...

diff --git a/helm/software/matita/nlibrary/nat/minus.ma b/helm/software/matita/nlibrary/nat/minus.ma
index f800cc0587c06799bb05848c216b56d6ae003b76..86a4b1df22866d3bfbd169de719e8889d4a79b13 100644 (file)
--- a/helm/software/matita/nlibrary/nat/minus.ma
+++ b/helm/software/matita/nlibrary/nat/minus.ma
@@ -22,5 +22,7 @@ nlet rec minus (n:nat) (m:nat) on m : nat ≝
       [ O ⇒ O
       | S n' ⇒ minus n' m']].
 
+interpretation "natural minus" 'minus x y = (minus x y).
+
 naxiom le_minus: ∀n,m,p. le n m → le (minus n p) m.
-naxiom lt_minus: ∀n,m,p. lt n m → lt (minus n p) m.
\ No newline at end of file
+naxiom lt_minus: ∀n,m,p. lt n m → lt (minus n p) m.

diff --git a/helm/software/matita/nlibrary/nat/plus.ma b/helm/software/matita/nlibrary/nat/plus.ma
index 75268ffcb4336b3ca709e5134dd51ae00c350a88..8bbc71b9cbbbc3b5d50ece774467c69ebf890048 100644 (file)
--- a/helm/software/matita/nlibrary/nat/plus.ma
+++ b/helm/software/matita/nlibrary/nat/plus.ma
@@ -21,6 +21,8 @@ nlet rec plus (n:nat) (m:nat) on n : nat ≝
   [ O ⇒ m
   | S n' ⇒ S (plus n' m) ].
 
+interpretation "natural plus" 'plus x y = (plus x y).
+
 ndefinition plus_magma_type: magma_type.
  napply mk_magma_type
   [ napply NAT
@@ -52,4 +54,4 @@ ndefinition plus_unital_magma_type: unital_magma_type.
   | #x; (* qua manca ancora l'hint *) napply (symm plus_abelian_magma_type) ]
 nqed.
 
-ndefinition big_plus ≝ λn.λf.big_op plus_magma_type n f O.
\ No newline at end of file
+ndefinition big_plus ≝ λn.λf.big_op plus_magma_type n f O.

diff --git a/helm/software/matita/nlibrary/sets/partitions.ma b/helm/software/matita/nlibrary/sets/partitions.ma
index f62c7281d1b074a068369cb9d71adfcca1c7b76d..d8ba34c5ccbec1849e60109915c688d52fa7e8cd 100644 (file)
--- a/helm/software/matita/nlibrary/sets/partitions.ma
+++ b/helm/software/matita/nlibrary/sets/partitions.ma
@@ -25,6 +25,9 @@ alias symbol "eq" = "setoid eq".
 alias symbol "eq" = "setoid1 eq".
 alias symbol "eq" = "setoid eq".
 alias symbol "eq" = "setoid1 eq".
+alias symbol "eq" = "setoid eq".
+alias symbol "eq" = "setoid1 eq".
+alias symbol "eq" = "setoid eq".
 nrecord partition (A: setoid) : Type[1] ≝ 
  { support: setoid;
    indexes: qpowerclass support;
@@ -65,7 +68,25 @@ naxiom minus_lt_to_lt: ∀n,m,p. n < p → minus n m < p.
 naxiom minus_O_n: ∀n. O = minus O n.
 naxiom le_O_to_eq: ∀n. n ≤ O → n=O.
 naxiom lt_to_minus_to_S: ∀n,m. m < n → ∃k. minus n m = S k.
-naxiom ltb_t: ∀n,m. n < m → ltb n m = true. 
+naxiom ltb_t: ∀n,m. n < m → ltb n m = true.
+naxiom ltb_f: ∀n,m. ¬ (n < m) → ltb n m = false.
+naxiom plus_n_O: ∀n. plus n O = n.
+naxiom not_lt_plus: ∀n,m. ¬ (plus n m < n).
+naxiom lt_to_lt_plus: ∀n,m,l. n < m → n < m + l.
+naxiom S_plus: ∀n,m. S (n + m) = n + S m.
+naxiom big_plus_ext: ∀n,f,f'. (∀i,p. f i p = f' i p) → big_plus n f = big_plus n f'.
+naxiom ad_hoc1: ∀n,m,l. n + (m + l) = l + (n + m).
+naxiom assoc: ∀n,m,l. n + m + l = n + (m + l).
+naxiom lt_canc: ∀n,m,p. n < m → p + n < p + m.
+naxiom ad_hoc2: ∀a,b. a < b → b - a - (b - S a) = S O.
+naxiom ad_hoc3: ∀a,b. b < a → S (O + (a - S b) + b) = a.
+naxiom ad_hoc4: ∀a,b. a - S b ≤ a - b.
+
+naxiom split_big_plus:
+  ∀n,m,f. m ≤ n →
+   big_plus n f = big_plus m (λi,p.f i ?) + big_plus (n - m) (λi.λp.f (i + m) ?).
+ nelim daemon.
+nqed.
 
 nlemma partition_splits_card:
  ∀A. ∀P:partition A. ∀n,s.
@@ -90,20 +111,50 @@ nlemma partition_splits_card:
     ngeneralize in match (f_sur ???? (fi nindex) y ?) in ⊢ ?
      [##2: napply (. #‡(†?));##[##3: napply Hni2 |##2: ##skip | nassumption]##]
     *; #nindex2; *; #Hni21; #Hni22;
-    nletin xxx ≝ (plus match minus n nindex return λ_.nat with [ O ⇒ O | S nn ⇒ big_plus nn (λi.λ_.s (S (plus i nindex)))] nindex2);
+    nletin xxx ≝ (plus (big_plus (minus n nindex) (λi.λ_.s (S (plus i nindex)))) nindex2);
     napply (ex_intro … xxx); napply conj
-     [ nwhd in Hni1; nwhd; nelim daemon
-     | nwhd in ⊢ (???%?);
+     [ nwhd in Hni1; nwhd; nwhd in ⊢ (?(? %)%);
+       nchange with (? < plus (s n) (big_plus n ?));
+       nelim (le_to_lt_or_eq … (le_S_S_to_le … Hni1))
+        [##2: #E; nrewrite < E; nrewrite < (minus_canc nindex);
+          nwhd in ⊢ (?%?); nrewrite < E; napply lt_to_lt_plus; nassumption
+        | #L; nrewrite > (split_big_plus n (S nindex) (λm.λ_.s m) L);
+          nrewrite > (split_big_plus (n - nindex) (n - S nindex) (λi.λ_.s (S (i+nindex))) ?)
+           [ ngeneralize in match (big_plus_ext (n - S nindex)
+              (λi,p.s (S (i+nindex))) (λi,p.s (i + S nindex)) ?) in ⊢ ?
+               [ #E;
+                 napply (eq_rect_CProp0_r ??
+                  (λx:nat.λ_. x + big_plus (n - nindex - (n - S nindex))
+                   (λi,p.s (S (i + (n - S nindex)+nindex))) + nindex2 <
+                   s n + (big_plus (S nindex) (λi,p.s i) +
+                    big_plus (n - S nindex) (λi,p. s (i + S nindex)))) ? ? E);
+                 nrewrite > (ad_hoc1 (s n) (big_plus (S nindex) (λi,p.s i))
+                  (big_plus (n - S nindex) (λi,p. s (i + S nindex))));
+                 napply (eq_rect_CProp0_r
+                  ?? (λx.λ_.x < ?) ?? (assoc
+                  (big_plus (n - S nindex) (λi,p.s (i + S nindex)))
+                  (big_plus (n - nindex - (n - S nindex))
+                   (λi,p.s (S (i + (n - S nindex)+nindex))))
+                  nindex2));
+                 napply lt_canc;
+                 nrewrite > (ad_hoc2 … L); nwhd in ⊢ (?(?%?)?);
+                 nrewrite > (ad_hoc3 … L);
+                 napply (eq_rect_CProp0_r ?? (λx.λ_.x < ?) ?? (assoc …));
+                 napply lt_canc; nnormalize in ⊢ (?%?); nwhd in ⊢ (??%);
+                 napply lt_to_lt_plus; nassumption
+             ##|##2: #i; #_; nrewrite > (S_plus i nindex); napply refl]
+         ##| napply ad_hoc4]##]
+   ##| nwhd in ⊢ (???%?);
        nchange in Hni1 with (nindex < S n);
        ngeneralize in match (le_S_S_to_le … Hni1) in ⊢ ?;
        nwhd in ⊢ (? → ???(???????%?)?);
        napply (nat_rect_CProp0
         (λx. nindex ≤ x →
-          partition_splits_card_map A P (S n) s f fi
+          eq_rel (carr A) (eq A)
+          (partition_splits_card_map A P (S n) s f fi
            (plus
-             match minus x nindex with
-              [ O ⇒ O | S nn ⇒ big_plus nn (λi.λ_.s (S (plus i nindex)))]
-             nindex2) x = y) ?? n)
+             (big_plus (minus x nindex) (λi.λ_:i < minus x nindex.s (S (plus i nindex))))
+             nindex2) x) y) ?? n)
         [ #K; nrewrite < (minus_O_n nindex); nwhd in ⊢ (???(???????%?)?);
           nwhd in ⊢ (???%?); nchange in Hni21 with (nindex2 < s nindex);
           ngeneralize in match (le_O_to_eq … K) in ⊢ ?; #K';
@@ -115,31 +166,28 @@ nlemma partition_splits_card:
         | #n'; #Hrec; #HH; nelim (le_to_lt_or_eq … HH)
            [##2: #K; nrewrite < K; nrewrite < (minus_canc nindex);
             nwhd in ⊢ (???(???????%?)?);
-            (*???????*)
-         ##| #K; nwhd in ⊢ (???%?);
+            nrewrite > K;
+            nwhd in ⊢ (???%?); nrewrite < K; nrewrite > (ltb_t … Hni21);
+            nwhd in ⊢ (???%?); nassumption
+         ##| #K; ngeneralize in match (le_S_S_to_le … K) in ⊢ ?; #K';
+             nwhd in ⊢ (???%?);
+             
+             
+             XXX;
              nrewrite > (minus_S n' nindex ?) [##2: napply le_S_S_to_le; nassumption]
              ngeneralize in match (? :
-              match S (minus n' nindex) with [O ⇒ O | S nn ⇒ big_plus nn (λi.λ_.s (S (plus i nindex)))]
-              = big_plus (minus n' nindex) (λi.λ_.s (S (plus i nindex)))) in ⊢ ? [##2: napply refl]
-             #He; napply (eq_rect_CProp0_r ??
-              (λx.λ_.
-                match ltb (plus x nindex2) (s (S n')) with
-                 [ true ⇒ iso_f ???? (fi (S n')) (plus x nindex2)
-                 | false ⇒ ?(*partition_splits_card_map A P (S n) s f fi
-                     (minus (plus x nindex2) (s (S n'))) n'*)
-                 ] = y)
-              ?? He);
-             ngeneralize in match (? :
-              ltb (plus (big_plus (minus n' nindex) (λi.λ_.s (S (plus i nindex)))) nindex2)
+              ltb (plus (big_plus (S (minus n' nindex)) (λi.λ_.s (S (plus i nindex)))) nindex2)
                (s (S n')) = false) in ⊢ ?
                [ #Hc; nrewrite > Hc; nwhd in ⊢ (???%?);
                  nelim (le_to_lt_or_eq … (le_S_S_to_le … K))
                   [
                 ##| #E; ngeneralize in match Hc in ⊢ ?;
                     nrewrite < E; nrewrite < (minus_canc nindex);
-                    nwhd in ⊢ (??(?%?)? → ?);
-                    nrewrite > E in Hni21; #E'; nchange in E' with (nindex2 < s n'); 
-                    ngeneralize in match Hni21 in ⊢ ?;
+                    nnormalize in ⊢ (??(?%?)? → ?);
+                    nrewrite > (plus_n_O (s (S nindex)));
+                    nrewrite > (ltb_f (plus (s (S nindex)) nindex2) (s (S nindex)) ?);
+                    
+                    XXX;
                     
                
                 ngeneralize in match (? :