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".
16 include "basics/core_notation/compose_2.ma".
18 inductive eq (A:Type[0]) (x:A) : A → CProp[0] ≝
21 notation "hvbox(a break = \sub \ID b)" non associative with precedence 45
24 notation > "hvbox(a break =_\ID b)" non associative with precedence 45
25 for @{ 'eqID ? $a $b }.
27 interpretation "ID eq" 'eqID x y = (eq ? x y).
29 record equivalence_relation (A:Type[0]) : Type[1] ≝
30 { eq_rel:2> A → A → CProp[0];
31 refl: reflexive ? eq_rel;
32 sym: symmetric ? eq_rel;
33 trans: transitive ? eq_rel
36 record setoid : Type[1] ≝
38 eq: equivalence_relation carr
41 record equivalence_relation1 (A:Type[1]) : Type[2] ≝
42 { eq_rel1:2> A → A → CProp[1];
43 refl1: reflexive1 ? eq_rel1;
44 sym1: symmetric1 ? eq_rel1;
45 trans1: transitive1 ? eq_rel1
48 record setoid1: Type[2] ≝
50 eq1: equivalence_relation1 carr1
53 definition setoid1_of_setoid: setoid → setoid1.
57 | @mk_equivalence_relation1
65 coercion setoid1_of_setoid.
67 prefer coercion Type_OF_setoid.
69 record equivalence_relation2 (A:Type[2]) : Type[3] ≝
70 { eq_rel2:2> A → A → CProp[2];
71 refl2: reflexive2 ? eq_rel2;
72 sym2: symmetric2 ? eq_rel2;
73 trans2: transitive2 ? eq_rel2
76 record setoid2: Type[3] ≝
78 eq2: equivalence_relation2 carr2
81 definition setoid2_of_setoid1: setoid1 → setoid2.
85 | @mk_equivalence_relation2
93 coercion setoid2_of_setoid1.
95 prefer coercion Type_OF_setoid2.
96 prefer coercion Type_OF_setoid.
97 prefer coercion Type_OF_setoid1.
98 (* we prefer 0 < 1 < 2 *)
100 record equivalence_relation3 (A:Type[3]) : Type[4] ≝
101 { eq_rel3:2> A → A → CProp[3];
102 refl3: reflexive3 ? eq_rel3;
103 sym3: symmetric3 ? eq_rel3;
104 trans3: transitive3 ? eq_rel3
107 record setoid3: Type[4] ≝
109 eq3: equivalence_relation3 carr3
112 interpretation "setoid3 eq" 'eq t x y = (eq_rel3 ? (eq3 t) x y).
113 interpretation "setoid2 eq" 'eq t x y = (eq_rel2 ? (eq2 t) x y).
114 interpretation "setoid1 eq" 'eq t x y = (eq_rel1 ? (eq1 t) x y).
115 interpretation "setoid eq" 'eq t x y = (eq_rel ? (eq t) x y).
117 notation > "hvbox(a break =_12 b)" non associative with precedence 45
118 for @{ eq_rel2 (carr2 (setoid2_of_setoid1 ?)) (eq2 (setoid2_of_setoid1 ?)) $a $b }.
119 notation > "hvbox(a break =_0 b)" non associative with precedence 45
120 for @{ eq_rel ? (eq ?) $a $b }.
121 notation > "hvbox(a break =_1 b)" non associative with precedence 45
122 for @{ eq_rel1 ? (eq1 ?) $a $b }.
123 notation > "hvbox(a break =_2 b)" non associative with precedence 45
124 for @{ eq_rel2 ? (eq2 ?) $a $b }.
125 notation > "hvbox(a break =_3 b)" non associative with precedence 45
126 for @{ eq_rel3 ? (eq3 ?) $a $b }.
128 interpretation "setoid3 symmetry" 'invert r = (sym3 ???? r).
129 interpretation "setoid2 symmetry" 'invert r = (sym2 ???? r).
130 interpretation "setoid1 symmetry" 'invert r = (sym1 ???? r).
131 interpretation "setoid symmetry" 'invert r = (sym ???? r).
132 notation ".= r" with precedence 55 for @{'trans $r}.
133 interpretation "trans3" 'trans r = (trans3 ????? r).
134 interpretation "trans2" 'trans r = (trans2 ????? r).
135 interpretation "trans1" 'trans r = (trans1 ????? r).
136 interpretation "trans" 'trans r = (trans ????? r).
138 record unary_morphism (A,B: setoid) : Type[0] ≝
140 prop1: ∀a,a'. eq ? a a' → eq ? (fun1 a) (fun1 a')
143 record unary_morphism1 (A,B: setoid1) : Type[1] ≝
145 prop11: ∀a,a'. eq1 ? a a' → eq1 ? (fun11 a) (fun11 a')
148 record unary_morphism2 (A,B: setoid2) : Type[2] ≝
150 prop12: ∀a,a'. eq2 ? a a' → eq2 ? (fun12 a) (fun12 a')
153 record unary_morphism3 (A,B: setoid3) : Type[3] ≝
155 prop13: ∀a,a'. eq3 ? a a' → eq3 ? (fun13 a) (fun13 a')
158 record binary_morphism (A,B,C:setoid) : Type[0] ≝
160 prop2: ∀a,a',b,b'. eq ? a a' → eq ? b b' → eq ? (fun2 a b) (fun2 a' b')
163 record binary_morphism1 (A,B,C:setoid1) : Type[1] ≝
164 { fun21:2> A → B → C;
165 prop21: ∀a,a',b,b'. eq1 ? a a' → eq1 ? b b' → eq1 ? (fun21 a b) (fun21 a' b')
168 record binary_morphism2 (A,B,C:setoid2) : Type[2] ≝
169 { fun22:2> A → B → C;
170 prop22: ∀a,a',b,b'. eq2 ? a a' → eq2 ? b b' → eq2 ? (fun22 a b) (fun22 a' b')
173 record binary_morphism3 (A,B,C:setoid3) : Type[3] ≝
174 { fun23:2> A → B → C;
175 prop23: ∀a,a',b,b'. eq3 ? a a' → eq3 ? b b' → eq3 ? (fun23 a b) (fun23 a' b')
178 notation "† c" with precedence 90 for @{'prop1 $c }.
179 notation "l ‡ r" with precedence 90 for @{'prop2 $l $r }.
180 notation "#" with precedence 90 for @{'refl}.
181 interpretation "prop1" 'prop1 c = (prop1 ????? c).
182 interpretation "prop11" 'prop1 c = (prop11 ????? c).
183 interpretation "prop12" 'prop1 c = (prop12 ????? c).
184 interpretation "prop13" 'prop1 c = (prop13 ????? c).
185 interpretation "prop2" 'prop2 l r = (prop2 ???????? l r).
186 interpretation "prop21" 'prop2 l r = (prop21 ???????? l r).
187 interpretation "prop22" 'prop2 l r = (prop22 ???????? l r).
188 interpretation "prop23" 'prop2 l r = (prop23 ???????? l r).
189 interpretation "refl" 'refl = (refl ???).
190 interpretation "refl1" 'refl = (refl1 ???).
191 interpretation "refl2" 'refl = (refl2 ???).
192 interpretation "refl3" 'refl = (refl3 ???).
194 notation > "A × term 74 B ⇒ term 19 C" non associative with precedence 72 for @{'binary_morphism0 $A $B $C}.
195 notation > "A × term 74 B ⇒_1 term 19 C" non associative with precedence 72 for @{'binary_morphism1 $A $B $C}.
196 notation > "A × term 74 B ⇒_2 term 19 C" non associative with precedence 72 for @{'binary_morphism2 $A $B $C}.
197 notation > "A × term 74 B ⇒_3 term 19 C" non associative with precedence 72 for @{'binary_morphism3 $A $B $C}.
198 notation > "B ⇒_1 C" right associative with precedence 72 for @{'arrows1_SET $B $C}.
199 notation > "B ⇒_1. C" right associative with precedence 72 for @{'arrows1_SETlow $B $C}.
200 notation > "B ⇒_2 C" right associative with precedence 72 for @{'arrows2_SET1 $B $C}.
201 notation > "B ⇒_2. C" right associative with precedence 72 for @{'arrows2_SET1low $B $C}.
203 notation "A × term 74 B ⇒\sub 1 term 19 C" non associative with precedence 72 for @{'binary_morphism1 $A $B $C}.
204 notation "A × term 74 B ⇒\sub 2 term 19 C" non associative with precedence 72 for @{'binary_morphism2 $A $B $C}.
205 notation "A × term 74 B ⇒\sub 3 term 19 C" non associative with precedence 72 for @{'binary_morphism3 $A $B $C}.
206 notation "B ⇒\sub 1 C" right associative with precedence 72 for @{'arrows1_SET $B $C}.
207 notation "B ⇒\sub 2 C" right associative with precedence 72 for @{'arrows2_SET1 $B $C}.
208 notation "B ⇒\sub 1. C" right associative with precedence 72 for @{'arrows1_SETlow $B $C}.
209 notation "B ⇒\sub 2. C" right associative with precedence 72 for @{'arrows2_SET1low $B $C}.
211 interpretation "'binary_morphism0" 'binary_morphism0 A B C = (binary_morphism A B C).
212 interpretation "'arrows2_SET1 low" 'arrows2_SET1 A B = (unary_morphism2 A B).
213 interpretation "'arrows2_SET1low" 'arrows2_SET1low A B = (unary_morphism2 A B).
214 interpretation "'binary_morphism1" 'binary_morphism1 A B C = (binary_morphism1 A B C).
215 interpretation "'binary_morphism2" 'binary_morphism2 A B C = (binary_morphism2 A B C).
216 interpretation "'binary_morphism3" 'binary_morphism3 A B C = (binary_morphism3 A B C).
217 interpretation "'arrows1_SET low" 'arrows1_SET A B = (unary_morphism1 A B).
218 interpretation "'arrows1_SETlow" 'arrows1_SETlow A B = (unary_morphism1 A B).
221 definition unary_morphism2_of_unary_morphism1:
222 ∀S,T:setoid1.unary_morphism1 S T → unary_morphism2 (setoid2_of_setoid1 S) T.
229 definition CPROP: setoid1.
232 | @mk_equivalence_relation1
234 | #x @mk_Iff #x1 assumption
235 | #x #y #i cases i #f #f1 @mk_Iff assumption
236 | #x #y #z #i1 #i cases i cases i1 #f #f1 #f2 #f3 @mk_Iff #x1
237 [ @(f2 (f x1)) | @(f1 (f3 x1))]]]
240 definition CProp0_of_CPROP: carr1 CPROP → CProp[0] ≝ λx.x.
241 coercion CProp0_of_CPROP.
243 alias symbol "eq" = "setoid1 eq".
244 definition fi': ∀A,B:CPROP. A = B → B → A.
245 #A #B #e #b @(fi ?? e) assumption.
248 notation ". r" with precedence 55 for @{'fi $r}.
249 interpretation "fi" 'fi r = (fi' ?? r).
251 definition and_morphism: binary_morphism1 CPROP CPROP CPROP.
254 | #a #a' #b #b' #e #e1 @mk_Iff #a1 cases a1 #a2 #b1 @Conj
261 interpretation "and_morphism" 'and a b = (fun21 ??? and_morphism a b).
263 definition or_morphism: binary_morphism1 CPROP CPROP CPROP.
266 | #a #a' #b #b' #e #e1 @mk_Iff #o cases o #a1 [1,3: @Left |2,4: @Right]
273 interpretation "or_morphism" 'or a b = (fun21 ??? or_morphism a b).
275 definition if_morphism: binary_morphism1 CPROP CPROP CPROP.
278 | #a #a' #b #b' #e #e1 @mk_Iff #f #a1
279 [ @(if ?? e1) @f @(fi ?? e) assumption
280 | @(fi ?? e1) @f @(if ?? e) assumption]]
283 notation > "hvbox(a break ∘ b)" left associative with precedence 60 for @{ comp ??? $a $b }.
285 record category : Type[1] ≝ {
287 arrows: objs → objs → setoid;
288 id: ∀o:objs. arrows o o;
289 comp: ∀o1,o2,o3. (arrows o1 o2) × (arrows o2 o3) ⇒ (arrows o1 o3);
290 comp_assoc: ∀o1,o2,o3,o4.∀a12:arrows o1 ?.∀a23:arrows o2 ?.∀a34:arrows o3 o4.
291 comp o1 o3 o4 (comp o1 o2 o3 a12 a23) a34 =_0 comp o1 o2 o4 a12 (comp o2 o3 o4 a23 a34);
292 id_neutral_left : ∀o1,o2. ∀a: arrows o1 o2. comp ??? (id o1) a =_0 a;
293 id_neutral_right: ∀o1,o2. ∀a: arrows o1 o2. comp ??? a (id o2) =_0 a
296 notation "hvbox(a break ∘ b)" left associative with precedence 60 for @{ 'compose $a $b }.
298 record category1 : Type[2] ≝
300 arrows1: objs1 → objs1 → setoid1;
301 id1: ∀o:objs1. arrows1 o o;
302 comp1: ∀o1,o2,o3. binary_morphism1 (arrows1 o1 o2) (arrows1 o2 o3) (arrows1 o1 o3);
303 comp_assoc1: ∀o1,o2,o3,o4. ∀a12,a23,a34.
304 comp1 o1 o3 o4 (comp1 o1 o2 o3 a12 a23) a34 =_1 comp1 o1 o2 o4 a12 (comp1 o2 o3 o4 a23 a34);
305 id_neutral_right1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? (id1 o1) a =_1 a;
306 id_neutral_left1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? a (id1 o2) =_1 a
309 record category2 : Type[3] ≝
311 arrows2: objs2 → objs2 → setoid2;
312 id2: ∀o:objs2. arrows2 o o;
313 comp2: ∀o1,o2,o3. binary_morphism2 (arrows2 o1 o2) (arrows2 o2 o3) (arrows2 o1 o3);
314 comp_assoc2: ∀o1,o2,o3,o4. ∀a12,a23,a34.
315 comp2 o1 o3 o4 (comp2 o1 o2 o3 a12 a23) a34 =_2 comp2 o1 o2 o4 a12 (comp2 o2 o3 o4 a23 a34);
316 id_neutral_right2: ∀o1,o2. ∀a: arrows2 o1 o2. comp2 ??? (id2 o1) a =_2 a;
317 id_neutral_left2: ∀o1,o2. ∀a: arrows2 o1 o2. comp2 ??? a (id2 o2) =_2 a
320 record category3 : Type[4] ≝
322 arrows3: objs3 → objs3 → setoid3;
323 id3: ∀o:objs3. arrows3 o o;
324 comp3: ∀o1,o2,o3. binary_morphism3 (arrows3 o1 o2) (arrows3 o2 o3) (arrows3 o1 o3);
325 comp_assoc3: ∀o1,o2,o3,o4. ∀a12,a23,a34.
326 comp3 o1 o3 o4 (comp3 o1 o2 o3 a12 a23) a34 =_3 comp3 o1 o2 o4 a12 (comp3 o2 o3 o4 a23 a34);
327 id_neutral_right3: ∀o1,o2. ∀a: arrows3 o1 o2. comp3 ??? (id3 o1) a =_3 a;
328 id_neutral_left3: ∀o1,o2. ∀a: arrows3 o1 o2. comp3 ??? a (id3 o2) =_3 a
331 notation "'ASSOC'" with precedence 90 for @{'assoc}.
333 interpretation "category2 composition" 'compose x y = (fun22 ??? (comp2 ????) y x).
334 interpretation "category2 assoc" 'assoc = (comp_assoc2 ????????).
335 interpretation "category1 composition" 'compose x y = (fun21 ??? (comp1 ????) y x).
336 interpretation "category1 assoc" 'assoc = (comp_assoc1 ????????).
337 interpretation "category composition" 'compose x y = (fun2 ??? (comp ????) y x).
338 interpretation "category assoc" 'assoc = (comp_assoc ????????).
340 definition category2_of_category1: category1 → category2.
344 | #o #o1 @(setoid2_of_setoid1 (arrows1 c o o1))
348 [ #c1 #c2 @(comp1 c o1 o2 o3 c1 c2)
349 | #a #a' #b #b' #e #e1 @(e‡e1) ]
350 | #o1 #o2 #o3 #o4 #a12 #a23 #a34 @comp_assoc1
351 | #o1 #o2 #a @id_neutral_right1
352 | #o1 #o2 #a @id_neutral_left1 ]
354 (*coercion category2_of_category1.*)
356 record functor2 (C1: category2) (C2: category2) : Type[3] ≝
357 { map_objs2:1> C1 → C2;
358 map_arrows2: ∀S,T. unary_morphism2 (arrows2 ? S T) (arrows2 ? (map_objs2 S) (map_objs2 T));
359 respects_id2: ∀o:C1. map_arrows2 ?? (id2 ? o) = id2 ? (map_objs2 o);
361 ∀o1,o2,o3.∀f1:arrows2 ? o1 o2.∀f2:arrows2 ? o2 o3.
362 map_arrows2 ?? (f2 ∘ f1) = map_arrows2 ?? f2 ∘ map_arrows2 ?? f1}.
364 notation > "F⎽⇒ x" left associative with precedence 65 for @{'map_arrows2 $F $x}.
365 notation "F\sub⇒ x" left associative with precedence 65 for @{'map_arrows2 $F $x}.
366 interpretation "map_arrows2" 'map_arrows2 F x = (fun12 ?? (map_arrows2 ?? F ??) x).
368 definition functor2_setoid: category2 → category2 → setoid3.
372 | @mk_equivalence_relation3
374 @(∀c:C1. cic:/matita/logic/equality/eq.ind#xpointer(1/1) ? (f c) (g c));
375 | simplify; intros; @cic:/matita/logic/equality/eq.ind#xpointer(1/1/1);
376 | simplify; intros; @cic:/matita/logic/equality/sym_eq.con; @H;
377 | simplify; intros; @cic:/matita/logic/equality/trans_eq.con;
378 [2: @H; | skip | @H1;]]]
381 definition functor2_of_functor2_setoid: ∀S,T. functor2_setoid S T → functor2 S T ≝ λS,T,x.x.
382 coercion functor2_of_functor2_setoid.
384 definition CAT2: category3.
388 | intros; constructor 1;
390 | intros; constructor 1;
392 | intros; assumption;]
395 | intros; constructor 1;
396 [ intros; constructor 1;
397 [ intros; @(c1 (c o));
398 | intros; constructor 1;
399 [ intro; @(map_arrows2 ?? c1 ?? (map_arrows2 ?? c ?? c2));
402 @(.= †(respects_id2 : ?));
405 @(.= †(respects_comp2 : ?));
406 @(respects_comp2 : ?); ]
407 | intros; intro; simplify;
408 @(cic:/matita/logic/equality/eq_ind.con ????? (e ?));
409 @(cic:/matita/logic/equality/eq_ind.con ????? (e1 ?));
411 | intros; intro; simplify; constructor 1;
412 | intros; intro; simplify; constructor 1;
413 | intros; intro; simplify; constructor 1; ]
416 definition category2_of_objs3_CAT2: objs3 CAT2 → category2 ≝ λx.x.
417 coercion category2_of_objs3_CAT2.
419 definition functor2_setoid_of_arrows3_CAT2: ∀S,T. arrows3 CAT2 S T → functor2_setoid S T ≝ λS,T,x.x.
420 coercion functor2_setoid_of_arrows3_CAT2.
422 notation > "B ⇒_\c3 C" right associative with precedence 72 for @{'arrows3_CAT $B $C}.
423 notation "B ⇒\sub (\c 3) C" right associative with precedence 72 for @{'arrows3_CAT $B $C}.
424 interpretation "'arrows3_CAT" 'arrows3_CAT A B = (arrows3 CAT2 A B).
426 definition unary_morphism_setoid: setoid → setoid → setoid.
429 [ @(unary_morphism s s1);
431 [ intros (f g); @(∀a:s. eq ? (f a) (g a));
432 | intros 1; simplify; intros; @refl;
433 | simplify; intros; @sym; @f;
434 | simplify; intros; @trans; [2: @f; | skip | @f1]]]
437 definition SET: category1.
440 | @rule (λS,T:setoid.setoid1_of_setoid (unary_morphism_setoid S T));
441 | intros; constructor 1; [ @(λx:carr o.x); | intros; assumption ]
442 | intros; constructor 1; [ intros; constructor 1; [ @(λx. c1 (c x)); | intros;
444 | intros; whd; intros; simplify; whd in H1; whd in H;
445 @trans; [ @(b (a' a1)); | l@(prop1 ?? b (a a1) (a' a1));
446 [ @Hletin | @(e a1); ] | @e1; ]]
447 | intros; whd; intros; simplify; @refl;
448 | intros; simplify; whd; intros; simplify; @refl;
449 | intros; simplify; whd; intros; simplify; @refl;
453 definition setoid_of_SET: objs1 SET → setoid ≝ λx.x.
454 coercion setoid_of_SET.
456 definition unary_morphism_setoid_of_arrows1_SET:
457 ∀P,Q.arrows1 SET P Q → unary_morphism_setoid P Q ≝ λP,Q,x.x.
458 coercion unary_morphism_setoid_of_arrows1_SET.
460 interpretation "'arrows1_SET" 'arrows1_SET A B = (arrows1 SET A B).
462 definition unary_morphism1_setoid1: setoid1 → setoid1 → setoid1.
465 [ @(unary_morphism1 s s1);
468 alias symbol "eq" = "setoid1 eq".
469 @(∀a: carr1 s. f a = g a);
470 | intros 1; simplify; intros; @refl1;
471 | simplify; intros; @sym1; @f;
472 | simplify; intros; @trans1; [2: @f; | skip | @f1]]]
475 definition unary_morphism1_of_unary_morphism1_setoid1 :
476 ∀S,T. unary_morphism1_setoid1 S T → unary_morphism1 S T ≝ λP,Q,x.x.
477 coercion unary_morphism1_of_unary_morphism1_setoid1.
479 definition SET1: objs3 CAT2.
482 | @rule (λS,T.setoid2_of_setoid1 (unary_morphism1_setoid1 S T));
483 | intros; constructor 1; [ @(λx.x); | intros; assumption ]
484 | intros; constructor 1; [ intros; constructor 1; [ @(λx. c1 (c x)); | intros;
486 | intros; whd; intros; simplify; whd in H1; whd in H;
487 @trans1; [ @(b (a' a1)); | l@(prop11 ?? b (a a1) (a' a1));
488 [ @Hletin | @(e a1); ] | @e1; ]]
489 | intros; whd; intros; simplify; @refl1;
490 | intros; simplify; whd; intros; simplify; @refl1;
491 | intros; simplify; whd; intros; simplify; @refl1;
495 interpretation "'arrows2_SET1" 'arrows2_SET1 A B = (arrows2 SET1 A B).
497 definition setoid1_of_SET1: objs2 SET1 → setoid1 ≝ λx.x.
498 coercion setoid1_of_SET1.
500 definition unary_morphism1_setoid1_of_arrows2_SET1:
501 ∀P,Q.arrows2 SET1 P Q → unary_morphism1_setoid1 P Q ≝ λP,Q,x.x.
502 coercion unary_morphism1_setoid1_of_arrows2_SET1.
504 variant objs2_of_category1: objs1 SET → objs2 SET1 ≝ setoid1_of_setoid.
505 coercion objs2_of_category1.
507 prefer coercion Type_OF_setoid. (* we prefer the lower carrier projection *)
508 prefer coercion Type_OF_objs1.
510 alias symbol "exists" (instance 1) = "CProp[2] exists".
512 λA,B:CAT2.λF:carr3 (arrows3 CAT2 A B).
513 ∀o1,o2:A.∀f.∃g:arrows2 A o1 o2.F⎽⇒ g =_2 f.
514 alias symbol "exists" (instance 1) = "CProp exists".
516 definition faithful2 ≝
517 λA,B:CAT2.λF:carr3 (arrows3 CAT2 A B).
518 ∀o1,o2:A.∀f,g:arrows2 A o1 o2.F⎽⇒ f =_2 F⎽⇒ g → f =_2 g.
521 notation "r \sup *" non associative with precedence 90 for @{'OR_f_star $r}.
522 notation > "r *" non associative with precedence 90 for @{'OR_f_star $r}.
524 notation "r \sup (⎻* )" non associative with precedence 90 for @{'OR_f_minus_star $r}.
525 notation > "r⎻*" non associative with precedence 90 for @{'OR_f_minus_star $r}.
527 notation "r \sup ⎻" non associative with precedence 90 for @{'OR_f_minus $r}.
528 notation > "r⎻" non associative with precedence 90 for @{'OR_f_minus $r}.