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);
+lemma len_concat: ∀T:Type.∀l1,l2:list T. len (l1@l2) = len l1 + len l2.
+intros; elim l1; [reflexivity] simplify; rewrite < H; reflexivity;
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.
+inductive non_empty_list (A:Type) : list A → Type :=
+| show_head: ∀x,l. non_empty_list A (x::l).
-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 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.
+
+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 nth_concat_lt_len:
+ ∀T:Type.∀l1,l2:list T.∀def.∀i.i < len l1 → nth (l1@l2) def i = nth l1 def i.
+intros 4; elim l1; [cases (not_le_Sn_O ? H)] cases i in H H1; simplify; intros;
+[reflexivity| rewrite < H;[reflexivity] apply le_S_S_to_le; apply H1]
+qed.
+
+lemma nth_concat_ge_len:
+ ∀T:Type.∀l1,l2:list T.∀def.∀i.
+ len l1 ≤ i → nth (l1@l2) def i = nth l2 def (i - len l1).
+intros 4; elim l1; [ rewrite < minus_n_O; reflexivity]
+cases i in H1; simplify; intros; [cases (not_le_Sn_O ? H1)]
+apply H; apply le_S_S_to_le; apply H1;
+qed.
+
+lemma nth_len:
+ ∀T:Type.∀l1,l2:list T.∀def,x.
+ nth (l1@x::l2) def (len l1) = x.
+intros 2; elim l1;[reflexivity] simplify; apply H; 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.
+
+lemma sorted_head_smaller:
+ ∀l,p. sorted (p::l) → ∀i.i < len l → \fst p < nth_base l i.
+intro l; elim l; intros; [cases (not_le_Sn_O ? H1)] cases i in H2; simplify; intros;
+[1: inversion H1; [1,2: simplify; intros; destruct H3] intros; destruct H6; assumption;
+|2: apply (H p ? n ?); [apply (sorted_skip ??? H1)] apply le_S_S_to_le; apply H2]
+qed.
-definition value : ∀f:q_f.∀i:ratio.∃p:ℚ.value_spec f (Qpos i) p.
+
+alias symbol "pi1" = "pair pi1".
+alias symbol "lt" (instance 6) = "Q less than".
+alias symbol "lt" (instance 2) = "Q less than".
+alias symbol "and" = "logical and".
+lemma sorted_pivot:
+ ∀l1,l2,p. sorted (l1@p::l2) →
+ (∀i. i < len l1 → nth_base l1 i < \fst p) ∧
+ (∀i. i < len l2 → \fst p < nth_base l2 i).
+intro l; elim l;
+[1: split; [intros; cases (not_le_Sn_O ? H1);] intros;
+ apply sorted_head_smaller; assumption;
+|2: cases (H ?? (sorted_tail a (l1@p::l2) H1));
+ lapply depth = 0 (sorted_head_smaller (l1@p::l2) a H1) as Hs;
+ split; simplify; intros;
+ [1: cases i in H4; simplify; intros;
+ [1: lapply depth = 0 (Hs (len l1)) as HS;
+ unfold nth_base in HS; rewrite > nth_len in HS; apply HS;
+ rewrite > len_concat; simplify; apply lt_n_plus_n_Sm;
+ |2: apply (H2 n); apply le_S_S_to_le; apply H4]
+ |2: apply H3; assumption]]
+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 "lt" (instance 5) = "Q less than".
+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.∃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. 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: ℚ) (l : list bar) on l : ℚ ≝
+ 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.
+ ∀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: ]]]
+
+
+
+
+
+
projects / helm.git / blobdiff