diff options
Diffstat (limited to 'mathcomp/ssreflect')
| -rw-r--r-- | mathcomp/ssreflect/seq.v | 99 |
1 files changed, 80 insertions, 19 deletions
diff --git a/mathcomp/ssreflect/seq.v b/mathcomp/ssreflect/seq.v index bb462a3..9747171 100644 --- a/mathcomp/ssreflect/seq.v +++ b/mathcomp/ssreflect/seq.v @@ -1271,6 +1271,17 @@ elim: s => //= y s IHs /andP[/negbTE s'y /IHs-> {IHs}]. by rewrite in_cons; case: (eqVneq y x) => // <-; rewrite s'y. Qed. +Lemma leq_uniq_countP x s1 s2 : uniq s1 -> + reflect (x \in s1 -> x \in s2) (count_mem x s1 <= count_mem x s2). +Proof. +move/count_uniq_mem->; case: (boolP (_ \in _)) => //= _; last by constructor. +by rewrite -has_pred1 has_count; apply: (iffP idP) => //; apply. +Qed. + +Lemma leq_uniq_count s1 s2 : uniq s1 -> {subset s1 <= s2} -> + (forall x, count_mem x s1 <= count_mem x s2). +Proof. by move=> s1_uniq s1_s2 x; apply/leq_uniq_countP/s1_s2. Qed. + Lemma filter_pred1_uniq s x : uniq s -> x \in s -> filter (pred1 x) s = [:: x]. Proof. move=> uniq_s s_x; rewrite (all_pred1P _ _ (filter_all _ _)). @@ -1395,6 +1406,9 @@ Proof. by move=> x; rewrite -[s in RHS](cat_take_drop n0) !mem_cat /= orbC. Qed. Lemma eqseq_rot s1 s2 : (rot n0 s1 == rot n0 s2) = (s1 == s2). Proof. exact/inj_eq/rot_inj. Qed. +Lemma drop_index s (n := index x0 s) : x0 \in s -> drop n s = x0 :: drop n.+1 s. +Proof. by move=> xs; rewrite (drop_nth x0) ?index_mem ?nth_index. Qed. + (* lemmas about the pivot pattern [_ ++ _ :: _] *) Lemma index_pivot x s1 s2 (s := s1 ++ x :: s2) : x \notin s1 -> @@ -1454,9 +1468,7 @@ Implicit Types x y z : T. Lemma rot_index s x (i := index x s) : x \in s -> rot i s = x :: (drop i.+1 s ++ take i s). -Proof. -by move=> x_s; rewrite /rot (drop_nth x) ?index_mem ?nth_index// cat_cons. -Qed. +Proof. by move=> x_s; rewrite /rot drop_index. Qed. Variant rot_to_spec s x := RotToSpec i s' of rot i s = x :: s'. @@ -2203,16 +2215,19 @@ Variables (T : eqType) (x : T). Fixpoint rem s := if s is y :: t then (if y == x then t else y :: rem t) else s. +Lemma rem_cons y s : rem (y :: s) = if y == x then s else y :: rem s. +Proof. by []. Qed. + +Lemma remE s : rem s = take (index x s) s ++ drop (index x s).+1 s. +Proof. by elim: s => //= y s ->; case: eqVneq; rewrite ?drop0. Qed. + Lemma rem_id s : x \notin s -> rem s = s. -Proof. -by elim: s => //= y s IHs /norP[neq_yx /IHs->]; rewrite eq_sym (negbTE neq_yx). -Qed. +Proof. by elim: s => //= y s IHs /norP[neq_yx /IHs->]; case: eqVneq neq_yx. Qed. Lemma perm_to_rem s : x \in s -> perm_eq s (x :: rem s). Proof. -elim: s => // y s IHs; rewrite inE /= eq_sym perm_sym. -case: eqP => [-> // | _ /IHs]. -by rewrite (perm_catCA [:: x] [:: y]) perm_cons perm_sym. +move=> xs; rewrite remE -[X in perm_eq X](cat_take_drop (index x s)). +by rewrite drop_index// -cat1s perm_catCA cat1s. Qed. Lemma size_rem s : x \in s -> size (rem s) = (size s).-1. @@ -2242,6 +2257,18 @@ Proof. by move/rem_filter=> -> y; rewrite mem_filter. Qed. Lemma mem_rem_uniqF s : uniq s -> x \in rem s = false. Proof. by move/mem_rem_uniq->; rewrite inE eqxx. Qed. +Lemma count_rem P s : count P (rem s) = count P s - (x \in s) && P x. +Proof. +have [/perm_to_rem/permP->|xNs]/= := boolP (x \in s); first by rewrite addKn. +by rewrite subn0 rem_id. +Qed. + +Lemma count_mem_rem y s : count_mem y (rem s) = count_mem y s - (x == y). +Proof. +rewrite count_rem; have []//= := boolP (x \in s). +by case: eqP => // <- /count_memPn->. +Qed. + End Rem. Section Map. @@ -2347,8 +2374,18 @@ Notation "[ 'seq' E : R | i : T <- s & C ]" := Lemma filter_mask T a (s : seq T) : filter a s = mask (map a s) s. Proof. by elim: s => //= x s <-; case: (a x). Qed. -Lemma mask_filter (T : eqType) (s : seq T) (m : bitseq) : - uniq s -> mask m s = [seq i <- s | i \in mask m s]. +Section MiscMask. + +Lemma leq_count_mask T (P : {pred T}) m s : count P (mask m s) <= count P s. +Proof. +by elim: s m => [|x s IHs] [|[] m]//=; + rewrite ?leq_add2l (leq_trans (IHs _)) ?leq_addl. +Qed. + +Variable (T : eqType). +Implicit Types (s : seq T) (m : bitseq). + +Lemma mask_filter s m : uniq s -> mask m s = [seq i <- s | i \in mask m s]. Proof. elim: m s => [|[] m ih] [|x s] //=. - by move=> _; elim: s. @@ -2357,6 +2394,33 @@ elim: m s => [|[] m ih] [|x s] //=. - by case: ifP => [/mem_mask -> // | _ /andP [] _ /ih]. Qed. +Lemma leq_count_subseq P s1 s2 : subseq s1 s2 -> count P s1 <= count P s2. +Proof. by move=> /subseqP[m _ ->]; rewrite leq_count_mask. Qed. + +Lemma count_maskP s1 s2 : + (forall x, count_mem x s1 <= count_mem x s2) <-> + exists2 m : bitseq, size m = size s2 & perm_eq s1 (mask m s2). +Proof. +split=> [s1_le|[m _ /permP s1ms2 x]]; last by rewrite s1ms2 leq_count_mask. +suff [m mP]: exists m, perm_eq s1 (mask m s2). + by have [m' sm' eqm] := resize_mask m s2; exists m'; rewrite -?eqm. +elim: s2 => [|x s2 IHs]//= in s1 s1_le *. + by exists [::]; apply/allP => x _/=; rewrite eqn_leq s1_le. +have [y|m s1s2] := IHs (rem x s1); first by rewrite count_mem_rem leq_subLR. +exists ((x \in s1) :: m); have [|/rem_id<-//] := boolP (x \in s1). +by move/perm_to_rem/permPl->; rewrite perm_cons. +Qed. + +Lemma count_subseqP s1 s2 : + (forall x, count_mem x s1 <= count_mem x s2) <-> + exists2 s, subseq s s2 & perm_eq s1 s. +Proof. +rewrite count_maskP; split=> [[m _]|[_/subseqP[m sm ->]]]; last by exists m. +by exists (mask m s2); rewrite ?mask_subseq. +Qed. + +End MiscMask. + Section FilterSubseq. Variable T : eqType. @@ -2403,15 +2467,12 @@ case: eqP => [-> | _] /IHs[s3 perm_s2] {IHs}. by exists (rcons s3 y); rewrite -cat_cons -perm_rcons -!cats1 catA perm_cat2r. Qed. -Lemma subset_maskP s1 s2 : uniq s1 -> {subset s1 <= s2} -> - exists2 m : seq bool, size m = size s2 & perm_eq s1 (mask m s2). +Lemma subseq_rem x : {homo rem x : s1 s2 / @subseq T s1 s2}. Proof. -move=> s1_uniq sub_s1_s2; pose s1' := [seq x <- undup s2 | x \in s1]. -have /subseqP[m sm s1'_eq] : subseq s1' s2. - by apply: subseq_trans (undup_subseq _); apply: filter_subseq. -exists m; rewrite // -s1'_eq; apply: uniq_perm => // [|x]. - by rewrite filter_uniq ?undup_uniq. -by rewrite mem_filter mem_undup; have [/sub_s1_s2|] := boolP (x \in s1). +move=> s1 s2; elim: s2 s1 => [|x2 s2 IHs2] [|x1 s1]; rewrite ?sub0seq //=. +have [->|_] := eqVneq x1 x2; first by case: eqP => //= _ /IHs2; rewrite eqxx. +move=> /IHs2/subseq_trans->//. +by have [->|_] := eqVneq x x2; [apply: rem_subseq|apply: subseq_cons]. Qed. End FilterSubseq. |