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
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
|
(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * Copyright INRIA, CNRS and contributors *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Set Primitive Projections.
(** An alternative to [zify] in ML parametrised by user-provided classes instances.
The framework has currently several limitations that are in place for simplicity.
For instance, we only consider binary operators of type [Op: S -> S -> S].
Another limitation is that our injection theorems e.g. [TBOpInj],
are using Leibniz equality; the payoff is that there is no need for morphisms...
*)
(** An injection [InjTyp S T] declares an injection
from source type S to target type T.
*)
Class InjTyp (S : Type) (T : Type) :=
mkinj {
(* [inj] is the injection function *)
inj : S -> T;
pred : T -> Prop;
(* [cstr] states that [pred] holds for any injected element.
[cstr (inj x)] is introduced in the goal for any leaf
term of the form [inj x]
*)
cstr : forall x, pred (inj x)
}.
(** [BinOp Op] declares a source operator [Op: S1 -> S2 -> S3].
*)
Class BinOp {S1 S2 S3 T1 T2 T3:Type} (Op : S1 -> S2 -> S3) {I1 : InjTyp S1 T1} {I2 : InjTyp S2 T2} {I3 : InjTyp S3 T3} :=
mkbop {
(* [TBOp] is the target operator after injection of operands. *)
TBOp : T1 -> T2 -> T3;
(* [TBOpInj] states the correctness of the injection. *)
TBOpInj : forall (n:S1) (m:S2), inj (Op n m) = TBOp (inj n) (inj m)
}.
(** [Unop Op] declares a source operator [Op : S1 -> S2]. *)
Class UnOp {S1 S2 T1 T2:Type} (Op : S1 -> S2) {I1 : InjTyp S1 T1} {I2 : InjTyp S2 T2} :=
mkuop {
(* [TUOp] is the target operator after injection of operands. *)
TUOp : T1 -> T2;
(* [TUOpInj] states the correctness of the injection. *)
TUOpInj : forall (x:S1), inj (Op x) = TUOp (inj x)
}.
(** [CstOp Op] declares a source constant [Op : S]. *)
Class CstOp {S T:Type} (Op : S) {I : InjTyp S T} :=
mkcst {
(* [TCst] is the target constant. *)
TCst : T;
(* [TCstInj] states the correctness of the injection. *)
TCstInj : inj Op = TCst
}.
(** In the framework, [Prop] is mapped to [Prop] and the injection is phrased in
terms of [=] instead of [<->].
*)
(** [BinRel R] declares the injection of a binary relation. *)
Class BinRel {S:Type} {T:Type} (R : S -> S -> Prop) {I : InjTyp S T} :=
mkbrel {
TR : T -> T -> Prop;
TRInj : forall n m : S, R n m <-> TR (@inj _ _ I n) (inj m)
}.
(** [PropOp Op] declares morphisms for [<->].
This will be used to deal with e.g. [and], [or],... *)
Class PropOp (Op : Prop -> Prop -> Prop) :=
mkprop {
op_iff : forall (p1 p2 q1 q2:Prop), (p1 <-> q1) -> (p2 <-> q2) -> (Op p1 p2 <-> Op q1 q2)
}.
Class PropUOp (Op : Prop -> Prop) :=
mkuprop {
uop_iff : forall (p1 q1 :Prop), (p1 <-> q1) -> (Op p1 <-> Op q1)
}.
(** Once the term is injected, terms can be replaced by their specification.
NB1: The Ltac code is currently limited to (Op: Z -> Z -> Z)
NB2: This is not sufficient to cope with [Z.div] or [Z.mod]
*)
Class BinOpSpec {T1 T2 T3: Type} (Op : T1 -> T2 -> T3) :=
mkbspec {
BPred : T1 -> T2 -> T3 -> Prop;
BSpec : forall x y, BPred x y (Op x y)
}.
Class UnOpSpec {T1 T2: Type} (Op : T1 -> T2) :=
mkuspec {
UPred : T1 -> T2 -> Prop;
USpec : forall x, UPred x (Op x)
}.
(** After injections, e.g. nat -> Z,
the fact that Z.of_nat x * Z.of_nat y is positive is lost.
This information can be recovered using instance of the [Saturate] class.
*)
Class Saturate {T: Type} (Op : T -> T -> T) :=
mksat {
(** Given [Op x y],
- [PArg1] is the pre-condition of x
- [PArg2] is the pre-condition of y
- [PRes] is the pos-condition of (Op x y) *)
PArg1 : T -> Prop;
PArg2 : T -> Prop;
PRes : T -> Prop;
(** [SatOk] states the correctness of the reasoning *)
SatOk : forall x y, PArg1 x -> PArg2 y -> PRes (Op x y)
}.
(* The [ZifyInst.saturate] iterates over all the instances
and for every pattern of the form
[H1 : PArg1 ?x , H2 : PArg2 ?y , T : context[Op ?x ?y] |- _ ]
[H1 : PArg1 ?x , H2 : PArg2 ?y |- context[Op ?x ?y] ]
asserts (SatOK x y H1 H2) *)
(** The rest of the file is for internal use by the ML tactic.
There are data-structures and lemmas used to inductively construct
the injected terms. *)
(** The data-structures [injterm] and [injected_prop]
are used to store source and target expressions together
with a correctness proof. *)
Record injterm {S T: Type} (I : S -> T) :=
mkinjterm { source : S ; target : T ; inj_ok : I source = target}.
Record injprop :=
mkinjprop {
source_prop : Prop ; target_prop : Prop ;
injprop_ok : source_prop <-> target_prop}.
(** Lemmas for building rewrite rules. *)
Definition PropOp_iff (Op : Prop -> Prop -> Prop) :=
forall (p1 p2 q1 q2:Prop), (p1 <-> q1) -> (p2 <-> q2) -> (Op p1 p2 <-> Op q1 q2).
Definition PropUOp_iff (Op : Prop -> Prop) :=
forall (p1 q1 :Prop), (p1 <-> q1) -> (Op p1 <-> Op q1).
Lemma mkapp2 (S1 S2 S3 T1 T2 T3 : Type) (Op : S1 -> S2 -> S3)
(I1 : S1 -> T1) (I2 : S2 -> T2) (I3 : S3 -> T3)
(TBOP : T1 -> T2 -> T3)
(TBOPINJ : forall n m, I3 (Op n m) = TBOP (I1 n) (I2 m))
(s1 : S1) (t1 : T1) (P1: I1 s1 = t1)
(s2 : S2) (t2 : T2) (P2: I2 s2 = t2): I3 (Op s1 s2) = TBOP t1 t2.
Proof.
subst. apply TBOPINJ.
Qed.
Lemma mkapp (S1 S2 T1 T2 : Type) (OP : S1 -> S2)
(I1 : S1 -> T1)
(I2 : S2 -> T2)
(TUOP : T1 -> T2)
(TUOPINJ : forall n, I2 (OP n) = TUOP (I1 n))
(s1: S1) (t1: T1) (P1: I1 s1 = t1): I2 (OP s1) = TUOP t1.
Proof.
subst. apply TUOPINJ.
Qed.
Lemma mkrel (S T : Type) (R : S -> S -> Prop)
(I : S -> T)
(TR : T -> T -> Prop)
(TRINJ : forall n m : S, R n m <-> TR (I n) (I m))
(s1 : S) (t1 : T) (P1 : I s1 = t1)
(s2 : S) (t2 : T) (P2 : I s2 = t2):
R s1 s2 <-> TR t1 t2.
Proof.
subst.
apply TRINJ.
Qed.
(** Hardcoded support and lemma for propositional logic *)
Lemma and_morph : forall (s1 s2 t1 t2:Prop), s1 <-> t1 -> s2 <-> t2 -> ((s1 /\ s2) <-> (t1 /\ t2)).
Proof.
intros. tauto.
Qed.
Lemma or_morph : forall (s1 s2 t1 t2:Prop), s1 <-> t1 -> s2 <-> t2 -> ((s1 \/ s2) <-> (t1 \/ t2)).
Proof.
intros. tauto.
Qed.
Lemma impl_morph : forall (s1 s2 t1 t2:Prop), s1 <-> t1 -> s2 <-> t2 -> ((s1 -> s2) <-> (t1 -> t2)).
Proof.
intros. tauto.
Qed.
Lemma iff_morph : forall (s1 s2 t1 t2:Prop), s1 <-> t1 -> s2 <-> t2 -> ((s1 <-> s2) <-> (t1 <-> t2)).
Proof.
intros. tauto.
Qed.
Lemma not_morph : forall (s1 t1:Prop), s1 <-> t1 -> (not s1) <-> (not t1).
Proof.
intros. tauto.
Qed.
Lemma eq_iff : forall (P Q : Prop), P = Q -> (P <-> Q).
Proof.
intros P Q H.
rewrite H.
apply iff_refl.
Defined.
Lemma rew_iff (P Q : Prop) (IFF : P <-> Q) : P -> Q.
Proof.
exact (fun H => proj1 IFF H).
Qed.
(** Registering constants for use by the plugin *)
Register eq_iff as ZifyClasses.eq_iff.
Register target_prop as ZifyClasses.target_prop.
Register mkrel as ZifyClasses.mkrel.
Register target as ZifyClasses.target.
Register mkapp2 as ZifyClasses.mkapp2.
Register mkapp as ZifyClasses.mkapp.
Register op_iff as ZifyClasses.op_iff.
Register uop_iff as ZifyClasses.uop_iff.
Register TR as ZifyClasses.TR.
Register TBOp as ZifyClasses.TBOp.
Register TUOp as ZifyClasses.TUOp.
Register TCst as ZifyClasses.TCst.
Register injprop_ok as ZifyClasses.injprop_ok.
Register inj_ok as ZifyClasses.inj_ok.
Register source as ZifyClasses.source.
Register source_prop as ZifyClasses.source_prop.
Register inj as ZifyClasses.inj.
Register TRInj as ZifyClasses.TRInj.
Register TUOpInj as ZifyClasses.TUOpInj.
Register not as ZifyClasses.not.
Register mkinjterm as ZifyClasses.mkinjterm.
Register eq_refl as ZifyClasses.eq_refl.
Register eq as ZifyClasses.eq.
Register mkinjprop as ZifyClasses.mkinjprop.
Register iff_refl as ZifyClasses.iff_refl.
Register rew_iff as ZifyClasses.rew_iff.
Register source_prop as ZifyClasses.source_prop.
Register injprop_ok as ZifyClasses.injprop_ok.
Register iff as ZifyClasses.iff.
Register InjTyp as ZifyClasses.InjTyp.
Register BinOp as ZifyClasses.BinOp.
Register UnOp as ZifyClasses.UnOp.
Register CstOp as ZifyClasses.CstOp.
Register BinRel as ZifyClasses.BinRel.
Register PropOp as ZifyClasses.PropOp.
Register PropUOp as ZifyClasses.PropUOp.
Register BinOpSpec as ZifyClasses.BinOpSpec.
Register UnOpSpec as ZifyClasses.UnOpSpec.
Register Saturate as ZifyClasses.Saturate.
(** Propositional logic *)
Register and as ZifyClasses.and.
Register and_morph as ZifyClasses.and_morph.
Register or as ZifyClasses.or.
Register or_morph as ZifyClasses.or_morph.
Register iff as ZifyClasses.iff.
Register iff_morph as ZifyClasses.iff_morph.
Register impl_morph as ZifyClasses.impl_morph.
Register not as ZifyClasses.not.
Register not_morph as ZifyClasses.not_morph.
Register True as ZifyClasses.True.
|
