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list.v
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(*
Copyright 2014 Cornell University
This file is part of VPrl (the Verified Nuprl project).
VPrl is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
VPrl is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with VPrl. If not, see <http://www.gnu.org/licenses/>.
Website: http://nuprl.org/html/verification/
Authors: Abhishek Anand & Vincent Rahli
*)
Require Import bin_rels.
Require Import eq_rel.
Require Import universe.
Require Import LibTactics.
Require Import tactics.
Require Import Coq.Bool.Bool.
Require Import Coq.Program.Tactics.
Require Import Omega.
Require Import Coq.Program.Basics.
Require Import Coq.Lists.List.
Require Import Coq.Init.Notations.
Require Import UsefulTypes.
Require Import Coq.Classes.DecidableClass.
Require Import Coq.Classes.Morphisms.
Notation LIn := (List.In).
(**
uncomment this and remove the line above if [univ]:=Type
Fixpoint LIn {A : Type} (a:A) (l:list A) : [univ] :=
match l with
| nil => False
| b :: m => (b = a) [+] LIn a m
end.
Definition In := 9.
*)
(* Move *)
Definition listPad {T} (def:T) (l: list T) (n: nat) : list T :=
l++(repeat def (n-length l)).
Lemma listPad_length {T} (def:T) (l: list T) (n: nat):
n <= length (listPad def l n).
Proof using.
setoid_rewrite app_length.
rewrite repeat_length.
omega.
Qed.
SearchAbout NoDup.
Fixpoint ball (l : list bool) : bool :=
match l with
| [] => true
| x :: xs => andb x (ball xs)
end.
Lemma ball_true :
forall l,
ball l = true <=> (forall x, LIn x l -> x = true).
Proof.
induction l; simpl; sp.
rw andb_eq_true.
trw IHl; split; sp.
Qed.
Lemma ball_map_true :
forall A,
forall f : A -> bool,
forall l : list A,
ball (map f l) = true <=> forall x, LIn x l -> f x = true.
Proof.
induction l; simpl; sp.
trw andb_eq_true.
trw IHl; split; sp; subst; sp.
Qed.
Fixpoint remove {A:Type } `{Deq A} (x : A) (l : list A) : list A :=
match l with
| [] => []
| y::tl => if (decide (x = y)) then remove x tl else y::(remove x tl)
end.
Theorem remove_In {A:Type } `{Deq A} : forall (l : list A) (x : A), ~ In x (remove x l).
Proof.
induction l as [|x l]; auto. simpl. intro y.
rewrite decide_decideP.
cases_if; simpl; auto.
firstorder.
Qed.
(** l \ lr -- removes the elements of lr from l *)
Fixpoint diff {T} {eqd:Deq T} (lr:list T) (l:list T) : list T :=
match lr with
| [] => l
| h::t => diff t (remove h l)
end.
Lemma remove_trivial :
forall T x eq,
forall l : list T,
(! LIn x l)
-> @remove T eq x l = l.
Proof.
induction l as [| a l]; simpl; intro H; sp.
rewrite decide_decideP.
cases_if as Heq.
destruct H. left. auto.
f_equal.
rewrite IHl. auto.
introv Hlin. apply H. auto.
Qed.
Lemma diff_nil :
forall T eq, forall l : list T, @diff T eq l [] = [].
Proof.
induction l; simpl; auto.
Qed.
Hint Rewrite diff_nil.
Typeclasses eauto := 2.
Lemma in_remove {T:Type} `{Deq T}:
forall x y, forall l : list T,
LIn x (remove y l) <=> (~ x = y) # LIn x l.
Proof.
induction l; simpl.
split; sp. rewrite decide_decideP.
cases_if; subst; allsimpl; allrw IHl; split; sp; subst; sp.
Qed.
Lemma in_diff :
forall T,
forall l1 l2 : list T,
forall x eq,
LIn x (@diff T eq l1 l2)
<=>
(LIn x l2 # (! LIn x l1)).
Proof.
induction l1; simpl; sp.
split; sp. introv. auto.
trw IHl1; trw in_remove; split; sp; auto.
Qed.
Lemma remove_app {T:Type} `{Deq T}:
forall x,
forall l1 l2 : list T,
remove x (l1 ++ l2) = remove x l1 ++ remove x l2.
Proof.
induction l1; simpl; sp. repeat rewrite decide_decideP.
cases_if; subst; rewrite IHl1; auto.
Qed.
(* Move *)
Lemma deq_refl {T:Type} `{Deq T} : forall (x:T), (decide (x=x)) = true.
Proof.
intros.
rewrite Decidable_complete; auto.
Qed.
Lemma deqP_refl {T:Type} `{Deq T} : forall (x:T), (decideP (x=x)) = left eq_refl.
Proof.
intros. destruct (decideP (x = x)); try discriminate.
- f_equal. apply Eqdep_dec.UIP_dec.
intros. apply decideP; auto.
- sp.
Qed.
Hint Rewrite (fun T d => @deq_refl T d) (fun T d => @deqP_refl T d) : SquiggleEq.
Lemma remove_comm {T:Type} `{Deq T} :
forall l : list T,
forall x y,
remove x (remove y l) = remove y (remove x l).
Proof.
induction l; simpl; sp.
repeat rewrite decide_decideP.
repeat (cases_if; subst; simpl in *; auto; autorewrite with SquiggleEq; try congruence;
repeat rewrite decide_decideP).
Qed.
Lemma diff_remove {T:Type} {eq:Deq T} :
forall l1 l2 : list T,
forall x,
@diff T eq l1 (@remove T eq x l2) = @remove _ eq x (@diff _ eq l1 l2).
Proof.
induction l1; simpl; sp.
repeat (rewrite IHl1).
rewrite remove_comm; auto.
Qed.
Lemma diff_comm :
forall T eq,
forall l1 l2 l3 : list T,
@diff _ eq l1 (@diff _ eq l2 l3) = @diff _ eq l2 (@diff _ eq l1 l3).
Proof.
induction l1; simpl; sp.
rewrite <- diff_remove.
rewrite IHl1; auto.
Qed.
Lemma diff_app_r :
forall T eq,
forall l1 l2 l3 : list T,
@diff _ eq l1 (l2 ++ l3) = @diff _ eq l1 l2 ++ @diff _ eq l1 l3.
Proof.
induction l1; simpl; sp.
rewrite remove_app.
rewrite IHl1; auto.
Qed.
Lemma diff_app_l :
forall T eq,
forall l1 l2 l3 : list T,
@diff _ eq l1 (@diff _ eq l2 l3) = @diff _ eq (l1 ++ l2) l3.
Proof.
induction l1; simpl; sp.
repeat (rewrite diff_remove).
rewrite IHl1; auto.
Qed.
Lemma remove_repeat :
forall T eq x,
forall l : list T,
@remove T eq x l = @remove T eq x (@remove T eq x l).
Proof.
induction l; simpl; sp. repeat rewrite decide_decideP.
repeat cases_if; auto. simpl.
repeat rewrite decide_decideP.
repeat cases_if; simpl; auto.
provefalse; apply H; auto.
rewrite <- IHl; auto.
Qed.
Lemma diff_repeat :
forall T eq,
forall l1 l2 : list T,
@diff _ eq l1 l2 = @diff _ eq l1 (@diff _ eq l1 l2).
Proof.
induction l1; simpl; sp.
repeat (rewrite diff_remove).
rewrite <- remove_repeat.
rewrite <- IHl1; auto.
Qed.
Lemma remove_dup :
forall T eq,
forall l1 l2 : list T,
forall x,
LIn x l1
-> @diff _ eq l1 l2 = @remove T eq x (@diff _ eq l1 l2).
Proof.
induction l1; simpl; sp; subst.
rewrite diff_remove.
rewrite <- remove_repeat; auto.
Qed.
Lemma diff_move :
forall T eq,
forall l1 l2 l3 : list T,
forall x,
@diff _ eq (l1 ++ x :: l2) l3 = @diff _ eq (x :: l1 ++ l2) l3.
Proof.
induction l1; simpl; sp.
rewrite IHl1; simpl.
rewrite remove_comm; auto.
Qed.
Lemma diff_dup :
forall T eq,
forall l1 l2 l3 : list T,
forall x,
LIn x (l1 ++ l2)
-> @diff _ eq (l1 ++ l2) l3 = @diff _ eq (l1 ++ x :: l2) l3.
Proof.
induction l1; simpl; sp; subst.
rewrite diff_remove.
apply remove_dup; auto.
rewrite diff_move; simpl.
rewrite <- remove_repeat; auto.
Qed.
Lemma in_app_iff :
forall A l l' (a:A),
LIn a (l++l') <=> (LIn a l) [+] (LIn a l').
Proof.
induction l as [| a l]; introv; simpl; try (rw IHl); split; sp.
Qed.
Lemma in_map_iff :
forall (A B : Type) (f : A -> B) l b,
LIn b (map f l) <=> {a : A $ LIn a l # b = f a}.
Proof.
induction l; simpl; sp; try (complete (split; sp)).
trw IHl; split; sp; subst; sp.
exists a; sp.
exists a0; sp.
right; exists a0; sp.
Qed.
Lemma diff_dup2 :
forall T eq,
forall l1 l2 l3 : list T,
forall x,
LIn x l1
-> @diff _ eq (l1 ++ l2) l3 = @diff _ eq (l1 ++ x :: l2) l3.
Proof.
intros; apply diff_dup.
apply in_app_iff; left; auto.
Qed.
Definition null {T} (l : list T) := forall x, !LIn x l.
Lemma null_nil : forall T, null ([] : list T).
Proof.
unfold null; sp.
Qed.
Hint Immediate null_nil.
Lemma null_nil_iff : forall T, null ([] : list T) <=> True.
Proof.
split; sp; apply null_nil.
Qed.
Hint Rewrite null_nil_iff.
Hint Resolve decideP : SquiggleEq.
Lemma null_diff :
forall T,
forall eq : Deq T,
forall l1 l2 : list T,
null (@diff _ eq l1 l2) <=> forall x, LIn x l2 -> LIn x l1.
Proof.
induction l1; simpl; sp.
trw IHl1; sp; split; sp.
assert ({a = x} + {a <> x}) by auto with SquiggleEq; sp.
right; apply_hyp.
trw in_remove; sp.
alltrewrite in_remove; sp.
apply_in_hyp p; sp; subst; sp.
Qed.
Lemma null_iff_nil : forall T, forall l : list T, null l <=> l = [].
Proof.
induction l; unfold null; simpl; split; sp.
assert ((a = a) [+] LIn a l) by (left; auto).
apply_in_hyp p; sp.
Qed.
Lemma null_cons :
forall T x,
forall l : list T,
!( null (x :: l)).
Proof.
unfold null; sp.
assert (LIn x (x :: l)) by (simpl; left; auto).
apply_in_hyp p; sp.
Qed.
Hint Immediate null_cons.
Lemma null_app :
forall T,
forall l1 l2 : list T,
null (l1 ++ l2) <=> null l1 # null l2.
Proof.
induction l1; simpl; sp; split; sp;
try (apply null_nil);
try(apply null_cons in H); sp;
try(apply null_cons in H0); sp.
Qed.
Lemma null_map :
forall A B,
forall f : A -> B,
forall l : list A,
null (map f l) <=> null l.
Proof.
induction l; simpl; sp; split; sp;
try (apply null_nil);
apply null_cons in H; sp.
Qed.
Definition nullb {T} (l : list T) := if l then true else false.
Lemma assert_nullb :
forall T,
forall l : list T,
assert (nullb l) <=> null l.
Proof.
destruct l; simpl; split; sp.
apply not_assert in H; sp.
apply null_cons in H; sp.
Qed.
Definition subsetb {T} (eq : Deq T) (l1 l2 : list T) : bool :=
nullb (@diff _ eq l2 l1).
Definition eqsetb {T} (eq : Deq T) (l1 l2 : list T) : bool :=
subsetb eq l1 l2 && subsetb eq l2 l1.
Lemma assert_subsetb :
forall T eq,
forall l1 l2 : list T,
assert (subsetb eq l1 l2)
<=>
forall x, LIn x l1 -> LIn x l2.
Proof.
sp; unfold subsetb.
trw assert_nullb; trw null_diff; split; sp.
Qed.
Lemma assert_eqsetb :
forall T eq,
forall l1 l2 : list T,
assert (eqsetb eq l1 l2)
<=>
forall x, LIn x l1 <=> LIn x l2.
Proof.
sp; unfold eqsetb; trw assert_of_andb;
repeat (trw assert_subsetb); repeat (split; sp);
apply_in_hyp p; auto.
Qed.
Fixpoint beq_list {A} (eq : Deq A) (l1 l2 : list A) : bool :=
match l1, l2 with
| [], [] => true
| [], _ => false
| _, [] => false
| x :: xs, y :: ys => if (decide (x=y)) then beq_list eq xs ys else false
end.
Lemma assert_beq_list :
forall A eq,
forall l1 l2 : list A,
assert (beq_list eq l1 l2) <=> l1 = l2.
Proof.
unfold assert.
induction l1; destruct l2; simpl; repeat rewrite decide_decideP;
split; sp; try (complete (inversion H)).
destruct (decideP (a=a0)); subst; sp.
f_equal. apply IHl1; auto.
inversion H. subst.
autorewrite with SquiggleEq.
rewrite IHl1. refl.
Qed.
Global Instance deq_list {A:Type} `{Deq A} : Deq (list A).
Proof.
intros l1 l2. exists (beq_list _ l1 l2).
apply assert_beq_list.
Defined.
Typeclasses eauto :=6.
Lemma beq_list_refl :
forall A eq,
forall l : list A,
beq_list eq l l = true.
Proof.
induction l; simpl; sp.
autorewrite with SquiggleEq.
auto.
Qed.
Lemma eq_lists :
forall A (l1 l2 : list A) x,
l1 = l2
<=>
(
length l1 = length l2
#
forall n, nth n l1 x = nth n l2 x
).
Proof.
induction l1; sp; destruct l2; sp; split; allsimpl; sp;
try(inversion H);try(inversion H0); subst; sp.
gen_some 0; subst.
assert (l1 = l2) as eq; try (rewrite eq; sp).
apply IHl1 with (x := x); sp.
gen_some (S n); sp.
Qed.
(*
Fixpoint memberb' {A} (eq : Deq A) (x : A) (l : list A) : { LIn x l } + { ! LIn x l} :=
match l return { LIn x l } + { ! LIn x l} with
| [] => right (fun x => x)
| y :: ys =>
match eq y x with
| left e => left (or_introl e)
| right _ =>
match memberb' eq x ys with
| left x => left (or_intror x)
| right y => right y
end
end
end.
*)
Fixpoint memberb {A : Type} (eq : Deq A) (x : A) (l : list A) : bool :=
match l with
| [] => false
| y :: ys => if decide (y=x) then true else memberb eq x ys
end.
Theorem assert_memberb :
forall {T:Type} {eq : Deq T} (a:T) (l: list T),
assert (memberb eq a l) <=> LIn a l.
Proof with (repeat rewrite decide_decideP).
intros. induction l. simpl.
try (unfold assert; repeat split; intros Hf; auto ; inversion Hf).
simpl ... cases_if. subst. unfold assert; repeat split; auto.
repeat split; intros Hlr. right. apply IHl; auto.
destruct Hlr as [Heq | Hin]; tauto.
Qed.
Lemma memberb_app :
forall A eq x,
forall l1 l2 : list A,
memberb eq x (l1 ++ l2) = memberb eq x l1 || memberb eq x l2.
Proof with (repeat rewrite decide_decideP).
induction l1; simpl; sp ...
destruct (decideP (a = x)); sp.
Qed.
Lemma in_app_deq :
forall A l1 l2 (a : A) (deq : Deq A),
LIn a (l1 ++ l2) -> (LIn a l1 + LIn a l2).
Proof.
introv deq i.
rw <- (@assert_memberb A deq) in i.
rw memberb_app in i.
apply assert_orb in i; sp; allrw (@assert_memberb A deq); sp.
Qed.
Lemma diff_cons_r :
forall A eq x,
forall l1 l2 : list A,
@diff _ eq l1 (x :: l2)
= if memberb eq x l1
then @diff _ eq l1 l2
else x :: @diff _ eq l1 l2.
Proof with (repeat rewrite decide_decideP).
induction l1; simpl; sp...
destruct (decideP (a=x)); subst; auto.
Qed.
Lemma diff_cons_r_ni :
forall A eq x,
forall l1 l2 : list A,
!LIn x l2 -> @diff _ eq (x :: l1) l2 = @diff _ eq l1 l2.
Proof with (repeat rewrite decide_decideP).
induction l1; simpl; sp.
induction l2; allsimpl; allrw not_over_or; sp...
destruct (decideP (x=a)); try subst; sp ;
allrw; sp.
Locate remove_trivial.
rw (remove_trivial A x); sp.
Qed.
Fixpoint maxl (ts : list nat) : nat :=
match ts with
| nil => 0
| n :: ns => max n (maxl ns)
end.
Lemma maxl_prop :
forall nats n,
LIn n nats -> n <= maxl nats.
Proof.
induction nats; simpl; sp; subst.
apply max_prop1.
allapply IHnats.
assert (maxl nats <= max a (maxl nats)) by apply max_prop2.
omega.
Qed.
Fixpoint addl (ts : list nat) : nat :=
match ts with
| nil => 0
| n :: ns => n + (addl ns)
end.
Theorem lin_flat_map :
forall (A B : Type) (f : A -> list B) (l : list A) (y : B),
LIn y (flat_map f l)
<=>
{x : A $ LIn x l # LIn y (f x)}.
Proof.
induction l; simpl; sp.
split; sp.
trw in_app_iff.
trw IHl.
split; sp; subst; sp.
exists a; sp.
exists x; sp.
right; exists x; sp.
Qed.
Theorem flat_map_empty:
forall A B (ll:list A) (f: A -> list B) , flat_map f ll =[]
<=> forall a, LIn a ll -> f a =[].
Proof. sp_iff Case.
Case "->".
intros Hmap a Hin; remember (f a) as fa; destruct fa.
auto. assert ({a: A $ LIn a ll # LIn b (f a)}) as Hass;
try (apply lin_flat_map in Hass;
rewrite Hmap in Hass; inversion Hass).
exists a. (split; auto). rewrite <- Heqfa.
constructor; auto.
Case "<-".
intros Hin.
remember (flat_map f ll) as flat; destruct flat ;auto.
assert ( LIn b (flat_map f ll)) as Hinbf by
(rewrite <- Heqflat; constructor; auto).
apply lin_flat_map in Hinbf. exrepnd.
apply Hin in Hinbf1.
rewrite Hinbf1 in Hinbf0.
inversion Hinbf0.
Qed.
Lemma flat_map_map :
forall A B C ,
forall f : B -> list C,
forall g : A -> B,
forall l : list A,
flat_map f (map g l) = flat_map (compose f g) l.
Proof.
induction l; simpl; sp.
rewrite IHl.
unfold compose; auto.
Qed.
Lemma map_map :
forall A B C ,
forall f : B -> C,
forall g : A -> B,
forall l : list A,
map f (map g l) = map (compose f g) l.
Proof.
induction l; simpl; sp.
rewrite IHl.
unfold compose; auto.
Qed.
Lemma eq_flat_maps :
forall A B,
forall f g : A -> list B,
forall l : list A,
(forall x, LIn x l -> f x = g x)
-> flat_map f l = flat_map g l.
Proof.
induction l; simpl; sp.
assert (f a = g a).
apply H; left; auto.
rewrite H0.
assert (flat_map f l = flat_map g l).
rewrite IHl; auto.
rewrite H1; auto.
Qed.
Lemma eq_maps :
forall A B,
forall f g : A -> B,
forall l : list A,
(forall x, LIn x l -> f x = g x)
-> map f l = map g l.
Proof.
induction l; simpl; sp.
assert (f a = g a).
apply H; left; auto.
rewrite H0.
rewrite IHl; auto.
Qed.
Lemma in_nth :
forall T a (l:list T),
LIn a l
-> {n : nat $ (n < length l) # a = nth n l a}.
Proof.
intros ? ? ?. induction l; intros Hin.
- simpl in Hin. contradiction.
- simpl in Hin. dorn Hin.
+ intros. subst. exists 0. split; auto. simpl. omega.
+ intros. apply IHl in Hin. exrepnd.
simpl.
exists (S n) ;split ; try (simpl; omega).
fold (app [a0] l);sp.
Qed.
(* stronger one above : no need for decidability
Lemma in_nth :
forall T a (l:list T),
Deq T
-> LIn a l
-> {n : nat $ (n < length l) # a = nth n l a}.
Proof.
intros ? ? ? deq. induction l; intros Hin.
- simpl in Hin. contradiction.
- case (deq a a0).
+ intros. subst. exists 0. split; auto. simpl. omega.
+ intros Hneq. simpl in Hin.
destruct Hin as [Heq | Hin].
* symmetry in Heq. apply Hneq in Heq. contradiction.
* apply IHl in Hin. clear Hneq. destruct Hin as [m Hp]. repnd.
exists (S m) ;split ; try (simpl; omega).
fold (app [a0] l).
rewrite app_nth2. simpl. assert (m-0 =m) as Hrew by omega.
rewrite Hrew. auto. simpl. omega.
Qed.
*)
Lemma nth_in :
forall A n (l : list A) d,
n < length l
-> LIn (nth n l d) l.
Proof.
intros A n l d H; revert n H.
induction l; simpl; sp.
destruct n; sp.
allapply S_lt_inj; sp.
Qed.
Fixpoint snoc {X:Type} (l:list X) (v:X) : (list X) :=
match l with
| nil => cons v nil
| cons h t => cons h (snoc t v)
end.
Lemma length_snoc :
forall T,
forall n : T,
forall l : list T,
length (snoc l n) = S (length l).
Proof.
intros; induction l; simpl; sp.
Qed.
Lemma snoc_as_append :
forall T, forall l : list T, forall n,
snoc l n = l ++ [n].
Proof.
intros; induction l; simpl; sp.
rewrite IHl; sp.
Qed.
Lemma snoc_append_r :
forall T, forall l1 l2 : list T, forall v : T,
(snoc l1 v) ++ l2 = l1 ++ (v :: l2).
Proof.
intros; induction l1; simpl; sp.
rewrite IHl1; sp.
Qed.
Lemma snoc_append_l :
forall T, forall l1 l2 : list T, forall v : T,
l2 ++ (snoc l1 v) = snoc (l2 ++ l1) v.
Proof.
intros; induction l2; simpl; sp.
rewrite IHl2; sp.
Qed.
Lemma in_snoc :
forall T,
forall l : list T,
forall x y : T,
LIn x (snoc l y) <=> (LIn x l [+] x = y).
Proof.
induction l; simpl; sp.
split; sp.
trw IHl.
apply sum_assoc.
Qed.
Hint Rewrite in_snoc.
Lemma snoc_cases :
forall T,
forall l : list T,
l = [] [+] {a : T $ {k : list T $ l = snoc k a}}.
Proof.
induction l.
left; auto.
sp; subst.
right; exists a; exists (@nil T); simpl; auto.
right.
exists a0; exists (a :: k); simpl; auto.
Qed.
Lemma snoc_inj :
forall T,
forall l1 l2 : list T,
forall x1 x2 : T,
snoc l1 x1 = snoc l2 x2 -> l1 = l2 # x1 = x2.
Proof.
induction l1; simpl; intros.
destruct l2; simpl in H; inversion H; subst; auto.
inversion H.
destruct l2; simpl in H1; inversion H1.
destruct l2; simpl in H.
inversion H.
destruct l1; simpl in H2; inversion H2.
inversion H.
apply IHl1 in H2. sp; subst; auto.
Qed.
Lemma map_snoc :
forall A B l x,
forall f : A -> B,
map f (snoc l x) = snoc (map f l) (f x).
Proof.
induction l; simpl; sp.
rewrite IHl; sp.
Qed.
Open Scope nat_scope.
Lemma length_app :
forall T,
forall l1 l2 : list T,
length (l1 ++ l2) = length l1 + length l2.
Proof.
induction l1; simpl; sp.
Qed.
Lemma nil_snoc_false :
forall T, forall a : list T, forall b : T, [] = snoc a b -> False.
Proof.
intros.
assert (length ([] : list T) = length (snoc a b)).
rewrite H; auto.
simpl in H0.
rewrite length_snoc in H0.
inversion H0.
Qed.
Definition subset {T} (l1 l2 : list T) :=
forall x, LIn x l1 -> LIn x l2.
Lemma fold_subset :
forall T l1 l2,
(forall x : T, LIn x l1 -> LIn x l2) = subset l1 l2.
Proof. sp. Qed.
Lemma null_diff_subset :
forall T,
forall eq : Deq T,
forall l1 l2 : list T,
null (@diff _ eq l1 l2) <=> subset l2 l1.
Proof.
sp; apply null_diff; unfold subset; split; sp.
Qed.
Lemma subsetb_subset :
forall T eq,
forall l1 l2 : list T,
assert (subsetb eq l1 l2) <=> subset l1 l2.
Proof.
intros.
apply assert_subsetb; unfold subset; split; sp.
Qed.
Lemma subset_refl :
forall T,
forall l : list T,
subset l l.
Proof.
unfold subset; sp.
Qed.
Hint Immediate subset_refl.
Lemma subset_refl_iff :
forall T,
forall l : list T,
subset l l <=> True.
Proof.
unfold subset; sp; split; sp.
Qed.
Hint Rewrite subset_refl_iff.
Lemma subset_nil_l :
forall T, forall s : list T, subset [] s.
Proof.
unfold subset; simpl; sp.
Qed.
Hint Immediate subset_nil_l.
Lemma subset_nil_l_iff :
forall T, forall s : list T, subset [] s <=> True.
Proof.
unfold subset; simpl; sp; split; sp.
Qed.
Hint Rewrite subset_nil_l_iff.
(* same as subset_nil_l *)
Lemma nil_subset :
forall T, forall l : list T, subset [] l.
Proof.
auto.
Qed.
(* same as subset_nil_l_iff *)
Lemma nil_subset_iff :
forall T,
forall l : list T,
subset [] l <=> True.
Proof.
sp; autorewrite with core; sp.
Qed.
Lemma cons_subset :
forall T,
forall x : T,
forall l1 l2 : list T,
subset (x :: l1) l2
<=> LIn x l2 # subset l1 l2.
Proof.
unfold subset; simpl; sp; split; sp; subst; auto.
Qed.
Tactic Notation "trewritec" constr(H) :=
build_and_rewrite H.
Lemma singleton_subset :
forall T,
forall x : T,
forall l : list T,
subset [x] l <=> LIn x l.
Proof.
intros.
remember (cons_subset T x [] l) as Htr.
trewrite Htr.
split; sp.
Qed.
Lemma app_subset :
forall T,
forall l1 l2 l3 : list T,
subset (l1 ++ l2) l3 <=> subset l1 l3 # subset l2 l3.
Proof.
induction l1; simpl; sp; try(split; sp; fail).
trw cons_subset. trw cons_subset.
split; introv Hlin; repnd;
try(trw IHl1); try(trw_h IHl1 Hlin; repnd);
repeat(auto;split;auto).
Qed.
Lemma subset_trans :
forall T,
forall l1 l2 l3 : list T,
subset l1 l2
-> subset l2 l3
-> subset l1 l3.
Proof.
unfold subset; sp.
Qed.
Lemma subset_cons_nil :
forall T x,
forall l : list T,
! subset (x :: l) [].
Proof.
unfold subset; sp.
assert ( LIn x (x :: l)) by (simpl; left; auto).
apply_in_hyp p; allsimpl; sp.
Qed.
Lemma subset_cons1 :
forall T,
forall x : T,