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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 "reals.ma".
+
+axiom F:Type.(*F=funzioni regolari*)
+axiom fplus:F→F→F.
+axiom fmult:F→F→F.
+axiom fcomp:F→F→F.
+
+axiom De: F→F. (*funzione derivata*)
+notation "a '"
+  non associative with precedence 80
+for @{ 'deriv $a }.
+interpretation "function derivative" 'deriv x =
+ (cic:/matita/didactic/deriv/De.con x). 
+interpretation "function mult" 'mult x y =
+ (cic:/matita/didactic/deriv/fmult.con x y). 
+interpretation "function compositon" 'compose x y =
+ (cic:/matita/didactic/deriv/fcomp.con x y).
+
+notation "hvbox(a break + b)"
+  right associative with precedence 45
+for @{ 'oplus $a $b }.
+
+interpretation "function plus" 'plus x y =
+ (cic:/matita/didactic/deriv/fplus.con x y).
+
+axiom i:R→F. (*mappatura  R in F*)
+coercion cic:/matita/didactic/deriv/i.con.
+axiom i_comm_plus: ∀x,y:R. (i (x+y)) = (i x) + (i y).
+axiom i_comm_mult: ∀x,y:R. (i (Rmult x y)) = (i x) · (i y).
+
+axiom freflex:F.
+notation "ρ"
+  non associative with precedence 100
+for @{ 'rho }.
+interpretation "function flip" 'rho =
+ cic:/matita/didactic/deriv/freflex.con.
+axiom reflex_ok: ∀f:F. ρ ∘ f = (i (-R1)) · f.
+axiom dereflex: ρ ' = i (-R1). (*Togliere*)
+
+axiom id:F. (* Funzione identita' *)
+axiom fcomp_id_neutral: ∀f:F. f ∘ id = f.
+axiom fcomp_id_commutative: ∀f:F. f ∘ id = id ∘ f.
+axiom deid: id ' = i R1.
+axiom rho_id: ρ ∘ ρ = id.
+
+lemma rho_disp: ρ = ρ ∘ (ρ ∘ ρ).
+ we need to prove (ρ = ρ ∘ (ρ ∘ ρ)).
+ by _ done.
+qed.
+
+lemma id_disp: id = ρ ∘ (id ∘ ρ).
+ we need to prove (id = ρ ∘ (id ∘ ρ)).
+ by _ done.
+qed.
+
+let rec felev (f:F) (n:nat) on n: F ≝
+ match n with
+ [ O ⇒ i R1
+ | S n ⇒ f · (felev f n)
+ ].
+
+(* Proprietà *)
+
+axiom fplus_commutative: ∀ f,g:F. f + g = g + f.
+axiom fplus_associative: ∀ f,g,h:F. f + (g + h) = (f + g) + h.
+axiom fplus_neutral: ∀f:F. (i R0) + f = f.
+axiom fmult_commutative: ∀ f,g:F. f · g = g · f.
+axiom fmult_associative: ∀ f,g,h:F. f · (g · h) = (f · g) · h.
+axiom fmult_neutral: ∀f:F. (i R1) · f = f.
+axiom fmult_assorb: ∀f:F. (i R0) · f = (i R0).
+axiom fdistr: ∀ f,g,h:F. (f + g) · h = (f · h) + (g · h).
+axiom fcomp_associative: ∀ f,g,h:F. f ∘ (g ∘ h) = (f ∘ g) ∘ h.
+
+axiom fcomp_distr1: ∀ f,g,h:F. (f + g) ∘ h = (f ∘ h) + (g ∘ h).
+axiom fcomp_distr2: ∀ f,g,h:F. (f · g) ∘ h = (f ∘ h) · (g ∘ h).
+
+axiom demult: ∀ f,g:F. (f · g) ' = (f ' · g) + (f · g ').
+axiom decomp: ∀ f,g:F. (f ∘ g) ' = (f ' ∘ g) · g '.
+axiom deplus: ∀ f,g:F. (f + g) ' = (f ') + (g ').
+
+axiom cost_assorb: ∀x:R. ∀f:F. (i x) ∘ f = i x.
+axiom cost_deriv: ∀x:R. (i x) ' = i R0.
+
+
+definition fpari ≝ λ f:F. f = f ∘ ρ.
+definition fdispari ≝ λ f:F. f = ρ ∘ (f ∘ ρ).
+axiom cost_pari: ∀ x:R. fpari (i x).
+
+axiom meno_piu_i: (i (-R1)) · (i (-R1)) = i R1.
+
+notation "hvbox(a break ^ b)"
+  right associative with precedence 75
+for @{ 'elev $a $b }.
+ 
+interpretation "function power" 'elev x y =
+ (cic:/matita/didactic/deriv/felev.con x y). 
+
+axiom tech1: ∀n,m. F_OF_nat n + F_OF_nat m = F_OF_nat (n + m).