From fa6d1f3df955464343b365114be1071fc0b0cd18 Mon Sep 17 00:00:00 2001
From: Claudio Sacerdoti Coen <claudio.sacerdoticoen@unibo.it>
Date: Fri, 15 Jan 2010 17:06:45 +0000
Subject: [PATCH] Finished!

---
 .../matita/nlibrary/topology/igft3.ma         | 38 +++++++++++--------
 1 file changed, 22 insertions(+), 16 deletions(-)

diff --git a/helm/software/matita/nlibrary/topology/igft3.ma b/helm/software/matita/nlibrary/topology/igft3.ma
index 32d0f8f84..3abe4af70 100644
--- a/helm/software/matita/nlibrary/topology/igft3.ma
+++ b/helm/software/matita/nlibrary/topology/igft3.ma
@@ -11,18 +11,16 @@
 (*        v         GNU General Public License Version 2                  *)
 (*                                                                        *)
 (**************************************************************************)
-(*
+
 include "arithmetics/nat.ma".
-*)
 include "datatypes/bool.ma".
 
-(*
 ndefinition two ≝ S (S O).
 ndefinition natone ≝ S O.
 ndefinition four ≝ two * two.
 ndefinition eight ≝ two * four.
 ndefinition natS ≝ S.
-*)
+
 include "topology/igft.ma".
 
 nlemma hint_auto2 : ∀T.∀U,V:Ω^T.(∀x.x ∈ U → x ∈ V) → U ⊆ V.
@@ -134,8 +132,6 @@ nrecord memdec (A: Type[0]) (U:Ω^A) : Type[0] ≝
 
 (*********** end from Cantor ********)
 
-naxiom daemon: False.
-
 nlemma csc_sym_eq: ∀A,x,y. eq A x y → eq A y x.
  #A; #x; #y; #H; ncases H; @1.
 nqed.
@@ -192,10 +188,13 @@ nlet rec cover_rect
 nqed.
 
 (********* Esempio:
-   let rec skipfact n =
+   let rec skip n =
      match n with
       [ O ⇒ 1
-      | S m ⇒ S m * skipfact (pred m) ]
+      | S m ⇒
+         match m with
+          [ O ⇒ skipfact O
+          | S _ ⇒ S m * skipfact (pred m) * skipfact O ]]
 **)
 
 ntheorem psym_plus: ∀n,m. n + m = m + n.
@@ -219,7 +218,7 @@ nlemma easy1: ∀n:nat. two * (S n) = two + two * n.
 nqed.
 
 ndefinition skipfact_dom: uuAx.
- @ nat; #n; ncases n [ napply None | #m; napply (Some … (pred m)) ]
+ @ nat; #n; ncases n [ napply None | #m; ncases m [ napply (Some … O) | #_; napply (Twice … (pred m) O) ]
 nqed.
 
 ntheorem skipfact_base_dec:
@@ -232,11 +231,15 @@ ntheorem skipfact_partial:
  ∀n: uuax skipfact_dom. two * n ◃ mk_powerclass ? (λx: uuax skipfact_dom.x=O).
  #n; nelim n
   [ @1; nnormalize; @1
-  | #m; #H; @2
-     [ nnormalize; @1
-     | nnormalize; #y; nchange in ⊢ (% → ?) with (y = pred (pred (two * (natone + m))));
-       nrewrite > (easy1 …); nchange in ⊢ (% → ?) with (y = two * m);
-       #E; nrewrite > E; nassumption ]##]
+  | #m; nelim m; nnormalize
+     [ #H; @2; nnormalize
+       [ //
+       | #y; *; #a; #E; nrewrite > E; ncases a; nnormalize; // ]
+   ##| #p; #H1; #H2; @2; nnormalize
+       [ //
+       | #y; *; #a; #E; nrewrite > E; ncases a; nnormalize
+          [ nrewrite < (plus_n_Sm …); //
+        ##| /2/ ]
 nqed.
 
 ndefinition skipfact: ∀n:nat. n ◃ mk_powerclass ? (λx: uuax skipfact_dom.x=O) → nat.
@@ -244,8 +247,11 @@ ndefinition skipfact: ∀n:nat. n ◃ mk_powerclass ? (λx: uuax skipfact_dom.x=
   [ #a; #_; napply natone
   | #a; ncases a
     [ nnormalize; #i; nelim i
-    | #m; #i; nnormalize in i; #d; #H;
-      napply (S m * H (pred m) …); //]
+    | #m; ncases m
+       [ nnormalize; #_; #_; #H; napply H; @1
+       | #p; #i; nnormalize in i; #K;
+         #H; nnormalize in H;
+         napply (S m * H true * H false) ]
 nqed.
 
 nlemma test: skipfact four ? = eight. ##[##2: napply (skipfact_partial two)]
-- 
2.39.2