300 lines
8.1 KiB
Coq
300 lines
8.1 KiB
Coq
|
(** Assorted facts about lists.
|
||
|
|
|
||
|
|
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.
|
||
|
|
Require Import AdditionalTactics.
|
||
|
|
|
||
|
|
Include AdditionalTactics.
|
||
|
|
|
||
|
|
(* ********************************************************************** *)
|
||
|
|
(** * List membership *)
|
||
|
|
|
||
|
|
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.
|
||
|
|
|
||
|
|
|
||
|
|
(* ********************************************************************** *)
|
||
|
|
(** * List inclusion *)
|
||
|
|
|
||
|
|
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.
|
||
|
|
|
||
|
|
|
||
|
|
(* ********************************************************************** *)
|
||
|
|
(** * Setoid facts *)
|
||
|
|
|
||
|
|
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.
|
||
|
|
Admitted.
|
||
|
|
|
||
|
|
|
||
|
|
(* ********************************************************************* *)
|
||
|
|
(** * Equality proofs for lists *)
|
||
|
|
|
||
|
|
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.
|
||
|
|
|
||
|
|
|
||
|
|
(* ********************************************************************** *)
|
||
|
|
(** * Decidable sorting and uniqueness of proofs *)
|
||
|
|
|
||
|
|
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.
|
||
|
|
(* case: nil_leA *)
|
||
|
|
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...
|
||
|
|
Admitted.
|
||
|
|
(*
|
||
|
|
(* case: cons_sort *)
|
||
|
|
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.
|
||
|
|
(* case: nil_sort *)
|
||
|
|
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...
|
||
|
|
Admitted.
|
||
|
|
(*
|
||
|
|
(* case: cons_sort *)
|
||
|
|
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.
|
||
|
|
|
||
|
|
|
||
|
|
(* ********************************************************************** *)
|
||
|
|
(** * Equality on sorted lists *)
|
||
|
|
|
||
|
|
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.
|
||
|
|
|
||
|
|
Create HintDb ListHints.
|
||
|
|
Hint Resolve ltA_trans :ListHints.
|
||
|
|
Hint Immediate ltA_eqA eqA_ltA :ListHints.
|
||
|
|
|
||
|
|
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.
|
||
|
|
Admitted.
|
||
|
|
|
||
|
|
(*
|
||
|
|
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.
|
||
|
|
Admitted.
|
||
|
|
(*
|
||
|
|
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.
|
||
|
|
(* xs -> nil, ys -> y :: ys *)
|
||
|
|
assert (Q : InA (@eq A) y nil) by firstorder.
|
||
|
|
inversion Q.
|
||
|
|
(* xs -> x :: xs, ys -> nil *)
|
||
|
|
assert (Q : InA (@eq A) x nil) by firstorder.
|
||
|
|
inversion Q.
|
||
|
|
(* xs -> x :: xs, ys -> y :: ys *)
|
||
|
|
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.
|