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Coq-MathComp

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edivp_map a b : edivp a^f b^f = (0, (a %/ b)^f, (a %% b)^f).
Proof. have [-> | bn0] := eqVneq b 0. rewrite (rmorph0 (map_poly f)) WeakIdomain.edivp_def !modp0 !divp0. by rewrite (rmorph0 (map_poly f)) scalp0. rewrite unlock redivp_map lead_coef_map rmorph_unit. by rewrite unitfE lead_coef_eq0. rewrite modpE divpE !map_polyZ [in RHS]rmorphV ?rmorphXn // unitfE. by rewrite e...
Lemma
edivp_map
algebra
algebra/polydiv.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...
[ "divp0", "divpE", "edivp", "edivp_def", "eqVneq", "expf_neq0", "lead_coef_eq0", "lead_coef_map", "map_poly", "map_polyZ", "modp0", "modpE", "redivp_map", "rmorph0", "rmorphV", "rmorphXn", "rmorph_unit", "scalp0", "unitfE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
scalp_map p q : scalp p^f q^f = scalp p q.
Proof. by rewrite /scalp edivp_map edivp_def. Qed.
Lemma
scalp_map
algebra
algebra/polydiv.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...
[ "edivp_def", "edivp_map", "scalp" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_divp p q : (p %/ q)^f = p^f %/ q^f.
Proof. by rewrite /divp edivp_map edivp_def. Qed.
Lemma
map_divp
algebra
algebra/polydiv.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...
[ "divp", "edivp_def", "edivp_map" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_modp p q : (p %% q)^f = p^f %% q^f.
Proof. by rewrite /modp edivp_map edivp_def. Qed.
Lemma
map_modp
algebra
algebra/polydiv.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...
[ "edivp_def", "edivp_map", "modp" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
egcdp_map p q : egcdp (map_poly f p) (map_poly f q) = (map_poly f (egcdp p q).1, map_poly f (egcdp p q).2).
Proof. wlog le_qp: p q / size q <= size p. move=> IH; have [/IH// | lt_qp] := leqP (size q) (size p). have /IH := ltnW lt_qp; rewrite /egcdp !size_map_poly ltnW // leqNgt lt_qp /=. by case: (egcdp_rec _ _ _) => u v [-> ->]. rewrite /egcdp !size_map_poly {}le_qp; move: (size q) => n. elim: n => /= [|n IHn] in p q ...
Lemma
egcdp_map
algebra
algebra/polydiv.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...
[ "egcdp", "egcdp_rec", "lead_coef_map", "leqNgt", "leqP", "ltnW", "map_divp", "map_modp", "map_poly", "map_polyZ", "map_poly_eq0", "rmorph0", "rmorph1", "rmorphB", "rmorphM", "rmorphXn", "scalp_map", "size", "size_map_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
dvdp_map p q : (p^f %| q^f) = (p %| q).
Proof. by rewrite /dvdp -map_modp map_poly_eq0. Qed.
Lemma
dvdp_map
algebra
algebra/polydiv.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...
[ "dvdp", "map_modp", "map_poly_eq0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
eqp_map p q : (p^f %= q^f) = (p %= q).
Proof. by rewrite /eqp !dvdp_map. Qed.
Lemma
eqp_map
algebra
algebra/polydiv.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...
[ "dvdp_map", "eqp" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
gcdp_map p q : (gcdp p q)^f = gcdp p^f q^f.
Proof. wlog lt_p_q: p q / size p < size q. move=> IHpq; case: (ltnP (size p) (size q)) => [|le_q_p]; first exact: IHpq. rewrite gcdpE (gcdpE p^f) !size_map_poly ltnNge le_q_p /= -map_modp. have [-> | q_nz] := eqVneq q 0; first by rewrite rmorph0 !gcdp0. by rewrite IHpq ?ltn_modp. have [m le_q_m] := ubnP (size q...
Lemma
gcdp_map
algebra
algebra/polydiv.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...
[ "eqVneq", "gcdp", "gcdp0", "gcdpE", "leq_trans", "ltnNge", "ltnP", "ltn_modp", "map_modp", "rmorph0", "size", "size_map_poly", "ubnP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coprimep_map p q : coprimep p^f q^f = coprimep p q.
Proof. by rewrite -!gcdp_eqp1 -eqp_map rmorph1 gcdp_map. Qed.
Lemma
coprimep_map
algebra
algebra/polydiv.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...
[ "coprimep", "eqp_map", "gcdp_eqp1", "gcdp_map", "rmorph1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
gdcop_rec_map p q n : (gdcop_rec p q n)^f = gdcop_rec p^f q^f n.
Proof. elim: n p q => [|n IH] => /= p q. by rewrite map_poly_eq0; case: eqP; rewrite ?rmorph1 ?rmorph0. rewrite /coprimep -gcdp_map size_map_poly. by case: eqP => Hq0 //; rewrite -map_divp -IH. Qed.
Lemma
gdcop_rec_map
algebra
algebra/polydiv.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...
[ "coprimep", "gcdp_map", "gdcop_rec", "map_divp", "map_poly_eq0", "rmorph0", "rmorph1", "size_map_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
gdcop_map p q : (gdcop p q)^f = gdcop p^f q^f.
Proof. by rewrite /gdcop gdcop_rec_map !size_map_poly. Qed.
Lemma
gdcop_map
algebra
algebra/polydiv.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...
[ "gdcop", "gdcop_rec_map", "size_map_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
root_coprimep (p q : {poly F}) : (forall x, root p x -> q.[x] != 0) -> coprimep p q.
Proof. move=> Ncmn; rewrite -gcdp_eqp1 -size_poly_eq1; apply/closed_rootP. by case=> r; rewrite root_gcd !rootE=> /andP [/Ncmn/negPf->]. Qed.
Lemma
root_coprimep
algebra
algebra/polydiv.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...
[ "apply", "closed_rootP", "coprimep", "gcdp_eqp1", "poly", "root", "rootE", "root_gcd", "size_poly_eq1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coprimepP (p q : {poly F}) : reflect (forall x, root p x -> q.[x] != 0) (coprimep p q).
Proof. by apply: (iffP idP)=> [/coprimep_root|/root_coprimep]. Qed.
Lemma
coprimepP
algebra
algebra/polydiv.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...
[ "apply", "coprimep", "coprimep_root", "poly", "root", "root_coprimep" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"p ^ f"
:= (map_poly f p) : ring_scope.
Notation
p ^ f
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "map_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
eval
:= horner_eval.
Notation
eval
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "horner_eval" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'Y"
:= 'X%:P : ring_scope.
Notation
'Y
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"p ^:P"
:= (p ^ polyC) (format "p ^:P") : ring_scope.
Notation
p ^:P
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "polyC" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"p .[ x , y ]"
:= (p.[x%:P].[y]) (left associativity, format "p .[ x , y ]") : ring_scope.
Notation
p .[ x , y ]
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
swapXY_key : unit.
Proof. by []. Qed.
Fact
swapXY_key
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "unit" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
swapXY_def u : {poly {poly R}}
:= (u ^ map_poly polyC).['Y].
Definition
swapXY_def
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "map_poly", "poly", "polyC" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
swapXY
:= locked_with swapXY_key swapXY_def.
Definition
swapXY
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "swapXY_def", "swapXY_key" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
swapXY_unlockable
:= [unlockable fun swapXY].
Canonical
swapXY_unlockable
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "swapXY" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sizeY u : nat
:= \max_(i < size u) (size u`_i).
Definition
sizeY
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "nat", "size" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
poly_XaY p : {poly {poly R}}
:= p^:P \Po ('X + 'Y).
Definition
poly_XaY
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
poly_XmY p : {poly {poly R}}
:= p^:P \Po ('X * 'Y).
Definition
poly_XmY
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_annihilant p q
:= resultant (poly_XaY p) q^:P.
Definition
sub_annihilant
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "poly_XaY", "resultant" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
div_annihilant p q
:= resultant (poly_XmY p) q^:P.
Definition
div_annihilant
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "poly_XmY", "resultant" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
swapXY_polyC p : swapXY p%:P = p^:P.
Proof. by rewrite unlock map_polyC hornerC. Qed.
Lemma
swapXY_polyC
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "hornerC", "map_polyC", "swapXY" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
swapXY_X : swapXY 'X = 'Y.
Proof. by rewrite unlock map_polyX hornerX. Qed.
Lemma
swapXY_X
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "hornerX", "map_polyX", "swapXY" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
swapXY_Y : swapXY 'Y = 'X.
Proof. by rewrite swapXY_polyC map_polyX. Qed.
Lemma
swapXY_Y
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "map_polyX", "swapXY", "swapXY_polyC" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
swapXY_is_zmod_morphism : zmod_morphism swapXY.
Proof. by move=> u v; rewrite unlock rmorphB !hornerE. Qed.
Lemma
swapXY_is_zmod_morphism
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "hornerE", "rmorphB", "swapXY", "zmod_morphism" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
swapXY_is_additive
:= swapXY_is_zmod_morphism.
Definition
swapXY_is_additive
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "swapXY_is_zmod_morphism" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coef_swapXY u i j : (swapXY u)`_i`_j = u`_j`_i.
Proof. elim/poly_ind: u => [|u p IHu] in i j *; first by rewrite raddf0 !coef0. rewrite raddfD !coefD /= swapXY_polyC coef_map /= !coefC coefMX. rewrite !(fun_if (fun q : {poly R} => q`_i)) coef0 -IHu; congr (_ + _). by rewrite unlock rmorphM /= map_polyX hornerMX coefMC coefMX. Qed.
Lemma
coef_swapXY
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "coef0", "coefC", "coefD", "coefMC", "coefMX", "coef_map", "hornerMX", "map_polyX", "poly", "poly_ind", "raddf0", "raddfD", "rmorphM", "swapXY", "swapXY_polyC" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
swapXYK : involutive swapXY.
Proof. by move=> u; apply/polyP=> i; apply/polyP=> j; rewrite !coef_swapXY. Qed.
Lemma
swapXYK
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "apply", "coef_swapXY", "polyP", "swapXY" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
swapXY_map_polyC p : swapXY p^:P = p%:P.
Proof. by rewrite -swapXY_polyC swapXYK. Qed.
Lemma
swapXY_map_polyC
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "swapXY", "swapXYK", "swapXY_polyC" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
swapXY_eq0 u : (swapXY u == 0) = (u == 0).
Proof. by rewrite (inv_eq swapXYK) raddf0. Qed.
Lemma
swapXY_eq0
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "inv_eq", "raddf0", "swapXY", "swapXYK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
swapXY_is_monoid_morphism : monoid_morphism swapXY.
Proof. split=> [|u v]; first by rewrite swapXY_polyC map_polyC. apply/polyP=> i; apply/polyP=> j; rewrite coef_swapXY !coefM !coef_sum. rewrite (eq_bigr _ (fun _ _ => coefM _ _ _)) exchange_big /=. apply: eq_bigr => j1 _; rewrite coefM; apply: eq_bigr=> i1 _. by rewrite !coef_swapXY. Qed.
Lemma
swapXY_is_monoid_morphism
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "apply", "coefM", "coef_sum", "coef_swapXY", "eq_bigr", "exchange_big", "map_polyC", "monoid_morphism", "polyP", "split", "swapXY", "swapXY_polyC" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
swapXY_is_multiplicative
:= (fun g => (g.2,g.1)) swapXY_is_monoid_morphism.
Definition
swapXY_is_multiplicative
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "swapXY_is_monoid_morphism" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
swapXY_is_scalable : scalable_for (map_poly polyC \; *%R) swapXY.
Proof. by move=> p u /=; rewrite -mul_polyC rmorphM /= swapXY_polyC. Qed.
Lemma
swapXY_is_scalable
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "map_poly", "mul_polyC", "polyC", "rmorphM", "scalable_for", "swapXY", "swapXY_polyC" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
swapXY_comp_poly p u : swapXY (p^:P \Po u) = p^:P \Po swapXY u.
Proof. rewrite -horner_map; congr _.[_]; rewrite -!map_poly_comp /=. by apply: eq_map_poly => x; rewrite /= swapXY_polyC map_polyC. Qed.
Lemma
swapXY_comp_poly
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "apply", "eq_map_poly", "horner_map", "map_polyC", "map_poly_comp", "swapXY", "swapXY_polyC" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
max_size_coefXY u i : size u`_i <= sizeY u.
Proof. have [ltiu | /(nth_default 0)->] := ltnP i (size u); last by rewrite size_poly0. exact: (bigmax_sup (Ordinal ltiu)). Qed.
Lemma
max_size_coefXY
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "bigmax_sup", "last", "ltnP", "nth_default", "size", "sizeY", "size_poly0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
max_size_lead_coefXY u : size (lead_coef u) <= sizeY u.
Proof. by rewrite lead_coefE max_size_coefXY. Qed.
Lemma
max_size_lead_coefXY
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "lead_coef", "lead_coefE", "max_size_coefXY", "size", "sizeY" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
max_size_evalX u : size u.['X] <= sizeY u + (size u).-1.
Proof. rewrite horner_coef (leq_trans (size_sum _ _ _)) //; apply/bigmax_leqP=> i _. rewrite (leq_trans (size_polyMleq _ _)) // size_polyXn addnS. by rewrite leq_add ?max_size_coefXY //= -ltnS (leq_trans _ (leqSpred _)). Qed.
Lemma
max_size_evalX
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "addnS", "apply", "bigmax_leqP", "horner_coef", "leqSpred", "leq_add", "leq_trans", "ltnS", "max_size_coefXY", "size", "sizeY", "size_polyMleq", "size_polyXn", "size_sum" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
max_size_evalC u x : size u.[x%:P] <= sizeY u.
Proof. rewrite horner_coef (leq_trans (size_sum _ _ _)) //; apply/bigmax_leqP=> i _. rewrite (leq_trans (size_polyMleq _ _)) // -polyC_exp size_polyC addnC -subn1. by rewrite (leq_trans _ (max_size_coefXY _ i)) // leq_subLR leq_add2r leq_b1. Qed.
Lemma
max_size_evalC
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "addnC", "apply", "bigmax_leqP", "horner_coef", "leq_add2r", "leq_b1", "leq_subLR", "leq_trans", "max_size_coefXY", "polyC_exp", "size", "sizeY", "size_polyC", "size_polyMleq", "size_sum", "subn1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sizeYE u : sizeY u = size (swapXY u).
Proof. apply/eqP; rewrite eqn_leq; apply/andP; split. apply/bigmax_leqP=> /= i _; apply/leq_sizeP => j /(nth_default 0) u_j_0. by rewrite -coef_swapXY u_j_0 coef0. apply/leq_sizeP=> j le_uY_j; apply/polyP=> i; rewrite coef_swapXY coef0. by rewrite nth_default // (leq_trans _ le_uY_j) ?max_size_coefXY. Qed.
Lemma
sizeYE
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "apply", "bigmax_leqP", "coef0", "coef_swapXY", "eqn_leq", "leq_sizeP", "leq_trans", "max_size_coefXY", "nth_default", "polyP", "size", "sizeY", "split", "swapXY" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sizeY_eq0 u : (sizeY u == 0) = (u == 0).
Proof. by rewrite sizeYE size_poly_eq0 swapXY_eq0. Qed.
Lemma
sizeY_eq0
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "sizeY", "sizeYE", "size_poly_eq0", "swapXY_eq0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sizeY_mulX u : sizeY (u * 'X) = sizeY u.
Proof. rewrite !sizeYE rmorphM /= swapXY_X rreg_size //. by have /monic_comreg[_ /rreg_lead] := monicX R. Qed.
Lemma
sizeY_mulX
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "monicX", "monic_comreg", "rmorphM", "rreg_lead", "rreg_size", "sizeY", "sizeYE", "swapXY_X" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
swapXY_poly_XaY p : swapXY (poly_XaY p) = poly_XaY p.
Proof. by rewrite swapXY_comp_poly rmorphD /= swapXY_X swapXY_Y addrC. Qed.
Lemma
swapXY_poly_XaY
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "addrC", "poly_XaY", "rmorphD", "swapXY", "swapXY_X", "swapXY_Y", "swapXY_comp_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
swapXY_poly_XmY p : swapXY (poly_XmY p) = poly_XmY p.
Proof. by rewrite swapXY_comp_poly rmorphM /= swapXY_X swapXY_Y commr_polyX. Qed.
Lemma
swapXY_poly_XmY
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "commr_polyX", "poly_XmY", "rmorphM", "swapXY", "swapXY_X", "swapXY_Y", "swapXY_comp_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
poly_XaY0 : poly_XaY 0 = 0.
Proof. by rewrite /poly_XaY rmorph0 comp_poly0. Qed.
Lemma
poly_XaY0
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "comp_poly0", "poly_XaY", "rmorph0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
poly_XmY0 : poly_XmY 0 = 0.
Proof. by rewrite /poly_XmY rmorph0 comp_poly0. Qed.
Lemma
poly_XmY0
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "comp_poly0", "poly_XmY", "rmorph0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
swapXY_map (R S : nzRingType) (f : {additive R -> S}) u : swapXY (u ^ map_poly f) = swapXY u ^ map_poly f.
Proof. by apply/polyP=> i; apply/polyP=> j; rewrite !(coef_map, coef_swapXY). Qed.
Lemma
swapXY_map
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "additive", "apply", "coef_map", "coef_swapXY", "map_poly", "polyP", "swapXY" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner_swapXY u x : (swapXY u).[x%:P] = u ^ eval x.
Proof. apply/polyP=> i /=; rewrite coef_map /= /eval horner_coef coef_sum -sizeYE. rewrite (horner_coef_wide _ (max_size_coefXY u i)); apply: eq_bigr=> j _. by rewrite -polyC_exp coefMC coef_swapXY. Qed.
Lemma
horner_swapXY
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "apply", "coefMC", "coef_map", "coef_sum", "coef_swapXY", "eq_bigr", "eval", "horner_coef", "horner_coef_wide", "max_size_coefXY", "polyC_exp", "polyP", "sizeYE", "swapXY" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner_polyC u x : u.[x%:P] = swapXY u ^ eval x.
Proof. by rewrite -horner_swapXY swapXYK. Qed.
Lemma
horner_polyC
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "eval", "horner_swapXY", "swapXY", "swapXYK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner2_swapXY u x y : (swapXY u).[x, y] = u.[y, x].
Proof. by rewrite horner_swapXY -{1}(hornerC y x) horner_map. Qed.
Lemma
horner2_swapXY
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "hornerC", "horner_map", "horner_swapXY", "swapXY" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner_poly_XaY p v : (poly_XaY p).[v] = p \Po (v + 'X).
Proof. by rewrite horner_comp !hornerE. Qed.
Lemma
horner_poly_XaY
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "hornerE", "horner_comp", "poly_XaY" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner_poly_XmY p v : (poly_XmY p).[v] = p \Po (v * 'X).
Proof. by rewrite horner_comp !hornerE. Qed.
Lemma
horner_poly_XmY
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "hornerE", "horner_comp", "poly_XmY" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_poly_XaY p : size (poly_XaY p) = size p.
Proof. by rewrite size_comp_poly2 ?size_XaddC // size_map_polyC. Qed.
Lemma
size_poly_XaY
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "poly_XaY", "size", "size_XaddC", "size_comp_poly2", "size_map_polyC" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
poly_XaY_eq0 p : (poly_XaY p == 0) = (p == 0).
Proof. by rewrite -!size_poly_eq0 size_poly_XaY. Qed.
Lemma
poly_XaY_eq0
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "poly_XaY", "size_poly_XaY", "size_poly_eq0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_poly_XmY p : size (poly_XmY p) = size p.
Proof. by rewrite size_comp_poly2 ?size_XmulC ?polyX_eq0 ?size_map_polyC. Qed.
Lemma
size_poly_XmY
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "polyX_eq0", "poly_XmY", "size", "size_XmulC", "size_comp_poly2", "size_map_polyC" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
poly_XmY_eq0 p : (poly_XmY p == 0) = (p == 0).
Proof. by rewrite -!size_poly_eq0 size_poly_XmY. Qed.
Lemma
poly_XmY_eq0
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "poly_XmY", "size_poly_XmY", "size_poly_eq0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lead_coef_poly_XaY p : lead_coef (poly_XaY p) = (lead_coef p)%:P.
Proof. rewrite lead_coef_comp ?size_XaddC // -['Y]opprK -polyCN lead_coefXsubC. by rewrite expr1n mulr1 lead_coef_map_inj //; apply: polyC_inj. Qed.
Lemma
lead_coef_poly_XaY
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "apply", "expr1n", "lead_coef", "lead_coefXsubC", "lead_coef_comp", "lead_coef_map_inj", "mulr1", "opprK", "polyCN", "polyC_inj", "poly_XaY", "size_XaddC" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_annihilant_in_ideal p q : 1 < size p -> 1 < size q -> {uv : {poly {poly R}} * {poly {poly R}} | size uv.1 < size q /\ size uv.2 < size p & forall x y, (sub_annihilant p q).[y] = uv.1.[x, y] * p.[x + y] + uv.2.[x, y] * q.[x]}.
Proof. rewrite -size_poly_XaY -(size_map_polyC q) => p1_gt1 q1_gt1. have [uv /= [ub_u ub_v Dr]] := resultant_in_ideal p1_gt1 q1_gt1. exists uv => // x y; rewrite -[r in r.[y] = _](hornerC _ x%:P) Dr. by rewrite !(hornerE, horner_comp). Qed.
Lemma
sub_annihilant_in_ideal
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "hornerC", "hornerE", "horner_comp", "poly", "resultant_in_ideal", "size", "size_map_polyC", "size_poly_XaY", "sub_annihilant" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_annihilantP p q x y : p != 0 -> q != 0 -> p.[x] = 0 -> q.[y] = 0 -> (sub_annihilant p q).[x - y] = 0.
Proof. move=> nz_p nz_q px0 qy0. have p_gt1: size p > 1 by have /rootP/root_size_gt1-> := px0. have q_gt1: size q > 1 by have /rootP/root_size_gt1-> := qy0. have [uv /= _ /(_ y)->] := sub_annihilant_in_ideal p_gt1 q_gt1. by rewrite (addrC y) subrK px0 qy0 !mulr0 addr0. Qed.
Lemma
sub_annihilantP
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "addr0", "addrC", "mulr0", "nz_p", "p_gt1", "q_gt1", "rootP", "root_size_gt1", "size", "sub_annihilant", "sub_annihilant_in_ideal", "subrK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_annihilant_neq0 p q : p != 0 -> q != 0 -> sub_annihilant p q != 0.
Proof. rewrite resultant_eq0; set p1 := poly_XaY p => nz_p nz_q. have [nz_p1 nz_q1]: p1 != 0 /\ q^:P != 0 by rewrite poly_XaY_eq0 map_polyC_eq0. rewrite -leqNgt eq_leq //; apply/eqP/Bezout_coprimepPn=> // [[[u v]]] /=. rewrite !size_poly_gt0 -andbA => /and4P[nz_u ltuq nz_v _] Duv. have /eqP/= := congr1 (size \o (lead_c...
Lemma
sub_annihilant_neq0
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "Bezout_coprimepPn", "addnC", "addnS", "apply", "eq_leq", "lead_coef", "lead_coefC", "lead_coefM", "lead_coef_eq0", "lead_coef_poly_XaY", "leqNgt", "leqRHS", "leq_addr", "leq_ltn_trans", "leq_trans", "ltnS", "ltn_eqF", "map_polyC_eq0", "max_size_lead_coefXY", "mul_polyC", "mu...
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
div_annihilant_in_ideal p q : 1 < size p -> 1 < size q -> {uv : {poly {poly R}} * {poly {poly R}} | size uv.1 < size q /\ size uv.2 < size p & forall x y, (div_annihilant p q).[y] = uv.1.[x, y] * p.[x * y] + uv.2.[x, y] * q.[x]}.
Proof. rewrite -size_poly_XmY -(size_map_polyC q) => p1_gt1 q1_gt1. have [uv /= [ub_u ub_v Dr]] := resultant_in_ideal p1_gt1 q1_gt1. exists uv => // x y; rewrite -[r in r.[y] = _](hornerC _ x%:P) Dr. by rewrite !(hornerE, horner_comp). Qed.
Lemma
div_annihilant_in_ideal
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "div_annihilant", "hornerC", "hornerE", "horner_comp", "poly", "resultant_in_ideal", "size", "size_map_polyC", "size_poly_XmY" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
div_annihilant_neq0 p q : p != 0 -> q.[0] != 0 -> div_annihilant p q != 0.
Proof. have factorX (S : nzRingType) (u : {poly S}) : u != 0 -> root u 0 -> exists2 v, v != 0 & u = v * 'X. move=> nz_u /factor_theorem[v]; rewrite subr0 => Du; exists v => //. by apply: contraNneq nz_u => v0; rewrite Du v0 mul0r. have nzX: 'X != 0 := monic_neq0 (monicX _); have rootC0 := root_polyC _ 0. rewrit...
Lemma
div_annihilant_neq0
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "Bezout_coprimepPn", "addnS", "apply", "comp_polyM", "comp_polyX", "contraNneq", "div_annihilant", "eq_leq", "factor_theorem", "hornerE", "hornerZ", "horner_coef0", "horner_comp", "horner_map", "last", "leqNgt", "leq_addr", "leq_trans", "linearZ", "ltnNge", "ltnS", "map_pol...
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pFtoE
:= (map_poly (GRing.RMorphism.sort FtoE)).
Notation
pFtoE
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "map_poly", "sort" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
div_annihilantP (p q : {poly E}) (x y : E) : p != 0 -> q != 0 -> y != 0 -> p.[x] = 0 -> q.[y] = 0 -> (div_annihilant p q).[x / y] = 0.
Proof. move=> nz_p nz_q nz_y px0 qy0. have p_gt1: size p > 1 by have /rootP/root_size_gt1-> := px0. have q_gt1: size q > 1 by have /rootP/root_size_gt1-> := qy0. have [uv /= _ /(_ y)->] := div_annihilant_in_ideal p_gt1 q_gt1. by rewrite (mulrC y) divfK // px0 qy0 !mulr0 addr0. Qed.
Lemma
div_annihilantP
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "addr0", "div_annihilant", "div_annihilant_in_ideal", "divfK", "mulr0", "mulrC", "nz_p", "p_gt1", "poly", "q_gt1", "rootP", "root_size_gt1", "size" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_sub_annihilantP (p q : {poly F}) (x y : E) : p != 0 -> q != 0 ->(p ^ FtoE).[x] = 0 -> (q ^ FtoE).[y] = 0 -> (sub_annihilant p q ^ FtoE).[x - y] = 0.
Proof. move=> nz_p nz_q px0 qy0; have pFto0 := map_poly_eq0 FtoE. rewrite map_resultant ?pFto0 ?lead_coef_eq0 ?map_poly_eq0 ?poly_XaY_eq0 //. rewrite map_comp_poly rmorphD /= map_polyC /= !map_polyX -!map_poly_comp /=. by rewrite !(eq_map_poly (map_polyC _)) !map_poly_comp sub_annihilantP ?pFto0. Qed.
Lemma
map_sub_annihilantP
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "eq_map_poly", "lead_coef_eq0", "map_comp_poly", "map_polyC", "map_polyX", "map_poly_comp", "map_poly_eq0", "map_resultant", "nz_p", "poly", "poly_XaY_eq0", "rmorphD", "sub_annihilant", "sub_annihilantP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_div_annihilantP (p q : {poly F}) (x y : E) : p != 0 -> q != 0 -> y != 0 -> (p ^ FtoE).[x] = 0 -> (q ^ FtoE).[y] = 0 -> (div_annihilant p q ^ FtoE).[x / y] = 0.
Proof. move=> nz_p nz_q nz_y px0 qy0; have pFto0 := map_poly_eq0 FtoE. rewrite map_resultant ?pFto0 ?lead_coef_eq0 ?map_poly_eq0 ?poly_XmY_eq0 //. rewrite map_comp_poly rmorphM /= map_polyC /= !map_polyX -!map_poly_comp /=. by rewrite !(eq_map_poly (map_polyC _)) !map_poly_comp div_annihilantP ?pFto0. Qed.
Lemma
map_div_annihilantP
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "div_annihilant", "div_annihilantP", "eq_map_poly", "lead_coef_eq0", "map_comp_poly", "map_polyC", "map_polyX", "map_poly_comp", "map_poly_eq0", "map_resultant", "nz_p", "poly", "poly_XmY_eq0", "rmorphM" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
root_annihilant x p (pEx := (p ^ pFtoE).[x%:P]) : pEx != 0 -> algebraicOver FtoE x -> exists2 r : {poly F}, r != 0 & forall y, root pEx y -> root (r ^ FtoE) y.
Proof. move=> nz_px [q nz_q qx0]. have [/size1_polyC Dp | p_gt1] := leqP (size p) 1. by rewrite {}/pEx Dp map_polyC hornerC map_poly_eq0 in nz_px *; exists p`_0. have nz_p: p != 0 by rewrite -size_poly_gt0 ltnW. have [m le_qm] := ubnP (size q); elim: m => // m IHm in q le_qm nz_q qx0 *. have nz_q1: q^:P != 0 by rewri...
Lemma
root_annihilant
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "Bezout_coprimepPn", "addr0", "algebraicOver", "apply", "coef0", "coefMC", "coef_map", "coef_sum", "coefp", "eq_bigr", "gtn_eqF", "hornerC", "hornerE", "hornerM", "horner_coef", "horner_swapXY", "last", "lead_coef", "lead_coef_eq0", "leqP", "leq_ltn_trans", "leq_trans", "...
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
algebraic_root_polyXY x y : (let pEx p := (p ^ map_poly FtoE).[x%:P] in exists2 p, pEx p != 0 & root (pEx p) y) -> algebraicOver FtoE x -> algebraicOver FtoE y.
Proof. by case=> p nz_px pxy0 /(root_annihilant nz_px)[r]; exists r; auto. Qed.
Lemma
algebraic_root_polyXY
algebra
algebra/polyXY.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "poly", "polydiv", "matrix", "mxpoly", "GRing.Theory" ]
[ "algebraicOver", "map_poly", "root", "root_annihilant" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
poly_of_size_pred
:= fun p : {poly R} => size p <= n.
Definition
poly_of_size_pred
algebra
algebra/qpoly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "nmodule", "rings_modules_and_algebras", "divalg", "countalg", "finalg", "poly", "polydiv", "matrix", "mxalgebra", "...
[ "poly", "size" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
poly_of_size
:= [qualify a p | poly_of_size_pred p].
Definition
poly_of_size
algebra
algebra/qpoly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "nmodule", "rings_modules_and_algebras", "divalg", "countalg", "finalg", "poly", "polydiv", "matrix", "mxalgebra", "...
[ "poly_of_size_pred" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
npoly_subsemimod_closed : subsemimod_closed poly_of_size.
Proof. split=> [|x q sq]; first split=> [|p q sp sq]; rewrite qualifE/= ?size_poly0//. by rewrite (leq_trans (size_polyD _ _)) // geq_max [_ <= _]sp. exact: leq_trans (size_scale_leq _ _) sq. Qed.
Fact
npoly_subsemimod_closed
algebra
algebra/qpoly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "nmodule", "rings_modules_and_algebras", "divalg", "countalg", "finalg", "poly", "polydiv", "matrix", "mxalgebra", "...
[ "geq_max", "leq_trans", "poly_of_size", "size_poly0", "size_polyD", "size_scale_leq", "sp", "split", "sq", "subsemimod_closed" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
npoly : predArgType
:= NPoly { polyn :> {poly R}; _ : polyn \is a poly_of_size }.
Record
npoly
algebra
algebra/qpoly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "nmodule", "rings_modules_and_algebras", "divalg", "countalg", "finalg", "poly", "polydiv", "matrix", "mxalgebra", "...
[ "poly", "poly_of_size" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
npoly_is_a_poly_of_size (p : npoly) : val p \is a poly_of_size.
Proof. by case: p. Qed.
Lemma
npoly_is_a_poly_of_size
algebra
algebra/qpoly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "nmodule", "rings_modules_and_algebras", "divalg", "countalg", "finalg", "poly", "polydiv", "matrix", "mxalgebra", "...
[ "npoly", "poly_of_size", "val" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_npoly (p : npoly) : size p <= n.
Proof. exact: npoly_is_a_poly_of_size. Qed.
Lemma
size_npoly
algebra
algebra/qpoly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "nmodule", "rings_modules_and_algebras", "divalg", "countalg", "finalg", "poly", "polydiv", "matrix", "mxalgebra", "...
[ "npoly", "npoly_is_a_poly_of_size", "size" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
npoly_rV : npoly -> 'rV[R]_n
:= poly_rV \o val.
Definition
npoly_rV
algebra
algebra/qpoly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "nmodule", "rings_modules_and_algebras", "divalg", "countalg", "finalg", "poly", "polydiv", "matrix", "mxalgebra", "...
[ "npoly", "poly_rV", "val" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rVnpoly : 'rV[R]_n -> npoly
:= insubd (0 : npoly) \o rVpoly.
Definition
rVnpoly
algebra
algebra/qpoly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "nmodule", "rings_modules_and_algebras", "divalg", "countalg", "finalg", "poly", "polydiv", "matrix", "mxalgebra", "...
[ "insubd", "npoly", "rVpoly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
npoly_rV_K : cancel npoly_rV rVnpoly.
Proof. move=> p /=; apply/val_inj. by rewrite val_insubd [_ \is a _]size_poly ?poly_rV_K. Qed.
Lemma
npoly_rV_K
algebra
algebra/qpoly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "nmodule", "rings_modules_and_algebras", "divalg", "countalg", "finalg", "poly", "polydiv", "matrix", "mxalgebra", "...
[ "apply", "npoly_rV", "poly_rV_K", "rVnpoly", "size_poly", "val_inj", "val_insubd" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rVnpolyK : cancel rVnpoly npoly_rV.
Proof. by move=> p /=; rewrite val_insubd [_ \is a _]size_poly rVpolyK. Qed.
Lemma
rVnpolyK
algebra
algebra/qpoly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "nmodule", "rings_modules_and_algebras", "divalg", "countalg", "finalg", "poly", "polydiv", "matrix", "mxalgebra", "...
[ "npoly_rV", "rVnpoly", "rVpolyK", "size_poly", "val_insubd" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
npoly_vect_axiom : SemiVector.axiom n npoly.
Proof. by exists npoly_rV; [exact: semilinearPZ | exists rVnpoly]. Qed.
Lemma
npoly_vect_axiom
algebra
algebra/qpoly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "nmodule", "rings_modules_and_algebras", "divalg", "countalg", "finalg", "poly", "polydiv", "matrix", "mxalgebra", "...
[ "axiom", "npoly", "npoly_rV", "rVnpoly", "semilinearPZ" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'{poly_' n R }"
:= (@npoly R n) : type_scope.
Notation
'{poly_' n R }
algebra
algebra/qpoly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "nmodule", "rings_modules_and_algebras", "divalg", "countalg", "finalg", "poly", "polydiv", "matrix", "mxalgebra", "...
[ "npoly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyn_is_semilinear : semilinear (@polyn _ _ : {poly_n R} -> _).
Proof. by []. Qed.
Fact
polyn_is_semilinear
algebra
algebra/qpoly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "nmodule", "rings_modules_and_algebras", "divalg", "countalg", "finalg", "poly", "polydiv", "matrix", "mxalgebra", "...
[ "semilinear" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mk_npoly (E : nat -> R) : {poly_n R}
:= @NPoly R _ (\poly_(i < n) E i) (size_poly _ _).
Canonical
mk_npoly
algebra
algebra/qpoly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "nmodule", "rings_modules_and_algebras", "divalg", "countalg", "finalg", "poly", "polydiv", "matrix", "mxalgebra", "...
[ "nat", "size_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_npoly0 : size (0 : {poly R}) <= n.
Proof. by rewrite size_poly0. Qed.
Fact
size_npoly0
algebra
algebra/qpoly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "nmodule", "rings_modules_and_algebras", "divalg", "countalg", "finalg", "poly", "polydiv", "matrix", "mxalgebra", "...
[ "poly", "size", "size_poly0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
npoly0
:= NPoly (size_npoly0).
Definition
npoly0
algebra
algebra/qpoly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "nmodule", "rings_modules_and_algebras", "divalg", "countalg", "finalg", "poly", "polydiv", "matrix", "mxalgebra", "...
[ "size_npoly0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
npolyp_key : unit.
Proof. exact: tt. Qed.
Fact
npolyp_key
algebra
algebra/qpoly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "nmodule", "rings_modules_and_algebras", "divalg", "countalg", "finalg", "poly", "polydiv", "matrix", "mxalgebra", "...
[ "unit" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
npolyp : {poly R} -> {poly_n R}
:= locked_with npolyp_key (mk_npoly \o (nth 0)).
Definition
npolyp
algebra
algebra/qpoly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "nmodule", "rings_modules_and_algebras", "divalg", "countalg", "finalg", "poly", "polydiv", "matrix", "mxalgebra", "...
[ "mk_npoly", "npolyp_key", "nth", "poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
npoly_of_seq
:= npolyp \o Poly.
Definition
npoly_of_seq
algebra
algebra/qpoly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "nmodule", "rings_modules_and_algebras", "divalg", "countalg", "finalg", "poly", "polydiv", "matrix", "mxalgebra", "...
[ "Poly", "npolyp" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
npolyP (p q : {poly_n R}) : nth 0 p =1 nth 0 q <-> p = q.
Proof. by split => [/polyP/val_inj|->]. Qed.
Lemma
npolyP
algebra
algebra/qpoly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "nmodule", "rings_modules_and_algebras", "divalg", "countalg", "finalg", "poly", "polydiv", "matrix", "mxalgebra", "...
[ "nth", "polyP", "split", "val_inj" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coef_npolyp (p : {poly R}) i : (npolyp p)`_i = if i < n then p`_i else 0.
Proof. by rewrite /npolyp unlock /= coef_poly. Qed.
Lemma
coef_npolyp
algebra
algebra/qpoly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "nmodule", "rings_modules_and_algebras", "divalg", "countalg", "finalg", "poly", "polydiv", "matrix", "mxalgebra", "...
[ "coef_poly", "npolyp", "poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
big_coef_npoly (p : {poly_n R}) i : n <= i -> p`_i = 0.
Proof. by move=> i_big; rewrite nth_default // (leq_trans _ i_big) ?size_npoly. Qed.
Lemma
big_coef_npoly
algebra
algebra/qpoly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "nmodule", "rings_modules_and_algebras", "divalg", "countalg", "finalg", "poly", "polydiv", "matrix", "mxalgebra", "...
[ "leq_trans", "nth_default", "size_npoly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
npolypK (p : {poly R}) : size p <= n -> npolyp p = p :> {poly R}.
Proof. move=> spn; apply/polyP=> i; rewrite coef_npolyp. by have [i_big|i_small] // := ltnP; rewrite nth_default ?(leq_trans spn). Qed.
Lemma
npolypK
algebra
algebra/qpoly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "nmodule", "rings_modules_and_algebras", "divalg", "countalg", "finalg", "poly", "polydiv", "matrix", "mxalgebra", "...
[ "apply", "coef_npolyp", "leq_trans", "ltnP", "npolyp", "nth_default", "poly", "polyP", "size" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coefn_sum (I : Type) (r : seq I) (P : pred I) (F : I -> {poly_n R}) (k : nat) : (\sum_(i <- r | P i) F i)`_k = \sum_(i <- r | P i) (F i)`_k.
Proof. by rewrite !raddf_sum //= coef_sum. Qed.
Lemma
coefn_sum
algebra
algebra/qpoly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "nmodule", "rings_modules_and_algebras", "divalg", "countalg", "finalg", "poly", "polydiv", "matrix", "mxalgebra", "...
[ "coef_sum", "nat", "raddf_sum", "seq" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
npoly_oppr_closed : oppr_closed (@poly_of_size R n).
Proof. by move=> p sp; rewrite qualifE/= size_polyN. Qed.
Fact
npoly_oppr_closed
algebra
algebra/qpoly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "nmodule", "rings_modules_and_algebras", "divalg", "countalg", "finalg", "poly", "polydiv", "matrix", "mxalgebra", "...
[ "oppr_closed", "poly_of_size", "size_polyN", "sp" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
npoly_enum : seq {poly_n R}
:= if n isn't n.+1 then [:: npoly0 _] else pmap insub [seq \poly_(i < n.+1) c (inord i) | c : (R ^ n.+1)%type].
Definition
npoly_enum
algebra
algebra/qpoly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "nmodule", "rings_modules_and_algebras", "divalg", "countalg", "finalg", "poly", "polydiv", "matrix", "mxalgebra", "...
[ "inord", "insub", "npoly0", "pmap", "seq", "type" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
npoly_enum_uniq : uniq npoly_enum.
Proof. rewrite /npoly_enum; case: n=> [|k] //. rewrite pmap_sub_uniq // map_inj_uniq => [f g eqfg|]; rewrite ?enum_uniq //. apply/ffunP => /= i; have /(congr1 (fun p : {poly _} => p`_i)) := eqfg. by rewrite !coef_poly ltn_ord inord_val. Qed.
Lemma
npoly_enum_uniq
algebra
algebra/qpoly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "nmodule", "rings_modules_and_algebras", "divalg", "countalg", "finalg", "poly", "polydiv", "matrix", "mxalgebra", "...
[ "apply", "coef_poly", "enum_uniq", "ffunP", "inord_val", "ltn_ord", "map_inj_uniq", "npoly_enum", "pmap_sub_uniq", "poly", "uniq" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d