X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fnlibrary%2Fsets%2Fpartitions.ma;h=f62c7281d1b074a068369cb9d71adfcca1c7b76d;hb=d70944c1513aa63e6494e58595fcc4214a2f6c68;hp=43940e1f9cc4b1d7831a841fe9abddff81b1bbf5;hpb=c6d3537eee27d05490a9555cc7326bc954b356c5;p=helm.git

diff --git a/helm/software/matita/nlibrary/sets/partitions.ma b/helm/software/matita/nlibrary/sets/partitions.ma
index 43940e1f9..f62c7281d 100644
--- a/helm/software/matita/nlibrary/sets/partitions.ma
+++ b/helm/software/matita/nlibrary/sets/partitions.ma
@@ -17,49 +17,234 @@ include "nat/plus.ma".
 include "nat/compare.ma".
 include "nat/minus.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".
+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".
+alias symbol "eq" = "setoid1 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 ? ? (λx.class x) = full_set A
+ }. napply indexes; nqed.
 
-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*)
- }.
-
-(*
-nlemma subset_of_finite:
- ∀A. ∃n. has_card ? (full_subset A) n → ∀S. ∃m. has_card ? S m.
-nqed.
-*)
+naxiom daemon: False.
 
 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 (H1: lt m (big_plus ? s)) index (p: lt index (card ? P)) on index : A ≝
- match index return λx. lt x (card ? P) → ? with
-  [ O ⇒ λp'.f ??? (H O p') m ?
-  | S index' ⇒ λp'.
-     match ltb m (s index p) with
-      [ or_introl K ⇒ f ??? (H index p) m K 
-      | or_intror _ ⇒ partition_splits_card_map A P s H (minus m (s index p)) ? index' ? ]] p.
-##[##2: napply lt_minus; nassumption
-  |##3: napply lt_Sn_m; nassumption
-  |
-nqed.
+ 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 ≝
+ match ltb m (s index) with
+  [ true ⇒ iso_f ???? (fi 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']].  
+
+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)).
+
+naxiom le_to_lt_or_eq: ∀n,m. n ≤ m → n < m ∨ n = m.
+alias symbol "eq" = "leibnitz's equality".
+naxiom minus_canc: ∀n. O = minus n n.
+naxiom lt_to_ltb_t: ∀n,m. ∀P: bool → CProp[0]. P true → n < m → P (ltb n m).
+naxiom lt_to_ltb_f: ∀n,m. ∀P: bool → CProp[0]. P false → ¬ (n < m) → P (ltb n m).
+naxiom lt_to_minus: ∀n,m. n < m →  S (minus (minus m n) (S O)) = minus m n.
+naxiom not_lt_O: ∀n. ¬ (n < O).
+naxiom minus_S: ∀n,m. m ≤ n → minus (S n) m = S (minus n m).
+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. 
 
 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; 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;
+    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 (S n) s f fi m n)
+     | #a; #a'; #H; nrewrite < H; napply refl ]
+##| #y; #_;
+    ngeneralize in match (covers ? P) in ⊢ ?; *; #_; #Hc;
+    ngeneralize in match (Hc y I) in ⊢ ?; *; #index; *; #Hi1; #Hi2;
+    ngeneralize in match (f_sur ???? f ? Hi1) in ⊢ ?; *; #nindex; *; #Hni1; #Hni2;
+    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);
+    napply (ex_intro … xxx); napply conj
+     [ nwhd in Hni1; nwhd; nelim daemon
+     | 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
+           (plus
+             match minus x nindex with
+              [ O ⇒ O | S nn ⇒ big_plus nn (λi.λ_.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';
+          ngeneralize in match Hni21 in ⊢ ?;
+          ngeneralize in match Hni22 in ⊢ ?;
+          nrewrite > K' in ⊢ (% → % → ?); #K1; #K2;
+          nrewrite > (ltb_t … K2);
+          nwhd in ⊢ (???%?); nassumption
+        | #n'; #Hrec; #HH; nelim (le_to_lt_or_eq … HH)
+           [##2: #K; nrewrite < K; nrewrite < (minus_canc nindex);
+            nwhd in ⊢ (???(???????%?)?);
+            (*???????*)
+         ##| #K; nwhd in ⊢ (???%?);
+             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)
+               (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 ⊢ ?;
+                    
+               
+                ngeneralize in match (? :
+                 minus (plus (big_plus (minus n' nindex) (λi.λ_.s (S (plus i nindex)))) nindex2)
+                  (s (S n'))
+                 =
+                  plus
+                   match minus n' nindex with
+                    [ O ⇒ O | S nn ⇒ big_plus nn (λi.λ_.s (S (plus i nindex)))] nindex2)
+                 in ⊢ ?
+                  [ #F; nrewrite > F; napply Hrec; napply le_S_S_to_le; nassumption
+                  | nelim (le_to_lt_or_eq … (le_S_S_to_le … K))
+                     [
+                   ##| #E; nrewrite < E; nrewrite < (minus_canc nindex); nnormalize;
+                       
+                       nwhd in ⊢ (???%);
+                     ]
+ 
+             
+              nrewrite > He;
+             
+               
+              nnormalize in ⊢ (???%?);
+          
+       
+       
+       nelim (le_to_lt_or_eq … K)
+        [##2: #K'; nrewrite > K'; nrewrite < (minus_canc n); nnormalize;
+         napply (eq_rect_CProp0 nat nindex (λx:nat.λ_.partition_splits_card_map A P (S n) s f fi nindex2 x = y) ? n K');
+         nchange in Hni21 with (nindex2 < s nindex); ngeneralize in match Hni21 in ⊢ ?;
+         ngeneralize in match Hni22 in ⊢ ?;
+         nelim nindex
+          [ #X1; #X2; nwhd in ⊢ (??? % ?);
+            napply (lt_to_ltb_t ???? X2); #D; nwhd in ⊢ (??? % ?); nassumption
+          | #n0; #_; #X1; #X2; nwhd in ⊢ (??? % ?);
+            napply (lt_to_ltb_t ???? X2); #D; nwhd in ⊢ (??? % ?); nassumption]
+      ##| #K'; ngeneralize in match (lt_to_minus … K') in ⊢ ?; #K2;
+          napply (eq_rect_CProp0 ?? (λx.λ_.?) ? ? K2); (* uffa, ancora??? *)
+          nwhd in ⊢ (??? (???????(?%?)?) ?);
+          ngeneralize in match K' in ⊢ ?;
+          napply (nat_rect_CProp0
+           (λx. nindex < x →
+             partition_splits_card_map A P (S n) s f fi
+              (plus (big_op plus_magma_type (minus (minus x nindex) (S O))
+                (λi.λ_.s (S (plus i nindex))) O) nindex2) x = y) ?? n)
+           [ #A; nelim (not_lt_O … A)
+           | #n'; #Hrec; #X; nwhd in ⊢ (???%?);
+             ngeneralize in match
+              (? : ¬ ((plus (big_op plus_magma_type (minus (minus (S n') nindex) (S O))
+                (λi.λ_.s (S (plus i nindex))) O) nindex2) < s (S n'))) in ⊢ ?
+              [ #B1; napply (lt_to_ltb_f ???? B1); #B1'; nwhd in ⊢ (???%?);
+                nrewrite > (minus_S n' nindex …) [##2: napply le_S_S_to_le; nassumption]
+                ngeneralize in match (le_S_S_to_le … X) in ⊢ ?; #X';
+                nelim (le_to_lt_or_eq … X')
+                 [##2: #X'';
+                  nchange in Hni21 with (nindex2 < s nindex); ngeneralize in match Hni21 in ⊢ ?;
+                  nrewrite > X''; nrewrite < (minus_canc n');
+                  nrewrite < (minus_canc (S O)); nnormalize in ⊢ (? → %);
+                  nelim n'
+                   [ #Y; nwhd in ⊢ (??? % ?);
+                     ngeneralize in match (minus_lt_to_lt ? (s (S O)) ? Y) in ⊢ ?; #Y';
+                     napply (lt_to_ltb_t … Y'); #H; nwhd in ⊢ (???%?);  
+                 
+                nrewrite > (minus_S (minus n' nindex) (S O) …) [##2: 
+              
+              XXX;
+          
+          nelim n in f K' ⊢ ?
+           [ #A; nelim daemon;
+         
+       (* BEL POSTO DOVE FARE UN LEMMA *)
+       (* invariante: Hni1; altre premesse: Hni1, Hni22 *)
+       nelim n in ⊢ (% → ??? (????????%) ?)
+        [ #A (* decompose *)
+        | #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 ⊢ (? → ? → %);
+    nelim daemon
   ]
+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
+  | #a; napply mk_qpowerclass
+     [ napply {x | R x a}
+     | #x; #x'; #H; nnormalize; napply mk_iff; #K; 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