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[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 7affc517f79d26aaa5dfcf06463ecf7881816ed0..1dce8d535f0926646958a8224412065e55dc1ea3 100644 (file)
--- a/helm/software/matita/nlibrary/sets/partitions.ma
+++ b/helm/software/matita/nlibrary/sets/partitions.ma
@@ -18,11 +18,11 @@ 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" = "setoid eq".
 alias symbol "eq" = "setoid1 eq".
 alias symbol "eq" = "setoid eq".
 alias symbol "eq" = "setoid1 eq".
@@ -32,10 +32,10 @@ nrecord partition (A: setoid) : Type[1] ≝
    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.
-
+   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 ≝
@@ -149,44 +149,47 @@ 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" = "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);
-    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
+          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 ∈ 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');
+    ##[ #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 (. ?‡#) [ 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''')
+       |##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);