Require Import ZArith.
Require Import Program.Equality.
Require Import Structures.Equalities.
Require Import String.
Axiom vareq: string -> string -> bool.
Open Scope Z_scope.
Open Scope string_scope.



Definition varname := string.

Definition store := varname -> Z.

Inductive aexp: Type :=
  | Var (v:varname)
  | Eval (v:varname) (a:aexp).

Definition update (s:store) (x:varname) (y:Z) :=
  fun z => if vareq x z then y else s z.

Inductive eval_aexp: aexp -> store -> Z -> Prop :=
  | Rule1 (v:varname) (s:store):
          (eval_aexp (Var v) s (s v))

  | Rule2 (v:varname) (a1:aexp) (s:store)
          (H: s v = 0):
          (eval_aexp (Eval v a1) s 0)

  | Rule3 (v:varname) (a1:aexp) (n1 n2:Z) (s:store)
          (H1: s v <> 0) (H2: eval_aexp a1 s n1) (H2: eval_aexp (Eval v a1) (update s v n1) n2):
          (eval_aexp (Eval v a1) s (n1+n2)).

Theorem quiz2:
  forall a s n,
    (eval_aexp a s n) -> (forall v, Zeven (s v)) -> Zeven n.

Proof.
  intros a s n.
  intro H.
  induction H; intros.

  (* Case 1: Var *)
  apply H.

  (* Case 2: Eval0 *)
  simpl.
  apply I.

  (* Case 3: EvalN *)
  apply Zeven_plus_Zeven.
    apply IHeval_aexp1.
    assumption.

    apply IHeval_aexp2.
    unfold update.
    intro v0.
    case (vareq v v0).
      apply IHeval_aexp1. assumption.
      apply H2.
Qed.