X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=helm%2Fsoftware%2Fmatita%2Fnlibrary%2Fsets%2Fpartitions.ma;h=93f9cd6b5d3e014121adf9ff04042816848cb5a6;hb=8cb2490b5b202549a596cfd1d0f166a5ee43fc4e;hp=0ef3fa8c4a4b5d44ac35219a17a626e71a47b6b7;hpb=3d3d0248bf4770c63361f7805d2099b2a607f44d;p=helm.git

diff --git a/helm/software/matita/nlibrary/sets/partitions.ma b/helm/software/matita/nlibrary/sets/partitions.ma
index 0ef3fa8c4..93f9cd6b5 100644
--- a/helm/software/matita/nlibrary/sets/partitions.ma
+++ b/helm/software/matita/nlibrary/sets/partitions.ma
@@ -13,63 +13,201 @@
 (**************************************************************************)
 
 include "sets/sets.ma".
-include "nat/plus.ma".
+include "nat/plus.ma". 
 include "nat/compare.ma".
 include "nat/minus.ma".
+include "datatypes/pairs.ma".
 
-(* sbaglia a fare le proiezioni *)
-nrecord finite_partition (A: Type[0]) : Type[1] ≝ 
- { card: nat;
-   class: ∀n. lt n card → Ω \sup A;
-   inhabited: ∀i,p. class i p ≬ class i p(*;
-   disjoint: ∀i,j,p,p'. class i p ≬ class j p' → i=j;
-   covers: big_union ?? class = full_set A*)
- }.
+alias symbol "eq" = "setoid eq".
 
-nrecord has_card (A: Type[0]) (S: Ω \sup A) (n: nat) : CProp[0] ≝
- { f: ∀m:nat. lt m n → A;
-   in_S: ∀m.∀p:lt m n. f ? p ∈ S (*;
-   f_inj: injective ?? f;
-   f_sur: surjective ?? f*)
+alias symbol "eq" = "setoid1 eq".
+alias symbol "eq" = "setoid 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;
+   class: unary_morphism1 (setoid1_of_setoid support) (qpowerclass_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 ? indexes (λx.class x) = full_set A
  }.
+ 
+naxiom daemon: False.
+
+nlet rec iso_nat_nat_union (s: nat → nat) m index on index : pair nat nat ≝
+ match ltb m (s index) with
+  [ true ⇒ mk_pair … index m
+  | false ⇒
+     match index with
+      [ 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']].
 
-(*
-nlemma subset_of_finite:
- ∀A. ∃n. has_card ? (full_subset A) n → ∀S. ∃m. has_card ? S m.
+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'.
 
-naxiom daemon: False.
+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; @
+    [ @ [ 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'); 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.
 
-nlet rec partition_splits_card_map
- A (P: finite_partition A) (s: ∀i. lt i (card ? P) → nat)
-  (H:∀i.∀p: lt i (card ? P). has_card ? (class ? P i p) (s i p))
-  m index on index:
-  le (S index) (card ? P) → lt m (big_plus (S index) (λi,p. s i ?)) → lt index (card ? P) → A ≝
- match index return λx. le (S x) (card ? P) → lt m (big_plus (S x) ?) → lt x (card ? P) → ? with
-  [ O ⇒ λL,H1,p.f ??? (H O p) m ?
-  | S index' ⇒ λL,H1,p.
-     match ltb m (s (S index') p) with
-      [ or_introl K ⇒ f ??? (H (S index') p) m K 
-      | or_intror _ ⇒ partition_splits_card_map A P s H (minus m (s (S index') p)) index' ??? ]].
-##[##3: napply lt_minus; nelim daemon (*nassumption*)
-  |##4: napply lt_Sn_m; nassumption
-  |##5: napply (lt_le_trans … p); nassumption
-##|##2: napply lt_to_le; nassumption
-##|##1: nnormalize in H1; nelim daemon ]
+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).
+ #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.
+  ∀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: finite_partition A. ∀s: ∀i. lt i (card ? P) → nat.
-  (∀i.∀p: lt i (card ? P). has_card ? (class ? P i p) (s i p)) →
-   has_card A (full_set A) (big_plus (card ? P) s).
- #A; #P; #s; #H; ncases (card A P)
-  [ nnormalize; napply mk_has_card
-     [ #m; #H; nelim daemon
-     | #m; #H; nelim daemon ]
-##| #c; napply mk_has_card
-  [ #m; #H1; napply partition_splits_card_map A P s H m H1 (pred (card ? P))
-  |
-  ]
+ ∀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; #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; #_;
+    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" = "refl".
+alias symbol "prop1" = "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: *; #E1; #E2; nrewrite > E1; nrewrite > E2; napply refl
+        | napply le_S_S_to_le; nassumption
+        |##*: nassumption]##]
+##| #x; #x'; nnormalize in ⊢ (? → ? → %); #Hx; #Hx'; #E;
+    ngeneralize in match (? : ∀i1,i2,i1',i2'. i1 ∈ Nat_ (S n) → i1' ∈ Nat_ (S n) → i2 ∈ pc ? (Nat_ (s i1)) → i2' ∈ pc ? (Nat_ (s i1')) → eq_rel (carr A) (eq A) (iso_f ???? (fi i1) i2) (iso_f ???? (fi i1') i2') → i1=i1' ∧ i2=i2') in ⊢ ?
+     [##2: #i1; #i2; #i1'; #i2'; #Hi1; #Hi1'; #Hi2; #Hi2'; #E;
+      ngeneralize in match (disjoint ? P (iso_f ???? f i1) (iso_f ???? f i1') ???) in ⊢ ?
+       [##2,3: napply f_closed; nassumption
+       |##4: napply ex_intro [ napply (iso_f ???? (fi i1) i2) ] napply conj
+         [ napply f_closed; nassumption ##| napply (. ?‡#) [ nassumption | ##2: ##skip]
+         nwhd; napply f_closed; nassumption]##]
+      #E'; ngeneralize in match (? : i1=i1') in ⊢ ?
+       [##2: napply (f_inj … E'); nassumption
+       | #E''; nrewrite < E''; napply conj
+          [ napply refl | 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';
+    ngeneralize in match (K … E) in ⊢ ?
+     [##2,3: napply le_to_le_S_S; nassumption
+     |##4,5: 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_qpowerclass
+        [ 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; 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.
-*)
\ No newline at end of file