- [ elim (H f' ? ? x);
- [ simplify in H5;
- clear Hcut;
- clear f';
- elim H5;
- clear H5;
- apply (ex_intro ? ? a);
- split;
- [ generalize in match H4;
- clear H4;
- rewrite < H6;
- clear H6;
- apply (ltb_elim (f (S n1)) (f a));
- [ (* TODO: caso impossibile (uso l'iniettivita') *)
- simplify;
- | simplify;
- intros;
- reflexivity
- ]
- | apply le_S;
+ [ cut (∀x. x ≤ n1 → f' x ≤ n1);
+ [ elim (H f' ? ? x);
+ [ simplify in H5;
+ clear Hcut;
+ clear Hcut1;
+ clear f';
+ elim H5;
+ clear H5;
+ apply (ex_intro ? ? a);
+ split;
+ [ generalize in match H4;
+ clear H4;
+ rewrite < H6;
+ clear H6;
+ apply (ltb_elim (f (S n1)) (f a));
+ [ (* TODO: caso impossibile (uso l'iniettivita') *)
+ simplify;
+ | simplify;
+ intros;
+ reflexivity
+ ]
+ | apply le_S;
+ assumption
+ ]
+ | apply Hcut
+ | apply Hcut1
+ | rewrite > (pred_Sn n1);
+ simplify;
+ generalize in match (H2 (S n1));
+ intro;
+ generalize in match (lt_to_le_to_lt ? ? ? H4 (H5 (le_n ?)));
+ intro;
+ unfold lt in H6;
+ apply le_S_S_to_le;