statement stringlengths 1 4.33k | proof stringlengths 0 37.9k | type stringclasses 25
values | symbolic_name stringlengths 1 67 | library stringclasses 10
values | filename stringclasses 112
values | imports listlengths 2 138 | deps listlengths 0 64 | docstring stringclasses 798
values | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
"n .-bseq" | := (bseq_of n) (format "n .-bseq") : type_scope. | Notation | n .-bseq | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"bseq_of"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"{ 'bseq' n 'of' T }" | := (n.-bseq T : predArgType)
(only parsing) : type_scope. | Notation | { 'bseq' n 'of' T } | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"bseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'bseq' 'of' s ]" | := (bseq (fun sP => @Bseq _ _ s sP))
(format "[ 'bseq' 'of' s ]") : form_scope. | Notation | [ 'bseq' 'of' s ] | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"bseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'bseq' x1 ; .. ; xn ]" | := [bseq of x1 :: .. [:: xn] ..]
(format "[ 'bseq' '[' x1 ; '/' .. ; '/' xn ']' ]") : form_scope. | Notation | [ 'bseq' x1 ; .. ; xn ] | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"bseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'bseq' ]" | := [bseq of [::]] (format "[ 'bseq' ]") : form_scope. | Notation | [ 'bseq' ] | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"bseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bseq_of_tuple n T (t : n.-tuple T) : n.-bseq T | :=
Bseq (eq_leq (size_tuple t)). | Coercion | bseq_of_tuple | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"bseq",
"eq_leq",
"size_tuple",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
insub_bseq n T (s : seq T) : n.-bseq T | := insubd [bseq] s. | Definition | insub_bseq | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"bseq",
"insubd",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_insub_bseq n T (s : seq T) : size (insub_bseq n s) <= size s. | Proof. by rewrite /insub_bseq /insubd; case: insubP => // ? ? ->. Qed. | Lemma | size_insub_bseq | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"insubP",
"insub_bseq",
"insubd",
"seq",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_bseq (s : seq T) : (size s).-bseq T | := Bseq (leqnn (size s)). | Definition | in_bseq | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"bseq",
"leqnn",
"seq",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cast_bseq m n (eq_mn : m = n) bs | :=
let: erefl in _ = n := eq_mn return n.-bseq T in bs. | Definition | cast_bseq | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"bseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
widen_bseq m n (lemn : m <= n) (bs : m.-bseq T) : n.-bseq T | :=
@Bseq n T bs (leq_trans (size_bseq bs) lemn). | Definition | widen_bseq | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"bseq",
"leq_trans",
"size_bseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cast_bseq_id n (eq_nn : n = n) bs : cast_bseq eq_nn bs = bs. | Proof. by rewrite (eq_axiomK eq_nn). Qed. | Lemma | cast_bseq_id | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"cast_bseq",
"eq_axiomK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cast_bseqK m n (eq_mn : m = n) :
cancel (cast_bseq eq_mn) (cast_bseq (esym eq_mn)). | Proof. by case: n / eq_mn. Qed. | Lemma | cast_bseqK | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"cast_bseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cast_bseqKV m n (eq_mn : m = n) :
cancel (cast_bseq (esym eq_mn)) (cast_bseq eq_mn). | Proof. by case: n / eq_mn. Qed. | Lemma | cast_bseqKV | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"cast_bseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cast_bseq_trans m 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 | cast_bseq_trans | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"cast_bseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_cast_bseq m 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 | size_cast_bseq | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"bseq",
"cast_bseq",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
widen_bseq_id n (lenn : n <= n) (bs : n.-bseq T) :
widen_bseq lenn bs = bs. | Proof. exact: val_inj. Qed. | Lemma | widen_bseq_id | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"bseq",
"val_inj",
"widen_bseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cast_bseqEwiden m 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 | cast_bseqEwiden | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"bseq",
"cast_bseq",
"eq_leq",
"widen_bseq",
"widen_bseq_id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
widen_bseqK m 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 | widen_bseqK | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"apply",
"val_inj",
"widen_bseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
widen_bseq_trans m 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 | widen_bseq_trans | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"bseq",
"leq_trans",
"val_inj",
"widen_bseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_widen_bseq m n (lemn : m <= n) (bs : m.-bseq T) :
size (widen_bseq lemn bs) = size bs. | Proof. by []. Qed. | Lemma | size_widen_bseq | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"bseq",
"size",
"widen_bseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_bseqE s : in_bseq s = s :> seq T. | Proof. by []. Qed. | Lemma | in_bseqE | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"in_bseq",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
widen_bseq_in_bseq n (bs : n.-bseq T) :
widen_bseq (size_bseq bs) (in_bseq bs) = bs. | Proof. exact: val_inj. Qed. | Lemma | widen_bseq_in_bseq | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"bseq",
"in_bseq",
"size_bseq",
"val_inj",
"widen_bseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rcons_bseqP s x : size (rcons s x) <= n.+1. | Proof. by rewrite size_rcons ltnS size_bseq. Qed. | Lemma | rcons_bseqP | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"ltnS",
"rcons",
"size",
"size_bseq",
"size_rcons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rcons_bseq s x | := Bseq (rcons_bseqP s x). | Canonical | rcons_bseq | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"rcons_bseqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
behead_bseqP s : size (behead s) <= n.-1. | Proof. rewrite size_behead -!subn1; apply/leq_sub2r/size_bseq. Qed. | Lemma | behead_bseqP | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"apply",
"behead",
"leq_sub2r",
"size",
"size_behead",
"size_bseq",
"subn1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
behead_bseq s | := Bseq (behead_bseqP s). | Canonical | behead_bseq | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"behead_bseqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
belast_bseqP x s : size (belast x s) <= n. | Proof. by rewrite size_belast; apply/size_bseq. Qed. | Lemma | belast_bseqP | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"apply",
"belast",
"size",
"size_belast",
"size_bseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
belast_bseq x s | := Bseq (belast_bseqP x s). | Canonical | belast_bseq | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"belast_bseqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cat_bseqP s (s' : m.-bseq T) : size (s ++ s') <= n + m. | Proof. by rewrite size_cat; apply/leq_add/size_bseq/size_bseq. Qed. | Lemma | cat_bseqP | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"apply",
"bseq",
"leq_add",
"size",
"size_bseq",
"size_cat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cat_bseq s (s' : m.-bseq T) | := Bseq (cat_bseqP s s'). | Canonical | cat_bseq | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"bseq",
"cat_bseqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
take_bseqP s : size (take m s) <= n. | Proof.
by rewrite size_take_min (leq_trans _ (size_bseq s)) // geq_minr.
Qed. | Lemma | take_bseqP | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"geq_minr",
"leq_trans",
"size",
"size_bseq",
"size_take_min",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
take_bseq s | := Bseq (take_bseqP s). | Canonical | take_bseq | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"take_bseqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
drop_bseqP s : size (drop m s) <= n - m. | Proof. by rewrite size_drop; apply/leq_sub2r/size_bseq. Qed. | Lemma | drop_bseqP | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"apply",
"drop",
"leq_sub2r",
"size",
"size_bseq",
"size_drop"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
drop_bseq s | := Bseq (drop_bseqP s). | Canonical | drop_bseq | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"drop_bseqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rev_bseqP s : size (rev s) <= n. | Proof. by rewrite size_rev size_bseq. Qed. | Lemma | rev_bseqP | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"rev",
"size",
"size_bseq",
"size_rev"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rev_bseq s | := Bseq (rev_bseqP s). | Canonical | rev_bseq | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"rev_bseqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rot_bseqP s : size (rot m s) <= n. | Proof. by rewrite size_rot size_bseq. Qed. | Lemma | rot_bseqP | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"rot",
"size",
"size_bseq",
"size_rot"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rot_bseq s | := Bseq (rot_bseqP s). | Canonical | rot_bseq | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"rot_bseqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rotr_bseqP s : size (rotr m s) <= n. | Proof. by rewrite size_rotr size_bseq. Qed. | Lemma | rotr_bseqP | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"rotr",
"size",
"size_bseq",
"size_rotr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rotr_bseq s | := Bseq (rotr_bseqP s). | Canonical | rotr_bseq | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"rotr_bseqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_bseqP f s : @size rT (map f s) <= n. | Proof. by rewrite size_map size_bseq. Qed. | Lemma | map_bseqP | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"map",
"size",
"size_bseq",
"size_map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_bseq f s | := Bseq (map_bseqP f s). | Canonical | map_bseq | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"map_bseqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scanl_bseqP f x s : @size rT (scanl f x s) <= n. | Proof. by rewrite size_scanl size_bseq. Qed. | Lemma | scanl_bseqP | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"scanl",
"size",
"size_bseq",
"size_scanl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scanl_bseq f x s | := Bseq (scanl_bseqP f x s). | Canonical | scanl_bseq | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"scanl_bseqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pairmap_bseqP f x s : @size rT (pairmap f x s) <= n. | Proof. by rewrite size_pairmap size_bseq. Qed. | Lemma | pairmap_bseqP | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"pairmap",
"size",
"size_bseq",
"size_pairmap"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pairmap_bseq f x s | := Bseq (pairmap_bseqP f x s). | Canonical | pairmap_bseq | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"pairmap_bseqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
allpairs_bseqP f 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 | allpairs_bseqP | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"allpairs",
"apply",
"bseq",
"leq_mul",
"size",
"size_allpairs",
"size_bseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
allpairs_bseq f s (s' : m.-bseq U) | := Bseq (allpairs_bseqP f s s'). | Canonical | allpairs_bseq | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"allpairs_bseqP",
"bseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sort_bseqP r s : size (sort r s) <= n. | Proof. by rewrite size_sort size_bseq. Qed. | Lemma | sort_bseqP | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"size",
"size_bseq",
"size_sort",
"sort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sort_bseq r s | := Bseq (sort_bseqP r s). | Canonical | sort_bseq | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"sort_bseqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bseq0 : all_equal_to ([bseq] : 0.-bseq T). | Proof. by move=> s; apply: val_inj; case: s => [[]]. Qed. | Lemma | bseq0 | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"apply",
"bseq",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bseq_predType n (T : eqType) | :=
Eval hnf in PredType (fun t : n.-bseq T => mem_seq t). | Canonical | bseq_predType | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"bseq",
"mem_seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
membsE n (T : eqType) (bs : n.-bseq T) : mem bs = mem (bseqval bs). | Proof. by []. Qed. | Lemma | membsE | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"bseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bseq_tagged_tuple n T (s : n.-bseq T) : {k : 'I_n.+1 & k.-tuple T} | :=
Tagged _ (in_tuple s : (Ordinal (size_bseq s : size s < n.+1)).-tuple _). | Definition | bseq_tagged_tuple | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"bseq",
"in_tuple",
"size",
"size_bseq",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tagged_tuple_bseq n T (t : {k : 'I_n.+1 & k.-tuple T}) : n.-bseq T | :=
widen_bseq (leq_ord (tag t)) (tagged t). | Definition | tagged_tuple_bseq | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"bseq",
"leq_ord",
"tuple",
"widen_bseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bseq_tagged_tupleK {n T} :
cancel (@bseq_tagged_tuple n T) tagged_tuple_bseq. | Proof. by move=> bs; apply/val_inj. Qed. | Lemma | bseq_tagged_tupleK | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"apply",
"bseq_tagged_tuple",
"tagged_tuple_bseq",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tagged_tuple_bseqK {n T} :
cancel (@tagged_tuple_bseq n T) bseq_tagged_tuple. | Proof.
move=> [[k lt_kn] t]; apply: eq_existT_curried => [|k_eq]; apply/val_inj.
by rewrite /= size_tuple.
by refine (let: erefl := k_eq in _).
Qed. | Lemma | tagged_tuple_bseqK | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"apply",
"bseq_tagged_tuple",
"size_tuple",
"tagged_tuple_bseq",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bseq_tagged_tuple_bij {n T} : bijective (@bseq_tagged_tuple n T). | Proof. exact/Bijective/tagged_tuple_bseqK/bseq_tagged_tupleK. Qed. | Lemma | bseq_tagged_tuple_bij | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"bseq_tagged_tuple",
"bseq_tagged_tupleK",
"tagged_tuple_bseqK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tagged_tuple_bseq_bij {n T} : bijective (@tagged_tuple_bseq n T). | Proof. exact/Bijective/bseq_tagged_tupleK/tagged_tuple_bseqK. Qed. | Lemma | tagged_tuple_bseq_bij | boot | boot/tuple.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"path"
] | [
"bseq_tagged_tupleK",
"tagged_tuple_bseq",
"tagged_tuple_bseqK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nz2: 2 != 0 :> L. | Proof.
apply/eqP=> pchar2; apply: conj_nt => e; apply/eqP/idPn=> eJ.
have opp_id x: - x = x :> L.
by apply/esym/eqP; rewrite -addr_eq0 -mulr2n -mulr_natl pchar2 mul0r.
have{} pchar2: 2%N \in [pchar L] by apply/eqP.
without loss{eJ} eJ: e / conj e = e + 1.
move/(_ (e / (e + conj e))); apply.
rewrite ... | Lemma | nz2 | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"addKr",
"addNKr",
"addr0",
"addrACA",
"addrC",
"addrCA",
"addrI",
"addr_eq0",
"apply",
"big_ord0",
"big_ord_recl",
"conj",
"conjK",
"conj_nt",
"divff",
"fmorph_div",
"mul0r",
"mul1r",
"mulN1r",
"mulf_eq0",
"mulr1",
"mulr2n",
"mulrC",
"mulrDl",
"mulrDr",
"mulr_natl"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul2I: injective (fun z : L => z *+ 2). | Proof.
have nz2 := nz2.
by move=> x y; rewrite /= -mulr_natl -(mulr_natl y) => /mulfI->.
Qed. | Lemma | mul2I | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"mulfI",
"mulr_natl",
"nz2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrt x : L | :=
sval (sig_eqW (@solve_monicpoly _ 2%N (nth 0 [:: x]) isT)). | Definition | sqrt | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"nth",
"sig_eqW",
"solve_monicpoly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrtK x: sqrt x ^+ 2 = x. | Proof.
rewrite /sqrt; case: sig_eqW => /= y ->.
by rewrite !big_ord_recl big_ord0 /= mulr1 mul0r !addr0.
Qed. | Lemma | sqrtK | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"addr0",
"big_ord0",
"big_ord_recl",
"mul0r",
"mulr1",
"sig_eqW",
"sqrt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrtE x y: y ^+ 2 = x -> {b : bool | y = (-1) ^+ b * sqrt x}. | Proof.
move=> Dx; exists (y != sqrt x); apply/eqP; rewrite mulr_sign if_neg.
by case: ifPn => //; apply/implyP; rewrite implyNb -eqf_sqr Dx sqrtK.
Qed. | Lemma | sqrtE | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Dx",
"apply",
"eqf_sqr",
"mulr_sign",
"sqrt",
"sqrtK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
i | := sqrt (- 1). | Definition | i | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"sqrt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrMi x: (i * x) ^+ 2 = - x ^+ 2. | Proof. by rewrite exprMn sqrtK mulN1r. Qed. | Lemma | sqrMi | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"exprMn",
"mulN1r",
"sqrtK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iJ : conj i = - i. | Proof.
have nz2 := nz2.
have /sqrtE[b]: conj i ^+ 2 = - 1 by rewrite -rmorphXn /= sqrtK rmorphN1.
rewrite mulr_sign -/i; case: b => // Ri.
case: conj_nt => z; wlog zJ: z / conj z = - z.
move/(_ (z - conj z)); rewrite !rmorphB conjK opprB => zJ.
by apply/mul2I/(canRL (subrK _)); rewrite addrAC zJ // subr... | Lemma | iJ | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"add0r",
"addrA",
"addrAC",
"addrACA",
"addrC",
"addr_eq0",
"apply",
"conj",
"conjK",
"conj_nt",
"divfK",
"divrN",
"eqVneq",
"eqf_sqr",
"fmorph_div",
"inj_eq",
"invfM",
"mul0rn",
"mul2I",
"mulNr",
"mulNrn",
"mulfK",
"mulr2n",
"mulrA",
"mulrC",
"mulrDl",
"mulrN",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
norm x | := sqrt x * conj (sqrt x). | Definition | norm | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"conj",
"sqrt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normK x : norm x ^+ 2 = x * conj x. | Proof. by rewrite exprMn -rmorphXn sqrtK. Qed. | Lemma | normK | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"conj",
"exprMn",
"norm",
"rmorphXn",
"sqrtK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normE x y : y ^+ 2 = x -> norm x = y * conj y. | Proof.
rewrite /norm => /sqrtE[b /(canLR (signrMK b)) <-].
by rewrite !rmorphM /= rmorph_sign mulrACA -mulrA signrMK.
Qed. | Lemma | normE | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"conj",
"mulrA",
"mulrACA",
"norm",
"rmorphM",
"rmorph_sign",
"signrMK",
"sqrtE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
norm_eq0 x : norm x = 0 -> x = 0. | Proof.
by move/eqP; rewrite mulf_eq0 fmorph_eq0 -mulf_eq0 -expr2 sqrtK => /eqP.
Qed. | Lemma | norm_eq0 | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"expr2",
"fmorph_eq0",
"mulf_eq0",
"norm",
"sqrtK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normM x y : norm (x * y) = norm x * norm y. | Proof.
by rewrite mulrACA -rmorphM; apply: normE; rewrite exprMn !sqrtK.
Qed. | Lemma | normM | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"apply",
"exprMn",
"mulrACA",
"norm",
"normE",
"rmorphM",
"sqrtK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normN x : norm (- x) = norm x. | Proof.
by rewrite -mulN1r normM {1}/norm iJ mulrN -expr2 sqrtK opprK mul1r.
Qed. | Lemma | normN | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"expr2",
"iJ",
"mul1r",
"mulN1r",
"mulrN",
"norm",
"normM",
"opprK",
"sqrtK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le x y | := norm (y - x) == y - x. | Definition | le | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"norm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt x y | := (y != x) && le x y. | Definition | lt | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"le"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
posE x: le 0 x = (norm x == x). | Proof. by rewrite /le subr0. Qed. | Lemma | posE | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"le",
"norm",
"subr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leB x y: le x y = le 0 (y - x). | Proof. by rewrite posE. Qed. | Lemma | leB | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"le",
"posE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
posP x : reflect (exists y, x = y * conj y) (le 0 x). | Proof.
rewrite posE; apply: (iffP eqP) => [Dx | [y {x}->]]; first by exists (sqrt x).
by rewrite (normE (normK y)) rmorphM /= conjK (mulrC (conj _)) -expr2 normK.
Qed. | Lemma | posP | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Dx",
"apply",
"conj",
"conjK",
"expr2",
"le",
"mulrC",
"normE",
"normK",
"posE",
"rmorphM",
"sqrt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
posJ x : le 0 x -> conj x = x. | Proof.
by case/posP=> {x}u ->; rewrite rmorphM /= conjK mulrC.
Qed. | Lemma | posJ | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"conj",
"conjK",
"le",
"mulrC",
"posP",
"rmorphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pos_linear x y : le 0 x -> le 0 y -> le x y || le y x. | Proof.
move=> pos_x pos_y; rewrite leB -opprB orbC leB !posE normN -eqf_sqr.
by rewrite normK rmorphB !posJ ?subrr.
Qed. | Lemma | pos_linear | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"eqf_sqr",
"le",
"leB",
"normK",
"normN",
"opprB",
"posE",
"posJ",
"rmorphB",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sposDl x y : lt 0 x -> le 0 y -> lt 0 (x + y). | Proof.
have sqrtJ z : le 0 z -> conj (sqrt z) = sqrt z.
rewrite posE -{2}[z]sqrtK -subr_eq0 -mulrBr mulf_eq0 subr_eq0.
by case/pred2P=> ->; rewrite ?rmorph0.
case/andP=> nz_x /sqrtJ uJ /sqrtJ vJ.
set u := sqrt x in uJ; set v := sqrt y in vJ; pose w := u + i * v.
have ->: x + y = w * conj w.
rewrite ... | Lemma | sposDl | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"addrA",
"apply",
"conj",
"contraNneq",
"expf_eq0",
"iJ",
"inj_eq",
"last",
"le",
"lt",
"mul2I",
"mulNr",
"mulf_eq0",
"mulr2n",
"mulrBr",
"mulrC",
"normK",
"norm_eq0",
"opprK",
"posE",
"posP",
"pred2P",
"rmorph0",
"rmorphD",
"rmorphM",
"split",
"sqrMi",
"sqrt",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sposD x y : lt 0 x -> lt 0 y -> lt 0 (x + y). | Proof.
by move=> x_gt0 /andP[_]; apply: sposDl.
Qed. | Lemma | sposD | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"apply",
"lt",
"sposDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normD x y : le (norm (x + y)) (norm x + norm y). | Proof.
have sposM u v: lt 0 u -> le 0 (u * v) -> le 0 v.
by rewrite /lt !posE normM andbC => /andP[/eqP-> /mulfI/inj_eq->].
have posD u v: le 0 u -> le 0 v -> le 0 (u + v).
have [-> | nz_u u_ge0 v_ge0] := eqVneq u 0; first by rewrite add0r.
by have /andP[]: lt 0 (u + v) by rewrite sposDl // /lt nz_u.
... | Lemma | normD | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"add0r",
"addr0",
"addrA",
"addrAC",
"addrACA",
"addrC",
"addrCA",
"addrKA",
"addr_eq0",
"apply",
"conj",
"conjK",
"eqVneq",
"exprMn",
"iJ",
"inj_eq",
"le",
"leB",
"lt",
"mulNr",
"mulfI",
"mulr0",
"mulr2n",
"mulrACA",
"mulrC",
"mulrDl",
"mulrDr",
"mulrN",
"mul... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
type : Type. | Parameter | type | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
conjMixin : Num.ClosedField type. | Parameter | conjMixin | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
isCountable : Countable type. | Parameter | isCountable | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
archimedean : Num.archimedean_axiom (Num.ClosedField.Pack conjMixin). | Axiom | archimedean | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"archimedean_axiom",
"conjMixin"
] | Note that this cannot be included in conjMixin since a few proofs
depend from nat_num being definitionally equal to (truncn x)%:R == x | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algebraic : integralRange (@ratr (Num.ClosedField.Pack conjMixin)). | Axiom | algebraic | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"conjMixin",
"integralRange",
"ratr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
L | := tag Fundamental_Theorem_of_Algebraics. | Definition | L | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Fundamental_Theorem_of_Algebraics"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjL : {rmorphism L -> L} | :=
s2val (tagged Fundamental_Theorem_of_Algebraics). | Definition | conjL | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Fundamental_Theorem_of_Algebraics"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjL_K : involutive conjL. | Proof. exact: s2valP (tagged Fundamental_Theorem_of_Algebraics). Qed. | Fact | conjL_K | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Fundamental_Theorem_of_Algebraics",
"conjL"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjL_nt : ~ conjL =1 id. | Proof. exact: s2valP' (tagged Fundamental_Theorem_of_Algebraics). Qed. | Fact | conjL_nt | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Fundamental_Theorem_of_Algebraics",
"conjL",
"id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
L' : Type | := eta L. | Definition | L' | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfType | := (L' : closedFieldType). | Notation | cfType | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"L'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
QtoL : {rmorphism _ -> _} | := @ratr cfType. | Definition | QtoL | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"cfType",
"ratr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pQtoL | := (map_poly QtoL). | Notation | pQtoL | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"QtoL",
"map_poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rootQtoL p_j | :=
if p_j.1 == 0 then 0 else
(sval (closed_field_poly_normal (pQtoL p_j.1)))`_p_j.2. | Definition | rootQtoL | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"closed_field_poly_normal",
"pQtoL"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_root p_j q_k | := rootQtoL p_j == rootQtoL q_k. | Definition | eq_root | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"rootQtoL"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_root_is_equiv : equiv_class_of eq_root. | Proof. by rewrite /eq_root; split=> [ ? | ? ? | ? ? ? ] // /eqP->. Qed. | Fact | eq_root_is_equiv | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"eq_root",
"equiv_class_of",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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