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Require Import Eqdep_dec.
Section MemoFunction.
Variable A: Type.
Variable f: nat -> A.
Variable g: A -> A.
Hypothesis Hg_correct: forall n, f (S n) = g (f n).
(* Memo Stream *)
CoInductive MStream: Type :=
MSeq : A -> MStream -> MStream.
(* Hd and Tl function *)
Definition mhd (x: MStream) :=
let (a,s) := x in a.
Definition mtl (x: MStream) :=
let (a,s) := x in s.
CoFixpoint memo_make (n: nat): MStream:= MSeq (f n) (memo_make (S n)).
Definition memo_list := memo_make 0.
Fixpoint memo_get (n: nat) (l: MStream) {struct n}: A :=
match n with O => mhd l | S n1 =>
memo_get n1 (mtl l) end.
Theorem memo_get_correct: forall n, memo_get n memo_list = f n.
Proof.
assert (F1: forall n m, memo_get n (memo_make m) = f (n + m)).
induction n as [| n Hrec]; try (intros m; refine (refl_equal _)).
intros m; simpl; rewrite Hrec.
rewrite plus_n_Sm; auto.
intros n; apply trans_equal with (f (n + 0)); try exact (F1 n 0).
rewrite <- plus_n_O; auto.
Qed.
(* Building with possible sharing using g *)
CoFixpoint imemo_make (fn: A): MStream :=
let fn1 := g fn in
MSeq fn1 (imemo_make fn1).
Definition imemo_list := let f0 := f 0 in
MSeq f0 (imemo_make f0).
Theorem imemo_get_correct: forall n, memo_get n imemo_list = f n.
Proof.
assert (F1: forall n m,
memo_get n (imemo_make (f m)) = f (S (n + m))).
induction n as [| n Hrec]; try (intros m; exact (sym_equal (Hg_correct m))).
simpl; intros m; rewrite <- Hg_correct; rewrite Hrec; rewrite <- plus_n_Sm; auto.
destruct n as [| n]; try apply refl_equal.
unfold imemo_list; simpl; rewrite F1.
rewrite <- plus_n_O; auto.
Qed.
End MemoFunction.
Section DependentMemoFunction.
Variable A: nat -> Type.
Variable f: forall n, A n.
Variable g: forall n, A n -> A (S n).
Hypothesis Hg_correct: forall n, f (S n) = g n (f n).
Inductive memo_val: Type :=
memo_mval: forall n, A n -> memo_val.
Fixpoint is_eq (n m : nat) {struct n}: {n = m} + {True} :=
match n, m return {n = m} + {True} with
| 0, 0 =>left True (refl_equal 0)
| 0, S m1 => right (0 = S m1) I
| S n1, 0 => right (S n1 = 0) I
| S n1, S m1 =>
match is_eq n1 m1 with
| left H => left True (f_equal S H)
| right _ => right (S n1 = S m1) I
end
end.
Definition memo_get_val n (v: memo_val): A n :=
match v with
| memo_mval m x =>
match is_eq n m with
| left H =>
match H in (@eq _ _ y) return (A y -> A n) with
| refl_equal => fun v1 : A n => v1
end
| right _ => fun _ : A m => f n
end x
end.
Let mf n := memo_mval n (f n).
Let mg v := match v with
memo_mval n1 v1 => memo_mval (S n1) (g n1 v1) end.
Definition dmemo_list := memo_list _ mf.
Definition dmemo_get n l := memo_get_val n (memo_get _ n l).
Definition dimemo_list := imemo_list _ mf mg.
Theorem dmemo_get_correct: forall n, dmemo_get n dmemo_list = f n.
Proof.
intros n; unfold dmemo_get, dmemo_list.
rewrite (memo_get_correct memo_val mf n); simpl.
case (is_eq n n); simpl; auto; intros e.
assert (e = refl_equal n).
apply eq_proofs_unicity.
induction x as [| x Hx]; destruct y as [| y].
left; auto.
right; intros HH; discriminate HH.
right; intros HH; discriminate HH.
case (Hx y).
intros HH; left; case HH; auto.
intros HH; right; intros HH1; case HH.
injection HH1; auto.
rewrite H; auto.
Qed.
Theorem dimemo_get_correct: forall n, dmemo_get n dimemo_list = f n.
Proof.
intros n; unfold dmemo_get, dimemo_list.
rewrite (imemo_get_correct memo_val mf mg); simpl.
case (is_eq n n); simpl; auto; intros e.
assert (e = refl_equal n).
apply eq_proofs_unicity.
induction x as [| x Hx]; destruct y as [| y].
left; auto.
right; intros HH; discriminate HH.
right; intros HH; discriminate HH.
case (Hx y).
intros HH; left; case HH; auto.
intros HH; right; intros HH1; case HH.
injection HH1; auto.
rewrite H; auto.
intros n1; unfold mf; rewrite Hg_correct; auto.
Qed.
End DependentMemoFunction.
(* An example with the memo function on factorial *)
(**
Require Import ZArith.
Fixpoint tfact (n: nat) {struct n} :=
match n with O => 1%Z |
S n1 => (Z_of_nat n * tfact n1)%Z end.
Definition lfact_list :=
dimemo_list _ tfact (fun n z => (Z_of_nat (S n) * z)%Z).
Definition lfact n := dmemo_get _ tfact n lfact_list.
Theorem lfact_correct n: lfact n = tfact n.
Proof.
intros n; unfold lfact, lfact_list.
rewrite dimemo_get_correct; auto.
Qed.
Fixpoint nop p := match p with
xH => 0 | xI p1 => nop p1 | xO p1 => nop p1 end.
Fixpoint test z := match z with
Z0 => 0 | Zpos p1 => nop p1 | Zneg p1 => nop p1 end.
Time Eval vm_compute in test (lfact 2000).
Time Eval vm_compute in test (lfact 2000).
Time Eval vm_compute in test (lfact 1500).
Time Eval vm_compute in (lfact 1500).
**)
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