X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fnlibrary%2Fsets%2Fpartitions.ma;h=de29e796c47141fc80cd8659aa35a64c68e01a82;hb=b2d91e46900424ce5eb5a058c33841e72cc4b229;hp=3b17099a371474318101168301daff9fe23a31ef;hpb=f5f35ef830b1335dad2fcc3c1aae2b57815f73b1;p=helm.git

diff --git a/helm/software/matita/nlibrary/sets/partitions.ma b/helm/software/matita/nlibrary/sets/partitions.ma
index 3b17099a3..de29e796c 100644
--- a/helm/software/matita/nlibrary/sets/partitions.ma
+++ b/helm/software/matita/nlibrary/sets/partitions.ma
@@ -13,28 +13,21 @@
 (**************************************************************************)
 
 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" = "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" (instance 7) = "setoid1 eq".
 nrecord partition (A: setoid) : Type[1] ≝ 
  { support: setoid;
-   indexes: qpowerclass support;
-   class: unary_morphism1 (setoid1_of_setoid support) (qpowerclass_setoid 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 iso_nat_nat_union (s: nat → nat) m index on index : pair nat nat ≝
@@ -62,7 +55,6 @@ 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_S: ∀a,b. S a - S b = a - 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:
@@ -80,23 +72,23 @@ ntheorem iso_nat_nat_union_char:
     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 conj [ napply conj [ napply refl | napply le_n ] ##| nassumption ]
+    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
+        [ 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));
-       ngeneralize in match (Hrec (m - s (S n')) ?) in ⊢ ?
-        [##2: napply ad_hoc9; nassumption] *; *; #Hrec1; #Hrec2; #Hrec3; napply conj
+       nlapply (Hrec (m - s (S n')) ?)
+        [ napply ad_hoc9; nassumption] *; *; #Hrec1; #Hrec2; #Hrec3; @
         [##2: nassumption
-        |napply conj
+        |@
          [nrewrite > (split_big_plus …); ##[##2:napply ad_hoc11;##|##3:##skip]
           nrewrite > (ad_hoc12 …); ##[##2: nassumption]
-          nwhd in ⊢ (????(?(??%)?));
+          nwhd in ⊢ (???(?(??%)?));
           nrewrite > (ad_hoc13 …);##[##2: nassumption]
           napply ad_hoc14 [ napply not_lt_to_le; nassumption ]
           nwhd in ⊢ (???(?(??%)?));
@@ -114,10 +106,7 @@ ntheorem iso_nat_nat_union_pre:
   [##2: napply le_to_le_S_S; nassumption]
  napply ad_hoc15
   [ nwhd in ⊢ (???(?%?));
-  
-    napply (eq_rect_CProp0_r nat (n - i1) (λx.λy.?) ?? (minus_S_S …)); STOP
-  
-    nrewrite > (minus_S_S n i1); napply big_plus_preserves_ext; #i; #_;
+    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) …);
@@ -138,58 +127,62 @@ nlemma partition_splits_card:
   ∀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)).
- STOP #A; #P; #Sn; ncases Sn
+#A; #P; #Sn; ncases Sn
   [ #s; #f; #fi;
-    ngeneralize in match (covers ? P) in ⊢ ?; *; #_; #H;
+    nlapply (covers ? P); *; #_; #H;
     (*
-    ngeneralize in match
-     (big_union_preserves_iso ??? (indexes A P) (Nat_ O) (λx.class ? P x) f) in ⊢ ?;
+    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 mk_isomorphism
-  [ napply mk_unary_morphism
-     [ napply (λm.let p ≝ iso_nat_nat_union s m n in iso_f ???? (fi (fst … p)) (snd … p))
+  | #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;
-    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]##]
+    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".
+alias symbol "prop2" = "prop21 mem".
+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);
-    napply (ex_intro … xxx); napply conj
+    @ xxx; @
      [ napply iso_nat_nat_union_pre [ napply le_S_S_to_le; nassumption | nassumption ]
-   ##| nwhd in ⊢ (???%%); napply (.= ?) [ nassumption|##skip]
-       ngeneralize in match (iso_nat_nat_union_char n s xxx ?) in ⊢ ?
-        [##2: 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;
-       ngeneralize in match
+       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)) ?????) in ⊢ ?
-        [ *; #E1; #E2; nrewrite > E1; nrewrite > E2; napply refl
-        | napply le_S_S_to_le; nassumption
+          nindex2 (snd … (iso_nat_nat_union s xxx n)) ?????)
+        [##6: *; #E1; #E2; nrewrite > E1; nrewrite > E2; napply refl
+        |##5: 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 ⊢ ?
+    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; nassumption
-       |##4: napply ex_intro [ napply (iso_f ???? (fi i1) i2) ] napply conj
-         [ napply f_closed; nassumption ##| napply (. ?‡#) [##2: nassumption | ##3: ##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''')
+       |##1: @ (fi i1 i2); @;
+         ##[ napply f_closed; nassumption ##| alias symbol "refl" = "refl1".
+napply (. E‡#);
+             nwhd; napply f_closed; nassumption]##]
+      #E'; ncut(i1 = i1'); ##[ napply (f_inj … E'); nassumption; ##]
+      #E''; nrewrite < E''; @; 
+      ##[ @;
+      ##| 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]
+    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);
@@ -204,7 +197,7 @@ ndefinition partition_of_compatible_equivalence_relation:
   [ napply (quotient ? R)
   | napply Full_set
   | napply mk_unary_morphism1
-     [ #a; napply mk_qpowerclass
+     [ #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]##]
@@ -213,4 +206,4 @@ ndefinition partition_of_compatible_equivalence_relation:
     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
+nqed.