(* *)
(**************************************************************************)
-include "models/q_bars.ma".
+include "nat_ordered_set.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 hide; 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);
-simplify in ⊢ (? ? ? (? ? ? %));
-cases (q_cmp input (start (mk_q_f init (〈w,OQ〉::bars l1)))) in H3;
-whd in ⊢ (% → ?); simplify in H3;
-[1: intro; cases H4; clear H4; rewrite > H3;
- cases (value l1 init); simplify; cases (q_cmp init (start l1)) in H4;
- [1: cases (?:False); apply (q_lt_corefl init); rewrite > H4 in ⊢ (?? %); apply H;
- |3: cases (?:False); apply (q_lt_antisym init (start l1)); assumption;
- |2: whd in ⊢ (% → ?); intro; rewrite > H8; clear H8 H4;
- rewrite > H7; clear H7; rewrite > (?:\fst w1 = O); [reflexivity]
- symmetry; apply le_n_O_to_eq;
- rewrite > (sum_bases_O (mk_q_f init (〈w,OQ〉::bars l1)) (\fst w1)); [apply le_n]
- clear H6 w2; simplify in H5:(? ? (? ? %));
- destruct H3; rewrite > q_d_x_x in H5; assumption;]
-|2: intros; cases (value l1 input); simplify in ⊢ (? ? (? ? ? %) ?);
- cases (q_cmp input (start l1)) in H5; whd in ⊢ (% → ?);
- [1: cases (?:False); clear w2 H4 w1 H2 w H1;
- apply (q_lt_antisym init (start l1)); [assumption] rewrite < H5; assumption
- |2: intros; rewrite > H6; clear H6; rewrite > H4; reflexivity;
- |3: cases (?:False); apply (q_lt_antisym input (start l1)); [2: assumption]
- apply (q_lt_trans ??? H3 H);]
-|3: intro; cases H4; clear H4;
- cases (value l1 input); simplify; cases (q_cmp input (start l1)) in H4; whd in ⊢ (% → ?);
- [1: intro; cases H8; clear H8; rewrite > H11; rewrite > H7; clear H11 H7;
- simplify in ⊢ (? ? ? (? ? ? (? ? % ? ?)));
- cut (\fst w1 = S (\fst w2)) as Key; [rewrite > Key; reflexivity;]
- cut (\fst w2 = O); [2: clear H10;
- symmetry; apply le_n_O_to_eq; rewrite > (sum_bases_O l1 (\fst w2)); [apply le_n]
- apply (q_le_trans ??? H9); rewrite < H4; rewrite > q_d_x_x;
- apply q_eq_to_le; reflexivity;]
- rewrite > Hcut; clear Hcut H10 H9; simplify in H5 H6;
- cut (ⅆ[input,init] = Qpos w) as E; [2:
- rewrite > H2; rewrite < H4; rewrite > q_d_sym;
- rewrite > q_d_noabs; [reflexivity] apply q_lt_to_le; assumption;]
- cases (\fst w1) in H5 H6; intros;
- [1: cases (?:False); clear H5; simplify in H6;
- apply (q_lt_corefl ⅆ[input,init]);
- rewrite > E in ⊢ (??%); rewrite < q_plus_OQ in ⊢ (??%);
- rewrite > q_plus_sym; assumption;
- |2: cases n in H5 H6; [intros; reflexivity] intros;
- cases (?:False); clear H6; cases (bars l1) in H5; simplify; intros;
- [apply (q_pos_OQ one);|apply (q_pos_OQ (\fst b));]
- apply (q_le_S ??? (sum_bases_ge_OQ (mk_q_f init ?) n1));[apply [];|3:apply l]
- simplify in ⊢ (? (? (? % ?) ?) ?); rewrite < (q_plus_minus w);
- apply q_le_minus_r; rewrite < q_minus_r;
- rewrite < E in ⊢ (??%); assumption;]
- |2: intros; rewrite > H8; rewrite > H7; clear H8 H7;
- simplify in H5 H6 ⊢ %;
- cases (\fst w1) in H5 H6; [intros; reflexivity]
- cases (bars l1);
- [1: intros; simplify; elim n [reflexivity] simplify; assumption;
- |2: simplify; intros; cases (?:False);
-
-STOP
-
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.
projects / helm.git / blobdiff