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.
-
-
-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]);
+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".
+alias symbol "Q" = "Rationals".
+coinductive value_spec (f : q_f) (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.
+
+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) ▭)));]
-exists [apply (pred (find ? P (bars f) ▭))] apply value_of;
+apply (value_of ?? (pred (find ? P (bars f) ▭)));
[1: reflexivity
|2: cases (cases_find bar P (bars f) ▭);
[1: cases i1 in H H1 H2 H3; simplify; intros;
unfold P 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_begin_OQ f); generalize in match (bars_sorted f);
-generalize in match (bars_end_OQ f);
-cases (len_gt_non_empty ?? (len_bases_gt_O f)); simplify;
-intros;
-[1:
-
-
-alias symbol "pi2" = "pair pi2".
-alias symbol "pi1" = "pair pi1".
-alias symbol "lt" (instance 7) = "Q less than".
-alias symbol "leq" = "Q less or equal than".
-letin value_spec_aux ≝ (
- λf,i,q. And4
- (\fst q < len f)
- (\snd q = nth_height f (\fst q))
- (nth_base f (\fst q) < i)
- (∀n.(\fst q) < n → n < len f → i ≤ nth_base f n));
-alias symbol "lt" (instance 5) = "Q less than".
-letin value ≝ (
- let rec value (acc: nat × ℚ) (l : list bar) on l : nat × ℚ ≝
- match l with
- [ nil ⇒ acc
- | cons x tl ⇒
- match q_cmp (\fst x) (Qpos i) with
- [ q_leq _ ⇒ value 〈S (\fst acc), \snd x〉 tl
- | q_gt _ ⇒ acc]]
- in value :
- ∀acc,l.∃p:nat × ℚ.
- ∀story. story @ l = bars f → S (\fst acc) = len story →
- value_spec_aux story (Qpos i) acc →
- value_spec_aux (story @ l) (Qpos i) p);
-[4: clearbody value; unfold value_spec;
- generalize in match (bars_begin_OQ f);
- generalize in match (bars_sorted f);
- cases (bars_not_nil f) in value; intros (value S); generalize in match (sorted_tail_bigger ?? S);
- clear S; cases (value 〈O,\snd x〉 l) (p Hp); intros;
- exists[apply (\snd p)];exists [apply (\fst p)] simplify;
- cases (Hp [x] (refl_eq ??) (refl_eq ??) ?) (Hg HV);
- [unfold; split; [apply le_n|reflexivity|rewrite > H; apply q_pos_OQ;]
- intros; cases n in H2 H3; [intro X; cases (not_le_Sn_O ? X)]
- intros; cases (not_le_Sn_O ? (le_S_S_to_le (S n1) O H3))]
- split;[rewrite > HV; reflexivity] split; [assumption;]
- intros; cases n in H4 H5; intros [cases (not_le_Sn_O ? H4)]
- apply (H3 (S n1)); assumption;
-|1: unfold value_spec_aux; clear value value_spec_aux H2; intros;
- cases H4; clear H4; split;
- [1: apply (trans_lt ??? H5); rewrite > len_concat; simplify; apply lt_n_plus_n_Sm;
- |2: unfold nth_height; rewrite > nth_concat_lt_len;[2:assumption]assumption;
- |3: unfold nth_base; rewrite > nth_concat_lt_len;[2:assumption]
- apply (q_le_lt_trans ???? H7); apply q_le_n;
- |4: intros; (*clear H6 H5 H4 H l;*) lapply (bars_sorted f) as HS;
- apply (all_bigger_can_concat_bigger story l1 (S (\fst p)));[6:apply q_lt_to_le]try assumption;
- [1: rewrite < H2 in HS; cases (sorted_pivot ??? HS); assumption
- |2: rewrite < H2 in HS; cases (sorted_pivot ??? HS);
- intros; apply q_lt_to_le; apply H11; assumption;
- |3: intros; apply H8; assumption;]]
-|3: intro; rewrite > append_nil; intros; assumption;
-|2: intros; cases (value 〈S (\fst p),\snd b〉 l1); unfold; simplify;
- cases (H6 (story@[b]) ???);
- [1: rewrite > associative_append; apply H3;
- |2: simplify; rewrite > H4; rewrite > len_concat; rewrite > sym_plus; reflexivity;
- |4: rewrite < (associative_append ? story [b] l1); split; assumption;
- |3: cases H5; clear H5; split; simplify in match (\snd ?); simplify in match (\fst ?);
- [1: rewrite > len_concat; simplify; rewrite < plus_n_SO; apply le_S_S; assumption;
- |2:
- |3:
- |4: ]]]
-
-
-
-
-
-
-
-
-
-
-[5: clearbody value;
- cases (q_cmp i (start f));
- [2: exists [apply 〈O,OQ〉] simplify; constructor 1; split; try assumption;
- try reflexivity; apply q_lt_to_le; assumption;
- |1: cases (bars f); [exists [apply 〈O,OQ〉] simplify; constructor 3; split;try assumption;reflexivity;]
- cases (value ⅆ[i,start f] (b::l)) (p Hp);
- cases (Hp (q_dist_ge_OQ ? ?)); clear Hp value; [cases H1; destruct H2]
- cases H1; clear H1; lapply (sum_bases_O (b::l) (\fst p)) as H1;
- [2: apply (q_le_trans ??? H2); rewrite > H; apply q_eq_to_le;
- rewrite > q_d_x_x; reflexivity;
- |1: exists [apply p] simplify; constructor 4; rewrite > H1; split;
- try split; try rewrite > q_d_x_x; try autobatch depth=2;
- [1: rewrite > H; rewrite > q_plus_sym; apply q_lt_plus;
- rewrite > q_plus_minus; apply q_lt_plus_trans; [apply sum_bases_ge_OQ]
- apply q_pos_lt_OQ;
- |2: rewrite > H; rewrite > q_d_x_x; apply q_eq_to_le; reflexivity;
- |3: rewrite > H; rewrite > q_d_x_x; apply q_lt_plus_trans;
- try apply sum_bases_ge_OQ; apply q_pos_lt_OQ;]]
- |3: cases (q_cmp i (start f+sum_bases (bars f) (len (bars f))));
- [1: exists [apply 〈O,OQ〉] simplify; constructor 2; split; try assumption;
- try reflexivity; rewrite > H1; apply q_eq_to_le; reflexivity;
- |3: exists [apply 〈O,OQ〉] simplify; constructor 2; split; try assumption;
- try reflexivity; apply q_lt_to_le; assumption;
- |2: generalize in match (refl_eq ? (bars f): bars f = bars f);
- generalize in match (bars f) in ⊢ (??? % → %); intro X; cases X; clear X;
- intros;
- [1: exists [apply 〈O,OQ〉] simplify; constructor 3; split; reflexivity;
- |2: cases (value ⅆ[i,start f] (b::l)) (p Hp);
- cases (Hp (q_dist_ge_OQ ? ?)); clear Hp value; [cases H3;destruct H4]
- cases H3; clear H3;
- exists [apply p]; constructor 4; split; try split; try assumption;
- [1: intro X; destruct X;
- |2: apply q_lt_to_le; assumption;
- |3: rewrite < H2; assumption;
- |4: cases (cmp_nat (\fst p) (len (bars f)));
- [1:apply lt_to_le;rewrite <H2; assumption|rewrite > H3;rewrite < H2;apply le_n]
- cases (?:False); cases (\fst p) in H3 H4 H6; clear H5;
- [1: intros; apply (not_le_Sn_O ? H5);
- |2: rewrite > q_d_sym; rewrite > q_d_noabs; [2: apply q_lt_to_le; assumption]
- intros; lapply (q_lt_inj_plus_r ?? (Qopp (start f)) H1); clear H1;
- generalize in match Hletin;
- rewrite > (q_plus_sym (start f)); rewrite < q_plus_assoc;
- do 2 rewrite < q_elim_minus; rewrite > q_plus_minus;
- rewrite > q_plus_OQ; intro K; apply (q_lt_corefl (i-start f));
- apply (q_lt_le_trans ???? H3); rewrite < H2;
- apply (q_lt_trans ??? K); apply sum_bases_increasing;
- assumption;]]]]]
-|1,3: intros; right; split;
- [1,4: clear H2; cases (value (q-Qpos (\fst b)) l1);
- cases (H2 (q_le_to_diff_ge_OQ ?? (? H1)));
- [1: intro; apply q_lt_to_le;assumption;
- |3: simplify; cases H4; apply q_le_minus; assumption;
- |2,5: simplify; cases H4; rewrite > H5; rewrite > H6;
- apply q_le_minus; apply sum_bases_empty_nat_of_q_le_q;
- |4: intro X; rewrite > X; apply q_eq_to_le; reflexivity;
- |*: simplify; apply q_le_minus; cases H4; assumption;]
- |2,5: cases (value (q-Qpos (\fst b)) l1);
- cases (H4 (q_le_to_diff_ge_OQ ?? (? H1)));
- [1,4: intros; [apply q_lt_to_le|apply q_eq_to_le;symmetry] assumption;
- |3,6: cases H5; simplify; change with (q < sum_bases l1 (S (\fst w)) + Qpos (\fst b));
- apply q_lt_plus; assumption;
- |2,5: simplify; cases H5; rewrite > H6; simplify; rewrite > H7;
- apply q_lt_plus; apply sum_bases_empty_nat_of_q_le_q_one;]
- |*: cases (value (q-Qpos (\fst b)) l1); simplify;
- cases (H4 (q_le_to_diff_ge_OQ ?? (? H1)));
- [1,4: intros; [apply q_lt_to_le|apply q_eq_to_le;symmetry] assumption;
- |3,6: cases H5; assumption;
- |*: cases H5; rewrite > H6; rewrite > H8;
- elim (\fst w); [1,3:reflexivity;] simplify; assumption;]]
-|2: clear value H2; simplify; intros; right; split; [assumption|3:reflexivity]
- rewrite > q_plus_sym; rewrite > q_plus_OQ; assumption;
-|4: intros; left; split; reflexivity;]
-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);
-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);
-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;]
-qed.
-
-definition same_values ≝
- λl1,l2:q_f.
- ∀input.\snd (\fst (value l1 input)) = \snd (\fst (value l2 input)).
-
-definition same_bases ≝
- λl1,l2:list bar. (∀i.\fst (nth l1 ▭ i) = \fst (nth l2 ▭ i)).
+ [1: generalize in match (bars_sorted f);
+ cases (list_break ??? H) in H1; rewrite > H6;
+ rewrite < H1; simplify; rewrite > nth_len; unfold P;
+ 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 (bars 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);]
+ 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]]]
+qed.
+
+lemma value : q_f → ratio → ℚ × ℚ.
+intros; cases (value_lemma q r); apply w; qed.
+
+lemma cases_value : ∀f,i. value_spec f (Qpos i) (value f i).
+intros; unfold value; cases (value_lemma 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).
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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