Definition instr_approx_rel (n:nat) (i1 i2:instruction) : Prop :=
  fmap_instr (K.approx n) i1 = fmap_instr (K.approx n) i2.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').

(* Specifications: *)
Lemma stack_approx_rel_refl : forall n stk, stack_approx_rel n stk stk.
Lemma intro_instr_approx_rel : forall m n i, m >= n -> instr_approx_rel n i (fmap_instr (K.approx m) i).
Lemma step_approx : forall n t t' p1 p2 s1 s1' s2 r1 r1' r2, level p2 < n -> termMeasure_incr t t' -> store_incr r1 r1' -> stack_approx_rel n s1 s2 -> step t p1 p2 r1 s1 r2 s1' -> exists r2', exists s2', step t' p1 p2 r1' s2 r2' s2' /\ store_incr r2 r2' /\ stack_approx_rel n s1' s2'.
Lemma step_prog_levels : forall t p p' r s r' s', step t p p' r s r' s' -> level p' <= level p.
Lemma stepstar_prog_levels : forall t p p' r s r' s', stepstar t p p' r s r' s' -> level p' <= level p.
Lemma stepstar_approx : forall n t t' p1 p2 s1 s1' s2 r1 r1' r2, level p1 < n -> termMeasure_incr t t' -> store_incr r1 r1' -> stack_approx_rel n s1 s2 -> stepstar t p1 p2 r1 s1 r2 s1' -> exists r2', exists s2', stepstar t' p1 p2 r1' s2 r2' s2' /\ store_incr r2 r2' /\ stack_approx_rel n s1' s2'.