X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fnlibrary%2Fsets%2Fpartitions.ma;h=b92fe9ab3011d9bcbcb75637b51c886c43de6d05;hb=c57405141d26ac2215a07b05d27a16a691dda50e;hp=40ecd43b43ec88377f22796f37a77555cfb54c53;hpb=8049c166a37789d7a1b1ca1c3a1174712bbf87ba;p=helm.git

diff --git a/helm/software/matita/nlibrary/sets/partitions.ma b/helm/software/matita/nlibrary/sets/partitions.ma
index 40ecd43b4..b92fe9ab3 100644
--- a/helm/software/matita/nlibrary/sets/partitions.ma
+++ b/helm/software/matita/nlibrary/sets/partitions.ma
@@ -18,23 +18,16 @@ include "nat/compare.ma".
 include "nat/minus.ma".
 include "datatypes/pairs.ma".
 
-alias symbol "eq" (instance 2) = "leibnitz's equality".
-alias symbol "eq" (instance 1) = "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" = "setoid1 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 ≝
@@ -79,46 +72,29 @@ 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 refl | napply le_n ] ##| nassumption ]
+    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'); 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 ]##]##]##]
+     [ 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).
- #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.
+/2/. nqed.
     
 ntheorem iso_nat_nat_union_uniq:
  ∀n:nat. ∀s: nat → nat.
@@ -136,7 +112,7 @@ nlemma partition_splits_card:
     (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;
+    nlapply (covers ? P); *; #_; #H;
     (*
     nlapply
      (big_union_preserves_iso ??? (indexes A P) (Nat_ O) (λx.class ? P x) f);
@@ -148,48 +124,50 @@ nlemma partition_splits_card:
      | #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 (. #‡(†?));##[##2: napply Hni2 |##1: ##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" (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);
-    napply (ex_intro … xxx); napply conj
+    @ xxx; @
      [ napply iso_nat_nat_union_pre [ napply le_S_S_to_le; nassumption | nassumption ]
    ##| nwhd in ⊢ (???%%); napply (.= ?) [##3: 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]##]
+       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
-        |##*: nassumption]##]
+          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;
-    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 ]##]##]
+    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';
-    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);
-    nrewrite > K1; nrewrite > K2; napply refl ]
+    nrewrite > K1; nrewrite > K2; napply refl.
 nqed.
 
 (************** equivalence relations vs partitions **********************)
@@ -200,13 +178,11 @@ 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]##]
-##| #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.
+##| #x; #_; nnormalize; /3/
+  | #x; #x'; #_; #_; nnormalize; *; #x''; *; /3/
+  | nnormalize; napply conj; /4/ ]
+nqed.
\ No newline at end of file