-sandwich.ma ordered_uniform.ma
property_sigma.ma ordered_uniform.ma russell_support.ma
-uniform.ma supremum.ma
bishop_set.ma ordered_set.ma
-sequence.ma nat/nat.ma
ordered_uniform.ma uniform.ma
-supremum.ma datatypes/constructors.ma nat/plus.ma nat_ordered_set.ma sequence.ma
-property_exhaustivity.ma ordered_uniform.ma property_sigma.ma
bishop_set_rewrite.ma bishop_set.ma
-cprop_connectives.ma datatypes/constructors.ma logic/equality.ma
+sequence.ma nat/nat.ma
nat_ordered_set.ma bishop_set.ma nat/compare.ma
lebesgue.ma property_exhaustivity.ma sandwich.ma
+property_exhaustivity.ma ordered_uniform.ma property_sigma.ma
+cprop_connectives.ma datatypes/constructors.ma logic/equality.ma
ordered_set.ma cprop_connectives.ma
+sandwich.ma ordered_uniform.ma
russell_support.ma cprop_connectives.ma nat/nat.ma
-models/nat_lebesgue.ma lebesgue.ma models/nat_order_continuous.ma
+uniform.ma supremum.ma
+supremum.ma datatypes/constructors.ma nat/plus.ma nat_ordered_set.ma sequence.ma
models/nat_ordered_uniform.ma bishop_set_rewrite.ma models/nat_uniform.ma ordered_uniform.ma
-models/q_support.ma Q/q/q.ma cprop_connectives.ma
-models/discrete_uniformity.ma bishop_set_rewrite.ma uniform.ma
-models/q_bars.ma cprop_connectives.ma models/list_support.ma models/q_support.ma nat_ordered_set.ma
-models/q_function.ma models/q_shift.ma nat_ordered_set.ma
models/nat_uniform.ma models/discrete_uniformity.ma nat_ordered_set.ma
-models/nat_dedekind_sigma_complete.ma models/nat_uniform.ma nat/le_arith.ma russell_support.ma supremum.ma
+models/q_support.ma Q/q/qplus.ma Q/q/qtimes.ma cprop_connectives.ma
models/q_shift.ma models/q_bars.ma
-models/list_support.ma list/list.ma
models/nat_order_continuous.ma models/nat_dedekind_sigma_complete.ma models/nat_ordered_uniform.ma
-Q/q/q.ma
+models/nat_lebesgue.ma lebesgue.ma models/nat_order_continuous.ma
+models/list_support.ma list/list.ma
+models/nat_dedekind_sigma_complete.ma models/nat_uniform.ma nat/le_arith.ma russell_support.ma supremum.ma
+models/discrete_uniformity.ma bishop_set_rewrite.ma uniform.ma
+models/q_function.ma Q/q/qtimes.ma models/q_shift.ma nat_ordered_set.ma
+models/q_bars.ma cprop_connectives.ma models/list_support.ma models/q_support.ma nat_ordered_set.ma
+models/q_value_skip.ma models/q_shift.ma
+Q/q/qplus.ma
+Q/q/qtimes.ma
datatypes/constructors.ma
list/list.ma
logic/equality.ma
record q_f : Type ≝ {
bars: list bar;
- increasing_bars : sorted bars;
+ 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.
-lemma all_bases_positives : ∀f:q_f.∀i.i < len (bars f) → OQ < nth_base (bars f) i.
-intro f; elim (increasing_bars f);
-[1: unfold nth_base; rewrite > nth_nil; apply (q_pos_OQ one);
-|2: cases i in H; [2: cases (?:False);
+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.
-definition eject_Q ≝
- λP.λp:∃x:ℚ.P x.match p with [ex_introT p _ ⇒ p].
-coercion eject_Q.
-definition inject_Q ≝ λP.λp:ℚ.λh:P p. ex_introT ? P p h.
-coercion inject_Q with 0 1 nocomposites.
+lemma sorted_tail: ∀x,l.sorted (x::l) → sorted l.
+intros; inversion H; intros; [destruct H1;|destruct H1;constructor 1;]
+destruct H4; assumption;
+qed.
-definition value_spec : q_f → ℚ → ℚ → Prop ≝
- λf,i,q.
- ∃j. q = nth_height (bars f) j ∧
- (nth_base (bars f) j < i ∧
- ∀n.j < n → n < len (bars f) → i ≤ nth_base (bars f) n).
+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 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 : ∀f:q_f.∀i:ratio.∃p:ℚ.value_spec f (Qpos i) p.
+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 "lt" (instance 5) = "Q less than".
+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.∃j. q = nth_height f j ∧
- (nth_base f j < i ∧ ∀n.j < n → n < len f → i ≤ nth_base f n));
+ 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 (acc: ℚ) (l : list bar) on l : ℚ ≝
+ 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 (\snd x) tl
+ [ q_leq _ ⇒ value 〈S (\fst acc), \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]
-
-
-
+ ∀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));
projects / helm.git / commitdiff
