P l1 l2.
intros (T1 T2 l1 l2 P Hl Pnil Pcons);
generalize in match Hl; clear Hl; generalize in match l2; clear l2;
-elim l1 1 (l2 x1); [ cases l2; intros (Hl); [assumption| destruct Hl]]
+elim l1 1 (l2 x1); [ cases l2; intros (Hl); [assumption| simplify in Hl; destruct Hl]]
intros 3 (tl1 IH l2); cases l2; [1: simplify; intros 1 (Hl); destruct Hl]
-intros 1 (Hl); apply Pcons; apply IH; destruct Hl; assumption;
+intros 1 (Hl); apply Pcons; apply IH; simplify in Hl; destruct Hl; assumption;
qed.
lemma eq_map : ∀A,B,f,g,l. (∀x.f x = g x) → map A B f l = map A B g l.
lcmp ? l1 l2 = true → length ? l1 = length ? l2.
intros 2 (d l1); elim l1 1 (l2 x1);
[1: cases l2; simplify; intros; [reflexivity|destruct H]
-|2: intros 3 (tl1 IH l2); cases (l2); intros; [1:destruct H]
+|2: intros 3 (tl1 IH l2); cases (l2); intros; [1:simplify in H; destruct H]
simplify; (* XXX la apply non fa simplify? *)
apply congr_S; apply (IH l);
(* XXX qualcosa di enorme e' rotto! la regola di convertibilita?! *)
rewrite > (IH H2); rewrite > (b2pT ? ? (eqP d ? ?) H1); reflexivity
| generalize in match H; clear H; generalize in match l2; clear l2;
elim l1 1 (l1 x1);
- [ cases l1; [intros; destruct H | unfold Not; intros; destruct H1;]
+ [ cases l1; simplify; [intros; destruct H | unfold Not; intros; destruct H1;]
| intros 3 (tl1 IH l2); cases l2;
[ unfold Not; intros; destruct H1;
| simplify; intros;
cases (b2pT ? ? (andbPF ? ?) (p2bT ? ? (negbP ?) H)); clear H;
[ intros; lapply (b2pF ? ? (eqP d ? ?) H1) as H'; clear H1;
- (* XXX destruct H; *)
- change in H' with (Not ((match (x1::tl1) with [nil⇒x1|(cons x1 _) ⇒ x1]) = s));
- rewrite > H in H'; simplify in H'; apply H'; reflexivity;
- | intros; lapply (IH ? H1) as H';
- (* XXX destruct H1 *)
- change in H' with (Not ((match (x1::tl1) with [nil⇒tl1|(cons _ l) ⇒ l]) = l));
- rewrite > H in H'; simplify in H'; apply H'; reflexivity;]]]]
+ destruct H; apply H'; reflexivity;
+ | intros; lapply (IH ? H1) as H'; destruct H;
+ apply H'; reflexivity;]]]]
qed.
definition list_eqType : eqType → eqType ≝ λd:eqType.mk_eqType ? ? (lcmpP d).