X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fnlibrary%2Fsets%2Fpartitions.ma;h=b92fe9ab3011d9bcbcb75637b51c886c43de6d05;hb=c8ec2f3e2aaa71efa702f86bacc95e393778a56f;hp=e588fd2bc2273d91b1e4122697ec7c0d0717da3b;hpb=5f77bf2f56b180e268b6acaa81a2fbb82a8fe026;p=helm.git

diff --git a/helm/software/matita/nlibrary/sets/partitions.ma b/helm/software/matita/nlibrary/sets/partitions.ma
index e588fd2bc..b92fe9ab3 100644
--- a/helm/software/matita/nlibrary/sets/partitions.ma
+++ b/helm/software/matita/nlibrary/sets/partitions.ma
@@ -13,49 +13,176 @@
 (**************************************************************************)
 
 include "sets/sets.ma".
-include "nat/plus.ma".
+include "nat/plus.ma". 
 include "nat/compare.ma".
 include "nat/minus.ma".
+include "datatypes/pairs.ma".
 
-alias symbol "eq" = "setoid eq".
-alias symbol "eq" = "setoid1 eq".
+alias symbol "eq" (instance 7) = "setoid1 eq".
 nrecord partition (A: setoid) : Type[1] ≝ 
  { support: setoid;
-   indexes: qpowerclass support;
-   class: support → qpowerclass A;
+   indexes: ext_powerclass support;
+   class: unary_morphism1 (setoid1_of_setoid support) (ext_powerclass_setoid A);
    inhabited: ∀i. i ∈ indexes → class i ≬ class i;
-   disjoint: ∀i,j. i ∈ indexes → j ∈ indexes → class i ≬ class j → i=j;
-   covers: big_union support ? ? (λx.class x) = full_set A
- }. napply indexes; nqed.
-
+   disjoint: ∀i,j. i ∈ indexes → j ∈ indexes → class i ≬ class j → i = j;
+   covers: big_union support ? indexes (λx.class x) = full_set A
+ }.
+ 
 naxiom daemon: False.
 
-nlet rec partition_splits_card_map
- A (P:partition A) n s (f:isomorphism ?? (Nat_ n) (indexes ? P))
- (fi: ∀i. isomorphism ?? (Nat_ (s i)) (class ? P (iso_f ???? f i))) m index
- on index : A ≝
+nlet rec iso_nat_nat_union (s: nat → nat) m index on index : pair nat nat ≝
  match ltb m (s index) with
-  [ or_introl _ ⇒ iso_f ???? (fi index) m
-  | or_intror _ ⇒
+  [ true ⇒ mk_pair … index m
+  | false ⇒
      match index with
-      [ O ⇒ (* dummy value: it could be an elim False: *) iso_f ???? (fi O) O
-      | S index' ⇒
-         partition_splits_card_map A P n s f fi (minus m (s index)) index']].  
+      [ O ⇒ (* dummy value: it could be an elim False: *) mk_pair … O O
+      | S index' ⇒ iso_nat_nat_union s (minus m (s index)) index']].
+
+alias symbol "eq" = "leibnitz's equality".
+naxiom plus_n_O: ∀n. n + O = n.
+naxiom plus_n_S: ∀n,m. n + S m = S (n + m).
+naxiom ltb_t: ∀n,m. n < m → ltb n m = true.
+naxiom ltb_f: ∀n,m. ¬ (n < m) → ltb n m = false.
+naxiom ltb_cases: ∀n,m. (n < m ∧ ltb n m = true) ∨ (¬ (n < m) ∧ ltb n m = false).
+naxiom minus_canc: ∀n. minus n n = O.
+naxiom ad_hoc9: ∀a,b,c. a < b + c → a - b < c.
+naxiom ad_hoc10: ∀a,b,c. a - b = c → a = b + c.
+naxiom ad_hoc11: ∀a,b. a - b ≤ S a - b.
+naxiom ad_hoc12: ∀a,b. b ≤ a → S a - b - (a - b) = S O.
+naxiom ad_hoc13: ∀a,b. b ≤ a → (O + (a - b)) + b = a.
+naxiom ad_hoc14: ∀a,b,c,d,e. c ≤ a → a - c = b + d + e → a = b + (c + d) + e.
+naxiom ad_hoc15: ∀a,a',b,c. a=a' → b < c → a + b < c + a'.
+naxiom ad_hoc16: ∀a,b,c. a < c → a < b + c.
+naxiom not_lt_to_le: ∀a,b. ¬ (a < b) → b ≤ a.
+naxiom le_to_le_S_S: ∀a,b. a ≤ b → S a ≤ S b.
+naxiom minus_S: ∀n. S n - n = S O.
+naxiom ad_hoc17: ∀a,b,c,d,d'. a+c+d=b+c+d' → a+d=b+d'.
+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.
+naxiom big_plus_preserves_ext:
+ ∀n,f,f'. (∀i,p. f i p = f' i p) → big_plus n f = big_plus n f'.
+
+ntheorem iso_nat_nat_union_char:
+ ∀n:nat. ∀s: nat → nat. ∀m:nat. m < big_plus (S n) (λi.λ_.s i) →
+  let p ≝ iso_nat_nat_union s m n in
+   m = big_plus (n - fst … p) (λi.λ_.s (S (i + fst … p))) + snd … p ∧
+    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/
+##| #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 …); // ##]##]
+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.
+    
+ntheorem iso_nat_nat_union_uniq:
+ ∀n:nat. ∀s: nat → nat.
+  ∀i1,i1',i2,i2'. i1 ≤ n → i1' ≤ n → i2 < s i1 → i2' < s i1' →
+   big_plus (n - i1) (λi.λ_.s (S (i + i1))) + i2 = big_plus (n - i1') (λi.λ_.s (S (i + i1'))) + i2' →
+    i1 = i1' ∧ i2 = i2'.
+ #n; #s; #i1; #i1'; #i2; #i2'; #H1; #H1'; #H2; #H2'; #E;
+ nelim daemon.
+nqed.
 
 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
-  [ napply mk_unary_morphism
-     [ napply (λm.partition_splits_card_map A P n s f fi m n)
+#A; #P; #Sn; ncases Sn
+  [ #s; #f; #fi;
+    nlapply (covers ? P); *; #_; #H;
+    (*
+    nlapply
+     (big_union_preserves_iso ??? (indexes A P) (Nat_ O) (λx.class ? P x) f);
+     *; #K; #_; nwhd in K: (? → ? → %);*)
+    nelim daemon (* impossibile *)
+  | #n; #s; #f; #fi; @
+  [ @
+     [ napply (λm.let p ≝ (iso_nat_nat_union s m n) in iso_f ???? (fi (fst … p)) (snd … p))
      | #a; #a'; #H; nrewrite < H; napply refl ]
+##| #x; #Hx; nwhd; napply I
 ##| #y; #_;
-    ngeneralize in match (covers ? P) in ⊢ ?; *; #_; #Hc;
-    ngeneralize in match (Hc y I) in ⊢ ?; *; #index; *; #Hi1; #Hi2;
-    nelim daemon
-  | #x; #x'; nnormalize in ⊢ (? → ? → %);
-    nelim daemon
-  ]
+    nlapply (covers ? P); *; #_; #Hc;
+    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".
+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);
+    @ xxx; @
+     [ napply iso_nat_nat_union_pre [ napply le_S_S_to_le; nassumption | nassumption ]
+   ##| nwhd in ⊢ (???%%); napply (.= ?) [##3: nassumption|##skip]
+       nlapply (iso_nat_nat_union_char n s xxx ?)
+        [napply iso_nat_nat_union_pre [ napply le_S_S_to_le; nassumption | 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 ]##]
+##| #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) (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; //
+       |##1: @ (fi i1 i2); @;
+         ##[ napply f_closed; // ##| 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''; // ]##]
+   ##] #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';
+    nlapply (K … E)  
+     [##1,2: nassumption;
+     ##|##3,4:napply le_to_le_S_S; nassumption; ##]
+    *; #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.
+nqed.
+
+(************** equivalence relations vs partitions **********************)
+
+ndefinition partition_of_compatible_equivalence_relation:
+ ∀A:setoid. compatible_equivalence_relation A → partition A.
+ #A; #R; napply mk_partition
+  [ napply (quotient ? R)
+  | napply Full_set
+  | napply mk_unary_morphism1
+     [ #a; napply mk_ext_powerclass
+        [ 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