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[helm.git] / matita / library / Q / Qaxioms.ma

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+(**************************************************************************)
+(*       ___                                                               *)
+(*      ||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