(* === is a morphism *)
Instance PEadd_ext : Proper (NPEequiv ==> NPEequiv ==> NPEequiv) (@PEadd C).

(* Specifications: *)
Lemma PE_1_l (e : PExpr C) : (1 * e === e)%poly.
Lemma PE_1_r (e : PExpr C) : (e * 1 === e)%poly.
Lemma PEpow_0_r (e : PExpr C) : (e ^ 0 === 1)%poly.
Lemma PEpow_1_r (e : PExpr C) : (e ^ 1 === e)%poly.
Lemma PEpow_1_l n : (1 ^ n === 1)%poly.
Lemma PEpow_add_r (e : PExpr C) n n' :
(e ^ (n+n') === e ^ n * e ^ n')%poly.
Lemma PEpow_mul_l (e e' : PExpr C) n :
((e * e') ^ n === e ^ n * e' ^ n)%poly.
Lemma PEpow_mul_r (e : PExpr C) n n' :
(e ^ (n * n') === (e ^ n) ^ n')%poly.
Lemma PEpow_nz l e n : ~ e @ l == 0 -> ~ (e^n) @ l == 0.
Local Notation "a &&& b" := (if a then b else false)
(at level 40, left associativity).