more theory for lists

[helm.git] / helm / software / matita / nlibrary / topology / igft2.ma

diff --git a/helm/software/matita/nlibrary/topology/igft2.ma b/helm/software/matita/nlibrary/topology/igft2.ma
index 1232642d8b03a816fe6cdae913225424a6709884..dc78b768079ef614c53908f3b3639d815c5fbcfd 100644 (file)
--- a/helm/software/matita/nlibrary/topology/igft2.ma
+++ b/helm/software/matita/nlibrary/topology/igft2.ma
@@ -12,8 +12,20 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "arithmetics/nat.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.
+nnormalize; /2/;
+nqed.
+
 alias symbol "covers" = "covers set".
 alias symbol "coverage" = "coverage cover".
 alias symbol "I" = "I".
@@ -21,20 +33,14 @@ nlemma cover_ind':
  ∀A:Ax.∀U,P:Ω^A.
    (U ⊆ P) → (∀a:A.∀j:𝐈 a. 𝐂 a j ◃ U → 𝐂 a j ⊆ P → a ∈ P) →
     ◃ U ⊆ P.
- #A; #U; #P; #refl; #infty; #a; #H; nelim H
-  [ nauto | #b; #j; #K1; #K2; napply (infty … j) [ nassumption | napply K2]##]
+ #A; #U; #P; #refl; #infty; #a; #H; nelim H; /3/. 
 nqed.
 
-alias symbol "covers" = "covers".
-alias symbol "covers" = "covers set".
-alias symbol "covers" = "covers".
-alias symbol "covers" = "covers set".
-alias symbol "covers" = "covers".
+alias symbol "covers" (instance 1) = "covers".
 nlemma cover_ind'':
  ∀A:Ax.∀U:Ω^A.∀P:A → CProp[0].
   (∀a. a ∈ U → P a) → (∀a:A.∀j:𝐈 a. 𝐂 a j ◃ U → (∀b. b ∈ 𝐂 a j → P b) → P a) →
    ∀b. b ◃ U → P b.
-   
  #A; #U; #P; nletin V ≝ {x | P x}; napply (cover_ind' … V).
 nqed.
 
@@ -70,7 +76,7 @@ ncoercion uuax : ∀u:uuAx. Ax ≝ uuax on _u : uuAx to Ax.
 
 nlemma eq_rect_Type0_r':
  ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → Type[0]. P a (refl A a) → P x p.
- #A; #a; #x; #p; ncases p; #P; #H; nassumption.
+ #A; #a; #x; #p; ncases p; //;
 nqed.
 
 nlemma eq_rect_Type0_r:
@@ -90,8 +96,8 @@ nrecord memdec (A: Type[0]) (U:Ω^A) : Type[0] ≝
 
 (*********** end from Cantor ********)
 
-alias symbol "covers" = "covers".
-alias symbol "covers" = "covers".
+alias symbol "covers" (instance 9) = "covers".
+alias symbol "covers" (instance 8) = "covers".
 nlet rec cover_rect
  (A:uuAx) (U:Ω^(uuax A)) (memdec: memdec … U) (P:uuax A → Type[0])
   (refl: ∀a:uuax A. a ∈ U → P a)
@@ -100,21 +106,68 @@ nlet rec cover_rect
 ≝ ?.
  nlapply (decide_mem_ok … memdec b); nlapply (decide_mem_ko … memdec b);
  ncases (decide_mem … memdec b)
-  [ #_; #H; napply refl; nauto
-  | #H; #_; ncut (uuC … b=uuC … b) [nauto] ncases (uuC … b) in ⊢ (???% → ?)
-    [ #E; napply False_rect_Type0; ncut (b=b) [nauto] ncases p in ⊢ (???% → ?)
-      [ #a; #K; #E2; napply H [ nauto | nrewrite > E2; nauto ]
+  [ #_; #H; napply refl; /2/
+  | #H; #_; ncut (uuC … b=uuC … b) [//] ncases (uuC … b) in ⊢ (???% → ?)
+    [ #E; napply False_rect_Type0; ncut (b=b) [//] ncases p in ⊢ (???% → ?)
+      [ #a; #K; #E2; napply H [ // | nrewrite > E2; // ]
     ##| #a; #i; #K; #E2; nrewrite < E2 in i; nnormalize; nrewrite > E; nnormalize;
-        nauto]
+        //]
   ##| #a; #E;
       ncut (a ◃ U)
-       [ nlapply E; nlapply (H ?) [nauto] ncases p
+       [ nlapply E; nlapply (H ?) [//] ncases p
           [ #x; #Hx; #K1; #_; ncases (K1 Hx)
         ##| #x; #i; #Hx; #K1; #E2; napply Hx; ngeneralize in match i; nnormalize;
-            nrewrite > E2; nnormalize; #_; nauto]##]
+            nrewrite > E2; nnormalize; #_; //]##]
       #Hcut; 
       nlapply (infty b); nnormalize; nrewrite > E; nnormalize; #H2;
-      napply (H2 one); #y; #E2; nrewrite > E2
+      napply (H2 one); #y; #E2; nrewrite > E2 
+      (* [##2: napply cover_rect] //; *)
        [ napply Hcut
-     ##| napply (cover_rect A U memdec P refl infty a); napply Hcut]##]
+     ##| napply (cover_rect A U memdec P refl infty a); // ]##]
+nqed.
+
+(********* Esempio:
+   let rec skipfact n =
+     match n with
+      [ O ⇒ 1
+      | S m ⇒ S m * skipfact (pred m) ]
+**)
+
+ntheorem psym_plus: ∀n,m. n + m = m + n.//.
+nqed.
+
+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)) ]
+nqed.
+
+ntheorem skipfact_base_dec:
+ memdec (uuax skipfact_dom) (mk_powerclass ? (λx: uuax skipfact_dom. x=O)).
+ nnormalize; @ (λx. match x with [ O ⇒ true | S _ ⇒ false ]); #n; nelim n;
+  nnormalize; //; #X; ndestruct; #Y; #Z; ndestruct; #W; ndestruct.
+nqed.
+
+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))));
+       nnormalize; nrewrite < (plus_n_Sm …); nnormalize;
+       #E; nrewrite > E; napply H ]##]
+nqed.
+
+ndefinition skipfact: ∀n:nat. n ◃ mk_powerclass ? (λx: uuax skipfact_dom.x=O) → nat.
+ #n; #D; napply (cover_rect … skipfact_base_dec … n D)
+  [ #a; #_; napply natone
+  | #a; ncases a
+    [ nnormalize; #i; nelim i
+    | #m; #i; nnormalize in i; #d; #H;
+      napply (S m * H (pred m) …); //]
+nqed.
+
+nlemma test: skipfact four ? = eight. ##[##2: napply (skipfact_partial two)] //.
 nqed.
\ No newline at end of file