Library stdpp.vector

This file collects general purpose definitions and theorems on vectors (lists of fixed length). It uses the definitions from the standard library, but renames or changes their notations, so that it becomes more consistent with the naming conventions in this development.
From stdpp Require Import countable.
From stdpp Require Export fin list.
From stdpp Require Import options.
Open Scope vector_scope.

The type vec n represents lists of consisting of exactly n elements. Whereas the standard library declares exactly the same notations for vectors as used for lists, we use slightly different notations so it becomes easier to use lists and vectors together.
Notation vec := Vector.t.
Notation vnil := Vector.nil.
Global Arguments vnil {_}.
Notation vcons := Vector.cons.
Notation vapp := Vector.append.
Global Arguments vcons {_} _ {_} _.

Infix ":::" := vcons (at level 60, right associativity) : vector_scope.
Notation "(:::)" := vcons (only parsing) : vector_scope.
Notation "( x :::.)" := (vcons x) (only parsing) : vector_scope.
Notation "(.::: v )" := (λ x, vcons x v) (only parsing) : vector_scope.
Notation "[# ] " := vnil : vector_scope.
Notation "[# x ] " := (vcons x vnil) : vector_scope.
Notation "[# x ; .. ; y ] " := (vcons x .. (vcons y vnil) ..) : vector_scope.
Infix "+++" := vapp (at level 60, right associativity) : vector_scope.
Notation "(+++)" := vapp (only parsing) : vector_scope.
Notation "( v +++.)" := (vapp v) (only parsing) : vector_scope.
Notation "(.+++ w )" := (λ v, vapp v w) (only parsing) : vector_scope.

Similar to fin, we provide an inversion principle that keeps the length fixed. We define a tactic inv_vec v to perform case analysis on v, using this inversion principle.
Notation vec_0_inv := Vector.case0.
Definition vec_S_inv {A n} (P : vec A (S n) Type)
  (Hcons : x v, P (x ::: v)) v : P v.
Proof.
  revert P Hcons.
  refine match v with [#]tt | x ::: vλ P Hcons, Hcons x v end.
Defined.

Global Instance vector_lookup_total A : m, LookupTotal (fin m) A (vec A m) :=
  fix go m i {struct i} := let _ : m, LookupTotal _ _ _ := @go in
  match i in fin m return vec A m A with
  | 0%finvec_S_inv (λ _, A) (λ x _, x)
  | FS jvec_S_inv (λ _, A) (λ _ v, v !!! j)
  end.

The tactic vec_double_ind v1 v2 performs double induction on v1 and v2 provided that they have the same length.
Notation vec_rect2 := Vector.rect2.
Ltac vec_double_ind v1 v2 :=
  match type of v1 with
  | vec _ ?n
    repeat match goal with
    | H' : context [ n ] |- _var_neq v1 H'; var_neq v2 H'; revert H'
    end;
    revert n v1 v2;
    match goal with |- n v1 v2, @?P n v1 v2apply (vec_rect2 P) end
  end.

Notation vcons_inj := VectorSpec.cons_inj.
Lemma vcons_inj_1 {A n} x y (v w : vec A n) : x ::: v = y ::: w x = y.
Proof. apply vcons_inj. Qed.
Lemma vcons_inj_2 {A n} x y (v w : vec A n) : x ::: v = y ::: w v = w.
Proof. apply vcons_inj. Qed.

Lemma vec_eq {A n} (v w : vec A n) : ( i, v !!! i = w !!! i) v = w.
Proof.
  vec_double_ind v w; [done|]. intros n v w IH x y Hi. f_equal.
  - apply (Hi 0%fin).
  - apply IH. intros i. apply (Hi (FS i)).
Qed.

Global Instance vec_dec {A} {dec : EqDecision A} {n} : EqDecision (vec A n).
Proof.
 refine (vec_rect2
  (λ n (v w : vec A n), { v = w } + { v w })
  (left _)
  (λ _ _ _ H x y, cast_if_and (dec x y) H));
  f_equal; eauto using vcons_inj_1, vcons_inj_2.
Defined.

Ltac inv_vec v :=
  let T := type of v in
  match eval hnf in T with
  | vec _ ?n
    match eval hnf in n with
    | 0 ⇒ revert dependent v; match goal with |- v, @?P vapply (vec_0_inv P) end
    | S ?n
      revert dependent v; match goal with |- v, @?P vapply (vec_S_inv P) end;
      
      try (let x := fresh "x" in intros x v; inv_vec v; revert x)
    end
  end.

The following tactic performs case analysis on all hypotheses of the shape fin 0, fin (S n), vec A 0 and vec A (S n) until no further case analyses are possible.
Ltac inv_all_vec_fin := block_goal;
  repeat match goal with
  | v : vec _ _ |- _inv_vec v; intros
  | i : fin _ |- _inv_fin i; intros
  end; unblock_goal.

We define a coercion from vec to list and show that it preserves the operations on vectors. We also define a function to go in the other way, but do not define it as a coercion, as it would otherwise introduce ambiguity.
Fixpoint vec_to_list {A n} (v : vec A n) : list A :=
  match v with [#][] | x ::: vx :: vec_to_list v end.
Coercion vec_to_list : vec >-> list.
Notation list_to_vec := Vector.of_list.

Lemma vec_to_list_cons {A n} x (v : vec A n) :
  vec_to_list (x ::: v) = x :: vec_to_list v.
Proof. done. Qed.
Lemma vec_to_list_app {A n m} (v : vec A n) (w : vec A m) :
  vec_to_list (v +++ w) = vec_to_list v ++ vec_to_list w.
Proof. by induction v; f_equal/=. Qed.
Lemma vec_to_list_to_vec {A} (l : list A): vec_to_list (list_to_vec l) = l.
Proof. by induction l; f_equal/=. Qed.
Lemma vec_to_list_length {A n} (v : vec A n) : length (vec_to_list v) = n.
Proof. induction v; simpl; by f_equal. Qed.
Lemma vec_to_list_same_length {A B n} (v : vec A n) (w : vec B n) :
  length v = length w.
Proof. by rewrite !vec_to_list_length. Qed.
Lemma vec_to_list_inj1 {A n m} (v : vec A n) (w : vec A m) :
  vec_to_list v = vec_to_list w n = m.
Proof.
  revert m w. induction v; intros ? [|???] ?;
    simplify_eq/=; f_equal; eauto.
Qed.
Lemma vec_to_list_inj2 {A n} (v : vec A n) (w : vec A n) :
  vec_to_list v = vec_to_list w v = w.
Proof.
  revert w. induction v; intros w; inv_vec w; intros;
    simplify_eq/=; f_equal; eauto.
Qed.
Lemma list_to_vec_to_list {A n} (v : vec A n) :
  list_to_vec (vec_to_list v) = eq_rect _ _ v _ (eq_sym (vec_to_list_length v)).
Proof.
  apply vec_to_list_inj2. rewrite vec_to_list_to_vec.
  by destruct (eq_sym (vec_to_list_length v)).
Qed.

Lemma vlookup_middle {A n m} (v : vec A n) (w : vec A m) x :
   i : fin (n + S m), x = (v +++ x ::: w) !!! i.
Proof.
  induction v as [|??? IHv]; simpl; [by eexists 0%fin|].
  destruct IHv as [i ?]. by (FS i).
Qed.
Lemma vec_to_list_lookup_middle {A n} (v : vec A n) (l k : list A) x :
  vec_to_list v = l ++ x :: k
     i : fin n, l = take i v x = v !!! i k = drop (S i) v.
Proof.
  intros H.
  rewrite <-(vec_to_list_to_vec l), <-(vec_to_list_to_vec k) in H.
  rewrite <-vec_to_list_cons, <-vec_to_list_app in H.
  pose proof (vec_to_list_inj1 _ _ H); subst.
  apply vec_to_list_inj2 in H; subst. induction l as [|?? IHl]; simpl.
  - eexists 0%fin. simpl. by rewrite vec_to_list_to_vec.
  - destruct IHl as [i ?]. (FS i). simpl. intuition congruence.
Qed.
Lemma vec_to_list_drop_lookup {A n} (v : vec A n) (i : fin n) :
  drop i v = v !!! i :: drop (S i) v.
Proof. induction i as [|?? IHi]; inv_vec v; simpl; intros; [done | by rewrite IHi]. Qed.
Lemma vec_to_list_take_drop_lookup {A n} (v : vec A n) (i : fin n) :
  vec_to_list v = take i v ++ v !!! i :: drop (S i) v.
Proof. rewrite <-(take_drop i v) at 1. by rewrite vec_to_list_drop_lookup. Qed.

Lemma vlookup_lookup {A n} (v : vec A n) (i : fin n) x :
  v !!! i = x (v : list A) !! (i : nat) = Some x.
Proof.
  induction v as [|? ? v IH]; inv_fin i.
  - simpl; split; congruence.
  - done.
Qed.
Lemma vlookup_lookup' {A n} (v : vec A n) (i : nat) x :
  ( H : i < n, v !!! nat_to_fin H = x) (v : list A) !! i = Some x.
Proof.
  split.
  - intros [Hlt ?]. rewrite <-(fin_to_nat_to_fin i n Hlt). by apply vlookup_lookup.
  - intros Hvix. assert (Hlt:=lookup_lt_Some _ _ _ Hvix).
    rewrite vec_to_list_length in Hlt. Hlt.
    apply vlookup_lookup. by rewrite fin_to_nat_to_fin.
Qed.
Lemma elem_of_vlookup {A n} (v : vec A n) x :
  x vec_to_list v i, v !!! i = x.
Proof.
  rewrite elem_of_list_lookup. setoid_rewrite <-vlookup_lookup'.
  split; [by intros (?&?&?); eauto|]. intros [i Hx].
   i, (fin_to_nat_lt _). by rewrite nat_to_fin_to_nat.
Qed.

Lemma Forall_vlookup {A} (P : A Prop) {n} (v : vec A n) :
  Forall P (vec_to_list v) i, P (v !!! i).
Proof. rewrite Forall_forall. setoid_rewrite elem_of_vlookup. naive_solver. Qed.
Lemma Forall_vlookup_1 {A} (P : A Prop) {n} (v : vec A n) i :
  Forall P (vec_to_list v) P (v !!! i).
Proof. by rewrite Forall_vlookup. Qed.
Lemma Forall_vlookup_2 {A} (P : A Prop) {n} (v : vec A n) :
  ( i, P (v !!! i)) Forall P (vec_to_list v).
Proof. by rewrite Forall_vlookup. Qed.
Lemma Exists_vlookup {A} (P : A Prop) {n} (v : vec A n) :
  Exists P (vec_to_list v) i, P (v !!! i).
Proof. rewrite Exists_exists. setoid_rewrite elem_of_vlookup. naive_solver. Qed.
Lemma Forall2_vlookup {A B} (P : A B Prop) {n}
    (v1 : vec A n) (v2 : vec B n) :
  Forall2 P (vec_to_list v1) (vec_to_list v2) i, P (v1 !!! i) (v2 !!! i).
Proof.
  split.
  - vec_double_ind v1 v2; [intros _ i; inv_fin i |].
    intros n v1 v2 IH a b; simpl. inversion_clear 1.
    intros i. inv_fin i; simpl; auto.
  - vec_double_ind v1 v2; [constructor|].
    intros ??? IH ?? H. constructor.
    + apply (H 0%fin).
    + apply IH, (λ i, H (FS i)).
Qed.

Given a function fin n A, we can construct a vector.
Fixpoint fun_to_vec {A n} {struct n} : (fin n A) vec A n :=
  match n with
  | 0 ⇒ λ f, [#]
  | S nλ f, f 0%fin ::: fun_to_vec (f FS)
  end.

Lemma lookup_fun_to_vec {A n} (f : fin n A) i : fun_to_vec f !!! i = f i.
Proof.
  revert f. induction i as [|n i IH]; intros f; simpl; [done|]. by rewrite IH.
Qed.

The function vmap f v applies a function f element wise to v.
Notation vmap := Vector.map.

Lemma vlookup_map `(f : A B) {n} (v : vec A n) i :
  vmap f v !!! i = f (v !!! i).
Proof. by induction v; inv_fin i; eauto. Qed.
Lemma vec_to_list_map `(f : A B) {n} (v : vec A n) :
  vec_to_list (vmap f v) = f <$> vec_to_list v.
Proof. induction v as [|??? IHv]; simpl; [done|]. by rewrite IHv. Qed.

The function vzip_with f v w combines the vectors v and w element wise using the function f.
Notation vzip_with := Vector.map2.

Lemma vlookup_zip_with `(f : A B C) {n} (v1 : vec A n) (v2 : vec B n) i :
  vzip_with f v1 v2 !!! i = f (v1 !!! i) (v2 !!! i).
Proof.
  vec_double_ind v1 v2.
  - intros i; inv_fin i.
  - intros n v1 v2 IH a b i.
    inv_fin i; eauto.
Qed.
Lemma vec_to_list_zip_with `(f : A B C) {n} (v1 : vec A n) (v2 : vec B n) :
  vec_to_list (vzip_with f v1 v2) =
    zip_with f (vec_to_list v1) (vec_to_list v2).
Proof.
  revert v2. induction v1 as [|??? IHv1]; intros v2; inv_vec v2; intros; simpl; [done|].
  by rewrite IHv1.
Qed.

Similar to vlookup, we cannot define vinsert as an instance of the Insert type class, as it has a dependent type.
Fixpoint vinsert {A n} (i : fin n) (x : A) : vec A n vec A n :=
  match i with
  | 0%finvec_S_inv _ (λ _ v, x ::: v)
  | FS ivec_S_inv _ (λ y v, y ::: vinsert i x v)
  end.

Lemma vec_to_list_insert {A n} i x (v : vec A n) :
  vec_to_list (vinsert i x v) = insert (fin_to_nat i) x (vec_to_list v).
Proof. induction v as [|??? IHv]; inv_fin i; [done|]. simpl. intros. by rewrite IHv. Qed.

Lemma vlookup_insert {A n} i x (v : vec A n) : vinsert i x v !!! i = x.
Proof. by induction i; inv_vec v. Qed.
Lemma vlookup_insert_ne {A n} i j x (v : vec A n) :
  i j vinsert i x v !!! j = v !!! j.
Proof.
  induction i as [|?? IHi]; inv_fin j; inv_vec v; simpl; try done.
  intros. apply IHi. congruence.
Qed.
Lemma vlookup_insert_self {A n} i (v : vec A n) : vinsert i (v !!! i) v = v.
Proof. by induction v; inv_fin i; intros; f_equal/=. Qed.

Lemma vmap_insert {A B} (f : A B) (n : nat) i x (v : vec A n) :
  vmap f (vinsert i x v) = vinsert i (f x) (vmap f v).
Proof. induction v; inv_fin i; intros; f_equal/=; auto. Qed.

The functions vtake i v and vdrop i v take the first i elements of a vector v, respectively remove the first i elements of a vector v.
Fixpoint vtake {A n} (i : fin n) : vec A n vec A i :=
  match i in fin n return vec A n vec A i with
  | 0%finλ _, [#]
  | FS ivec_S_inv _ (λ x v, x ::: vtake i v)
  end.
Fixpoint vdrop {A n} (i : fin n) : vec A n vec A (n - i) :=
  match i in fin n return vec A n vec A (n - i) with
  | 0%finid
  | FS ivec_S_inv _ (λ _, vdrop i)
  end.

Lemma vec_to_list_take {A n} i (v : vec A n) :
  vec_to_list (vtake i v) = take (fin_to_nat i) (vec_to_list v).
Proof. induction i; inv_vec v; intros; f_equal/=; auto. Qed.
Lemma vec_to_list_drop {A n} i (v : vec A n) :
  vec_to_list (vdrop i v) = drop (fin_to_nat i) (vec_to_list v).
Proof. induction i; inv_vec v; intros; f_equal/=; auto. Qed.

The function vreplicate n x generates a vector with length n of elements with value x.
Fixpoint vreplicate {A} (n : nat) (x : A) : vec A n :=
  match n with 0 ⇒ [#] | S nx ::: vreplicate n x end.

Lemma vec_to_list_replicate {A} n (x : A) :
  vec_to_list (vreplicate n x) = replicate n x.
Proof. induction n; by f_equal/=. Qed.

Lemma vlookup_replicate {A} n (x : A) i : vreplicate n x !!! i = x.
Proof. induction i; f_equal/=; auto. Qed.

Lemma vmap_replicate {A B} (f : A B) n (x : A) :
  vmap f (vreplicate n x) = vreplicate n (f x).
Proof. induction n; f_equal/=; auto. Qed.

Vectors are inhabited and countable
Global Instance vec_0_inhabited T : Inhabited (vec T 0) := populate [#].
Global Instance vec_inhabited `{Inhabited T} n : Inhabited (vec T n) :=
  populate (vreplicate n inhabitant).

Global Instance vec_countable `{Countable A} n : Countable (vec A n).
Proof.
  apply (inj_countable vec_to_list (λ l,
    guard (n = length l) as H; Some (eq_rect _ _ (list_to_vec l) _ (eq_sym H)))).
  intros v. case_option_guard as Hn.
  - rewrite list_to_vec_to_list.
    rewrite (proof_irrel (eq_sym _) Hn). by destruct Hn.
  - by rewrite vec_to_list_length in Hn.
Qed.