Section AlmostField.
Variable AFth : almost_field_theory.
Let ARth := (AF_AR AFth).
Let rI_neq_rO := (AF_1_neq_0 AFth).
Let rdiv_def := (AFdiv_def AFth).
Let rinv_l := (AFinv_l AFth).
Add Morphism radd with signature (req ==> req ==> req) as radd_ext.
(* Power coefficients : Cpow *)
Variable Cpow : Type.
Variable Cp_phi : N -> Cpow.
Variable rpow : R -> Cpow -> R.
Variable pow_th : power_theory rI rmul req Cp_phi rpow.
(* sign function *)
Variable get_sign : C -> option C.
Variable get_sign_spec : sign_theory copp ceqb get_sign.
Variable cdiv:C -> C -> C*C.
Variable cdiv_th : div_theory req cadd cmul phi cdiv.
Let rpow_pow := (rpow_pow_N pow_th).
(* Polynomial expressions : (PExpr C) *)
Declare Scope PE_scope.
Bind Scope PE_scope with PExpr.
Delimit Scope PE_scope with poly.
Notation NPEeval := (PEeval rO rI radd rmul rsub ropp phi Cp_phi rpow).
Notation "P @ l" := (NPEeval l P) (at level 10, no associativity).
Arguments PEc _ _%coef.
Notation "0" := (PEc 0) : PE_scope.
Notation "1" := (PEc 1) : PE_scope.
Infix "+" := PEadd : PE_scope.
Infix "-" := PEsub : PE_scope.
Infix "*" := PEmul : PE_scope.
Notation "- e" := (PEopp e) : PE_scope.
Infix "^" := PEpow : PE_scope.
Definition NPEequiv e e' := forall l, e@l == e'@l.
Infix "===" := NPEequiv (at level 70, no associativity) : PE_scope.
Instance NPEequiv_eq : Equivalence NPEequiv.
Instance NPEeval_ext : Proper (eq ==> NPEequiv ==> req) NPEeval.
Notation Nnorm :=
(norm_subst cO cI cadd cmul csub copp ceqb cdiv).
Notation NPphi_dev :=
(Pphi_dev rO rI radd rmul rsub ropp cO cI ceqb phi get_sign).
Notation NPphi_pow :=
(Pphi_pow rO rI radd rmul rsub ropp cO cI ceqb phi Cp_phi rpow get_sign).
(* add abstract semi-ring to help with some proofs *)
Add Ring Rring : (ARth_SRth ARth).
(* additional ring properties *)
(***************************************************************************  Properties of division  ***************************************************************************)
Instance rdiv_ext: Proper (req ==> req ==> req) rdiv.
(* Results about [pow_pos] and [pow_N] *)
Instance pow_ext : Proper (req ==> eq ==> req) (pow_pos rmul).
Instance pow_N_ext : Proper (req ==> eq ==> req) (pow_N rI rmul).
(* === is a morphism *)
Instance PEadd_ext : Proper (NPEequiv ==> NPEequiv ==> NPEequiv) (@PEadd C).
(***************************************************************************  Some equality test  ***************************************************************************)
Local Notation "a &&& b" := (if a then b else false)
(at level 40, left associativity).
(* equality test *)
Fixpoint PExpr_eq (e e' : PExpr C) {struct e} : bool :=
match e, e' with
| PEc c, PEc c' => ceqb c c'
| PEX _ p, PEX _ p' => Pos.eqb p p'
| e1 + e2, e1' + e2' => PExpr_eq e1 e1' &&& PExpr_eq e2 e2'
| e1 - e2, e1' - e2' => PExpr_eq e1 e1' &&& PExpr_eq e2 e2'
| e1 * e2, e1' * e2' => PExpr_eq e1 e1' &&& PExpr_eq e2 e2'
| - e, - e' => PExpr_eq e e'
| e ^ n, e' ^ n' => N.eqb n n' &&& PExpr_eq e e'
| _, _ => false
end%poly.
(** Smart constructors for polynomial expression, with reduction of constants *)
Definition NPEadd e1 e2 :=
match e1, e2 with
| PEc c1, PEc c2 => PEc (c1 + c2)
| PEc c, _ => if (c =? 0)%coef then e2 else e1 + e2
|  _, PEc c => if (c =? 0)%coef then e1 else e1 + e2
(* Peut t'on factoriser ici ??? *)
| _, _ => (e1 + e2)
end%poly.
Infix "++" := NPEadd (at level 60, right associativity).
Definition NPEsub e1 e2 :=
match e1, e2 with
| PEc c1, PEc c2 => PEc (c1 - c2)
| PEc c, _ => if (c =? 0)%coef then - e2 else e1 - e2
|  _, PEc c => if (c =? 0)%coef then e1 else e1 - e2
(* Peut-on factoriser ici *)
| _, _ => e1 - e2
end%poly.
Infix "--" := NPEsub (at level 50, left associativity).
Definition NPEopp e1 :=
match e1 with PEc c1 => PEc (- c1) | _ => - e1 end%poly.
Definition NPEpow x n :=
match n with
| N0 => 1
| Npos p =>
if (p =? 1)%positive then x else
match x with
| PEc c =>
if (c =? 1)%coef then 1
else if (c =? 0)%coef then 0
else PEc (pow_pos cmul c p)
| _ => x ^ n
end
end%poly.
Infix "^^" := NPEpow (at level 35, right associativity).
Fixpoint NPEmul (x y : PExpr C) {struct x} : PExpr C :=
match x, y with
| PEc c1, PEc c2 => PEc (c1 * c2)
| PEc c, _ => if (c =? 1)%coef then y else if (c =? 0)%coef then 0 else x * y
| _, PEc c => if (c =? 1)%coef then x else if (c =? 0)%coef then 0 else x * y
| e1 ^ n1, e2 ^ n2 => if (n1 =? n2)%N then (NPEmul e1 e2)^^n1 else x * y
| _, _ => x * y
end%poly.
Infix "**" := NPEmul (at level 40, left associativity).

Theorem NPEmul_ok e1 e2 : (e1 ** e2 === e1 * e2)%poly.