// used everywhere!

[helm.git] / helm / software / matita / nlibrary / sets / partitions.ma

diff --git a/helm/software/matita/nlibrary/sets/partitions.ma b/helm/software/matita/nlibrary/sets/partitions.ma
index 1dce8d535f0926646958a8224412065e55dc1ea3..de29e796c47141fc80cd8659aa35a64c68e01a82 100644 (file)
--- a/helm/software/matita/nlibrary/sets/partitions.ma
+++ b/helm/software/matita/nlibrary/sets/partitions.ma
@@ -18,19 +18,11 @@ 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" = "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 ? indexes (λx.class x) = full_set A
@@ -86,7 +78,7 @@ ntheorem iso_nat_nat_union_char:
     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));
@@ -96,7 +88,7 @@ ntheorem iso_nat_nat_union_char:
         |@
          [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 ⊢ (???(?(??%)?));
@@ -155,6 +147,7 @@ nlemma partition_splits_card:
     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);
@@ -171,7 +164,7 @@ napply (. #‡(†?));##[##2: napply Hni2 |##1: ##skip | nassumption]##]
         |##5: napply le_S_S_to_le; nassumption
         |##*: nassumption]##]
 ##| #x; #x'; nnormalize in ⊢ (? → ? → %); #Hx; #Hx'; #E;
-    ncut(∀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');
+    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
@@ -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]##]