Require Import ZArith.

Inductive Exp : N -> Type :=
| Const : forall M, N -> Exp M
| Add : forall M, Exp M -> Exp M -> Exp M
| Mul : forall M, Exp M -> Exp M -> Exp M
| Mod : forall M, Exp M -> Exp M.

Arguments Const {M}.
Arguments Add {M}.
Arguments Mul {M}.
Arguments Mod {M}.

Infix "+"    := Add.
Infix "*"    := Mul.

Fixpoint eval {M}(e : Exp M) : N :=
  match e with
  | Const x => x
  | Add e1 e2 => eval e1 + eval e2
  | Mul e1 e2 => eval e1 * eval e2
  | Mod ep    => eval ep mod M
  end.

Ltac reify e M :=
  match e with
  | (?e1 + ?e2)%N =>
  		let e1p := reify constr:(e1) M in
      let e2p := reify constr:(e2) M in
      constr:(Add (M:=M) e1p e2p)
  | (?e1 * ?e2)%N =>
			let e1p := reify constr:(e1) M in
      let e2p := reify constr:(e2) M in
      constr:(Mul (M:=M) e1p e2p)
  | (?ep mod ?M)%N =>
			let epp := reify ep M in
      constr:(Mod (M:=M) epp)
  | _  => constr:(Const (M:=M) e)
  end.