X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=inline;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fdemo%2Fpower_derivative.ma;h=bca5bbf0181a941d650e4bd24190706673f4ac36;hb=93cc2a2254a2620000377dfc99a7aaedf2b8ec63;hp=e8ab55c3595108eb3e08a8d60c49819752831832;hpb=ca41435a6021292ccba239aa173651c0be705b45;p=helm.git

diff --git a/helm/software/matita/library/demo/power_derivative.ma b/helm/software/matita/library/demo/power_derivative.ma
index e8ab55c35..bca5bbf01 100644
--- a/helm/software/matita/library/demo/power_derivative.ma
+++ b/helm/software/matita/library/demo/power_derivative.ma
@@ -24,28 +24,17 @@ 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)).
+interpretation "Rzero" 'zero = (R0).
+interpretation "Nzero" 'zero = (O).
 
 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 "Rone" 'one = (R1).
+interpretation "None" 'one = (S O).
 
-interpretation "Rplus" 'plus x y =
- (cic:/matita/demo/power_derivative/Rplus.con x y).
+interpretation "Rplus" 'plus x y = (Rplus 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).
+interpretation "Rmult" 'middot x y = (Rmult x y).
 
 definition Fplus ≝
  λf,g:R→R.λx:R.f x + g x.
@@ -53,21 +42,13 @@ definition Fplus ≝
 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).
+interpretation "Fplus" 'plus x y = (Fplus x y).
+interpretation "Fmult" 'middot x y = (Fmult 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)))).
+interpretation "Rtwo" 'two = (Rplus R1 R1).
+interpretation "Ntwo" 'two = (S (S O)).
 
 let rec Rpower (x:R) (n:nat) on n ≝
  match n with
@@ -75,8 +56,7 @@ let rec Rpower (x:R) (n:nat) on n ≝
   | S n ⇒ x · (Rpower x n)
   ].
 
-interpretation "Rpower" 'exp x n =
- (cic:/matita/demo/power_derivative/Rpower.con x n).
+interpretation "Rpower" 'exp x n = (Rpower x n).
 
 let rec inj (n:nat) on n : R ≝
  match n with
@@ -93,13 +73,13 @@ coercion inj.
 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_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 "middot" = "Rmult".
 alias symbol "plus" = "natural plus".
 
 definition monomio ≝
@@ -227,59 +207,61 @@ notation "hvbox('D'[f])"
   non associative with precedence 90
 for @{ 'derivative $f }.
 
-interpretation "Rderivative" 'derivative f =
- (cic:/matita/demo/power_derivative/derivative.con f).
+interpretation "Rderivative" 'derivative f = (derivative f).
 
-notation "hvbox('x' \sup n)"
+(*  FG: we definitely do not want 'x' as a keyward! 
+ *  Any file that includes this one can not use 'x' as an identifier
+ *)  
+notation "hvbox('X' \sup n)"
   non associative with precedence 60
 for @{ 'monomio $n }.
 
-notation "hvbox('x')"
+notation "hvbox('X')"
   non associative with precedence 60
 for @{ 'monomio 1 }.
 
-interpretation "Rmonomio" 'monomio n =
- (cic:/matita/demo/power_derivative/monomio.con n).
+interpretation "Rmonomio" 'monomio n = (monomio n).
+
+axiom derivative_x0: D[X \sup 0] = 0.
+axiom derivative_x1: D[X] = 1.
 
-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".
+alias symbol "middot" = "Fmult".
 
-theorem derivative_power: ∀n:nat. D[x \sup n] = n·x \sup (pred n).
+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)).*)
+ (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)).
+   the thesis becomes (D[X \sup 0] = 0·X \sup (pred 0)).
   done.
  case S (m:nat).
   by induction hypothesis we know
-   (D[x \sup m] = m·x \sup (pred m)) (H).
+   (D[X \sup m] = m·X \sup (pred m)) (H).
   the thesis becomes
-   (D[x \sup (1+m)] = (1+m) · x \sup m).
+   (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).
+   (m · (X \sup (1+ pred m)) = m · X \sup m) (Ppred).
    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).
+   to prove (m · (X \sup (1+ pred m)) = m · X \sup m).
     case left.
       suppose (0 < m) (m_pos).
       using (S_pred ? m_pos) we proved (m = 1 + pred m) (H1).
-     done.
+      by H1 done.
     case right.
       suppose (0=m) (m_zero). 
-    done.
+    by m_zero, Fmult_zero_f 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))) timeout=30.
-   = (x \sup m + m · (x \sup (1 + pred m))).
-   = (x \sup m + m · x \sup m).
-   = ((1+m) · x \sup m) timeout=30 by Fmult_one_f, Fmult_commutative, Fmult_Fplus_distr, costante_sum
+     (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))) timeout=30.
+   = (X \sup m + m · (X \sup (1 + pred m))).
+   = (X \sup m + m · X \sup m).
+   = ((1+m) · X \sup m) timeout=30 by Fmult_one_f, Fmult_commutative, Fmult_Fplus_distr, costante_sum
   done.
 qed.
 
@@ -290,35 +272,33 @@ 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).
+interpretation "Rderivative" 'derivative \eta.f = (derivative f).
 *)
 
-notation "hvbox(\frac 'd' ('d' 'x') break p)" with precedence 90
+notation "hvbox(\frac 'd' ('d' 'X') break p)" with precedence 90
 for @{ 'derivative $p}.
 
-interpretation "Rderivative" 'derivative f =
- (cic:/matita/demo/power_derivative/derivative.con f).
+interpretation "Rderivative" 'derivative f = (derivative f).
 
-theorem derivative_power': ∀n:nat. D[x \sup (1+n)] = (1+n) · x \sup n.
+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.
+ (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).
+   the thesis becomes (D[X \sup 1] = 1 · X \sup 0).
   done.
  case S (m:nat).
   by induction hypothesis we know
-   (D[x \sup (1+m)] = (1+m) · x \sup m) (H).
+   (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)).
+   (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)) timeout=30 by Fmult_one_f, Fmult_commutative,
+     (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)) timeout=30 by Fmult_one_f, Fmult_commutative,
        Fmult_Fplus_distr, assoc_plus, plus_n_SO, costante_sum
   done.
 qed.