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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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