1 | (* *********************************************************************) |
---|
2 | (* *) |
---|
3 | (* The Compcert verified compiler *) |
---|
4 | (* *) |
---|
5 | (* Xavier Leroy, INRIA Paris-Rocquencourt *) |
---|
6 | (* *) |
---|
7 | (* Copyright Institut National de Recherche en Informatique et en *) |
---|
8 | (* Automatique. All rights reserved. This file is distributed *) |
---|
9 | (* under the terms of the GNU General Public License as published by *) |
---|
10 | (* the Free Software Foundation, either version 2 of the License, or *) |
---|
11 | (* (at your option) any later version. This file is also distributed *) |
---|
12 | (* under the terms of the INRIA Non-Commercial License Agreement. *) |
---|
13 | (* *) |
---|
14 | (* *********************************************************************) |
---|
15 | |
---|
16 | (* * This file defines a number of data types and operations used in |
---|
17 | the abstract syntax trees of many of the intermediate languages. *) |
---|
18 | |
---|
19 | include "basics/types.ma". |
---|
20 | include "common/Integers.ma". |
---|
21 | include "common/Floats.ma". |
---|
22 | include "ASM/Arithmetic.ma". |
---|
23 | include "common/Identifiers.ma". |
---|
24 | |
---|
25 | |
---|
26 | (* * * Syntactic elements *) |
---|
27 | |
---|
28 | (* Global variables and functions are represented by identifiers with the |
---|
29 | tag for symbols. Note that Clight also uses them for locals due to |
---|
30 | the ambiguous syntax. *) |
---|
31 | |
---|
32 | axiom SymbolTag : String. |
---|
33 | |
---|
34 | definition ident ≝ identifier SymbolTag. |
---|
35 | |
---|
36 | definition ident_eq : ∀x,y:ident. (x=y) + (x≠y) ≝ identifier_eq ?. |
---|
37 | |
---|
38 | definition ident_of_nat : nat → ident ≝ identifier_of_nat ?. |
---|
39 | |
---|
40 | definition Immediate ≝ nat. (* XXX is this the best place to put this? *) |
---|
41 | |
---|
42 | (* dpm: not needed |
---|
43 | inductive quantity: Type[0] ≝ |
---|
44 | | q_int: Byte → quantity |
---|
45 | | q_offset: quantity |
---|
46 | | q_ptr: quantity. |
---|
47 | |
---|
48 | inductive abstract_size: Type[0] ≝ |
---|
49 | | size_q: quantity → abstract_size |
---|
50 | | size_prod: list abstract_size → abstract_size |
---|
51 | | size_sum: list abstract_size → abstract_size |
---|
52 | | size_array: nat → abstract_size → abstract_size. |
---|
53 | *) |
---|
54 | |
---|
55 | |
---|
56 | (* Memory spaces *) |
---|
57 | |
---|
58 | inductive region : Type[0] ≝ |
---|
59 | | Any : region |
---|
60 | | Data : region |
---|
61 | | IData : region |
---|
62 | | PData : region |
---|
63 | | XData : region |
---|
64 | | Code : region. |
---|
65 | |
---|
66 | definition eq_region : region → region → bool ≝ |
---|
67 | λr1,r2. |
---|
68 | match r1 with |
---|
69 | [ Any ⇒ match r2 with [ Any ⇒ true | _ ⇒ false ] |
---|
70 | | Data ⇒ match r2 with [ Data ⇒ true | _ ⇒ false ] |
---|
71 | | IData ⇒ match r2 with [ IData ⇒ true | _ ⇒ false ] |
---|
72 | | PData ⇒ match r2 with [ PData ⇒ true | _ ⇒ false ] |
---|
73 | | XData ⇒ match r2 with [ XData ⇒ true | _ ⇒ false ] |
---|
74 | | Code ⇒ match r2 with [ Code ⇒ true | _ ⇒ false ] |
---|
75 | ]. |
---|
76 | |
---|
77 | lemma eq_region_elim : ∀P:bool → Type[0]. ∀r1,r2. |
---|
78 | (r1 = r2 → P true) → (r1 ≠ r2 → P false) → |
---|
79 | P (eq_region r1 r2). |
---|
80 | #P #r1 #r2 cases r1; cases r2; #Ptrue #Pfalse |
---|
81 | try ( @Ptrue // ) |
---|
82 | @Pfalse % #E destruct |
---|
83 | qed. |
---|
84 | |
---|
85 | lemma reflexive_eq_region: ∀r. eq_region r r = true. |
---|
86 | * // |
---|
87 | qed. |
---|
88 | |
---|
89 | definition eq_region_dec : ∀r1,r2:region. (r1=r2)+(r1≠r2). |
---|
90 | #r1 #r2 @(eq_region_elim ? r1 r2) /2/; qed. |
---|
91 | |
---|
92 | definition size_pointer : region → nat ≝ |
---|
93 | λsp. match sp with [ Data ⇒ 1 | IData ⇒ 1 | PData ⇒ 1 | XData ⇒ 2 | Code ⇒ 2 | Any ⇒ 3 ]. |
---|
94 | |
---|
95 | (* We maintain some reasonable type information through the front end of the |
---|
96 | compiler. *) |
---|
97 | |
---|
98 | inductive signedness : Type[0] ≝ |
---|
99 | | Signed: signedness |
---|
100 | | Unsigned: signedness. |
---|
101 | |
---|
102 | inductive intsize : Type[0] ≝ |
---|
103 | | I8: intsize |
---|
104 | | I16: intsize |
---|
105 | | I32: intsize. |
---|
106 | |
---|
107 | (* * Float types come in two sizes: 32 bits (single precision) |
---|
108 | and 64-bit (double precision). *) |
---|
109 | |
---|
110 | inductive floatsize : Type[0] ≝ |
---|
111 | | F32: floatsize |
---|
112 | | F64: floatsize. |
---|
113 | |
---|
114 | inductive typ : Type[0] ≝ |
---|
115 | | ASTint : intsize → signedness → typ |
---|
116 | | ASTptr : region → typ |
---|
117 | | ASTfloat : floatsize → typ. |
---|
118 | |
---|
119 | (* XXX aliases *) |
---|
120 | definition SigType ≝ typ. |
---|
121 | definition SigType_Int ≝ ASTint I32 Unsigned. |
---|
122 | (* |
---|
123 | | SigType_Float: SigType |
---|
124 | *) |
---|
125 | definition SigType_Ptr ≝ ASTptr Any. |
---|
126 | |
---|
127 | (* Define these carefully so that we always know that the result is nonzero, |
---|
128 | and can be used in dependent types of the form (S n). |
---|
129 | (At the time of writing this is only used for bitsize_of_intsize.) *) |
---|
130 | |
---|
131 | definition size_intsize : intsize → nat ≝ |
---|
132 | λsz. S match sz with [ I8 ⇒ 0 | I16 ⇒ 1 | I32 ⇒ 3]. |
---|
133 | |
---|
134 | definition bitsize_of_intsize : intsize → nat ≝ |
---|
135 | λsz. S match sz with [ I8 ⇒ 7 | I16 ⇒ 15 | I32 ⇒ 31]. |
---|
136 | |
---|
137 | definition eq_intsize : intsize → intsize → bool ≝ |
---|
138 | λsz1,sz2. |
---|
139 | match sz1 with |
---|
140 | [ I8 ⇒ match sz2 with [ I8 ⇒ true | _ ⇒ false ] |
---|
141 | | I16 ⇒ match sz2 with [ I16 ⇒ true | _ ⇒ false ] |
---|
142 | | I32 ⇒ match sz2 with [ I32 ⇒ true | _ ⇒ false ] |
---|
143 | ]. |
---|
144 | |
---|
145 | lemma eq_intsize_elim : ∀sz1,sz2. ∀P:bool → Type[0]. |
---|
146 | (sz1 ≠ sz2 → P false) → (sz1 = sz2 → P true) → P (eq_intsize sz1 sz2). |
---|
147 | * * #P #Hne #Heq whd in ⊢ (?%) try (@Heq @refl) @Hne % #E destruct |
---|
148 | qed. |
---|
149 | |
---|
150 | lemma eq_intsize_true : ∀sz. eq_intsize sz sz = true. |
---|
151 | * @refl |
---|
152 | qed. |
---|
153 | |
---|
154 | lemma eq_intsize_false : ∀sz,sz'. sz ≠ sz' → eq_intsize sz sz' = false. |
---|
155 | * * * #NE try @refl @False_ind @NE @refl |
---|
156 | qed. |
---|
157 | |
---|
158 | (* [intsize_eq_elim ? sz1 sz2 ? n (λn.e1) e2] checks if [sz1] equals [sz2] and |
---|
159 | if it is returns [e1] where the type of [n] has its dependency on [sz1] |
---|
160 | changed to [sz2], and if not returns [e2]. *) |
---|
161 | let rec intsize_eq_elim (A:Type[0]) (sz1,sz2:intsize) (P:intsize → Type[0]) on sz1 : |
---|
162 | P sz1 → (P sz2 → A) → A → A ≝ |
---|
163 | match sz1 return λsz. P sz → (P sz2 → A) → A → A with |
---|
164 | [ I8 ⇒ λx. match sz2 return λsz. (P sz → A) → A → A with [ I8 ⇒ λf,d. f x | _ ⇒ λf,d. d ] |
---|
165 | | I16 ⇒ λx. match sz2 return λsz. (P sz → A) → A → A with [ I16 ⇒ λf,d. f x | _ ⇒ λf,d. d ] |
---|
166 | | I32 ⇒ λx. match sz2 return λsz. (P sz → A) → A → A with [ I32 ⇒ λf,d. f x | _ ⇒ λf,d. d ] |
---|
167 | ]. |
---|
168 | |
---|
169 | lemma intsize_eq_elim_true : ∀A,sz,P,p,f,d. |
---|
170 | intsize_eq_elim A sz sz P p f d = f p. |
---|
171 | #A * // |
---|
172 | qed. |
---|
173 | |
---|
174 | lemma intsize_eq_elim_elim : ∀A,sz1,sz2,P,p,f,d. ∀Q:A → Type[0]. |
---|
175 | (sz1 ≠ sz2 → Q d) → (∀E:sz1 = sz2. match sym_eq ??? E return λx.λ_.P x → Type[0] with [ refl ⇒ λp. Q (f p) ] p ) → Q (intsize_eq_elim A sz1 sz2 P p f d). |
---|
176 | #A * * #P #p #f #d #Q #Hne #Heq |
---|
177 | [ 1,5,9: whd in ⊢ (?%) @(Heq (refl ??)) |
---|
178 | | *: whd in ⊢ (?%) @Hne % #E destruct |
---|
179 | ] qed. |
---|
180 | |
---|
181 | |
---|
182 | (* A type for the integers that appear in the semantics. *) |
---|
183 | definition bvint : intsize → Type[0] ≝ λsz. BitVector (bitsize_of_intsize sz). |
---|
184 | |
---|
185 | definition repr : ∀sz:intsize. nat → bvint sz ≝ |
---|
186 | λsz,x. bitvector_of_nat (bitsize_of_intsize sz) x. |
---|
187 | |
---|
188 | definition size_floatsize : floatsize → nat ≝ |
---|
189 | λsz. S match sz with [ F32 ⇒ 3 | F64 ⇒ 7 ]. |
---|
190 | |
---|
191 | definition typesize : typ → nat ≝ λty. |
---|
192 | match ty with |
---|
193 | [ ASTint sz _ ⇒ size_intsize sz |
---|
194 | | ASTptr r ⇒ size_pointer r |
---|
195 | | ASTfloat sz ⇒ size_floatsize sz ]. |
---|
196 | |
---|
197 | lemma typesize_pos: ∀ty. typesize ty > 0. |
---|
198 | *; try *; try * /2/ qed. |
---|
199 | |
---|
200 | lemma typ_eq: ∀t1,t2: typ. (t1=t2) + (t1≠t2). |
---|
201 | * *; try *; try *; try *; try *; try (%1 @refl) %2 @nmk #H destruct qed. |
---|
202 | |
---|
203 | lemma opt_typ_eq: ∀t1,t2: option typ. (t1=t2) + (t1≠t2). |
---|
204 | #t1 #t2 cases t1 cases t2 |
---|
205 | [ %1 @refl |
---|
206 | | 2,3: #ty %2 % #H destruct |
---|
207 | | #ty1 #ty2 elim (typ_eq ty1 ty2) #E [ %1 >E @refl | %2 % #E' destruct cases E /2/ |
---|
208 | ] |
---|
209 | qed. |
---|
210 | |
---|
211 | (* * Additionally, function definitions and function calls are annotated |
---|
212 | by function signatures indicating the number and types of arguments, |
---|
213 | as well as the type of the returned value if any. These signatures |
---|
214 | are used in particular to determine appropriate calling conventions |
---|
215 | for the function. *) |
---|
216 | |
---|
217 | record signature : Type[0] ≝ { |
---|
218 | sig_args: list typ; |
---|
219 | sig_res: option typ |
---|
220 | }. |
---|
221 | |
---|
222 | (* XXX aliases *) |
---|
223 | definition Signature ≝ signature. |
---|
224 | definition signature_args ≝ sig_args. |
---|
225 | definition signature_return ≝ sig_res. |
---|
226 | |
---|
227 | definition proj_sig_res : signature → typ ≝ λs. |
---|
228 | match sig_res s with |
---|
229 | [ None ⇒ ASTint I32 Unsigned |
---|
230 | | Some t ⇒ t |
---|
231 | ]. |
---|
232 | |
---|
233 | (* * Memory accesses (load and store instructions) are annotated by |
---|
234 | a ``memory chunk'' indicating the type, size and signedness of the |
---|
235 | chunk of memory being accessed. *) |
---|
236 | |
---|
237 | inductive memory_chunk : Type[0] ≝ |
---|
238 | | Mint8signed : memory_chunk (*r 8-bit signed integer *) |
---|
239 | | Mint8unsigned : memory_chunk (*r 8-bit unsigned integer *) |
---|
240 | | Mint16signed : memory_chunk (*r 16-bit signed integer *) |
---|
241 | | Mint16unsigned : memory_chunk (*r 16-bit unsigned integer *) |
---|
242 | | Mint32 : memory_chunk (*r 32-bit integer, or pointer *) |
---|
243 | | Mfloat32 : memory_chunk (*r 32-bit single-precision float *) |
---|
244 | | Mfloat64 : memory_chunk (*r 64-bit double-precision float *) |
---|
245 | | Mpointer : region → memory_chunk. (* pointer addressing given region *) |
---|
246 | |
---|
247 | definition typ_of_memory_chunk : memory_chunk → typ ≝ |
---|
248 | λc. match c with |
---|
249 | [ Mint8signed ⇒ ASTint I8 Signed |
---|
250 | | Mint8unsigned ⇒ ASTint I8 Unsigned |
---|
251 | | Mint16signed ⇒ ASTint I16 Signed |
---|
252 | | Mint16unsigned ⇒ ASTint I16 Unsigned |
---|
253 | | Mint32 ⇒ ASTint I32 Unsigned (* XXX signed? *) |
---|
254 | | Mfloat32 ⇒ ASTfloat F32 |
---|
255 | | Mfloat64 ⇒ ASTfloat F64 |
---|
256 | | Mpointer r ⇒ ASTptr r |
---|
257 | ]. |
---|
258 | |
---|
259 | (* * Initialization data for global variables. *) |
---|
260 | |
---|
261 | inductive init_data: Type[0] ≝ |
---|
262 | | Init_int8: bvint I8 → init_data |
---|
263 | | Init_int16: bvint I16 → init_data |
---|
264 | | Init_int32: bvint I32 → init_data |
---|
265 | | Init_float32: float → init_data |
---|
266 | | Init_float64: float → init_data |
---|
267 | | Init_space: nat → init_data |
---|
268 | | Init_null: region → init_data |
---|
269 | | Init_addrof: region → ident → nat → init_data. (*r address of symbol + offset *) |
---|
270 | |
---|
271 | (* * Whole programs consist of: |
---|
272 | - a collection of function definitions (name and description); |
---|
273 | - the name of the ``main'' function that serves as entry point in the program; |
---|
274 | - a collection of global variable declarations, consisting of |
---|
275 | a name, initialization data, and additional information. |
---|
276 | |
---|
277 | The type of function descriptions and that of additional information |
---|
278 | for variables vary among the various intermediate languages and are |
---|
279 | taken as parameters to the [program] type. The other parts of whole |
---|
280 | programs are common to all languages. *) |
---|
281 | |
---|
282 | record program (F: list ident → Type[0]) (V: Type[0]) : Type[0] := { |
---|
283 | prog_vars: list (ident × region × V); |
---|
284 | prog_funct: list (ident × (F (map … (λx. \fst (\fst x)) prog_vars))); |
---|
285 | prog_main: ident |
---|
286 | }. |
---|
287 | |
---|
288 | |
---|
289 | definition prog_funct_names ≝ λF,V. λp: program F V. |
---|
290 | map … \fst (prog_funct … p). |
---|
291 | |
---|
292 | definition prog_var_names ≝ λF,V. λp: program F V. |
---|
293 | map … (λx. \fst (\fst x)) (prog_vars … p). |
---|
294 | |
---|
295 | (* * * Generic transformations over programs *) |
---|
296 | |
---|
297 | (* * We now define a general iterator over programs that applies a given |
---|
298 | code transformation function to all function descriptions and leaves |
---|
299 | the other parts of the program unchanged. *) |
---|
300 | (* |
---|
301 | Section TRANSF_PROGRAM. |
---|
302 | |
---|
303 | Variable A B V: Type. |
---|
304 | Variable transf: A -> B. |
---|
305 | *) |
---|
306 | |
---|
307 | definition transf_program : ∀A,B. (A → B) → list (ident × A) → list (ident × B) ≝ |
---|
308 | λA,B,transf,l. |
---|
309 | map ?? (λid_fn. 〈fst ?? id_fn, transf (snd ?? id_fn)〉) l. |
---|
310 | |
---|
311 | definition transform_program : ∀A,B,V. ∀p:program A V. (A (prog_var_names … p) → B (prog_var_names … p)) → program B V ≝ |
---|
312 | λA,B,V,p,transf. |
---|
313 | mk_program B V |
---|
314 | (prog_vars A V p) |
---|
315 | (transf_program ?? transf (prog_funct A V p)) |
---|
316 | (prog_main A V p). |
---|
317 | (* |
---|
318 | lemma transform_program_function: |
---|
319 | ∀A,B,V,transf,p,i,tf. |
---|
320 | in_list ? 〈i, tf〉 (prog_funct ?? (transform_program A B V transf p)) → |
---|
321 | ∃f. in_list ? 〈i, f〉 (prog_funct ?? p) ∧ transf f = tf. |
---|
322 | normalize; #A #B #V #transf #p #i #tf #H elim (list_in_map_inv ????? H); |
---|
323 | #x elim x; #i' #tf' *; #e #H destruct; %{tf'} /2/; |
---|
324 | qed. |
---|
325 | *) |
---|
326 | (* |
---|
327 | End TRANSF_PROGRAM. |
---|
328 | |
---|
329 | (** The following is a variant of [transform_program] where the |
---|
330 | code transformation function can fail and therefore returns an |
---|
331 | option type. *) |
---|
332 | |
---|
333 | Open Local Scope error_monad_scope. |
---|
334 | Open Local Scope string_scope. |
---|
335 | |
---|
336 | Section MAP_PARTIAL. |
---|
337 | |
---|
338 | Variable A B C: Type. |
---|
339 | Variable prefix_errmsg: A -> errmsg. |
---|
340 | Variable f: B -> res C. |
---|
341 | *) |
---|
342 | definition map_partial : ∀A,B,C:Type[0]. (B → res C) → |
---|
343 | list (A × B) → res (list (A × C)) ≝ |
---|
344 | λA,B,C,f. mmap ?? (λab. let 〈a,b〉 ≝ ab in do c ← f b; OK ? 〈a,c〉). |
---|
345 | |
---|
346 | lemma map_partial_preserves_first: |
---|
347 | ∀A,B,C:Type[0]. ∀f: B → res C. ∀l: list (A × B). ∀l': list (A × C). |
---|
348 | map_partial … f l = OK ? l' → |
---|
349 | map … \fst l = map … \fst l'. |
---|
350 | #A #B #C #f #l elim l |
---|
351 | [ #l' #E normalize in E; destruct % |
---|
352 | | * #a #b #tl #IH #l' normalize in ⊢ (??%? → ?) cases (f b) normalize in ⊢ (? → ??%? → ?) |
---|
353 | [2: #err #E destruct |
---|
354 | | #c change with (match map_partial … f tl with [ OK x ⇒ ? | Error y ⇒ ?] = ? → ?) |
---|
355 | cases (map_partial … f tl) in IH ⊢ % |
---|
356 | #x #IH normalize in ⊢ (??%? → ?) #ABS destruct normalize |
---|
357 | >(IH x) // ]] |
---|
358 | qed. |
---|
359 | |
---|
360 | (* |
---|
361 | Fixpoint map_partial (l: list (A * B)) : res (list (A * C)) := |
---|
362 | match l with |
---|
363 | | nil => OK nil |
---|
364 | | (a, b) :: rem => |
---|
365 | match f b with |
---|
366 | | Error msg => Error (prefix_errmsg a ++ msg)%list |
---|
367 | | OK c => |
---|
368 | do rem' <- map_partial rem; |
---|
369 | OK ((a, c) :: rem') |
---|
370 | end |
---|
371 | end. |
---|
372 | |
---|
373 | Remark In_map_partial: |
---|
374 | forall l l' a c, |
---|
375 | map_partial l = OK l' -> |
---|
376 | In (a, c) l' -> |
---|
377 | exists b, In (a, b) l /\ f b = OK c. |
---|
378 | Proof. |
---|
379 | induction l; simpl. |
---|
380 | intros. inv H. elim H0. |
---|
381 | intros until c. destruct a as [a1 b1]. |
---|
382 | caseEq (f b1); try congruence. |
---|
383 | intro c1; intros. monadInv H0. |
---|
384 | elim H1; intro. inv H0. exists b1; auto. |
---|
385 | exploit IHl; eauto. intros [b [P Q]]. exists b; auto. |
---|
386 | Qed. |
---|
387 | |
---|
388 | Remark map_partial_forall2: |
---|
389 | forall l l', |
---|
390 | map_partial l = OK l' -> |
---|
391 | list_forall2 |
---|
392 | (fun (a_b: A * B) (a_c: A * C) => |
---|
393 | fst a_b = fst a_c /\ f (snd a_b) = OK (snd a_c)) |
---|
394 | l l'. |
---|
395 | Proof. |
---|
396 | induction l; simpl. |
---|
397 | intros. inv H. constructor. |
---|
398 | intro l'. destruct a as [a b]. |
---|
399 | caseEq (f b). 2: congruence. intro c; intros. monadInv H0. |
---|
400 | constructor. simpl. auto. auto. |
---|
401 | Qed. |
---|
402 | |
---|
403 | End MAP_PARTIAL. |
---|
404 | |
---|
405 | Remark map_partial_total: |
---|
406 | forall (A B C: Type) (prefix: A -> errmsg) (f: B -> C) (l: list (A * B)), |
---|
407 | map_partial prefix (fun b => OK (f b)) l = |
---|
408 | OK (List.map (fun a_b => (fst a_b, f (snd a_b))) l). |
---|
409 | Proof. |
---|
410 | induction l; simpl. |
---|
411 | auto. |
---|
412 | destruct a as [a1 b1]. rewrite IHl. reflexivity. |
---|
413 | Qed. |
---|
414 | |
---|
415 | Remark map_partial_identity: |
---|
416 | forall (A B: Type) (prefix: A -> errmsg) (l: list (A * B)),cmp |
---|
417 | map_partial prefix (fun b => OK b) l = OK l. |
---|
418 | Proof. |
---|
419 | induction l; simpl. |
---|
420 | auto. |
---|
421 | destruct a as [a1 b1]. rewrite IHl. reflexivity. |
---|
422 | Qed. |
---|
423 | |
---|
424 | Section TRANSF_PARTIAL_PROGRAM. |
---|
425 | |
---|
426 | Variable A B V: Type. |
---|
427 | Variable transf_partial: A -> res B. |
---|
428 | |
---|
429 | Definition prefix_funct_name (id: ident) : errmsg := |
---|
430 | MSG "In function " :: CTX id :: MSG ": " :: nil. |
---|
431 | *) |
---|
432 | definition transform_partial_program : ∀A,B,V. ∀p:program A V. (A (prog_var_names … p) → res (B (prog_var_names … p))) → res (program B V) ≝ |
---|
433 | λA,B,V,p,transf_partial. |
---|
434 | do fl ← map_partial … transf_partial (prog_funct … p); |
---|
435 | OK (program B V) (mk_program … (prog_vars … p) fl (prog_main ?? p)). |
---|
436 | |
---|
437 | (* |
---|
438 | Lemma transform_partial_program_function: |
---|
439 | forall p tp i tf, |
---|
440 | transform_partial_program p = OK tp -> |
---|
441 | In (i, tf) tp.(prog_funct) -> |
---|
442 | exists f, In (i, f) p.(prog_funct) /\ transf_partial f = OK tf. |
---|
443 | Proof. |
---|
444 | intros. monadInv H. simpl in H0. |
---|
445 | eapply In_map_partial; eauto. |
---|
446 | Qed. |
---|
447 | |
---|
448 | Lemma transform_partial_program_main: |
---|
449 | forall p tp, |
---|
450 | transform_partial_program p = OK tp -> |
---|
451 | tp.(prog_main) = p.(prog_main). |
---|
452 | Proof. |
---|
453 | intros. monadInv H. reflexivity. |
---|
454 | Qed. |
---|
455 | |
---|
456 | Lemma transform_partial_program_vars: |
---|
457 | forall p tp, |
---|
458 | transform_partial_program p = OK tp -> |
---|
459 | tp.(prog_vars) = p.(prog_vars). |
---|
460 | Proof. |
---|
461 | intros. monadInv H. reflexivity. |
---|
462 | Qed. |
---|
463 | |
---|
464 | End TRANSF_PARTIAL_PROGRAM. |
---|
465 | |
---|
466 | (** The following is a variant of [transform_program_partial] where |
---|
467 | both the program functions and the additional variable information |
---|
468 | are transformed by functions that can fail. *) |
---|
469 | |
---|
470 | Section TRANSF_PARTIAL_PROGRAM2. |
---|
471 | |
---|
472 | Variable A B V W: Type. |
---|
473 | Variable transf_partial_function: A -> res B. |
---|
474 | Variable transf_partial_variable: V -> res W. |
---|
475 | |
---|
476 | Definition prefix_var_name (id_init: ident * list init_data) : errmsg := |
---|
477 | MSG "In global variable " :: CTX (fst id_init) :: MSG ": " :: nil. |
---|
478 | *) |
---|
479 | |
---|
480 | (* CSC: ad hoc lemma, move away? *) |
---|
481 | lemma map_fst: |
---|
482 | ∀A,B,C,C':Type[0].∀l:list (A × B × C).∀l':list (A × B × C'). |
---|
483 | map … \fst l = map … \fst l' → |
---|
484 | map … (λx. \fst (\fst x)) l = map … (λx. \fst (\fst x)) l'. |
---|
485 | #A #B #C #C' #l elim l |
---|
486 | [ #l' elim l' // #he #tl #IH #ABS normalize in ABS; destruct |
---|
487 | | #he1 #tl1 #IH #l' cases l' [ #ABS normalize in ABS; destruct ] |
---|
488 | #he2 #tl2 #EQ whd in EQ:(??%%) ⊢ (??%%) >(IH tl2) destruct normalize in e1 ⊢ % >e0 // |
---|
489 | >e0 in e1; normalize #H @H ] |
---|
490 | qed. |
---|
491 | |
---|
492 | definition transform_partial_program2 : |
---|
493 | ∀A,B,V,W. ∀p: program A V. |
---|
494 | (A (prog_var_names … p) → res (B (prog_var_names ?? p))) |
---|
495 | → (V → res W) → res (program B W) ≝ |
---|
496 | λA,B,V,W,p, transf_partial_function, transf_partial_variable. |
---|
497 | do fl ← map_partial … (*prefix_funct_name*) transf_partial_function (prog_funct ?? p); ?. |
---|
498 | (*CSC: interactive mode because of dependent types *) |
---|
499 | generalize in match (map_partial_preserves_first … transf_partial_variable (prog_vars … p)) |
---|
500 | cases (map_partial … transf_partial_variable (prog_vars … p)) |
---|
501 | [ #vl #EQ |
---|
502 | @(OK (program B W) (mk_program … vl … (prog_main … p))) |
---|
503 | <(map_fst … (EQ vl (refl …))) @fl |
---|
504 | | #err #_ @(Error … err)] |
---|
505 | qed. |
---|
506 | |
---|
507 | (* |
---|
508 | Lemma transform_partial_program2_function: |
---|
509 | forall p tp i tf, |
---|
510 | transform_partial_program2 p = OK tp -> |
---|
511 | In (i, tf) tp.(prog_funct) -> |
---|
512 | exists f, In (i, f) p.(prog_funct) /\ transf_partial_function f = OK tf. |
---|
513 | Proof. |
---|
514 | intros. monadInv H. |
---|
515 | eapply In_map_partial; eauto. |
---|
516 | Qed. |
---|
517 | |
---|
518 | Lemma transform_partial_program2_variable: |
---|
519 | forall p tp i tv, |
---|
520 | transform_partial_program2 p = OK tp -> |
---|
521 | In (i, tv) tp.(prog_vars) -> |
---|
522 | exists v, In (i, v) p.(prog_vars) /\ transf_partial_variable v = OK tv. |
---|
523 | Proof. |
---|
524 | intros. monadInv H. |
---|
525 | eapply In_map_partial; eauto. |
---|
526 | Qed. |
---|
527 | |
---|
528 | Lemma transform_partial_program2_main: |
---|
529 | forall p tp, |
---|
530 | transform_partial_program2 p = OK tp -> |
---|
531 | tp.(prog_main) = p.(prog_main). |
---|
532 | Proof. |
---|
533 | intros. monadInv H. reflexivity. |
---|
534 | Qed. |
---|
535 | |
---|
536 | End TRANSF_PARTIAL_PROGRAM2. |
---|
537 | |
---|
538 | (** The following is a relational presentation of |
---|
539 | [transform_program_partial2]. Given relations between function |
---|
540 | definitions and between variable information, it defines a relation |
---|
541 | between programs stating that the two programs have the same shape |
---|
542 | (same global names, etc) and that identically-named function definitions |
---|
543 | are variable information are related. *) |
---|
544 | |
---|
545 | Section MATCH_PROGRAM. |
---|
546 | |
---|
547 | Variable A B V W: Type. |
---|
548 | Variable match_fundef: A -> B -> Prop. |
---|
549 | Variable match_varinfo: V -> W -> Prop. |
---|
550 | |
---|
551 | Definition match_funct_entry (x1: ident * A) (x2: ident * B) := |
---|
552 | match x1, x2 with |
---|
553 | | (id1, fn1), (id2, fn2) => id1 = id2 /\ match_fundef fn1 fn2 |
---|
554 | end. |
---|
555 | |
---|
556 | Definition match_var_entry (x1: ident * list init_data * V) (x2: ident * list init_data * W) := |
---|
557 | match x1, x2 with |
---|
558 | | (id1, init1, info1), (id2, init2, info2) => id1 = id2 /\ init1 = init2 /\ match_varinfo info1 info2 |
---|
559 | end. |
---|
560 | |
---|
561 | Definition match_program (p1: program A V) (p2: program B W) : Prop := |
---|
562 | list_forall2 match_funct_entry p1.(prog_funct) p2.(prog_funct) |
---|
563 | /\ p1.(prog_main) = p2.(prog_main) |
---|
564 | /\ list_forall2 match_var_entry p1.(prog_vars) p2.(prog_vars). |
---|
565 | |
---|
566 | End MATCH_PROGRAM. |
---|
567 | |
---|
568 | Remark transform_partial_program2_match: |
---|
569 | forall (A B V W: Type) |
---|
570 | (transf_partial_function: A -> res B) |
---|
571 | (transf_partial_variable: V -> res W) |
---|
572 | (p: program A V) (tp: program B W), |
---|
573 | transform_partial_program2 transf_partial_function transf_partial_variable p = OK tp -> |
---|
574 | match_program |
---|
575 | (fun fd tfd => transf_partial_function fd = OK tfd) |
---|
576 | (fun info tinfo => transf_partial_variable info = OK tinfo) |
---|
577 | p tp. |
---|
578 | Proof. |
---|
579 | intros. monadInv H. split. |
---|
580 | apply list_forall2_imply with |
---|
581 | (fun (ab: ident * A) (ac: ident * B) => |
---|
582 | fst ab = fst ac /\ transf_partial_function (snd ab) = OK (snd ac)). |
---|
583 | eapply map_partial_forall2. eauto. |
---|
584 | intros. destruct v1; destruct v2; simpl in *. auto. |
---|
585 | split. auto. |
---|
586 | apply list_forall2_imply with |
---|
587 | (fun (ab: ident * list init_data * V) (ac: ident * list init_data * W) => |
---|
588 | fst ab = fst ac /\ transf_partial_variable (snd ab) = OK (snd ac)). |
---|
589 | eapply map_partial_forall2. eauto. |
---|
590 | intros. destruct v1; destruct v2; simpl in *. destruct p0; destruct p1. intuition congruence. |
---|
591 | Qed. |
---|
592 | *) |
---|
593 | (* * * External functions *) |
---|
594 | |
---|
595 | (* * For most languages, the functions composing the program are either |
---|
596 | internal functions, defined within the language, or external functions |
---|
597 | (a.k.a. system calls) that emit an event when applied. We define |
---|
598 | a type for such functions and some generic transformation functions. *) |
---|
599 | |
---|
600 | record external_function : Type[0] ≝ { |
---|
601 | ef_id: ident; |
---|
602 | ef_sig: signature |
---|
603 | }. |
---|
604 | |
---|
605 | definition ExternalFunction ≝ external_function. |
---|
606 | definition external_function_tag ≝ ef_id. |
---|
607 | definition external_function_sig ≝ ef_sig. |
---|
608 | |
---|
609 | inductive fundef (F: Type[0]): Type[0] ≝ |
---|
610 | | Internal: F → fundef F |
---|
611 | | External: external_function → fundef F. |
---|
612 | |
---|
613 | (* Implicit Arguments External [F]. *) |
---|
614 | (* |
---|
615 | Section TRANSF_FUNDEF. |
---|
616 | |
---|
617 | Variable A B: Type. |
---|
618 | Variable transf: A -> B. |
---|
619 | *) |
---|
620 | definition transf_fundef : ∀A,B. (A→B) → fundef A → fundef B ≝ |
---|
621 | λA,B,transf,fd. |
---|
622 | match fd with |
---|
623 | [ Internal f ⇒ Internal ? (transf f) |
---|
624 | | External ef ⇒ External ? ef |
---|
625 | ]. |
---|
626 | |
---|
627 | (* |
---|
628 | End TRANSF_FUNDEF. |
---|
629 | |
---|
630 | Section TRANSF_PARTIAL_FUNDEF. |
---|
631 | |
---|
632 | Variable A B: Type. |
---|
633 | Variable transf_partial: A -> res B. |
---|
634 | *) |
---|
635 | |
---|
636 | definition transf_partial_fundef : ∀A,B. (A → res B) → fundef A → res (fundef B) ≝ |
---|
637 | λA,B,transf_partial,fd. |
---|
638 | match fd with |
---|
639 | [ Internal f ⇒ do f' ← transf_partial f; OK ? (Internal ? f') |
---|
640 | | External ef ⇒ OK ? (External ? ef) |
---|
641 | ]. |
---|
642 | (* |
---|
643 | End TRANSF_PARTIAL_FUNDEF. |
---|
644 | *) |
---|
645 | |
---|
646 | |
---|
647 | |
---|
648 | (* Partially merged stuff derived from the prototype cerco compiler. *) |
---|
649 | |
---|
650 | (* |
---|
651 | definition bool_to_Prop ≝ |
---|
652 | λb. match b with [ true ⇒ True | false ⇒ False ]. |
---|
653 | |
---|
654 | coercion bool_to_Prop: ∀b:bool. Prop ≝ bool_to_Prop on _b:bool to Type[0]. |
---|
655 | *) |
---|
656 | |
---|
657 | (* dpm: should go to standard library *) |
---|
658 | let rec member (i: ident) (eq_i: ident → ident → bool) |
---|
659 | (g: list ident) on g: Prop ≝ |
---|
660 | match g with |
---|
661 | [ nil ⇒ False |
---|
662 | | cons hd tl ⇒ |
---|
663 | bool_to_Prop (eq_i hd i) ∨ member i eq_i tl |
---|
664 | ]. |
---|
665 | |
---|
666 | (* TODO: merge in comparison definitions from Integers.ma. *) |
---|
667 | inductive Compare: Type[0] ≝ |
---|
668 | Compare_Equal: Compare |
---|
669 | | Compare_NotEqual: Compare |
---|
670 | | Compare_Less: Compare |
---|
671 | | Compare_Greater: Compare |
---|
672 | | Compare_LessEqual: Compare |
---|
673 | | Compare_GreaterEqual: Compare. |
---|
674 | |
---|
675 | inductive op1: Type[0] ≝ |
---|
676 | | op_cast: nat → signedness → nat → op1 |
---|
677 | | op_neg_int: op1 (**r integer opposite *) |
---|
678 | | op_not_bool: op1 (**r boolean negation *) |
---|
679 | | op_not_int: op1 (**r bitwise complement *) |
---|
680 | | op_id: op1 (**r identity *) |
---|
681 | | op_ptr_of_int: op1 (**r int to pointer *) |
---|
682 | | op_int_of_ptr: op1. (**r pointer to int *) |
---|
683 | |
---|
684 | inductive op2: Type[0] ≝ |
---|
685 | | op_add: op2 (**r integer addition *) |
---|
686 | | op_sub: op2 (**r integer subtraction *) |
---|
687 | | op_mul: op2 (**r integer multiplication *) |
---|
688 | | op_div: op2 (**r integer signed division *) |
---|
689 | | op_divu: op2 (**r integer unsigned division *) |
---|
690 | | op_mod: op2 (**r integer signed modulus *) |
---|
691 | | op_modu: op2 (**r integer unsigned modulus *) |
---|
692 | | op_and: op2 (**r bitwise ``and'' *) |
---|
693 | | op_or: op2 (**r bitwise ``or'' *) |
---|
694 | | op_xor: op2 (**r bitwise ``xor'' *) |
---|
695 | | op_shl: op2 (**r left shift *) |
---|
696 | | op_shr: op2 (**r right signed shift *) |
---|
697 | | op_shru: op2 (**r right unsigned shift *) |
---|
698 | | op_cmp: Compare → op2 (**r integer signed comparison *) |
---|
699 | | op_cmpu: Compare → op2 (**r integer unsigned comparison *) |
---|
700 | | op_addp: op2 (**r addition for a pointer and an integer *) |
---|
701 | | op_subp: op2 (**r substraction for a pointer and a integer *) |
---|
702 | | op_subpp: op2 |
---|
703 | | op_cmpp: Compare → op2. (**r pointer comparaison *) |
---|