-definition value : ∀f:q_f.∀i:ratio.∃p:ℚ.value_spec f (Qpos i) p.
-intros;
-alias symbol "lt" (instance 5) = "Q less than".
-alias symbol "leq" = "Q less or equal than".
-letin value_spec_aux ≝ (
- λf,i,q.∃j. q = nth_height f j ∧
- (nth_base f j < i ∧ ∀n.j < n → n < len f → i ≤ nth_base f n));
-letin value ≝ (
- let rec value (acc: ℚ) (l : list bar) on l : ℚ ≝
- match l with
- [ nil ⇒ acc
- | cons x tl ⇒
- match q_cmp (\fst x) (Qpos i) with
- [ q_leq _ ⇒ value (\snd x) tl
- | q_gt _ ⇒ acc]]
- in value :
- ∀acc,l.∃p:ℚ. OQ ≤ acc → value_spec_aux l (Qpos i) p);
-[4: clearbody value; cases (value OQ (bars f)) (p Hp); exists[apply p];
- cases (Hp (q_le_n ?)) (j Hj); cases Hj (Hjp H); cases H (Hin Hmax);
- clear Hp value value_spec_aux Hj H; exists [apply j]; split[2:split;intros;]
- try apply Hmax; assumption;
-|1: intro Hacc; clear H2; cases (value (\snd b) l1) (j Hj);
- cases (q_cmp (\snd b) (Qpos i)) (Hib Hib);
- [1: cases (Hj Hib) (w Hw); simplify in ⊢ (? ? ? %); clear Hib Hj;
- exists [apply (S w)] cases Hw; cases H3; clear Hw H3;
- split; try assumption; split; try assumption; intros;
- apply (q_le_trans ??? (H5 (pred n) ??)); [3: apply q_le_n]
-
-
-
-
-[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.