left; trivial.
elim (eq_nat_dec n n0); intuition.
right; intro; injection H; trivial.
Defined.
Definition update (s : store) (c : cell) (v : nat) : store :=
fun c' : cell =>
match cell_eq_dec c' c with
| left _ => v
| right _ => s c'
end.
(* semantics of target language *)
Definition Si (s : store) (i : instr) : store :=
match i with
| LI n => update s Acc n
| LOAD r => update s Acc (s (Reg r))
| STO r => update s (Reg r) (s Acc)
| ADD r => update s Acc (s (Reg r) + s Acc)
end.
Definition list1 (i : instr) := i :: nil.
Fixpoint Sl (s : store) (l : list instr) {struct l} : store :=
match l with
| nil => s
| i :: l' => Sl (Si s i) l'
end.
(* the compiler *)
Definition symt := string -> nat.
Fixpoint C (m : symt) (r : nat) (e : expr) {struct e} :
list instr :=
match e with
| Lit n => list1 (LI n)
| Var s => list1 (LOAD (m s))
| Plus e1 e2 =>
(C m r e1 ++ list1 (STO r)) ++ C m (S r) e2 ++ list1 (ADD r)
end.
(* lemmas *)

Lemma Sl_invariant :
forall (m : symt) (e : expr) (r r' : nat),
r' < r -> forall s : store, Sl s (C m r e) (Reg r') = s (Reg r').