(***************************************************************************

                       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.

Lemma if_true (a b : bool) : a &&& b = true -> a = true /\ b = true.