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 include "formal_topology/cprop_connectives.ma".
17 notation "hvbox(a break = \sub \ID b)" non associative with precedence 45
20 notation > "hvbox(a break =_\ID b)" non associative with precedence 45
21 for @{ cic:/matita/logic/equality/eq.ind#xpointer(1/1) ? $a $b }.
23 interpretation "ID eq" 'eqID x y = (cic:/matita/logic/equality/eq.ind#xpointer(1/1) ? x y).
25 record equivalence_relation (A:Type0) : Type1 ≝
26 { eq_rel:2> A → A → CProp0;
27 refl: reflexive ? eq_rel;
28 sym: symmetric ? eq_rel;
29 trans: transitive ? eq_rel
32 record setoid : Type1 ≝
34 eq: equivalence_relation carr
37 record equivalence_relation1 (A:Type1) : Type2 ≝
38 { eq_rel1:2> A → A → CProp1;
39 refl1: reflexive1 ? eq_rel1;
40 sym1: symmetric1 ? eq_rel1;
41 trans1: transitive1 ? eq_rel1
44 record setoid1: Type2 ≝
46 eq1: equivalence_relation1 carr1
49 definition setoid1_of_setoid: setoid → setoid1.
61 coercion setoid1_of_setoid.
62 prefer coercion Type_OF_setoid.
64 record equivalence_relation2 (A:Type2) : Type3 ≝
65 { eq_rel2:2> A → A → CProp2;
66 refl2: reflexive2 ? eq_rel2;
67 sym2: symmetric2 ? eq_rel2;
68 trans2: transitive2 ? eq_rel2
71 record setoid2: Type3 ≝
73 eq2: equivalence_relation2 carr2
76 definition setoid2_of_setoid1: setoid1 → setoid2.
88 coercion setoid2_of_setoid1.
89 prefer coercion Type_OF_setoid2.
90 prefer coercion Type_OF_setoid.
91 prefer coercion Type_OF_setoid1.
92 (* we prefer 0 < 1 < 2 *)
94 record equivalence_relation3 (A:Type3) : Type4 ≝
95 { eq_rel3:2> A → A → CProp3;
96 refl3: reflexive3 ? eq_rel3;
97 sym3: symmetric3 ? eq_rel3;
98 trans3: transitive3 ? eq_rel3
101 record setoid3: Type4 ≝
103 eq3: equivalence_relation3 carr3
106 interpretation "setoid3 eq" 'eq t x y = (eq_rel3 ? (eq3 t) x y).
107 interpretation "setoid2 eq" 'eq t x y = (eq_rel2 ? (eq2 t) x y).
108 interpretation "setoid1 eq" 'eq t x y = (eq_rel1 ? (eq1 t) x y).
109 interpretation "setoid eq" 'eq t x y = (eq_rel ? (eq t) x y).
111 notation > "hvbox(a break =_12 b)" non associative with precedence 45
112 for @{ eq_rel2 (carr2 (setoid2_of_setoid1 ?)) (eq2 (setoid2_of_setoid1 ?)) $a $b }.
113 notation > "hvbox(a break =_0 b)" non associative with precedence 45
114 for @{ eq_rel ? (eq ?) $a $b }.
115 notation > "hvbox(a break =_1 b)" non associative with precedence 45
116 for @{ eq_rel1 ? (eq1 ?) $a $b }.
117 notation > "hvbox(a break =_2 b)" non associative with precedence 45
118 for @{ eq_rel2 ? (eq2 ?) $a $b }.
119 notation > "hvbox(a break =_3 b)" non associative with precedence 45
120 for @{ eq_rel3 ? (eq3 ?) $a $b }.
122 interpretation "setoid3 symmetry" 'invert r = (sym3 ???? r).
123 interpretation "setoid2 symmetry" 'invert r = (sym2 ???? r).
124 interpretation "setoid1 symmetry" 'invert r = (sym1 ???? r).
125 interpretation "setoid symmetry" 'invert r = (sym ???? r).
126 notation ".= r" with precedence 50 for @{'trans $r}.
127 interpretation "trans3" 'trans r = (trans3 ????? r).
128 interpretation "trans2" 'trans r = (trans2 ????? r).
129 interpretation "trans1" 'trans r = (trans1 ????? r).
130 interpretation "trans" 'trans r = (trans ????? r).
132 record unary_morphism (A,B: setoid) : Type0 ≝
134 prop1: ∀a,a'. eq ? a a' → eq ? (fun1 a) (fun1 a')
137 record unary_morphism1 (A,B: setoid1) : Type1 ≝
139 prop11: ∀a,a'. eq1 ? a a' → eq1 ? (fun11 a) (fun11 a')
142 record unary_morphism2 (A,B: setoid2) : Type2 ≝
144 prop12: ∀a,a'. eq2 ? a a' → eq2 ? (fun12 a) (fun12 a')
147 record unary_morphism3 (A,B: setoid3) : Type3 ≝
149 prop13: ∀a,a'. eq3 ? a a' → eq3 ? (fun13 a) (fun13 a')
152 record binary_morphism (A,B,C:setoid) : Type0 ≝
154 prop2: ∀a,a',b,b'. eq ? a a' → eq ? b b' → eq ? (fun2 a b) (fun2 a' b')
157 record binary_morphism1 (A,B,C:setoid1) : Type1 ≝
158 { fun21:2> A → B → C;
159 prop21: ∀a,a',b,b'. eq1 ? a a' → eq1 ? b b' → eq1 ? (fun21 a b) (fun21 a' b')
162 record binary_morphism2 (A,B,C:setoid2) : Type2 ≝
163 { fun22:2> A → B → C;
164 prop22: ∀a,a',b,b'. eq2 ? a a' → eq2 ? b b' → eq2 ? (fun22 a b) (fun22 a' b')
167 record binary_morphism3 (A,B,C:setoid3) : Type3 ≝
168 { fun23:2> A → B → C;
169 prop23: ∀a,a',b,b'. eq3 ? a a' → eq3 ? b b' → eq3 ? (fun23 a b) (fun23 a' b')
172 notation "† c" with precedence 90 for @{'prop1 $c }.
173 notation "l ‡ r" with precedence 90 for @{'prop2 $l $r }.
174 notation "#" with precedence 90 for @{'refl}.
175 interpretation "prop1" 'prop1 c = (prop1 ????? c).
176 interpretation "prop11" 'prop1 c = (prop11 ????? c).
177 interpretation "prop12" 'prop1 c = (prop12 ????? c).
178 interpretation "prop13" 'prop1 c = (prop13 ????? c).
179 interpretation "prop2" 'prop2 l r = (prop2 ???????? l r).
180 interpretation "prop21" 'prop2 l r = (prop21 ???????? l r).
181 interpretation "prop22" 'prop2 l r = (prop22 ???????? l r).
182 interpretation "prop23" 'prop2 l r = (prop23 ???????? l r).
183 interpretation "refl" 'refl = (refl ???).
184 interpretation "refl1" 'refl = (refl1 ???).
185 interpretation "refl2" 'refl = (refl2 ???).
186 interpretation "refl3" 'refl = (refl3 ???).
188 notation > "A × term 74 B ⇒ term 19 C" non associative with precedence 72 for @{'binary_morphism0 $A $B $C}.
189 notation > "A × term 74 B ⇒_1 term 19 C" non associative with precedence 72 for @{'binary_morphism1 $A $B $C}.
190 notation > "A × term 74 B ⇒_2 term 19 C" non associative with precedence 72 for @{'binary_morphism2 $A $B $C}.
191 notation > "A × term 74 B ⇒_3 term 19 C" non associative with precedence 72 for @{'binary_morphism3 $A $B $C}.
192 notation > "B ⇒_1 C" right associative with precedence 72 for @{'arrows1_SET $B $C}.
193 notation > "B ⇒_1. C" right associative with precedence 72 for @{'arrows1_SETlow $B $C}.
194 notation > "B ⇒_2 C" right associative with precedence 72 for @{'arrows2_SET1 $B $C}.
195 notation > "B ⇒_2. C" right associative with precedence 72 for @{'arrows2_SET1low $B $C}.
197 notation "A × term 74 B ⇒ term 19 C" non associative with precedence 72 for @{'binary_morphism0 $A $B $C}.
198 notation "A × term 74 B ⇒\sub 1 term 19 C" non associative with precedence 72 for @{'binary_morphism1 $A $B $C}.
199 notation "A × term 74 B ⇒\sub 2 term 19 C" non associative with precedence 72 for @{'binary_morphism2 $A $B $C}.
200 notation "A × term 74 B ⇒\sub 3 term 19 C" non associative with precedence 72 for @{'binary_morphism3 $A $B $C}.
201 notation "B ⇒\sub 1 C" right associative with precedence 72 for @{'arrows1_SET $B $C}.
202 notation "B ⇒\sub 2 C" right associative with precedence 72 for @{'arrows2_SET1 $B $C}.
203 notation "B ⇒\sub 1. C" right associative with precedence 72 for @{'arrows1_SETlow $B $C}.
204 notation "B ⇒\sub 2. C" right associative with precedence 72 for @{'arrows2_SET1low $B $C}.
206 interpretation "'binary_morphism0" 'binary_morphism0 A B C = (binary_morphism A B C).
207 interpretation "'arrows2_SET1 low" 'arrows2_SET1 A B = (unary_morphism2 A B).
208 interpretation "'arrows2_SET1low" 'arrows2_SET1low A B = (unary_morphism2 A B).
209 interpretation "'binary_morphism1" 'binary_morphism1 A B C = (binary_morphism1 A B C).
210 interpretation "'binary_morphism2" 'binary_morphism2 A B C = (binary_morphism2 A B C).
211 interpretation "'binary_morphism3" 'binary_morphism3 A B C = (binary_morphism3 A B C).
212 interpretation "'arrows1_SET low" 'arrows1_SET A B = (unary_morphism1 A B).
213 interpretation "'arrows1_SETlow" 'arrows1_SETlow A B = (unary_morphism1 A B).
216 definition unary_morphism2_of_unary_morphism1:
217 ∀S,T:setoid1.unary_morphism1 S T → unary_morphism2 (setoid2_of_setoid1 S) T.
220 [ apply (fun11 ?? u);
221 | apply (prop11 ?? u); ]
224 definition CPROP: setoid1.
229 | intros 1; split; intro; assumption
230 | intros 3; cases i; split; assumption
231 | intros 5; cases i; cases i1; split; intro;
232 [ apply (f2 (f x1)) | apply (f1 (f3 z1))]]]
235 definition CProp0_of_CPROP: carr1 CPROP → CProp0 ≝ λx.x.
236 coercion CProp0_of_CPROP.
238 alias symbol "eq" = "setoid1 eq".
239 definition fi': ∀A,B:CPROP. A = B → B → A.
240 intros; apply (fi ?? e); assumption.
243 notation ". r" with precedence 50 for @{'fi $r}.
244 interpretation "fi" 'fi r = (fi' ?? r).
246 definition and_morphism: binary_morphism1 CPROP CPROP CPROP.
249 | intros; split; intro; cases a1; split;
251 | apply (if ?? e1 b1)
253 | apply (fi ?? e1 b1)]]
256 interpretation "and_morphism" 'and a b = (fun21 ??? and_morphism a b).
258 definition or_morphism: binary_morphism1 CPROP CPROP CPROP.
261 | intros; split; intro; cases o; [1,3:left |2,4: right]
264 | apply (if ?? e1 b1)
265 | apply (fi ?? e1 b1)]]
268 interpretation "or_morphism" 'or a b = (fun21 ??? or_morphism a b).
270 definition if_morphism: binary_morphism1 CPROP CPROP CPROP.
272 [ apply (λA,B. A → B)
273 | intros; split; intros;
274 [ apply (if ?? e1); apply f; apply (fi ?? e); assumption
275 | apply (fi ?? e1); apply f; apply (if ?? e); assumption]]
278 notation > "hvbox(a break ∘ b)" left associative with precedence 55 for @{ comp ??? $a $b }.
279 record category : Type1 ≝ {
281 arrows: objs → objs → setoid;
282 id: ∀o:objs. arrows o o;
283 comp: ∀o1,o2,o3. (arrows o1 o2) × (arrows o2 o3) ⇒ (arrows o1 o3);
284 comp_assoc: ∀o1,o2,o3,o4.∀a12:arrows o1 ?.∀a23:arrows o2 ?.∀a34:arrows o3 o4.
285 (a12 ∘ a23) ∘ a34 =_0 a12 ∘ (a23 ∘ a34);
286 id_neutral_left : ∀o1,o2. ∀a: arrows o1 o2. (id o1) ∘ a =_0 a;
287 id_neutral_right: ∀o1,o2. ∀a: arrows o1 o2. a ∘ (id o2) =_0 a
289 notation "hvbox(a break ∘ b)" left associative with precedence 55 for @{ 'compose $a $b }.
291 record category1 : Type2 ≝
293 arrows1: objs1 → objs1 → setoid1;
294 id1: ∀o:objs1. arrows1 o o;
295 comp1: ∀o1,o2,o3. binary_morphism1 (arrows1 o1 o2) (arrows1 o2 o3) (arrows1 o1 o3);
296 comp_assoc1: ∀o1,o2,o3,o4. ∀a12,a23,a34.
297 comp1 o1 o3 o4 (comp1 o1 o2 o3 a12 a23) a34 =_1 comp1 o1 o2 o4 a12 (comp1 o2 o3 o4 a23 a34);
298 id_neutral_right1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? (id1 o1) a =_1 a;
299 id_neutral_left1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? a (id1 o2) =_1 a
302 record category2 : Type3 ≝
304 arrows2: objs2 → objs2 → setoid2;
305 id2: ∀o:objs2. arrows2 o o;
306 comp2: ∀o1,o2,o3. binary_morphism2 (arrows2 o1 o2) (arrows2 o2 o3) (arrows2 o1 o3);
307 comp_assoc2: ∀o1,o2,o3,o4. ∀a12,a23,a34.
308 comp2 o1 o3 o4 (comp2 o1 o2 o3 a12 a23) a34 =_2 comp2 o1 o2 o4 a12 (comp2 o2 o3 o4 a23 a34);
309 id_neutral_right2: ∀o1,o2. ∀a: arrows2 o1 o2. comp2 ??? (id2 o1) a =_2 a;
310 id_neutral_left2: ∀o1,o2. ∀a: arrows2 o1 o2. comp2 ??? a (id2 o2) =_2 a
313 record category3 : Type4 ≝
315 arrows3: objs3 → objs3 → setoid3;
316 id3: ∀o:objs3. arrows3 o o;
317 comp3: ∀o1,o2,o3. binary_morphism3 (arrows3 o1 o2) (arrows3 o2 o3) (arrows3 o1 o3);
318 comp_assoc3: ∀o1,o2,o3,o4. ∀a12,a23,a34.
319 comp3 o1 o3 o4 (comp3 o1 o2 o3 a12 a23) a34 =_3 comp3 o1 o2 o4 a12 (comp3 o2 o3 o4 a23 a34);
320 id_neutral_right3: ∀o1,o2. ∀a: arrows3 o1 o2. comp3 ??? (id3 o1) a =_3 a;
321 id_neutral_left3: ∀o1,o2. ∀a: arrows3 o1 o2. comp3 ??? a (id3 o2) =_3 a
324 notation "'ASSOC'" with precedence 90 for @{'assoc}.
326 interpretation "category2 composition" 'compose x y = (fun22 ??? (comp2 ????) y x).
327 interpretation "category2 assoc" 'assoc = (comp_assoc2 ????????).
328 interpretation "category1 composition" 'compose x y = (fun21 ??? (comp1 ????) y x).
329 interpretation "category1 assoc" 'assoc = (comp_assoc1 ????????).
330 interpretation "category composition" 'compose x y = (fun2 ??? (comp ????) y x).
331 interpretation "category assoc" 'assoc = (comp_assoc ????????).
333 definition category2_of_category1: category1 → category2.
337 | intros; apply (setoid2_of_setoid1 (arrows1 c o o1));
341 [ intros; apply (comp1 c o1 o2 o3 c1 c2);
342 | intros; unfold setoid2_of_setoid1 in e e1 a a' b b'; simplify in e e1 a a' b b';
343 change with ((b∘a) =_1 (b'∘a')); apply (e‡e1); ]
344 | intros; simplify; whd in a12 a23 a34; whd; apply rule (ASSOC);
345 | intros; simplify; whd in a; whd; apply id_neutral_right1;
346 | intros; simplify; whd in a; whd; apply id_neutral_left1; ]
348 (*coercion category2_of_category1.*)
350 record functor2 (C1: category2) (C2: category2) : Type3 ≝
351 { map_objs2:1> C1 → C2;
352 map_arrows2: ∀S,T. unary_morphism2 (arrows2 ? S T) (arrows2 ? (map_objs2 S) (map_objs2 T));
353 respects_id2: ∀o:C1. map_arrows2 ?? (id2 ? o) = id2 ? (map_objs2 o);
355 ∀o1,o2,o3.∀f1:arrows2 ? o1 o2.∀f2:arrows2 ? o2 o3.
356 map_arrows2 ?? (f2 ∘ f1) = map_arrows2 ?? f2 ∘ map_arrows2 ?? f1}.
358 notation > "F⎽⇒ x" left associative with precedence 60 for @{'map_arrows2 $F $x}.
359 notation "F\sub⇒ x" left associative with precedence 60 for @{'map_arrows2 $F $x}.
360 interpretation "map_arrows2" 'map_arrows2 F x = (fun12 ?? (map_arrows2 ?? F ??) x).
362 definition functor2_setoid: category2 → category2 → setoid3.
365 [ apply (functor2 C1 C2);
368 apply (∀c:C1. cic:/matita/logic/equality/eq.ind#xpointer(1/1) ? (f c) (g c));
369 | simplify; intros; apply cic:/matita/logic/equality/eq.ind#xpointer(1/1/1);
370 | simplify; intros; apply cic:/matita/logic/equality/sym_eq.con; apply H;
371 | simplify; intros; apply cic:/matita/logic/equality/trans_eq.con;
372 [2: apply H; | skip | apply H1;]]]
375 definition functor2_of_functor2_setoid: ∀S,T. functor2_setoid S T → functor2 S T ≝ λS,T,x.x.
376 coercion functor2_of_functor2_setoid.
378 definition CAT2: category3.
381 | apply functor2_setoid;
382 | intros; constructor 1;
384 | intros; constructor 1;
386 | intros; assumption;]
387 | intros; apply rule #;
388 | intros; apply rule #; ]
389 | intros; constructor 1;
390 [ intros; constructor 1;
391 [ intros; apply (c1 (c o));
392 | intros; constructor 1;
393 [ intro; apply (map_arrows2 ?? c1 ?? (map_arrows2 ?? c ?? c2));
394 | intros; apply (††e); ]
396 apply (.= †(respects_id2 : ?));
397 apply (respects_id2 : ?);
399 apply (.= †(respects_comp2 : ?));
400 apply (respects_comp2 : ?); ]
401 | intros; intro; simplify;
402 apply (cic:/matita/logic/equality/eq_ind.con ????? (e ?));
403 apply (cic:/matita/logic/equality/eq_ind.con ????? (e1 ?));
405 | intros; intro; simplify; constructor 1;
406 | intros; intro; simplify; constructor 1;
407 | intros; intro; simplify; constructor 1; ]
410 definition category2_of_objs3_CAT2: objs3 CAT2 → category2 ≝ λx.x.
411 coercion category2_of_objs3_CAT2.
413 definition functor2_setoid_of_arrows3_CAT2: ∀S,T. arrows3 CAT2 S T → functor2_setoid S T ≝ λS,T,x.x.
414 coercion functor2_setoid_of_arrows3_CAT2.
416 notation > "B ⇒_\c3 C" right associative with precedence 72 for @{'arrows3_CAT $B $C}.
417 notation "B ⇒\sub (\c 3) C" right associative with precedence 72 for @{'arrows3_CAT $B $C}.
418 interpretation "'arrows3_CAT" 'arrows3_CAT A B = (arrows3 CAT2 A B).
420 definition unary_morphism_setoid: setoid → setoid → setoid.
423 [ apply (unary_morphism s s1);
425 [ intros (f g); apply (∀a:s. eq ? (f a) (g a));
426 | intros 1; simplify; intros; apply refl;
427 | simplify; intros; apply sym; apply f;
428 | simplify; intros; apply trans; [2: apply f; | skip | apply f1]]]
431 definition SET: category1.
434 | apply rule (λS,T:setoid.setoid1_of_setoid (unary_morphism_setoid S T));
435 | intros; constructor 1; [ apply (λx:carr o.x); | intros; assumption ]
436 | intros; constructor 1; [ intros; constructor 1; [ apply (λx. c1 (c x)); | intros;
438 | intros; whd; intros; simplify; whd in H1; whd in H;
439 apply trans; [ apply (b (a' a1)); | lapply (prop1 ?? b (a a1) (a' a1));
440 [ apply Hletin | apply (e a1); ] | apply e1; ]]
441 | intros; whd; intros; simplify; apply refl;
442 | intros; simplify; whd; intros; simplify; apply refl;
443 | intros; simplify; whd; intros; simplify; apply refl;
447 definition setoid_of_SET: objs1 SET → setoid ≝ λx.x.
448 coercion setoid_of_SET.
450 definition unary_morphism_setoid_of_arrows1_SET:
451 ∀P,Q.arrows1 SET P Q → unary_morphism_setoid P Q ≝ λP,Q,x.x.
452 coercion unary_morphism_setoid_of_arrows1_SET.
454 interpretation "'arrows1_SET" 'arrows1_SET A B = (arrows1 SET A B).
456 definition unary_morphism1_setoid1: setoid1 → setoid1 → setoid1.
459 [ apply (unary_morphism1 s s1);
462 alias symbol "eq" = "setoid1 eq".
463 apply (∀a: carr1 s. f a = g a);
464 | intros 1; simplify; intros; apply refl1;
465 | simplify; intros; apply sym1; apply f;
466 | simplify; intros; apply trans1; [2: apply f; | skip | apply f1]]]
469 definition unary_morphism1_of_unary_morphism1_setoid1 :
470 ∀S,T. unary_morphism1_setoid1 S T → unary_morphism1 S T ≝ λP,Q,x.x.
471 coercion unary_morphism1_of_unary_morphism1_setoid1.
473 definition SET1: objs3 CAT2.
476 | apply rule (λS,T.setoid2_of_setoid1 (unary_morphism1_setoid1 S T));
477 | intros; constructor 1; [ apply (λx.x); | intros; assumption ]
478 | intros; constructor 1; [ intros; constructor 1; [ apply (λx. c1 (c x)); | intros;
480 | intros; whd; intros; simplify; whd in H1; whd in H;
481 apply trans1; [ apply (b (a' a1)); | lapply (prop11 ?? b (a a1) (a' a1));
482 [ apply Hletin | apply (e a1); ] | apply e1; ]]
483 | intros; whd; intros; simplify; apply refl1;
484 | intros; simplify; whd; intros; simplify; apply refl1;
485 | intros; simplify; whd; intros; simplify; apply refl1;
489 interpretation "'arrows2_SET1" 'arrows2_SET1 A B = (arrows2 SET1 A B).
491 definition setoid1_of_SET1: objs2 SET1 → setoid1 ≝ λx.x.
492 coercion setoid1_of_SET1.
494 definition unary_morphism1_setoid1_of_arrows2_SET1:
495 ∀P,Q.arrows2 SET1 P Q → unary_morphism1_setoid1 P Q ≝ λP,Q,x.x.
496 coercion unary_morphism1_setoid1_of_arrows2_SET1.
498 variant objs2_of_category1: objs1 SET → objs2 SET1 ≝ setoid1_of_setoid.
499 coercion objs2_of_category1.
501 prefer coercion Type_OF_setoid. (* we prefer the lower carrier projection *)
502 prefer coercion Type_OF_objs1.
504 alias symbol "exists" (instance 1) = "CProp2 exists".
506 λA,B:CAT2.λF:carr3 (arrows3 CAT2 A B).
507 ∀o1,o2:A.∀f.∃g:arrows2 A o1 o2.F⎽⇒ g =_2 f.
508 alias symbol "exists" (instance 1) = "CProp exists".
510 definition faithful2 ≝
511 λA,B:CAT2.λF:carr3 (arrows3 CAT2 A B).
512 ∀o1,o2:A.∀f,g:arrows2 A o1 o2.F⎽⇒ f =_2 F⎽⇒ g → f =_2 g.
515 notation "r \sup *" non associative with precedence 90 for @{'OR_f_star $r}.
516 notation > "r *" non associative with precedence 90 for @{'OR_f_star $r}.
518 notation "r \sup (⎻* )" non associative with precedence 90 for @{'OR_f_minus_star $r}.
519 notation > "r⎻*" non associative with precedence 90 for @{'OR_f_minus_star $r}.
521 notation "r \sup ⎻" non associative with precedence 90 for @{'OR_f_minus $r}.
522 notation > "r⎻" non associative with precedence 90 for @{'OR_f_minus $r}.