Require Import ZArith.
Require Import String.
Open Scope Z.

Ltac instexist H P := elim H;clear H;intro P;intro H.


Inductive iarith : Type :=
  Const : Z -> iarith
| Var : string -> iarith
| Plus : iarith -> iarith -> iarith
| Minus : iarith -> iarith -> iarith
| Times : iarith -> iarith -> iarith.

Definition store : Set := string -> Z.

Fixpoint eval_arith (s:store) (a:iarith) : Z :=
  match a with
    Const n => n
  | Var v => s v
  | Plus a1 a2 => (eval_arith s a1)+(eval_arith s a2)
  | Minus a1 a2 => (eval_arith s a1)-(eval_arith s a2)
  | Times a1 a2 => (eval_arith s a1)*(eval_arith s a2)
  end.

Definition Divides (x y : Z) : Prop := exists q:Z, y=x*q.

Fixpoint adivides (m:Z) (a:iarith) : Prop :=
  match a with
    Const n => Divides m n
  | Var _ => True
  | Plus a1 a2 | Minus a1 a2 | Times a1 a2 =>
     (adivides m a1) /\ (adivides m a2)
  end.

Definition sdivides (m:Z) (s:store) : Prop :=
  forall (v:string) (n:Z), (s v)=n -> Divides m n.



Theorem mytheorem:
  forall (m:Z) (a:iarith) (s:store),
    (sdivides m s) -> (adivides m a) -> (Divides m (eval_arith s a)).

Proof.
(* Complete the proof here. *)