- (*???????*)
- ##| #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 ⊢ (? → ? → %);
+ 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 ⊢ (? → ? → %);