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 |
|---|---|---|---|---|---|---|
inj_homo_ltn_in: {in D & D', injective f} ->
{in D & D', {homo f : m n / m <= n}} ->
{in D & D', {homo f : m n / m < n}}.
Proof. exact: inj_homo_in. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | inj_homo_ltn_in | |
inj_nhomo_ltn_in: {in D & D', injective f} ->
{in D & D', {homo f : m n /~ m <= n}} ->
{in D & D', {homo f : m n /~ m < n}}.
Proof. exact: inj_homo_in. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | inj_nhomo_ltn_in | |
incn_inj_in: {in D &, {mono f : m n / m <= n}} ->
{in D &, injective f}.
Proof. exact: mono_inj_in. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | incn_inj_in | |
decn_inj_in: {in D &, {mono f : m n /~ m <= n}} ->
{in D &, injective f}.
Proof. exact: mono_inj_in. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | decn_inj_in | |
leqW_mono_in: {in D &, {mono f : m n / m <= n}} ->
{in D &, {mono f : m n / m < n}}.
Proof. exact: anti_mono_in. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | leqW_mono_in | |
leqW_nmono_in: {in D &, {mono f : m n /~ m <= n}} ->
{in D &, {mono f : m n /~ m < n}}.
Proof. exact: anti_mono_in. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | leqW_nmono_in | |
leq_mono_in: {in D &, {homo f : m n / m < n}} ->
{in D &, {mono f : m n / m <= n}}.
Proof. exact: total_homo_mono_in. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | leq_mono_in | |
leq_nmono_in: {in D &, {homo f : m n /~ m < n}} ->
{in D &, {mono f : m n /~ m <= n}}.
Proof. exact: total_homo_mono_in. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | leq_nmono_in | |
leq_pfact: {in [pred n | 0 < n] &, {mono factorial : m n / m <= n}}.
Proof. by apply: leq_mono_in => n m n0 m0; apply: ltn_fact. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | leq_pfact | |
leq_fact: {homo factorial : m n / m <= n}.
Proof.
by move=> [m|m n mn]; rewrite ?fact_gt0// leq_pfact// inE (leq_trans _ mn).
Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | leq_fact | |
ltn_pfact: {in [pred n | 0 < n] &, {mono factorial : m n / m < n}}.
Proof. exact/leqW_mono_in/leq_pfact. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | ltn_pfact | |
addm n := if m is m'.+1 then m' + n.+1 else n
where "n + m" := (add n m) : nat_scope. | Fixpoint | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | add | |
add_mulm n s := if m is m'.+1 then add_mul m' n (n + s) else s. | Fixpoint | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | add_mul | |
mulm n := if m is m'.+1 then add_mul m' n n else 0. | Definition | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | mul | |
mul_expm n p := if n is n'.+1 then mul_exp m n' (m * p) else p. | Fixpoint | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | mul_exp | |
expm n := if n is n'.+1 then mul_exp m n' m else 1. | Definition | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | exp | |
oddn := if n is n'.+2 then odd n' else eqn n 1.
Local Notation doublen := double. | Fixpoint | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | odd | |
doublen := if n is n'.+1 then n' + n.+1 else 0. | Definition | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | double | |
addE: add =2 addn.
Proof. by elim=> //= n IHn m; rewrite IHn addSnnS. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | addE | |
doubleE: double =1 doublen.
Proof. by case=> // n; rewrite -addnn -addE. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | doubleE | |
add_mulEn m s : add_mul n m s = addn (muln n m) s.
Proof. by elim: n => //= n IHn in m s *; rewrite IHn addE addnCA addnA. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | add_mulE | |
mulE: mul =2 muln.
Proof. by case=> //= n m; rewrite add_mulE addnC. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | mulE | |
mul_expEm n p : mul_exp m n p = muln (expn m n) p.
Proof.
by elim: n => [|n IHn] in p *; rewrite ?mul1n //= expnS IHn mulE mulnCA mulnA.
Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | mul_expE | |
expE: exp =2 expn.
Proof. by move=> m [|n] //=; rewrite mul_expE expnS mulnC. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | expE | |
oddE: odd =1 oddn.
Proof.
move=> n; rewrite -[n in LHS]odd_double_half addnC.
by elim: n./2 => //=; case (oddn n).
Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | oddE | |
trecE:= (addE, (doubleE, oddE), (mulE, add_mulE, (expE, mul_expE))). | Definition | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | trecE | |
natTrecE:= NatTrec.trecE. | Notation | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | natTrecE | |
N_eqbn m :=
match n, m with
| N0, N0 => true
| Npos p, Npos q => Pos.eqb p q
| _, _ => false
end. | Definition | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | N_eqb | |
eq_binP: Equality.axiom N_eqb.
Proof.
move=> p q; apply: (iffP idP) => [|<-]; last by case: p => //; elim.
by case: q; case: p => //; elim=> [p IHp|p IHp|] [q|q|] //= /IHp [->].
Qed.
HB.instance Definition _ := hasDecEq.Build N eq_binP.
Arguments N_eqb !n !m. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | eq_binP | |
nat_of_posp0 :=
match p0 with
| xO p => (nat_of_pos p).*2
| xI p => (nat_of_pos p).*2.+1
| xH => 1
end. | Fixpoint | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | nat_of_pos | |
nat_of_binb := if b is Npos p then p : nat else 0. | Coercion | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | nat_of_bin | |
pos_of_natn0 m0 :=
match n0, m0 with
| n.+1, m.+2 => pos_of_nat n m
| n.+1, 1 => xO (pos_of_nat n n)
| n.+1, 0 => xI (pos_of_nat n n)
| 0, _ => xH
end. | Fixpoint | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | pos_of_nat | |
bin_of_natn0 := if n0 is n.+1 then Npos (pos_of_nat n n) else N0. | Definition | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | bin_of_nat | |
bin_of_natK: cancel bin_of_nat nat_of_bin.
Proof.
have sub2nn n : n.*2 - n = n by rewrite -addnn addKn.
case=> //= n; rewrite -[n in RHS]sub2nn.
by elim: n {2 4}n => // m IHm [|[|n]] //=; rewrite IHm // natTrecE sub2nn.
Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | bin_of_natK | |
nat_of_binK: cancel nat_of_bin bin_of_nat.
Proof.
case=> //=; elim=> //= p; case: (nat_of_pos p) => //= n [<-].
by rewrite natTrecE !addnS {2}addnn; elim: {1 3}n.
by rewrite natTrecE addnS /= addnS {2}addnn; elim: {1 3}n.
Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | nat_of_binK | |
nat_of_succ_posp : Pos.succ p = p.+1 :> nat.
Proof. by elim: p => //= p ->; rewrite !natTrecE. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | nat_of_succ_pos | |
nat_of_add_posp q : Pos.add p q = p + q :> nat.
Proof.
apply: @fst _ (Pos.add_carry p q = (p + q).+1 :> nat) _.
elim: p q => [p IHp|p IHp|] [q|q|] //=; rewrite !natTrecE //;
by rewrite ?IHp ?nat_of_succ_pos ?(doubleS, doubleD, addn1, addnS).
Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | nat_of_add_pos | |
nat_of_mul_posp q : Pos.mul p q = p * q :> nat.
Proof.
elim: p => [p IHp|p IHp|] /=; rewrite ?mul1n //;
by rewrite ?nat_of_add_pos /= !natTrecE IHp doubleMl.
Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | nat_of_mul_pos | |
number: Type := Num {bin_of_number :> N}. | Record | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | number | |
number_subType:= Eval hnf in [isNew for bin_of_number].
HB.instance Definition _ := number_subType.
HB.instance Definition _ := [Equality of number by <:]. | Definition | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | number_subType | |
pop_succne := if e is e'.+1 then fun n => pop_succn e' n.+1 else id. | Fixpoint | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | pop_succn | |
pop_succne := eval lazy beta iota delta [pop_succn] in (pop_succn e 1). | Ltac | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | pop_succn | |
succn_to_add:=
match goal with
| |- context G [?e.+1] =>
let x := fresh "NatLit0" in
match pop_succn e with
| ?n.+1 => pose x := n.+1; let G' := context G [x] in change G'
| _ ?e' ?n => pose x := n; let G' := context G [x + e'] in change G'
end; succn_to_add; rewrite {}/x
| _ => idtac
end. | Ltac | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | succn_to_add | |
nat_norm:=
succn_to_add; rewrite ?add0n ?addn0 -?addnA ?(addSn, addnS, add0n, addn0). | Ltac | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | nat_norm | |
nat_congr:= first
[ apply: (congr1 succn _)
| apply: (congr1 predn _)
| apply: (congr1 (addn _) _)
| apply: (congr1 (subn _) _)
| apply: (congr1 (addn^~ _) _)
| match goal with |- (?X1 + ?X2 = ?X3) =>
symmetry;
rewrite -1?(addnC X1) -?(addnCA X1);
apply: (congr1 (addn X1) _);
symmetry
end ]. | Ltac | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | nat_congr | |
tuple_of: Type := Tuple {tval :> seq T; _ : size tval == n}.
HB.instance Definition _ := [isSub for tval].
Implicit Type t : tuple_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 | tuple_of | |
tsizeof tuple_of := 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 | tsize | |
size_tuplet : size t = n.
Proof. exact: (eqP (valP 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 | size_tuple | |
tnth_defaultt : 'I_n -> T.
Proof. by rewrite -(size_tuple t); case: (tval 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 | tnth_default | |
tntht i := nth (tnth_default t i) t i. | 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 | tnth | |
tnth_nthx t i : tnth t i = nth x t i.
Proof. by apply: set_nth_default; 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_nth | |
tnth_onthx t i : tnth t i = x <-> onth t i = Some x.
Proof.
rewrite (tnth_nth x) onthE (nth_map x) ?size_tuple//.
by split; [move->|case].
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_onth | |
map_tnth_enumt : map (tnth t) (enum 'I_n) = t.
Proof.
case def_t: {-}(val t) => [|x0 t'].
by rewrite [enum _]size0nil // -cardE card_ord -(size_tuple t) def_t.
apply: (@eq_from_nth _ x0) => [|i]; rewrite size_map.
by rewrite -cardE size_tuple card_ord.
move=> lt_i_e; have lt_i_n: i < n by rewrite -cardE card_ord in lt_i_e.
by rewrite (nth_map (Ordinal lt_i_n)) // (tnth_nth x0) nth_enum_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 | map_tnth_enum | |
eq_from_tntht1 t2 : tnth t1 =1 tnth t2 -> t1 = t2.
Proof.
by move/eq_map=> eq_t; apply: val_inj; rewrite /= -!map_tnth_enum eq_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 | eq_from_tnth | |
tuplet mkT : tuple_of :=
mkT (let: Tuple _ tP := t return size t == n in tP). | 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 | tuple | |
tupleEt : tuple (fun sP => @Tuple t sP) = t.
Proof. by 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 | tupleE | |
nil_tupleT := Tuple (isT : @size T [::] == 0). | 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_tuple | |
cons_tuplen T x (t : n.-tuple T) :=
Tuple (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_tuple | |
in_tuple(s : seq T) := Tuple (eqxx (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_tuple | |
tcastm n (eq_mn : m = n) t :=
let: erefl in _ = n := eq_mn return n.-tuple T in t. | 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 | tcast | |
tcastEm n (eq_mn : m = n) t i :
tnth (tcast eq_mn t) i = tnth t (cast_ord (esym eq_mn) i).
Proof. by case: n / eq_mn in i *; rewrite cast_ord_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 | tcastE | |
tcast_idn (eq_nn : n = n) t : tcast eq_nn t = t.
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 | tcast_id | |
tcastKm n (eq_mn : m = n) : cancel (tcast eq_mn) (tcast (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 | tcastK | |
tcastKVm n (eq_mn : m = n) : cancel (tcast (esym eq_mn)) (tcast 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 | tcastKV | |
tcast_transm n p (eq_mn : m = n) (eq_np : n = p) t:
tcast (etrans eq_mn eq_np) t = tcast eq_np (tcast eq_mn t).
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 | tcast_trans | |
tvalKn (t : n.-tuple T) : in_tuple t = tcast (esym (size_tuple t)) t.
Proof. by apply: val_inj => /=; case: _ / (esym _). 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 | tvalK | |
val_tcastm n (eq_mn : m = n) (t : m.-tuple T) :
tcast eq_mn t = t :> seq T.
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 | val_tcast | |
in_tupleEs : in_tuple 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_tupleE | |
rcons_tuplePt x : size (rcons t x) == n.+1.
Proof. by rewrite size_rcons 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 | rcons_tupleP | |
rcons_tuplet x := Tuple (rcons_tupleP t 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_tuple | |
nseq_tuplePx : @size T (nseq n x) == n.
Proof. by rewrite size_nseq. 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 | nseq_tupleP | |
nseq_tuplex := Tuple (nseq_tupleP 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 | nseq_tuple | |
iota_tupleP: size (iota m n) == n.
Proof. by rewrite size_iota. 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 | iota_tupleP | |
iota_tuple:= Tuple iota_tupleP. | 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 | iota_tuple | |
behead_tuplePt : size (behead t) == n.-1.
Proof. by rewrite size_behead 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 | behead_tupleP | |
behead_tuplet := Tuple (behead_tupleP 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 | behead_tuple | |
belast_tuplePx t : size (belast x t) == n.
Proof. by rewrite size_belast 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 | belast_tupleP | |
belast_tuplex t := Tuple (belast_tupleP x 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 | belast_tuple | |
cat_tuplePt (u : m.-tuple T) : size (t ++ u) == n + m.
Proof. by rewrite size_cat !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 | cat_tupleP | |
cat_tuplet u := Tuple (cat_tupleP t u). | 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_tuple | |
take_tuplePt : size (take m t) == minn m n.
Proof. by rewrite size_take size_tuple eqxx. 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_tupleP | |
take_tuplet := Tuple (take_tupleP 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 | take_tuple | |
drop_tuplePt : size (drop m t) == n - m.
Proof. by rewrite size_drop 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 | drop_tupleP | |
drop_tuplet := Tuple (drop_tupleP 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 | drop_tuple | |
rev_tuplePt : size (rev t) == n.
Proof. by rewrite size_rev 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 | rev_tupleP | |
rev_tuplet := Tuple (rev_tupleP 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 | rev_tuple | |
rot_tuplePt : size (rot m t) == n.
Proof. by rewrite size_rot 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 | rot_tupleP | |
rot_tuplet := Tuple (rot_tupleP 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 | rot_tuple | |
rotr_tuplePt : size (rotr m t) == n.
Proof. by rewrite size_rotr 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 | rotr_tupleP | |
rotr_tuplet := Tuple (rotr_tupleP 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 | rotr_tuple | |
map_tuplePf t : @size rT (map f t) == n.
Proof. by rewrite size_map 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 | map_tupleP | |
map_tuplef t := Tuple (map_tupleP f 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 | map_tuple | |
scanl_tuplePf x t : @size rT (scanl f x t) == n.
Proof. by rewrite size_scanl 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 | scanl_tupleP | |
scanl_tuplef x t := Tuple (scanl_tupleP f x 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 | scanl_tuple | |
pairmap_tuplePf x t : @size rT (pairmap f x t) == n.
Proof. by rewrite size_pairmap 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 | pairmap_tupleP | |
pairmap_tuplef x t := Tuple (pairmap_tupleP f x 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 | pairmap_tuple | |
zip_tuplePt (u : n.-tuple U) : size (zip t u) == n.
Proof. by rewrite size1_zip !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 | zip_tupleP | |
zip_tuplet u := Tuple (zip_tupleP t u). | 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 | zip_tuple | |
allpairs_tuplePf t (u : m.-tuple U) : @size rT (allpairs f t u) == n * m.
Proof. by rewrite size_allpairs !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 | allpairs_tupleP | |
allpairs_tuplef t u := Tuple (allpairs_tupleP f t u). | 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_tuple |
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