X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fnlibrary%2Fsets%2Fpartitions.ma;h=de29e796c47141fc80cd8659aa35a64c68e01a82;hb=a90c31c1b53222bd6d57360c5ba5c2d0fe7d5207;hp=3eea5010166d4d61c5ff51e98646fd3c8ca94e36;hpb=4377e950998c9c63937582952a79975947aa9a45;p=helm.git

diff --git a/helm/software/matita/nlibrary/sets/partitions.ma b/helm/software/matita/nlibrary/sets/partitions.ma
index 3eea50101..de29e796c 100644
--- a/helm/software/matita/nlibrary/sets/partitions.ma
+++ b/helm/software/matita/nlibrary/sets/partitions.ma
@@ -72,29 +72,46 @@ ntheorem iso_nat_nat_union_char:
     fst … p ≤ n ∧ snd … p < s (fst … p).
  #n; #s; nelim n
   [ #m; nwhd in ⊢ (??% → let p ≝ % in ?); nwhd in ⊢ (??(??%) → ?);
-    nrewrite > (plus_n_O (s O)); #H; nrewrite > (ltb_t … H); nnormalize; @; /2/
+    nrewrite > (plus_n_O (s O)); #H; nrewrite > (ltb_t … H); nnormalize; @
+    [ @ [ napply refl | napply le_n ] ##| nassumption ]
 ##| #n'; #Hrec; #m; nwhd in ⊢ (??% → let p ≝ % in ?); #H;
     ncases (ltb_cases m (s (S n'))); *; #H1; #H2; nrewrite > H2;
     nwhd in ⊢ (let p ≝ % in ?); nwhd
-     [ napply conj [napply conj; //;
-       nwhd in ⊢ (???(?(?%(λ_.λ_:(??%).?))%)); nrewrite > (minus_canc n'); //
-   ##| nnormalize; // ]
-##| nchange in H with (m < s (S n') + big_plus (S n') (λi.λ_.s i));
-    nlapply (Hrec (m - s (S n')) ?); /2/; *; *; #Hrec1; #Hrec2; #Hrec3; @; //; @; /2/;
-    nrewrite > (split_big_plus …); ##[##2:napply ad_hoc11;##|##3:##skip]
-    nrewrite > (ad_hoc12 …); //;
-    nwhd in ⊢ (???(?(??%)?));
-    nrewrite > (ad_hoc13 …); //;
-    napply ad_hoc14; /2/;
-    nwhd in ⊢ (???(?(??%)?));
-    nrewrite > (plus_n_O …); // ##]##]
+     [ napply conj [napply conj
+        [ nwhd in ⊢ (???(?(?%(λ_.λ_:(??%).?))%)); nrewrite > (minus_canc n'); napply refl
+        | nnormalize; napply le_n]
+      ##| nnormalize; nassumption ]
+   ##| nchange in H with (m < s (S n') + big_plus (S n') (λi.λ_.s i));
+       nlapply (Hrec (m - s (S n')) ?)
+        [ napply ad_hoc9; nassumption] *; *; #Hrec1; #Hrec2; #Hrec3; @
+        [##2: nassumption
+        |@
+         [nrewrite > (split_big_plus …); ##[##2:napply ad_hoc11;##|##3:##skip]
+          nrewrite > (ad_hoc12 …); ##[##2: nassumption]
+          nwhd in ⊢ (???(?(??%)?));
+          nrewrite > (ad_hoc13 …);##[##2: nassumption]
+          napply ad_hoc14 [ napply not_lt_to_le; nassumption ]
+          nwhd in ⊢ (???(?(??%)?));
+          nrewrite > (plus_n_O …);
+          nassumption;
+        ##| napply le_S; nassumption ]##]##]##]
 nqed.
 
 ntheorem iso_nat_nat_union_pre:
  ∀n:nat. ∀s: nat → nat.
   ∀i1,i2. i1 ≤ n → i2 < s i1 →
    big_plus (n - i1) (λi.λ_.s (S (i + i1))) + i2 < big_plus (S n) (λi.λ_.s i).
-/2/. nqed.
+ #n; #s; #i1; #i2; #H1; #H2;
+ nrewrite > (split_big_plus (S n) (S i1) (λi.λ_.s i) ?)
+  [##2: napply le_to_le_S_S; nassumption]
+ napply ad_hoc15
+  [ nwhd in ⊢ (???(?%?));
+    napply big_plus_preserves_ext; #i; #_;
+    nrewrite > (plus_n_S i i1); napply refl
+  | nrewrite > (split_big_plus (S i1) i1 (λi.λ_.s i) ?) [##2: napply le_S; napply le_n]
+    napply ad_hoc16; nrewrite > (minus_S i1); nnormalize; nrewrite > (plus_n_O (s i1) …);
+    nassumption ]
+nqed.
     
 ntheorem iso_nat_nat_union_uniq:
  ∀n:nat. ∀s: nat → nat.
@@ -128,9 +145,9 @@ nlemma partition_splits_card:
     nlapply (Hc y I); *; #index; *; #Hi1; #Hi2;
     nlapply (f_sur ???? f ? Hi1); *; #nindex; *; #Hni1; #Hni2;
     nlapply (f_sur ???? (fi nindex) y ?)
-     [ alias symbol "refl" (instance 3) = "refl".
-alias symbol "prop2" (instance 2) = "prop21".
-alias symbol "prop1" (instance 4) = "prop11".
+     [ alias symbol "refl" = "refl".
+alias symbol "prop1" = "prop11".
+alias symbol "prop2" = "prop21 mem".
 napply (. #‡(†?));##[##2: napply Hni2 |##1: ##skip | nassumption]##]
     *; #nindex2; *; #Hni21; #Hni22;
     nletin xxx ≝ (plus (big_plus (minus n nindex) (λi.λ_.s (S (plus i nindex)))) nindex2);
@@ -142,22 +159,24 @@ napply (. #‡(†?));##[##2: napply Hni2 |##1: ##skip | nassumption]##]
        *; *; #K1; #K2; #K3;
        nlapply
         (iso_nat_nat_union_uniq n s nindex (fst … (iso_nat_nat_union s xxx n))
-          nindex2 (snd … (iso_nat_nat_union s xxx n)) ?????); /2/
-        [##2: *; #E1; #E2; nrewrite > E1; nrewrite > E2; //
-        | nassumption ]##]
+          nindex2 (snd … (iso_nat_nat_union s xxx n)) ?????)
+        [##6: *; #E1; #E2; nrewrite > E1; nrewrite > E2; napply refl
+        |##5: napply le_S_S_to_le; nassumption
+        |##*: nassumption]##]
 ##| #x; #x'; nnormalize in ⊢ (? → ? → %); #Hx; #Hx'; #E;
-    ncut(∀i1,i2,i1',i2'. i1 ∈ Nat_ (S n) → i1' ∈ Nat_ (S n) → i2 ∈ Nat_ (s i1) → i2' ∈ Nat_ (s i1') → eq_rel (carr A) (eq0 A) (fi i1 i2) (fi i1' i2') → i1=i1' ∧ i2=i2');
+    ncut(∀i1,i2,i1',i2'. i1 ∈ Nat_ (S n) → i1' ∈ Nat_ (S n) → i2 ∈ Nat_ (s i1) → i2' ∈ Nat_ (s i1') → eq_rel (carr A) (eq A) (fi i1 i2) (fi i1' i2') → i1=i1' ∧ i2=i2');
     ##[ #i1; #i2; #i1'; #i2'; #Hi1; #Hi1'; #Hi2; #Hi2'; #E;
        nlapply(disjoint … P (f i1) (f i1') ???)
-       [##2,3: napply f_closed; //
+       [##2,3: napply f_closed; nassumption
        |##1: @ (fi i1 i2); @;
-         ##[ napply f_closed; // ##| alias symbol "refl" = "refl1".
+         ##[ napply f_closed; nassumption ##| alias symbol "refl" = "refl1".
 napply (. E‡#);
-             nwhd; napply f_closed; //]##]
-      #E'; ncut(i1 = i1'); ##[ napply (f_inj … E'); // ##]
-      #E''; nrewrite < E''; @; //;
-      nrewrite < E'' in E; #E'''; napply (f_inj … E'''); //;
-      nrewrite > E''; // ]##]
+             nwhd; napply f_closed; nassumption]##]
+      #E'; ncut(i1 = i1'); ##[ napply (f_inj … E'); nassumption; ##]
+      #E''; nrewrite < E''; @; 
+      ##[ @;
+      ##| nrewrite < E'' in E; #E'''; napply (f_inj … E''')
+             [ nassumption | nrewrite > E''; nassumption ]##]##]
    ##] #K;
     nelim (iso_nat_nat_union_char n s x Hx); *; #i1x; #i2x; #i3x;
     nelim (iso_nat_nat_union_char n s x' Hx'); *; #i1x'; #i2x'; #i3x';
@@ -167,7 +186,7 @@ napply (. E‡#);
     *; #K1; #K2;
     napply (eq_rect_CProp0_r ?? (λX.λ_.? = X) ?? i1x');
     napply (eq_rect_CProp0_r ?? (λX.λ_.X = ?) ?? i1x);
-    nrewrite > K1; nrewrite > K2; napply refl.
+    nrewrite > K1; nrewrite > K2; napply refl ]
 nqed.
 
 (************** equivalence relations vs partitions **********************)
@@ -179,10 +198,12 @@ ndefinition partition_of_compatible_equivalence_relation:
   | napply Full_set
   | napply mk_unary_morphism1
      [ #a; napply mk_ext_powerclass
-        [ napply {x | rel ? R x a}
+        [ napply {x | R x a}
         | #x; #x'; #H; nnormalize; napply mk_iff; #K; nelim daemon]
    ##| #a; #a'; #H; napply conj; #x; nnormalize; #K [ nelim daemon | nelim daemon]##]
-##| #x; #_; nnormalize; /3/
-  | #x; #x'; #_; #_; nnormalize; *; #x''; *; /3/
-  | nnormalize; napply conj; /4/ ]
-nqed.
\ No newline at end of file
+##| #x; #_; nnormalize; napply (ex_intro … x); napply conj; napply refl
+  | #x; #x'; #_; #_; nnormalize; *; #x''; *; #H1; #H2; napply (trans ?????? H2);
+    napply sym; nassumption
+  | nnormalize; napply conj
+     [ #a; #_; napply I | #a; #_; napply (ex_intro … a); napply conj [ napply I | napply refl]##]
+nqed.