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dynamic_dependent_program.v
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(* Xuanrui Qi, Kazunari Tanaka, Jacques Garrigue *)
From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq.
From mathcomp Require Import choice fintype prime tuple finfun finset bigop.
Require Import compact_data_structures rank_select insert_delete Program.
Require Import set_clear Compare_dec ExtrOcamlNatInt dynamic.
Set Implicit Arguments.
Section dynamic_dependent.
Variable w : nat.
Hypothesis wordsize_gt1: w > 1.
Section insert.
(*
* Translated from https://github.com/xuanruiqi/dtp/blob/master/RedBlack.idr
* which in turn is translated from dynamic_dependent.v
*)
Program Definition balanceL {lnum lones rnum rones d cl cr} (c : color)
(l : near_tree w lnum lones d cl)
(r : tree w rnum rones d cr)
(ok_l : color_ok c (fix_color l))
(ok_r : color_ok c cr) :
{ t' : near_tree w (lnum + rnum) (lones + rones) (incr_black d c) c
| dflatteni t' = dflatteni l ++ dflatten r } :=
match c with
| Black =>
match l with
| Bad _ _ _ _ _ _ _ t1 t2 t3 => Good Black (rnode (bnode t1 t2) (bnode t3 r))
| Good _ _ _ _ _ l' => Good Black (bnode l' r)
end
| Red => match cr with
| Red => _ (* impossible *)
| Black => match l with
| Bad _ _ _ _ _ _ _ _ _ _ => _ (* impossible *)
| @Good _ _ _ _ c' _ l' =>
match c' with
| Red =>
match l' with
| Leaf _ _ _ => _ (* impossible *)
| @Node _ _ _ _ _ _ cll clr _ _ _ t1 t2 =>
match cll with
| Black => match clr with
| Black => Bad t1 t2 r
| Red => _ (* impossible *)
end
| Red => _ (* impossible *)
end
end
| Black => Good Red (rnode l' r)
end
end
end
end.
Next Obligation. by rewrite addnA. Qed.
Next Obligation. by rewrite addnA. Qed.
Next Obligation.
rewrite /eq_rect. destruct balanceL_obligation_2 => //=.
destruct balanceL_obligation_3 => //=.
by rewrite -Heq_l //= -!catA.
Qed.
Next Obligation.
rewrite /eq_rect. destruct balanceL_obligation_7 => //=.
by rewrite -Heq_l.
Qed.
Next Obligation. subst l; by move: ok_l. Qed.
Next Obligation.
rewrite /eq_rect. destruct balanceL_obligation_18 => //=.
by rewrite -Heq_l -Heq_l' //= catA.
Qed.
Next Obligation.
rewrite /eq_rect. destruct balanceL_obligation_28 => //=.
by rewrite -Heq_l.
Qed.
Program Definition balanceR {lnum lones rnum rones d cl cr} (c : color)
(l : tree w lnum lones d cl)
(r : near_tree w rnum rones d cr)
(ok_l : color_ok c cl)
(ok_r : color_ok c (fix_color r)) :
{ t' : near_tree w (lnum + rnum) (lones + rones) (incr_black d c) c |
dflatteni t' = dflatten l ++ dflatteni r } :=
match c with
| Black =>
match r with
| Bad _ _ _ _ _ _ _ t1 t2 t3 => Good Black (rnode (bnode l t1) (bnode t2 t3))
| Good _ _ _ _ _ r' => Good Black (bnode l r')
end
| Red => match cl with
| Red => _ (* impossible *)
| Black => match r with
| Bad _ _ _ _ _ _ _ _ _ _ => _ (* impossible *)
| @Good _ _ _ _ c' _ r' =>
match c' with
| Red =>
match r' with
| Leaf _ _ _ => _ (* impossible *)
| @Node _ _ _ _ _ _ cll clr _ _ _ t1 t2 =>
match cll with
| Black => match clr with
| Black => Bad l t1 t2
| Red => _ (* impossible *)
end
| Red => _ (* impossible *)
end
end
| Black => Good Red (rnode l r')
end
end
end
end.
Next Obligation. by rewrite !addnA. Qed.
Next Obligation. by rewrite !addnA. Qed.
Next Obligation.
rewrite /eq_rect.
destruct balanceR_obligation_3, balanceR_obligation_4 => //=.
by rewrite -Heq_r //= !catA.
Qed.
Next Obligation.
rewrite /eq_rect. destruct balanceR_obligation_7 => //=.
by rewrite -Heq_r.
Qed.
Next Obligation. subst r; by move: ok_r. Qed.
Next Obligation. by rewrite addnA. Qed.
Next Obligation. by rewrite addnA. Qed.
Next Obligation.
rewrite /eq_rect.
destruct balanceR_obligation_18, balanceR_obligation_19 => //=.
by rewrite -Heq_r -Heq_r'.
Qed.
Next Obligation.
rewrite /eq_rect. destruct balanceR_obligation_27 => //=.
by rewrite -Heq_r.
Qed.
Program Fixpoint dins {num ones d c}
(B : tree w num ones d c)
(b : bool) (i : nat) {measure (size_of_tree B) } :
{ B' : near_tree w num.+1 (ones + b) d c |
dflatteni B' = insert1 (dflatten B) b i } :=
match B with
| Leaf s _ _ =>
let s' := insert1 s b i in
(* cannot use "if" due to bugs in Program *)
match (size s' == 2 * (w ^ 2)) with
| true => (* split the node *)
let n := (size s') %/ 2 in
let sl := take n s' in
let sr := drop n s' in
Good c (rnode (Leaf _ sl _ _) (Leaf _ sr _ _))
| false => Good c (Leaf _ s' _ _)
end
| Node lnum _ _ _ _ _ _ _ ok_l ok_r l r =>
if i < lnum
then (` (balanceL c (dins l b i) r _ ok_r))
else (` (balanceR c l (dins r b (i - lnum)) ok_l _))
end.
Next Obligation.
move/eqP/eqnP: Heq_anonymous => -> //=.
rewrite size_take size_insert1 mulKn //=.
case: ifP => H. exact: leq_div.
apply: leq_trans. apply: wildcard'.
by rewrite leq_eqVlt ltnSn orbT.
Qed.
Next Obligation.
move/eqP/eqnP: Heq_anonymous => -> //=.
rewrite size_take size_insert1 mulKn //=.
case: ifP => H. by rewrite ltn_Pmull //= expn_gt0 wordsize_gt0.
move/negbT: H. rewrite -leqNgt => H.
apply: leq_ltn_trans. apply: H.
by rewrite ltn_Pmull //= expn_gt0 wordsize_gt0.
Qed.
Next Obligation.
move/eqP/eqnP: Heq_anonymous => //=.
rewrite size_drop size_insert1 => ->.
by rewrite mulKn //= mulSn mul1n -addnBA // subnKC // leq_div.
Qed.
Next Obligation.
move/eqP/eqnP: Heq_anonymous => //=.
rewrite size_drop size_insert1 => ->.
rewrite mulKn //= mulSn mul1n -addnBA // subnKC //.
by rewrite -{1}[w ^ 2]addn0 ltn_add2l expn_gt0 wordsize_gt0.
Qed.
Next Obligation. by rewrite -size_cat cat_take_drop size_insert1. Qed.
Next Obligation.
rewrite -count_cat cat_take_drop /count_one count_insert1. by destruct b.
Qed.
Next Obligation.
rewrite /eq_rect.
destruct dins_func_obligation_7, dins_func_obligation_6, dins_func_obligation_5 => //=.
by rewrite cat_take_drop.
Qed.
Next Obligation.
rewrite size_insert1. exact: leq_trans.
Qed.
Next Obligation.
move/eqP/eqnP/eqP: Heq_anonymous => /=.
rewrite size_insert1 neq_ltn. case/orP => //.
by rewrite ltnNge wildcard'0.
Qed.
Next Obligation. by rewrite size_insert1. Qed.
Next Obligation. rewrite /count_one count_insert1. by destruct b. Qed.
Next Obligation.
rewrite /eq_rect.
by destruct dins_func_obligation_13, dins_func_obligation_12, dins_func_obligation_11.
Qed.
Next Obligation.
apply/ltP. by rewrite -Heq_B /= -[X in X < _]addn0 ltn_add2l size_of_tree_pos.
Qed.
Next Obligation. by destruct dins, x, c. Qed.
Next Obligation.
apply/ltP. by rewrite -Heq_B /= -[X in X < _]add0n ltn_add2r size_of_tree_pos.
Qed.
Next Obligation. by destruct dins, x, c. Qed.
Next Obligation. by rewrite -addn1 -[in RHS]addn1 addnAC addnA. Qed.
Next Obligation. by rewrite addnAC addnA. Qed.
Next Obligation. by rewrite addnAC. Qed.
Next Obligation.
case: ifP => /= H.
- set B' := balanceL _ _ _ _ _.
destruct dins_func_obligation_23.
rewrite (proj2_sig B') {B'}.
destruct dins => /=.
by rewrite e /insert1 /insert take_cat drop_cat size_dflatten H -!catA.
- set B' := balanceR _ _ _ _ _.
rewrite /dflatteni /eq_rect => //=.
destruct dins_func_obligation_23, dins_func_obligation_22, dins_func_obligation_21.
rewrite -/(dflatteni (proj1_sig B')) (proj2_sig B') {B'}.
destruct dins => //=.
by rewrite e /insert1 /insert take_cat drop_cat size_dflatten H -!catA.
Qed.
Program Definition paint_black {num ones d c} (B : tree w num ones d c) :
{ B' : tree w num ones (incr_black d (inv c)) Black |
dflatten B' = dflatten B } :=
match B with
| Leaf _ _ _ => B
| Node _ _ _ _ _ _ _ _ _ _ l r => bnode l r
end.
Next Obligation.
rewrite /eq_rect.
destruct paint_black_obligation_4 => //=. by rewrite -Heq_B.
Qed.
Next Obligation. by destruct c. Qed.
Next Obligation. rewrite /eq_rect. by destruct paint_black_obligation_6. Qed.
Definition dinsert {num ones d c}
(B : tree w num ones d c) (b : bool) (i : nat) :=
(` (paint_black (fix_near_tree (` (dins B b i))))).
Theorem dinsertK {num ones d c} (B : tree w num ones d c) (b : bool) (i : nat) :
dflatten (dinsert B b i) = insert1 (dflatten B) b i.
Proof.
rewrite /dinsert. destruct dins, paint_black => //=.
by rewrite e0 //= fix_near_treeK e.
Qed.
End insert.
Section query.
Fixpoint daccess {n m d c} (tr : tree w n m d c) i :=
match tr with
| Leaf s _ _ => nth false s i
| Node lnum _ _ _ _ _ _ _ _ _ l r =>
if i < lnum
then daccess l i
else daccess r (i - lnum)
end.
Fixpoint drank {n m d c} (tr : tree w n m d c) i :=
match tr with
| Leaf s _ _ => rank true i s
| Node lnum lones rnum rones _ _ _ _ _ _ l r =>
if i < lnum
then drank l i
else lones + drank r (i - lnum)
end.
Fixpoint dselect_0 {n m d c} (tr : tree w n m d c) i :=
match tr with
| Leaf s _ _ => select false i s
| Node s1 o1 s2 o2 _ _ _ _ _ _ l r =>
let zeroes := s1 - o1
in if i <= zeroes
then dselect_0 l i
else s1 + dselect_0 r (i - zeroes)
end.
Fixpoint dselect_1 {n m d c} (tr : tree w n m d c) i :=
match tr with
| Leaf s _ _ => select true i s
| Node s1 o1 s2 o2 _ _ _ _ _ _ l r =>
if i <= o1
then dselect_1 l i
else s1 + dselect_1 r (i - o1)
end.
Lemma daccessK nums ones d c (B : tree w nums ones d c) :
daccess B =1 access (dflatten B).
Proof.
rewrite /access.
elim: B => //= lnum o1 s2 o2 d0 cl cr c0 i i0 l IHl r IHr x.
by rewrite nth_cat size_dflatten -IHl -IHr.
Qed.
Lemma drankK nums ones d c (B : tree w nums ones d c) i :
drank B i = rank true i (dflatten B).
Proof.
elim: B i => //= lnum o1 s2 o2 d0 cl cr c0 i i0 l IHl r IHr x.
by rewrite rank_cat size_dflatten IHl -IHr -dflatten_rank.
Qed.
Lemma drank_ones num ones d c (B : tree w num ones d c) :
drank B num = ones.
Proof.
by rewrite [in RHS](dflatten_rank B) drankK.
Qed.
Lemma dselect1K nums ones d c (B : tree w nums ones d c) i :
dselect_1 B i = select true i (dflatten B).
Proof.
elim: B i => //= lnum o1 s2 o2 d0 cl cr c0 i i0 l IHl r IHr x.
by rewrite select_cat -dflatten_ones IHl IHr size_dflatten.
Qed.
Lemma dselect0K nums ones d c (B : tree w nums ones d c) i :
dselect_0 B i = select false i (dflatten B).
Proof.
elim: B i => //= lnum o1 s2 o2 d0 cl cr c0 i i0 l IHl r IHr x.
by rewrite select_cat -dflatten_zeroes IHl IHr size_dflatten.
Qed.
Lemma access_leq_count (s : seq bool) i : i < size s -> access s i <= count_one s.
Proof.
rewrite /access.
elim: s i => //= h s IHs i H.
case_eq i => [| i'] //= Hi'. by case: h.
apply: leq_trans. apply: IHs. move: H. by rewrite Hi' ltnS.
by rewrite leq_addl.
Qed.
Lemma daccess_leq_ones {num ones d c} (B : tree w num ones d c) i :
i < num -> daccess B i <= ones.
Proof.
elim: B i => [s w_wf size_wf | lnum lones rnum rones d' cl cr c' ok_l ok_r l IHl r IHr] //= i H.
by rewrite /count_one access_leq_count.
case: ifP => Hi.
apply: leq_trans. exact: IHl. exact: leq_addr.
apply: leq_trans. apply: IHr.
rewrite -(ltn_add2r lnum) addnC addnBA.
by rewrite addnC addnK addnC. by rewrite leqNgt Hi.
by rewrite leq_addl.
Qed.
End query.
Section set_clear.
Obligation Tactic := idtac.
Program Fixpoint bset {num ones d c} (B : tree w num ones d c) i
{measure (size_of_tree B)} :
{ B'b : tree w num (ones + (~~ (daccess B i)) && (i < num)) d c * bool
| dflatten (fst B'b) = bit_set (dflatten B) i/\snd B'b = ~~ daccess B i } :=
match B with
| Leaf s _ _ => (Leaf _ (bit_set s i) _ _, ~~ (access s i))
| Node lnum lones rnum rones _ _ _ _ col cor l r =>
match lt_dec i lnum with
| left H =>
let x := bset l i
in (Node col cor x.1 r, x.2)
| right H =>
let x := bset r (i - lnum)
in (Node col cor l x.1, x.2)
end
end.
Next Obligation. intros. by rewrite size_bit_set. Qed.
Next Obligation. intros. by rewrite size_bit_set. Qed.
Next Obligation. intros; apply: size_bit_set. Qed.
Next Obligation.
intros; case Hi: (i < size s).
rewrite /count_one /daccess (count_bit_set false). by rewrite andbT addnC.
by rewrite Hi.
rewrite andbF addn0. by rewrite /count_one /daccess bit_set_over //= leqNgt Hi.
Qed.
Next Obligation.
intros; subst; split => //.
by destruct bset_func_obligation_4 , bset_func_obligation_3 => /=.
Qed.
Next Obligation.
intros; subst. apply /ltP.
rewrite -Heq_B /=.
by rewrite -addn1 leq_add2l size_of_tree_pos.
Qed.
Next Obligation.
intros; move/ltP: (H) => Hi /=.
by rewrite Hi (ltn_addr _ Hi) addnAC.
Qed.
Next Obligation.
split; subst; last first.
destruct bset as [[l' flip][Hl' Hf]] => /=.
move/ltP: (H) => ->.
by rewrite -Hf.
move=> /=.
move: (lones + rones + _) (bset_func_obligation_7 _ _ _ _ _ _) => ones Ho.
destruct Ho => /=.
destruct bset as [[l' flip][Hl' Hf]] => /=.
rewrite /= in Hl'.
move/ltP: (H).
rewrite Hl' /bit_set update_cat {1}size_dflatten => Hi.
by rewrite ifT.
Qed.
Next Obligation.
intros; subst. apply /ltP.
rewrite -Heq_B /=.
by rewrite -add1n leq_add2r size_of_tree_pos.
Qed.
Next Obligation.
intros; move/ltP: (H) => Hi /=.
rewrite -if_neg Hi !addnA.
by rewrite -(ltn_add2l lnum) subnKC // leqNgt.
Qed.
Next Obligation.
split; subst; last first.
destruct bset as [[r' flip][Hr' Hf]] => /=.
move/ltP: (H) => Hi.
by rewrite -if_neg Hi -Hf.
move=> /=.
move: (lones + rones + _) (bset_func_obligation_10 _ _ _ _ _ _) => ones Ho.
destruct Ho => /=.
destruct bset as [[r' flip][Hr' Hf]] => /=.
rewrite /= in Hr'.
move/ltP: (H).
rewrite Hr' /bit_set update_cat size_dflatten => Hi.
by rewrite -if_neg Hi.
Qed.
Next Obligation. intuition. Qed.
Program Fixpoint bclear {num ones d c}
(B : tree w num ones d c) i
{ measure (size_of_tree B) } :
{ B'b : tree w num (ones - (daccess B i) && (i < num)) d c * bool |
dflatten B'b.1 = bit_clear (dflatten B) i /\ snd B'b = daccess B i } :=
match B with
| Leaf s _ _ => (Leaf _ (bit_clear s i) _ _, access s i)
| Node lnum lones rnum rones _ _ _ _ col cor l r =>
match lt_dec i lnum with
| left H =>
let l'b := bclear l i
in (Node col cor l'b.1 r, l'b.2)
| right H =>
let r'b := bclear r (i - lnum)
in (Node col cor l r'b.1, r'b.2)
end
end.
Next Obligation. intros. by rewrite size_bit_clear. Qed.
Next Obligation. intros. by rewrite size_bit_clear. Qed.
Next Obligation. intros. by rewrite size_bit_clear. Qed.
Next Obligation.
intros. case Hi: (i < size s).
+ by rewrite /count_one //= (count_bit_clear false) //= andbT.
+ rewrite bit_clear_over //=. by rewrite andbF subn0.
by rewrite leqNgt Hi.
Qed.
Next Obligation.
intros; subst. rewrite /eq_rect.
by destruct bclear_func_obligation_4, bclear_func_obligation_3.
Qed.
Next Obligation.
intros; subst. apply/ltP.
by rewrite -Heq_B //= -addn1 leq_add2l size_of_tree_pos.
Qed.
Next Obligation.
intros; subst => //=.
move/ltP: (H) => Hi /=.
rewrite Hi ltn_addr //= andbT.
rewrite addnC addnBA. by rewrite addnC. by rewrite daccess_leq_ones.
Qed.
Next Obligation.
intros; subst. rewrite /eq_rect.
destruct bclear_func_obligation_7 => //=.
move/ltP : (H) => Hi /=.
rewrite /proj1_sig; subst l'b. destruct bclear => //=.
split.
+ rewrite (proj1 a) /bit_clear update_cat size_dflatten.
by rewrite Hi.
+ by rewrite (proj2 a) Hi.
Qed.
Next Obligation.
intros; subst. rewrite -Heq_B //=. apply/ltP.
by rewrite -[X in X < _]add0n ltn_add2r size_of_tree_pos.
Qed.
Next Obligation.
intros; subst => //=.
move/ltP: (H) => Hi /=. rewrite -if_neg Hi.
case Hi': (i - lnum < rnum); move: (Hi'); rewrite -(ltn_add2l lnum) subnKC;
try move => ->.
+ rewrite andbT addnBA //= daccess_leq_ones //= Hi' //=.
by rewrite leqNgt Hi.
+ by rewrite andbF !subn0.
by rewrite leqNgt Hi.
Qed.
Next Obligation.
split; subst; last first.
destruct bclear as [[r' tgt][Hr' Hf]] => /=.
move/ltP: (H) => Hi.
by rewrite -if_neg Hi -Hf.
move=> /=.
move: (lones + rones - _) (bclear_func_obligation_10 _ _ _ _ _ _) => ones Ho.
destruct Ho => /=.
destruct bclear as [[r' tgt][Hr' Hf]] => /=.
rewrite /= in Hr'.
move/ltP: (H).
rewrite Hr' /bit_clear update_cat (size_dflatten l) => Hi.
by rewrite -if_neg Hi.
Qed.
Next Obligation. intuition. Qed.
End set_clear.
Section delete.
Inductive del_tree : nat -> nat -> nat -> color -> Type :=
| Stay : forall {num ones d c} pc,
color_ok c (inv pc) -> tree w num ones d c -> del_tree num ones d pc
| Down : forall {num ones d}, tree w num ones d Black -> del_tree num ones d.+1 Black.
Definition dflattend {num ones d c} (tr : del_tree num ones d c) :=
match tr with
| Stay _ _ _ _ _ _ t => dflatten t
| Down _ _ _ t => dflatten t
end.
(* NB: move to dynamic_dependent.v? *)
Lemma wordsize_sqrn_div2_gt0 : 0 < w ^ 2 %/ 2.
Proof.
rewrite lt0n. apply/eqP. exact: wordsize_sqrn_div2_neq0.
Qed.
Lemma wordsize_del_lt_twice : (w ^ 2 %/ 2 + (w ^ 2 %/ 2).-1 < 2 * w ^ 2).
Proof.
have Hw: (w ^ 2 %/ 2 + (w ^ 2 %/ 2).-1 < w ^ 2).
rewrite -subn1 addnBA divn2. rewrite addnn.
rewrite -[X in _ < X]odd_double_half subn1.
have Hw': (((w ^ 2)./2).*2.-1 < ((w ^ 2)./2).*2).
rewrite -subn1 -[X in _ < X]subn0 ltn_sub2l //=.
by rewrite -[X in X < _]double0 ltn_double -divn2 wordsize_sqrn_div2_gt0.
apply: leq_trans. apply: Hw'. apply: leq_addl.
by rewrite -divn2 wordsize_sqrn_div2_gt0.
apply: leq_trans. apply: Hw. exact: leq_pmull.
Qed.
Lemma cons_del_head s : size s > 0 -> access s 0 :: delete s 0 = s.
Proof.
case: s => //= h s H.
by rewrite /delete //= drop0.
Qed.
Program Definition merge_arrays
(s1 s2 : seq bool) (i : nat)
(w_ok1 : w ^ 2 %/ 2 == size s1)
(w_ok2 : w ^ 2 %/ 2 == size s2)
(Hi : i < size s1 + size s2) :
{ tr : tree w (size s1 + size s2 - 1)
(count_one s1 + count_one s2 - access (s1 ++ s2) i) 0 Black |
dflatten tr = delete (s1 ++ s2) i } :=
if i < size s1 is true
then Leaf _ ((rcons (delete s1 i) (access s2 0)) ++ (delete s2 0)) _ _
else Leaf _ (s1 ++ (delete s2 (i - size s1))) _ _.
Next Obligation.
rewrite size_cat size_rcons.
case Hi': (i < size s1).
+ rewrite !size_delete. rewrite -(eqP w_ok1) -(eqP w_ok2).
rewrite prednK. exact: leq_addr. exact: wordsize_sqrn_div2_gt0.
by rewrite -(eqP w_ok2) wordsize_sqrn_div2_gt0. by rewrite Hi'.
+ rewrite delete_oversize //=. rewrite size_delete.
rewrite -[X in _ <= X + _]addn1 -(eqP w_ok1). rewrite addnAC -addnA.
apply: leq_addr. by rewrite -(eqP w_ok2) wordsize_sqrn_div2_gt0.
by rewrite leqNgt Hi'.
Qed.
Next Obligation.
rewrite size_cat size_rcons.
case Hi': (i < size s1).
+ rewrite !size_delete. rewrite -(eqP w_ok1) -(eqP w_ok2).
rewrite prednK. apply: wordsize_del_lt_twice.
apply: wordsize_sqrn_div2_gt0.
by rewrite -(eqP w_ok2) wordsize_sqrn_div2_gt0.
by rewrite Hi'.
+ rewrite delete_oversize. rewrite size_delete.
rewrite -(eqP w_ok1) -(eqP w_ok2).
rewrite addSnnS prednK.
by rewrite addnn mul2n ltn_double ltn_Pdiv //= wordsize_sqrn_gt0.
apply: wordsize_sqrn_div2_gt0. by rewrite -(eqP w_ok2) wordsize_sqrn_div2_gt0.
by rewrite leqNgt Hi'.
Qed.
Next Obligation.
rewrite size_cat size_rcons !size_delete ?prednK.
by rewrite -subn1 addnBA //= -(eqP w_ok2) wordsize_sqrn_div2_gt0.
by rewrite -(eqP w_ok1) wordsize_sqrn_div2_gt0.
by rewrite -?(eqP w_ok2) wordsize_sqrn_div2_gt0.
by rewrite -Heq_anonymous.
Qed.
Next Obligation.
rewrite /count_one. rewrite -[in LHS]cats1 count_cat count_cat.
rewrite -/count_one -!count_delete.
rewrite /access nth_cat -Heq_anonymous //= addn0.
rewrite -addnA eqb_id subnKC.
by rewrite addnC [in RHS]addnC addnBA //= -/access access_leq_count.
have Hmatch: (match s2 with
| []%list => false
| x :: _ => x
end = access s2 0). by rewrite /access.
by rewrite Hmatch access_leq_count //= -(eqP w_ok2) wordsize_sqrn_div2_gt0.
Qed.
Next Obligation.
rewrite /eq_rect.
destruct merge_arrays_obligation_3, merge_arrays_obligation_4 => //=.
have Hmatch: (match s2 with
| []%list => false
| x :: _ => x
end = access s2 0). by rewrite /access.
rewrite Hmatch -cats1 -catA cat1s. rewrite cons_del_head.
rewrite /delete take_cat -Heq_anonymous -catA drop_cat.
case: ifP => Hi' //=.
rewrite drop_oversize //=.
have H: (i.+1 - size s1 == 0). by rewrite subn_eq0 -Heq_anonymous.
by rewrite (eqP H) drop0.
move/negbT: Hi'. by rewrite -leqNgt => ->.
by rewrite -(eqP w_ok2) wordsize_sqrn_div2_gt0.
Qed.
Next Obligation.
by rewrite size_cat -(eqP w_ok1) leq_addr.
Qed.
Next Obligation.
rewrite size_cat. rewrite size_delete.
rewrite -(eqP w_ok1) -(eqP w_ok2).
apply: wordsize_del_lt_twice.
rewrite -(ltn_add2r (size s1)) subnK. by rewrite addnC.
move/eqP/eqnP/eqP: H. by rewrite subn_eq0 -leqNgt.
Qed.
Next Obligation.
rewrite size_cat. rewrite size_delete.
by rewrite -subn1 addnBA //= -(eqP w_ok2) wordsize_sqrn_div2_gt0.
rewrite -(ltn_add2r (size s1)) subnK. by rewrite addnC.
move/eqP/eqnP/eqP: H. by rewrite subn_eq0 -leqNgt.
Qed.
Next Obligation.
rewrite /count_one count_cat -/count_one. rewrite -count_delete.
rewrite /access. rewrite nth_cat.
case: ifP => H' //=. move: H. by rewrite H'.
rewrite addnBA //= -/access access_leq_count //=.
rewrite -(ltn_add2r (size s1)) subnK. by rewrite addnC.
move/eqP/eqnP/eqP: H. by rewrite subn_eq0 -leqNgt.
Qed.
Next Obligation.
rewrite /eq_rect.
destruct merge_arrays_obligation_9, merge_arrays_obligation_8 => //=.
rewrite /delete take_cat.
move/eqP/eqnP/eqP: H. rewrite subn_eq0. move/negbTE => H.
rewrite H //=. rewrite drop_cat.
move/negbT: (H). rewrite -ltnNge => H'.
rewrite ltnNge leq_eqVlt H' orbT //= subSn.
rewrite catA //=. by rewrite leqNgt H.
Qed.
Definition delete_last (s : seq bool) := delete s (size s).-1.
Definition blast (s : seq bool) := access s (size s).-1.
Program Definition delete_from_leaves {lnum lones rnum rones : nat}
(pc : color)
(l : tree w lnum lones 0 Black)
(r : tree w rnum rones 0 Black)
(i : nat) :
{ B' : del_tree (lnum + rnum - (i < lnum + rnum))
(lones + rones - access (dflatten l ++ dflatten r) i)
(incr_black 0 pc) pc
| dflattend B' = delete (dflatten l ++ dflatten r) i} :=
match l with
| Leaf arr1 _ _ =>
match r with
| Leaf arr2 _ _ =>
if i < size arr1 is true
then if w ^ 2 %/ 2 == size arr1 is true
then if w ^ 2 %/ 2 == size arr2 is true
then
let ret := (` (merge_arrays arr1 arr2 i _ _ _)) in
match pc with
| Red => Stay Red _ ret
| Black => Down ret
end
else
match pc with
| Red => Stay Red _ (rnode
(Leaf _ (rcons (delete arr1 i)
(access arr2 0)) _ _)
(Leaf _ (delete arr2 0) _ _))
| Black => Stay Black _ (bnode
(Leaf _ (rcons (delete arr1 i)
(access arr2 0)) _ _)
(Leaf _ (delete arr2 0) _ _))
end
else
match pc with
| Red => Stay Red _ (rnode
(Leaf _ (delete arr1 i) _ _)
(Leaf _ arr2 _ _))
| Black => Stay Black _ (bnode
(Leaf _ (delete arr1 i) _ _)
(Leaf _ arr2 _ _))
end
else
if i < size arr1 + size arr2 is true
then if w ^ 2 %/ 2 == size arr2 is true
then if w ^ 2 %/ 2 == size arr1 is true
then
let ret := (` (merge_arrays arr1 arr2 i _ _ _)) in
match pc with
| Red => Stay Red _ ret
| Black => Down ret
end
else
match pc with
| Red => Stay Red _
(rnode (Leaf _ (delete_last arr1) _ _)
(Leaf _ ((blast arr1) ::
(delete arr2 (i - size arr1)))
_ _))
| Black => Stay Black _
(bnode (Leaf _ (delete_last arr1) _ _)
(Leaf _ ((blast arr1) ::
(delete arr2 (i - size arr1)))
_ _))
end
else
match pc with
| Red => Stay Red _ (rnode (Leaf _ arr1 _ _)
(Leaf _ (delete arr2 (i - size arr1)) _ _))
| Black => Stay Black _ (bnode (Leaf _ arr1 _ _)
(Leaf _ (delete arr2 (i - size arr1)) _ _))
end
else
match pc with
| Red => Stay Red _ (rnode (Leaf _ arr1 _ _) (Leaf _ arr2 _ _))
| Black => Stay Black _ (bnode (Leaf _ arr1 _ _) (Leaf _ arr2 _ _))
end
| Node _ _ _ _ _ _ _ _ _ _ _ _ => _
end
| Node _ _ _ _ _ _ _ _ _ _ _ _ => _
end.
Next Obligation.
have H: i < size arr1. by rewrite -Heq_anonymous4.
apply: leq_trans. apply: H. apply: leq_addr.
Qed.
Next Obligation.
have H: i < size arr1. by rewrite -Heq_anonymous4.
have H': (i < size arr1 + size arr2).
apply: leq_trans. apply: H. apply: leq_addr. by rewrite H'.
Qed.
Next Obligation.
by rewrite /access !nth_cat -Heq_l -Heq_anonymous4.
Qed.
Next Obligation.
rewrite /eq_rect.
destruct delete_from_leaves_obligation_6, delete_from_leaves_obligation_5 => //=.
destruct merge_arrays. by rewrite /proj1_sig //= -Heq_l -Heq_r.
Qed.
Next Obligation.
have H: i < size arr1. by rewrite -Heq_anonymous4.
have H': (i < size arr1 + size arr2).
apply: leq_trans. apply: H. apply: leq_addr. by rewrite H'.
Qed.
Next Obligation.
by rewrite /access !nth_cat -Heq_l -Heq_anonymous4.
Qed.
Next Obligation.
rewrite /eq_rect.
destruct delete_from_leaves_obligation_9, delete_from_leaves_obligation_8 => //=.
destruct merge_arrays. by rewrite /proj1_sig //= -Heq_l -Heq_r.
Qed.
Next Obligation.
rewrite -cats1 size_cat //= size_delete. rewrite addn1 prednK //=.
move/eqP/eqnP: Heq_anonymous5 => <-. apply: wordsize_sqrn_div2_gt0.
by rewrite -Heq_anonymous4.
Qed.
Next Obligation.
rewrite -cats1 size_cat //= size_delete. rewrite addn1 prednK //=.
move/eqP/eqnP: Heq_anonymous5 => <-. apply: wordsize_sqrn_div2_gt0.
by rewrite -Heq_anonymous4.
Qed.
Next Obligation.
move: (wildcard'3). rewrite leq_eqVlt.
move/eqP/eqnP/eqP/negbTE: H => -> //=.
have Harr2: 0 < size arr2.
apply: leq_trans. apply: wordsize_sqrn_div2_gt0. exact.
rewrite size_delete //=. move => Hw.
by rewrite -(leq_add2r 1) !addn1 prednK.
Qed.
Next Obligation.
have Harr2: 0 < size arr2.
apply: leq_trans. apply: wordsize_sqrn_div2_gt0. exact.
rewrite size_delete //=. apply: leq_ltn_trans.
apply: leq_pred. exact.
Qed.
Next Obligation.
have Harr2: 0 < size arr2.
apply: leq_trans. apply: wordsize_sqrn_div2_gt0. exact.
rewrite -cats1 size_cat //= !size_delete //= addn1 prednK //=.
have H': (i < size arr1 + size arr2).
move/eqP/eqnP/eqP: Heq_anonymous4 => H'.
apply: leq_trans. apply: H'. apply: leq_addr.
rewrite H' -subn1 addnBA //=.
move/eqP/eqnP: Heq_anonymous5 => <-. apply: wordsize_sqrn_div2_gt0.
Qed.
Next Obligation.
rewrite -cats1. rewrite /count_one count_cat /=.
have Hmatch: (match arr2 with
| []%list => false
| x :: _ => x
end = access arr2 0). by rewrite /access.
rewrite /access nth_cat -Heq_l /= -Heq_anonymous4 Hmatch.
rewrite eqb_id /access addn0.
rewrite [in RHS]addnC -/access -/count_one.
rewrite -addnBA. rewrite count_delete.
rewrite -[X in _ + _ + X = _]count_delete.
rewrite -addnA subnKC. by rewrite addnC.
apply: access_leq_count. apply: leq_trans.
apply: wordsize_sqrn_div2_gt0. exact.
apply: access_leq_count. by rewrite -Heq_anonymous4.
Qed.
Next Obligation.
rewrite /eq_rect.
destruct delete_from_leaves_obligation_17, delete_from_leaves_obligation_16 => //=.
rewrite -cats1 delete_catL -Heq_l //= -Heq_r //= -catA.
destruct arr2 => //=.
have Habsurd: (~~ ((w ^ 2 %/ 2) <= 0)).
rewrite -ltnNge. apply: wordsize_sqrn_div2_gt0.
move: (wildcard'3) => //=. move/negbTE: Habsurd => -> //=.
by rewrite /delete /= drop0.
Qed.
Next Obligation.
rewrite -cats1 size_cat //= size_delete //= addn1 prednK //=.
move/eqP/eqnP: Heq_anonymous5 => <-. apply: wordsize_sqrn_div2_gt0.
Qed.
Next Obligation.
rewrite -cats1 size_cat //= size_delete //= addn1 prednK //=.
move/eqP/eqnP: Heq_anonymous5 => <-. apply: wordsize_sqrn_div2_gt0.
Qed.
Next Obligation.
move: (wildcard'3). rewrite leq_eqVlt.
move/eqP/eqnP/eqP/negbTE: H => -> //=.
have Harr2: 0 < size arr2.
apply: leq_trans. apply: wordsize_sqrn_div2_gt0. exact.
rewrite size_delete //=. move => Hw.
by rewrite -(leq_add2r 1) !addn1 prednK.
Qed.
Next Obligation.
rewrite size_delete. apply: leq_ltn_trans. apply: leq_pred. exact.
apply: leq_trans. apply: wordsize_sqrn_div2_gt0. exact.
Qed.
Next Obligation.
have H': i < size arr1. by rewrite -Heq_anonymous4.
have H1: 0 < size arr1.
move/eqP/eqnP: Heq_anonymous5 => <-. apply: wordsize_sqrn_div2_gt0.
have H2: 0 < size arr2. apply: leq_trans. apply: wordsize_sqrn_div2_gt0. exact.
rewrite -cats1 size_cat //= !size_delete //= addn1 prednK //=.
have H'': (i < size arr1 + size arr2).
apply: leq_trans. apply: H'. apply: leq_addr.
by rewrite H'' -subn1 //= addnBA.
Qed.
Next Obligation.
rewrite -cats1. rewrite /count_one count_cat /=.
have Hmatch: (match arr2 with
| []%list => false
| x :: _ => x
end = access arr2 0). by rewrite /access.
rewrite /access nth_cat -Heq_l /= -Heq_anonymous4 Hmatch.
rewrite eqb_id /access addn0.
rewrite [in RHS]addnC -/access -/count_one.
rewrite -addnBA. rewrite count_delete.
rewrite -[X in _ + _ + X = _]count_delete.
rewrite -addnA subnKC. by rewrite addnC.
apply: access_leq_count. apply: leq_trans.
apply: wordsize_sqrn_div2_gt0. exact.
apply: access_leq_count. by rewrite -Heq_anonymous4.
Qed.
Next Obligation.
rewrite /eq_rect.
destruct delete_from_leaves_obligation_25, delete_from_leaves_obligation_24 => //=.
rewrite -cats1 delete_catL -Heq_l //= -Heq_r //= -catA.
destruct arr2 => //=.
have Habsurd: (~~ ((w ^ 2 %/ 2) <= 0)).
rewrite -ltnNge. apply: wordsize_sqrn_div2_gt0.
move: (wildcard'3) => //=. move/negbTE: Habsurd => -> //=.
by rewrite /delete /= drop0.
Qed.
Next Obligation.
move: (wildcard'1). rewrite leq_eqVlt.
move/eqP/eqnP/eqP/negbTE: H => -> //=.
have Harr2: 0 < size arr1.
apply: leq_trans. apply: wordsize_sqrn_div2_gt0. exact.