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.
+definition bar ≝ ℚ × (ℚ × ℚ).
notation < "\rationals \sup 2" non associative with precedence 90 for @{'q2}.
interpretation "Q x Q" 'q2 = (Prod Q Q).
-definition empty_bar : bar ≝ 〈Qpos one,OQ〉.
+definition empty_bar : bar ≝ 〈Qpos one,〈OQ,OQ〉〉.
notation "\rect" with precedence 90 for @{'empty_bar}.
interpretation "q0" 'empty_bar = empty_bar.
bars: list bar;
bars_sorted : sorted q2_lt bars;
bars_begin_OQ : nth_base bars O = OQ;
- bars_end_OQ : nth_height bars (pred (\len bars)) = OQ
+ bars_end_OQ : nth_height bars (pred (\len bars)) = 〈OQ,OQ〉
}.
lemma len_bases_gt_O: ∀f.O < \len (bars f).
apply (H2 n1); simplify in H3; apply (le_S_S_to_le ?? H3);]
qed.
-(*
-lemma lt_n_plus_n_Sm : ∀n,m:nat.n < n + S m.
-intros; rewrite > sym_plus; apply (le_S_S n (m+n)); apply (le_plus_n m n); qed.
-*)
-
-(*
-lemma all_bigger_can_concat_bigger:
- ∀l1,l2,start,b,x,n.
- (∀i.i< len l1 → nth_base l1 i < \fst b) →
- (∀i.i< len l2 → \fst b ≤ nth_base l2 i) →
- (∀i.i< len l1 → start ≤ i → x ≤ nth_base l1 i) →
- start ≤ n → n < len (l1@b::l2) → x ≤ \fst b → x ≤ nth_base (l1@b::l2) n.
-intros; cases (cmp_nat n (len l1));
-[1: unfold nth_base; rewrite > (nth_concat_lt_len ????? H6);
- apply (H2 n); assumption;
-|2: rewrite > H6; unfold nth_base; rewrite > nth_len; assumption;
-|3: unfold nth_base; rewrite > nth_concat_ge_len; [2: apply lt_to_le; assumption]
- rewrite > len_concat in H4; simplify in H4; rewrite < plus_n_Sm in H4;
- lapply linear le_S_S_to_le to H4 as K; rewrite > sym_plus in K;
- lapply linear le_plus_to_minus to K as X;
- generalize in match X; generalize in match (n - len l1); intro W; cases W; clear W X;
- [intros; assumption] intros;
- apply (q_le_trans ??? H5); apply (H1 n1); assumption;]
-qed.
-*)
-
-
-inductive value_spec (f : q_f) (i : ℚ) : ℚ → nat → CProp ≝
- value_of : ∀q,j.
- 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 j.
-
-
-inductive break_spec (T : Type) (n : nat) (l : list T) : list T → CProp ≝
-| break_to: ∀l1,x,l2. \len l1 = n → l = l1 @ [x] @ l2 → break_spec T n l l.
-
-lemma list_break: ∀T,n,l. n < \len l → break_spec T n l l.
-intros 2; elim n;
-[1: elim l in H; [cases (not_le_Sn_O ? H)]
- apply (break_to ?? ? [] a l1); reflexivity;
-|2: cases (H l); [2: apply lt_S_to_lt; assumption;] cases l2 in H3; intros;
- [1: rewrite < H2 in H1; rewrite > H3 in H1; rewrite > append_nil in H1;
- rewrite > len_append in H1; rewrite > plus_n_SO in H1;
- cases (not_le_Sn_n ? H1);
- |2: apply (break_to ?? ? (l1@[x]) t l3);
- [2: simplify; rewrite > associative_append; assumption;
- |1: rewrite < H2; rewrite > len_append; rewrite > plus_n_SO; reflexivity]]]
-qed.
-
-definition value : ∀f:q_f.∀i:ratio.∃p:ℚ.∃j.value_spec f (Qpos i) p j.
-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) ▭)));]
-exists [apply (pred (find ? P (bars f) ▭))] apply value_of;
+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 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).
+
+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 value_OQ_l:
- ∀l,i.i < start l → \snd (\fst (value l i)) = OQ.
-intros; cases (value l i) (q Hq); cases Hq; clear Hq; simplify; cases H1; clear H1;
-try assumption; cases H2; cases (?:False); apply (q_lt_le_incompat ?? H H6);
-qed.
-
-lemma value_OQ_r:
- ∀l,i.start l + sum_bases (bars l) (len (bars l)) ≤ i → \snd (\fst (value l i)) = OQ.
-intros; cases (value l i) (q Hq); cases Hq; clear Hq; simplify; cases H1; clear H1;
-try assumption; cases H2; cases (?:False); apply (q_lt_le_incompat ?? H7 H);
+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_OQ_e:
- ∀l,i.bars l = [] → \snd (\fst (value l i)) = OQ.
-intros; cases (value l i) (q Hq); cases Hq; clear Hq; simplify; cases H1; clear H1;
-try assumption; cases H2; cases (?:False); apply (H1 H);
+
+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.
-inductive value_ok_spec (f : q_f) (i : ℚ) : nat × ℚ → Type ≝
- | value_ok : ∀n,q. n ≤ (len (bars f)) →
- q = \snd (nth (bars f) ▭ n) →
- sum_bases (bars f) n ≤ ⅆ[i,start f] →
- ⅆ[i, start f] < sum_bases (bars f) (S n) → value_ok_spec f i 〈n,q〉.
-
-lemma value_ok:
- ∀f,i.bars f ≠ [] → start f ≤ i → i < start f + sum_bases (bars f) (len (bars f)) →
- value_ok_spec f i (\fst (value f i)).
-intros; cases (value f i); simplify;
-cases H3; simplify; clear H3; cases H4; clear H4;
-[1,2,3: cases (?:False);
- [1: apply (q_lt_le_incompat ?? H3 H1);
- |2: apply (q_lt_le_incompat ?? H2 H3);
- |3: apply (H H3);]
-|4: cases H7; clear H7; cases w in H3 H4 H5 H6 H8; simplify; intros;
- constructor 1; assumption;]
+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.\snd (\fst (value l1 input)) = \snd (\fst (value l2 input)).
+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).
-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]
cases (?:False);
-[ apply (q_lt_corefl ? H)|apply (q_neg_gt ? H)]
+[ apply (q_lt_corefl ? H)| cases (q_lt_le_incompat ?? (q_neg_gt ?) (q_lt_to_le ?? H))]
qed.
notation < "x \blacksquare" non associative with precedence 50 for @{'unpos $x}.
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