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+(* 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.
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