Inductive stack_approx_rel (n:nat) : stack -> stack -> Prop :=
  | stack_approx_nil : stack_approx_rel n nil nil
  | stack_approx_cons : forall i i' stk stk',
     instr_approx_rel n i i' ->
     stack_approx_rel n stk stk' ->
     stack_approx_rel n (i::stk) (i'::stk').Inductive stack_approx_rel (n:nat) : stack -> stack -> Prop :=
  | stack_approx_nil : stack_approx_rel n nil nil
  | stack_approx_cons : forall i i' stk stk',
     instr_approx_rel n i i' ->
     stack_approx_rel n stk stk' ->
     stack_approx_rel n (i::stk) (i'::stk').Inductive stack_approx_rel (n:nat) : stack -> stack -> Prop :=
  | stack_approx_nil : stack_approx_rel n nil nil
  | stack_approx_cons : forall i i' stk stk',
     instr_approx_rel n i i' ->
     stack_approx_rel n stk stk' ->
     stack_approx_rel n (i::stk) (i'::stk').

Lemma stack_approx_downward : forall n m s1 s2, n <= m -> stack_approx_rel m s1 s2 -> stack_approx_rel n s1 s2.