1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
17 (* Grammar ******************************************************************)
20 non associative with precedence 55
23 notation "hvbox( ⓪ { term 46 I } )"
24 non associative with precedence 55
28 non associative with precedence 46
31 notation "hvbox( ⋆ term 90 k )"
32 non associative with precedence 55
35 notation "hvbox( # term 90 i )"
36 non associative with precedence 55
39 notation "hvbox( § term 90 p )"
40 non associative with precedence 55
43 notation "hvbox( ② term 55 T1 . break term 55 T )"
44 non associative with precedence 55
45 for @{ 'SnItem2 $T1 $T }.
47 notation "hvbox( ② { term 46 I } break term 55 T1 . break term 55 T )"
48 non associative with precedence 55
49 for @{ 'SnItem2 $I $T1 $T }.
51 notation "hvbox( ⓑ { term 46 a , term 46 I } break term 55 T1 . break term 55 T )"
52 non associative with precedence 55
53 for @{ 'SnBind2 $a $I $T1 $T }.
55 notation "hvbox( + ⓑ { term 46 I } break term 55 T1 . break term 55 T )"
56 non associative with precedence 55
57 for @{ 'SnBind2Pos $I $T1 $T }.
59 notation "hvbox( - ⓑ { term 46 I } break term 55 T1 . break term 55 T )"
60 non associative with precedence 55
61 for @{ 'SnBind2Neg $I $T1 $T }.
63 notation "hvbox( ⓕ { term 46 I } break term 55 T1 . break term 55 T )"
64 non associative with precedence 55
65 for @{ 'SnFlat2 $I $T1 $T }.
67 notation "hvbox( ⓓ { term 46 a } term 55 T1 . break term 55 T2 )"
68 non associative with precedence 55
69 for @{ 'SnAbbr $a $T1 $T2 }.
71 notation "hvbox( + ⓓ term 55 T1 . break term 55 T2 )"
72 non associative with precedence 55
73 for @{ 'SnAbbrPos $T1 $T2 }.
75 notation "hvbox( - ⓓ term 55 T1 . break term 55 T2 )"
76 non associative with precedence 55
77 for @{ 'SnAbbrNeg $T1 $T2 }.
79 notation "hvbox( ⓛ { term 46 a } term 55 T1 . break term 55 T2 )"
80 non associative with precedence 55
81 for @{ 'SnAbst $a $T1 $T2 }.
83 notation "hvbox( + ⓛ term 55 T1 . break term 55 T2 )"
84 non associative with precedence 55
85 for @{ 'SnAbstPos $T1 $T2 }.
87 notation "hvbox( - ⓛ term 55 T1 . break term 55 T2 )"
88 non associative with precedence 55
89 for @{ 'SnAbstNeg $T1 $T2 }.
91 notation "hvbox( ⓐ term 55 T1 . break term 55 T2 )"
92 non associative with precedence 55
93 for @{ 'SnAppl $T1 $T2 }.
95 notation "hvbox( ⓝ term 55 T1 . break term 55 T2 )"
96 non associative with precedence 55
97 for @{ 'SnCast $T1 $T2 }.
99 notation "hvbox( Ⓐ term 55 T1 . break term 55 T )"
100 non associative with precedence 55
101 for @{ 'SnApplV $T1 $T }.
103 notation > "hvbox( T . break ②{ term 46 I } break term 47 T1 )"
104 non associative with precedence 46
105 for @{ 'DxBind2 $T $I $T1 }.
107 notation "hvbox( T . break ⓑ { term 46 I } break term 48 T1 )"
108 non associative with precedence 47
109 for @{ 'DxBind2 $T $I $T1 }.
111 notation "hvbox( T1 . break ⓓ T2 )"
112 left associative with precedence 48
113 for @{ 'DxAbbr $T1 $T2 }.
115 notation "hvbox( T1 . break ⓛ T2 )"
116 left associative with precedence 49
117 for @{ 'DxAbst $T1 $T2 }.
119 notation "hvbox( T . break ④ { term 46 I } break { term 46 T1 , break term 46 T2 , break term 46 T3 } )"
120 non associative with precedence 50
121 for @{ 'DxItem4 $T $I $T1 $T2 $T3 }.
123 notation "hvbox( ♯ { term 46 x } )"
124 non associative with precedence 90
127 notation "hvbox( ♯ { term 46 x , break term 46 y } )"
128 non associative with precedence 90
129 for @{ 'Weight $x $y }.
131 notation "hvbox( 𝐒 ⦃ term 46 T ⦄ )"
132 non associative with precedence 45
135 notation "hvbox( L ⊢ break term 46 T1 ≈ break term 46 T2 )"
136 non associative with precedence 45
137 for @{ 'Hom $L $T1 $T2 }.
139 notation "hvbox( T1 ≃ break term 46 T2 )"
140 non associative with precedence 45
141 for @{ 'Iso $T1 $T2 }.
143 (* Substitution *************************************************************)
145 notation "hvbox( ⇧ [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )"
146 non associative with precedence 45
147 for @{ 'RLift $d $e $T1 $T2 }.
149 notation "hvbox( L1 break ⊑ [ term 46 d , break term 46 e ] break term 46 L2 )"
150 non associative with precedence 45
151 for @{ 'SubEq $L1 $d $e $L2 }.
153 notation "hvbox( ⊒ [ term 46 d , break term 46 e ] break term 46 L2 )"
154 non associative with precedence 45
155 for @{ 'SubEqBottom $d $e $L2 }.
157 notation "hvbox( ⇩ [ term 46 e ] break term 46 L1 ≡ break term 46 L2 )"
158 non associative with precedence 45
159 for @{ 'RDrop $e $L1 $L2 }.
161 notation "hvbox( ⇩ [ term 46 d , break term 46 e ] break term 46 L1 ≡ break term 46 L2 )"
162 non associative with precedence 45
163 for @{ 'RDrop $d $e $L1 $L2 }.
165 notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⊃ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
166 non associative with precedence 45
167 for @{ 'SupTerm $L1 $T1 $L2 $T2 }.
169 notation "hvbox( L ⊢ break ⌘ ⦃ term 46 T ⦄ ≡ break term 46 k )"
170 non associative with precedence 45
171 for @{ 'ICM $L $T $k }.
173 (* Unfold *******************************************************************)
175 notation "hvbox( @ ⦃ term 46 T1 , break term 46 f ⦄ ≡ break term 46 T2 )"
176 non associative with precedence 45
177 for @{ 'RAt $T1 $f $T2 }.
179 notation "hvbox( T1 ▭ break term 46 T2 ≡ break term 46 T )"
180 non associative with precedence 45
181 for @{ 'RMinus $T1 $T2 $T }.
183 notation "hvbox( ⇧ * [ term 46 e ] break term 46 T1 ≡ break term 46 T2 )"
184 non associative with precedence 45
185 for @{ 'RLiftStar $e $T1 $T2 }.
187 notation "hvbox( ⇩ * [ term 46 e ] break term 46 L1 ≡ break term 46 L2 )"
188 non associative with precedence 45
189 for @{ 'RDropStar $e $L1 $L2 }.
191 notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⊃ + break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
192 non associative with precedence 45
193 for @{ 'SupTermPlus $L1 $T1 $L2 $T2 }.
195 notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⊃ * break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
196 non associative with precedence 45
197 for @{ 'SupTermStar $L1 $T1 $L2 $T2 }.
199 notation "hvbox( L ⊢ break term 46 T1 ▶* break term 46 T2 )"
200 non associative with precedence 45
201 for @{ 'PSubstStar $L $T1 $T2 }.
203 notation "hvbox( T1 ⊢ ▶ * break term 46 T2 )"
204 non associative with precedence 45
205 for @{ 'PSubstStarSn $T1 $T2 }.
207 notation "hvbox( ▼ * [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )"
208 non associative with precedence 45
209 for @{ 'TSubst $T1 $d $e $T2 }.
211 notation "hvbox( L ⊢ break ▼ * [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )"
212 non associative with precedence 45
213 for @{ 'TSubst $L $T1 $d $e $T2 }.
215 notation "hvbox( ▼ ▼ * [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )"
216 non associative with precedence 45
217 for @{ 'TSubstAlt $T1 $d $e $T2 }.
219 notation "hvbox( L ⊢ break ▼ ▼ * [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )"
220 non associative with precedence 45
221 for @{ 'TSubstAlt $L $T1 $d $e $T2 }.
223 (* Static typing ************************************************************)
225 notation "hvbox( L ⊢ break term 46 T ⁝ break term 46 A )"
226 non associative with precedence 45
227 for @{ 'AtomicArity $L $T $A }.
229 notation "hvbox( T1 ⁝ ⊑ break term 46 T2 )"
230 non associative with precedence 45
231 for @{ 'CrSubEqA $T1 $T2 }.
233 notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T ÷ break term 46 A )"
234 non associative with precedence 45
235 for @{ 'BinaryArity $h $L $T $A }.
237 notation "hvbox( h ⊢ break term 46 L1 ÷ ⊑ break term 46 L2 )"
238 non associative with precedence 45
239 for @{ 'CrSubEqB $h $L1 $L2 }.
241 notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 • break [ term 46 g ] break ⦃ term 46 l , break term 46 T2 ⦄ )"
242 non associative with precedence 45
243 for @{ 'StaticType $h $g $L $T1 $T2 $l }.
245 (* Unwind *******************************************************************)
247 notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 •* break [ term 46 g ] break term 46 T2 )"
248 non associative with precedence 45
249 for @{ 'StaticTypeStar $h $g $L $T1 $T2 }.
251 notation "hvbox( L1 ⊢ ⧫ * break term 46 T ≡ break term 46 L2 )"
252 non associative with precedence 45
253 for @{ 'Unwind $L1 $T $L2 }.
255 (* Restricted ***************************************************************)
257 notation "hvbox( L ⊢ break term 46 T1 ➤ * break term 46 T2 )"
258 non associative with precedence 45
259 for @{ 'PRestStar $L $T1 $T2 }.
261 notation "hvbox( T1 ⊢ ➤ * break term 46 T2 )"
262 non associative with precedence 45
263 for @{ 'PRestStarSn $T1 $T2 }.
265 (* Reducibility *************************************************************)
267 notation "hvbox( L ⊢ break 𝐑 ⦃ term 46 T ⦄ )"
268 non associative with precedence 45
269 for @{ 'Reducible $L $T }.
271 notation "hvbox( L ⊢ break 𝐈 ⦃ term 46 T ⦄ )"
272 non associative with precedence 45
273 for @{ 'NotReducible $L $T }.
275 notation "hvbox( L ⊢ break 𝐍 ⦃ term 46 T ⦄ )"
276 non associative with precedence 45
277 for @{ 'Normal $L $T }.
279 (* this might be removed *)
280 notation "hvbox( 𝐇𝐑 ⦃ term 46 T ⦄ )"
281 non associative with precedence 45
282 for @{ 'HdReducible $T }.
284 (* this might be removed *)
285 notation "hvbox( L ⊢ break 𝐇𝐑 ⦃ term 46 T ⦄ )"
286 non associative with precedence 45
287 for @{ 'HdReducible $L $T }.
289 (* this might be removed *)
290 notation "hvbox( 𝐇𝐈 ⦃ term 46 T ⦄ )"
291 non associative with precedence 45
292 for @{ 'NotHdReducible $T }.
294 (* this might be removed *)
295 notation "hvbox( L ⊢ break 𝐇𝐈 ⦃ term 46 T ⦄ )"
296 non associative with precedence 45
297 for @{ 'NotHdReducible $L $T }.
299 (* this might be removed *)
300 notation "hvbox( 𝐇𝐍 ⦃ term 46 T ⦄ )"
301 non associative with precedence 45
302 for @{ 'HdNormal $T }.
304 (* this might be removed *)
305 notation "hvbox( L ⊢ break 𝐇𝐍 ⦃ term 46 T ⦄ )"
306 non associative with precedence 45
307 for @{ 'HdNormal $L $T }.
309 notation "hvbox( L ⊢ break term 46 T1 ➡ break term 46 T2 )"
310 non associative with precedence 45
311 for @{ 'PRed $L $T1 $T2 }.
313 notation "hvbox( L1 ⊢ ➡ break term 46 L2 )"
314 non associative with precedence 45
315 for @{ 'PRedSn $L1 $L2 }.
317 notation "hvbox( L1 ⊢ ➡ ➡ break term 46 L2 )"
318 non associative with precedence 45
319 for @{ 'PRedSnAlt $L1 $L2 }.
321 notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ➡ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
322 non associative with precedence 45
323 for @{ 'FocalizedPRed $L1 $T1 $L2 $T2 }.
325 notation "hvbox( L ⊢ break ⦃ term 46 L1, break term 46 T1 ⦄ ➡ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
326 non associative with precedence 45
327 for @{ 'FocalizedPRed $L $L1 $T1 $L2 $T2 }.
329 (* Computation **************************************************************)
331 notation "hvbox( T1 ➡ * break term 46 T2 )"
332 non associative with precedence 45
333 for @{ 'PRedStar $T1 $T2 }.
335 notation "hvbox( L ⊢ break term 46 T1 ➡ * break term 46 T2 )"
336 non associative with precedence 45
337 for @{ 'PRedStar $L $T1 $T2 }.
339 notation "hvbox( T1 ➡ ➡ * break term 46 T2 )"
340 non associative with precedence 45
341 for @{ 'PRedStarAlt $T1 $T2 }.
343 notation "hvbox( ⦃ term 46 L1 ⦄ ➡ * break ⦃ term 46 L2 ⦄ )"
344 non associative with precedence 45
345 for @{ 'FocalizedPRedStar $L1 $L2 }.
347 notation "hvbox( ⦃ term 46 L1 , term 46 T1 ⦄ ➡ * break ⦃ term 46 L2 , term 46 T2 ⦄ )"
348 non associative with precedence 45
349 for @{ 'FocalizedPRedStar $L1 $T1 $L2 $T2 }.
351 notation "hvbox( ⦃ term 46 L1 ⦄ ➡ ➡ * break ⦃ term 46 L2 ⦄ )"
352 non associative with precedence 45
353 for @{ 'FocalizedPRedStarAlt $L1 $L2 }.
355 notation "hvbox( ⦃ term 46 L1 , term 46 T1 ⦄ ➡ ➡ * break ⦃ term 46 L2 , term 46 T2 ⦄ )"
356 non associative with precedence 45
357 for @{ 'FocalizedPRedStarAlt $L1 $T1 $L2 $T2 }.
359 notation "hvbox( L ⊢ break term 46 T1 ➡ * break 𝐍 ⦃ term 46 T2 ⦄ )"
360 non associative with precedence 45
361 for @{ 'PEval $L $T1 $T2 }.
363 notation "hvbox( ⬊ * term 46 T )"
364 non associative with precedence 45
367 notation "hvbox( L ⊢ ⬊ * break term 46 T )"
368 non associative with precedence 45
371 notation "hvbox( L ⊢ ⬊ ⬊ * break term 46 T )"
372 non associative with precedence 45
373 for @{ 'SNAlt $L $T }.
375 notation "hvbox( ⦃ term 46 L, break term 46 T ⦄ ϵ break [ term 46 R ] break 〚term 46 A 〛 )"
376 non associative with precedence 45
377 for @{ 'InEInt $R $L $T $A }.
379 notation "hvbox( T1 ⊑ break [ term 46 R ] break term 46 T2 )"
380 non associative with precedence 45
381 for @{ 'CrSubEq $T1 $R $T2 }.
383 notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 • * ➡ * break [ term 46 g ] break term 46 T2 )"
384 non associative with precedence 45
385 for @{ 'DecomposedXPRedStar $h $g $L $T1 $T2 }.
387 notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ • * ⬊ * break [ term 46 g ] break term 46 T2 )"
388 non associative with precedence 45
389 for @{ 'DecomposedXSN $h $g $L $T }.
391 (* Conversion ***************************************************************)
393 notation "hvbox( L ⊢ break term 46 T1 ⬌ break term 46 T2 )"
394 non associative with precedence 45
395 for @{ 'PConv $L $T1 $T2 }.
397 notation "hvbox( ⦃ term 46 L1 ⦄ ⬌ break ⦃ term 46 L2 ⦄ )"
398 non associative with precedence 45
399 for @{ 'FocalizedPConv $L1 $L2 }.
401 notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
402 non associative with precedence 45
403 for @{ 'FocalizedPConv $L1 $T1 $L2 $T2 }.
405 notation "hvbox( ⦃ term 46 L1 ⦄ ⬌ ⬌ break ⦃ term 46 L2 ⦄ )"
406 non associative with precedence 45
407 for @{ 'FocalizedPConvAlt $L1 $L2 }.
409 notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ ⬌ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
410 non associative with precedence 45
411 for @{ 'FocalizedPConvAlt $L1 $T1 $L2 $T2 }.
413 (* Equivalence **************************************************************)
415 notation "hvbox( L ⊢ break term 46 T1 ⬌* break term 46 T2 )"
416 non associative with precedence 45
417 for @{ 'PConvStar $L $T1 $T2 }.
419 notation "hvbox( h ⊢ break term 46 L1 • ⊑ break [ term 46 g ] break term 46 L2 )"
420 non associative with precedence 45
421 for @{ 'CrSubEqS $h $g $L1 $L2 }.
423 notation "hvbox( ⦃ term 46 L1 ⦄ ⬌ * break ⦃ term 46 L2 ⦄ )"
424 non associative with precedence 45
425 for @{ 'FocalizedPConvStar $L1 $L2 }.
427 notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ * break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
428 non associative with precedence 45
429 for @{ 'FocalizedPConvStar $L1 $T1 $L2 $T2 }.
431 notation "hvbox( ⦃ term 46 L1 ⦄ ⬌ ⬌ * break ⦃ term 46 L2 ⦄ )"
432 non associative with precedence 45
433 for @{ 'FocalizedPConvStarAlt $L1 $L2 }.
435 notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ ⬌ * break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
436 non associative with precedence 45
437 for @{ 'FocalizedPConvStarAlt $L1 $T1 $L2 $T2 }.
439 (* Dynamic typing ***********************************************************)
441 notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T ¡ break [ term 46 g ] )"
442 non associative with precedence 45
443 for @{ 'NativeValid $h $g $L $T }.
445 notation "hvbox( h ⊢ break term 46 L1 ¡ ⊑ break [ term 46 g ] break term 46 L2 )"
446 non associative with precedence 45
447 for @{ 'CrSubEqV $h $g $L1 $L2 }.
449 notation "hvbox( h ⊢ break ⦃ term 46 L1, break term 46 T1 ⦄ ≽ break [ term 46 g ] break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
450 non associative with precedence 45
451 for @{ 'BTPRed $h $g $L1 $T1 $L2 $T2 }.
453 notation "hvbox( h ⊢ break ⦃ term 46 L1, break term 46 T1 ⦄ ≻ break [ term 46 g ] break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
454 non associative with precedence 45
455 for @{ 'BTPRedProper $h $g $L1 $T1 $L2 $T2 }.
457 notation "hvbox( h ⊢ break ⦃ term 46 L1, break term 46 T1 ⦄ ≥ break [ term 46 g ] break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
458 non associative with precedence 45
459 for @{ 'BTPRedStar $h $g $L1 $T1 $L2 $T2 }.
461 notation "hvbox( h ⊢ break ⦃ term 46 L1, break term 46 T1 ⦄ > break [ term 46 g ] break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
462 non associative with precedence 45
463 for @{ 'BTPRedStarProper $h $g $L1 $T1 $L2 $T2 }.
465 notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 : break term 46 T2 )"
466 non associative with precedence 45
467 for @{ 'NativeType $h $L $T1 $T2 }.
469 notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 : : break term 46 T2 )"
470 non associative with precedence 45
471 for @{ 'NativeTypeAlt $h $L $T1 $T2 }.
473 (* Higher order dynamic typing **********************************************)
475 notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 : * break term 46 T2 )"
476 non associative with precedence 45
477 for @{ 'NativeTypeStar $h $L $T1 $T2 }.