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Start of Featherweight Ur semantics
author Adam Chlipala <adamc@hcoop.net>
date Tue, 17 Feb 2009 14:49:28 -0500
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1 (* Copyright (c) 2009, Adam Chlipala
2 * All rights reserved.
3 *
4 * Redistribution and use in source and binary forms, with or without
5 * modification, are permitted provided that the following conditions are met:
6 *
7 * - Redistributions of source code must retain the above copyright notice,
8 * this list of conditions and the following disclaimer.
9 * - Redistributions in binary form must reproduce the above copyright notice,
10 * this list of conditions and the following disclaimer in the documentation
11 * and/or other materials provided with the distribution.
12 * - The names of contributors may not be used to endorse or promote products
13 * derived from this software without specific prior written permission.
14 *
15 * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
16 * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
17 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
18 * ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
19 * LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
20 * CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
21 * SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
22 * INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
23 * CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
24 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
25 * POSSIBILITY OF SUCH DAMAGE.
26 *)
27
28 Set Implicit Arguments.
29
30
31 Definition name := nat.
32
33
34 (** Syntax of Featherweight Ur *)
35
36 Inductive kind : Type :=
37 | KType : kind
38 | KName : kind
39 | KArrow : kind -> kind -> kind
40 | KRecord : kind -> kind.
41
42 Section vars.
43 Variable cvar : kind -> Type.
44
45 Inductive con : kind -> Type :=
46 | CVar : forall k, cvar k -> con k
47 | Arrow : con KType -> con KType -> con KType
48 | Poly : forall k, (cvar k -> con KType) -> con KType
49 | CAbs : forall k1 k2, (cvar k1 -> con k2) -> con (KArrow k1 k2)
50 | CApp : forall k1 k2, con (KArrow k1 k2) -> con k1 -> con k2
51 | Name : name -> con KName
52 | TRecord : con (KRecord KType) -> con KType
53 | CEmpty : forall k, con (KRecord k)
54 | CSingle : forall k, con KName -> con k -> con (KRecord k)
55 | CConcat : forall k, con (KRecord k) -> con (KRecord k) -> con (KRecord k)
56 | CFold : forall k1 k2, con (KArrow (KArrow KName (KArrow k1 (KArrow k2 k2)))
57 (KArrow k2 (KArrow (KRecord k1) k2)))
58 | CGuarded : forall k1 k2, con (KRecord k1) -> con (KRecord k1) -> con k2 -> con k2.
59
60 Variable dvar : forall k, con (KRecord k) -> con (KRecord k) -> Type.
61
62 Section subs.
63 Variable k1 : kind.
64 Variable c1 : con k1.
65
66 Inductive subs : forall k2, (cvar k1 -> con k2) -> con k2 -> Type :=
67 | S_Unchanged : forall k2 (c2 : con k2),
68 subs (fun _ => c2) c2
69 | S_CVar : subs (fun x => CVar x) c1
70 | S_Arrow : forall c2 c3 c2' c3',
71 subs c2 c2'
72 -> subs c3 c3'
73 -> subs (fun x => Arrow (c2 x) (c3 x)) (Arrow c2' c3')
74 | S_Poly : forall k (c2 : cvar k1 -> cvar k -> _) (c2' : cvar k -> _),
75 (forall x', subs (fun x => c2 x x') (c2' x'))
76 -> subs (fun x => Poly (c2 x)) (Poly c2')
77 | S_CAbs : forall k2 k3 (c2 : cvar k1 -> cvar k2 -> con k3) (c2' : cvar k2 -> _),
78 (forall x', subs (fun x => c2 x x') (c2' x'))
79 -> subs (fun x => CAbs (c2 x)) (CAbs c2')
80 | S_CApp : forall k1 k2 (c2 : _ -> con (KArrow k1 k2)) c3 c2' c3',
81 subs c2 c2'
82 -> subs c3 c3'
83 -> subs (fun x => CApp (c2 x) (c3 x)) (CApp c2' c3')
84 | S_TRecord : forall c2 c2',
85 subs c2 c2'
86 -> subs (fun x => TRecord (c2 x)) (TRecord c2')
87 | S_CSingle : forall k2 c2 (c3 : _ -> con k2) c2' c3',
88 subs c2 c2'
89 -> subs c3 c3'
90 -> subs (fun x => CSingle (c2 x) (c3 x)) (CSingle c2' c3')
91 | S_CConcat : forall k2 (c2 c3 : _ -> con (KRecord k2)) c2' c3',
92 subs c2 c2'
93 -> subs c3 c3'
94 -> subs (fun x => CConcat (c2 x) (c3 x)) (CConcat c2' c3')
95 | S_CGuarded : forall k2 k3 (c2 c3 : _ -> con (KRecord k2)) (c4 : _ -> con k3) c2' c3' c4',
96 subs c2 c2'
97 -> subs c3 c3'
98 -> subs c4 c4'
99 -> subs (fun x => CGuarded (c2 x) (c3 x) (c4 x)) (CGuarded c2' c3' c4').
100 End subs.
101
102 Inductive disj : forall k, con (KRecord k) -> con (KRecord k) -> Prop :=
103 | DVar : forall k (c1 c2 : con (KRecord k)),
104 dvar c1 c2 -> disj c1 c2
105 | DComm : forall k (c1 c2 : con (KRecord k)),
106 disj c1 c2 -> disj c2 c1
107
108 | DEmpty : forall k c2,
109 disj (CEmpty k) c2
110 | DSingleKeys : forall k X1 X2 (c1 c2 : con k),
111 X1 <> X2
112 -> disj (CSingle (Name X1) c1) (CSingle (Name X2) c2)
113 | DSingleValues : forall k n1 n2 (c1 c2 : con k) k' (c1' c2' : con k'),
114 disj (CSingle n1 c1') (CSingle n2 c2')
115 -> disj (CSingle n1 c1) (CSingle n2 c2)
116
117 | DConcat : forall k (c1 c2 c : con (KRecord k)),
118 disj c1 c
119 -> disj c2 c
120 -> disj (CConcat c1 c2) c
121
122 | DEq : forall k (c1 c2 c1' : con (KRecord k)),
123 disj c1 c2
124 -> deq c1 c1'
125 -> disj c1' c2
126
127 with deq : forall k, con k -> con k -> Prop :=
128 | Eq_Beta : forall k1 k2 (c1 : cvar k1 -> con k2) c2 c1',
129 subs c2 c1 c1'
130 -> deq (CApp (CAbs c1) c2) c1'
131 | Eq_Refl : forall k (c : con k),
132 deq c c
133 | Eq_Comm : forall k (c1 c2 : con k),
134 deq c2 c1
135 -> deq c1 c2
136 | Eq_Trans : forall k (c1 c2 c3 : con k),
137 deq c1 c2
138 -> deq c2 c3
139 -> deq c1 c3
140 | Eq_Cong : forall k1 k2 c1 c1' (c2 : cvar k1 -> con k2) c2' c2'',
141 deq c1 c1'
142 -> subs c1 c2 c2'
143 -> subs c1' c2 c2''
144 -> deq c2' c2''
145
146 | Eq_Concat_Empty : forall k c,
147 deq (CConcat (CEmpty k) c) c
148 | Eq_Concat_Comm : forall k (c1 c2 : con (KRecord k)),
149 deq (CConcat c1 c2) (CConcat c2 c1)
150 | Eq_Concat_Assoc : forall k (c1 c2 c3 : con (KRecord k)),
151 deq (CConcat c1 (CConcat c2 c3)) (CConcat (CConcat c1 c2) c3)
152
153 | Eq_Fold_Empty : forall k1 k2 f i,
154 deq (CApp (CApp (CApp (CFold k1 k2) f) i) (CEmpty _)) i
155 | Eq_Fold_Cons : forall k1 k2 f i c1 c2 c3,
156 deq (CApp (CApp (CApp (CFold k1 k2) f) i) (CConcat (CSingle c1 c2) c3))
157 (CApp (CApp (CApp f c1) c2) (CApp (CApp (CApp (CFold k1 k2) f) i) c3))
158
159 | Eq_Guarded : forall k1 k2 (c1 c2 : con (KRecord k1)) (c : con k2),
160 disj c1 c2
161 -> deq (CGuarded c1 c2 c) c
162
163 | Eq_Map_Ident : forall k c,
164 deq (CApp (CApp (CApp (CFold k (KRecord k))
165 (CAbs (fun x1 => CAbs (fun x2 => CAbs (fun x3 => CConcat (CSingle (CVar x1) (CVar x2)) (CVar x3))))))
166 (CEmpty _)) c) c
167 | Eq_Map_Dist : forall k1 k2 f c1 c2,
168 deq (CApp (CApp (CApp (CFold k1 (KRecord k2))
169 (CAbs (fun x1 => CAbs (fun x2 => CAbs (fun x3 => CConcat (CSingle (CVar x1) (CApp f (CVar x2))) (CVar x3))))))
170 (CEmpty _)) (CConcat c1 c2))
171 (CConcat
172 (CApp (CApp (CApp (CFold k1 (KRecord k2))
173 (CAbs (fun x1 => CAbs (fun x2 => CAbs (fun x3 => CConcat (CSingle (CVar x1) (CApp f (CVar x2))) (CVar x3))))))
174 (CEmpty _)) c1)
175 (CApp (CApp (CApp (CFold k1 (KRecord k2))
176 (CAbs (fun x1 => CAbs (fun x2 => CAbs (fun x3 => CConcat (CSingle (CVar x1) (CApp f (CVar x2))) (CVar x3))))))
177 (CEmpty _)) c2))
178
179 | Eq_Fold_Fuse : forall k1 k2 k3 f i f' c,
180 deq (CApp (CApp (CApp (CFold k1 k2) f) i)
181 (CApp (CApp (CApp (CFold k3 (KRecord k1))
182 (CAbs (fun x1 => CAbs (fun x2 => CAbs (fun x3 => CConcat (CSingle (CVar x1) (CApp f' (CVar x2))) (CVar x3))))))
183 (CEmpty _)) c))
184 (CApp (CApp (CApp (CFold k3 k2)
185 (CAbs (fun x1 => CAbs (fun x2 => CApp (CApp f (CVar x1)) (CApp f' (CVar x2))))))
186 i) c).
187
188 Inductive wf : forall k, con k -> Type :=
189 | HK_CVar : forall k (x : cvar k),
190 wf (CVar x)
191 | HK_Arrow : forall c1 c2,
192 wf c1 -> wf c2 -> wf (Arrow c1 c2)
193 | HK_Poly : forall k (c1 : cvar k -> _),
194 (forall x, wf (c1 x)) -> wf (Poly c1)
195 | HK_CAbs : forall k1 k2 (c1 : cvar k1 -> con k2),
196 (forall x, wf (c1 x)) -> wf (CAbs c1)
197 | HK_CApp : forall k1 k2 (c1 : con (KArrow k1 k2)) c2,
198 wf c1 -> wf c2 -> wf (CApp c1 c2)
199 | HK_Name : forall X,
200 wf (Name X)
201 | HK_TRecord : forall c,
202 wf c -> wf (TRecord c)
203 | HK_CEmpty : forall k,
204 wf (CEmpty k)
205 | HK_CSingle : forall k c1 (c2 : con k),
206 wf c1 -> wf c2 -> wf (CSingle c1 c2)
207 | HK_CConcat : forall k (c1 c2 : con (KRecord k)),
208 wf c2 -> wf c2 -> disj c1 c2 -> wf (CConcat c1 c2)
209 | HK_CFold : forall k1 k2,
210 wf (CFold k1 k2)
211 | HK_CGuarded : forall k1 k2 (c1 c2 : con (KRecord k1)) (c : con k2),
212 wf c1 -> wf c2 -> (disj c1 c2 -> wf c) -> wf (CGuarded c1 c2 c).
213 End vars.