include "nat_ordered_set.ma".
include "models/q_support.ma".
-include "models/list_support.ma".
-include "cprop_connectives.ma".
+include "models/list_support.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.
notation < "\ldots\rect\square\EmptySmallSquare\ldots" with precedence 90 for @{'lq2}.
interpretation "lq2" 'lq2 = (list bar).
-inductive sorted : list bar → Prop ≝
-| sorted_nil : sorted []
-| sorted_one : ∀x. sorted [x]
-| sorted_cons : ∀x,y,tl. \fst x < \fst y → sorted (y::tl) → sorted (x::y::tl).
+definition q2_lt := mk_rel bar (λx,y:bar.\fst x < \fst y).
-definition nth_base ≝ λf,n. \fst (nth f ▭ n).
-definition nth_height ≝ λf,n. \snd (nth f ▭ n).
+interpretation "bar lt" 'lt x y = (rel_op _ q2_lt x y).
-record q_f : Type ≝ {
- bars: list bar;
- bars_sorted : sorted bars;
- bars_begin_OQ : nth_base bars O = OQ;
- bars_tail_OQ : nth_height bars (pred (len bars)) = OQ
-}.
+lemma q2_trans : ∀a,b,c:bar. a < b → b < c → a < c.
+intros 3; cases a; cases b; cases c; unfold q2_lt; simplify; intros;
+apply (q_lt_trans ??? H H1);
+qed.
-lemma nth_nil: ∀T,i.∀def:T. nth [] def i = def.
-intros; elim i; simplify; [reflexivity;] assumption; qed.
+definition q2_trel := mk_trans_rel bar q2_lt q2_trans.
-inductive non_empty_list (A:Type) : list A → Type :=
-| show_head: ∀x,l. non_empty_list A (x::l).
+interpretation "bar lt" 'lt x y = (FunClass_2_OF_trans_rel q2_trel x y).
-lemma bars_not_nil: ∀f:q_f.non_empty_list ? (bars f).
-intro f; generalize in match (bars_begin_OQ f); cases (bars f);
-[1: intro X; normalize in X; destruct X;
-|2: intros; constructor 1;]
-qed.
+definition canonical_q_lt : rel bar → trans_rel ≝ λx:rel bar.q2_trel.
-lemma sorted_tail: ∀x,l.sorted (x::l) → sorted l.
-intros; inversion H; intros; [destruct H1;|destruct H1;constructor 1;]
-destruct H4; assumption;
-qed.
+coercion canonical_q_lt with nocomposites.
-lemma sorted_skip: ∀x,y,l. sorted (x::y::l) → sorted (x::l).
-intros; inversion H; intros; [1,2: destruct H1]
-destruct H4; inversion H2; intros; [destruct H4]
-[1: destruct H4; constructor 2;
-|2: destruct H7; constructor 3; [apply (q_lt_trans ??? H1 H4);]
- apply (sorted_tail ?? H2);]
-qed.
+interpretation "bar lt" 'lt x y = (FunClass_2_OF_trans_rel (canonical_q_lt _) x y).
-lemma sorted_tail_bigger : ∀x,l.sorted (x::l) → ∀i. i < len l → \fst x < nth_base l i.
-intros 2; elim l; [ cases (not_le_Sn_O i H1);]
-cases i in H2;
-[2: intros; apply (H ? n);[apply (sorted_skip ??? H1)|apply le_S_S_to_le; apply H2]
-|1: intros; inversion H1; intros; [1,2: destruct H3]
- destruct H6; simplify; assumption;]
+definition nth_base ≝ λf,n. \fst (\nth f ▭ n).
+definition nth_height ≝ λf,n. \snd (\nth f ▭ n).
+
+record q_f : Type ≝ {
+ 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,OQ〉
+}.
+
+lemma len_bases_gt_O: ∀f.O < \len (bars f).
+intros; generalize in match (bars_begin_OQ f); cases (bars f); intros;
+[2: simplify; apply le_S_S; apply le_O_n;
+|1: normalize in H; destruct H;]
qed.
lemma all_bases_positive : ∀f:q_f.∀i. OQ < nth_base (bars f) (S i).
intro f; generalize in match (bars_begin_OQ f); generalize in match (bars_sorted f);
-cases (bars_not_nil f); intros;
-cases (cmp_nat i (len l));
-[1: lapply (sorted_tail_bigger ?? H ? H2) as K; simplify in H1;
- rewrite > H1 in K; apply K;
-|2: rewrite > H2; simplify; elim l; simplify; [apply (q_pos_OQ one)]
- assumption;
-|3: simplify; elim l in i H2;[simplify; rewrite > nth_nil; apply (q_pos_OQ one)]
- cases n in H3; intros; [cases (not_le_Sn_O ? H3)] apply (H2 n1);
- apply (le_S_S_to_le ?? H3);]
+cases (len_gt_non_empty ?? (len_bases_gt_O f)); intros;
+cases (cmp_nat (\len l) i);
+[2: lapply (sorted_tail_bigger q2_lt ?? ▭ H ? H2) as K;
+ simplify in H1; rewrite < H1; apply K;
+|1: simplify; elim l in i H2;[simplify; rewrite > nth_nil; apply (q_pos_OQ one)]
+ cases n in H3; intros; [simplify in H3; cases (not_le_Sn_O ? H3)]
+ apply (H2 n1); simplify in H3; apply (le_S_S_to_le ?? H3);]
qed.
-definition eject_NxQ ≝
- λP.λp:∃x:nat × ℚ.P x.match p with [ex_introT p _ ⇒ p].
-coercion eject_NxQ.
-definition inject_NxQ ≝ λP.λp:nat × ℚ.λh:P p. ex_introT ? P p h.
-coercion inject_NxQ with 0 1 nocomposites.
-
-definition value_spec : q_f → ℚ → nat × ℚ → Prop ≝
- λf,i,q. nth_height (bars f) (\fst q) = \snd q ∧
- (nth_base (bars f) (\fst q) < i ∧
- ∀n.\fst q < n → n < len (bars f) → i ≤ nth_base (bars f) n).
-
-definition value : ∀f:q_f.∀i:ratio.∃p:ℚ.∃j.value_spec f (Qpos i) 〈j,p〉.
-intros;
-alias symbol "pi2" = "pair pi2".
-alias symbol "pi1" = "pair pi1".
-alias symbol "lt" (instance 6) = "Q less than".
-alias symbol "leq" = "Q less or equal than".
-letin value_spec_aux ≝ (
- λf,i,q.
- \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 ≝ (
- METTERE IN ACC LA LISTA PROCESSATA SO FAR
- E DIRE CHE QUELLA@L=BARS
- 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.
- (∀i.i < len l → nth_base (bars f) (\fst acc) < nth_base l i) →
- nth_height (bars f) (\fst acc) = \snd acc →
- value_spec_aux l (Qpos i) p);
-[3: intros; unfold;
-[4: clearbody value; unfold value_spec;
- generalize in match (bars_begin_OQ f);
- generalize in match (bars_sorted f);
- cases (bars_not_nil f); intro S; generalize in match (sorted_tail_bigger ?? S);
- clear S; cases (value 〈O,\snd x〉 (x::l)) (p Hp); intros;
- exists[apply (\snd p)];exists [apply (\fst p)]
- cases (Hp ?) (Hg HV);
- [unfold; split[reflexivity]simplify;split;
- [rewrite > H1;apply q_pos_OQ;
- |intros; cases n in H2 H3; [intro X; cases (not_le_Sn_O ? X)]
- intros;
- rewrite > H1; apply q_pos_OQ;
- cases HV (Hi Hm); clear Hp value value_spec_aux HV;
- exists [apply (\fst p)]; split;[rewrite > Hg;reflexivity|split;[assumption]intros]
- apply Hm; assumption;
-|1: unfold value_spec_aux; clear value value_spec_aux H2;intros; split[2:split]
- [1: apply (q_lt_le_trans ??? (H4 (\fst p))); clear H4 H5;
-
-[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)).
+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) ▭)));]
+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;
+ [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;
+ 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 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_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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