+ [ 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 ⊢ (???%?);