statement stringlengths 1 4.33k | proof stringlengths 0 37.9k | type stringclasses 25
values | symbolic_name stringlengths 1 67 | library stringclasses 10
values | filename stringclasses 112
values | imports listlengths 2 138 | deps listlengths 0 64 | docstring stringclasses 798
values | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
set_set_nth s n1 y1 n2 y2 (s2 := set_nth s n2 y2) :
set_nth (set_nth s n1 y1) n2 y2 = if n1 == n2 then s2 else set_nth s2 n1 y1. | Proof.
have [-> | ne_n12] := eqVneq.
apply: eq_from_nth => [|i _]; first by rewrite !size_set_nth maxnA maxnn.
by do 2!rewrite !nth_set_nth /=; case: eqP.
apply: eq_from_nth => [|i _]; first by rewrite !size_set_nth maxnCA.
by do 2!rewrite !nth_set_nth /=; case: eqP => // ->; case: eqVneq ne_n12.
Qed. | Lemma | set_set_nth | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"eqVneq",
"eq_from_nth",
"maxnA",
"maxnCA",
"maxnn",
"nth_set_nth",
"s2",
"set_nth",
"size_set_nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
find s | := if s is x :: s' then if a x then 0 else (find s').+1 else 0. | Fixpoint | find | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
filter s | :=
if s is x :: s' then if a x then x :: filter s' else filter s' else [::]. | Fixpoint | filter | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
count s | := if s is x :: s' then a x + count s' else 0. | Fixpoint | count | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has s | := if s is x :: s' then a x || has s' else false. | Fixpoint | has | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all s | := if s is x :: s' then a x && all s' else true. | Fixpoint | all | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_filter s : size (filter s) = count s. | Proof. by elim: s => //= x s <-; case (a x). Qed. | Lemma | size_filter | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"count",
"filter",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_count s : has s = (0 < count s). | Proof. by elim: s => //= x s ->; case (a x). Qed. | Lemma | has_count | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"count",
"has"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_filter_gt0 s : (size (filter s) > 0) = (has s). | Proof. by rewrite size_filter -has_count. Qed. | Lemma | size_filter_gt0 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"filter",
"has",
"has_count",
"size",
"size_filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
count_size s : count s <= size s. | Proof. by elim: s => //= x s; case: (a x); last apply: leqW. Qed. | Lemma | count_size | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"count",
"last",
"leqW",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_count s : all s = (count s == size s). | Proof.
elim: s => //= x s; case: (a x) => _ //=.
by rewrite add0n eqn_leq andbC ltnNge count_size.
Qed. | Lemma | all_count | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"add0n",
"all",
"count",
"count_size",
"eqn_leq",
"ltnNge",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
filter_all s : all (filter s). | Proof. by elim: s => //= x s IHs; case: ifP => //= ->. Qed. | Lemma | filter_all | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_filterP s : reflect (filter s = s) (all s). | Proof.
apply: (iffP idP) => [| <-]; last exact: filter_all.
by elim: s => //= x s IHs /andP[-> Hs]; rewrite IHs.
Qed. | Lemma | all_filterP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"apply",
"filter",
"filter_all",
"last"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
filter_id s : filter (filter s) = filter s. | Proof. by apply/all_filterP; apply: filter_all. Qed. | Lemma | filter_id | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all_filterP",
"apply",
"filter",
"filter_all"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_find s : has s = (find s < size s). | Proof. by elim: s => //= x s IHs; case (a x); rewrite ?leqnn. Qed. | Lemma | has_find | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"find",
"has",
"leqnn",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
find_size s : find s <= size s. | Proof. by elim: s => //= x s IHs; case (a x). Qed. | Lemma | find_size | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"find",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
find_cat s1 s2 :
find (s1 ++ s2) = if has s1 then find s1 else size s1 + find s2. | Proof.
by elim: s1 => //= x s1 IHs; case: (a x) => //; rewrite IHs (fun_if succn).
Qed. | Lemma | find_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"find",
"has",
"s1",
"s2",
"size",
"succn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_nil : has [::] = false. | Proof. by []. Qed. | Lemma | has_nil | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"has"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_seq1 x : has [:: x] = a x. | Proof. exact: orbF. Qed. | Lemma | has_seq1 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"has"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_nseq n x : has (nseq n x) = (0 < n) && a x. | Proof. by elim: n => //= n ->; apply: andKb. Qed. | Lemma | has_nseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"andKb",
"apply",
"has",
"nseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_seqb (b : bool) x : has (nseq b x) = b && a x. | Proof. by rewrite has_nseq lt0b. Qed. | Lemma | has_seqb | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"has",
"has_nseq",
"lt0b",
"nseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_nil : all [::] = true. | Proof. by []. Qed. | Lemma | all_nil | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_seq1 x : all [:: x] = a x. | Proof. exact: andbT. Qed. | Lemma | all_seq1 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_nseq n x : all (nseq n x) = (n == 0) || a x. | Proof. by elim: n => //= n ->; apply: orKb. Qed. | Lemma | all_nseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"apply",
"nseq",
"orKb"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_nseqb (b : bool) x : all (nseq b x) = b ==> a x. | Proof. by rewrite all_nseq eqb0 implybE. Qed. | Lemma | all_nseqb | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"all_nseq",
"eqb0",
"nseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
filter_nseq n x : filter (nseq n x) = nseq (a x * n) x. | Proof. by elim: n => /= [|n ->]; case: (a x). Qed. | Lemma | filter_nseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"filter",
"nseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
count_nseq n x : count (nseq n x) = a x * n. | Proof. by rewrite -size_filter filter_nseq size_nseq. Qed. | Lemma | count_nseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"count",
"filter_nseq",
"nseq",
"size_filter",
"size_nseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
find_nseq n x : find (nseq n x) = ~~ a x * n. | Proof. by elim: n => /= [|n ->]; case: (a x). Qed. | Lemma | find_nseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"find",
"nseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_find s : has s -> a (nth s (find s)). | Proof. by elim: s => //= x s IHs; case a_x: (a x). Qed. | Lemma | nth_find | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"find",
"has",
"nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
before_find s i : i < find s -> a (nth s i) = false. | Proof. by elim: s i => //= x s IHs; case: ifP => // a'x [|i] // /(IHs i). Qed. | Lemma | before_find | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"find",
"nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hasNfind s : ~~ has s -> find s = size s. | Proof. by rewrite has_find; case: ltngtP (find_size s). Qed. | Lemma | hasNfind | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"find",
"find_size",
"has",
"has_find",
"ltngtP",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
filter_cat s1 s2 : filter (s1 ++ s2) = filter s1 ++ filter s2. | Proof. by elim: s1 => //= x s1 ->; case (a x). Qed. | Lemma | filter_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"filter",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
filter_rcons s x :
filter (rcons s x) = if a x then rcons (filter s) x else filter s. | Proof. by rewrite -!cats1 filter_cat /=; case (a x); rewrite /= ?cats0. Qed. | Lemma | filter_rcons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cats0",
"cats1",
"filter",
"filter_cat",
"rcons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
count_cat s1 s2 : count (s1 ++ s2) = count s1 + count s2. | Proof. by rewrite -!size_filter filter_cat size_cat. Qed. | Lemma | count_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"count",
"filter_cat",
"s1",
"s2",
"size_cat",
"size_filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_cat s1 s2 : has (s1 ++ s2) = has s1 || has s2. | Proof. by elim: s1 => [|x s1 IHs] //=; rewrite IHs orbA. Qed. | Lemma | has_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"has",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_rcons s x : has (rcons s x) = a x || has s. | Proof. by rewrite -cats1 has_cat has_seq1 orbC. Qed. | Lemma | has_rcons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cats1",
"has",
"has_cat",
"has_seq1",
"rcons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_cat s1 s2 : all (s1 ++ s2) = all s1 && all s2. | Proof. by elim: s1 => [|x s1 IHs] //=; rewrite IHs andbA. Qed. | Lemma | all_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_rcons s x : all (rcons s x) = a x && all s. | Proof. by rewrite -cats1 all_cat all_seq1 andbC. Qed. | Lemma | all_rcons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"all_cat",
"all_seq1",
"cats1",
"rcons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
find_pred0 s : find pred0 s = size s. | Proof. by []. Qed. | Lemma | find_pred0 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"find",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
find_predT s : find predT s = 0. | Proof. by case: s. Qed. | Lemma | find_predT | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"find"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_find a1 a2 : a1 =1 a2 -> find a1 =1 find a2. | Proof. by move=> Ea; elim=> //= x s IHs; rewrite Ea IHs. Qed. | Lemma | eq_find | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"a1",
"a2",
"find"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_filter a1 a2 : a1 =1 a2 -> filter a1 =1 filter a2. | Proof. by move=> Ea; elim=> //= x s IHs; rewrite Ea IHs. Qed. | Lemma | eq_filter | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"a1",
"a2",
"filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_count a1 a2 : a1 =1 a2 -> count a1 =1 count a2. | Proof. by move=> Ea s; rewrite -!size_filter (eq_filter Ea). Qed. | Lemma | eq_count | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"a1",
"a2",
"count",
"eq_filter",
"size_filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_has a1 a2 : a1 =1 a2 -> has a1 =1 has a2. | Proof. by move=> Ea s; rewrite !has_count (eq_count Ea). Qed. | Lemma | eq_has | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"a1",
"a2",
"eq_count",
"has",
"has_count"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_all a1 a2 : a1 =1 a2 -> all a1 =1 all a2. | Proof. by move=> Ea s; rewrite !all_count (eq_count Ea). Qed. | Lemma | eq_all | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"a1",
"a2",
"all",
"all_count",
"eq_count"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_filter (p q : pred T) xs :
all p (filter q xs) = all [pred i | q i ==> p i] xs. | Proof. by elim: xs => //= x xs <-; case: (q x). Qed. | Lemma | all_filter | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
s12 : subpred a1 a2. | Hypothesis | s12 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"a1",
"a2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
sub_find s : find a2 s <= find a1 s. | Proof. by elim: s => //= x s IHs; case: ifP => // /(contraFF (@s12 x))->. Qed. | Lemma | sub_find | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"a1",
"a2",
"find",
"s12"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_has s : has a1 s -> has a2 s. | Proof. by rewrite !has_find; apply: leq_ltn_trans (sub_find s). Qed. | Lemma | sub_has | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"a1",
"a2",
"apply",
"has",
"has_find",
"leq_ltn_trans",
"sub_find"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_count s : count a1 s <= count a2 s. | Proof.
by elim: s => //= x s; apply: leq_add; case a1x: (a1 x); rewrite // s12.
Qed. | Lemma | sub_count | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"a1",
"a2",
"apply",
"count",
"leq_add",
"s12"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_all s : all a1 s -> all a2 s. | Proof.
by rewrite !all_count !eqn_leq !count_size => /leq_trans-> //; apply: sub_count.
Qed. | Lemma | sub_all | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"a1",
"a2",
"all",
"all_count",
"apply",
"count_size",
"eqn_leq",
"leq_trans",
"sub_count"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
filter_pred0 s : filter pred0 s = [::]. | Proof. by elim: s. Qed. | Lemma | filter_pred0 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
filter_predT s : filter predT s = s. | Proof. by elim: s => //= x s ->. Qed. | Lemma | filter_predT | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
filter_predI a1 a2 s : filter (predI a1 a2) s = filter a1 (filter a2 s). | Proof. by elim: s => //= x s ->; rewrite andbC; case: (a2 x). Qed. | Lemma | filter_predI | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"a1",
"a2",
"filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
count_pred0 s : count pred0 s = 0. | Proof. by rewrite -size_filter filter_pred0. Qed. | Lemma | count_pred0 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"count",
"filter_pred0",
"size_filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
count_predT s : count predT s = size s. | Proof. by rewrite -size_filter filter_predT. Qed. | Lemma | count_predT | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"count",
"filter_predT",
"size",
"size_filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
count_predUI a1 a2 s :
count (predU a1 a2) s + count (predI a1 a2) s = count a1 s + count a2 s. | Proof.
elim: s => //= x s IHs; rewrite /= addnACA [RHS]addnACA IHs.
by case: (a1 x) => //; rewrite addn0.
Qed. | Lemma | count_predUI | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"a1",
"a2",
"addn0",
"addnACA",
"count"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
count_predC a s : count a s + count (predC a) s = size s. | Proof. by elim: s => //= x s IHs; rewrite addnACA IHs; case: (a _). Qed. | Lemma | count_predC | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addnACA",
"count",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
count_filter a1 a2 s : count a1 (filter a2 s) = count (predI a1 a2) s. | Proof. by rewrite -!size_filter filter_predI. Qed. | Lemma | count_filter | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"a1",
"a2",
"count",
"filter",
"filter_predI",
"size_filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_pred0 s : has pred0 s = false. | Proof. by rewrite has_count count_pred0. Qed. | Lemma | has_pred0 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"count_pred0",
"has",
"has_count"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_predT s : has predT s = (0 < size s). | Proof. by rewrite has_count count_predT. Qed. | Lemma | has_predT | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"count_predT",
"has",
"has_count",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_predC a s : has (predC a) s = ~~ all a s. | Proof. by elim: s => //= x s ->; case (a x). Qed. | Lemma | has_predC | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"has"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_predU a1 a2 s : has (predU a1 a2) s = has a1 s || has a2 s. | Proof. by elim: s => //= x s ->; rewrite -!orbA; do !bool_congr. Qed. | Lemma | has_predU | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"a1",
"a2",
"has"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_pred0 s : all pred0 s = (size s == 0). | Proof. by rewrite all_count count_pred0 eq_sym. Qed. | Lemma | all_pred0 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"all_count",
"count_pred0",
"eq_sym",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_predT s : all predT s. | Proof. by rewrite all_count count_predT. Qed. | Lemma | all_predT | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"all_count",
"count_predT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
allT (a : pred T) s : (forall x, a x) -> all a s. | Proof. by move/eq_all->; apply/all_predT. Qed. | Lemma | allT | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"all_predT",
"apply",
"eq_all"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_predC a s : all (predC a) s = ~~ has a s. | Proof. by elim: s => //= x s ->; case (a x). Qed. | Lemma | all_predC | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"has"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_predI a1 a2 s : all (predI a1 a2) s = all a1 s && all a2 s. | Proof.
apply: (can_inj negbK); rewrite negb_and -!has_predC -has_predU.
by apply: eq_has => x; rewrite /= negb_and.
Qed. | Lemma | all_predI | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"a1",
"a2",
"all",
"apply",
"eq_has",
"has_predC",
"has_predU"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
drop n s {struct s} | :=
match s, n with
| _ :: s', n'.+1 => drop n' s'
| _, _ => s
end. | Fixpoint | drop | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"n'"
] | Surgery: drop, take, rot, rotr. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
drop_behead : drop n0 =1 iter n0 behead. | Proof. by elim: n0 => [|n IHn] [|x s] //; rewrite iterSr -IHn. Qed. | Lemma | drop_behead | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"behead",
"drop",
"iter",
"iterSr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
drop0 s : drop 0 s = s. | Proof. by case: s. Qed. | Lemma | drop0 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"drop"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
drop1 : drop 1 =1 behead. | Proof. by case=> [|x [|y s]]. Qed. | Lemma | drop1 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"behead",
"drop"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
drop_oversize n s : size s <= n -> drop n s = [::]. | Proof. by elim: s n => [|x s IHs] []. Qed. | Lemma | drop_oversize | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"drop",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
drop_size s : drop (size s) s = [::]. | Proof. by rewrite drop_oversize // leqnn. Qed. | Lemma | drop_size | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"drop",
"drop_oversize",
"leqnn",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
drop_cons x s :
drop n0 (x :: s) = if n0 is n.+1 then drop n s else x :: s. | Proof. by []. Qed. | Lemma | drop_cons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"drop"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_drop s : size (drop n0 s) = size s - n0. | Proof. by elim: s n0 => [|x s IHs] []. Qed. | Lemma | size_drop | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"drop",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
drop_cat s1 s2 :
drop n0 (s1 ++ s2) =
if n0 < size s1 then drop n0 s1 ++ s2 else drop (n0 - size s1) s2. | Proof. by elim: s1 n0 => [|x s1 IHs] []. Qed. | Lemma | drop_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"drop",
"s1",
"s2",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
drop_size_cat n s1 s2 : size s1 = n -> drop n (s1 ++ s2) = s2. | Proof. by move <-; elim: s1 => //=; rewrite drop0. Qed. | Lemma | drop_size_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"drop",
"drop0",
"s1",
"s2",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nconsK n x : cancel (ncons n x) (drop n). | Proof. by elim: n => // -[]. Qed. | Lemma | nconsK | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"drop",
"ncons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
drop_drop s n1 n2 : drop n1 (drop n2 s) = drop (n1 + n2) s. | Proof. by elim: s n2 => // x s ihs [|n2]; rewrite ?drop0 ?addn0 ?addnS /=. Qed. | Lemma | drop_drop | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addn0",
"addnS",
"drop",
"drop0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
take n s {struct s} | :=
match s, n with
| x :: s', n'.+1 => x :: take n' s'
| _, _ => [::]
end. | Fixpoint | take | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
take0 s : take 0 s = [::]. | Proof. by case: s. Qed. | Lemma | take0 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
take_oversize n s : size s <= n -> take n s = s. | Proof. by elim: s n => [|x s IHs] [|n] //= /IHs->. Qed. | Lemma | take_oversize | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"size",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
take_size s : take (size s) s = s. | Proof. exact: take_oversize. Qed. | Lemma | take_size | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"size",
"take",
"take_oversize"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
take_cons x s :
take n0 (x :: s) = if n0 is n.+1 then x :: (take n s) else [::]. | Proof. by []. Qed. | Lemma | take_cons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
drop_rcons s : n0 <= size s ->
forall x, drop n0 (rcons s x) = rcons (drop n0 s) x. | Proof. by elim: s n0 => [|y s IHs] []. Qed. | Lemma | drop_rcons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"drop",
"rcons",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cat_take_drop s : take n0 s ++ drop n0 s = s. | Proof. by elim: s n0 => [|x s IHs] [|n] //=; rewrite IHs. Qed. | Lemma | cat_take_drop | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"drop",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_takel s : n0 <= size s -> size (take n0 s) = n0. | Proof.
by move/subKn; rewrite -size_drop -[in size s](cat_take_drop s) size_cat addnK.
Qed. | Lemma | size_takel | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addnK",
"cat_take_drop",
"size",
"size_cat",
"size_drop",
"subKn",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_take s : size (take n0 s) = if n0 < size s then n0 else size s. | Proof.
have [le_sn | lt_ns] := leqP (size s) n0; first by rewrite take_oversize.
by rewrite size_takel // ltnW.
Qed. | Lemma | size_take | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"leqP",
"ltnW",
"size",
"size_takel",
"take",
"take_oversize"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_take_min s : size (take n0 s) = minn n0 (size s). | Proof. exact: size_take. Qed. | Lemma | size_take_min | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"minn",
"size",
"size_take",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
take_cat s1 s2 :
take n0 (s1 ++ s2) =
if n0 < size s1 then take n0 s1 else s1 ++ take (n0 - size s1) s2. | Proof.
elim: s1 n0 => [|x s1 IHs] [|n] //=.
by rewrite ltnS subSS -(fun_if (cons x)) -IHs.
Qed. | Lemma | take_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"ltnS",
"s1",
"s2",
"size",
"subSS",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
take_size_cat n s1 s2 : size s1 = n -> take n (s1 ++ s2) = s1. | Proof. by move <-; elim: s1 => [|x s1 IHs]; rewrite ?take0 //= IHs. Qed. | Lemma | take_size_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"s1",
"s2",
"size",
"take",
"take0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
takel_cat s1 s2 : n0 <= size s1 -> take n0 (s1 ++ s2) = take n0 s1. | Proof.
by rewrite take_cat; case: ltngtP => // ->; rewrite subnn take0 take_size cats0.
Qed. | Lemma | takel_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cats0",
"ltngtP",
"s1",
"s2",
"size",
"subnn",
"take",
"take0",
"take_cat",
"take_size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_drop s i : nth (drop n0 s) i = nth s (n0 + i). | Proof.
rewrite -[s in RHS]cat_take_drop nth_cat size_take ltnNge.
case: ltnP => [?|le_s_n0]; rewrite ?(leq_trans le_s_n0) ?leq_addr ?addKn //=.
by rewrite drop_oversize // !nth_default.
Qed. | Lemma | nth_drop | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addKn",
"cat_take_drop",
"drop",
"drop_oversize",
"leq_addr",
"leq_trans",
"ltnNge",
"ltnP",
"nth",
"nth_cat",
"nth_default",
"size_take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
find_ltn p s i : has p (take i s) -> find p s < i. | Proof. by elim: s i => [|y s ihs] [|i]//=; case: (p _) => //= /ihs. Qed. | Lemma | find_ltn | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"find",
"has",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_take p s i : has p s -> has p (take i s) = (find p s < i). | Proof. by elim: s i => [|y s ihs] [|i]//=; case: (p _) => //= /ihs ->. Qed. | Lemma | has_take | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"find",
"has",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_take_leq (p : pred T) (s : seq T) i : i <= size s ->
has p (take i s) = (find p s < i). | Proof. by elim: s i => [|y s ihs] [|i]//=; case: (p _) => //= /ihs ->. Qed. | Lemma | has_take_leq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"find",
"has",
"seq",
"size",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_take i : i < n0 -> forall s, nth (take n0 s) i = nth s i. | Proof.
move=> lt_i_n0 s; case lt_n0_s: (n0 < size s).
by rewrite -[s in RHS]cat_take_drop nth_cat size_take lt_n0_s /= lt_i_n0.
by rewrite -[s in LHS]cats0 take_cat lt_n0_s /= cats0.
Qed. | Lemma | nth_take | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cat_take_drop",
"cats0",
"nth",
"nth_cat",
"size",
"size_take",
"take",
"take_cat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
take_min i j s : take (minn i j) s = take i (take j s). | Proof. by elim: s i j => //= a l IH [|i] [|j] //=; rewrite minnSS IH. Qed. | Lemma | take_min | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"minn",
"minnSS",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
take_takel i j s : i <= j -> take i (take j s) = take i s. | Proof. by move=> ?; rewrite -take_min (minn_idPl _). Qed. | Lemma | take_takel | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"minn_idPl",
"take",
"take_min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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