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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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/demo/power_derivative".
+
+include "nat/plus.ma".
+include "nat/orders.ma".
+include "nat/compare.ma".
+
+axiom R: Type.
+axiom R0: R.
+axiom R1: R.
+axiom Rplus: R→R→R.
+axiom Rmult: R→R→R.
+
+notation "0" with precedence 89
+for @{ 'zero }.
+interpretation "Rzero" 'zero =
+ (cic:/matita/demo/power_derivative/R0.con).
+interpretation "Nzero" 'zero =
+ (cic:/matita/nat/nat/nat.ind#xpointer(1/1/1)).
+
+notation "1" with precedence 89
+for @{ 'one }.
+interpretation "Rone" 'one =
+ (cic:/matita/demo/power_derivative/R1.con).
+interpretation "None" 'one =
+ (cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+   cic:/matita/nat/nat/nat.ind#xpointer(1/1/1)).
+
+interpretation "Rplus" 'plus x y =
+ (cic:/matita/demo/power_derivative/Rplus.con x y).
+
+notation "hvbox(a break \middot b)" 
+  left associative with precedence 55
+for @{ 'times $a $b }.
+
+interpretation "Rmult" 'times x y =
+ (cic:/matita/demo/power_derivative/Rmult.con x y).
+
+definition Fplus ≝
+ λf,g:R→R.λx:R.f x + g x.
+ 
+definition Fmult ≝
+ λf,g:R→R.λx:R.f x · g x.
+
+interpretation "Fplus" 'plus x y =
+ (cic:/matita/demo/power_derivative/Fplus.con x y).
+interpretation "Fmult" 'times x y =
+ (cic:/matita/demo/power_derivative/Fmult.con x y).
+
+notation "2" with precedence 89
+for @{ 'two }.
+interpretation "Rtwo" 'two =
+ (cic:/matita/demo/power_derivative/Rplus.con
+   cic:/matita/demo/power_derivative/R1.con
+   cic:/matita/demo/power_derivative/R1.con).
+interpretation "Ntwo" 'two =
+ (cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+   (cic:/matita/nat/nat/nat.ind#xpointer(1/1/2)
+     (cic:/matita/nat/nat/nat.ind#xpointer(1/1/1)))).
+
+let rec Rpower (x:R) (n:nat) on n ≝
+ match n with
+  [ O ⇒ 1
+  | S n ⇒ x · (Rpower x n)
+  ].
+
+interpretation "Rpower" 'exp x n =
+ (cic:/matita/demo/power_derivative/Rpower.con x n).
+
+let rec inj (n:nat) on n : R ≝
+ match n with
+  [ O ⇒ 0
+  | S n ⇒
+     match n with
+      [ O ⇒ 1
+      | S m ⇒ 1 + inj n
+      ]
+  ].
+
+coercion cic:/matita/demo/power_derivative/inj.con.
+
+axiom Rplus_Rzero_x: ∀x:R.0+x=x.
+axiom Rplus_comm: symmetric ? Rplus.
+axiom Rplus_assoc: associative ? Rplus.
+axiom Rmult_Rone_x: ∀x:R.1*x=x.
+axiom Rmult_Rzero_x: ∀x:R.0*x=0.
+axiom Rmult_assoc: associative ? Rmult.
+axiom Rmult_comm: symmetric ? Rmult.
+axiom Rmult_Rplus_distr: distributive ? Rmult Rplus.
+
+alias symbol "times" = "Rmult".
+alias symbol "plus" = "natural plus".
+
+definition monomio ≝
+ λn.λx:R.x\sup n.
+
+definition costante : nat → R → R ≝
+ λa:nat.λx:R.inj a.
+
+coercion cic:/matita/demo/power_derivative/costante.con 1.
+
+axiom f_eq_extensional:
+ ∀f,g:R→R.(∀x:R.f x = g x) → f=g.
+
+lemma Fmult_one_f: ∀f:R→R.1·f=f.
+ intro;
+ unfold Fmult;
+ simplify;
+ apply f_eq_extensional;
+ intro;
+ autobatch.
+qed.
+
+lemma Fmult_zero_f: ∀f:R→R.0·f=0.
+ intro;
+ unfold Fmult;
+ simplify;
+ apply f_eq_extensional;
+ intro;
+ autobatch.
+qed.
+
+lemma Fmult_commutative: symmetric ? Fmult.
+ unfold;
+ intros;
+ unfold Fmult;
+ apply f_eq_extensional;
+ intros;
+ autobatch.
+qed.
+
+lemma Fmult_associative: associative ? Fmult.
+ unfold;
+ intros;
+ unfold Fmult;
+ unfold Fmult;
+ apply f_eq_extensional;
+ intros;
+ autobatch.
+qed.
+
+lemma Fmult_Fplus_distr: distributive ? Fmult Fplus.
+ unfold;
+ intros;
+ unfold Fmult;
+ unfold Fplus;
+ apply f_eq_extensional;
+ intros;
+ simplify;
+ autobatch.
+qed.
+
+lemma monomio_product:
+ ∀n,m.monomio (n+m) = monomio n · monomio m.
+ intros;
+ unfold monomio;
+ unfold Fmult;
+ simplify;
+ elim n;
+  [ simplify;
+    apply f_eq_extensional;
+    intro;
+    autobatch
+  | simplify;
+    apply f_eq_extensional;
+    intro;
+    cut (x\sup (n1+m) = x \sup n1 · x \sup m);
+     [ rewrite > Hcut;
+       autobatch
+     | change in ⊢ (? ? % ?) with ((λx:R.x\sup(n1+m)) x);
+       rewrite > H;
+       reflexivity
+     ]
+  ].
+qed.
+
+lemma costante_sum:
+ ∀n,m.costante n + costante m = costante (n+m).
+ intros;
+ unfold Fplus;
+ unfold costante;
+ apply f_eq_extensional;
+ intros;
+ elim n;
+  [ simplify;
+    autobatch
+  | simplify;
+    clear x;
+    clear H;
+    clear n;
+    elim n1;
+     [ simplify;
+       elim m;
+        [ simplify;
+          autobatch
+        | simplify;
+          rewrite < H;
+          autobatch
+        ]
+     | simplify;
+       rewrite < H;
+       clear H;
+       elim n;
+        [ simplify;
+          autobatch
+        | simplify;
+          autobatch
+        ]
+     ]
+   ].
+qed.
+
+axiom derivative: (R→R) → R → R.
+
+notation "hvbox('D'[f])"
+  non associative with precedence 90
+for @{ 'derivative $f }.
+
+interpretation "Rderivative" 'derivative f =
+ (cic:/matita/demo/power_derivative/derivative.con f).
+
+notation "hvbox('x' \sup n)"
+  non associative with precedence 60
+for @{ 'monomio $n }.
+
+notation "hvbox('x')"
+  non associative with precedence 60
+for @{ 'monomio 1 }.
+
+interpretation "Rmonomio" 'monomio n =
+ (cic:/matita/demo/power_derivative/monomio.con n).
+
+axiom derivative_x0: D[x \sup 0] = 0.
+axiom derivative_x1: D[x] = 1.
+axiom derivative_mult: ∀f,g:R→R. D[f·g] = D[f]·g + f·D[g].
+
+alias symbol "times" = "Fmult".
+
+theorem derivative_power: ∀n:nat. D[x \sup n] = n·x \sup (pred n).
+ assume n:nat.
+ (*we proceed by induction on n to prove
+ (D[x \sup n] = n · x \sup (pred n)).*)
+ elim n 0.
+ case O.
+   the thesis becomes (D[x \sup 0] = 0·x \sup (pred 0)).
+   by _
+  done.
+ case S (m:nat).
+  by induction hypothesis we know
+   (D[x \sup m] = m·x \sup (pred m)) (H).
+  the thesis becomes
+   (D[x \sup (1+m)] = (1+m) · x \sup m).
+  we need to prove
+   (m · (x \sup (1+ pred m)) = m · x \sup m) (Ppred).
+   by _ we proved (0 < m ∨ 0=m) (cases).
+   we proceed by induction on cases
+   to prove (m · (x \sup (1+ pred m)) = m · x \sup m).
+    case left.
+      suppose (0 < m) (m_pos).
+      by (S_pred m m_pos) we proved (m = 1 + pred m) (H1).
+      by _
+     done.
+    case right.
+      suppose (0=m) (m_zero). by _ done.
+  conclude
+     (D[x \sup (1+m)])
+   = (D[x · x \sup m]).
+   = (D[x] · x \sup m + x · D[x \sup m]).
+   = (x \sup m + x · (m · x \sup (pred m))).
+   = (x \sup m + m · (x \sup (1 + pred m))).
+   = (x \sup m + m · x \sup m).
+   = ((1+m) · x \sup m) by Fmult_one_f Fmult_commutative Fmult_Fplus_distr costante_sum
+  done.
+qed.
+
+(*
+notation "hvbox(\frac 'd' ('d' ident i) break p)"
+  right associative with precedence 90
+for @{ 'derivative ${default
+  @{\lambda ${ident i} : $ty. $p)}
+  @{\lambda ${ident i} . $p}}}.
+
+interpretation "Rderivative" 'derivative \eta.f =
+ (cic:/matita/demo/power_derivative/derivative.con f).
+*)
+
+notation "hvbox(\frac 'd' ('d' 'x') break p)"
+  right associative with precedence 90
+for @{ 'derivative $p}.
+
+interpretation "Rderivative" 'derivative f =
+ (cic:/matita/demo/power_derivative/derivative.con f).
+
+theorem derivative_power': ∀n:nat. D[x \sup (1+n)] = (1+n) · x \sup n.
+ assume n:nat.
+ (*we proceed by induction on n to prove
+ (D[x \sup (1+n)] = (1+n) · x \sup n).*) elim n 0.
+ case O.
+   the thesis becomes (D[x \sup 1] = 1 · x \sup 0).
+   by _
+  done.
+ case S (m:nat).
+  by induction hypothesis we know
+   (D[x \sup (1+m)] = (1+m) · x \sup m) (H).
+  the thesis becomes
+   (D[x \sup (2+m)] = (2+m) · x \sup (1+m)).
+  conclude
+     (D[x \sup (2+m)])
+   = (D[x · x \sup (1+m)]).
+   = (D[x] · x \sup (1+m) + x · D[x \sup (1+m)]).
+   = (x \sup (1+m) + x · (costante (1+m) · x \sup m)).
+   = (x \sup (1+m) + costante (1+m) · x \sup (1+m)).
+   = ((2+m) · x \sup (1+m)) by Fmult_one_f Fmult_commutative
+     Fmult_Fplus_distr assoc_plus plus_n_SO costante_sum
+  done.
+qed.
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