From 2c81e4168d6cec784100b0e3cd3f377aa257daee Mon Sep 17 00:00:00 2001
From: Claudio Sacerdoti Coen <claudio.sacerdoticoen@unibo.it>
Date: Wed, 19 Aug 2009 10:15:59 +0000
Subject: [PATCH] ...

---
 .../matita/nlibrary/sets/partitions.ma        | 35 +++++++++++++++----
 1 file changed, 29 insertions(+), 6 deletions(-)

diff --git a/helm/software/matita/nlibrary/sets/partitions.ma b/helm/software/matita/nlibrary/sets/partitions.ma
index f1f64f881..93172d617 100644
--- a/helm/software/matita/nlibrary/sets/partitions.ma
+++ b/helm/software/matita/nlibrary/sets/partitions.ma
@@ -44,14 +44,26 @@ nlet rec partition_splits_card_map
       | S index' ⇒
          partition_splits_card_map A P n s f fi (minus m (s index)) index']].  
 
+naxiom big_union_preserves_iso:
+ ∀A,A',B,T,T',f.
+  ∀g: isomorphism A' A T' T.
+   big_union A B T f = big_union A' B T' (λx.f (iso_f ???? g x)).
+
 nlemma partition_splits_card:
  ∀A. ∀P:partition A. ∀n,s.
   ∀f:isomorphism ?? (Nat_ n) (indexes ? P).
    (∀i. isomorphism ?? (Nat_ (s i)) (class ? P (iso_f ???? f i))) →
     (isomorphism ?? (Nat_ (big_plus n (λi.λ_.s i))) (Full_set A)).
- #A; #P; #n; #s; #f; #fi; napply mk_isomorphism
+ #A; #P; #Sn; ncases Sn
+  [ #s; #f; #fi;
+    ngeneralize in match (covers ? P) in ⊢ ?; *; #_; #H;
+    ngeneralize in match
+     (big_union_preserves_iso ??? (indexes A P) (Nat_ O) (λx.class ? P x) f) in ⊢ ?;
+     *; #K; #_; nwhd in K: (? → ? → %);
+    nelim daemon (* impossibile *)
+  | #n; #s; #f; #fi; napply mk_isomorphism
   [ napply mk_unary_morphism
-     [ napply (λm.partition_splits_card_map A P n s f fi m n)
+     [ napply (λm.partition_splits_card_map A P (S n) s f fi m n)
      | #a; #a'; #H; nrewrite < H; napply refl ]
 ##| #y; #_;
     ngeneralize in match (covers ? P) in ⊢ ?; *; #_; #Hc;
@@ -60,15 +72,26 @@ nlemma partition_splits_card:
     ngeneralize in match (f_sur ???? (fi nindex) y ?) in ⊢ ?
      [##2: napply (. #‡(†?));##[##3: napply Hni2 |##2: ##skip | nassumption]##]
     *; #nindex2; *; #Hni21; #Hni22;
-    napply (ex_intro … (plus (big_plus nindex (λi.λ_.s i)) nindex2)); napply conj
+    nletin xxx ≝ (plus match nindex return λ_.nat with [ O ⇒ O | S nn ⇒ big_plus nn (λi.λ_.s i)] nindex2);
+    napply (ex_intro … xxx); napply conj
      [ nwhd in Hni1; nelim daemon
      | nwhd in ⊢ (???%?);
        (* BEL POSTO DOVE FARE UN LEMMA *)
        (* invariante: Hni1; altre premesse: Hni1, Hni22 *)
-       nchange in Hni1 with (nindex < n); ngeneralize in match Hni1 in ⊢ ?;
-       nelim n
+       nchange in Hni1 with (nindex < S n); ngeneralize in match Hni1 in ⊢ ?;
+       nelim n in ⊢ (% → ??? (????????%) ?)
         [ #A (* decompose *)
-        | #index'; #Hrec; 
+        | #index'; #Hrec; #K; nwhd in ⊢ (???%?);
+          nelim (ltb xxx (s (S index')));
+          #K1; nwhd in ⊢ (???%?)
+           [
+           
+           nindex < S index' + 1
+           +^{nindex} (s i) w < s (S index') 
+           S index' == nindex
+           
+           |
+           ]
         ]
      ]
   | #x; #x'; nnormalize in ⊢ (? → ? → %);
-- 
2.39.2