', + subs c2 c2' + -> subs c3 c3' + -> subs (fun x => CApp (c2 x) (c3 x)) (CApp c2' c3') + | S_TRecord : forall c2 c2', + subs c2 c2' + -> subs (fun x => TRecord (c2 x)) (TRecord c2') + | S_CSingle : forall k2 c2 (c3 : _ -> con k2) c2' c3', + subs c2 c2' + -> subs c3 c3' + -> subs (fun x => CSingle (c2 x) (c3 x)) (CSingle c2' c3') + | S_CConcat : forall k2 (c2 c3 : _ -> con (KRecord k2)) c2' c3', + subs c2 c2' + -> subs c3 c3' + -> subs (fun x => CConcat (c2 x) (c3 x)) (CConcat c2' c3') + | S_CGuarded : forall k2 k3 (c2 c3 : _ -> con (KRecord k2)) (c4 : _ -> con k3) c2' c3' c4', + subs c2 c2' + -> subs c3 c3' + -> subs c4 c4' + -> subs (fun x => CGuarded (c2 x) (c3 x) (c4 x)) (CGuarded c2' c3' c4'). + End subs. + + Inductive disj : forall k, con (KRecord k) -> con (KRecord k) -> Prop := + | DVar : forall k (c1 c2 : con (KRecord k)), + dvar c1 c2 -> disj c1 c2 + | DComm : forall k (c1 c2 : con (KRecord k)), + disj c1 c2 -> disj c2 c1 + + | DEmpty : forall k c2, + disj (CEmpty k) c2 + | DSingleKeys : forall k X1 X2 (c1 c2 : con k), + X1 <> X2 + -> disj (CSingle (Name X1) c1) (CSingle (Name X2) c2) + | DSingleValues : forall k n1 n2 (c1 c2 : con k) k' (c1' c2' : con k'), + disj (CSingle n1 c1') (CSingle n2 c2') + -> disj (CSingle n1 c1) (CSingle n2 c2) + + | DConcat : forall k (c1 c2 c : con (KRecord k)), + disj c1 c + -> disj c2 c + -> disj (CConcat c1 c2) c + + | DEq : forall k (c1 c2 c1' : con (KRecord k)), + disj c1 c2 + -> deq c1 c1' + -> disj c1' c2 + + with deq : forall k, con k -> con k -> Prop := + | Eq_Beta : forall k1 k2 (c1 : cvar k1 -> con k2) c2 c1', + subs c2 c1 c1' + -> deq (CApp (CAbs c1) c2) c1' + | Eq_Refl : forall k (c : con k), + deq c c + | Eq_Comm : forall k (c1 c2 : con k), + deq c2 c1 + -> deq c1 c2 + | Eq_Trans : forall k (c1 c2 c3 : con k), + deq c1 c2 + -> deq c2 c3 + -> deq c1 c3 + | Eq_Cong : forall k1 k2 c1 c1' (c2 : cvar k1 -> con k2) c2' c2'', + deq c1 c1' + -> subs c1 c2 c2' + -> subs c1' c2 c2'' + -> deq c2' c2'' + + | Eq_Concat_Empty : forall k c, + deq (CConcat (CEmpty k) c) c + | Eq_Concat_Comm : forall k (c1 c2 : con (KRecord k)), + deq (CConcat c1 c2) (CConcat c2 c1) + | Eq_Concat_Assoc : forall k (c1 c2 c3 : con (KRecord k)), + deq (CConcat c1 (CConcat c2 c3)) (CConcat (CConcat c1 c2) c3) + + | Eq_Fold_Empty : forall k1 k2 f i, + deq (CApp (CApp (CApp (CFold k1 k2) f) i) (CEmpty _)) i + | Eq_Fold_Cons : forall k1 k2 f i c1 c2 c3, + deq (CApp (CApp (CApp (CFold k1 k2) f) i) (CConcat (CSingle c1 c2) c3)) + (CApp (CApp (CApp f c1) c2) (CApp (CApp (CApp (CFold k1 k2) f) i) c3)) + + | Eq_Guarded : forall k1 k2 (c1 c2 : con (KRecord k1)) (c : con k2), + disj c1 c2 + -> deq (CGuarded c1 c2 c) c + + | Eq_Map_Ident : forall k c, + deq (CApp (CApp (CApp (CFold k (KRecord k)) + (CAbs (fun x1 => CAbs (fun x2 => CAbs (fun x3 => CConcat (CSingle (CVar x1) (CVar x2)) (CVar x3)))))) + (CEmpty _)) c) c + | Eq_Map_Dist : forall k1 k2 f c1 c2, + deq (CApp (CApp (CApp (CFold k1 (KRecord k2)) + (CAbs (fun x1 => CAbs (fun x2 => CAbs (fun x3 => CConcat (CSingle (CVar x1) (CApp f (CVar x2))) (CVar x3)))))) + (CEmpty _)) (CConcat c1 c2)) + (CConcat + (CApp (CApp (CApp (CFold k1 (KRecord k2)) + (CAbs (fun x1 => CAbs (fun x2 => CAbs (fun x3 => CConcat (CSingle (CVar x1) (CApp f (CVar x2))) (CVar x3)))))) + (CEmpty _)) c1) + (CApp (CApp (CApp (CFold k1 (KRecord k2)) + (CAbs (fun x1 => CAbs (fun x2 => CAbs (fun x3 => CConcat (CSingle (CVar x1) (CApp f (CVar x2))) (CVar x3)))))) + (CEmpty _)) c2)) + + | Eq_Fold_Fuse : forall k1 k2 k3 f i f' c, + deq (CApp (CApp (CApp (CFold k1 k2) f) i) + (CApp (CApp (CApp (CFold k3 (KRecord k1)) + (CAbs (fun x1 => CAbs (fun x2 => CAbs (fun x3 => CConcat (CSingle (CVar x1) (CApp f' (CVar x2))) (CVar x3)))))) + (CEmpty _)) c)) + (CApp (CApp (CApp (CFold k3 k2) + (CAbs (fun x1 => CAbs (fun x2 => CApp (CApp f (CVar x1)) (CApp f' (CVar x2)))))) + i) c). + + Inductive wf : forall k, con k -> Type := + | HK_CVar : forall k (x : cvar k), + wf (CVar x) + | HK_Arrow : forall c1 c2, + wf c1 -> wf c2 -> wf (Arrow c1 c2) + | HK_Poly : forall k (c1 : cvar k -> _), + (forall x, wf (c1 x)) -> wf (Poly c1) + | HK_CAbs : forall k1 k2 (c1 : cvar k1 -> con k2), + (forall x, wf (c1 x)) -> wf (CAbs c1) + | HK_CApp : forall k1 k2 (c1 : con (KArrow k1 k2)) c2, + wf c1 -> wf c2 -> wf (CApp c1 c2) + | HK_Name : forall X, + wf (Name X) + | HK_TRecord : forall c, + wf c -> wf (TRecord c) + | HK_CEmpty : forall k, + wf (CEmpty k) + | HK_CSingle : forall k c1 (c2 : con k), + wf c1 -> wf c2 -> wf (CSingle c1 c2) + | HK_CConcat : forall k (c1 c2 : con (KRecord k)), + wf c2 -> wf c2 -> disj c1 c2 -> wf (CConcat c1 c2) + | HK_CFold : forall k1 k2, + wf (CFold k1 k2) + | HK_CGuarded : forall k1 k2 (c1 c2 : con (KRecord k1)) (c : con k2), + wf c1 -> wf c2 -> (disj c1 c2 -> wf c) -> wf (CGuarded c1 c2 c). +End vars. -- cgit v1.2.3