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 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/CoRN-Decl/algebra/CSetoids".
21 (* $Id.v,v 1.18 2002/11/25 14:43:42 lcf Exp $ *)
23 (*#* printing [=] %\ensuremath{\equiv}% #≡# *)
25 (*#* printing [~=] %\ensuremath{\mathrel{\not\equiv}}% #≠# *)
27 (*#* printing [#] %\ensuremath{\mathrel\#}% *)
29 (*#* printing ex_unq %\ensuremath{\exists^1}% #∃<sup>1</sup># *)
31 (*#* printing [o] %\ensuremath\circ% #⋅# *)
33 (*#* printing [-C-] %\ensuremath\diamond% *)
38 Definition of a constructive setoid with apartness,
39 i.e.%\% a set with an equivalence relation and an apartness relation compatible with it.
42 include "algebra/CLogic.ma".
44 include "tactics/Step.ma".
46 inline "cic:/CoRN/algebra/CSetoids/Relation.con".
51 Implicit Arguments Treflexive [A].
55 Implicit Arguments Creflexive [A].
61 Implicit Arguments Tsymmetric [A].
65 Implicit Arguments Csymmetric [A].
69 Implicit Arguments Ttransitive [A].
73 Implicit Arguments Ctransitive [A].
79 Set Implicit Arguments.
83 Unset Strict Implicit.
88 (*#* **Relations necessary for Setoids
89 %\begin{convention}% Let [A:Type].
92 Notice that their type depends on the main logical connective.
96 Section Properties_of_relations
99 alias id "A" = "cic:/CoRN/algebra/CSetoids/Properties_of_relations/A.var".
101 inline "cic:/CoRN/algebra/CSetoids/irreflexive.con".
103 inline "cic:/CoRN/algebra/CSetoids/cotransitive.con".
105 inline "cic:/CoRN/algebra/CSetoids/tight_apart.con".
107 inline "cic:/CoRN/algebra/CSetoids/antisymmetric.con".
110 End Properties_of_relations
120 Unset Implicit Arguments.
125 (*#* **Definition of Setoid
127 Apartness, being the main relation, needs to be [CProp]-valued. Equality,
128 as it is characterized by a negative statement, lives in [Prop]. *)
130 inline "cic:/CoRN/algebra/CSetoids/is_CSetoid.ind".
132 inline "cic:/CoRN/algebra/CSetoids/CSetoid.ind".
134 coercion cic:/matita/CoRN-Decl/algebra/CSetoids/cs_crr.con 0 (* compounds *).
137 Implicit Arguments cs_eq [c].
141 Implicit Arguments cs_ap [c].
145 Infix "[=]" := cs_eq (at level 70, no associativity).
149 Infix "[#]" := cs_ap (at level 70, no associativity).
154 inline "cic:/CoRN/algebra/CSetoids/cs_neq.con".
157 Implicit Arguments cs_neq [S].
161 Infix "[~=]" := cs_neq (at level 70, no associativity).
165 %\begin{nameconvention}%
166 In the names of lemmas, we refer to [ [=] ] by [eq], [ [~=] ] by
167 [neq], and [ [#] ] by [ap].
168 %\end{nameconvention}%
171 We want concrete lemmas that state the axiomatic properties of a setoid.
177 (* Begin_SpecReals *)
180 Section CSetoid_axioms
183 alias id "S" = "cic:/CoRN/algebra/CSetoids/CSetoid_axioms/S.var".
185 inline "cic:/CoRN/algebra/CSetoids/CSetoid_is_CSetoid.con".
187 inline "cic:/CoRN/algebra/CSetoids/ap_irreflexive.con".
189 inline "cic:/CoRN/algebra/CSetoids/ap_symmetric.con".
191 inline "cic:/CoRN/algebra/CSetoids/ap_cotransitive.con".
193 inline "cic:/CoRN/algebra/CSetoids/ap_tight.con".
201 (*#* **Setoid basics%\label{section:setoid-basics}%
202 %\begin{convention}% Let [S] be a setoid.
206 (* Begin_SpecReals *)
209 Section CSetoid_basics
212 alias id "S" = "cic:/CoRN/algebra/CSetoids/CSetoid_basics/S.var".
217 In `there exists a unique [a:S] such that %\ldots%#...#', we now mean unique with respect to the setoid equality. We use [ex_unq] to denote unique existence.
220 inline "cic:/CoRN/algebra/CSetoids/ex_unq.con".
222 inline "cic:/CoRN/algebra/CSetoids/eq_reflexive.con".
224 inline "cic:/CoRN/algebra/CSetoids/eq_symmetric.con".
226 inline "cic:/CoRN/algebra/CSetoids/eq_transitive.con".
229 %\begin{shortcoming}%
230 The lemma [eq_reflexive] above is convertible to
231 [eq_reflexive_unfolded] below. We need the second version too,
232 because the first cannot be applied when an instance of reflexivity is needed.
233 (``I have complained bitterly about this.'' RP)
236 %\begin{nameconvention}%
237 If lemma [a] is just an unfolding of lemma [b], the name of [a] is the name
238 [b] with the suffix ``[_unfolded]''.
239 %\end{nameconvention}%
242 inline "cic:/CoRN/algebra/CSetoids/eq_reflexive_unfolded.con".
244 inline "cic:/CoRN/algebra/CSetoids/eq_symmetric_unfolded.con".
246 inline "cic:/CoRN/algebra/CSetoids/eq_transitive_unfolded.con".
248 inline "cic:/CoRN/algebra/CSetoids/eq_wdl.con".
250 (* Begin_SpecReals *)
252 inline "cic:/CoRN/algebra/CSetoids/ap_irreflexive_unfolded.con".
256 inline "cic:/CoRN/algebra/CSetoids/ap_cotransitive_unfolded.con".
258 inline "cic:/CoRN/algebra/CSetoids/ap_symmetric_unfolded.con".
261 %\begin{shortcoming}%
262 We would like to write
264 Lemma eq_equiv_not_ap : forall (x y:S), x [=] y Iff ~(x [#] y).
266 In Coq, however, this lemma cannot be easily applied.
267 Therefore we have to split the lemma into the following two lemmas [eq_imp_not_ap] and [not_ap_imp_eq].
271 inline "cic:/CoRN/algebra/CSetoids/eq_imp_not_ap.con".
273 inline "cic:/CoRN/algebra/CSetoids/not_ap_imp_eq.con".
275 inline "cic:/CoRN/algebra/CSetoids/neq_imp_notnot_ap.con".
277 inline "cic:/CoRN/algebra/CSetoids/notnot_ap_imp_neq.con".
279 inline "cic:/CoRN/algebra/CSetoids/ap_imp_neq.con".
281 inline "cic:/CoRN/algebra/CSetoids/not_neq_imp_eq.con".
283 inline "cic:/CoRN/algebra/CSetoids/eq_imp_not_neq.con".
285 (* Begin_SpecReals *)
294 Section product_csetoid
297 (*#* **The product of setoids *)
299 inline "cic:/CoRN/algebra/CSetoids/prod_ap.con".
301 inline "cic:/CoRN/algebra/CSetoids/prod_eq.con".
303 inline "cic:/CoRN/algebra/CSetoids/prodcsetoid_is_CSetoid.con".
305 inline "cic:/CoRN/algebra/CSetoids/ProdCSetoid.con".
312 Implicit Arguments ex_unq [S].
316 Hint Resolve eq_reflexive_unfolded: algebra_r.
320 Hint Resolve eq_symmetric_unfolded: algebra_s.
324 Declare Left Step eq_wdl.
328 Declare Right Step eq_transitive_unfolded.
331 (* Begin_SpecReals *)
333 (*#* **Relations and predicates
334 Here we define the notions of well-definedness and strong extensionality
335 on predicates and relations.
337 %\begin{convention}% Let [S] be a setoid.
340 %\begin{nameconvention}%
341 - ``well-defined'' is abbreviated to [well_def] (or [wd]).
342 - ``strongly extensional'' is abbreviated to [strong_ext] (or [strext]).
344 %\end{nameconvention}%
348 Section CSetoid_relations_and_predicates
351 alias id "S" = "cic:/CoRN/algebra/CSetoids/CSetoid_relations_and_predicates/S.var".
357 At this stage, we consider [CProp]- and [Prop]-valued predicates on setoids.
359 %\begin{convention}% Let [P] be a predicate on (the carrier of) [S].
364 Section CSetoidPredicates
367 alias id "P" = "cic:/CoRN/algebra/CSetoids/CSetoid_relations_and_predicates/CSetoidPredicates/P.var".
369 inline "cic:/CoRN/algebra/CSetoids/pred_strong_ext.con".
371 inline "cic:/CoRN/algebra/CSetoids/pred_wd.con".
374 End CSetoidPredicates
377 inline "cic:/CoRN/algebra/CSetoids/CSetoid_predicate.ind".
379 coercion cic:/matita/CoRN-Decl/algebra/CSetoids/csp_pred.con 0 (* compounds *).
381 inline "cic:/CoRN/algebra/CSetoids/csp_wd.con".
383 (*#* Similar, with [Prop] instead of [CProp]. *)
386 Section CSetoidPPredicates
389 alias id "P" = "cic:/CoRN/algebra/CSetoids/CSetoid_relations_and_predicates/CSetoidPPredicates/P.var".
391 inline "cic:/CoRN/algebra/CSetoids/pred_strong_ext'.con".
393 inline "cic:/CoRN/algebra/CSetoids/pred_wd'.con".
396 End CSetoidPPredicates
399 (*#* ***Definition of a setoid predicate *)
401 inline "cic:/CoRN/algebra/CSetoids/CSetoid_predicate'.ind".
403 coercion cic:/matita/CoRN-Decl/algebra/CSetoids/csp'_pred.con 0 (* compounds *).
405 inline "cic:/CoRN/algebra/CSetoids/csp'_wd.con".
407 (* Begin_SpecReals *)
411 Let [R] be a relation on (the carrier of) [S].
412 %\end{convention}% *)
415 Section CsetoidRelations
418 alias id "R" = "cic:/CoRN/algebra/CSetoids/CSetoid_relations_and_predicates/CsetoidRelations/R.var".
420 inline "cic:/CoRN/algebra/CSetoids/rel_wdr.con".
422 inline "cic:/CoRN/algebra/CSetoids/rel_wdl.con".
424 inline "cic:/CoRN/algebra/CSetoids/rel_strext.con".
428 inline "cic:/CoRN/algebra/CSetoids/rel_strext_lft.con".
430 inline "cic:/CoRN/algebra/CSetoids/rel_strext_rht.con".
432 inline "cic:/CoRN/algebra/CSetoids/rel_strext_imp_lftarg.con".
434 inline "cic:/CoRN/algebra/CSetoids/rel_strext_imp_rhtarg.con".
436 inline "cic:/CoRN/algebra/CSetoids/rel_strextarg_imp_strext.con".
438 (* Begin_SpecReals *)
444 (*#* ***Definition of a setoid relation
445 The type of relations over a setoid. *)
447 inline "cic:/CoRN/algebra/CSetoids/CSetoid_relation.ind".
449 coercion cic:/matita/CoRN-Decl/algebra/CSetoids/csr_rel.con 0 (* compounds *).
451 (*#* ***[CProp] Relations
453 Let [R] be a relation on (the carrier of) [S].
458 Section CCsetoidRelations
461 alias id "R" = "cic:/CoRN/algebra/CSetoids/CSetoid_relations_and_predicates/CCsetoidRelations/R.var".
463 inline "cic:/CoRN/algebra/CSetoids/Crel_wdr.con".
465 inline "cic:/CoRN/algebra/CSetoids/Crel_wdl.con".
467 inline "cic:/CoRN/algebra/CSetoids/Crel_strext.con".
471 inline "cic:/CoRN/algebra/CSetoids/Crel_strext_lft.con".
473 inline "cic:/CoRN/algebra/CSetoids/Crel_strext_rht.con".
475 inline "cic:/CoRN/algebra/CSetoids/Crel_strext_imp_lftarg.con".
477 inline "cic:/CoRN/algebra/CSetoids/Crel_strext_imp_rhtarg.con".
479 inline "cic:/CoRN/algebra/CSetoids/Crel_strextarg_imp_strext.con".
481 (* Begin_SpecReals *)
484 End CCsetoidRelations
487 (*#* ***Definition of a [CProp] setoid relation
489 The type of relations over a setoid. *)
491 inline "cic:/CoRN/algebra/CSetoids/CCSetoid_relation.ind".
493 coercion cic:/matita/CoRN-Decl/algebra/CSetoids/Ccsr_rel.con 0 (* compounds *).
495 inline "cic:/CoRN/algebra/CSetoids/Ccsr_wdr.con".
497 inline "cic:/CoRN/algebra/CSetoids/Ccsr_wdl.con".
501 inline "cic:/CoRN/algebra/CSetoids/ap_wdr.con".
503 inline "cic:/CoRN/algebra/CSetoids/ap_wdl.con".
505 inline "cic:/CoRN/algebra/CSetoids/ap_wdr_unfolded.con".
507 inline "cic:/CoRN/algebra/CSetoids/ap_wdl_unfolded.con".
509 inline "cic:/CoRN/algebra/CSetoids/ap_strext.con".
511 inline "cic:/CoRN/algebra/CSetoids/predS_well_def.con".
513 (* Begin_SpecReals *)
516 End CSetoid_relations_and_predicates
520 Declare Left Step ap_wdl_unfolded.
524 Declare Right Step ap_wdr_unfolded.
529 (*#* **Functions between setoids
530 Such functions must preserve the setoid equality
531 and be strongly extensional w.r.t.%\% the apartness, i.e.%\%
532 if [f(x,y) [#] f(x1,y1)], then [x [#] x1 + y [#] y1].
533 For every arity this has to be defined separately.
535 Let [S1], [S2] and [S3] be setoids.
538 First we consider unary functions. *)
540 (* Begin_SpecReals *)
543 Section CSetoid_functions
546 alias id "S1" = "cic:/CoRN/algebra/CSetoids/CSetoid_functions/S1.var".
548 alias id "S2" = "cic:/CoRN/algebra/CSetoids/CSetoid_functions/S2.var".
550 alias id "S3" = "cic:/CoRN/algebra/CSetoids/CSetoid_functions/S3.var".
553 Section unary_functions
557 In the following two definitions,
558 [f] is a function from (the carrier of) [S1] to
559 (the carrier of) [S2]. *)
561 alias id "f" = "cic:/CoRN/algebra/CSetoids/CSetoid_functions/unary_functions/f.var".
563 inline "cic:/CoRN/algebra/CSetoids/fun_wd.con".
565 inline "cic:/CoRN/algebra/CSetoids/fun_strext.con".
569 inline "cic:/CoRN/algebra/CSetoids/fun_strext_imp_wd.con".
571 (* Begin_SpecReals *)
577 inline "cic:/CoRN/algebra/CSetoids/CSetoid_fun.ind".
579 coercion cic:/matita/CoRN-Decl/algebra/CSetoids/csf_fun.con 0 (* compounds *).
581 inline "cic:/CoRN/algebra/CSetoids/csf_wd.con".
585 inline "cic:/CoRN/algebra/CSetoids/Const_CSetoid_fun.con".
587 (* Begin_SpecReals *)
590 Section binary_functions
594 Now we consider binary functions.
595 In the following two definitions,
596 [f] is a function from [S1] and [S2] to [S3].
599 alias id "f" = "cic:/CoRN/algebra/CSetoids/CSetoid_functions/binary_functions/f.var".
601 inline "cic:/CoRN/algebra/CSetoids/bin_fun_wd.con".
603 inline "cic:/CoRN/algebra/CSetoids/bin_fun_strext.con".
607 inline "cic:/CoRN/algebra/CSetoids/bin_fun_strext_imp_wd.con".
609 (* Begin_SpecReals *)
615 inline "cic:/CoRN/algebra/CSetoids/CSetoid_bin_fun.ind".
617 coercion cic:/matita/CoRN-Decl/algebra/CSetoids/csbf_fun.con 0 (* compounds *).
619 inline "cic:/CoRN/algebra/CSetoids/csbf_wd.con".
621 inline "cic:/CoRN/algebra/CSetoids/csf_wd_unfolded.con".
623 inline "cic:/CoRN/algebra/CSetoids/csf_strext_unfolded.con".
625 inline "cic:/CoRN/algebra/CSetoids/csbf_wd_unfolded.con".
628 End CSetoid_functions
634 Hint Resolve csf_wd_unfolded csbf_wd_unfolded: algebra_c.
638 Implicit Arguments fun_wd [S1 S2].
642 Implicit Arguments fun_strext [S1 S2].
645 (* Begin_SpecReals *)
647 (*#* **The unary and binary (inner) operations on a csetoid
648 An operation is a function with domain(s) and co-domain equal.
650 %\begin{nameconvention}%
651 The word ``unary operation'' is abbreviated to [un_op];
652 ``binary operation'' is abbreviated to [bin_op].
653 %\end{nameconvention}%
661 Section csetoid_inner_ops
664 alias id "S" = "cic:/CoRN/algebra/CSetoids/csetoid_inner_ops/S.var".
666 (*#* Properties of binary operations *)
668 inline "cic:/CoRN/algebra/CSetoids/commutes.con".
670 inline "cic:/CoRN/algebra/CSetoids/associative.con".
672 (*#* Well-defined unary operations on a setoid. *)
674 inline "cic:/CoRN/algebra/CSetoids/un_op_wd.con".
676 inline "cic:/CoRN/algebra/CSetoids/un_op_strext.con".
678 inline "cic:/CoRN/algebra/CSetoids/CSetoid_un_op.con".
680 inline "cic:/CoRN/algebra/CSetoids/Build_CSetoid_un_op.con".
684 inline "cic:/CoRN/algebra/CSetoids/id_strext.con".
686 inline "cic:/CoRN/algebra/CSetoids/id_pres_eq.con".
688 inline "cic:/CoRN/algebra/CSetoids/id_un_op.con".
692 coercion cic:/matita/CoRN-Decl/algebra/CSetoids/un_op_fun.con 0 (* compounds *).
696 (* Begin_SpecReals *)
698 inline "cic:/CoRN/algebra/CSetoids/cs_un_op_strext.con".
702 inline "cic:/CoRN/algebra/CSetoids/un_op_wd_unfolded.con".
704 inline "cic:/CoRN/algebra/CSetoids/un_op_strext_unfolded.con".
706 (*#* Well-defined binary operations on a setoid. *)
708 inline "cic:/CoRN/algebra/CSetoids/bin_op_wd.con".
710 inline "cic:/CoRN/algebra/CSetoids/bin_op_strext.con".
712 (* Begin_SpecReals *)
714 inline "cic:/CoRN/algebra/CSetoids/CSetoid_bin_op.con".
716 inline "cic:/CoRN/algebra/CSetoids/Build_CSetoid_bin_op.con".
720 inline "cic:/CoRN/algebra/CSetoids/cs_bin_op_wd.con".
722 inline "cic:/CoRN/algebra/CSetoids/cs_bin_op_strext.con".
724 (* Begin_SpecReals *)
728 coercion cic:/matita/CoRN-Decl/algebra/CSetoids/bin_op_bin_fun.con 0 (* compounds *).
734 inline "cic:/CoRN/algebra/CSetoids/bin_op_wd_unfolded.con".
736 inline "cic:/CoRN/algebra/CSetoids/bin_op_strext_unfolded.con".
738 inline "cic:/CoRN/algebra/CSetoids/bin_op_is_wd_un_op_lft.con".
740 inline "cic:/CoRN/algebra/CSetoids/bin_op_is_wd_un_op_rht.con".
742 inline "cic:/CoRN/algebra/CSetoids/bin_op_is_strext_un_op_lft.con".
744 inline "cic:/CoRN/algebra/CSetoids/bin_op_is_strext_un_op_rht.con".
746 inline "cic:/CoRN/algebra/CSetoids/bin_op2un_op_rht.con".
748 inline "cic:/CoRN/algebra/CSetoids/bin_op2un_op_lft.con".
750 (* Begin_SpecReals *)
753 End csetoid_inner_ops
759 Implicit Arguments commutes [S].
762 (* Begin_SpecReals *)
765 Implicit Arguments associative [S].
771 Hint Resolve bin_op_wd_unfolded un_op_wd_unfolded: algebra_c.
774 (*#* **The binary outer operations on a csetoid
776 Let [S1] and [S2] be setoids.
781 Section csetoid_outer_ops
784 alias id "S1" = "cic:/CoRN/algebra/CSetoids/csetoid_outer_ops/S1.var".
786 alias id "S2" = "cic:/CoRN/algebra/CSetoids/csetoid_outer_ops/S2.var".
789 Well-defined outer operations on a setoid.
792 inline "cic:/CoRN/algebra/CSetoids/outer_op_well_def.con".
794 inline "cic:/CoRN/algebra/CSetoids/outer_op_strext.con".
796 inline "cic:/CoRN/algebra/CSetoids/CSetoid_outer_op.con".
798 inline "cic:/CoRN/algebra/CSetoids/Build_CSetoid_outer_op.con".
800 inline "cic:/CoRN/algebra/CSetoids/csoo_wd.con".
802 inline "cic:/CoRN/algebra/CSetoids/csoo_strext.con".
806 coercion cic:/matita/CoRN-Decl/algebra/CSetoids/outer_op_bin_fun.con 0 (* compounds *).
810 inline "cic:/CoRN/algebra/CSetoids/csoo_wd_unfolded.con".
813 End csetoid_outer_ops
817 Hint Resolve csoo_wd_unfolded: algebra_c.
820 (* Begin_SpecReals *)
824 Let [S] be a setoid, and [P] a predicate on the carrier of [S].
832 alias id "S" = "cic:/CoRN/algebra/CSetoids/SubCSetoids/S.var".
834 alias id "P" = "cic:/CoRN/algebra/CSetoids/SubCSetoids/P.var".
836 inline "cic:/CoRN/algebra/CSetoids/subcsetoid_crr.ind".
838 coercion cic:/matita/CoRN-Decl/algebra/CSetoids/scs_elem.con 0 (* compounds *).
840 (*#* Though [scs_elem] is declared as a coercion, it does not satisfy the
841 uniform inheritance condition and will not be inserted. However it will
842 also not be printed, which is handy.
845 inline "cic:/CoRN/algebra/CSetoids/restrict_relation.con".
847 inline "cic:/CoRN/algebra/CSetoids/Crestrict_relation.con".
849 inline "cic:/CoRN/algebra/CSetoids/subcsetoid_eq.con".
851 inline "cic:/CoRN/algebra/CSetoids/subcsetoid_ap.con".
855 inline "cic:/CoRN/algebra/CSetoids/subcsetoid_equiv.con".
857 (* Begin_SpecReals *)
859 inline "cic:/CoRN/algebra/CSetoids/subcsetoid_is_CSetoid.con".
861 inline "cic:/CoRN/algebra/CSetoids/Build_SubCSetoid.con".
865 (*#* ***Subsetoid unary operations
867 Let [f] be a unary setoid operation on [S].
872 Section SubCSetoid_unary_operations
875 alias id "f" = "cic:/CoRN/algebra/CSetoids/SubCSetoids/SubCSetoid_unary_operations/f.var".
877 inline "cic:/CoRN/algebra/CSetoids/un_op_pres_pred.con".
881 Assume [pr:un_op_pres_pred].
882 %\end{convention}% *)
884 alias id "pr" = "cic:/CoRN/algebra/CSetoids/SubCSetoids/SubCSetoid_unary_operations/pr.var".
886 inline "cic:/CoRN/algebra/CSetoids/restr_un_op.con".
888 inline "cic:/CoRN/algebra/CSetoids/restr_un_op_wd.con".
890 inline "cic:/CoRN/algebra/CSetoids/restr_un_op_strext.con".
892 inline "cic:/CoRN/algebra/CSetoids/Build_SubCSetoid_un_op.con".
895 End SubCSetoid_unary_operations
898 (*#* ***Subsetoid binary operations
900 Let [f] be a binary setoid operation on [S].
905 Section SubCSetoid_binary_operations
908 alias id "f" = "cic:/CoRN/algebra/CSetoids/SubCSetoids/SubCSetoid_binary_operations/f.var".
910 inline "cic:/CoRN/algebra/CSetoids/bin_op_pres_pred.con".
914 Assume [bin_op_pres_pred].
918 alias id "pr" = "cic:/CoRN/algebra/CSetoids/SubCSetoids/SubCSetoid_binary_operations/pr.var".
920 inline "cic:/CoRN/algebra/CSetoids/restr_bin_op.con".
922 inline "cic:/CoRN/algebra/CSetoids/restr_bin_op_well_def.con".
924 inline "cic:/CoRN/algebra/CSetoids/restr_bin_op_strext.con".
926 inline "cic:/CoRN/algebra/CSetoids/Build_SubCSetoid_bin_op.con".
928 inline "cic:/CoRN/algebra/CSetoids/restr_f_assoc.con".
931 End SubCSetoid_binary_operations
934 (* Begin_SpecReals *)
945 Ltac Step_final x := apply eq_transitive_unfolded with x; Algebra.
951 Tactic Notation "Step_final" constr(c) := Step_final c.
957 inline "cic:/CoRN/algebra/CSetoids/proper_caseZ_diff_CS.con".
960 Finally, we characterize functions defined on the natural numbers also as setoid functions, similarly to what we already did for predicates.
963 inline "cic:/CoRN/algebra/CSetoids/nat_less_n_fun.con".
965 inline "cic:/CoRN/algebra/CSetoids/nat_less_n_fun'.con".
968 Implicit Arguments nat_less_n_fun [S n].
972 Implicit Arguments nat_less_n_fun' [S n].