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 , break 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 } break 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 } break 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( ♯ { term 46 x } )"
120 non associative with precedence 90
123 notation "hvbox( ♯ { term 46 x , break term 46 y } )"
124 non associative with precedence 90
125 for @{ 'Weight $x $y }.
127 notation "hvbox( 𝐒 ⦃ term 46 T ⦄ )"
128 non associative with precedence 45
131 notation "hvbox( T1 ≃ break term 46 T2 )"
132 non associative with precedence 45
133 for @{ 'Iso $T1 $T2 }.
135 (* Relocation ***************************************************************)
137 notation "hvbox( ⇧ [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )"
138 non associative with precedence 45
139 for @{ 'RLift $d $e $T1 $T2 }.
141 notation "hvbox( ⇩ [ term 46 e ] break term 46 L1 ≡ break term 46 L2 )"
142 non associative with precedence 45
143 for @{ 'RDrop $e $L1 $L2 }.
145 notation "hvbox( ⇩ [ term 46 d , break term 46 e ] break term 46 L1 ≡ break term 46 L2 )"
146 non associative with precedence 45
147 for @{ 'RDrop $d $e $L1 $L2 }.
149 notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⊃ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
150 non associative with precedence 45
151 for @{ 'SupTerm $L1 $T1 $L2 $T2 }.
153 notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⊃⸮ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
154 non associative with precedence 45
155 for @{ 'SupTermOpt $L1 $T1 $L2 $T2 }.
157 notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⊃⊃⸮ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
158 non associative with precedence 45
159 for @{ 'SupTermOptAlt $L1 $T1 $L2 $T2 }.
161 notation "hvbox( L ⊢ break ⌘ ⦃ term 46 T ⦄ ≡ break term 46 k )"
162 non associative with precedence 45
163 for @{ 'ICM $L $T $k }.
165 (* Substitution *************************************************************)
167 notation "hvbox( @ ⦃ term 46 T1 , break term 46 f ⦄ ≡ break term 46 T2 )"
168 non associative with precedence 45
169 for @{ 'RAt $T1 $f $T2 }.
171 notation "hvbox( T1 ▭ break term 46 T2 ≡ break term 46 T )"
172 non associative with precedence 45
173 for @{ 'RMinus $T1 $T2 $T }.
175 notation "hvbox( ⇧ * [ term 46 e ] break term 46 T1 ≡ break term 46 T2 )"
176 non associative with precedence 45
177 for @{ 'RLiftStar $e $T1 $T2 }.
179 notation "hvbox( ⇩ * [ term 46 e ] break term 46 L1 ≡ break term 46 L2 )"
180 non associative with precedence 45
181 for @{ 'RDropStar $e $L1 $L2 }.
183 notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⊃ + break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
184 non associative with precedence 45
185 for @{ 'SupTermPlus $L1 $T1 $L2 $T2 }.
187 notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⊃ * break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
188 non associative with precedence 45
189 for @{ 'SupTermStar $L1 $T1 $L2 $T2 }.
191 notation "hvbox( L1 ⊑ break term 46 L2 )"
192 non associative with precedence 45
193 for @{ 'SubEq $L1 $L2 }.
195 notation "hvbox( L ⊢ break term 46 T1 ▶* break term 46 T2 )"
196 non associative with precedence 45
197 for @{ 'PSubstStar $L $T1 $T2 }.
199 notation "hvbox( T1 ⊢ ▶ * break term 46 T2 )"
200 non associative with precedence 45
201 for @{ 'PSubstStarSn $T1 $T2 }.
203 (* Static typing ************************************************************)
205 notation "hvbox( L ⊢ break term 46 T ⁝ break term 46 A )"
206 non associative with precedence 45
207 for @{ 'AtomicArity $L $T $A }.
209 notation "hvbox( T1 ⁝ ⊑ break term 46 T2 )"
210 non associative with precedence 45
211 for @{ 'CrSubEqA $T1 $T2 }.
213 notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T ÷ break term 46 A )"
214 non associative with precedence 45
215 for @{ 'BinaryArity $h $L $T $A }.
217 notation "hvbox( h ⊢ break term 46 L1 ÷ ⊑ break term 46 L2 )"
218 non associative with precedence 45
219 for @{ 'CrSubEqB $h $L1 $L2 }.
221 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 ⦄ )"
222 non associative with precedence 45
223 for @{ 'StaticType $h $g $L $T1 $T2 $l }.
225 (* Unfold *******************************************************************)
227 notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 •* break [ term 46 g ] break term 46 T2 )"
228 non associative with precedence 45
229 for @{ 'StaticTypeStar $h $g $L $T1 $T2 }.
231 notation "hvbox( L1 ⊢ ⧫ * break term 46 T ≡ break term 46 L2 )"
232 non associative with precedence 45
233 for @{ 'Unfold $L1 $T $L2 }.
235 notation "hvbox( L ⊢ break term 46 T1 ➤ * break term 46 T2 )"
236 non associative with precedence 45
237 for @{ 'PRestStar $L $T1 $T2 }.
239 notation "hvbox( T1 ⊢ ➤ * break term 46 T2 )"
240 non associative with precedence 45
241 for @{ 'PRestStarSn $T1 $T2 }.
243 (* Reduction ****************************************************************)
245 notation "hvbox( L ⊢ break 𝐑 ⦃ term 46 T ⦄ )"
246 non associative with precedence 45
247 for @{ 'Reducible $L $T }.
249 notation "hvbox( L ⊢ break 𝐈 ⦃ term 46 T ⦄ )"
250 non associative with precedence 45
251 for @{ 'NotReducible $L $T }.
253 notation "hvbox( L ⊢ break 𝐍 ⦃ term 46 T ⦄ )"
254 non associative with precedence 45
255 for @{ 'Normal $L $T }.
257 notation "hvbox( L ⊢ break term 46 T1 ➡ break term 46 T2 )"
258 non associative with precedence 45
259 for @{ 'PRed $L $T1 $T2 }.
261 notation "hvbox( ⦃ term 46 h, break term 46 L ⦄ ⊢ break term 46 T1 ➡ break [ term 46 g ] break term 46 T2 )"
262 non associative with precedence 45
263 for @{ 'PRed $h $g $L $T1 $T2 }.
265 notation "hvbox( L1 ⊢ ➡ break term 46 L2 )"
266 non associative with precedence 45
267 for @{ 'PRedSn $L1 $L2 }.
269 notation "hvbox( ⦃ term 46 h, break term 46 L1 ⦄ ⊢ ➡ break [ term 46 g ] break term 46 L2 )"
270 non associative with precedence 45
271 for @{ 'PRedSn $h $g $L1 $L2 }.
273 (* Computation **************************************************************)
275 notation "hvbox( L ⊢ break term 46 T1 ➡ * break term 46 T2 )"
276 non associative with precedence 45
277 for @{ 'PRedStar $L $T1 $T2 }.
279 notation "hvbox( L1 ⊢ ➡* break term 46 L2 )"
280 non associative with precedence 45
281 for @{ 'PRedSnStar $L1 $L2 }.
283 notation "hvbox( L1 ⊢ ➡➡* break term 46 L2 )"
284 non associative with precedence 45
285 for @{ 'PRedSnStarAlt $L1 $L2 }.
287 notation "hvbox( L ⊢ break term 46 T1 ➡ * break 𝐍 ⦃ term 46 T2 ⦄ )"
288 non associative with precedence 45
289 for @{ 'PEval $L $T1 $T2 }.
291 notation "hvbox( L ⊢ ⬊ * break term 46 T )"
292 non associative with precedence 45
295 notation "hvbox( L ⊢ ⬊ ⬊ * break term 46 T )"
296 non associative with precedence 45
297 for @{ 'SNAlt $L $T }.
299 notation "hvbox( ⦃ term 46 h, break term 46 L ⦄ ⊢ break term 46 T1 ➡ * break [ term 46 g ] break term 46 T2 )"
300 non associative with precedence 45
301 for @{ 'PRedStar $h $g $L $T1 $T2 }.
303 notation "hvbox( ⦃ term 46 h, break term 46 L ⦄ ⊢ ⬊ * break [ term 46 g ] break term 46 T )"
304 non associative with precedence 45
305 for @{ 'SN $h $g $L $T }.
307 notation "hvbox( ⦃ term 46 h, break term 46 L ⦄ ⊢ ⬊ ⬊ * break [ term 46 g ] break term 46 T )"
308 non associative with precedence 45
309 for @{ 'SNAlt $h $g $L $T }.
311 notation "hvbox( ⦃ term 46 L, break term 46 T ⦄ ϵ break [ term 46 R ] break 〚term 46 A 〛 )"
312 non associative with precedence 45
313 for @{ 'InEInt $R $L $T $A }.
315 notation "hvbox( T1 ⊑ break [ term 46 R ] break term 46 T2 )"
316 non associative with precedence 45
317 for @{ 'CrSubEq $T1 $R $T2 }.
319 notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 • * ➡ * break [ term 46 g ] break term 46 T2 )"
320 non associative with precedence 45
321 for @{ 'DecomposedXPRedStar $h $g $L $T1 $T2 }.
323 notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ • * ⬊ * break [ term 46 g ] break term 46 T2 )"
324 non associative with precedence 45
325 for @{ 'DecomposedXSN $h $g $L $T }.
327 (* Conversion ***************************************************************)
329 notation "hvbox( L ⊢ break term 46 T1 ⬌ break term 46 T2 )"
330 non associative with precedence 45
331 for @{ 'PConv $L $T1 $T2 }.
333 notation "hvbox( L1 ⊢ ⬌ break term 46 L2 )"
334 non associative with precedence 45
335 for @{ 'PConvSn $L1 $L2 }.
337 (* Equivalence **************************************************************)
339 notation "hvbox( L ⊢ break term 46 T1 ⬌* break term 46 T2 )"
340 non associative with precedence 45
341 for @{ 'PConvStar $L $T1 $T2 }.
343 notation "hvbox( h ⊢ break term 46 L1 • ⊑ break [ term 46 g ] break term 46 L2 )"
344 non associative with precedence 45
345 for @{ 'CrSubEqS $h $g $L1 $L2 }.
347 notation "hvbox( L1 ⊢ ⬌* break term 46 L2 )"
348 non associative with precedence 45
349 for @{ 'PConvSnStar $L1 $L2 }.
351 (* Dynamic typing ***********************************************************)
353 notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T ¡ break [ term 46 g ] )"
354 non associative with precedence 45
355 for @{ 'NativeValid $h $g $L $T }.
357 notation "hvbox( h ⊢ break term 46 L1 ¡ ⊑ break [ term 46 g ] break term 46 L2 )"
358 non associative with precedence 45
359 for @{ 'CrSubEqV $h $g $L1 $L2 }.
361 notation "hvbox( h ⊢ break ⦃ term 46 L1, break term 46 T1 ⦄ ≽ break [ term 46 g ] break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
362 non associative with precedence 45
363 for @{ 'BTPRed $h $g $L1 $T1 $L2 $T2 }.
365 notation "hvbox( h ⊢ break ⦃ term 46 L1, break term 46 T1 ⦄ ≻ break [ term 46 g ] break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
366 non associative with precedence 45
367 for @{ 'BTPRedProper $h $g $L1 $T1 $L2 $T2 }.
369 notation "hvbox( h ⊢ break ⦃ term 46 L1, break term 46 T1 ⦄ ≥ break [ term 46 g ] break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
370 non associative with precedence 45
371 for @{ 'BTPRedStar $h $g $L1 $T1 $L2 $T2 }.
373 notation "hvbox( h ⊢ break ⦃ term 46 L1, break term 46 T1 ⦄ > break [ term 46 g ] break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
374 non associative with precedence 45
375 for @{ 'BTPRedStarProper $h $g $L1 $T1 $L2 $T2 }.
377 notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 : break term 46 T2 )"
378 non associative with precedence 45
379 for @{ 'NativeType $h $L $T1 $T2 }.
381 notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 : : break term 46 T2 )"
382 non associative with precedence 45
383 for @{ 'NativeTypeAlt $h $L $T1 $T2 }.
385 (* Higher order dynamic typing **********************************************)
387 notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 : * break term 46 T2 )"
388 non associative with precedence 45
389 for @{ 'NativeTypeStar $h $L $T1 $T2 }.