(* *)
(**************************************************************************)
+include "nat_ordered_set.ma".
include "models/q_support.ma".
include "models/list_support.ma".
include "cprop_connectives.ma".
-definition bar ≝ ratio × ℚ. (* base (Qpos) , height *)
-record q_f : Type ≝ { start : ℚ; bars: list bar }.
+definition bar ≝ 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 ≝ 〈one,OQ〉.
+definition empty_bar : bar ≝ 〈Qpos one,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).
-
-let rec sum_bases (l:list bar) (i:nat)on i ≝
- match i with
- [ O ⇒ OQ
- | S m ⇒
- match l with
- [ nil ⇒ sum_bases l m + Qpos one
- | cons x tl ⇒ sum_bases tl m + Qpos (\fst x)]].
-
-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 ≝
+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 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 bars;
+ bars_begin_OQ : nth_base bars O = OQ;
+ bars_tail_OQ : nth_height bars (pred (len bars)) = OQ
+}.
+
+lemma nth_nil: ∀T,i.∀def:T. nth [] def i = def.
+intros; elim i; simplify; [reflexivity;] assumption; qed.
+
+inductive non_empty_list (A:Type) : list A → Type :=
+| show_head: ∀x,l. non_empty_list A (x::l).
+
+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.
+
+lemma sorted_tail: ∀x,l.sorted (x::l) → sorted l.
+intros; inversion H; intros; [destruct H1;|destruct H1;constructor 1;]
+destruct H4; assumption;
+qed.
+
+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.
+
+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;]
+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);]
+qed.
+
+definition eject_NxQ ≝
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)))].
+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 ≝ (
- let rec value (p: ℚ) (l : list bar) on l ≝
+ 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 ⇒ 〈nat_of_q p,OQ〉
+ [ nil ⇒ acc
| 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〉]]
+ 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. OQ ≤ acc →
- And3
- (sum_bases l (\fst p) ≤ acc)
- (acc < sum_bases l (S (\fst p)))
- (\snd p = \snd (nth l ▭ (\fst p))));
+ ∀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; 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);
+ [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)));
- [1,3: intros; [apply q_lt_to_le|apply q_eq_to_le;symmetry] assumption]
- simplify; apply q_le_minus; assumption;
+ [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)));
- [1,3: 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;
+ [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)));
- [1,3: intros; [apply q_lt_to_le|apply q_eq_to_le;symmetry] assumption]
- assumption;]
-|2: clear value H2; simplify; intros; split; [assumption|3:reflexivity]
+ [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: 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]]
+|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:q_f.
- (∀i.\fst (nth (bars l1) ▭ i) = \fst (nth (bars l2) ▭ i)).
+ 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.
[ apply (q_lt_corefl ? H)|apply (q_neg_gt ? H)]
qed.
-notation < "\blacksquare" non associative with precedence 90 for @{'hide}.
-definition hide ≝ λT:Type.λx:T.x.
-interpretation "hide" 'hide = (hide _ _).
-
-lemma sum_bases_ge_OQ:
- ∀l,n. OQ ≤ sum_bases l n.
-intro; elim l; simplify; intros;
-[1: elim n; [apply q_eq_to_le;reflexivity] simplify;
- apply q_le_plus_trans; try assumption; apply q_lt_to_le; apply q_pos_lt_OQ;
-|2: cases n; [apply q_eq_to_le;reflexivity] simplify;
- apply q_le_plus_trans; [apply H| apply q_lt_to_le; apply q_pos_lt_OQ;]]
-qed.
-
-lemma sum_bases_O:
- ∀l:q_f.∀x.sum_bases (bars l) x ≤ OQ → x = O.
-intros; cases x in H; [intros; reflexivity] intro; cases (?:False);
-cases (q_le_cases ?? H);
-[1: apply (q_lt_corefl OQ); rewrite < H1 in ⊢ (?? %);
-|2: apply (q_lt_antisym ??? H1);] clear H H1; cases (bars l);
-simplify; apply q_lt_plus_trans;
-try apply q_pos_lt_OQ;
-try apply (sum_bases_ge_OQ []);
-apply (sum_bases_ge_OQ l1);
-qed.
+notation < "x \blacksquare" non associative with precedence 50 for @{'unpos $x}.
+interpretation "hide unpos proof" 'unpos x = (unpos x _).
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