--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/CoRN-Decl/algebra/CAbGroups".
+
+include "CoRN.ma".
+
+include "algebra/CGroups.ma".
+
+(* UNEXPORTED
+Section Abelian_Groups
+*)
+
+(*#*
+* Abelian Groups
+Now we introduce commutativity and add some results.
+*)
+
+inline "cic:/CoRN/algebra/CAbGroups/is_CAbGroup.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/CAbGroup.ind".
+
+coercion cic:/matita/CoRN-Decl/algebra/CAbGroups/cag_crr.con 0 (* compounds *).
+
+(* UNEXPORTED
+Section AbGroup_Axioms
+*)
+
+alias id "G" = "cic:/CoRN/algebra/CAbGroups/Abelian_Groups/AbGroup_Axioms/G.var".
+
+(*#*
+%\begin{convention}% Let [G] be an Abelian Group.
+%\end{convention}%
+*)
+
+inline "cic:/CoRN/algebra/CAbGroups/CAbGroup_is_CAbGroup.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/cag_commutes.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/cag_commutes_unfolded.con".
+
+(* UNEXPORTED
+End AbGroup_Axioms
+*)
+
+(* UNEXPORTED
+Section SubCAbGroups
+*)
+
+(*#*
+** Subgroups of an Abelian Group
+*)
+
+alias id "G" = "cic:/CoRN/algebra/CAbGroups/Abelian_Groups/SubCAbGroups/G.var".
+
+alias id "P" = "cic:/CoRN/algebra/CAbGroups/Abelian_Groups/SubCAbGroups/P.var".
+
+alias id "Punit" = "cic:/CoRN/algebra/CAbGroups/Abelian_Groups/SubCAbGroups/Punit.var".
+
+alias id "op_pres_P" = "cic:/CoRN/algebra/CAbGroups/Abelian_Groups/SubCAbGroups/op_pres_P.var".
+
+alias id "inv_pres_P" = "cic:/CoRN/algebra/CAbGroups/Abelian_Groups/SubCAbGroups/inv_pres_P.var".
+
+(*#*
+%\begin{convention}%
+Let [G] be an Abelian Group and [P] be a ([CProp]-valued) predicate on [G]
+that contains [Zero] and is closed under [[+]] and [[--]].
+%\end{convention}%
+*)
+
+inline "cic:/CoRN/algebra/CAbGroups/Abelian_Groups/SubCAbGroups/subcrr.con" "Abelian_Groups__SubCAbGroups__".
+
+inline "cic:/CoRN/algebra/CAbGroups/isabgrp_scrr.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/Build_SubCAbGroup.con".
+
+(* UNEXPORTED
+End SubCAbGroups
+*)
+
+(* UNEXPORTED
+Section Various
+*)
+
+(*#*
+** Basic properties of Abelian groups
+*)
+
+(* UNEXPORTED
+Hint Resolve cag_commutes_unfolded: algebra.
+*)
+
+alias id "G" = "cic:/CoRN/algebra/CAbGroups/Abelian_Groups/Various/G.var".
+
+(*#*
+%\begin{convention}% Let [G] be an Abelian Group.
+%\end{convention}%
+*)
+
+inline "cic:/CoRN/algebra/CAbGroups/cag_op_inv.con".
+
+(* UNEXPORTED
+Hint Resolve cag_op_inv: algebra.
+*)
+
+inline "cic:/CoRN/algebra/CAbGroups/assoc_1.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/minus_plus.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/op_lft_resp_ap.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/cag_ap_cancel_lft.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/plus_cancel_ap_lft.con".
+
+(* UNEXPORTED
+End Various
+*)
+
+(* UNEXPORTED
+End Abelian_Groups
+*)
+
+(* UNEXPORTED
+Hint Resolve cag_commutes_unfolded: algebra.
+*)
+
+(* UNEXPORTED
+Hint Resolve cag_op_inv assoc_1 zero_minus minus_plus op_lft_resp_ap: algebra.
+*)
+
+(* UNEXPORTED
+Section Nice_Char
+*)
+
+(*#*
+** Building an abelian group
+
+In order to actually define concrete abelian groups,
+it is not in general practical to construct first a semigroup,
+then a monoid, then a group and finally an abelian group. The
+presence of commutativity, for example, makes many of the monoid
+proofs trivial. In this section, we provide a constructor that
+will allow us to go directly from a setoid to an abelian group.
+
+We start from a setoid S with an element [unit], a
+commutative and associative binary operation [plus] which
+is strongly extensional in its first argument and admits [unit]
+as a left unit, and a unary setoid
+function [inv] which inverts elements respective to [plus].
+*)
+
+alias id "S" = "cic:/CoRN/algebra/CAbGroups/Nice_Char/S.var".
+
+alias id "unit" = "cic:/CoRN/algebra/CAbGroups/Nice_Char/unit.var".
+
+alias id "plus" = "cic:/CoRN/algebra/CAbGroups/Nice_Char/plus.var".
+
+(*#*
+%\begin{convention}%
+Let [S] be a Setoid and [unit:S], [plus:S->S->S] and [inv] a unary
+setoid operation on [S].
+Assume that [plus] is commutative, associative and `left-strongly-extensional
+([(plus x z) [#] (plus y z) -> x [#] y]), that [unit] is a left-unit
+for [plus] and [(inv x)] is a right-inverse of [x] w.r.t.%\% [plus].
+%\end{convention}%
+*)
+
+alias id "plus_lext" = "cic:/CoRN/algebra/CAbGroups/Nice_Char/plus_lext.var".
+
+alias id "plus_lunit" = "cic:/CoRN/algebra/CAbGroups/Nice_Char/plus_lunit.var".
+
+alias id "plus_comm" = "cic:/CoRN/algebra/CAbGroups/Nice_Char/plus_comm.var".
+
+alias id "plus_assoc" = "cic:/CoRN/algebra/CAbGroups/Nice_Char/plus_assoc.var".
+
+alias id "inv" = "cic:/CoRN/algebra/CAbGroups/Nice_Char/inv.var".
+
+alias id "inv_inv" = "cic:/CoRN/algebra/CAbGroups/Nice_Char/inv_inv.var".
+
+inline "cic:/CoRN/algebra/CAbGroups/plus_rext.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/plus_runit.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/plus_is_fun.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/inv_inv'.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/plus_fun.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/Build_CSemiGroup'.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/Build_CMonoid'.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/Build_CGroup'.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/Build_CAbGroup'.con".
+
+(* UNEXPORTED
+End Nice_Char
+*)
+
+(*#* ** Iteration
+
+For reflection the following is needed; hopefully it is also useful.
+*)
+
+(* UNEXPORTED
+Section Group_Extras
+*)
+
+alias id "G" = "cic:/CoRN/algebra/CAbGroups/Group_Extras/G.var".
+
+inline "cic:/CoRN/algebra/CAbGroups/nmult.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/nmult_wd.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/nmult_one.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/nmult_Zero.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/nmult_plus.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/nmult_mult.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/nmult_inv.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/nmult_plus'.con".
+
+(* UNEXPORTED
+Hint Resolve nmult_wd nmult_Zero nmult_inv nmult_plus nmult_plus': algebra.
+*)
+
+inline "cic:/CoRN/algebra/CAbGroups/zmult.con".
+
+(*
+Lemma Zeq_imp_nat_eq : forall m n:nat, m = n -> m = n.
+auto.
+intro m; induction m.
+intro n; induction n; auto.
+
+intro; induction n.
+intro. inversion H.
+intros.
+rewrite (IHm n).
+auto.
+repeat rewrite inj_S in H.
+auto with zarith.
+Qed.
+*)
+
+inline "cic:/CoRN/algebra/CAbGroups/zmult_char.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/zmult_wd.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/zmult_one.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/zmult_min_one.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/zmult_zero.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/zmult_Zero.con".
+
+(* UNEXPORTED
+Hint Resolve zmult_zero: algebra.
+*)
+
+inline "cic:/CoRN/algebra/CAbGroups/zmult_plus.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/zmult_mult.con".
+
+inline "cic:/CoRN/algebra/CAbGroups/zmult_plus'.con".
+
+(* UNEXPORTED
+End Group_Extras
+*)
+
+(* UNEXPORTED
+Hint Resolve nmult_wd nmult_one nmult_Zero nmult_plus nmult_inv nmult_mult
+ nmult_plus' zmult_wd zmult_one zmult_min_one zmult_zero zmult_Zero
+ zmult_plus zmult_mult zmult_plus': algebra.
+*)
+
+(* UNEXPORTED
+Implicit Arguments nmult [G].
+*)
+
+(* UNEXPORTED
+Implicit Arguments zmult [G].
+*)
+
projects / helm.git / blobdiff