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 inductive eq (A:Type[0]) (x:A) : A → CProp[0] ≝
20 notation "hvbox(a break = \sub \ID b)" non associative with precedence 45
23 notation > "hvbox(a break =_\ID b)" non associative with precedence 45
24 for @{ 'eqID ? $a $b }.
26 interpretation "ID eq" 'eqID x y = (eq ? x y).
28 record equivalence_relation (A:Type[0]) : Type[1] ≝
29 { eq_rel:2> A → A → CProp[0];
30 refl: reflexive ? eq_rel;
31 sym: symmetric ? eq_rel;
32 trans: transitive ? eq_rel
35 record setoid : Type[1] ≝
37 eq: equivalence_relation carr
40 record equivalence_relation1 (A:Type[1]) : Type[2] ≝
41 { eq_rel1:2> A → A → CProp[1];
42 refl1: reflexive1 ? eq_rel1;
43 sym1: symmetric1 ? eq_rel1;
44 trans1: transitive1 ? eq_rel1
47 record setoid1: Type[2] ≝
49 eq1: equivalence_relation1 carr1
52 definition setoid1_of_setoid: setoid → setoid1.
56 | @mk_equivalence_relation1
64 coercion setoid1_of_setoid.
66 prefer coercion Type_OF_setoid.
68 record equivalence_relation2 (A:Type[2]) : Type[3] ≝
69 { eq_rel2:2> A → A → CProp[2];
70 refl2: reflexive2 ? eq_rel2;
71 sym2: symmetric2 ? eq_rel2;
72 trans2: transitive2 ? eq_rel2
75 record setoid2: Type[3] ≝
77 eq2: equivalence_relation2 carr2
80 definition setoid2_of_setoid1: setoid1 → setoid2.
84 | @mk_equivalence_relation2
92 coercion setoid2_of_setoid1.
94 prefer coercion Type_OF_setoid2.
95 prefer coercion Type_OF_setoid.
96 prefer coercion Type_OF_setoid1.
97 (* we prefer 0 < 1 < 2 *)
99 record equivalence_relation3 (A:Type[3]) : Type[4] ≝
100 { eq_rel3:2> A → A → CProp[3];
101 refl3: reflexive3 ? eq_rel3;
102 sym3: symmetric3 ? eq_rel3;
103 trans3: transitive3 ? eq_rel3
106 record setoid3: Type[4] ≝
108 eq3: equivalence_relation3 carr3
111 interpretation "setoid3 eq" 'eq t x y = (eq_rel3 ? (eq3 t) x y).
112 interpretation "setoid2 eq" 'eq t x y = (eq_rel2 ? (eq2 t) x y).
113 interpretation "setoid1 eq" 'eq t x y = (eq_rel1 ? (eq1 t) x y).
114 interpretation "setoid eq" 'eq t x y = (eq_rel ? (eq t) x y).
116 notation > "hvbox(a break =_12 b)" non associative with precedence 45
117 for @{ eq_rel2 (carr2 (setoid2_of_setoid1 ?)) (eq2 (setoid2_of_setoid1 ?)) $a $b }.
118 notation > "hvbox(a break =_0 b)" non associative with precedence 45
119 for @{ eq_rel ? (eq ?) $a $b }.
120 notation > "hvbox(a break =_1 b)" non associative with precedence 45
121 for @{ eq_rel1 ? (eq1 ?) $a $b }.
122 notation > "hvbox(a break =_2 b)" non associative with precedence 45
123 for @{ eq_rel2 ? (eq2 ?) $a $b }.
124 notation > "hvbox(a break =_3 b)" non associative with precedence 45
125 for @{ eq_rel3 ? (eq3 ?) $a $b }.
127 interpretation "setoid3 symmetry" 'invert r = (sym3 ???? r).
128 interpretation "setoid2 symmetry" 'invert r = (sym2 ???? r).
129 interpretation "setoid1 symmetry" 'invert r = (sym1 ???? r).
130 interpretation "setoid symmetry" 'invert r = (sym ???? r).
131 notation ".= r" with precedence 55 for @{'trans $r}.
132 interpretation "trans3" 'trans r = (trans3 ????? r).
133 interpretation "trans2" 'trans r = (trans2 ????? r).
134 interpretation "trans1" 'trans r = (trans1 ????? r).
135 interpretation "trans" 'trans r = (trans ????? r).
137 record unary_morphism (A,B: setoid) : Type[0] ≝
139 prop1: ∀a,a'. eq ? a a' → eq ? (fun1 a) (fun1 a')
142 record unary_morphism1 (A,B: setoid1) : Type[1] ≝
144 prop11: ∀a,a'. eq1 ? a a' → eq1 ? (fun11 a) (fun11 a')
147 record unary_morphism2 (A,B: setoid2) : Type[2] ≝
149 prop12: ∀a,a'. eq2 ? a a' → eq2 ? (fun12 a) (fun12 a')
152 record unary_morphism3 (A,B: setoid3) : Type[3] ≝
154 prop13: ∀a,a'. eq3 ? a a' → eq3 ? (fun13 a) (fun13 a')
157 record binary_morphism (A,B,C:setoid) : Type[0] ≝
159 prop2: ∀a,a',b,b'. eq ? a a' → eq ? b b' → eq ? (fun2 a b) (fun2 a' b')
162 record binary_morphism1 (A,B,C:setoid1) : Type[1] ≝
163 { fun21:2> A → B → C;
164 prop21: ∀a,a',b,b'. eq1 ? a a' → eq1 ? b b' → eq1 ? (fun21 a b) (fun21 a' b')
167 record binary_morphism2 (A,B,C:setoid2) : Type[2] ≝
168 { fun22:2> A → B → C;
169 prop22: ∀a,a',b,b'. eq2 ? a a' → eq2 ? b b' → eq2 ? (fun22 a b) (fun22 a' b')
172 record binary_morphism3 (A,B,C:setoid3) : Type[3] ≝
173 { fun23:2> A → B → C;
174 prop23: ∀a,a',b,b'. eq3 ? a a' → eq3 ? b b' → eq3 ? (fun23 a b) (fun23 a' b')
177 notation "† c" with precedence 90 for @{'prop1 $c }.
178 notation "l ‡ r" with precedence 90 for @{'prop2 $l $r }.
179 notation "#" with precedence 90 for @{'refl}.
180 interpretation "prop1" 'prop1 c = (prop1 ????? c).
181 interpretation "prop11" 'prop1 c = (prop11 ????? c).
182 interpretation "prop12" 'prop1 c = (prop12 ????? c).
183 interpretation "prop13" 'prop1 c = (prop13 ????? c).
184 interpretation "prop2" 'prop2 l r = (prop2 ???????? l r).
185 interpretation "prop21" 'prop2 l r = (prop21 ???????? l r).
186 interpretation "prop22" 'prop2 l r = (prop22 ???????? l r).
187 interpretation "prop23" 'prop2 l r = (prop23 ???????? l r).
188 interpretation "refl" 'refl = (refl ???).
189 interpretation "refl1" 'refl = (refl1 ???).
190 interpretation "refl2" 'refl = (refl2 ???).
191 interpretation "refl3" 'refl = (refl3 ???).
193 notation > "A × term 74 B ⇒ term 19 C" non associative with precedence 72 for @{'binary_morphism0 $A $B $C}.
194 notation > "A × term 74 B ⇒_1 term 19 C" non associative with precedence 72 for @{'binary_morphism1 $A $B $C}.
195 notation > "A × term 74 B ⇒_2 term 19 C" non associative with precedence 72 for @{'binary_morphism2 $A $B $C}.
196 notation > "A × term 74 B ⇒_3 term 19 C" non associative with precedence 72 for @{'binary_morphism3 $A $B $C}.
197 notation > "B ⇒_1 C" right associative with precedence 72 for @{'arrows1_SET $B $C}.
198 notation > "B ⇒_1. C" right associative with precedence 72 for @{'arrows1_SETlow $B $C}.
199 notation > "B ⇒_2 C" right associative with precedence 72 for @{'arrows2_SET1 $B $C}.
200 notation > "B ⇒_2. C" right associative with precedence 72 for @{'arrows2_SET1low $B $C}.
202 notation "A × term 74 B ⇒\sub 1 term 19 C" non associative with precedence 72 for @{'binary_morphism1 $A $B $C}.
203 notation "A × term 74 B ⇒\sub 2 term 19 C" non associative with precedence 72 for @{'binary_morphism2 $A $B $C}.
204 notation "A × term 74 B ⇒\sub 3 term 19 C" non associative with precedence 72 for @{'binary_morphism3 $A $B $C}.
205 notation "B ⇒\sub 1 C" right associative with precedence 72 for @{'arrows1_SET $B $C}.
206 notation "B ⇒\sub 2 C" right associative with precedence 72 for @{'arrows2_SET1 $B $C}.
207 notation "B ⇒\sub 1. C" right associative with precedence 72 for @{'arrows1_SETlow $B $C}.
208 notation "B ⇒\sub 2. C" right associative with precedence 72 for @{'arrows2_SET1low $B $C}.
210 interpretation "'binary_morphism0" 'binary_morphism0 A B C = (binary_morphism A B C).
211 interpretation "'arrows2_SET1 low" 'arrows2_SET1 A B = (unary_morphism2 A B).
212 interpretation "'arrows2_SET1low" 'arrows2_SET1low A B = (unary_morphism2 A B).
213 interpretation "'binary_morphism1" 'binary_morphism1 A B C = (binary_morphism1 A B C).
214 interpretation "'binary_morphism2" 'binary_morphism2 A B C = (binary_morphism2 A B C).
215 interpretation "'binary_morphism3" 'binary_morphism3 A B C = (binary_morphism3 A B C).
216 interpretation "'arrows1_SET low" 'arrows1_SET A B = (unary_morphism1 A B).
217 interpretation "'arrows1_SETlow" 'arrows1_SETlow A B = (unary_morphism1 A B).
220 definition unary_morphism2_of_unary_morphism1:
221 ∀S,T:setoid1.unary_morphism1 S T → unary_morphism2 (setoid2_of_setoid1 S) T.
228 definition CPROP: setoid1.
231 | @mk_equivalence_relation1
233 | #x @mk_Iff #x1 assumption
234 | #x #y #i cases i #f #f1 @mk_Iff assumption
235 | #x #y #z #i1 #i cases i cases i1 #f #f1 #f2 #f3 @mk_Iff #x1
236 [ @(f2 (f x1)) | @(f1 (f3 x1))]]]
239 definition CProp0_of_CPROP: carr1 CPROP → CProp[0] ≝ λx.x.
240 coercion CProp0_of_CPROP.
242 alias symbol "eq" = "setoid1 eq".
243 definition fi': ∀A,B:CPROP. A = B → B → A.
244 #A #B #e #b @(fi ?? e) assumption.
247 notation ". r" with precedence 55 for @{'fi $r}.
248 interpretation "fi" 'fi r = (fi' ?? r).
250 definition and_morphism: binary_morphism1 CPROP CPROP CPROP.
253 | #a #a' #b #b' #e #e1 @mk_Iff #a1 cases a1 #a2 #b1 @Conj
260 interpretation "and_morphism" 'and a b = (fun21 ??? and_morphism a b).
262 definition or_morphism: binary_morphism1 CPROP CPROP CPROP.
265 | #a #a' #b #b' #e #e1 @mk_Iff #o cases o #a1 [1,3: @Left |2,4: @Right]
272 interpretation "or_morphism" 'or a b = (fun21 ??? or_morphism a b).
274 definition if_morphism: binary_morphism1 CPROP CPROP CPROP.
277 | #a #a' #b #b' #e #e1 @mk_Iff #f #a1
278 [ @(if ?? e1) @f @(fi ?? e) assumption
279 | @(fi ?? e1) @f @(if ?? e) assumption]]
282 notation > "hvbox(a break ∘ b)" left associative with precedence 60 for @{ comp ??? $a $b }.
284 record category : Type[1] ≝ {
286 arrows: objs → objs → setoid;
287 id: ∀o:objs. arrows o o;
288 comp: ∀o1,o2,o3. (arrows o1 o2) × (arrows o2 o3) ⇒ (arrows o1 o3);
289 comp_assoc: ∀o1,o2,o3,o4.∀a12:arrows o1 ?.∀a23:arrows o2 ?.∀a34:arrows o3 o4.
290 comp o1 o3 o4 (comp o1 o2 o3 a12 a23) a34 =_0 comp o1 o2 o4 a12 (comp o2 o3 o4 a23 a34);
291 id_neutral_left : ∀o1,o2. ∀a: arrows o1 o2. comp ??? (id o1) a =_0 a;
292 id_neutral_right: ∀o1,o2. ∀a: arrows o1 o2. comp ??? a (id o2) =_0 a
295 notation "hvbox(a break ∘ b)" left associative with precedence 60 for @{ 'compose $a $b }.
297 record category1 : Type[2] ≝
299 arrows1: objs1 → objs1 → setoid1;
300 id1: ∀o:objs1. arrows1 o o;
301 comp1: ∀o1,o2,o3. binary_morphism1 (arrows1 o1 o2) (arrows1 o2 o3) (arrows1 o1 o3);
302 comp_assoc1: ∀o1,o2,o3,o4. ∀a12,a23,a34.
303 comp1 o1 o3 o4 (comp1 o1 o2 o3 a12 a23) a34 =_1 comp1 o1 o2 o4 a12 (comp1 o2 o3 o4 a23 a34);
304 id_neutral_right1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? (id1 o1) a =_1 a;
305 id_neutral_left1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? a (id1 o2) =_1 a
308 record category2 : Type[3] ≝
310 arrows2: objs2 → objs2 → setoid2;
311 id2: ∀o:objs2. arrows2 o o;
312 comp2: ∀o1,o2,o3. binary_morphism2 (arrows2 o1 o2) (arrows2 o2 o3) (arrows2 o1 o3);
313 comp_assoc2: ∀o1,o2,o3,o4. ∀a12,a23,a34.
314 comp2 o1 o3 o4 (comp2 o1 o2 o3 a12 a23) a34 =_2 comp2 o1 o2 o4 a12 (comp2 o2 o3 o4 a23 a34);
315 id_neutral_right2: ∀o1,o2. ∀a: arrows2 o1 o2. comp2 ??? (id2 o1) a =_2 a;
316 id_neutral_left2: ∀o1,o2. ∀a: arrows2 o1 o2. comp2 ??? a (id2 o2) =_2 a
319 record category3 : Type[4] ≝
321 arrows3: objs3 → objs3 → setoid3;
322 id3: ∀o:objs3. arrows3 o o;
323 comp3: ∀o1,o2,o3. binary_morphism3 (arrows3 o1 o2) (arrows3 o2 o3) (arrows3 o1 o3);
324 comp_assoc3: ∀o1,o2,o3,o4. ∀a12,a23,a34.
325 comp3 o1 o3 o4 (comp3 o1 o2 o3 a12 a23) a34 =_3 comp3 o1 o2 o4 a12 (comp3 o2 o3 o4 a23 a34);
326 id_neutral_right3: ∀o1,o2. ∀a: arrows3 o1 o2. comp3 ??? (id3 o1) a =_3 a;
327 id_neutral_left3: ∀o1,o2. ∀a: arrows3 o1 o2. comp3 ??? a (id3 o2) =_3 a
330 notation "'ASSOC'" with precedence 90 for @{'assoc}.
332 interpretation "category2 composition" 'compose x y = (fun22 ??? (comp2 ????) y x).
333 interpretation "category2 assoc" 'assoc = (comp_assoc2 ????????).
334 interpretation "category1 composition" 'compose x y = (fun21 ??? (comp1 ????) y x).
335 interpretation "category1 assoc" 'assoc = (comp_assoc1 ????????).
336 interpretation "category composition" 'compose x y = (fun2 ??? (comp ????) y x).
337 interpretation "category assoc" 'assoc = (comp_assoc ????????).
339 definition category2_of_category1: category1 → category2.
343 | #o #o1 @(setoid2_of_setoid1 (arrows1 c o o1))
347 [ #c1 #c2 @(comp1 c o1 o2 o3 c1 c2)
348 | #a #a' #b #b' #e #e1 @(e‡e1) ]
349 | #o1 #o2 #o3 #o4 #a12 #a23 #a34 @comp_assoc1
350 | #o1 #o2 #a @id_neutral_right1
351 | #o1 #o2 #a @id_neutral_left1 ]
353 (*coercion category2_of_category1.*)
355 record functor2 (C1: category2) (C2: category2) : Type[3] ≝
356 { map_objs2:1> C1 → C2;
357 map_arrows2: ∀S,T. unary_morphism2 (arrows2 ? S T) (arrows2 ? (map_objs2 S) (map_objs2 T));
358 respects_id2: ∀o:C1. map_arrows2 ?? (id2 ? o) = id2 ? (map_objs2 o);
360 ∀o1,o2,o3.∀f1:arrows2 ? o1 o2.∀f2:arrows2 ? o2 o3.
361 map_arrows2 ?? (f2 ∘ f1) = map_arrows2 ?? f2 ∘ map_arrows2 ?? f1}.
363 notation > "F⎽⇒ x" left associative with precedence 65 for @{'map_arrows2 $F $x}.
364 notation "F\sub⇒ x" left associative with precedence 65 for @{'map_arrows2 $F $x}.
365 interpretation "map_arrows2" 'map_arrows2 F x = (fun12 ?? (map_arrows2 ?? F ??) x).
367 definition functor2_setoid: category2 → category2 → setoid3.
371 | @mk_equivalence_relation3
373 @(∀c:C1. cic:/matita/logic/equality/eq.ind#xpointer(1/1) ? (f c) (g c));
374 | simplify; intros; @cic:/matita/logic/equality/eq.ind#xpointer(1/1/1);
375 | simplify; intros; @cic:/matita/logic/equality/sym_eq.con; @H;
376 | simplify; intros; @cic:/matita/logic/equality/trans_eq.con;
377 [2: @H; | skip | @H1;]]]
380 definition functor2_of_functor2_setoid: ∀S,T. functor2_setoid S T → functor2 S T ≝ λS,T,x.x.
381 coercion functor2_of_functor2_setoid.
383 definition CAT2: category3.
387 | intros; constructor 1;
389 | intros; constructor 1;
391 | intros; assumption;]
394 | intros; constructor 1;
395 [ intros; constructor 1;
396 [ intros; @(c1 (c o));
397 | intros; constructor 1;
398 [ intro; @(map_arrows2 ?? c1 ?? (map_arrows2 ?? c ?? c2));
401 @(.= †(respects_id2 : ?));
404 @(.= †(respects_comp2 : ?));
405 @(respects_comp2 : ?); ]
406 | intros; intro; simplify;
407 @(cic:/matita/logic/equality/eq_ind.con ????? (e ?));
408 @(cic:/matita/logic/equality/eq_ind.con ????? (e1 ?));
410 | intros; intro; simplify; constructor 1;
411 | intros; intro; simplify; constructor 1;
412 | intros; intro; simplify; constructor 1; ]
415 definition category2_of_objs3_CAT2: objs3 CAT2 → category2 ≝ λx.x.
416 coercion category2_of_objs3_CAT2.
418 definition functor2_setoid_of_arrows3_CAT2: ∀S,T. arrows3 CAT2 S T → functor2_setoid S T ≝ λS,T,x.x.
419 coercion functor2_setoid_of_arrows3_CAT2.
421 notation > "B ⇒_\c3 C" right associative with precedence 72 for @{'arrows3_CAT $B $C}.
422 notation "B ⇒\sub (\c 3) C" right associative with precedence 72 for @{'arrows3_CAT $B $C}.
423 interpretation "'arrows3_CAT" 'arrows3_CAT A B = (arrows3 CAT2 A B).
425 definition unary_morphism_setoid: setoid → setoid → setoid.
428 [ @(unary_morphism s s1);
430 [ intros (f g); @(∀a:s. eq ? (f a) (g a));
431 | intros 1; simplify; intros; @refl;
432 | simplify; intros; @sym; @f;
433 | simplify; intros; @trans; [2: @f; | skip | @f1]]]
436 definition SET: category1.
439 | @rule (λS,T:setoid.setoid1_of_setoid (unary_morphism_setoid S T));
440 | intros; constructor 1; [ @(λx:carr o.x); | intros; assumption ]
441 | intros; constructor 1; [ intros; constructor 1; [ @(λx. c1 (c x)); | intros;
443 | intros; whd; intros; simplify; whd in H1; whd in H;
444 @trans; [ @(b (a' a1)); | l@(prop1 ?? b (a a1) (a' a1));
445 [ @Hletin | @(e a1); ] | @e1; ]]
446 | intros; whd; intros; simplify; @refl;
447 | intros; simplify; whd; intros; simplify; @refl;
448 | intros; simplify; whd; intros; simplify; @refl;
452 definition setoid_of_SET: objs1 SET → setoid ≝ λx.x.
453 coercion setoid_of_SET.
455 definition unary_morphism_setoid_of_arrows1_SET:
456 ∀P,Q.arrows1 SET P Q → unary_morphism_setoid P Q ≝ λP,Q,x.x.
457 coercion unary_morphism_setoid_of_arrows1_SET.
459 interpretation "'arrows1_SET" 'arrows1_SET A B = (arrows1 SET A B).
461 definition unary_morphism1_setoid1: setoid1 → setoid1 → setoid1.
464 [ @(unary_morphism1 s s1);
467 alias symbol "eq" = "setoid1 eq".
468 @(∀a: carr1 s. f a = g a);
469 | intros 1; simplify; intros; @refl1;
470 | simplify; intros; @sym1; @f;
471 | simplify; intros; @trans1; [2: @f; | skip | @f1]]]
474 definition unary_morphism1_of_unary_morphism1_setoid1 :
475 ∀S,T. unary_morphism1_setoid1 S T → unary_morphism1 S T ≝ λP,Q,x.x.
476 coercion unary_morphism1_of_unary_morphism1_setoid1.
478 definition SET1: objs3 CAT2.
481 | @rule (λS,T.setoid2_of_setoid1 (unary_morphism1_setoid1 S T));
482 | intros; constructor 1; [ @(λx.x); | intros; assumption ]
483 | intros; constructor 1; [ intros; constructor 1; [ @(λx. c1 (c x)); | intros;
485 | intros; whd; intros; simplify; whd in H1; whd in H;
486 @trans1; [ @(b (a' a1)); | l@(prop11 ?? b (a a1) (a' a1));
487 [ @Hletin | @(e a1); ] | @e1; ]]
488 | intros; whd; intros; simplify; @refl1;
489 | intros; simplify; whd; intros; simplify; @refl1;
490 | intros; simplify; whd; intros; simplify; @refl1;
494 interpretation "'arrows2_SET1" 'arrows2_SET1 A B = (arrows2 SET1 A B).
496 definition setoid1_of_SET1: objs2 SET1 → setoid1 ≝ λx.x.
497 coercion setoid1_of_SET1.
499 definition unary_morphism1_setoid1_of_arrows2_SET1:
500 ∀P,Q.arrows2 SET1 P Q → unary_morphism1_setoid1 P Q ≝ λP,Q,x.x.
501 coercion unary_morphism1_setoid1_of_arrows2_SET1.
503 variant objs2_of_category1: objs1 SET → objs2 SET1 ≝ setoid1_of_setoid.
504 coercion objs2_of_category1.
506 prefer coercion Type_OF_setoid. (* we prefer the lower carrier projection *)
507 prefer coercion Type_OF_objs1.
509 alias symbol "exists" (instance 1) = "CProp[2] exists".
511 λA,B:CAT2.λF:carr3 (arrows3 CAT2 A B).
512 ∀o1,o2:A.∀f.∃g:arrows2 A o1 o2.F⎽⇒ g =_2 f.
513 alias symbol "exists" (instance 1) = "CProp exists".
515 definition faithful2 ≝
516 λA,B:CAT2.λF:carr3 (arrows3 CAT2 A B).
517 ∀o1,o2:A.∀f,g:arrows2 A o1 o2.F⎽⇒ f =_2 F⎽⇒ g → f =_2 g.
520 notation "r \sup *" non associative with precedence 90 for @{'OR_f_star $r}.
521 notation > "r *" non associative with precedence 90 for @{'OR_f_star $r}.
523 notation "r \sup (⎻* )" non associative with precedence 90 for @{'OR_f_minus_star $r}.
524 notation > "r⎻*" non associative with precedence 90 for @{'OR_f_minus_star $r}.
526 notation "r \sup ⎻" non associative with precedence 90 for @{'OR_f_minus $r}.
527 notation > "r⎻" non associative with precedence 90 for @{'OR_f_minus $r}.