--- /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/model/structures/Qsec".
+
+include "CoRN.ma".
+
+(* $Id: Qsec.v,v 1.7 2004/04/08 08:20:35 lcf Exp $ *)
+
+(*#* printing Q %\ensuremath{\mathbb{Q}}% *)
+
+(*#* printing QZERO %\ensuremath{0_\mathbb{Q}}% #0<sub>Q</sub># *)
+
+(*#* printing QONE %\ensuremath{1_\mathbb{Q}}% #1<sub>Q</sub># *)
+
+(*#* printing QTWO %\ensuremath{2_\mathbb{Q}}% #2<sub>Q</sub># *)
+
+(*#* printing QFOUR %\ensuremath{4_\mathbb{Q}}% #4<sub>Q</sub># *)
+
+include "algebra/CLogic.ma".
+
+include "model/structures/Zsec.ma".
+
+(*#* *[Q]
+**About [Q]
+We define the structure of rational numbers as follows. First of all,
+it consists of the set of rational numbers, defined as the set of
+pairs $\langle a,n\rangle$#⟨a,n⟩# with [a:Z] and
+[n:positive]. Intuitively, $\langle a,n\rangle$#⟨a,n⟩#
+represents the rational number [a[/]n]. Then there is the equality on
+[Q]: $\langle a,m\rangle=\langle
+b,n\rangle$#⟨a,m⟩=⟨b,n⟩# iff [an [=] bm]. We
+also define apartness, order, addition, multiplication, opposite,
+inverse an de constants 0 and 1. *)
+
+(* UNEXPORTED
+Section Q
+*)
+
+inline "cic:/CoRN/model/structures/Qsec/Q.ind".
+
+inline "cic:/CoRN/model/structures/Qsec/Qeq.con".
+
+inline "cic:/CoRN/model/structures/Qsec/Qap.con".
+
+inline "cic:/CoRN/model/structures/Qsec/Qlt.con".
+
+inline "cic:/CoRN/model/structures/Qsec/Qplus.con".
+
+inline "cic:/CoRN/model/structures/Qsec/Qmult.con".
+
+inline "cic:/CoRN/model/structures/Qsec/Qopp.con".
+
+inline "cic:/CoRN/model/structures/Qsec/QZERO.con".
+
+inline "cic:/CoRN/model/structures/Qsec/QONE.con".
+
+inline "cic:/CoRN/model/structures/Qsec/Qinv.con".
+
+(* UNEXPORTED
+End Q
+*)
+
+(* NOTATION
+Infix "{=Q}" := Qeq (no associativity, at level 90).
+*)
+
+(* NOTATION
+Infix "{#Q}" := Qap (no associativity, at level 90).
+*)
+
+(* NOTATION
+Infix "{<Q}" := Qlt (no associativity, at level 90).
+*)
+
+(* NOTATION
+Infix "{+Q}" := Qplus (no associativity, at level 85).
+*)
+
+(* NOTATION
+Infix "{*Q}" := Qmult (no associativity, at level 80).
+*)
+
+(* NOTATION
+Notation "{-Q}" := Qopp (at level 1, left associativity).
+*)
+
+(*#* ***Constants
+*)
+
+inline "cic:/CoRN/model/structures/Qsec/QTWO.con".
+
+inline "cic:/CoRN/model/structures/Qsec/QFOUR.con".
+
+(*#* ***Equality
+Here we prove that [QONE] is #<i>#%\emph{%not equal%}%#</i># to [QZERO]:
+*)
+
+inline "cic:/CoRN/model/structures/Qsec/ONEQ_neq_ZEROQ.con".
+
+inline "cic:/CoRN/model/structures/Qsec/refl_Qeq.con".
+
+inline "cic:/CoRN/model/structures/Qsec/sym_Qeq.con".
+
+inline "cic:/CoRN/model/structures/Qsec/trans_Qeq.con".
+
+(*#*
+ The equality is decidable:
+*)
+
+inline "cic:/CoRN/model/structures/Qsec/dec_Qeq.con".
+
+(*#* ***Apartness
+*)
+
+inline "cic:/CoRN/model/structures/Qsec/Q_non_zero.con".
+
+inline "cic:/CoRN/model/structures/Qsec/ap_Q_irreflexive0.con".
+
+inline "cic:/CoRN/model/structures/Qsec/ap_Q_symmetric0.con".
+
+inline "cic:/CoRN/model/structures/Qsec/ap_Q_cotransitive0.con".
+
+inline "cic:/CoRN/model/structures/Qsec/ap_Q_tight0.con".
+
+(*#* ***Addition
+*)
+
+inline "cic:/CoRN/model/structures/Qsec/Qplus_simpl.con".
+
+(*#*
+ Addition is associative:
+*)
+
+inline "cic:/CoRN/model/structures/Qsec/Qplus_assoc.con".
+
+(*#*
+ [QZERO] as the neutral element for addition:
+*)
+
+inline "cic:/CoRN/model/structures/Qsec/QZERO_right.con".
+
+(*#*
+ Commutativity of addition:
+*)
+
+inline "cic:/CoRN/model/structures/Qsec/Qplus_sym.con".
+
+inline "cic:/CoRN/model/structures/Qsec/Qplus_strext0.con".
+
+inline "cic:/CoRN/model/structures/Qsec/ZEROQ_as_rht_unit0.con".
+
+inline "cic:/CoRN/model/structures/Qsec/ZEROQ_as_lft_unit0.con".
+
+inline "cic:/CoRN/model/structures/Qsec/Qplus_is_commut0.con".
+
+(*#* ***Opposite
+ [{-Q}] is a well defined unary operation:
+*)
+
+inline "cic:/CoRN/model/structures/Qsec/Qopp_simpl.con".
+
+(*#*
+ The group equation for [{-Q}]
+*)
+
+inline "cic:/CoRN/model/structures/Qsec/Qplus_inverse_r.con".
+
+(*#* ***Multiplication
+Next we shall prove the properties of multiplication. First we prove
+that [{*Q}] is well-defined
+*)
+
+inline "cic:/CoRN/model/structures/Qsec/Qmult_simpl.con".
+
+(*#*
+ and it is associative:
+*)
+
+inline "cic:/CoRN/model/structures/Qsec/Qmult_assoc.con".
+
+(*#*
+ [QONE] is the neutral element for multiplication:
+*)
+
+inline "cic:/CoRN/model/structures/Qsec/Qmult_n_1.con".
+
+(*#*
+ The commutativity for [{*Q}]:
+*)
+
+inline "cic:/CoRN/model/structures/Qsec/Qmult_sym.con".
+
+inline "cic:/CoRN/model/structures/Qsec/Qmult_plus_distr_r.con".
+
+(*#*
+ And a property of multiplication which says if [x [~=] Zero] and [xy [=] Zero] then [y [=] Zero]:
+*)
+
+inline "cic:/CoRN/model/structures/Qsec/Qmult_eq.con".
+
+inline "cic:/CoRN/model/structures/Qsec/Qmult_strext0.con".
+
+inline "cic:/CoRN/model/structures/Qsec/nonZero.con".
+
+(*#* ***Inverse
+*)
+
+inline "cic:/CoRN/model/structures/Qsec/Qinv_strext.con".
+
+inline "cic:/CoRN/model/structures/Qsec/Qinv_is_inv.con".
+
+(*#* ***Less-than
+*)
+
+inline "cic:/CoRN/model/structures/Qsec/Qlt_wd_right.con".
+
+inline "cic:/CoRN/model/structures/Qsec/Qlt_wd_left.con".
+
+inline "cic:/CoRN/model/structures/Qsec/Qlt_eq_gt_dec.con".
+
+inline "cic:/CoRN/model/structures/Qsec/Qlt_is_transitive_unfolded.con".
+
+inline "cic:/CoRN/model/structures/Qsec/Qlt_strext_unfolded.con".
+
+inline "cic:/CoRN/model/structures/Qsec/Qlt_is_irreflexive_unfolded.con".
+
+inline "cic:/CoRN/model/structures/Qsec/Qlt_is_antisymmetric_unfolded.con".
+
+inline "cic:/CoRN/model/structures/Qsec/Qplus_resp_Qlt.con".
+
+inline "cic:/CoRN/model/structures/Qsec/Qmult_resp_pos_Qlt.con".
+
+inline "cic:/CoRN/model/structures/Qsec/Qlt_gives_apartness.con".
+
+(*#* ***Miscellaneous
+We consider the injection [inject_Z] from [Z] to [Q] as a coercion.
+*)
+
+inline "cic:/CoRN/model/structures/Qsec/inject_Z.con".
+
+coercion cic:/matita/CoRN-Decl/model/structures/Qsec/inject_Z.con 0 (* compounds *).
+
+inline "cic:/CoRN/model/structures/Qsec/injz_plus.con".
+
+inline "cic:/CoRN/model/structures/Qsec/injZ_One.con".
+
+(*#* We can always find a natural number that is bigger than a given rational
+number.
+*)
+
+inline "cic:/CoRN/model/structures/Qsec/Q_is_archemaedian0.con".
+
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