fixed notation

[helm.git] / helm / software / matita / library / demo / power_derivative.ma

diff --git a/helm/software/matita/library/demo/power_derivative.ma b/helm/software/matita/library/demo/power_derivative.ma
index 24656c1d3da9f53f43d170b4837cc765ce1fb138..e8ab55c3595108eb3e08a8d60c49819752831832 100644 (file)
--- 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".
@@ -90,7 +88,7 @@ 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.
@@ -110,7 +108,7 @@ 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.
@@ -121,7 +119,7 @@ lemma Fmult_one_f: ∀f:R→R.1·f=f.
  simplify;
  apply f_eq_extensional;
  intro;
- auto.
+ autobatch.
 qed.
 
 lemma Fmult_zero_f: ∀f:R→R.0·f=0.
@@ -130,7 +128,7 @@ lemma Fmult_zero_f: ∀f:R→R.0·f=0.
  simplify;
  apply f_eq_extensional;
  intro;
- auto.
+ autobatch.
 qed.
 
 lemma Fmult_commutative: symmetric ? Fmult.
@@ -139,7 +137,7 @@ lemma Fmult_commutative: symmetric ? Fmult.
  unfold Fmult;
  apply f_eq_extensional;
  intros;
- auto.
+ autobatch.
 qed.
 
 lemma Fmult_associative: associative ? Fmult.
@@ -149,7 +147,7 @@ lemma Fmult_associative: associative ? Fmult.
  unfold Fmult;
  apply f_eq_extensional;
  intros;
- auto.
+ autobatch.
 qed.
 
 lemma Fmult_Fplus_distr: distributive ? Fmult Fplus.
@@ -160,7 +158,7 @@ lemma Fmult_Fplus_distr: distributive ? Fmult Fplus.
  apply f_eq_extensional;
  intros;
  simplify;
- auto.
+ autobatch.
 qed.
 
 lemma monomio_product:
@@ -173,13 +171,13 @@ 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);
      [ rewrite > Hcut;
-       auto
+       autobatch
      | change in ⊢ (? ? % ?) with ((λx:R.x\sup(n1+m)) x);
        rewrite > H;
        reflexivity
@@ -196,7 +194,7 @@ lemma costante_sum:
  intros;
  elim n;
   [ simplify;
-    auto
+    autobatch
   | simplify;
     clear x;
     clear H;
@@ -205,19 +203,19 @@ 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
         ]
      ]
    ].
@@ -251,11 +249,11 @@ 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)).
+ (*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
@@ -264,25 +262,24 @@ theorem derivative_power: ∀n:nat. D[x \sup n] = n·x \sup (pred n).
    (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 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 _
+      using (S_pred ? m_pos) we proved (m = 1 + pred m) (H1).
      done.
     case right.
-      suppose (0=m) (m_zero). by _ done.
+      suppose (0=m) (m_zero). 
+    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 · 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.
 
@@ -297,8 +294,7 @@ 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
+notation "hvbox(\frac 'd' ('d' 'x') break p)" with precedence 90
 for @{ 'derivative $p}.
 
 interpretation "Rderivative" 'derivative f =
@@ -306,11 +302,10 @@ interpretation "Rderivative" 'derivative 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).
+ (*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
@@ -319,11 +314,11 @@ theorem derivative_power': ∀n:nat. D[x \sup (1+n)] = (1+n) · x \sup n.
    (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 · 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.
\ No newline at end of file
+qed.