1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "nat_ordered_set.ma".
16 include "models/q_bars.ma".
18 lemma initial_shift_same_values:
19 ∀l1:q_f.∀init.init < start l1 →
21 (mk_q_f init (〈\fst (unpos (start l1 - init) ?),OQ〉:: bars l1)).
22 [apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ; assumption]
23 intros; generalize in ⊢ (? ? (? ? (? ? (? ? ? (? ? ? (? ? %)) ?) ?))); intro;
24 cases (unpos (start l1-init) H1); intro input;
25 simplify in ⊢ (? ? ? (? ? ? (? ? ? (? (? ? (? ? (? ? ? % ?) ?)) ?))));
26 cases (value (mk_q_f init (〈w,OQ〉::bars l1)) input) (v1 Hv1);
27 cases Hv1 (HV1 HV1 HV1 HV1); cases HV1 (Hi1 Hv11 Hv12); clear HV1 Hv1;
28 [1: cut (input < start l1) as K;[2: apply (q_lt_trans ??? Hi1 H)]
29 rewrite > (value_OQ_l ?? K); simplify; symmetry; assumption;
30 |2: cut (start l1 + sum_bases (bars l1) (len (bars l1)) ≤ input) as K;[2:
31 simplify in Hi1; apply (q_le_trans ???? Hi1); rewrite > H2;
32 rewrite > q_plus_sym in ⊢ (? ? (? ? %));
33 rewrite > q_plus_assoc; rewrite > q_elim_minus;
34 rewrite > q_plus_sym in ⊢ (? ? (? (? ? %) ?));
35 rewrite > q_plus_assoc; rewrite < q_elim_minus;
36 rewrite > q_plus_minus; rewrite > q_plus_sym in ⊢ (? ? (? % ?));
37 rewrite > q_plus_OQ; apply q_eq_to_le; reflexivity;]
38 rewrite > (value_OQ_r ?? K); simplify; symmetry; assumption;
39 |3: simplify in Hi1; destruct Hi1;
48 alias symbol "pi2" = "pair pi2".
49 alias symbol "pi1" = "pair pi1".
50 definition rebase_spec ≝
51 ∀l1,l2:q_f.∃p:q_f × q_f.
53 (*len (bars (\fst p)) = len (bars (\snd p))*)
54 (start (\fst p) = start (\snd p))
55 (same_bases (\fst p) (\snd p))
56 (same_values l1 (\fst p))
57 (same_values l2 (\snd p)).
59 definition rebase_spec_simpl ≝
60 λstart.λl1,l2:list bar.λp:(list bar) × (list bar).
62 (same_bases (mk_q_f start (\fst p)) (mk_q_f start (\snd p)))
63 (same_values (mk_q_f start l1) (mk_q_f start (\fst p)))
64 (same_values (mk_q_f start l2) (mk_q_f start (\snd p))).
66 (* a local letin makes russell fail *)
67 definition cb0h : list bar → list bar ≝
68 λl.mk_list (λi.〈\fst (nth l ▭ i),OQ〉) (len l).
71 λP.λp:∃x:(list bar) × (list bar).P x.match p with [ex_introT p _ ⇒ p].
73 definition inject ≝ λP.λp:(list bar) × (list bar).λh:P p. ex_introT ? P p h.
74 coercion inject with 0 1 nocomposites.
76 definition rebase: rebase_spec.
77 intros 2 (f1 f2); cases f1 (s1 l1); cases f2 (s2 l2); clear f1 f2;
79 λs.λl1,l2.λm.λz.len l1 + len l2 < m → rebase_spec_simpl s l1 l2 z);
80 alias symbol "pi1" (instance 34) = "exT \fst".
81 alias symbol "pi1" (instance 21) = "exT \fst".
83 let rec aux (l1,l2:list bar) (n:nat) on n : (list bar) × (list bar) ≝
85 [ O ⇒ 〈 nil ? , nil ? 〉
93 let base1 ≝ Qpos (\fst he1) in
94 let base2 ≝ Qpos (\fst he2) in
95 let height1 ≝ (\snd he1) in
96 let height2 ≝ (\snd he2) in
97 match q_cmp base1 base2 with
99 let rc ≝ aux tl1 tl2 m in
100 〈he1 :: \fst rc,he2 :: \snd rc〉
102 let rest ≝ base2 - base1 in
103 let rc ≝ aux tl1 (〈\fst (unpos rest ?),height2〉 :: tl2) m in
104 〈〈\fst he1,height1〉 :: \fst rc,〈\fst he1,height2〉 :: \snd rc〉
106 let rest ≝ base1 - base2 in
107 let rc ≝ aux (〈\fst (unpos rest ?),height1〉 :: tl1) tl2 m in
108 〈〈\fst he2,height1〉 :: \fst rc,〈\fst he2,height2〉 :: \snd rc〉
110 in aux : ∀l1,l2,m.∃z.∀s.spec s l1 l2 m z); unfold spec;
111 [9: clearbody aux; unfold spec in aux; clear spec;
113 [1: cases (aux l1 l2 (S (len l1 + len l2)));
114 cases (H1 s1 (le_n ?)); clear H1;
115 exists [apply 〈mk_q_f s1 (\fst w), mk_q_f s2 (\snd w)〉] split;
117 |3: intro; apply (H3 input);
118 |4: intro; rewrite > H in H4;
119 rewrite > (H4 input); reflexivity;]
120 |2: letin l2' ≝ (〈\fst (unpos (s2-s1) ?),OQ〉::l2);[
121 apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
123 cases (aux l1 l2' (S (len l1 + len l2')));
124 cases (H1 s1 (le_n ?)); clear H1 aux;
125 exists [apply 〈mk_q_f s1 (\fst w), mk_q_f s1 (\snd w)〉] split;
129 |4: intro; rewrite < (H4 input); clear H3 H4 H2 w;
130 cases (value (mk_q_f s1 l2') input);
131 cases (q_cmp input (start (mk_q_f s1 l2'))) in H1;
133 [1: intros; cases H2; clear H2; whd in ⊢ (??? %);
134 cases (value (mk_q_f s2 l2) input);
135 cases (q_cmp input (start (mk_q_f s2 l2))) in H2;
137 [1: intros; cases H6; clear H6; change with (w1 = w);
140 |1,2: unfold rest; apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
147 |8: intros; cases (?:False); apply (not_le_Sn_O ? H1);]