Library ListFacts
Assorted facts about lists.
Author: Brian Aydemir.
Implicit arguments are declared by default in this library.
Author: Brian Aydemir.
Implicit arguments are declared by default in this library.
Set Implicit Arguments.
Require Import Eqdep_dec.
Require Import List.
Require Import SetoidList.
Require Import Sorting.
Require Import Relations.
Lemma not_in_cons :
forall (A : Type) (ys : list A) x y,
x <> y -> ~ In x ys -> ~ In x (y :: ys).
Proof.
induction ys; simpl; intuition.
Qed.
Lemma not_In_app :
forall (A : Type) (xs ys : list A) x,
~ In x xs -> ~ In x ys -> ~ In x (xs ++ ys).
Proof.
intros A xs ys x H J K.
destruct (in_app_or _ _ _ K); auto.
Qed.
Lemma elim_not_In_cons :
forall (A : Type) (y : A) (ys : list A) (x : A),
~ In x (y :: ys) -> x <> y /\ ~ In x ys.
Proof.
intros. simpl in *. auto.
Qed.
Lemma elim_not_In_app :
forall (A : Type) (xs ys : list A) (x : A),
~ In x (xs ++ ys) -> ~ In x xs /\ ~ In x ys.
Proof.
split; auto using in_or_app.
Qed.
Lemma incl_nil :
forall (A : Type) (xs : list A), incl nil xs.
Proof.
unfold incl.
intros A xs a H; inversion H.
Qed.
Lemma incl_trans :
forall (A : Type) (xs ys zs : list A),
incl xs ys -> incl ys zs -> incl xs zs.
Proof.
unfold incl; firstorder.
Qed.
Lemma In_incl :
forall (A : Type) (x : A) (ys zs : list A),
In x ys -> incl ys zs -> In x zs.
Proof.
unfold incl; auto.
Qed.
Lemma elim_incl_cons :
forall (A : Type) (x : A) (xs zs : list A),
incl (x :: xs) zs -> In x zs /\ incl xs zs.
Proof.
unfold incl. auto with datatypes.
Qed.
Lemma elim_incl_app :
forall (A : Type) (xs ys zs : list A),
incl (xs ++ ys) zs -> incl xs zs /\ incl ys zs.
Proof.
unfold incl. auto with datatypes.
Qed.
Lemma InA_iff_In :
forall (A : Set) x xs, InA (@eq A) x xs <-> In x xs.
Proof.
split. 2:auto using In_InA.
induction xs as [ | y ys IH ].
intros H. inversion H.
intros H. inversion H; subst; auto with datatypes.
Qed.
Section EqRectList.
Variable A : Type.
Variable eq_A_dec : forall (x y : A), {x = y} + {x <> y}.
Lemma eq_rect_eq_list :
forall (p : list A) (Q : list A -> Type) (x : Q p) (h : p = p),
eq_rect p Q x p h = x.
Proof with auto.
intros.
apply K_dec with (p := h)...
decide equality. destruct (eq_A_dec a a0)...
Qed.
End EqRectList.
Section DecidableSorting.
Variable A : Set.
Variable leA : relation A.
Hypothesis leA_dec : forall x y, {leA x y} + {~ leA x y}.
Theorem lelistA_dec :
forall a xs, {lelistA leA a xs} + {~ lelistA leA a xs}.
Proof.
induction xs as [ | x xs IH ]; auto with datatypes.
destruct (leA_dec a x); auto with datatypes.
right. intros J. inversion J. auto.
Qed.
Theorem sort_dec :
forall xs, {sort leA xs} + {~ sort leA xs}.
Proof.
induction xs as [ | x xs IH ]; auto with datatypes.
destruct IH; destruct (lelistA_dec x xs); auto with datatypes.
right. intros K. inversion K. auto.
right. intros K. inversion K. auto.
right. intros K. inversion K. auto.
Qed.
Section UniqueSortingProofs.
Hypothesis eq_A_dec : forall (x y : A), {x = y} + {x <> y}.
Hypothesis leA_unique : forall (x y : A) (p q : leA x y), p = q.
Scheme lelistA_ind' := Induction for lelistA Sort Prop.
Scheme sort_ind' := Induction for sort Sort Prop.
Theorem lelistA_unique :
forall (x : A) (xs : list A) (p q : lelistA leA x xs), p = q.
Proof with auto.
induction p using lelistA_ind'; intros q.
replace (nil_leA leA x) with (eq_rect _ (fun xs => lelistA leA x xs)
(nil_leA leA x) _ (refl_equal (@nil A)))...
generalize (refl_equal (@nil A)).
pattern (@nil A) at 1 3 4 6, q. case q; [ | intros; discriminate ].
intros. rewrite eq_rect_eq_list...
replace (cons_leA leA x b l l0) with (eq_rect _ (fun xs => lelistA leA x xs)
(cons_leA leA x b l l0) _ (refl_equal (b :: l)))...
generalize (refl_equal (b :: l)).
pattern (b :: l) at 1 3 4 6, q. case q; [ intros; discriminate | ].
intros. inversion e; subst.
rewrite eq_rect_eq_list...
rewrite (leA_unique l0 l2)...
Qed.
Theorem sort_unique :
forall (xs : list A) (p q : sort leA xs), p = q.
Proof with auto.
induction p using sort_ind'; intros q.
replace (nil_sort leA) with (eq_rect _ (fun xs => sort leA xs)
(nil_sort leA) _ (refl_equal (@nil A)))...
generalize (refl_equal (@nil A)).
pattern (@nil A) at 1 3 4 6, q. case q; [ | intros; discriminate ].
intros. rewrite eq_rect_eq_list...
replace (cons_sort p l0) with (eq_rect _ (fun xs => sort leA xs)
(cons_sort p l0) _ (refl_equal (a :: l)))...
generalize (refl_equal (a :: l)).
pattern (a :: l) at 1 3 4 6, q. case q; [ intros; discriminate | ].
intros. inversion e; subst.
rewrite eq_rect_eq_list...
rewrite (lelistA_unique l0 l2).
rewrite (IHp s)...
Qed.
End UniqueSortingProofs.
End DecidableSorting.
Section Equality_ext.
Variable A : Set.
Variable ltA : relation A.
Hypothesis ltA_trans : forall x y z, ltA x y -> ltA y z -> ltA x z.
Hypothesis ltA_not_eqA : forall x y, ltA x y -> x <> y.
Hypothesis ltA_eqA : forall x y z, ltA x y -> y = z -> ltA x z.
Hypothesis eqA_ltA : forall x y z, x = y -> ltA y z -> ltA x z.
Hint Resolve ltA_trans.
Hint Immediate ltA_eqA eqA_ltA.
Notation Inf := (lelistA ltA).
Notation Sort := (sort ltA).
Lemma not_InA_if_Sort_Inf :
forall xs a, Sort xs -> Inf a xs -> ~ InA (@eq A) a xs.
Proof.
induction xs as [ | x xs IH ]; intros a Hsort Hinf H.
inversion H.
inversion H; subst.
inversion Hinf; subst.
assert (x <> x) by auto; intuition.
inversion Hsort; inversion Hinf; subst.
assert (Inf a xs) by eauto using InfA_ltA.
assert (~ InA (@eq A) a xs) by auto.
intuition.
Qed.
Lemma Sort_eq_head :
forall x xs y ys,
Sort (x :: xs) ->
Sort (y :: ys) ->
(forall a, InA (@eq A) a (x :: xs) <-> InA (@eq A) a (y :: ys)) ->
x = y.
Proof.
intros x xs y ys SortXS SortYS H.
inversion SortXS; inversion SortYS; subst.
assert (Q3 : InA (@eq A) x (y :: ys)) by firstorder.
assert (Q4 : InA (@eq A) y (x :: xs)) by firstorder.
inversion Q3; subst; auto.
inversion Q4; subst; auto.
assert (ltA y x) by (refine (SortA_InfA_InA _ _ _ _ _ H6 H7 H1); auto).
assert (ltA x y) by (refine (SortA_InfA_InA _ _ _ _ _ H2 H3 H4); auto).
assert (y <> y) by eauto.
intuition.
Qed.
Lemma Sort_InA_eq_ext :
forall xs ys,
Sort xs ->
Sort ys ->
(forall a, InA (@eq A) a xs <-> InA (@eq A) a ys) ->
xs = ys.
Proof.
induction xs as [ | x xs IHxs ]; induction ys as [ | y ys IHys ];
intros SortXS SortYS H; auto.
assert (Q : InA (@eq A) y nil) by firstorder.
inversion Q.
assert (Q : InA (@eq A) x nil) by firstorder.
inversion Q.
inversion SortXS; inversion SortYS; subst.
assert (x = y) by eauto using Sort_eq_head.
cut (forall a, InA (@eq A) a xs <-> InA (@eq A) a ys).
intros. assert (xs = ys) by auto. subst. auto.
intros a; split; intros L.
assert (Q2 : InA (@eq A) a (y :: ys)) by firstorder.
inversion Q2; subst; auto.
assert (Q5 : ~ InA (@eq A) y xs) by auto using not_InA_if_Sort_Inf.
intuition.
assert (Q2 : InA (@eq A) a (x :: xs)) by firstorder.
inversion Q2; subst; auto.
assert (Q5 : ~ InA (@eq A) y ys) by auto using not_InA_if_Sort_Inf.
intuition.
Qed.
End Equality_ext.