Structure eparams: Type := {
 E:> Set;                      (* the vector type *)
 stype:> fparams;              (* the scalar field *)
 E0: E;                        (* 0 *)
 eqE: E -> E -> bool;          (* = as a boolean function *)
 addE : E -> E -> E;          (* + *)
 scalE : (K stype) -> E -> E   (* scalar *)
}.Structure eparams: Type := {
 E:> Set;                      (* the vector type *)
 stype:> fparams;              (* the scalar field *)
 E0: E;                        (* 0 *)
 eqE: E -> E -> bool;          (* = as a boolean function *)
 addE : E -> E -> E;          (* + *)
 scalE : (K stype) -> E -> E   (* scalar *)
}.Structure eparams: Type := {
 E:> Set;                      (* the vector type *)
 stype:> fparams;              (* the scalar field *)
 E0: E;                        (* 0 *)
 eqE: E -> E -> bool;          (* = as a boolean function *)
 addE : E -> E -> E;          (* + *)
 scalE : (K stype) -> E -> E   (* scalar *)
}.Structure eparams: Type := {
 E:> Set;                      (* the vector type *)
 stype:> fparams;              (* the scalar field *)
 E0: E;                        (* 0 *)
 eqE: E -> E -> bool;          (* = as a boolean function *)
 addE : E -> E -> E;          (* + *)
 scalE : (K stype) -> E -> E   (* scalar *)
}.Structure eparams: Type := {
 E:> Set;                      (* the vector type *)
 stype:> fparams;              (* the scalar field *)
 E0: E;                        (* 0 *)
 eqE: E -> E -> bool;          (* = as a boolean function *)
 addE : E -> E -> E;          (* + *)
 scalE : (K stype) -> E -> E   (* scalar *)
}.Structure eparams: Type := {
 E:> Set;                      (* the vector type *)
 stype:> fparams;              (* the scalar field *)
 E0: E;                        (* 0 *)
 eqE: E -> E -> bool;          (* = as a boolean function *)
 addE : E -> E -> E;          (* + *)
 scalE : (K stype) -> E -> E   (* scalar *)
}.Structure eparams: Type := {
 E:> Set;                      (* the vector type *)
 stype:> fparams;              (* the scalar field *)
 E0: E;                        (* 0 *)
 eqE: E -> E -> bool;          (* = as a boolean function *)
 addE : E -> E -> E;          (* + *)
 scalE : (K stype) -> E -> E   (* scalar *)
}.Notation "x ?= y" := (eqE _ x y) (at level 70): vector_scope.Notation "0" := (E0 _): vector_scope.Notation "x + y" := (addE _ x y): vector_scope.Notation "x .* y" := (scalE _ x%f y) (at level 31, no associativity): vector_scope.Implicit Type v x y z: p.Implicit Type k: stype p.Notation " x *X* y " := (mprod _ x y) (at level 40, no associativity): vector_scope.(* What it means for a sub vector space (i.e a list of vectors)
   to be free *)
Definition free vs := 
  forall ks, length ks = length vs -> ks *X* vs = 0 -> 
    (forall k, In k ks -> k = 0%f).

(* Specifications: *)
Lemma cbl_trans l1 l2 x: (forall i, In i l2 -> cbl l1 i) -> cbl l2 x -> cbl l1 x.