(**************************************************************************)
include "nat_ordered_set.ma".
-include "models/q_bars.ma".
+include "models/q_shift.ma".
-lemma initial_shift_same_values:
- ∀l1:q_f.∀init.init < start l1 →
- same_values l1
- (mk_q_f init (〈\fst (unpos (start l1 - init) ?),OQ〉:: bars l1)).
-[apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ; assumption]
-intros; generalize in ⊢ (? ? (? ? (? ? (? ? ? (? ? ? (? ? %)) ?) ?))); intro;
-cases (unpos (start l1-init) H1); intro input;
-simplify in ⊢ (? ? ? (? ? ? (? ? ? (? (? ? (? ? (? ? ? % ?) ?)) ?))));
-cases (value (mk_q_f init (〈w,OQ〉::bars l1)) input) (v1 Hv1);
-cases Hv1 (HV1 HV1 HV1 HV1); cases HV1 (Hi1 Hv11 Hv12); clear HV1 Hv1;
-[1: cut (input < start l1) as K;[2: apply (q_lt_trans ??? Hi1 H)]
- rewrite > (value_OQ_l ?? K); simplify; symmetry; assumption;
-|2: cut (start l1 + sum_bases (bars l1) (len (bars l1)) ≤ input) as K;[2:
- simplify in Hi1; apply (q_le_trans ???? Hi1); rewrite > H2;
- rewrite > q_plus_sym in ⊢ (? ? (? ? %));
- rewrite > q_plus_assoc; rewrite > q_elim_minus;
- rewrite > q_plus_sym in ⊢ (? ? (? (? ? %) ?));
- rewrite > q_plus_assoc; rewrite < q_elim_minus;
- rewrite > q_plus_minus; rewrite > q_plus_sym in ⊢ (? ? (? % ?));
- rewrite > q_plus_OQ; apply q_eq_to_le; reflexivity;]
- rewrite > (value_OQ_r ?? K); simplify; symmetry; assumption;
-|3: simplify in Hi1; destruct Hi1;
-|4:
-
-STOP
-
-qed.
-
-
-
alias symbol "pi2" = "pair pi2".
alias symbol "pi1" = "pair pi1".
definition rebase_spec ≝
[1: reflexivity
|2: assumption;
|3: assumption;
- |4: intro; rewrite < (H4 input); clear H3 H4 H2 w;
- cases (value (mk_q_f s1 l2') input);
- cases (q_cmp input (start (mk_q_f s1 l2'))) in H1;
- whd in ⊢ (% → ?);
- [1: intros; cases H2; clear H2; whd in ⊢ (??? %);
- cases (value (mk_q_f s2 l2) input);
- cases (q_cmp input (start (mk_q_f s2 l2))) in H2;
- whd in ⊢ (% → ?);
- [1: intros; cases H6; clear H6; change with (w1 = w);
-
- (* TODO *) ]]
+ |4: intro; rewrite > (initial_shift_same_values (mk_q_f s2 l2) s1 H input);
+ rewrite < (H4 input); reflexivity;]
+ |3: letin l1' ≝ (〈\fst (unpos (s1-s2) ?),OQ〉::l1);[
+ apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
+ assumption]
+ cases (aux l1' l2 (S (len l1' + len l2)));
+ cases (H1 s2 (le_n ?)); clear H1 aux;
+ exists [apply 〈mk_q_f s2 (\fst w), mk_q_f s2 (\snd w)〉] split;
+ [1: reflexivity
+ |2: assumption;
+ |4: assumption;
+ |3: intro; rewrite > (initial_shift_same_values (mk_q_f s1 l1) s2 H input);
+ rewrite < (H3 input); reflexivity;]]
|1,2: unfold rest; apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
assumption;
-|3:(* TODO *)
-|4:(* TODO *)
-|5:(* TODO *)
-|6:(* TODO *)
-|7:(* TODO *)
-|8: intros; cases (?:False); apply (not_le_Sn_O ? H1);]
+|8: intros; cases (?:False); apply (not_le_Sn_O ? H1);
+|3: intros; generalize in match (unpos ??); intro X; cases X; clear X;
+ simplify in ⊢ (???? (??? (??? (??? (?? (? (?? (??? % ?) ?) ??)))) ?));
+ simplify in ⊢ (???? (???? (??? (??? (?? (? (?? (??? % ?) ?) ??))))));
+ clear H4; cases (aux (〈w,\snd b〉::l4) l5 n1); clear aux;
+ cut (len (〈w,\snd b〉::l4) + len l5 < n1) as K;[2:
+ simplify in H5; simplify; rewrite > sym_plus in H5; simplify in H5;
+ rewrite > sym_plus in H5; apply le_S_S_to_le; apply H5;]
+ split;
+ [1: simplify in ⊢ (? % ?); simplify in ⊢ (? ? %);
+ cases (H4 s K); clear K H4; intro input; cases input; [reflexivity]
+ simplify; apply H7;
+ |2: simplify in ⊢ (? ? %); cases (H4 s K); clear H4 K H5 spec;
+ intro;
+ (* input < s + b1 || input >= s + b1 *)
+ |3: simplify in ⊢ (? ? %);]
+|4: intros; generalize in match (unpos ??); intro X; cases X; clear X;
+ (* duale del 3 *)
+|5: intros; (* triviale, caso in cui non fa nulla *)
+|6,7: (* casi base in cui allunga la lista più corta *)
+]
qed.