Use a `list RPNOp` instead of a `RPN`

1 files changed, 18 insertions, 35 deletions

diff --git a/expr.v b/expr.v
index 089ac03..7c6cab3 100644
--- a/expr.v
+++ b/expr.v
@@ -13,25 +13,16 @@ Fixpoint eval (env : nat -> nat) (e : Exp) :=
   | 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.
+Inductive RPNOp :=
+  | rpnlit : nat -> RPNOp
+  | rpnvar : nat -> RPNOp
+  | rpnadd : RPNOp.
 
 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))
+  | elit n => cons (rpnlit n) nil
+  | evar n => cons (rpnvar n) nil
+  | eadd a b => cons rpnadd (app (rpn a) (rpn b))
   end.
 
 Definition _fmap A B (f : A -> B) (o : option A) :=
@@ -41,18 +32,18 @@ Definition _fmap A B (f : A -> B) (o : option A) :=
   end.
 Notation fmap := (_fmap _ _).
 
-Fixpoint rpn_eval' (env : nat -> nat) (rpn : RPN) :=
+Fixpoint rpn_eval' (env : nat -> nat) (rpn : list RPNOp) :=
   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
+  | nil => Some nil
+  | cons (rpnlit n) a => fmap (cons n) (rpn_eval' env a)
+  | cons (rpnvar n) a => fmap (cons (env n)) (rpn_eval' env a)
+  | cons (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) :=
+Function rpn_eval (env : nat -> nat) (rpn : list RPNOp) :=
   match rpn_eval' env rpn with
   | Some (cons x nil) => Some x
   | _                 => None
@@ -70,22 +61,19 @@ 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).
+  Some (eval env e1 :: results)%list = rpn_eval' env (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.
+rewrite (List.app_assoc_reverse (rpn e1_1) (rpn e1_2) rest).
 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))).
+    = rpn_eval' env (app (rpn e1_1) (app (rpn e1_2) rest))).
   apply H. apply exprsize_left.
   apply H. apply exprsize_right.
   assumption.
@@ -93,16 +81,11 @@ 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.
+assert (app (rpn e) nil = rpn e).
+apply List.app_nil_r.
 rewrite <- H.
 apply free_append'.
 simpl. reflexivity.