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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+
+
+include "nat/plus.ma".
+
+axiom R:Type.
+axiom R0:R.
+axiom R1:R.
+axiom Rplus: R→R→R.
+axiom Ropp:R→R. (*funzione da x → -x*)
+axiom Rmult: R→R→R.
+axiom Rdiv: R→R→R.
+axiom Relev: R→R→R.
+axiom Rle: R→R→Prop.
+axiom Rge: R→R→Prop.
+
+interpretation "real plus" 'plus x y =
+ (cic:/matita/didactic/reals/Rplus.con x y).
+
+interpretation "real opp" 'uminus x =
+ (cic:/matita/didactic/reals/Ropp.con x). 
+
+notation "hvbox(a break · b)"
+  right associative with precedence 55
+for @{ 'mult $a $b }.
+
+interpretation "real mult" 'mult x y =
+ (cic:/matita/didactic/reals/Rmult.con x y).
+
+interpretation "real leq" 'leq x y =
+ (cic:/matita/didactic/reals/Rle.con x y).
+
+interpretation "real geq" 'geq x y =
+ (cic:/matita/didactic/reals/Rge.con x y).
+
+let rec elev (x:R) (n:nat) on n: R ≝
+ match n with
+  [O ⇒ R1
+  | S n ⇒ Rmult x (elev x n)
+  ].
+
+let rec real_of_nat (n:nat) : R ≝
+ match n with
+  [ O ⇒ R0
+  | S n ⇒
+     match n with
+      [ O ⇒ R1
+      | _ ⇒ real_of_nat n + R1
+      ]
+  ].
+
+coercion cic:/matita/didactic/reals/real_of_nat.con.
+
+axiom Rplus_commutative: ∀x,y:R. x+y = y+x.
+axiom R0_neutral: ∀x:R. x+R0=x.
+axiom Rdiv_le: ∀x,y:R. R1 ≤ y → Rdiv x y ≤ x.
+axiom R2_1: R1 ≤ S (S O).
+(* assioma falso! *)
+axiom Rmult_Rle: ∀x,y,z,w. x ≤ y → z ≤ w → Rmult x z ≤ Rmult y w.
+
+axiom Rdiv_pos: ∀ x,y:R. R0 ≤ x → R1 ≤ y → R0 ≤ Rdiv x y.
+axiom Rle_R0_R1: R0 ≤ R1.
+axiom div: ∀x:R. x = Rdiv x (S (S O)) → x = O.
+(* Proprieta' elevamento a potenza NATURALE *)
+axiom elev_incr: ∀x:R.∀n:nat. R1 ≤ x → elev x (S n) ≥ elev x n.
+axiom elev_decr: ∀x:R.∀n:nat. R0 ≤ x ∧ x ≤ R1 → elev x (S n) ≤ elev x n.
+axiom Rle_to_Rge: ∀x,y:R. x ≤ y → y ≥ x.
+axiom Rge_to_Rle: ∀x,y:R. x ≥ y → y ≤ x.
+
+(* Proprieta' elevamento a potenza TRA REALI *)
+(*
+axiom Relev_ge: ∀x,y:R.
+ (x ≥ R1 ∧ y ≥ R1) ∨ (x ≤ R1 ∧ y ≤ R1) → Relev x y ≥ x.
+axiom Relev_le: ∀x,y:R.
+ (x ≥ R1 ∧ y ≤ R1) ∨ (x ≤ R1 ∧ y ≥ R1) → Relev x y ≤ x.
+*)
+
+lemma stupido: ∀x:R.R0+x=x.
+ assume x:R.
+ conclude (R0+x) = (x+R0) by _.
+               = x by _
+ done.
+qed.
+
+axiom opposto1: ∀x:R. x + -x = R0.
+axiom opposto2: ∀x:R. -x = Rmult x (-R1).
+axiom meno_piu: Rmult (-R1) (-R1) = R1.
+axiom R1_neutral: ∀x:R.Rmult R1 x = x.
+(* assioma falso *)
+axiom uffa: ∀x,y:R. R1 ≤ x → y ≤ x · y.