From: Andrea Asperti <andrea.asperti@unibo.it>
Date: Mon, 8 Oct 2007 13:02:33 +0000 (+0000)
Subject: Some axioms for Q.
X-Git-Tag: make_still_working~5992
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=962891404ac84c5f91425b89611ac9d8eca13f46;p=helm.git

Some axioms for Q.
---

diff --git a/helm/software/matita/library/Q/Qaxioms.ma b/helm/software/matita/library/Q/Qaxioms.ma
new file mode 100644
index 000000000..6cf6b6994
--- /dev/null
+++ b/helm/software/matita/library/Q/Qaxioms.ma
@@ -0,0 +1,71 @@
+(**************************************************************************)
+(*       ___	                                                            *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||       A.Asperti, C.Sacerdoti Coen,                          *)
+(*      ||A||       E.Tassi, S.Zacchiroli                                 *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU Lesser General Public License Version 2.1         *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/Q/Qaxioms".
+
+include "Z/compare.ma".
+include "Z/times.ma".
+include "nat/iteration2.ma".
+
+(* a fraction is a list of Z-coefficients for primes, in natural
+order. The last coefficient must eventually be different from 0 *)
+
+axiom Q:Type.
+axiom Qopp:Q \to Q.
+axiom Qinv:Q \to Q.
+axiom Qplus:Q \to Q \to Q.
+axiom Qtimes:Q \to Q \to Q.
+axiom QO:Q.
+axiom Q1:Q.
+axiom Qlt:Q \to Q \to Prop.
+
+axiom num: Q \to Z.
+axiom denom: Q \to nat.
+axiom frac: Z \to nat \to Q.
+
+(* plus *)
+axiom symmetric_Qplus: symmetric ? Qplus.
+axiom associative_Qplus: associative ? Qplus.
+axiom Qplus_QO: \forall q:Q.Qplus q QO = q.
+axiom Qplus_Qopp: \forall q:Q.Qplus q (Qopp q) = QO.
+
+(* times *)
+axiom symmetric_Qtimes: symmetric ? Qtimes.
+axiom associative_Qtimes: associative ? Qtimes.
+axiom Qtimes_Q1: \forall q:Q.Qtimes q Q1 = q.
+axiom Qtimes_Qinv: \forall q:Q.q \neq QO \to Qtimes q (Qinv q) = Q1.
+
+(* plus times *)
+axiom distributive_Qtimes_Qplus: distributive ? Qtimes Qplus.
+
+axiom frac_num_denom: \forall q.frac (num q) (denom q) = q.
+axiom frac_O: \forall n. frac O n = QO.
+axiom frac_n: \forall n. n\neq O \to frac n n = Q1.
+axiom Qtimes_frac : \forall a,b,c,d.Qtimes (frac a b) (frac c d) =
+(frac (a * c) (b * d)). 
+alias symbol "times" = "natural times".
+axiom Qplus_frac:\forall a,b,c,d.Qplus (frac a b) (frac c d) =
+(frac (a * d + b * c) (b * d)).
+axiom Qlt_fracl:\forall a,b,c,d.Qlt (frac a b) (frac c d) \to
+a*d \lt b*c. 
+axiom Qlt_fracr:\forall a,b,c,d.a*d \lt b*c \to Qlt (frac a b) (frac c d).
+axiom frac_Qopp: \forall a,b.Qopp(frac a b) = frac (Zopp a) b.
+axiom frac_Qinv1: \forall a,b:nat.Qinv(frac a b) = frac b a.
+axiom frac_Qinv2: \forall a,b:nat.Qinv(frac (Zopp a) b) = frac (Zopp b) a.
+
+definition sigma_Q \def \lambda n,p,f.iter_p_gen n p Q f QO Qplus. 
+
+theorem geometric: \forall q.\exists k. 
+Qlt q (sigma_Q k (\lambda x.true) (\lambda x. frac (S O) x)).
+    
\ No newline at end of file
diff --git a/helm/software/matita/library/Q/q.ma b/helm/software/matita/library/Q/q.ma
index 07a992926..23ef52551 100644
--- a/helm/software/matita/library/Q/q.ma
+++ b/helm/software/matita/library/Q/q.ma
@@ -319,3 +319,23 @@ intro.elim r.
 reflexivity.
 simplify.apply ftimes_finv.
 qed.
+
+definition Qtimes : Q \to Q \to Q \def
+\lambda p,q.
+  match p with
+  [OQ \Rightarrow OQ
+  |Qpos p1 \Rightarrow 
+    match q with
+    [OQ \Rightarrow OQ
+    |Qpos q1 \Rightarrow Qpos (rtimes p1 q1)
+    |Qneg q1 \Rightarrow Qneg (rtimes p1 q1)
+    ]
+  |neg p1 \Rightarrow 
+    match q with
+    [OQ \Rightarrow OQ
+    |Qpos q1 \Rightarrow Qneg (rtimes p1 q1)
+    |Qneg q1 \Rightarrow Qpos (rtimes p1 q1)
+    ]
+  ]
+.
+    
\ No newline at end of file