Injectivity proved! What a mess...

diff --git a/helm/software/matita/nlibrary/sets/partitions.ma b/helm/software/matita/nlibrary/sets/partitions.ma
index d8ba34c5ccbec1849e60109915c688d52fa7e8cd..acdd6a687cf49874f18f2fd23206bec5043d398b 100644 (file)
--- a/helm/software/matita/nlibrary/sets/partitions.ma
+++ b/helm/software/matita/nlibrary/sets/partitions.ma
@@ -81,6 +81,11 @@ 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 split_big_plus:
   ∀n,m,f. m ≤ n →
@@ -171,111 +176,29 @@ nlemma partition_splits_card:
             nwhd in ⊢ (???%?); nassumption
          ##| #K; ngeneralize in match (le_S_S_to_le … K) in ⊢ ?; #K';
              nwhd in ⊢ (???%?);
-             
-             
-             XXX;
-             nrewrite > (minus_S n' nindex ?) [##2: napply le_S_S_to_le; nassumption]
-             ngeneralize in match (? :
-              ltb (plus (big_plus (S (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);
-                    nnormalize in ⊢ (??(?%?)? → ?);
-                    nrewrite > (plus_n_O (s (S nindex)));
-                    nrewrite > (ltb_f (plus (s (S nindex)) nindex2) (s (S nindex)) ?);
-                    
-                    XXX;
-                    
-               
-                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 ⊢ (? → ? → %);
+             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 ⊢ (? → ? → %);
     nelim daemon
   ]
 nqed.
@@ -287,9 +210,11 @@ 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