X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fnlibrary%2Fsets%2Fpartitions.ma;h=93f9cd6b5d3e014121adf9ff04042816848cb5a6;hb=f1c4852a4359cf278ed00d73d608856ff46bafbb;hp=f62c7281d1b074a068369cb9d71adfcca1c7b76d;hpb=d70944c1513aa63e6494e58595fcc4214a2f6c68;p=helm.git

diff --git a/helm/software/matita/nlibrary/sets/partitions.ma b/helm/software/matita/nlibrary/sets/partitions.ma
index f62c7281d..93f9cd6b5 100644
--- a/helm/software/matita/nlibrary/sets/partitions.ma
+++ b/helm/software/matita/nlibrary/sets/partitions.ma
@@ -13,223 +13,184 @@
 (**************************************************************************)
 
 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".
 nrecord partition (A: setoid) : Type[1] ≝ 
  { support: setoid;
    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 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 ≝
+nlet rec iso_nat_nat_union (s: nat → nat) m index on index : pair nat nat ≝
  match ltb m (s index) with
-  [ true ⇒ iso_f ???? (fi index) m
+  [ true ⇒ mk_pair … 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)).
+      [ O ⇒ (* dummy value: it could be an elim False: *) mk_pair … O O
+      | S index' ⇒ iso_nat_nat_union s (minus m (s index)) index']].
 
-naxiom le_to_lt_or_eq: ∀n,m. n ≤ m → n < m ∨ n = m.
 alias symbol "eq" = "leibnitz's equality".
-naxiom minus_canc: ∀n. O = minus n n.
-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 ltb_t: ∀n,m. n < m → ltb n m = true. 
+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).
+naxiom minus_canc: ∀n. minus n n = O.
+naxiom ad_hoc9: ∀a,b,c. a < b + c → a - b < c.
+naxiom ad_hoc10: ∀a,b,c. a - b = c → a = b + c.
+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) →
+  let p ≝ iso_nat_nat_union s m n in
+   m = big_plus (n - fst … p) (λi.λ_.s (S (i + fst … p))) + snd … p ∧
+    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 ]
+##| #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 ]##]##]##]
+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.
+    
+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:
  ∀A. ∀P:partition A. ∀n,s.
   ∀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 match minus n nindex return λ_.nat with [ O ⇒ O | S nn ⇒ big_plus nn (λi.λ_.s (S (plus i nindex)))] nindex2);
-    napply (ex_intro … xxx); napply conj
-     [ nwhd in Hni1; nwhd; nelim daemon
-     | 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 →
-          partition_splits_card_map A P (S n) s f fi
-           (plus
-             match minus x nindex with
-              [ O ⇒ O | S nn ⇒ big_plus nn (λi.λ_.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 ⊢ (???(???????%?)?);
-            (*???????*)
-         ##| #K; nwhd in ⊢ (???%?);
-             nrewrite > (minus_S n' nindex ?) [##2: napply le_S_S_to_le; nassumption]
-             ngeneralize in match (? :
-              match S (minus n' nindex) with [O ⇒ O | S nn ⇒ big_plus nn (λi.λ_.s (S (plus i nindex)))]
-              = big_plus (minus n' nindex) (λi.λ_.s (S (plus i nindex)))) in ⊢ ? [##2: napply refl]
-             #He; napply (eq_rect_CProp0_r ??
-              (λx.λ_.
-                match ltb (plus x nindex2) (s (S n')) with
-                 [ true ⇒ iso_f ???? (fi (S n')) (plus x nindex2)
-                 | false ⇒ ?(*partition_splits_card_map A P (S n) s f fi
-                     (minus (plus x nindex2) (s (S n'))) n'*)
-                 ] = y)
-              ?? He);
-             ngeneralize in match (? :
-              ltb (plus (big_plus (minus n' nindex) (λi.λ_.s (S (plus i nindex)))) nindex2)
-               (s (S n')) = false) in ⊢ ?
-               [ #Hc; nrewrite > Hc; nwhd in ⊢ (???%?);
-                 nelim (le_to_lt_or_eq … (le_S_S_to_le … K))
-                  [
-                ##| #E; ngeneralize in match Hc in ⊢ ?;
-                    nrewrite < E; nrewrite < (minus_canc nindex);
-                    nwhd in ⊢ (??(?%?)? → ?);
-                    nrewrite > E in Hni21; #E'; nchange in E' with (nindex2 < s n'); 
-                    ngeneralize in match Hni21 in ⊢ ?;
-                    
-               
-                ngeneralize in match (? :
-                 minus (plus (big_plus (minus n' nindex) (λi.λ_.s (S (plus i nindex)))) nindex2)
-                  (s (S n'))
-                 =
-                  plus
-                   match minus n' nindex with
-                    [ O ⇒ O | S nn ⇒ big_plus nn (λi.λ_.s (S (plus i nindex)))] nindex2)
-                 in ⊢ ?
-                  [ #F; nrewrite > F; napply Hrec; napply le_S_S_to_le; nassumption
-                  | nelim (le_to_lt_or_eq … (le_S_S_to_le … K))
-                     [
-                   ##| #E; nrewrite < E; nrewrite < (minus_canc nindex); nnormalize;
-                       
-                       nwhd in ⊢ (???%);
-                     ]
- 
-             
-              nrewrite > He;
-             
-               
-              nnormalize in ⊢ (???%?);
-          
-       
-       
-       nelim (le_to_lt_or_eq … K)
-        [##2: #K'; nrewrite > K'; nrewrite < (minus_canc n); nnormalize;
-         napply (eq_rect_CProp0 nat nindex (λx:nat.λ_.partition_splits_card_map A P (S n) s f fi nindex2 x = y) ? n K');
-         nchange in Hni21 with (nindex2 < s nindex); ngeneralize in match Hni21 in ⊢ ?;
-         ngeneralize in match Hni22 in ⊢ ?;
-         nelim nindex
-          [ #X1; #X2; nwhd in ⊢ (??? % ?);
-            napply (lt_to_ltb_t ???? X2); #D; nwhd in ⊢ (??? % ?); nassumption
-          | #n0; #_; #X1; #X2; nwhd in ⊢ (??? % ?);
-            napply (lt_to_ltb_t ???? X2); #D; nwhd in ⊢ (??? % ?); nassumption]
-      ##| #K'; ngeneralize in match (lt_to_minus … K') in ⊢ ?; #K2;
-          napply (eq_rect_CProp0 ?? (λx.λ_.?) ? ? K2); (* uffa, ancora??? *)
-          nwhd in ⊢ (??? (???????(?%?)?) ?);
-          ngeneralize in match K' in ⊢ ?;
-          napply (nat_rect_CProp0
-           (λx. nindex < x →
-             partition_splits_card_map A P (S n) s f fi
-              (plus (big_op plus_magma_type (minus (minus x nindex) (S O))
-                (λi.λ_.s (S (plus i nindex))) O) nindex2) x = y) ?? n)
-           [ #A; nelim (not_lt_O … A)
-           | #n'; #Hrec; #X; nwhd in ⊢ (???%?);
-             ngeneralize in match
-              (? : ¬ ((plus (big_op plus_magma_type (minus (minus (S n') nindex) (S O))
-                (λi.λ_.s (S (plus i nindex))) O) nindex2) < s (S n'))) in ⊢ ?
-              [ #B1; napply (lt_to_ltb_f ???? B1); #B1'; nwhd in ⊢ (???%?);
-                nrewrite > (minus_S n' nindex …) [##2: napply le_S_S_to_le; nassumption]
-                ngeneralize in match (le_S_S_to_le … X) in ⊢ ?; #X';
-                nelim (le_to_lt_or_eq … X')
-                 [##2: #X'';
-                  nchange in Hni21 with (nindex2 < s nindex); ngeneralize in match Hni21 in ⊢ ?;
-                  nrewrite > X''; nrewrite < (minus_canc n');
-                  nrewrite < (minus_canc (S O)); nnormalize in ⊢ (? → %);
-                  nelim n'
-                   [ #Y; nwhd in ⊢ (??? % ?);
-                     ngeneralize in match (minus_lt_to_lt ? (s (S O)) ? Y) in ⊢ ?; #Y';
-                     napply (lt_to_ltb_t … Y'); #H; nwhd in ⊢ (???%?);  
-                 
-                nrewrite > (minus_S (minus n' nindex) (S O) …) [##2: 
-              
-              XXX;
-          
-          nelim n in f K' ⊢ ?
-           [ #A; nelim daemon;
-         
-       (* BEL POSTO DOVE FARE UN LEMMA *)
-       (* invariante: Hni1; altre premesse: Hni1, Hni22 *)
-       nelim n in ⊢ (% → ??? (????????%) ?)
-        [ #A (* decompose *)
-        | #index'; #Hrec; #K; nwhd in ⊢ (???%?);
-          nelim (ltb xxx (s (S index')));
-          #K1; nwhd in ⊢ (???%?)
-           [
-           
-           nindex < S index' + 1
-           +^{nindex} (s i) w < s (S index') 
-           S index' == nindex
-           
-           |
-           ]
-        ]
-     ]
-  | #x; #x'; nnormalize in ⊢ (? → ? → %);
-    nelim daemon
-  ]
+    nletin xxx ≝ (plus (big_plus (minus n nindex) (λi.λ_.s (S (plus i nindex)))) nindex2);
+    @ 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)) ?????)
+        [##2: *; #E1; #E2; nrewrite > E1; nrewrite > E2; napply refl
+        | 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 ⊢ ?
+       [##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 ]##]##]
+   ##] #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]
+    *; #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 **********************)
@@ -239,12 +200,14 @@ ndefinition partition_of_compatible_equivalence_relation:
  #A; #R; napply mk_partition
   [ napply (quotient ? R)
   | napply Full_set
-  | #a; napply mk_qpowerclass
-     [ napply {x | R x a}
-     | #x; #x'; #H; nnormalize; napply mk_iff; #K; nelim daemon]
+  | napply mk_unary_morphism1
+     [ #a; napply mk_qpowerclass
+        [ 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.
\ No newline at end of file
+nqed.