include "nat_ordered_set.ma".
include "models/q_support.ma".
include "models/list_support.ma".
-include "cprop_connectives.ma".
+include "logic/cprop_connectives.ma".
definition bar ≝ ℚ × (ℚ × ℚ).
apply (H2 n1); simplify in H3; apply (le_S_S_to_le ?? H3);]
qed.
-coinductive value_spec (f : q_f) (i : ℚ) : ℚ × ℚ → CProp ≝
+alias symbol "lt" (instance 9) = "Q less than".
+alias symbol "lt" (instance 7) = "natural 'less than'".
+alias symbol "lt" (instance 6) = "natural 'less than'".
+alias symbol "lt" (instance 5) = "Q less than".
+alias symbol "lt" (instance 4) = "natural 'less than'".
+alias symbol "lt" (instance 2) = "natural 'less than'".
+alias symbol "leq" = "Q less or equal than".
+coinductive value_spec (f : list bar) (i : ℚ) : ℚ × ℚ → CProp ≝
| value_of : ∀j,q.
- nth_height (bars f) j = q → nth_base (bars f) j < i →
- (∀n.j < n → n < \len (bars f) → i ≤ nth_base (bars f) n) → value_spec f i q.
+ nth_height f j = q → nth_base f j < i → j < \len f →
+ (∀n.n<j → nth_base f n < i) →
+ (∀n.j < n → n < \len f → i ≤ nth_base f n) → value_spec f i q.
+
+definition match_pred ≝
+ λi.λx:bar.match q_cmp (Qpos i) (\fst x) with[ q_leq _ ⇒ true| q_gt _ ⇒ false].
+
+definition match_domain ≝
+ λf: list bar.λi:ratio. pred (find ? (match_pred i) f ▭).
+
+definition value_simple ≝
+ λf: list bar.λi:ratio. nth_height f (match_domain f i).
-definition value_lemma : ∀f:q_f.∀i:ratio.∃p:ℚ×ℚ.value_spec f (Qpos i) p.
-intros;
-letin P ≝
- (λx:bar.match q_cmp (Qpos i) (\fst x) with[ q_leq _ ⇒ true| q_gt _ ⇒ false]);
-exists [apply (nth_height (bars f) (pred (find ? P (bars f) ▭)));]
-apply (value_of ?? (pred (find ? P (bars f) ▭)));
+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: cases (cases_find bar P (bars f) ▭);
+|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;
- rewrite > H4; apply q_pos_OQ;
- |2: cases (len_gt_non_empty ?? (len_bases_gt_O f)) in H3;
- intros; lapply (H3 n (le_n ?)) as K; unfold P in K;
+ [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;
+ |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 P in K; cases (q_cmp (Qpos i) (\fst (\nth (x::l) ▭ (\len l)))) in K;
+ unfold match_pred in K; cases (q_cmp (Qpos i) (\fst (\nth (x::l) ▭ (\len l)))) in K;
simplify; intros; [destruct H2] assumption;]
-|3: intro; cases (cases_find bar P (bars f) ▭); intros;
- [1: generalize in match (bars_sorted f);
+|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 P;
+ 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);
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 (bars f) - \len (l1 @ [x]));[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 P i;
- elim (\len (bars f)) in i1 n H5; [cases (not_le_Sn_O ? H);]
+ 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;]
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]]]
+ [ 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 q r); apply w; qed.
+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 f (Qpos i) (value f i).
-intros; unfold value; cases (value_lemma f i); assumption; 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;
+qed.
definition same_values ≝ λl1,l2:q_f.∀input. value l1 input = value l2 input.
definition same_bases ≝ λl1,l2:list bar. ∀i.\fst (\nth l1 ▭ i) = \fst (\nth l2 ▭ i).
+lemma same_bases_cons: ∀a,b,l1,l2.
+ same_bases l1 l2 → \fst a = \fst b → same_bases (a::l1) (b::l2).
+intros; intro; cases i; simplify; [assumption;] apply (H n);
+qed.
+
alias symbol "lt" = "Q less than".
lemma unpos: ∀x:ℚ.OQ < x → ∃r:ratio.Qpos r = x.
intro; cases x; intros; [2:exists [apply r] reflexivity]
projects / helm.git / blobdiff