X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fnlibrary%2Fsets%2Fpartitions.ma;h=1dce8d535f0926646958a8224412065e55dc1ea3;hb=d93c422814484956482b1e70d90bb443eb99af21;hp=d87ccd3a9c5e6a1b20aa224d5708d7e936aa69b8;hpb=7508af658c85b26d0af929c85069bae18bc6546a;p=helm.git

diff --git a/helm/software/matita/nlibrary/sets/partitions.ma b/helm/software/matita/nlibrary/sets/partitions.ma
index d87ccd3a9..1dce8d535 100644
--- a/helm/software/matita/nlibrary/sets/partitions.ma
+++ b/helm/software/matita/nlibrary/sets/partitions.ma
@@ -13,19 +13,15 @@
 (**************************************************************************)
 
 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" = "setoid1 eq".
 alias symbol "eq" = "setoid eq".
 alias symbol "eq" = "setoid1 eq".
 alias symbol "eq" = "setoid eq".
@@ -36,10 +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 ≝
@@ -51,7 +47,8 @@ nlet rec iso_nat_nat_union (s: nat → nat) m index on index : pair nat nat ≝
       | S index' ⇒ iso_nat_nat_union s (minus m (s index)) index']].
 
 alias symbol "eq" = "leibnitz's equality".
-naxiom plus_n_O: ∀n. plus n O = n.
+naxiom plus_n_O: ∀n. n + O = n.
+naxiom plus_n_S: ∀n,m. n + S m = S (n + m).
 naxiom ltb_t: ∀n,m. n < m → ltb n m = true.
 naxiom ltb_f: ∀n,m. ¬ (n < m) → ltb n m = false.
 naxiom ltb_cases: ∀n,m. (n < m ∧ ltb n m = true) ∨ (¬ (n < m) ∧ ltb n m = false).
@@ -62,12 +59,19 @@ naxiom ad_hoc11: ∀a,b. a - b ≤ S a - b.
 naxiom ad_hoc12: ∀a,b. b ≤ a → S a - b - (a - b) = S O.
 naxiom ad_hoc13: ∀a,b. b ≤ a → (O + (a - b)) + b = a.
 naxiom ad_hoc14: ∀a,b,c,d,e. c ≤ a → a - c = b + d + e → a = b + (c + d) + e.
+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: ∀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:
   ∀n,m,f. m ≤ n →
    big_plus n f = big_plus m (λi,p.f i ?) + big_plus (n - m) (λi.λp.f (i + m) ?).
  nelim daemon.
 nqed.
+naxiom big_plus_preserves_ext:
+ ∀n,f,f'. (∀i,p. f i p = f' i p) → big_plus n f = big_plus n f'.
 
 ntheorem iso_nat_nat_union_char:
  ∀n:nat. ∀s: nat → nat. ∀m:nat. m < big_plus (S n) (λi.λ_.s i) →
@@ -76,8 +80,8 @@ 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
@@ -86,92 +90,44 @@ ntheorem iso_nat_nat_union_char:
         | 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
-         [napply (eq_rect_CProp0_r ?? (λx.λ_. m = x + snd … (iso_nat_nat_union s (m - s (S n')) n')) ??
-           (split_big_plus
-            (S n' - fst … (iso_nat_nat_union s (m - s (S n')) n'))
-            (n' - fst … (iso_nat_nat_union s (m - s (S n')) n'))
-            (λi.λ_.s (S (i + fst … (iso_nat_nat_union s (m - s (S n')) n'))))?))
-           [##2: napply ad_hoc11]
-          napply (eq_rect_CProp0_r ?? (λx.λ_. ? = ? + big_plus x (λ_.λ_:? < x.?) + ?)
-           ?? (ad_hoc12 n' (fst … (iso_nat_nat_union s (m - s (S n')) n')) ?))
-            [##2: nassumption]
-          nwhd in ⊢ (???(?(??%)?));
-          nrewrite > (ad_hoc13 n' (fst … (iso_nat_nat_union s (m - s (S n')) n')) ?)
-           [##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 ⊢ (???(?(??%)?));
-          napply (eq_rect_CProp0_r ?? (λx.λ_. ? = x + ?) ??
-           (plus_n_O (big_plus (n' - fst … (iso_nat_nat_union s (m - s (S n')) n'))
-            (λi.λ_.s (S (i + fst … (iso_nat_nat_union s (m - s (S n')) n')))))));
-          nassumption
-        | napply le_S; nassumption ]##]##]##]
+          nrewrite > (plus_n_O …);
+          nassumption;
+        ##| napply le_S; nassumption ]##]##]##]
 nqed.
 
-
-nlet rec partition_splits_card_map
- 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 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 not_lt_plus: ∀n,m. ¬ (plus n m < n).
-naxiom lt_to_lt_plus: ∀n,m,l. n < m → n < m + l.
-naxiom S_plus: ∀n,m. S (n + m) = n + S m.
-naxiom big_plus_ext: ∀n,f,f'. (∀i,p. f i p = f' i p) → big_plus n f = big_plus n f'.
-naxiom ad_hoc1: ∀n,m,l. n + (m + l) = l + (n + m).
-naxiom assoc: ∀n,m,l. n + m + l = n + (m + l).
-naxiom lt_canc: ∀n,m,p. n < m → p + n < p + m.
-naxiom ad_hoc2: ∀a,b. a < b → b - a - (b - S a) = S O.
-naxiom ad_hoc3: ∀a,b. b < a → S (O + (a - S b) + b) = a.
-naxiom ad_hoc4: ∀a,b. a - S b ≤ a - b.
-naxiom ad_hoc5: ∀a. S a - a = S O.
-naxiom ad_hoc6: ∀a,b. b ≤ a → a - b + b = a.
-naxiom ad_hoc7: ∀a,b,c. a + (b + O) + c - b = a + c.
-naxiom ad_hoc8: ∀a,b,c. ¬ (a + (b + O) + c < b).
-naxiom ltb_elim_CProp0: ∀n,m. ∀P: bool → CProp[0]. (n < m → P true) → (¬ (n < m) → P false) → P (ltb n m).               
-
-nlemma partition_splits_card_output:
- ∀A. ∀P:partition A. ∀n,s.
-  ∀f:isomorphism ?? (Nat_ (S n)) (indexes ? P).
-   ∀fi:∀i. isomorphism ?? (Nat_ (s i)) (class ? P (iso_f ???? f i)).
-    ∀x. x ∈ Nat_ (big_plus (S n) (λi.λ_.s i)) →
-     ∃i1.∃i2. partition_splits_card_map A P (S n) s f fi x n = iso_f ???? (fi i1) i2.
- #A; #P; #n; #s; #f; #fi;
- nelim n in ⊢ (? → % → ??(λ_.??(λ_.???(????????%)?)))
-  [ #x; nnormalize in ⊢ (% → ?); nrewrite > (plus_n_O (s O)); nchange in ⊢ (% → ?) with (x < s O);
-    #H; napply (ex_intro … O); napply (ex_intro … x); nwhd in ⊢ (???%?);
-    nrewrite > (ltb_t … H); nwhd in ⊢ (???%?); napply refl
-  | #n'; #Hrec; #x; #Hx; nwhd in ⊢ (??(λ_.??(λ_.???%?))); nwhd in Hx; nwhd in Hx: (??%);
-    nelim (ltb_cases x (s (S n'))); *; #K1; #K2; nrewrite > K2; nwhd in ⊢ (??(λ_.??(λ_.???%?)))
-     [ napply (ex_intro … (S n')); napply (ex_intro … x); napply refl
-     | napply (Hrec (x - s (S n')) ?); nwhd; nrewrite < (minus_S x (s (S n')) ?)
-       [ napply ad_hoc9; nassumption | napply not_lt_to_le; nassumption ]##]
+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.
+    
+ntheorem iso_nat_nat_union_uniq:
+ ∀n:nat. ∀s: nat → nat.
+  ∀i1,i1',i2,i2'. i1 ≤ n → i1' ≤ n → i2 < s i1 → i2' < s i1' →
+   big_plus (n - i1) (λi.λ_.s (S (i + i1))) + i2 = big_plus (n - i1') (λi.λ_.s (S (i + i1'))) + i2' →
+    i1 = i1' ∧ i2 = i2'.
+ #n; #s; #i1; #i1'; #i2; #i2'; #H1; #H1'; #H2; #H2'; #E;
+ nelim daemon.
 nqed.
 
 nlemma partition_splits_card:
@@ -179,122 +135,65 @@ 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)).
- #A; #P; #Sn; ncases Sn
+#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: (? → ? → %);
+    nlapply (covers ? P); *; #_; #H;
+    (*
+    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.partition_splits_card_map A P (S n) s f fi m n)
+  | #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".
+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
-     [ nwhd in Hni1; nwhd; nwhd in ⊢ (?(? %)%);
-       nchange with (? < plus (s n) (big_plus n ?));
-       nelim (le_to_lt_or_eq … (le_S_S_to_le … Hni1))
-        [##2: #E; nrewrite < E; nrewrite < (minus_canc nindex);
-          nwhd in ⊢ (?%?); nrewrite < E; napply lt_to_lt_plus; nassumption
-        | #L; nrewrite > (split_big_plus n (S nindex) (λm.λ_.s m) L);
-          nrewrite > (split_big_plus (n - nindex) (n - S nindex) (λi.λ_.s (S (i+nindex))) ?)
-           [ ngeneralize in match (big_plus_ext (n - S nindex)
-              (λi,p.s (S (i+nindex))) (λi,p.s (i + S nindex)) ?) in ⊢ ?
-               [ #E;
-                 napply (eq_rect_CProp0_r ??
-                  (λx:nat.λ_. x + big_plus (n - nindex - (n - S nindex))
-                   (λi,p.s (S (i + (n - S nindex)+nindex))) + nindex2 <
-                   s n + (big_plus (S nindex) (λi,p.s i) +
-                    big_plus (n - S nindex) (λi,p. s (i + S nindex)))) ? ? E);
-                 nrewrite > (ad_hoc1 (s n) (big_plus (S nindex) (λi,p.s i))
-                  (big_plus (n - S nindex) (λi,p. s (i + S nindex))));
-                 napply (eq_rect_CProp0_r
-                  ?? (λx.λ_.x < ?) ?? (assoc
-                  (big_plus (n - S nindex) (λi,p.s (i + S nindex)))
-                  (big_plus (n - nindex - (n - S nindex))
-                   (λi,p.s (S (i + (n - S nindex)+nindex))))
-                  nindex2));
-                 napply lt_canc;
-                 nrewrite > (ad_hoc2 … L); nwhd in ⊢ (?(?%?)?);
-                 nrewrite > (ad_hoc3 … L);
-                 napply (eq_rect_CProp0_r ?? (λx.λ_.x < ?) ?? (assoc …));
-                 napply lt_canc; nnormalize in ⊢ (?%?); nwhd in ⊢ (??%);
-                 napply lt_to_lt_plus; nassumption
-             ##|##2: #i; #_; nrewrite > (S_plus i nindex); napply refl]
-         ##| napply ad_hoc4]##]
-   ##| 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 →
-          eq_rel (carr A) (eq A)
-          (partition_splits_card_map A P (S n) s f fi
-           (plus
-             (big_plus (minus x nindex) (λi.λ_:i < minus x nindex.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 ⊢ (???(???????%?)?);
-            nrewrite > K;
-            nwhd in ⊢ (???%?); nrewrite < K; nrewrite > (ltb_t … Hni21);
-            nwhd in ⊢ (???%?); nassumption
-         ##| #K; ngeneralize in match (le_S_S_to_le … K) in ⊢ ?; #K';
-             nwhd in ⊢ (???%?);
-             ngeneralize in match (?:
-              ¬ (big_plus (S n' - nindex) (λi,p.s (S (i+nindex))) + nindex2 < s (S n'))) in ⊢ ?
-              [ #N; nrewrite > (ltb_f … N); nwhd in ⊢ (???%?);
-                ngeneralize in match (Hrec K') in ⊢ ?; #Hrec';
-                napply (eq_rect_CProp0_r ??
-                 (λx,p. eq_rel (carr A) (eq A) (partition_splits_card_map A P (S n) s f fi
-                  (big_plus x ? + ? - ?) n') y) ?? (minus_S n' nindex K'));
-                nrewrite > (split_big_plus (S (n' - nindex)) (n' - nindex)
-                 (λi,p.s (S (i+nindex))) (le_S ?? (le_n ?)));
-                nrewrite > (ad_hoc5 (n' - nindex));
-                nnormalize in ⊢ (???(???????(?(?(??%)?)?)?)?);
-                nrewrite > (ad_hoc6 … K');
-                nrewrite > (ad_hoc7 (big_plus (n' - nindex) (λi,p.s (S (i+nindex))))
-                 (s (S n')) nindex2);
-                nassumption
-              | nrewrite > (minus_S … K');
-                nrewrite > (split_big_plus (S (n' - nindex)) (n' - nindex)
-                 (λi,p.s (S (i+nindex))) (le_S ?? (le_n ?)));
-                nrewrite > (ad_hoc5 (n' - nindex));
-                nnormalize in ⊢ (?(?(?(??%)?)?));
-                nrewrite > (ad_hoc6 … K');
-                napply ad_hoc8]##]##]##]
-##| #x; #x'; nnormalize in ⊢ (? → ? → %); #Hx; #Hx';
-    nelim (partition_splits_card_output A P n s f fi x Hx); #i1x; *; #i2x; #Ex;
-    nelim (partition_splits_card_output A P n s f fi x' Hx'); #i1x'; *; #i2x'; #Ex';
-    ngeneralize in match (? :
-     iso_f ???? fi i1x(* ≬ iso_f ???? (fi i1x'))*)) in ⊢ ?;
-    #E; napply (f_inj ???? (fi i1x));
-    
-    nelim n in ⊢ (% → % → (???(????????%)(????????%)) → ?)
-     [ nnormalize in ⊢ (% → % → ?); nrewrite > (plus_n_O (s O));
-       nchange in ⊢ (% → ?) with (x < s O);
-       nchange in ⊢ (? → % → ?) with (x' < s O);
-       #H1; #H2; nwhd in ⊢ (???%% → ?);
-       nrewrite > (ltb_t … H1); nrewrite > (ltb_t … H2); nwhd in ⊢ (???%% → ?);
-       napply f_inj; nassumption
-     | #n'; #Hrec; #Hx; #Hx'; nwhd in ⊢ (???%% → ?);
-  ]
+    @ xxx; @
+     [ 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;
+       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)) ?????)
+        [##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;
+    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
+       |##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';
+    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 ]
 nqed.
 
 (************** equivalence relations vs partitions **********************)
@@ -314,4 +213,4 @@ 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.