1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
|
(* 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.
*)
Require Import Name.
Export Name.
Set Implicit Arguments.
(** 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)
| CMap : forall k1 k2, con (KArrow (KArrow k1 k2) (KArrow (KRecord k1) (KRecord k2)))
| TGuarded : forall k, con (KRecord k) -> con (KRecord k) -> con KType -> con KType.
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_TGuarded : forall k2 (c2 c3 : _ -> con (KRecord k2)) c4 c2' c3' c4',
subs c2 c2'
-> subs c3 c3'
-> subs c4 c4'
-> subs (fun x => TGuarded (c2 x) (c3 x) (c4 x)) (TGuarded 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 c3 : con (KRecord k)),
disj c1 c2
-> 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_Map_Empty : forall k1 k2 f,
deq (CApp (CApp (CMap k1 k2) f) (CEmpty _)) (CEmpty _)
| Eq_Map_Cons : forall k1 k2 f c1 c2 c3,
disj (CSingle c1 c2) c3
-> deq (CApp (CApp (CMap k1 k2) f) (CConcat (CSingle c1 c2) c3))
(CConcat (CSingle c1 (CApp f c2)) (CApp (CApp (CMap k1 k2) f) c3))
| Eq_Map_Ident : forall k c,
deq (CApp (CApp (CMap k k) (CAbs (fun x => CVar x))) c) c
| Eq_Map_Dist : forall k1 k2 f c1 c2,
deq (CApp (CApp (CMap k1 k2) f) (CConcat c1 c2))
(CConcat (CApp (CApp (CMap k1 k2) f) c1) (CApp (CApp (CMap k1 k2) f) c2))
| Eq_Map_Fuse : forall k1 k2 k3 f f' c,
deq (CApp (CApp (CMap k2 k3) f')
(CApp (CApp (CMap k1 k2) f) c))
(CApp (CApp (CMap k1 k3) (CAbs (fun x => CApp f' (CApp f (CVar x))))) c).
Variable evar : con KType -> Type.
Inductive exp : con KType -> Type :=
| Var : forall t, evar t -> exp t
| App : forall dom ran, exp (Arrow dom ran) -> exp dom -> exp ran
| Abs : forall dom ran, (evar dom -> exp ran) -> exp (Arrow dom ran)
| ECApp : forall k (dom : con k) ran ran', exp (Poly ran) -> subs dom ran ran' -> exp ran'
| ECAbs : forall k (ran : cvar k -> _), (forall X, exp (ran X)) -> exp (Poly ran)
| Cast : forall t1 t2, deq t1 t2 -> exp t1 -> exp t2
| Empty : exp (TRecord (CEmpty _))
| Single : forall c t, exp t -> exp (TRecord (CConcat (CSingle c t) (CEmpty _)))
| Proj : forall c t c', exp (TRecord (CConcat (CSingle c t) c')) -> exp t
| Cut : forall c t c', disj (CSingle c t) c' -> exp (TRecord (CConcat (CSingle c t) c')) -> exp (TRecord c')
| Concat : forall c1 c2, exp (TRecord c1) -> exp (TRecord c2) -> exp (TRecord (CConcat c1 c2))
| Guarded : forall k (c1 c2 : con (KRecord k)) c, (dvar c1 c2 -> exp c) -> exp (TGuarded c1 c2 c)
| GuardedApp : forall k (c1 c2 : con (KRecord k)) t, exp (TGuarded c1 c2 t) -> disj c1 c2 -> exp t.
End vars.
|
