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Set Implicit Arguments.
Require Import Lia.
Require Import Coquplib.
From Coq Require Import Lists.List Datatypes.
Import ListNotations.
Local Open Scope nat_scope.
Record Array (A : Type) : Type :=
mk_array
{ arr_contents : list A
; arr_length : nat
; arr_wf : length arr_contents = arr_length
}.
Definition make_array {A : Type} (l : list A) : Array A :=
@mk_array A l (length l) eq_refl.
Fixpoint list_set {A : Type} (i : nat) (x : A) (l : list A) {struct l} : list A :=
match i, l with
| _, nil => nil
| S n, h :: t => h :: list_set n x t
| O, h :: t => x :: t
end.
Lemma list_set_spec1 {A : Type} :
forall l i (x : A),
i < length l -> nth_error (list_set i x l) i = Some x.
Proof.
induction l; intros; destruct i; simplify; try lia; try reflexivity; firstorder.
Qed.
Hint Resolve list_set_spec1 : array.
Lemma list_set_spec2 {A : Type} :
forall l i (x : A) d,
i < length l -> nth i (list_set i x l) d = x.
Proof.
induction l; intros; destruct i; simplify; try lia; try reflexivity; firstorder.
Qed.
Hint Resolve list_set_spec2 : array.
Lemma array_set_wf {A : Type} :
forall l ln i (x : A),
length l = ln -> length (list_set i x l) = ln.
Proof.
induction l; intros; destruct i; auto.
invert H; simplify; auto.
Qed.
Definition array_set {A : Type} (i : nat) (x : A) (a : Array A) :=
let l := a.(arr_contents) in
let ln := a.(arr_length) in
let WF := a.(arr_wf) in
@mk_array A (list_set i x l) ln (@array_set_wf A l ln i x WF).
Lemma array_set_spec1 {A : Type} :
forall a i (x : A),
i < a.(arr_length) -> nth_error ((array_set i x a).(arr_contents)) i = Some x.
Proof.
intros.
rewrite <- a.(arr_wf) in H.
unfold array_set. simplify.
eauto with array.
Qed.
Hint Resolve array_set_spec1 : array.
Lemma array_set_spec2 {A : Type} :
forall a i (x : A) d,
i < a.(arr_length) -> nth i ((array_set i x a).(arr_contents)) d = x.
Proof.
intros.
rewrite <- a.(arr_wf) in H.
unfold array_set. simplify.
eauto with array.
Qed.
Hint Resolve array_set_spec2 : array.
Definition array_get_error {A : Type} (i : nat) (a : Array A) : option A :=
nth_error a.(arr_contents) i.
Lemma array_get_error_bound {A : Type} :
forall (a : Array A) i,
i < a.(arr_length) -> exists x, array_get_error i a = Some x.
Proof.
intros.
rewrite <- a.(arr_wf) in H.
assert (~ length (arr_contents a) <= i) by lia.
pose proof (nth_error_None a.(arr_contents) i).
apply not_iff_compat in H1.
apply <- H1 in H0.
destruct (nth_error (arr_contents a) i ) eqn:EQ; try contradiction; eauto.
Qed.
Lemma array_get_error_set_bound {A : Type} :
forall (a : Array A) i x,
i < a.(arr_length) -> array_get_error i (array_set i x a) = Some x.
Proof.
intros.
unfold array_get_error.
eauto with array.
Qed.
Definition array_get {A : Type} (i : nat) (x : A) (a : Array A) : A :=
nth i a.(arr_contents) x.
Lemma array_get_set_bound {A : Type} :
forall (a : Array A) i x d,
i < a.(arr_length) -> array_get i d (array_set i x a) = x.
Proof.
intros.
unfold array_get.
info_eauto with array.
Qed.
(** Tail recursive version of standard library function. *)
Fixpoint list_repeat' {A : Type} (acc : list A) (a : A) (n : nat) : list A :=
match n with
| O => acc
| S n => list_repeat' (a::acc) a n
end.
Lemma list_repeat'_len {A : Type} : forall (a : A) n l,
length (list_repeat' l a n) = (n + Datatypes.length l)%nat.
Proof.
induction n; intros; simplify; try reflexivity.
specialize (IHn (a :: l)).
rewrite IHn.
simplify.
lia.
Qed.
Lemma list_repeat'_app {A : Type} : forall (a : A) n l,
list_repeat' l a n = list_repeat' [] a n ++ l.
Proof.
induction n; intros; simplify; try reflexivity.
pose proof IHn.
specialize (H (a :: l)).
rewrite H. clear H.
specialize (IHn (a :: nil)).
rewrite IHn. clear IHn.
remember (list_repeat' [] a n) as l0.
rewrite <- app_assoc.
f_equal.
Qed.
Lemma list_repeat'_head_tail {A : Type} : forall n (a : A),
a :: list_repeat' [] a n = list_repeat' [] a n ++ [a].
Proof.
induction n; intros; simplify; try reflexivity.
rewrite list_repeat'_app.
replace (a :: list_repeat' [] a n ++ [a]) with (list_repeat' [] a n ++ [a] ++ [a]).
2: { rewrite app_comm_cons. rewrite IHn; auto.
rewrite app_assoc. reflexivity. }
rewrite app_assoc. reflexivity.
Qed.
Lemma list_repeat'_cons {A : Type} : forall (a : A) n,
list_repeat' [a] a n = a :: list_repeat' [] a n.
Proof.
intros.
rewrite list_repeat'_head_tail; auto.
apply list_repeat'_app.
Qed.
Definition list_repeat {A : Type} : A -> nat -> list A := list_repeat' nil.
Lemma list_repeat_len {A : Type} : forall n (a : A), length (list_repeat a n) = n.
Proof.
intros.
unfold list_repeat.
rewrite list_repeat'_len.
simplify. lia.
Qed.
Lemma dec_list_repeat_spec {A : Type} : forall n (a : A) a',
(forall x x' : A, {x' = x} + {~ x' = x}) ->
In a' (list_repeat a n) -> a' = a.
Proof.
induction n; intros; simplify; try contradiction.
unfold list_repeat in *.
simplify.
rewrite list_repeat'_app in H.
pose proof (X a a').
destruct H0; auto.
(* This is actually a degenerate case, not an unprovable goal. *)
pose proof (in_app_or (list_repeat' [] a n) ([a])).
apply H0 in H. invert H.
- eapply IHn in X; eassumption.
- invert H1; contradiction.
Qed.
Lemma list_repeat_head_tail {A : Type} : forall n (a : A),
a :: list_repeat a n = list_repeat a n ++ [a].
Proof.
unfold list_repeat. apply list_repeat'_head_tail.
Qed.
Lemma list_repeat_cons {A : Type} : forall n (a : A),
list_repeat a (S n) = a :: list_repeat a n.
Proof.
intros.
unfold list_repeat.
apply list_repeat'_cons.
Qed.
Definition arr_repeat {A : Type} (a : A) (n : nat) : Array A := make_array (list_repeat a n).
Fixpoint list_combine {A B C : Type} (f : A -> B -> C) (x : list A) (y : list B) : list C :=
match x, y with
| a :: t, b :: t' => f a b :: list_combine f t t'
| _, _ => nil
end.
Lemma list_combine_length {A B C : Type} (f : A -> B -> C) : forall (x : list A) (y : list B),
length (list_combine f x y) = min (length x) (length y).
Proof.
induction x; intros; simplify; try reflexivity.
destruct y; simplify; auto.
Qed.
Definition combine {A B C : Type} (f : A -> B -> C) (x : Array A) (y : Array B) : Array C :=
make_array (list_combine f x.(arr_contents) y.(arr_contents)).
Lemma combine_length {A B C: Type} : forall x y (f : A -> B -> C),
x.(arr_length) = y.(arr_length) -> arr_length (combine f x y) = x.(arr_length).
Proof.
intros.
unfold combine.
unfold make_array.
simplify.
rewrite <- (arr_wf x) in *.
rewrite <- (arr_wf y) in *.
destruct (arr_contents x); simplify.
- reflexivity.
- destruct (arr_contents y); simplify.
f_equal.
rewrite list_combine_length.
destruct (Min.min_dec (length l) (length l0)); congruence.
Qed.
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