-axiom sum_bases_empty_nat_of_q_ge_OQ:
- ∀q:ℚ.OQ ≤ sum_bases [] (nat_of_q q).
-axiom sum_bases_empty_nat_of_q_le_q:
- ∀q:ℚ.sum_bases [] (nat_of_q q) ≤ q.
-axiom sum_bases_empty_nat_of_q_le_q_one:
- ∀q:ℚ.q < sum_bases [] (nat_of_q q) + Qpos one.
-
-definition eject1 ≝
- λP.λp:∃x:nat × ℚ.P x.match p with [ex_introT p _ ⇒ p].
-coercion eject1.
-definition inject1 ≝ λP.λp:nat × ℚ.λh:P p. ex_introT ? P p h.
-coercion inject1 with 0 1 nocomposites.
-
-definition value :
- ∀f:q_f.∀i:ℚ.∃p:nat × ℚ.
- match q_cmp i (start f) with
- [ q_lt _ ⇒ \snd p = OQ
- | _ ⇒
- And3
- (sum_bases (bars f) (\fst p) ≤ ⅆ[i,start f])
- (ⅆ[i, start f] < sum_bases (bars f) (S (\fst p)))
- (\snd p = \snd (nth (bars f) ▭ (\fst p)))].
-intros;
-alias symbol "pi2" = "pair pi2".
-alias symbol "pi1" = "pair pi1".
-letin value ≝ (
- let rec value (p: ℚ) (l : list bar) on l ≝
- match l with
- [ nil ⇒ 〈nat_of_q p,OQ〉
- | cons x tl ⇒
- match q_cmp p (Qpos (\fst x)) with
- [ q_lt _ ⇒ 〈O, \snd x〉
- | _ ⇒
- let rc ≝ value (p - Qpos (\fst x)) tl in
- 〈S (\fst rc),\snd rc〉]]
- in value :
- ∀acc,l.∃p:nat × ℚ. OQ ≤ acc →
- And3
- (sum_bases l (\fst p) ≤ acc)
- (acc < sum_bases l (S (\fst p)))
- (\snd p = \snd (nth l ▭ (\fst p))));
-[5: clearbody value;
- cases (q_cmp i (start f));
- [2: exists [apply 〈O,OQ〉] simplify; reflexivity;
- |*: cases (value ⅆ[i,start f] (bars f)) (p Hp);
- cases (Hp (q_dist_ge_OQ ? ?)); clear Hp value;
- exists[1,3:apply p]; simplify; split; assumption;]
-|1,3: intros; split;
- [1,4: clear H2; cases (value (q-Qpos (\fst b)) l1);
- cases (H2 (q_le_to_diff_ge_OQ ?? (? H1)));
- [1,3: intros; [apply q_lt_to_le|apply q_eq_to_le;symmetry] assumption]
- simplify; apply q_le_minus; assumption;
- |2,5: cases (value (q-Qpos (\fst b)) l1);
- cases (H4 (q_le_to_diff_ge_OQ ?? (? H1)));
- [1,3: intros; [apply q_lt_to_le|apply q_eq_to_le;symmetry] assumption]
- clear H3 H2 value;
- change with (q < sum_bases l1 (S (\fst w)) + Qpos (\fst b));
- apply q_lt_plus; assumption;
- |*: cases (value (q-Qpos (\fst b)) l1); simplify;
- cases (H4 (q_le_to_diff_ge_OQ ?? (? H1)));
- [1,3: intros; [apply q_lt_to_le|apply q_eq_to_le;symmetry] assumption]
- assumption;]
-|2: clear value H2; simplify; intros; split; [assumption|3:reflexivity]
- rewrite > q_plus_sym; rewrite > q_plus_OQ; assumption;
-|4: simplify; intros; split;
- [1: apply sum_bases_empty_nat_of_q_le_q;
- |2: apply sum_bases_empty_nat_of_q_le_q_one;
- |3: elim (nat_of_q q); [reflexivity] simplify; assumption]]
+alias symbol "lt" (instance 5) = "Q less than".
+alias symbol "lt" (instance 6) = "natural 'less than'".
+definition value_lemma :
+ ∀f:list bar.sorted q2_lt f → O < length bar f →
+ ∀i:ratio.nth_base f O < Qpos i → ∃p:ℚ×ℚ.value_spec f (Qpos i) p.
+intros (f bars_sorted_f len_bases_gt_O_f i bars_begin_OQ_f);
+exists [apply (value_simple f i);]
+apply (value_of ?? (match_domain f i));
+[1: reflexivity
+|2: unfold match_domain; cases (cases_find bar (match_pred i) f ▭);
+ [1: cases i1 in H H1 H2 H3; simplify; intros;
+ [1: generalize in match (bars_begin_OQ_f);
+ cases (len_gt_non_empty ?? (len_bases_gt_O_f)); simplify; intros;
+ assumption;
+ |2: cases (len_gt_non_empty ?? (len_bases_gt_O_f)) in H3;
+ intros; lapply (H3 n (le_n ?)) as K; unfold match_pred in K;
+ cases (q_cmp (Qpos i) (\fst (\nth (x::l) ▭ n))) in K;
+ simplify; intros; [destruct H5] assumption]
+ |2: destruct H; cases (len_gt_non_empty ?? (len_bases_gt_O_f)) in H2;
+ simplify; intros; lapply (H (\len l) (le_n ?)) as K; clear H;
+ unfold match_pred in K; cases (q_cmp (Qpos i) (\fst (\nth (x::l) ▭ (\len l)))) in K;
+ simplify; intros; [destruct H2] assumption;]
+|5: intro; unfold match_domain; cases (cases_find bar (match_pred i) f ▭); intros;
+ [1: generalize in match (bars_sorted_f);
+ cases (list_break ??? H) in H1; rewrite > H6;
+ rewrite < H1; simplify; rewrite > nth_len; unfold match_pred;
+ cases (q_cmp (Qpos i) (\fst x)); simplify;
+ intros (X Hs); [2: destruct X] clear X;
+ cases (sorted_pivot q2_lt ??? ▭ Hs);
+ cut (\len l1 ≤ n) as Hn; [2:
+ rewrite > H1; cases i1 in H4; simplify; intro X; [2: assumption]
+ apply lt_to_le; assumption;]
+ unfold nth_base; rewrite > (nth_append_ge_len ????? Hn);
+ cut (n - \len l1 < \len (x::l2)) as K; [2:
+ simplify; rewrite > H1; rewrite > (?:\len l2 = \len f - \len (l1 @ [x]));[2:
+ rewrite > H6; repeat rewrite > len_append; simplify;
+ repeat rewrite < plus_n_Sm; rewrite < plus_n_O; simplify;
+ rewrite > sym_plus; rewrite < minus_plus_m_m; reflexivity;]
+ rewrite > len_append; rewrite > H1; simplify; rewrite < plus_n_SO;
+ apply le_S_S; clear H1 H6 H7 Hs H8 H9 Hn x l2 l1 H4 H3 H2 H;
+ elim (\len f) in i1 n H5; [cases (not_le_Sn_O ? H);]
+ simplify; cases n2; [ repeat rewrite < minus_n_O; apply le_S_S_to_le; assumption]
+ cases n1 in H1; [intros; rewrite > eq_minus_n_m_O; apply le_O_n]
+ intros; simplify; apply H; apply le_S_S_to_le; assumption;]
+ cases (n - \len l1) in K; simplify; intros; [ assumption]
+ lapply (H9 ? (le_S_S_to_le ?? H10)) as W; apply (q_le_trans ??? H7);
+ apply q_lt_to_le; apply W;
+ |2: cases (not_le_Sn_n i1); rewrite > H in ⊢ (??%);
+ apply (trans_le ??? ? H4); cases i1 in H3; intros; apply le_S_S;
+ [ apply le_O_n; | assumption]]
+|3: unfold match_domain; cases (cases_find bar (match_pred i) f ▭); [
+ cases i1 in H; intros; simplify; [assumption]
+ apply lt_S_to_lt; assumption;]
+ rewrite > H; cases (\len f) in len_bases_gt_O_f; intros; [cases (not_le_Sn_O ? H3)]
+ simplify; apply le_n;
+|4: intros; unfold match_domain in H; cases (cases_find bar (match_pred i) f ▭) in H; simplify; intros;
+ [1: lapply (H3 n); [2: cases i1 in H4; intros [assumption] apply le_S; assumption;]
+ unfold match_pred in Hletin; cases (q_cmp (Qpos i) (\fst (\nth f ▭ n))) in Hletin;
+ simplify; intros; [destruct H6] assumption;
+ |2: destruct H; cases f in len_bases_gt_O_f H2 H3; clear H1; simplify; intros;
+ [cases (not_le_Sn_O ? H)] lapply (H1 n); [2: apply le_S; assumption]
+ unfold match_pred in Hletin; cases (q_cmp (Qpos i) (\fst (\nth (b::l) ▭ n))) in Hletin;
+ simplify; intros; [destruct H4] assumption;]]
+qed.
+
+lemma bars_begin_lt_Qpos : ∀q,r. nth_base (bars q) O<Qpos r.
+intros; rewrite > bars_begin_OQ; apply q_pos_OQ;
+qed.
+
+lemma value : q_f → ratio → ℚ × ℚ.
+intros; cases (value_lemma (bars q) ?? r);
+[ apply bars_sorted.
+| apply len_bases_gt_O;
+| apply w;
+| apply bars_begin_lt_Qpos;]
+qed.
+
+lemma cases_value : ∀f,i. value_spec (bars f) (Qpos i) (value f i).
+intros; unfold value;
+cases (value_lemma (bars f) (bars_sorted f) (len_bases_gt_O f) i (bars_begin_lt_Qpos f i));
+assumption;