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3805fe5 bff4ec7 3805fe5 bff4ec7 3805fe5 bff4ec7 3805fe5 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 | Require Export PVbench.Algorithms.maximum_subarray.rocq.helper_lib.
Require Import Coq.ZArith.ZArith.
Require Import Coq.Lists.List.
From AUXLib Require Import ListLib.
From MaxMinLib Require Import MaxMin Interface.
Import ListNotations.
Local Open Scope Z_scope.
Local Open Scope list_scope.
Definition SubarraySum (l : list Z) (lo hi s : Z) : Prop :=
0 <= lo /\ lo < hi /\ hi <= Zlength l /\ s = sum (sublist lo hi l).
Require Import Coq.micromega.Lia.
Require Import SetsClass.SetsClass.
Import SetsNotation.
Lemma sum_sublist_single :
forall (l : list Z) (n : Z),
0 <= n < Zlength l ->
sum (sublist n (n + 1) l) = Znth n l 0.
Proof.
intros l n Hn.
rewrite (sublist_single 0 n l) by lia.
simpl; lia.
Qed.
Lemma sum_sublist_snoc :
forall (l : list Z) (lo hi : Z),
0 <= lo <= hi ->
hi + 1 <= Zlength l ->
sum (sublist lo (hi + 1) l) =
sum (sublist lo hi l) + Znth hi l 0.
Proof.
intros l lo hi Hlo Hhi.
rewrite (sublist_split lo (hi + 1) hi l) by lia.
rewrite sum_app.
rewrite sum_sublist_single by lia.
lia.
Qed.
Lemma MaxSuffixSumPrefix_single :
forall (l : list Z),
1 <= Zlength l ->
MaxSuffixSumPrefix l 1 (Znth 0 l 0).
Proof.
intros l Hlen.
unfold MaxSuffixSumPrefix.
replace (Znth 0 l 0) with (sum (sublist 0 1 l)).
- eapply (@max_1 Z Z.le Zle_TotalOrder Z 0).
intros lo; split; intros Hlo; lia.
- rewrite sum_sublist_single by lia. reflexivity.
Qed.
Lemma MaxSubarraySumPrefix_single :
forall (l : list Z),
1 <= Zlength l ->
MaxSubarraySumPrefix l 1 (Znth 0 l 0).
Proof.
intros l Hlen.
unfold MaxSubarraySumPrefix.
replace (Znth 0 l 0) with (sum (sublist 0 1 l)).
- eapply (@max_1 Z Z.le Zle_TotalOrder (Z * Z)%type (0, 1)).
intros [lo hi]; simpl; split; intros H.
+ destruct H as [? [? ?]].
f_equal; lia.
+ inversion H; subst; lia.
- rewrite sum_sublist_single by lia. reflexivity.
Qed.
Lemma MaxSuffixSumPrefix_step :
forall (l : list Z) (i cur : Z),
0 <= i ->
i + 1 <= Zlength l ->
MaxSuffixSumPrefix l i cur ->
MaxSuffixSumPrefix l (i + 1)
(max_Z (Znth i l 0) (cur + Znth i l 0)).
Proof.
intros l i cur Hi Hlen Hcur.
unfold MaxSuffixSumPrefix in *.
destruct Hcur as [a [[Ha_in Ha_max] Ha_eq]].
simpl in Ha_in, Ha_max.
sets_unfold in Ha_in.
sets_unfold in Ha_max.
set (x := Znth i l 0).
destruct (Z_le_dec x (cur + x)) as [Hle | Hnle].
- exists a. split.
+ unfold max_object_of_subset; simpl.
sets_unfold.
split; [lia |].
intros b Hb.
simpl in Hb.
destruct (Z.eq_dec b i) as [-> | Hbneq].
* rewrite sum_sublist_single by lia.
rewrite (sum_sublist_snoc l a i) by lia.
rewrite Ha_eq.
unfold x in *; lia.
* assert (Hb_old : 0 <= b /\ b < i) by lia.
rewrite sum_sublist_snoc by lia.
rewrite (sum_sublist_snoc l a i) by lia.
specialize (Ha_max b Hb_old).
unfold x in *; lia.
+ rewrite sum_sublist_snoc by lia.
rewrite Ha_eq.
unfold max_Z. rewrite Z.max_r by lia. reflexivity.
- exists i. split.
+ unfold max_object_of_subset; simpl.
sets_unfold.
split; [lia |].
intros b Hb.
simpl in Hb.
destruct (Z.eq_dec b i) as [-> | Hbneq].
* rewrite sum_sublist_single by lia. lia.
* assert (Hb_old : 0 <= b /\ b < i) by lia.
rewrite sum_sublist_snoc by lia.
rewrite sum_sublist_single by lia.
specialize (Ha_max b Hb_old).
rewrite Ha_eq in Ha_max.
unfold x in *.
lia.
+ rewrite sum_sublist_single by lia.
unfold max_Z, x. rewrite Z.max_l by lia. reflexivity.
Qed.
Lemma MaxSubarraySumPrefix_step :
forall (l : list Z) (i res cur : Z),
MaxSubarraySumPrefix l i res ->
MaxSuffixSumPrefix l (i + 1) cur ->
MaxSubarraySumPrefix l (i + 1) (max_Z res cur).
Proof.
intros l i res cur Hres Hcur.
unfold MaxSubarraySumPrefix in *.
unfold MaxSuffixSumPrefix in Hcur.
destruct Hres as [[lo hi] [[Hobj_in Hobj_max] Hobj_eq]].
destruct Hcur as [s [[Hs_in Hs_max] Hs_eq]].
simpl in Hobj_in, Hobj_max, Hs_in, Hs_max.
sets_unfold in Hobj_in.
sets_unfold in Hobj_max.
sets_unfold in Hs_in.
sets_unfold in Hs_max.
destruct (Z_le_dec cur res) as [Hle | Hnle].
- exists (lo, hi). split.
+ unfold max_object_of_subset; simpl.
sets_unfold.
split; [lia |].
intros [blo bhi] Hb.
simpl in Hb.
destruct (Z.eq_dec bhi (i + 1)) as [Hbhi | Hbhi].
* subst bhi.
specialize (Hs_max blo ltac:(lia)).
rewrite Hs_eq in Hs_max.
rewrite Hobj_eq.
lia.
* assert (Hb_old : 0 <= blo /\ blo < bhi /\ bhi <= i) by lia.
specialize (Hobj_max (blo, bhi) Hb_old).
exact Hobj_max.
+ rewrite Hobj_eq.
unfold max_Z. rewrite Z.max_l by lia. reflexivity.
- exists (s, i + 1). split.
+ unfold max_object_of_subset; simpl.
sets_unfold.
split; [lia |].
intros [blo bhi] Hb.
simpl in Hb.
destruct (Z.eq_dec bhi (i + 1)) as [Hbhi | Hbhi].
* subst bhi.
specialize (Hs_max blo ltac:(lia)).
exact Hs_max.
* assert (Hb_old : 0 <= blo /\ blo < bhi /\ bhi <= i) by lia.
specialize (Hobj_max (blo, bhi) Hb_old).
rewrite Hobj_eq in Hobj_max.
rewrite Hs_eq.
lia.
+ rewrite Hs_eq.
unfold max_Z. rewrite Z.max_r by lia. reflexivity.
Qed.
Lemma sum_upper_bound :
forall (l : list Z) (b : Z),
0 <= b ->
(forall i : Z, 0 <= i < Zlength l -> Znth i l 0 <= b) ->
sum l <= Zlength l * b.
Proof.
induction l as [| a l IH]; intros b Hb Hbound.
- simpl. change (Zlength (@nil Z)) with 0. lia.
- simpl.
rewrite Zlength_cons.
assert (Ha : a <= b).
{
specialize (Hbound 0).
rewrite Znth0_cons in Hbound.
apply Hbound.
rewrite Zlength_cons.
pose proof (Zlength_nonneg l); lia.
}
assert (Htail : forall i : Z, 0 <= i < Zlength l -> Znth i l 0 <= b).
{
intros i Hi.
specialize (Hbound (i + 1)).
rewrite Znth_cons in Hbound by lia.
replace (i + 1 - 1) with i in Hbound by lia.
apply Hbound.
rewrite Zlength_cons.
pose proof (Zlength_nonneg l); lia.
}
specialize (IH b Hb Htail).
lia.
Qed.
Lemma sum_sublist_upper_bound :
forall (l : list Z) (lo hi n b : Z),
0 <= b ->
0 <= lo <= hi ->
hi <= n ->
n <= Zlength l ->
(forall k : Z, 0 <= k < n -> Znth k l 0 <= b) ->
sum (sublist lo hi l) <= (hi - lo) * b.
Proof.
intros l lo hi n b Hb Hlo Hhi Hn Hbound.
eapply Z.le_trans with (m := Zlength (sublist lo hi l) * b).
- apply sum_upper_bound; [lia |].
intros j Hj.
assert (Hj' : 0 <= j < hi - lo).
{
rewrite Zlength_sublist in Hj by lia.
lia.
}
rewrite Znth_sublist by lia.
apply Hbound; lia.
- rewrite Zlength_sublist by lia.
lia.
Qed.
Lemma MaxSuffixSumPrefix_upper_bound :
forall (l : list Z) (i ans n : Z),
0 <= i <= n ->
n <= Zlength l ->
n <= 100000 ->
(forall k : Z, 0 <= k < n -> Znth k l 0 <= 10000) ->
MaxSuffixSumPrefix l i ans ->
ans <= 1000000000.
Proof.
intros l i ans n Hi Hn Hn_bound Hbound Hmax.
unfold MaxSuffixSumPrefix in Hmax.
destruct Hmax as [lo [[Hlo_in _] Hsum]].
simpl in Hlo_in.
sets_unfold in Hlo_in.
rewrite <- Hsum.
eapply Z.le_trans.
- eapply sum_sublist_upper_bound with (n := n) (b := 10000); try lia.
exact Hbound.
- lia.
Qed. |