fact stringlengths 8 1.54k | type stringclasses 19 values | library stringclasses 8 values | imports listlengths 1 10 | filename stringclasses 98 values | symbolic_name stringlengths 1 42 | docstring stringclasses 1 value |
|---|---|---|---|---|---|---|
sort_tuplePr t : size (sort r t) == n.
Proof. by rewrite size_sort size_tuple. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | sort_tupleP | |
sort_tupler t := Tuple (sort_tupleP r t). | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | sort_tuple | |
thead(u : n.+1.-tuple T) := tnth u ord0. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | thead | |
tnth0x t : tnth [tuple of x :: t] ord0 = x.
Proof. by []. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | tnth0 | |
tnthSx t i : tnth [tuple of x :: t] (lift ord0 i) = tnth t i.
Proof. by rewrite (tnth_nth (tnth_default t i)). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | tnthS | |
theadEx t : thead [tuple of x :: t] = x.
Proof. by []. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | theadE | |
tuple0: all_equal_to ([tuple] : 0.-tuple T).
Proof. by move=> t; apply: val_inj; case: t => [[]]. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | tuple0 | |
tuple1_spec: n.+1.-tuple T -> Type :=
Tuple1spec x t : tuple1_spec [tuple of x :: t]. | Variant | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | tuple1_spec | |
tuplePu : tuple1_spec u.
Proof.
case: u => [[|x s] //= sz_s]; pose t := @Tuple n _ s sz_s.
by rewrite (_ : Tuple _ = [tuple of x :: t]) //; apply: val_inj.
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | tupleP | |
tnth_mapf t i : tnth [tuple of map f t] i = f (tnth t i) :> rT.
Proof. by apply: nth_map; rewrite size_tuple. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | tnth_map | |
tnth_nseqx i : tnth [tuple of nseq n x] i = x.
Proof.
by rewrite !(tnth_nth (tnth_default (nseq_tuple x) i)) nth_nseq ltn_ord.
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | tnth_nseq | |
tnth_beheadn T (t : n.+1.-tuple T) i :
tnth [tuple of behead t] i = tnth t (inord i.+1).
Proof. by case/tupleP: t => x t; rewrite !(tnth_nth x) inordK ?ltnS. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | tnth_behead | |
tuple_etan T (t : n.+1.-tuple T) : t = [tuple of thead t :: behead t].
Proof. by case/tupleP: t => x t; apply: val_inj. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | tuple_eta | |
tnth_lshifti : tnth [tuple of t1 ++ t2] (lshift n2 i) = tnth t1 i.
Proof.
have x0 := tnth_default t1 i; rewrite !(tnth_nth x0).
by rewrite nth_cat size_tuple /= ltn_ord.
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | tnth_lshift | |
tnth_rshiftj : tnth [tuple of t1 ++ t2] (rshift n1 j) = tnth t2 j.
Proof.
have x0 := tnth_default t2 j; rewrite !(tnth_nth x0).
by rewrite nth_cat size_tuple ltnNge leq_addr /= addKn.
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | tnth_rshift | |
forallb_tntha t : [forall i, a (tnth t i)] = all a t.
Proof.
apply: negb_inj; rewrite -has_predC -has_map negb_forall.
apply/existsP/(has_nthP true) => [[i a_t_i] | [i lt_i_n a_t_i]].
by exists i; rewrite ?size_tuple // -tnth_nth tnth_map.
rewrite size_tuple in lt_i_n; exists (Ordinal lt_i_n).
by rewrite -tnth_map (tnth_nth true).
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | forallb_tnth | |
existsb_tntha t : [exists i, a (tnth t i)] = has a t.
Proof. by apply: negb_inj; rewrite negb_exists -all_predC -forallb_tnth. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | existsb_tnth | |
all_tnthPa t : reflect (forall i, a (tnth t i)) (all a t).
Proof. by rewrite -forallb_tnth; apply: forallP. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | all_tnthP | |
has_tnthPa t : reflect (exists i, a (tnth t i)) (has a t).
Proof. by rewrite -existsb_tnth; apply: existsP. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | has_tnthP | |
Definition_ : hasDecEq (n.-tuple T) :=
[Equality of n.-tuple T by <:]. | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | Definition | |
tuple_predType:= PredType (pred_of_seq : n.-tuple T -> pred T). | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | tuple_predType | |
eqEtuple(t1 t2 : n.-tuple T) :
(t1 == t2) = [forall i, tnth t1 i == tnth t2 i].
Proof. by apply/eqP/'forall_eqP => [->|/eq_from_tnth]. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | eqEtuple | |
memtE(t : n.-tuple T) : mem t = mem (tval t).
Proof. by []. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | memtE | |
mem_tnthi (t : n.-tuple T) : tnth t i \in t.
Proof. by rewrite mem_nth ?size_tuple. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | mem_tnth | |
memt_nthx0 (t : n.-tuple T) i : i < n -> nth x0 t i \in t.
Proof. by move=> i_lt_n; rewrite mem_nth ?size_tuple. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | memt_nth | |
tnthP(t : n.-tuple T) x : reflect (exists i, x = tnth t i) (x \in t).
Proof.
apply: (iffP idP) => [/(nthP x)[i ltin <-] | [i ->]]; last exact: mem_tnth.
by rewrite size_tuple in ltin; exists (Ordinal ltin); rewrite (tnth_nth x).
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | tnthP | |
seq_tnthP(s : seq T) x : x \in s -> {i | x = tnth (in_tuple s) i}.
Proof.
move=> s_x; pose i := index x s; have lt_i: i < size s by rewrite index_mem.
by exists (Ordinal lt_i); rewrite (tnth_nth x) nth_index.
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | seq_tnthP | |
tuple_uniqP(t : n.-tuple T) : reflect (injective (tnth t)) (uniq t).
Proof.
case: {+}n => [|m] in t *; first by rewrite tuple0; constructor => -[].
pose x0 := tnth t ord0; apply/(equivP (uniqP x0)); split=> tinj i j.
by rewrite !(tnth_nth x0) => /tinj/val_inj; apply; rewrite size_tuple inE.
rewrite !size_tuple !inE => im jm; have := tinj (Ordinal im) (Ordinal jm).
by rewrite !(tnth_nth x0) => /[apply]-[].
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | tuple_uniqP | |
Definition_ n (T : choiceType) :=
[Choice of n.-tuple T by <:].
HB.instance Definition _ n (T : countType) :=
[Countable of n.-tuple T by <:]. | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | Definition | |
enum: seq (n.-tuple T). | Parameter | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | enum | |
enumP: Finite.axiom enum. | Axiom | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | enumP | |
size_enum: size enum = #|T| ^ n. | Axiom | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | size_enum | |
enum: seq (n.-tuple T) :=
let extend e := flatten (codom (fun x => map (cons x) e)) in
pmap insub (iter n extend [::[::]]). | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | enum | |
enumP: Finite.axiom enum.
Proof.
case=> /= t t_n; rewrite -(count_map _ (pred1 t)) (pmap_filter (insubK _)).
rewrite count_filter -(@eq_count _ (pred1 t)) => [|s /=]; last first.
by rewrite isSome_insub; case: eqP=> // ->.
elim: n t t_n => [|m IHm] [|x t] //= {}/IHm; move: (iter m _ _) => em IHm.
transitivity (x \in T : nat); rewrite // -mem_enum codomE.
elim: (fintype.enum T) (enum_uniq T) => //= y e IHe /andP[/negPf ney].
rewrite count_cat count_map inE /preim /= [in LHS]/eq_op /= eq_sym => /IHe->.
by case: eqP => [->|_]; rewrite ?(ney, count_pred0, IHm).
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | enumP | |
size_enum: size enum = #|T| ^ n.
Proof.
rewrite /= cardE size_pmap_sub; elim: n => //= m IHm.
rewrite expnS /codom /image_mem; elim: {2 3}(fintype.enum T) => //= x e IHe.
by rewrite count_cat {}IHe count_map IHm.
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | size_enum | |
Definition_ := isFinite.Build (n.-tuple T) (@FinTuple.enumP n T). | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | Definition | |
card_tuple: #|{:n.-tuple T}| = #|T| ^ n.
Proof. by rewrite [#|_|]cardT enumT unlock FinTuple.size_enum. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | card_tuple | |
enum_tupleP(A : {pred T}) : size (enum A) == #|A|.
Proof. by rewrite -cardE. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | enum_tupleP | |
enum_tupleA := Tuple (enum_tupleP A). | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | enum_tuple | |
ord_tuple: n.-tuple 'I_n := Tuple (introT eqP (size_enum_ord n)). | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | ord_tuple | |
val_ord_tuple: val ord_tuple = enum 'I_n. Proof. by []. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | val_ord_tuple | |
tuple_map_ordU (t : n.-tuple U) : t = [tuple of map (tnth t) ord_tuple].
Proof. by apply: val_inj => /=; rewrite map_tnth_enum. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | tuple_map_ord | |
tnth_ord_tuplei : tnth ord_tuple i = i.
Proof. by rewrite (tnth_nth i) val_ord_tuple nth_ord_enum. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | tnth_ord_tuple | |
image_tuple: #|A|.-tuple T' := [tuple of image f A]. | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | image_tuple | |
codom_tuple: #|T|.-tuple T' := [tuple of codom f]. | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | codom_tuple | |
mktuple:= map_tuple f ord_tuple. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | mktuple | |
tnth_mktuplei : tnth mktuple i = f i.
Proof. by rewrite tnth_map tnth_ord_tuple. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | tnth_mktuple | |
nth_mktuplex0 (i : 'I_n) : nth x0 mktuple i = f i.
Proof. by rewrite -tnth_nth tnth_mktuple. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | nth_mktuple | |
eq_mktupleT' (f1 f2 : 'I_n -> T') :
f1 =1 f2 -> mktuple f1 = mktuple f2.
Proof. by move=> eq_f; apply eq_from_tnth=> i; rewrite !tnth_map eq_f. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | eq_mktuple | |
bseq_of: Type := Bseq {bseqval :> seq T; _ : size bseqval <= n}.
HB.instance Definition _ := [isSub for bseqval].
Implicit Type bs : bseq_of. | Structure | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | bseq_of | |
size_bseqbs : size bs <= n.
Proof. by case: bs. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | size_bseq | |
bseqbs mkB : bseq_of :=
mkB (let: Bseq _ bsP := bs return size bs <= n in bsP). | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | bseq | |
bseqEbs : bseq (fun sP => @Bseq bs sP) = bs.
Proof. by case: bs. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | bseqE | |
nil_bseqn T := Bseq (isT : @size T [::] <= n). | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | nil_bseq | |
cons_bseqn T x (t : bseq_of n T) :=
Bseq (valP t : size (x :: t) <= n.+1). | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | cons_bseq | |
bseq_of_tuplen T (t : n.-tuple T) : n.-bseq T :=
Bseq (eq_leq (size_tuple t)). | Coercion | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | bseq_of_tuple | |
insub_bseqn T (s : seq T) : n.-bseq T := insubd [bseq] s. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | insub_bseq | |
size_insub_bseqn T (s : seq T) : size (insub_bseq n s) <= size s.
Proof. by rewrite /insub_bseq /insubd; case: insubP => // ? ? ->. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | size_insub_bseq | |
in_bseq(s : seq T) : (size s).-bseq T := Bseq (leqnn (size s)). | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | in_bseq | |
cast_bseqm n (eq_mn : m = n) bs :=
let: erefl in _ = n := eq_mn return n.-bseq T in bs. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | cast_bseq | |
widen_bseqm n (lemn : m <= n) (bs : m.-bseq T) : n.-bseq T :=
@Bseq n T bs (leq_trans (size_bseq bs) lemn). | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | widen_bseq | |
cast_bseq_idn (eq_nn : n = n) bs : cast_bseq eq_nn bs = bs.
Proof. by rewrite (eq_axiomK eq_nn). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | cast_bseq_id | |
cast_bseqKm n (eq_mn : m = n) :
cancel (cast_bseq eq_mn) (cast_bseq (esym eq_mn)).
Proof. by case: n / eq_mn. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | cast_bseqK | |
cast_bseqKVm n (eq_mn : m = n) :
cancel (cast_bseq (esym eq_mn)) (cast_bseq eq_mn).
Proof. by case: n / eq_mn. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | cast_bseqKV | |
cast_bseq_transm n p (eq_mn : m = n) (eq_np : n = p) bs :
cast_bseq (etrans eq_mn eq_np) bs = cast_bseq eq_np (cast_bseq eq_mn bs).
Proof. by case: n / eq_mn eq_np; case: p /. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | cast_bseq_trans | |
size_cast_bseqm n (eq_mn : m = n) (bs : m.-bseq T) :
size (cast_bseq eq_mn bs) = size bs.
Proof. by case: n / eq_mn. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | size_cast_bseq | |
widen_bseq_idn (lenn : n <= n) (bs : n.-bseq T) :
widen_bseq lenn bs = bs.
Proof. exact: val_inj. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | widen_bseq_id | |
cast_bseqEwidenm n (eq_mn : m = n) (bs : m.-bseq T) :
cast_bseq eq_mn bs = widen_bseq (eq_leq eq_mn) bs.
Proof. by case: n / eq_mn; rewrite widen_bseq_id. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | cast_bseqEwiden | |
widen_bseqKm n (lemn : m <= n) (lenm : n <= m) :
cancel (@widen_bseq m n lemn) (widen_bseq lenm).
Proof. by move=> t; apply: val_inj. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | widen_bseqK | |
widen_bseq_transm n p (lemn : m <= n) (lenp : n <= p) (bs : m.-bseq T) :
widen_bseq (leq_trans lemn lenp) bs = widen_bseq lenp (widen_bseq lemn bs).
Proof. exact/val_inj. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | widen_bseq_trans | |
size_widen_bseqm n (lemn : m <= n) (bs : m.-bseq T) :
size (widen_bseq lemn bs) = size bs.
Proof. by []. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | size_widen_bseq | |
in_bseqEs : in_bseq s = s :> seq T. Proof. by []. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | in_bseqE | |
widen_bseq_in_bseqn (bs : n.-bseq T) :
widen_bseq (size_bseq bs) (in_bseq bs) = bs.
Proof. exact: val_inj. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | widen_bseq_in_bseq | |
rcons_bseqPs x : size (rcons s x) <= n.+1.
Proof. by rewrite size_rcons ltnS size_bseq. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | rcons_bseqP | |
rcons_bseqs x := Bseq (rcons_bseqP s x). | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | rcons_bseq | |
behead_bseqPs : size (behead s) <= n.-1.
Proof. rewrite size_behead -!subn1; apply/leq_sub2r/size_bseq. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | behead_bseqP | |
behead_bseqs := Bseq (behead_bseqP s). | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | behead_bseq | |
belast_bseqPx s : size (belast x s) <= n.
Proof. by rewrite size_belast; apply/size_bseq. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | belast_bseqP | |
belast_bseqx s := Bseq (belast_bseqP x s). | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | belast_bseq | |
cat_bseqPs (s' : m.-bseq T) : size (s ++ s') <= n + m.
Proof. by rewrite size_cat; apply/leq_add/size_bseq/size_bseq. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | cat_bseqP | |
cat_bseqs (s' : m.-bseq T) := Bseq (cat_bseqP s s'). | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | cat_bseq | |
take_bseqPs : size (take m s) <= n.
Proof.
by rewrite size_take_min (leq_trans _ (size_bseq s)) // geq_minr.
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | take_bseqP | |
take_bseqs := Bseq (take_bseqP s). | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | take_bseq | |
drop_bseqPs : size (drop m s) <= n - m.
Proof. by rewrite size_drop; apply/leq_sub2r/size_bseq. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | drop_bseqP | |
drop_bseqs := Bseq (drop_bseqP s). | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | drop_bseq | |
rev_bseqPs : size (rev s) <= n.
Proof. by rewrite size_rev size_bseq. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | rev_bseqP | |
rev_bseqs := Bseq (rev_bseqP s). | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | rev_bseq | |
rot_bseqPs : size (rot m s) <= n.
Proof. by rewrite size_rot size_bseq. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | rot_bseqP | |
rot_bseqs := Bseq (rot_bseqP s). | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | rot_bseq | |
rotr_bseqPs : size (rotr m s) <= n.
Proof. by rewrite size_rotr size_bseq. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | rotr_bseqP | |
rotr_bseqs := Bseq (rotr_bseqP s). | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | rotr_bseq | |
map_bseqPf s : @size rT (map f s) <= n.
Proof. by rewrite size_map size_bseq. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | map_bseqP | |
map_bseqf s := Bseq (map_bseqP f s). | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | map_bseq | |
scanl_bseqPf x s : @size rT (scanl f x s) <= n.
Proof. by rewrite size_scanl size_bseq. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | scanl_bseqP | |
scanl_bseqf x s := Bseq (scanl_bseqP f x s). | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | scanl_bseq | |
pairmap_bseqPf x s : @size rT (pairmap f x s) <= n.
Proof. by rewrite size_pairmap size_bseq. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | pairmap_bseqP | |
pairmap_bseqf x s := Bseq (pairmap_bseqP f x s). | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | pairmap_bseq | |
allpairs_bseqPf s (s' : m.-bseq U) : @size rT (allpairs f s s') <= n * m.
Proof. by rewrite size_allpairs; apply/leq_mul/size_bseq/size_bseq. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | allpairs_bseqP | |
allpairs_bseqf s (s' : m.-bseq U) := Bseq (allpairs_bseqP f s s'). | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | allpairs_bseq | |
sort_bseqPr s : size (sort r s) <= n.
Proof. by rewrite size_sort size_bseq. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | sort_bseqP |
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