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Adapted to more general truncated power series. #2

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2fd9fb8
Compat with Florents generalized truncpowerseries
hivert Nov 9, 2019
4ab6214
Geometric series
hivert Nov 10, 2019
8ea173d
Minor improvement + reformating
hivert Nov 10, 2019
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18 changes: 12 additions & 6 deletions expansiblefracpoly.v
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Original file line number Diff line number Diff line change
Expand Up @@ -33,7 +33,7 @@ Variable K : fieldType.
Definition fracpoly_of of phant K := {fracpoly K}.
Notation "{ 'fracpoly' K }" := (fracpoly_of (Phant K)).

Definition devs : pred_class := fun f : {fracpoly K} => (~~ (0.-fppole) f).
Definition devs := fun f : {fracpoly K} => (~~ (0.-fppole) f).

Lemma devs_invf (p : {poly K}) : p != 0 -> (0.-fppole) p%:F^-1 = (p`_0 == 0).
Proof. by move => p_neq0; rewrite fpoleV -horner_coef0 froot_tofrac. Qed.
Expand Down Expand Up @@ -80,6 +80,12 @@ Proof. by move => dx dy; rewrite devsD // devsN. Qed.

Variable n : nat.

Notation Tfpsp := trXn.

Lemma Tfpsp_is_unit (p : {poly K}) :
((Tfpsp n p) \is a GRing.unit) = (p`_0 != 0).
Proof. by rewrite unit_tfpsE unitfE coef0_trXn. Qed.

Definition Tfpsfp (x : {fracpoly K}) :=
if (0.-fppole) x then 0
else let (x1, x2) := projT1 (fracpolyE x) in (Tfpsp n x1) / (Tfpsp n x2).
Expand Down Expand Up @@ -111,7 +117,7 @@ have v0_neq0 : v`_0 != 0.
apply/negbTE; move: q0_neq0; apply/contra.
exact: devs_frac.
by [].
apply/eqP ; rewrite eq_divr ?Tfpsp_is_unit // -!rmorphM.
apply/eqP; rewrite (@eq_divr _ (Tfpsp n u)) ?Tfpsp_is_unit // -!rmorphM.
apply/eqP ; apply: (congr1 _) ; apply/eqP.
apply/eqP/tofrac_inj/eqP.
rewrite !rmorphM /= -eq_divf ?raddf_eq0 //; last 2 first.
Expand Down Expand Up @@ -237,8 +243,8 @@ Proof.
move: (fracpolyE x) => [ [u v] /= Hx ] /andP [ v_neq0 coprime_u_v ].
rewrite Hx fmorph_div /= !map_fracE /=.
have [ v0_eq0 | v0_neq0 ] := eqVneq v`_0 0; last first.
+ rewrite Tfpsfp_frac // Tfpsfp_frac; last first.
- rewrite Tfpsfp_frac // Tfpsfp_frac; last first.
by rewrite coef_map raddf_eq0 //; apply: fmorph_inj.
+ by rewrite rmorph_div /= ?Tfpsp_is_unit // !Tfps_map_poly.
by rewrite !Tfpsfp_eq0 // ?coef_map ?v0_eq0 /= ?rmorph0 // coprimep_map.
Qed.
- by rewrite rmorph_div /= ?Tfpsp_is_unit // !trXn_map_poly.
by rewrite !Tfpsfp_eq0 // ?coef_map ?v0_eq0 /= ?rmorph0 // coprimep_map.
Qed.
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