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Require Import Omega.
Require Import Wellfounded.
Inductive Exp : Set :=
| elit : nat -> Exp
| evar : nat -> Exp
| eadd : Exp -> Exp -> Exp.
Fixpoint eval (env : nat -> nat) (e : Exp) :=
match e with
| elit n => n
| evar n => env n
| eadd a b => eval env a + eval env b
end.
Inductive RPN :=
| rpnnil : RPN
| rpnlit : nat -> RPN -> RPN
| rpnvar : nat -> RPN -> RPN
| rpnadd : RPN -> RPN.
Fixpoint rpn_app (a : RPN) (b : RPN) :=
match a with
| rpnnil => b
| rpnlit n a => rpnlit n (rpn_app a b)
| rpnvar n a => rpnvar n (rpn_app a b)
| rpnadd a => rpnadd (rpn_app a b)
end.
Fixpoint rpn (e : Exp) :=
match e with
| elit n => rpnlit n rpnnil
| evar n => rpnvar n rpnnil
| eadd a b => rpnadd (rpn_app (rpn a) (rpn b))
end.
Definition _fmap A B (f : A -> B) (o : option A) :=
match o with
| Some x => Some (f x)
| None => None
end.
Notation fmap := (_fmap _ _).
Fixpoint rpn_eval' (env : nat -> nat) (rpn : RPN) :=
match rpn with
| rpnnil => Some nil
| rpnlit n a => fmap (cons n) (rpn_eval' env a)
| rpnvar n a => fmap (cons (env n)) (rpn_eval' env a)
| rpnadd a => match rpn_eval' env a with
| Some (cons x (cons y stack)) => Some (cons (x+y) stack)
| _ => None
end
end.
Function rpn_eval (env : nat -> nat) (rpn : RPN) :=
match rpn_eval' env rpn with
| Some (cons x nil) => Some x
| _ => None
end.
Fixpoint exprsize e :=
match e with
| elit _ => 1
| evar _ => 1
| eadd a b => 1 + exprsize a + exprsize b
end.
Lemma exprsize_left e1 e2 : exprsize e1 < exprsize (eadd e1 e2).
Proof. simpl. omega. Qed.
Lemma exprsize_right e1 e2 : exprsize e1 < exprsize (eadd e2 e1).
Proof. simpl. omega. Qed.
Lemma append_assoc a b c : rpn_app (rpn_app a b) c = rpn_app a (rpn_app b c).
Proof. induction a; simpl; try rewrite IHa; reflexivity. Qed.
Lemma free_append' e1 :
forall env rest results,
Some results = rpn_eval' env rest ->
Some (eval env e1 :: results)%list = rpn_eval' env (rpn_app (rpn e1) rest).
Proof.
induction e1 using (well_founded_induction
(wf_inverse_image _ nat _ exprsize PeanoNat.Nat.lt_wf_0)).
intros.
destruct e1; simpl; try rewrite <- H0; try reflexivity.
(* Only eadd is left now *)
rewrite append_assoc.
assert (Some (eval env e1_1 :: eval env e1_2 :: results)%list
= rpn_eval' env (rpn_app (rpn e1_1) (rpn_app (rpn e1_2) rest))).
apply H. apply exprsize_left.
apply H. apply exprsize_right.
assumption.
rewrite <- H1.
reflexivity.
Qed.
Lemma append_nil_at_end r : rpn_app r rpnnil = r.
Proof.
induction r; simpl; try rewrite IHr; reflexivity.
Qed.
Lemma correctness e env : Some (eval env e) = rpn_eval env (rpn e).
Proof.
assert (Some (eval env e :: nil)%list = rpn_eval' env (rpn e)).
assert (rpn_app (rpn e) rpnnil = rpn e).
apply append_nil_at_end.
rewrite <- H.
apply free_append'.
simpl. reflexivity.
unfold rpn_eval.
rewrite <- H.
reflexivity.
Qed.
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