X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fdemo%2Fpower_derivative.ma;h=a425dfcc357783ee52a23f8215be3d91265d29b5;hb=070b44c9c2344967ca8c4531909614a0d4da2fbe;hp=879f55d7e4976227c48756d39a626a1120d12147;hpb=9a27004d7665a5659d1959392d3d0ff5b23b3603;p=helm.git

diff --git a/helm/software/matita/library/demo/power_derivative.ma b/helm/software/matita/library/demo/power_derivative.ma
index 879f55d7e..a425dfcc3 100644
--- a/helm/software/matita/library/demo/power_derivative.ma
+++ b/helm/software/matita/library/demo/power_derivative.ma
@@ -12,8 +12,6 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/demo/power_derivative".
-
 include "nat/plus.ma".
 include "nat/orders.ma".
 include "nat/compare.ma".
@@ -26,54 +24,39 @@ 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 = (Rplus x y).
 
-interpretation "Rplus" 'plus x y =
- (cic:/matita/demo/power_derivative/Rplus.con x y).
-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.
  
 definition Fmult ≝
- λf,g:R→R.λx:R.f x * g x.
+ 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
   [ O ⇒ 1
-  | S n ⇒ x * (Rpower x 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
@@ -85,18 +68,18 @@ let rec inj (n:nat) on n : R ≝
       ]
   ].
 
-coercion cic:/matita/demo/power_derivative/inj.con.
+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 ≝
@@ -105,27 +88,27 @@ definition monomio ≝
 definition costante : nat → R → R ≝
  λa:nat.λx:R.inj a.
 
-coercion cic:/matita/demo/power_derivative/costante.con 1.
+coercion costante with 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.
+lemma Fmult_one_f: ∀f:R→R.1·f=f.
  intro;
  unfold Fmult;
  simplify;
  apply f_eq_extensional;
  intro;
- auto.
+ autobatch.
 qed.
 
-lemma Fmult_zero_f: ∀f:R→R.0*f=0.
+lemma Fmult_zero_f: ∀f:R→R.0·f=0.
  intro;
  unfold Fmult;
  simplify;
  apply f_eq_extensional;
  intro;
- auto.
+ autobatch.
 qed.
 
 lemma Fmult_commutative: symmetric ? Fmult.
@@ -134,7 +117,7 @@ lemma Fmult_commutative: symmetric ? Fmult.
  unfold Fmult;
  apply f_eq_extensional;
  intros;
- auto.
+ autobatch.
 qed.
 
 lemma Fmult_associative: associative ? Fmult.
@@ -144,7 +127,7 @@ lemma Fmult_associative: associative ? Fmult.
  unfold Fmult;
  apply f_eq_extensional;
  intros;
- auto.
+ autobatch.
 qed.
 
 lemma Fmult_Fplus_distr: distributive ? Fmult Fplus.
@@ -155,11 +138,11 @@ lemma Fmult_Fplus_distr: distributive ? Fmult Fplus.
  apply f_eq_extensional;
  intros;
  simplify;
- auto.
+ autobatch.
 qed.
 
 lemma monomio_product:
- ∀n,m.monomio (n+m) = monomio n * monomio m.
+ ∀n,m.monomio (n+m) = monomio n · monomio m.
  intros;
  unfold monomio;
  unfold Fmult;
@@ -168,14 +151,14 @@ lemma monomio_product:
   [ simplify;
     apply f_eq_extensional;
     intro;
-    auto
+    autobatch
   | simplify;
     apply f_eq_extensional;
     intro;
-    cut (x\sup (n1+m) = x \sup n1 * x \sup m);
+    cut (x\sup (n1+m) = x \sup n1 · x \sup m);
      [ rewrite > Hcut;
-       auto
-     | change in ⊢ (? ? % ?) with ((λx:R.(x)\sup(n1+m)) x);
+       autobatch
+     | change in ⊢ (? ? % ?) with ((λx:R.x\sup(n1+m)) x);
        rewrite > H;
        reflexivity
      ]
@@ -191,7 +174,7 @@ lemma costante_sum:
  intros;
  elim n;
   [ simplify;
-    auto
+    autobatch
   | simplify;
     clear x;
     clear H;
@@ -200,19 +183,19 @@ lemma costante_sum:
      [ simplify;
        elim m;
         [ simplify;
-          auto
+          autobatch
         | simplify;
           rewrite < H;
-          auto
+          autobatch
         ]
      | simplify;
        rewrite < H;
        clear H;
        elim n;
         [ simplify;
-          auto
+          autobatch
         | simplify;
-          auto
+          autobatch
         ]
      ]
    ].
@@ -224,60 +207,65 @@ 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_mult: ∀f,g:R→R. D[f*g] = D[f]*g + f*D[g].
+axiom derivative_x0: D[X \sup 0] = 0.
+axiom derivative_x1: D[X] = 1.
 
-alias symbol "times" = "Fmult".
+axiom derivative_mult: ∀f,g:R→R. D[f·g] = D[f]·g + f·D[g].
 
-theorem derivative_power: ∀n:nat. D[x \sup n] = n*x \sup (pred n).
+alias symbol "middot" = "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)).
+ (*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 _
+   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).
-   by _ we proved (0 < m ∨ 0=m) (cases).
+   (m · (X \sup (1+ pred m)) = m · X \sup m) (Ppred).
+   lapply depth=0 le_n;
+   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).
-      by (S_pred m m_pos) we proved (m = 1 + pred m) (H1).
-      by _
-     done.
+      using (S_pred ? m_pos) we proved (m = 1 + pred m) (H1).
+      by H1 done.
     case right.
-      suppose (0=m) (m_zero). by _ done.
+      suppose (0=m) (m_zero). 
+    by m_zero, Fmult_zero_f done.
   conclude
-     (D[x \sup (1+m)])
-   = (D[x * x \sup m]) by _.
-   = (D[x] * x \sup m + x * D[x \sup m]) by _.
-   = (x \sup m + x * (m * x \sup (pred m))) by _.
-clear H.
-   = (x \sup m + m * (x \sup (1 + pred m))) by _.
-   = (x \sup m + m * x \sup m) by _.
-   = ((1+m)*x \sup m) by _ (timeout=30)
+     (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))). 
+   lapply depth=0 Fmult_associative;
+   lapply depth=0 Fmult_commutative;
+   = (X \sup m + m · (X · X \sup (pred m))) by Fmult_associative, Fmult_commutative.
+   = (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.
 
@@ -288,37 +276,33 @@ 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)"
-  right associative 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).
+ (*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 _
+   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 \sup 1 * x \sup (1+m)]) by _.
-   = (D[x \sup 1] * x \sup (1+m) + x * D[x \sup (1+m)]) by _.
-   = (x \sup (1+m) + x * (costante (1+m) * x \sup m)) by _.
-clear H.
-   = (x \sup (1+m) + costante (1+m) * x \sup (1+m)) by _.
-   = (x \sup (1+m) * (costante (2 + m))) by _
+     (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.