statement stringlengths 1 4.33k | proof stringlengths 0 37.9k | type stringclasses 25
values | symbolic_name stringlengths 1 67 | library stringclasses 10
values | filename stringclasses 112
values | imports listlengths 2 138 | deps listlengths 0 64 | docstring stringclasses 798
values | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
contra_leqN b m n : (b -> m < n) -> (n <= m -> ~~ b). | Proof. by rewrite ltnNge; apply: contraTN. Qed. | Lemma | contra_leqN | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
contra_ltnN b m n : (b -> m <= n) -> (n < m -> ~~ b). | Proof. by rewrite ltnNge; apply: contraNN. Qed. | Lemma | contra_ltnN | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
contra_leq_not P m n : (P -> m < n) -> (n <= m -> ~ P). | Proof. by rewrite ltnNge; apply: contraTnot. Qed. | Lemma | contra_leq_not | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
contra_ltn_not P m n : (P -> m <= n) -> (n < m -> ~ P). | Proof. by rewrite ltnNge; apply: contraNnot. Qed. | Lemma | contra_ltn_not | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
contra_leqF b m n : (b -> m < n) -> (n <= m -> b = false). | Proof. by rewrite ltnNge; apply: contraTF. Qed. | Lemma | contra_leqF | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
contra_ltnF b m n : (b -> m <= n) -> (n < m -> b = false). | Proof. by rewrite ltnNge; apply: contraNF. Qed. | Lemma | contra_ltnF | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
contra_leq m n p q : (q < p -> n < m) -> (m <= n -> p <= q). | Proof. by rewrite !ltnNge; apply: contraTT. Qed. | Lemma | contra_leq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
contra_leq_ltn m n p q : (q <= p -> n < m) -> (m <= n -> p < q). | Proof. by rewrite !ltnNge; apply: contraTN. Qed. | Lemma | contra_leq_ltn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
contra_ltn_leq m n p q : (q < p -> n <= m) -> (m < n -> p <= q). | Proof. by rewrite !ltnNge; apply: contraNT. Qed. | Lemma | contra_ltn_leq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
contra_ltn m n p q : (q <= p -> n <= m) -> (m < n -> p < q). | Proof. by rewrite !ltnNge; apply: contraNN. Qed. | Lemma | contra_ltn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
homo_ltn_in (D : {pred nat}) (f : nat -> T) (r : T -> T -> Prop) :
(forall y x z, r x y -> r y z -> r x z) ->
{in D &, forall i j k, i < k < j -> k \in D} ->
{in D, forall i, i.+1 \in D -> r (f i) (f i.+1)} ->
{in D &, {homo f : i j / i < j >-> r i j}}. | Proof.
move=> r_trans Dcx r_incr i j iD jD lt_ij; move: (lt_ij) (jD) => /subnKC<-.
elim: (_ - _) => [|k ihk]; first by rewrite addn0 => Dsi; apply: r_incr.
move=> DSiSk [: DSik]; apply: (r_trans _ _ _ (ihk _)); rewrite ?addnS.
by abstract: DSik; apply: (Dcx _ _ iD DSiSk); rewrite ltn_addr ?addnS /=.
by apply: r_incr;... | Lemma | homo_ltn_in | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addn0",
"addnS",
"apply",
"ltn_addr",
"nat",
"subnKC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
homo_ltn (f : nat -> T) (r : T -> T -> Prop) :
(forall y x z, r x y -> r y z -> r x z) ->
(forall i, r (f i) (f i.+1)) -> {homo f : i j / i < j >-> r i j}. | Proof. by move=> /(@homo_ltn_in predT f) fr fS i j; apply: fr. Qed. | Lemma | homo_ltn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"homo_ltn_in",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
homo_leq_in (D : {pred nat}) (f : nat -> T) (r : T -> T -> Prop) :
(forall x, r x x) -> (forall y x z, r x y -> r y z -> r x z) ->
{in D &, forall i j k, i < k < j -> k \in D} ->
{in D, forall i, i.+1 \in D -> r (f i) (f i.+1)} ->
{in D &, {homo f : i j / i <= j >-> r i j}}. | Proof.
move=> r_refl r_trans Dcx /(homo_ltn_in r_trans Dcx) lt_r i j iD jD.
case: ltngtP => [? _||->] //; exact: lt_r.
Qed. | Lemma | homo_leq_in | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"homo_ltn_in",
"ltngtP",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
homo_leq (f : nat -> T) (r : T -> T -> Prop) :
(forall x, r x x) -> (forall y x z, r x y -> r y z -> r x z) ->
(forall i, r (f i) (f i.+1)) -> {homo f : i j / i <= j >-> r i j}. | Proof. by move=> rrefl /(@homo_leq_in predT f r) fr fS i j; apply: fr. Qed. | Lemma | homo_leq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"homo_leq_in",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_neqAle | := ltn_neqAle. | Let | ltn_neqAle | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gtn_neqAge x y : (y < x) = (x != y) && (y <= x). | Proof. by rewrite ltn_neqAle eq_sym. Qed. | Let | gtn_neqAge | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"eq_sym",
"ltn_neqAle"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
anti_leq | := anti_leq. | Let | anti_leq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
anti_geq : antisymmetric geq. | Proof. by move=> m n /=; rewrite andbC => /anti_leq. Qed. | Let | anti_geq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"anti_leq",
"geq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_total | := leq_total. | Let | leq_total | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltnW_homo : {homo f : m n / m < n} -> {homo f : m n / m <= n}. | Proof. exact: homoW. Qed. | Lemma | ltnW_homo | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"homoW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inj_homo_ltn : injective f -> {homo f : m n / m <= n} ->
{homo f : m n / m < n}. | Proof. exact: inj_homo. Qed. | Lemma | inj_homo_ltn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"inj_homo"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltnW_nhomo : {homo f : m n /~ m < n} -> {homo f : m n /~ m <= n}. | Proof. exact: homoW. Qed. | Lemma | ltnW_nhomo | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"homoW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inj_nhomo_ltn : injective f -> {homo f : m n /~ m <= n} ->
{homo f : m n /~ m < n}. | Proof. exact: inj_homo. Qed. | Lemma | inj_nhomo_ltn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"inj_homo"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
incn_inj : {mono f : m n / m <= n} -> injective f. | Proof. exact: mono_inj. Qed. | Lemma | incn_inj | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"mono_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
decn_inj : {mono f : m n /~ m <= n} -> injective f. | Proof. exact: mono_inj. Qed. | Lemma | decn_inj | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"mono_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leqW_mono : {mono f : m n / m <= n} -> {mono f : m n / m < n}. | Proof. exact: anti_mono. Qed. | Lemma | leqW_mono | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"anti_mono"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leqW_nmono : {mono f : m n /~ m <= n} -> {mono f : m n /~ m < n}. | Proof. exact: anti_mono. Qed. | Lemma | leqW_nmono | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"anti_mono"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_mono : {homo f : m n / m < n} -> {mono f : m n / m <= n}. | Proof. exact: total_homo_mono. Qed. | Lemma | leq_mono | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"total_homo_mono"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_nmono : {homo f : m n /~ m < n} -> {mono f : m n /~ m <= n}. | Proof. exact: total_homo_mono. Qed. | Lemma | leq_nmono | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"total_homo_mono"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltnW_homo_in : {in D & D', {homo f : m n / m < n}} ->
{in D & D', {homo f : m n / m <= n}}. | Proof. exact: homoW_in. Qed. | Lemma | ltnW_homo_in | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"homoW_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltnW_nhomo_in : {in D & D', {homo f : m n /~ m < n}} ->
{in D & D', {homo f : m n /~ m <= n}}. | Proof. exact: homoW_in. Qed. | Lemma | ltnW_nhomo_in | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"homoW_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | inj_homo_ltn_in | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"inj_homo_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | inj_nhomo_ltn_in | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"inj_homo_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
incn_inj_in : {in D &, {mono f : m n / m <= n}} ->
{in D &, injective f}. | Proof. exact: mono_inj_in. Qed. | Lemma | incn_inj_in | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"mono_inj_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
decn_inj_in : {in D &, {mono f : m n /~ m <= n}} ->
{in D &, injective f}. | Proof. exact: mono_inj_in. Qed. | Lemma | decn_inj_in | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"mono_inj_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | leqW_mono_in | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"anti_mono_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | leqW_nmono_in | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"anti_mono_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | leq_mono_in | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"total_homo_mono_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | leq_nmono_in | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"total_homo_mono_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | leq_pfact | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"factorial",
"leq_mono_in",
"ltn_fact"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | leq_fact | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"fact_gt0",
"factorial",
"inE",
"leq_pfact",
"leq_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_pfact : {in [pred n | 0 < n] &, {mono factorial : m n / m < n}}. | Proof. exact/leqW_mono_in/leq_pfact. Qed. | Lemma | ltn_pfact | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"factorial",
"leqW_mono_in",
"leq_pfact"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add m n | := if m is m'.+1 then m' + n.+1 else n
where "n + m" := (add n m) : nat_scope. | Fixpoint | add | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | in the correctness proof, restores operators | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
add_mul m n s | := if m is m'.+1 then add_mul m' n (n + s) else s. | Fixpoint | add_mul | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul m n | := if m is m'.+1 then add_mul m' n n else 0. | Definition | mul | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"add_mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"n * m" | := (mul n m) : nat_scope. | Notation | n * m | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_exp m n p | := if n is n'.+1 then mul_exp m n' (m * p) else p. | Fixpoint | mul_exp | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exp m n | := if n is n'.+1 then mul_exp m n' m else 1. | Definition | exp | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"mul_exp",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"n ^ m" | := (exp n m) : nat_scope. | Notation | n ^ m | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"exp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oddn | := odd. | Notation | oddn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"odd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
odd n | := if n is n'.+2 then odd n' else eqn n 1. | Fixpoint | odd | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"eqn",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
doublen | := double. | Notation | doublen | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"double"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
double n | := if n is n'.+1 then n' + n.+1 else 0. | Definition | double | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addE : add =2 addn. | Proof. by elim=> //= n IHn m; rewrite IHn addSnnS. Qed. | Lemma | addE | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"add",
"addSnnS",
"addn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
doubleE : double =1 doublen. | Proof. by case=> // n; rewrite -addnn -addE. Qed. | Lemma | doubleE | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addE",
"addnn",
"double",
"doublen"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add_mulE n 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 | add_mulE | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addE",
"add_mul",
"addn",
"addnA",
"addnCA",
"muln"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulE : mul =2 muln. | Proof. by case=> //= n m; rewrite add_mulE addnC. Qed. | Lemma | mulE | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"add_mulE",
"addnC",
"mul",
"muln"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_expE m 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 | mul_expE | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"expn",
"expnS",
"mul1n",
"mulE",
"mul_exp",
"muln",
"mulnA",
"mulnCA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expE : exp =2 expn. | Proof. by move=> m [|n] //=; rewrite mul_expE expnS mulnC. Qed. | Lemma | expE | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"exp",
"expn",
"expnS",
"mul_expE",
"mulnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oddE : odd =1 oddn. | Proof.
move=> n; rewrite -[n in LHS]odd_double_half addnC.
by elim: n./2 => //=; case (oddn n).
Qed. | Lemma | oddE | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC",
"odd",
"odd_double_half",
"oddn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trecE | := (addE, (doubleE, oddE), (mulE, add_mulE, (expE, mul_expE))). | Definition | trecE | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addE",
"add_mulE",
"doubleE",
"expE",
"mulE",
"mul_expE",
"oddE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natTrecE | := NatTrec.trecE. | Notation | natTrecE | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"trecE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
N_eqb n m | :=
match n, m with
| N0, N0 => true
| Npos p, Npos q => Pos.eqb p q
| _, _ => false
end. | Definition | N_eqb | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"eqb"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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. | Lemma | eq_binP | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"N_eqb",
"apply",
"axiom",
"last"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_of_pos p0 | :=
match p0 with
| xO p => (nat_of_pos p).*2
| xI p => (nat_of_pos p).*2.+1
| xH => 1
end. | Fixpoint | nat_of_pos | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"p0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_of_pos : positive >-> nat. | Coercion | nat_of_pos | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
nat_of_bin b | := if b is Npos p then p : nat else 0. | Coercion | nat_of_bin | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pos_of_nat n0 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 | pos_of_nat | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bin_of_nat n0 | := if n0 is n.+1 then Npos (pos_of_nat n n) else N0. | Definition | bin_of_nat | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"pos_of_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | bin_of_natK | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addKn",
"addnn",
"bin_of_nat",
"natTrecE",
"nat_of_bin"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | nat_of_binK | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnS",
"addnn",
"bin_of_nat",
"natTrecE",
"nat_of_bin",
"nat_of_pos"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_of_succ_pos p : Pos.succ p = p.+1 :> nat. | Proof. by elim: p => //= p ->; rewrite !natTrecE. Qed. | Lemma | nat_of_succ_pos | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"nat",
"natTrecE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_of_add_pos p 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 | nat_of_add_pos | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"add",
"addn1",
"addnS",
"apply",
"doubleD",
"doubleS",
"nat",
"natTrecE",
"nat_of_succ_pos"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_of_mul_pos p 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 | nat_of_mul_pos | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"doubleMl",
"mul",
"mul1n",
"nat",
"natTrecE",
"nat_of_add_pos"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
number : Type | := Num {bin_of_number :> N}. | Record | number | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | returns a larger integer. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
number_subType | := Eval hnf in [isNew for bin_of_number]. | Definition | number_subType | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'Num' 'of' e ]" | := (Num (bin_of_nat e))
(format "[ 'Num' 'of' e ]") : nat_scope. | Notation | [ 'Num' 'of' e ] | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"bin_of_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pop_succn e | := if e is e'.+1 then fun n => pop_succn e' n.+1 else id. | Fixpoint | pop_succn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"e'",
"id"
] | normalization tactic. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
pop_succn e | := eval lazy beta iota delta [pop_succn] in (pop_succn e 1). | Ltac | pop_succn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"delta",
"eval",
"iota"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | succn_to_add | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"G'",
"e'",
"pop_succn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_norm | :=
succn_to_add; rewrite ?add0n ?addn0 -?addnA ?(addSn, addnS, add0n, addn0). | Ltac | nat_norm | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"add0n",
"addSn",
"addn0",
"addnA",
"addnS",
"succn_to_add"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | nat_congr | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addn",
"addnC",
"addnCA",
"apply",
"predn",
"subn",
"succn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tuple_of : Type | := Tuple {tval :> seq T; _ : size tval == n}. | Structure | tuple_of | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"seq",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tsize & tuple_of | := n. | Definition | tsize | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"tuple_of"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_tuple t : size t = n. | Proof. exact: (eqP (valP t)). Qed. | Lemma | size_tuple | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"size",
"valP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tnth_default t : 'I_n -> T. | Proof. by rewrite -(size_tuple t); case: (tval t) => [|//] []. Qed. | Lemma | tnth_default | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"size_tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tnth t i | := nth (tnth_default t i) t i. | Definition | tnth | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"nth",
"tnth_default"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tnth_nth x t i : tnth t i = nth x t i. | Proof. by apply: set_nth_default; rewrite size_tuple. Qed. | Lemma | tnth_nth | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"apply",
"nth",
"set_nth_default",
"size_tuple",
"tnth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tnth_onth x 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 | tnth_onth | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"nth_map",
"onth",
"onthE",
"size_tuple",
"split",
"tnth",
"tnth_nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_tnth_enum t : 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)) ... | Lemma | map_tnth_enum | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"apply",
"cardE",
"card_ord",
"enum",
"eq_from_nth",
"map",
"nth_enum_ord",
"nth_map",
"size0nil",
"size_map",
"size_tuple",
"tnth",
"tnth_nth",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_from_tnth t1 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 | eq_from_tnth | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"apply",
"eq_map",
"map_tnth_enum",
"tnth",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tuple t mkT : tuple_of | :=
mkT (let: Tuple _ tP := t return size t == n in tP). | Definition | tuple | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"size",
"tuple_of"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tupleE t : tuple (fun sP => @Tuple t sP) = t. | Proof. by case: t. Qed. | Lemma | tupleE | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"n .-tuple" | := (tuple_of n) (format "n .-tuple") : type_scope. | Notation | n .-tuple | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"tuple_of"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"{ 'tuple' n 'of' T }" | := (n.-tuple T : predArgType)
(only parsing) : type_scope. | Notation | { 'tuple' n 'of' T } | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'tuple' 'of' s ]" | := (tuple (fun sP => @Tuple _ _ s sP))
(format "[ 'tuple' 'of' s ]") : form_scope. | Notation | [ 'tuple' 'of' s ] | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'tnth' t i ]" | := (tnth t (@Ordinal (tsize t) i (erefl true)))
(t, i at level 8, format "[ 'tnth' t i ]") : form_scope. | Notation | [ 'tnth' t i ] | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"tnth",
"tsize"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nil_tuple T | := Tuple (isT : @size T [::] == 0). | Canonical | nil_tuple | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cons_tuple n T x (t : n.-tuple T) | :=
Tuple (valP t : size (x :: t) == n.+1). | Canonical | cons_tuple | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"size",
"tuple",
"valP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'tuple' x1 ; .. ; xn ]" | := [tuple of x1 :: .. [:: xn] ..]
(format "[ 'tuple' '[' x1 ; '/' .. ; '/' xn ']' ]") : form_scope. | Notation | [ 'tuple' x1 ; .. ; xn ] | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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