| assumption
]
]
- | (* TODO: pred (f x1) = f y assurdo per iniettivita'
- poiche' y ≠ S n1 da cui f y ≠ f (S n1) da cui f y < f (S n1) < f x1
- da cui assurdo pred (f x1) = f y *)
- | (* TODO: f x1 = pred (f y) assurdo per iniettivita' *)
+ | (* pred (f x1) = f y absurd since y ≠ S n1 and thus f y ≠ f (S n1)
+ so that f y < f (S n1) < f x1; hence pred (f x1) = f y is absurd *)
+ cut (y < S n1);
+ [ generalize in match (lt_to_not_eq ? ? Hcut);
+ intro;
+ cut (f y ≠ f (S n1));
+ [ cut (f y < f (S n1));
+ [ rewrite < H9 in Hcut2;
+ unfold lt in Hcut2;
+ unfold lt in H8;
+ generalize in match (le_S_S ? ? Hcut2);
+ intro;
+ generalize in match (transitive_le ? ? ? H11 H8);
+ intros;
+ rewrite < (S_pred (f x1)) in H12;
+ [ elim (not_le_Sn_n ? H12)
+ | fold simplify ((f (S n1)) < (f x1)) in H8;
+ apply (ltn_to_ltO ? ? H8)
+ ]
+ | apply not_eq_to_le_to_lt;
+ [ assumption
+ | apply not_lt_to_le;
+ assumption
+ ]
+ ]
+ | unfold Not;
+ intro;
+ apply H10;
+ apply (H1 ? ? ? ? H11);
+ [ apply lt_to_le;
+ assumption
+ | constructor 1
+ ]
+ ]
+ | unfold lt;
+ apply le_S_S;
+ assumption
+ ]
+ | (* f x1 = pred (f y) absurd since it implies S (f x1) = f y and
+ f x1 ≤ f (S n1) < f y = S (f x1) so that f x1 = f (S n1); by
+ injectivity x1 = S n1 that is absurd since x1 ≤ n1 *)
+ generalize in match (eq_f ? ? S ? ? H9);
+ intro;
+ rewrite < S_pred in H10;
+ [ rewrite < H10 in H7;
+ generalize in match (not_lt_to_le ? ? H8);
+ intro;
+ unfold lt in H7;
+ generalize in match (le_S_S ? ? H11);
+ intro;
+ generalize in match (antisym_le ? ? H12 H7);
+ intro;
+ generalize in match (inj_S ? ? H13);
+ intro;
+ generalize in match (H1 ? ? ? ? H14);
+ intro;
+ rewrite > H15 in H5;
+ elim (not_le_Sn_n ? H5)
+ | apply (ltn_to_ltO ?? H7)
+ ]
| apply (H1 ? ? ? ? H9);
apply le_S;
assumption