Section FinGroup.
Variables (R : numDomainType) (gT : finGroupType).
Implicit Types G : {group gT}.
Definition midf_lte := (midf_le, midf_lt).
(* The AGM, unscaled but without the nth root. *)
Implicit Type p : {poly F}.
Variant sgr_val x : R -> bool -> bool -> bool -> bool -> bool -> bool
-> bool -> bool -> bool -> bool -> bool -> bool -> R -> Set :=
| SgrNull of x = 0 : sgr_val x 0 true true true true false false
true false false true false false 0
| SgrPos of x > 0 : sgr_val x x false false true false false true
false false true false false true 1
| SgrNeg of x < 0 : sgr_val x (- x) false true false false true false
false true false false true false (-1).

Lemma monic_Cauchy_bound : p \is monic -> {b | forall x, x >= b -> p.[x] > 0}.