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
value |
|---|---|---|---|---|---|---|---|---|---|---|
take_taker i j s : j <= i -> take i (take j s) = take j s. | Proof. by move=> ?; rewrite -take_min (minn_idPr _). Qed. | Lemma | take_taker | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"minn_idPr",
"take",
"take_min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
take_drop i j s : take i (drop j s) = drop j (take (i + j) s). | Proof. by rewrite addnC; elim: s i j => // x s IHs [|i] [|j] /=. Qed. | Lemma | take_drop | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addnC",
"drop",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
takeD i j s : take (i + j) s = take i s ++ take j (drop i s). | Proof.
elim: i j s => [|i IHi] [|j] [|a s] //; first by rewrite take0 addn0 cats0.
by rewrite addSn /= IHi.
Qed. | Lemma | takeD | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addSn",
"addn0",
"cats0",
"drop",
"take",
"take0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
takeC i j s : take i (take j s) = take j (take i s). | Proof. by rewrite -!take_min minnC. Qed. | Lemma | takeC | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"minnC",
"take",
"take_min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
take_nseq i j x : i <= j -> take i (nseq j x) = nseq i x. | Proof. by move=>/subnKC <-; rewrite nseqD take_size_cat // size_nseq. Qed. | Lemma | take_nseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"nseq",
"nseqD",
"size_nseq",
"subnKC",
"take",
"take_size_cat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
drop_nseq i j x : drop i (nseq j x) = nseq (j - i) x. | Proof.
case: (leqP i j) => [/subnKC {1}<-|/ltnW j_le_i].
by rewrite nseqD drop_size_cat // size_nseq.
by rewrite drop_oversize ?size_nseq // (eqP j_le_i).
Qed. | Lemma | drop_nseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"drop",
"drop_oversize",
"drop_size_cat",
"leqP",
"ltnW",
"nseq",
"nseqD",
"size_nseq",
"subnKC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
drop_nth n s : n < size s -> drop n s = nth s n :: drop n.+1 s. | Proof. by elim: s n => [|x s IHs] [|n] Hn //=; rewrite ?drop0 1?IHs. Qed. | Lemma | drop_nth | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"drop",
"drop0",
"nth",
"size"
] | will have to be given explicitly (and this will provide "d" as well). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
take_nth n s : n < size s -> take n.+1 s = rcons (take n s) (nth s n). | Proof. by elim: s n => [|x s IHs] //= [|n] Hn /=; rewrite ?take0 -?IHs. Qed. | Lemma | take_nth | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"nth",
"rcons",
"size",
"take",
"take0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rot n s | := drop n s ++ take n s. | Definition | rot | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"drop",
"take"
] | Rotation | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
rot0 s : rot 0 s = s. | Proof. by rewrite /rot drop0 take0 cats0. Qed. | Lemma | rot0 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cats0",
"drop0",
"rot",
"take0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_rot s : size (rot n0 s) = size s. | Proof. by rewrite -[s in RHS]cat_take_drop /rot !size_cat addnC. Qed. | Lemma | size_rot | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addnC",
"cat_take_drop",
"rot",
"size",
"size_cat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rot_oversize n s : size s <= n -> rot n s = s. | Proof. by move=> le_s_n; rewrite /rot take_oversize ?drop_oversize. Qed. | Lemma | rot_oversize | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"drop_oversize",
"rot",
"size",
"take_oversize"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rot_size s : rot (size s) s = s. | Proof. exact: rot_oversize. Qed. | Lemma | rot_size | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"rot",
"rot_oversize",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_rot s a : has a (rot n0 s) = has a s. | Proof. by rewrite has_cat orbC -has_cat cat_take_drop. Qed. | Lemma | has_rot | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cat_take_drop",
"has",
"has_cat",
"rot"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rot_size_cat s1 s2 : rot (size s1) (s1 ++ s2) = s2 ++ s1. | Proof. by rewrite /rot take_size_cat ?drop_size_cat. Qed. | Lemma | rot_size_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"drop_size_cat",
"rot",
"s1",
"s2",
"size",
"take_size_cat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rotr n s | := rot (size s - n) s. | Definition | rotr | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"rot",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rotK : cancel (rot n0) (rotr n0). | Proof.
move=> s; rewrite /rotr size_rot -size_drop {2}/rot.
by rewrite rot_size_cat cat_take_drop.
Qed. | Lemma | rotK | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cat_take_drop",
"rot",
"rot_size_cat",
"rotr",
"size_drop",
"size_rot"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rot_inj : injective (rot n0). | Proof. exact (can_inj rotK). Qed. | Lemma | rot_inj | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"rot",
"rotK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
catrev s1 s2 | := if s1 is x :: s1' then catrev s1' (x :: s2) else s2. | Fixpoint | catrev | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"s1",
"s2"
] | (efficient) reversal | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
rev s | := catrev s [::]. | Definition | rev | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"catrev"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
catrev_catl s t u : catrev (s ++ t) u = catrev t (catrev s u). | Proof. by elim: s u => /=. Qed. | Lemma | catrev_catl | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"catrev"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
catrev_catr s t u : catrev s (t ++ u) = catrev s t ++ u. | Proof. by elim: s t => //= x s IHs t; rewrite -IHs. Qed. | Lemma | catrev_catr | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"catrev"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
catrevE s t : catrev s t = rev s ++ t. | Proof. by rewrite -catrev_catr. Qed. | Lemma | catrevE | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"catrev",
"catrev_catr",
"rev"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rev_cons x s : rev (x :: s) = rcons (rev s) x. | Proof. by rewrite -cats1 -catrevE. Qed. | Lemma | rev_cons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"catrevE",
"cats1",
"rcons",
"rev"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_rev s : size (rev s) = size s. | Proof. by elim: s => // x s IHs; rewrite rev_cons size_rcons IHs. Qed. | Lemma | size_rev | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"rev",
"rev_cons",
"size",
"size_rcons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rev_nilp s : nilp (rev s) = nilp s. | Proof. by rewrite /nilp size_rev. Qed. | Lemma | rev_nilp | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"nilp",
"rev",
"size_rev"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rev_cat s t : rev (s ++ t) = rev t ++ rev s. | Proof. by rewrite -catrev_catr -catrev_catl. Qed. | Lemma | rev_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"catrev_catl",
"catrev_catr",
"rev"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rev_rcons s x : rev (rcons s x) = x :: rev s. | Proof. by rewrite -cats1 rev_cat. Qed. | Lemma | rev_rcons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cats1",
"rcons",
"rev",
"rev_cat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
revK : involutive rev. | Proof. by elim=> //= x s IHs; rewrite rev_cons rev_rcons IHs. Qed. | Lemma | revK | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"rev",
"rev_cons",
"rev_rcons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_rev n s : n < size s -> nth (rev s) n = nth s (size s - n.+1). | Proof.
elim/last_ind: s => // s x IHs in n *.
rewrite rev_rcons size_rcons ltnS subSS -cats1 nth_cat /=.
case: n => [|n] lt_n_s; first by rewrite subn0 ltnn subnn.
by rewrite subnSK //= leq_subr IHs.
Qed. | Lemma | nth_rev | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cats1",
"last_ind",
"leq_subr",
"ltnS",
"ltnn",
"nth",
"nth_cat",
"rev",
"rev_rcons",
"size",
"size_rcons",
"subSS",
"subn0",
"subnSK",
"subnn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
filter_rev a s : filter a (rev s) = rev (filter a s). | Proof. by elim: s => //= x s IH; rewrite fun_if !rev_cons filter_rcons IH. Qed. | Lemma | filter_rev | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"filter",
"filter_rcons",
"rev",
"rev_cons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
count_rev a s : count a (rev s) = count a s. | Proof. by rewrite -!size_filter filter_rev size_rev. Qed. | Lemma | count_rev | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"count",
"filter_rev",
"rev",
"size_filter",
"size_rev"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_rev a s : has a (rev s) = has a s. | Proof. by rewrite !has_count count_rev. Qed. | Lemma | has_rev | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"count_rev",
"has",
"has_count",
"rev"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_rev a s : all a (rev s) = all a s. | Proof. by rewrite !all_count count_rev size_rev. Qed. | Lemma | all_rev | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"all_count",
"count_rev",
"rev",
"size_rev"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rev_nseq n x : rev (nseq n x) = nseq n x. | Proof. by elim: n => // n IHn; rewrite -[in LHS]addn1 nseqD rev_cat IHn. Qed. | Lemma | rev_nseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addn1",
"nseq",
"nseqD",
"rev",
"rev_cat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
count_mem x | := (count (pred_of_simpl (pred1 x))). | Notation | count_mem | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"count",
"pred1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'seq' x <- s | C ]" | := (filter (fun x => C%B) s)
(x at level 99,
format "[ '[hv' 'seq' x <- s '/ ' | C ] ']'") : seq_scope. | Notation | [ 'seq' x <- s | C ] | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'seq' x <- s | C1 & C2 ]" | := [seq x <- s | C1 && C2]
(format "[ '[hv' 'seq' x <- s '/ ' | C1 '/ ' & C2 ] ']'") : seq_scope. | Notation | [ 'seq' x <- s | C1 & C2 ] | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'seq' ' x <- s | C ]" | := (filter (fun x => C%B) s)
(x strict pattern,
format "[ '[hv' 'seq' ' x <- s '/ ' | C ] ']'") : seq_scope. | Notation | [ 'seq' ' x <- s | C ] | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"filter",
"pattern"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'seq' ' x <- s | C1 & C2 ]" | := [seq x <- s | C1 && C2]
(x strict pattern,
format "[ '[hv' 'seq' ' x <- s '/ ' | C1 '/ ' & C2 ] ']'") : seq_scope. | Notation | [ 'seq' ' x <- s | C1 & C2 ] | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"pattern",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'seq' x : T <- s | C ]" | := (filter (fun x : T => C%B) s)
(only parsing). | Notation | [ 'seq' x : T <- s | C ] | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'seq' x : T <- s | C1 & C2 ]" | := [seq x : T <- s | C1 && C2]
(only parsing). | Notation | [ 'seq' x : T <- s | C1 & C2 ] | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
seq_ind2 {S T} (P : seq S -> seq T -> Type) :
P [::] [::] ->
(forall x y s t, size s = size t -> P s t -> P (x :: s) (y :: t)) ->
forall s t, size s = size t -> P s t. | Proof.
by move=> Pnil Pcons; elim=> [|x s IHs] [|y t] //= [eq_sz]; apply/Pcons/IHs.
Qed. | Lemma | seq_ind2 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"seq",
"size"
] | Double induction/recursion. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
all_iff_and (P Q : Prop) : Prop | := AllIffConj of P & Q. | Inductive | all_iff_and | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | items in a circular list of implications | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
all_iff (P0 : Prop) (Ps : seq Prop) : Prop | :=
let fix loop (P : Prop) (Qs : seq Prop) : Prop :=
if Qs is Q :: Qs then all_iff_and (P -> Q) (loop Q Qs) else P -> P0 in
loop P0 Ps. | Definition | all_iff | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"P0",
"all_iff_and",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_iffLR P0 Ps : all_iff P0 Ps ->
forall m n, nth P0 (P0 :: Ps) m -> nth P0 (P0 :: Ps) n. | Proof.
move=> iffPs; have PsS n: nth P0 Ps n -> nth P0 Ps n.+1.
elim: n P0 Ps iffPs => [|n IHn] P0 [|P [|Q Ps]] //= [iP0P] //; first by case.
by rewrite nth_nil.
by case=> iPQ iffPs; apply: IHn; split=> // /iP0P.
have{PsS} lePs: {homo nth P0 Ps : m n / m <= n >-> (m -> n)}.
by move=> m n /subnK<-; elim: {n}(n... | Lemma | all_iffLR | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"P0",
"all_iff",
"apply",
"leq0n",
"leq_maxl",
"leq_maxr",
"nth",
"nth_default",
"nth_nil",
"size",
"split",
"subnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_iffP P0 Ps :
all_iff P0 Ps -> forall m n, nth P0 (P0 :: Ps) m <-> nth P0 (P0 :: Ps) n. | Proof. by move=> /all_iffLR-iffPs m n; split => /iffPs. Qed. | Lemma | all_iffP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"P0",
"all_iff",
"all_iffLR",
"nth",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_iffP : all_iff >-> Funclass. | Coercion | all_iffP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all_iff"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
"[ '<->' P0 ; P1 ; .. ; Pn ]" | :=
(all_iff P0 (@cons Prop P1 (.. (@cons Prop Pn nil) ..))) : form_scope. | Notation | [ '<->' P0 ; P1 ; .. ; Pn ] | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"P0",
"P1",
"all_iff"
] | This means "the following are all equivalent: P0, ... Pn" | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
tfae | := do !apply: AllIffConj. | Ltac | tfae | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
find_spec : bool -> nat -> Type | :=
| NotFound of ~~ has a s : find_spec false (size s)
| Found (i : nat) of i < size s & (forall x0, a (nth x0 s i)) &
(forall x0 j, j < i -> a (nth x0 s j) = false) : find_spec true i. | Variant | find_spec | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"has",
"nat",
"nth",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
findP : find_spec (has a s) (find a s). | Proof.
have [a_s|aNs] := boolP (has a s); last by rewrite hasNfind//; constructor.
by constructor=> [|x0|x0]; rewrite -?has_find ?nth_find//; apply: before_find.
Qed. | Lemma | findP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"before_find",
"find",
"find_spec",
"has",
"hasNfind",
"has_find",
"last",
"nth_find"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rot1_cons x s : rot 1 (x :: s) = rcons s x. | Proof. by rewrite /rot /= take0 drop0 -cats1. Qed. | Lemma | rot1_cons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cats1",
"drop0",
"rcons",
"rot",
"take0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rcons_inj s1 s2 x1 x2 :
rcons s1 x1 = rcons s2 x2 :> seq T -> (s1, x1) = (s2, x2). | Proof. by rewrite -!rot1_cons => /rot_inj[-> ->]. Qed. | Lemma | rcons_inj | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"rcons",
"rot1_cons",
"rot_inj",
"s1",
"s2",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rcons_injl x : injective (rcons^~ x). | Proof. by move=> s1 s2 /rcons_inj[]. Qed. | Lemma | rcons_injl | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"rcons",
"rcons_inj",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rcons_injr s : injective (rcons s). | Proof. by move=> x1 x2 /rcons_inj[]. Qed. | Lemma | rcons_injr | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"rcons",
"rcons_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth | := (nth x0). | Notation | nth | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqseq s1 s2 {struct s2} | :=
match s1, s2 with
| [::], [::] => true
| x1 :: s1', x2 :: s2' => (x1 == x2) && eqseq s1' s2'
| _, _ => false
end. | Fixpoint | eqseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqseqP : Equality.axiom eqseq. | Proof.
move; elim=> [|x1 s1 IHs] [|x2 s2]; do [by constructor | simpl].
have [<-|neqx] := x1 =P x2; last by right; case.
by apply: (iffP (IHs s2)) => [<-|[]].
Qed. | Lemma | eqseqP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"axiom",
"eqseq",
"last",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqseqE : eqseq = eq_op. | Proof. by []. Qed. | Lemma | eqseqE | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"eqseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqseq_cons x1 x2 s1 s2 :
(x1 :: s1 == x2 :: s2) = (x1 == x2) && (s1 == s2). | Proof. by []. Qed. | Lemma | eqseq_cons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqseq_cat s1 s2 s3 s4 :
size s1 = size s2 -> (s1 ++ s3 == s2 ++ s4) = (s1 == s2) && (s3 == s4). | Proof.
elim: s1 s2 => [|x1 s1 IHs] [|x2 s2] //= [sz12].
by rewrite !eqseq_cons -andbA IHs.
Qed. | Lemma | eqseq_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"eqseq_cons",
"s1",
"s2",
"s3",
"s4",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqseq_rcons s1 s2 x1 x2 :
(rcons s1 x1 == rcons s2 x2) = (s1 == s2) && (x1 == x2). | Proof. by rewrite -(can_eq revK) !rev_rcons eqseq_cons andbC (can_eq revK). Qed. | Lemma | eqseq_rcons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"can_eq",
"eqseq_cons",
"rcons",
"revK",
"rev_rcons",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_eq0 s : (size s == 0) = (s == [::]). | Proof. exact: (sameP nilP eqP). Qed. | Lemma | size_eq0 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"nilP",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nilpE s : nilp s = (s == [::]). | Proof. by case: s. Qed. | Lemma | nilpE | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"nilp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_filter a s : has a s = (filter a s != [::]). | Proof. by rewrite -size_eq0 size_filter has_count lt0n. Qed. | Lemma | has_filter | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"filter",
"has",
"has_count",
"lt0n",
"size_eq0",
"size_filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_seq (s : seq T) | :=
if s is y :: s' then xpredU1 y (mem_seq s') else xpred0. | Fixpoint | mem_seq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"seq",
"xpredU1"
] | mem_seq defines a predType for seq. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
seq_eqclass | := seq T. | Definition | seq_eqclass | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pred_of_seq (s : seq_eqclass) : {pred T} | := mem_seq s. | Coercion | pred_of_seq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"mem_seq",
"seq_eqclass"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
seq_predType | := PredType (pred_of_seq : seq T -> pred T). | Canonical | seq_predType | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"pred_of_seq",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_seq_predType | := PredType mem_seq. | Canonical | mem_seq_predType | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"mem_seq"
] | The line below makes mem_seq a canonical instance of topred. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
in_cons y s x : (x \in y :: s) = (x == y) || (x \in s). | Proof. by []. Qed. | Lemma | in_cons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_nil x : (x \in [::]) = false. | Proof. by []. Qed. | Lemma | in_nil | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_seq1 x y : (x \in [:: y]) = (x == y). | Proof. by rewrite in_cons orbF. Qed. | Lemma | mem_seq1 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"in_cons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inE | := (mem_seq1, in_cons, inE). | Let | inE | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"in_cons",
"mem_seq1"
] | to be repeated after the Section discharge. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
forall_cons {P : T -> Prop} {a s} :
{in a::s, forall x, P x} <-> P a /\ {in s, forall x, P x}. | Proof.
split=> [A|[A B]]; last by move => x /predU1P [-> //|]; apply: B.
by split=> [|b Hb]; apply: A; rewrite !inE ?eqxx ?Hb ?orbT.
Qed. | Lemma | forall_cons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"eqxx",
"inE",
"last",
"predU1P",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exists_cons {P : T -> Prop} {a s} :
(exists2 x, x \in a::s & P x) <-> P a \/ exists2 x, x \in s & P x. | Proof.
split=> [[x /predU1P[->|x_s] Px]|]; [by left| by right; exists x|].
by move=> [?|[x x_s ?]]; [exists a|exists x]; rewrite ?inE ?eqxx ?x_s ?orbT.
Qed. | Lemma | exists_cons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"Px",
"eqxx",
"inE",
"predU1P",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_seq2 x y z : (x \in [:: y; z]) = xpred2 y z x. | Proof. by rewrite !inE. Qed. | Lemma | mem_seq2 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"inE",
"xpred2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_seq3 x y z t : (x \in [:: y; z; t]) = xpred3 y z t x. | Proof. by rewrite !inE. Qed. | Lemma | mem_seq3 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"inE",
"xpred3"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_seq4 x y z t u : (x \in [:: y; z; t; u]) = xpred4 y z t u x. | Proof. by rewrite !inE. Qed. | Lemma | mem_seq4 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"inE",
"xpred4"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_cat x s1 s2 : (x \in s1 ++ s2) = (x \in s1) || (x \in s2). | Proof. by elim: s1 => //= y s1 IHs; rewrite !inE /= -orbA -IHs. Qed. | Lemma | mem_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"inE",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_catC x s1 s2 : (x \in s1 ++ s2) = (x \in s2 ++ s1). | Proof. by rewrite !mem_cat orbC. Qed. | Lemma | mem_catC | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"mem_cat",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_rcons s y : rcons s y =i y :: s. | Proof. by move=> x; rewrite -cats1 /= mem_cat mem_seq1 orbC in_cons. Qed. | Lemma | mem_rcons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cats1",
"in_cons",
"mem_cat",
"mem_seq1",
"rcons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_head x s : x \in x :: s. | Proof. exact: predU1l. Qed. | Lemma | mem_head | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"predU1l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_last x s : last x s \in x :: s. | Proof. by rewrite lastI mem_rcons mem_head. Qed. | Lemma | mem_last | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"last",
"lastI",
"mem_head",
"mem_rcons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_behead s : {subset behead s <= s}. | Proof. by case: s => // y s x; apply: predU1r. Qed. | Lemma | mem_behead | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"behead",
"predU1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_belast s y : {subset belast y s <= y :: s}. | Proof. by move=> x ys'x; rewrite lastI mem_rcons mem_behead. Qed. | Lemma | mem_belast | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"belast",
"lastI",
"mem_behead",
"mem_rcons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_nth s n : n < size s -> nth s n \in s. | Proof.
by elim: s n => // x s IHs [_|n sz_s]; rewrite ?mem_head // mem_behead ?IHs.
Qed. | Lemma | mem_nth | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"mem_behead",
"mem_head",
"nth",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_nthE s n : (nth s n \in s) = ((n >= size s) ==> (x0 \in s)). | Proof.
apply/idP/idP; first by case: ltnP => // ?; rewrite nth_default.
by case: ltnP => //= ns x0s; [rewrite mem_nth|rewrite nth_default].
Qed. | Lemma | mem_nthE | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"ltnP",
"mem_nth",
"nth",
"nth_default",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_take s x : x \in take n0 s -> x \in s. | Proof. by move=> s0x; rewrite -(cat_take_drop n0 s) mem_cat /= s0x. Qed. | Lemma | mem_take | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cat_take_drop",
"mem_cat",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_drop s x : x \in drop n0 s -> x \in s. | Proof. by move=> s0'x; rewrite -(cat_take_drop n0 s) mem_cat /= s0'x orbT. Qed. | Lemma | mem_drop | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cat_take_drop",
"drop",
"mem_cat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
last_eq s z x y : x != y -> z != y -> (last x s == y) = (last z s == y). | Proof. by move=> /negPf xz /negPf yz; case: s => [|t s]//; rewrite xz yz. Qed. | Lemma | last_eq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"last"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subset_cons2 x s s' : {subset s <= s'} -> {subset x :: s <= x :: s'}. | Proof. by move=> ss' y; rewrite !in_cons => /orP[->//|/ss'->]; rewrite orbT. Qed. | Lemma | subset_cons2 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"in_cons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subset_cons x s s' : {subset s <= s'} -> {subset s <= x :: s'}. | Proof. by move=> ss' y; rewrite !in_cons => /ss'-> /[!orbT]. Qed. | Lemma | subset_cons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"in_cons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hasP {a s} : reflect (exists2 x, x \in s & a x) (has a s). | Proof.
elim: s => [|y s IHs] /=; first by right; case.
exact: equivP (orPP idP IHs) (iff_sym exists_cons).
Qed. | Lemma | hasP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"exists_cons",
"has"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
allP {a s} : reflect {in s, forall x, a x} (all a s). | Proof.
elim: s => [|/= y s IHs]; first by left.
exact: equivP (andPP idP IHs) (iff_sym forall_cons).
Qed. | Lemma | allP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"forall_cons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hasPn a s : reflect {in s, forall x, ~~ a x} (~~ has a s). | Proof. by rewrite -all_predC; apply: allP. Qed. | Lemma | hasPn | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"allP",
"all_predC",
"apply",
"has"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
allPn a s : reflect (exists2 x, x \in s & ~~ a x) (~~ all a s). | Proof. by rewrite -has_predC; apply: hasP. Qed. | Lemma | allPn | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"apply",
"hasP",
"has_predC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
allss s : all [in s] s. | Proof. exact/allP. Qed. | Lemma | allss | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"allP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_filter a x s : (x \in filter a s) = a x && (x \in s). | Proof.
rewrite andbC; elim: s => //= y s IHs.
rewrite (fun_if (fun s' : seq T => x \in s')) !in_cons {}IHs.
by case: eqP => [->|_]; case (a y); rewrite /= ?andbF.
Qed. | Lemma | mem_filter | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"filter",
"in_cons",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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