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Prelude.v
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Prelude.v
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Require Import HahnBase.
Require Import List.
Require Import ListSet.
Require Import Permutation.
Require Import PermutationTactic.
Require Import Setoid.
Require Import ZArith.
Import ListNotations.
Set Implicit Arguments.
(** * Domains *)
Module Type Domains.
(** Syntactic and semantic domains used in the formalisation: *)
Parameter Act : Type.
Parameter ProcVar : Type.
Parameter Var : Type.
Parameter Val : Type.
(** Operations and relations used in the formalisation: *)
Parameter val_op : Val -> Val -> Val.
Parameter val_unit : Val.
(** Decidable equality: *)
Parameter procvar_eq_dec : forall x y: ProcVar, { x = y } + { x <> y }.
Parameter val_eq_dec : forall x y: Val, { x = y } + { x <> y }.
Parameter var_eq_dec : forall x y: Var, { x = y } + { x <> y }.
Lemma procvars_eq_dec :
forall xs ys: list ProcVar, { xs = ys } + { xs <> ys }.
Proof.
decide equality.
apply procvar_eq_dec.
Qed.
Lemma vals_eq_dec :
forall x y: list Val, { x = y } + { x <> y }.
Proof.
decide equality.
apply val_eq_dec.
Qed.
(** Infinite domains *)
Parameter val_inf : forall xs: list Val, exists v: Val, ~ In v xs.
End Domains.
(** * Preliminaries *)
Lemma ex_iff :
forall A p q (EQ: forall x : A, p x <-> q x),
(exists x, p x) <-> (exists x, q x).
Proof.
firstorder.
Qed.
Lemma all_iff :
forall A p q (EQ: forall x : A, p x <-> q x),
(forall x, p x) <-> (forall x, q x).
Proof.
firstorder.
Qed.
Lemma option_not_none {T} :
forall x : option T,
~ x = None <-> exists v, x = Some v.
Proof.
intros x. split; intro H1.
- intuition. destruct x; vauto.
- intro. desf.
Qed.
(** ** Functions *)
(** The following section provides an update function for
arbitrary functions [f : A -> B] for which members of
[A] have decidable equality. *)
Section Update.
Hypothesis A : Type.
Hypothesis B : Type.
Hypothesis eq_dec : forall x y: A, { x = y } + { x <> y }.
Definition update (f : A -> B)(x : A)(v : B) : A -> B :=
fun y: A => if eq_dec x y then v else f y.
Lemma update_apply :
forall f x v, update f x v x = v.
Proof.
intros f x v. unfold update.
destruct (eq_dec x x); intuition.
Qed.
Lemma update_eq_dep :
forall f g x v,
update f x v = update g x v ->
forall y, x <> y -> f y = g y.
Proof.
intros f g x v H1 y H2.
unfold update in H1.
apply equal_f with y in H1.
desf.
Qed.
Lemma update_idle :
forall f x, f = update f x (f x).
Proof.
intros f x. unfold update.
extensionality y. desf.
Qed.
Lemma update_idle2 :
forall f x v, f x = v -> update f x v = f.
Proof.
intros f x v H. unfold update.
extensionality y. desf.
Qed.
Lemma update_comm :
forall f x1 v1 x2 v2,
x1 <> x2 ->
update (update f x1 v1) x2 v2 =
update (update f x2 v2) x1 v1.
Proof.
intros f x1 v1 x2 v2 H1.
extensionality y.
unfold update. desf.
Qed.
Lemma update_redundant :
forall f x v v',
update (update f x v) x v' = (update f x v').
Proof.
intros f x v v'. extensionality y.
unfold update. desf.
Qed.
Fixpoint updates (f : A -> B)(xs : list A)(vs : list B) : A -> B :=
match xs, vs with
| x :: xs', v :: vs' => update (updates f xs' vs') x v
| _, _ => f
end.
Lemma updates_empty :
forall f, updates f [] [] = f.
Proof. ins. Qed.
End Update.
(** ** Lists *)
Definition disjoint A (X Y : A -> Prop) :=
forall x, X x -> Y x -> False.
Definition disjoint_list A (xl yl : list A) :=
forall x (IN1 : In x xl) (IN2 : In x yl), False.
Definition pred_of_list A (l : list A) (x : A) := In x l.
Coercion pred_of_list : list >-> Funclass.
Lemma disjoint_conv :
forall A (l1 l2 : list A),
disjoint l1 l2 -> disjoint_list l1 l2.
Proof.
done.
Qed.
(** Auxiliary results and lemmas on lists. *)
Fixpoint list_zip {A B} (xs : list A) (ys : list B) : list (A * B) :=
match (xs, ys) with
| (x::xs', y::ys') => (x, y) :: list_zip xs' ys'
| _ => nil
end.
Fixpoint list_zipped {A B} (xs : list A) (ys : list B) (zs : list (A * B)) : Prop :=
match (xs, ys, zs) with
| (x::xs', y::ys', (z1, z2)::zs') => x = z1 /\ y = z2 /\ list_zipped xs' ys' zs'
| (nil, nil, nil) => True
| _ => False
end.
Fixpoint list_disj {A} (xs ys : list A) : Prop :=
match xs with
| nil => True
| x::xs' => ~ In x ys /\ list_disj xs' ys
end.
Lemma in_app {T} :
forall (xs ys : list T)(x : T),
In x xs -> In x (xs ++ ys).
Proof.
intros xs ys x H.
assert (In x xs \/ In x ys).
by left.
by apply in_or_app in H0.
Qed.
Lemma list_member_not_split {T} :
forall (x : T)(xs1 xs2 : list T),
~ In x (xs1 ++ xs2) ->
~ In x xs1 /\ ~ In x xs2.
Proof.
intros x xs1 xs2 H.
intuition.
Qed.
Lemma remove_In_neq {T} (eq_dec : forall x y : T, { x = y } + { x <> y }) :
forall xs x y,
x <> y ->
In x xs <-> In x (remove eq_dec y xs).
Proof.
induction xs; intros x y H1; simpls.
split; intro H2.
(* left to right *)
- destruct H2 as [H2 | H2]; clarify.
desf; vauto.
desf. by apply IHxs.
by apply in_cons, IHxs.
(* right to left *)
- desf; vauto.
right. by apply IHxs in H2.
simpls.
destruct H2 as [H2 | H2]; clarify.
by left. right.
rewrite <- IHxs in H2; auto.
Qed.
Lemma map_eq_In {T U} :
forall (xs : list T)(x : T)(f1 f2 : T -> U),
In x xs ->
map f1 xs = map f2 xs ->
f1 x = f2 x.
Proof.
induction xs; intros x f1 f2 H1 H2; simpls.
destruct H1 as [H1 | H1]. clarify.
apply IHxs; auto.
repeat rewrite <- map_cons in H2.
desf.
Qed.
(** *** List partitioning *)
(** Properties of the [partition] operation that are not in the standard library. *)
Lemma partition_permut {T} :
forall (xs ys1 ys2 : list T)(f : T -> bool),
partition f xs = (ys1, ys2) ->
Permutation xs (ys1 ++ ys2).
Proof.
induction xs as [|x xs IH]; intros ys1 ys2 f H1.
(* base case *)
- simpls. vauto.
(* induction case *)
- simpls. desf; simpls.
+ apply perm_skip. by apply IH with f.
+ transitivity (x :: ys1 ++ l0); [|by list_permutation].
apply perm_skip. by apply IH with f.
Qed.
Lemma partition_f_left {T} :
forall (xs ys1 ys2 : list T)(f : T -> bool),
partition f xs = (ys1, ys2) ->
forall x : T, In x ys1 -> f x = true.
Proof.
induction xs as [|x xs IH]; intros ys1 ys2 f H1 y H2.
(* base case *)
- simpls. vauto.
(* induction case *)
- simpls. desf; simpls; desf; vauto.
+ by apply IH with l ys2.
+ by apply IH with ys1 l0.
Qed.
Lemma partition_f_right {T} :
forall (xs ys1 ys2 : list T)(f : T -> bool),
partition f xs = (ys1, ys2) ->
forall x : T, In x ys2 -> f x = false.
Proof.
induction xs as [|x xs IH]; intros ys1 ys2 f H1 y H2.
(* base case *)
- simpls. vauto.
(* induction case *)
- simpls. desf; simpls; desf; vauto.
+ by apply IH with l ys2.
+ by apply IH with ys1 l0.
Qed.
Lemma partition_exists {T} :
forall (xs : list T)(f : T -> bool),
exists ys, partition f xs = ys.
Proof.
induction xs as [|x xs IH]. vauto.
intros f. specialize IH with f.
destruct IH as ((ys1 & ys2) & H1). simpls. desf.
(* [f x] is true *)
- by exists (x :: ys1, ys2).
(* [f x] is false *)
- by exists (ys1, x :: ys2).
Qed.
Lemma partition_res {T} :
forall (xs : list T)(f : T -> bool),
partition f xs = (fst (partition f xs), snd (partition f xs)).
Proof.
induction xs as [|x xs IH]. vauto.
intros f. simpl. desf.
Qed.
(** *** Lists of natural numbers *)
Open Scope nat_scope.
Fixpoint list_nat_max (xs : list nat) : nat :=
match xs with
| nil => 0
| x :: xs' => max x (list_nat_max xs')
end.
Lemma list_nat_max_app :
forall xs x,
list_nat_max (x :: xs) = max x (list_nat_max xs).
Proof.
induction xs; intro x; simpls.
Qed.
Lemma list_nat_max_tail :
forall xs x,
list_nat_max xs <= list_nat_max (x :: xs).
Proof.
induction xs; intro x; simpls.
rewrite Max.max_0_r.
apply Peano.le_0_n.
apply Nat.le_trans with (max a (max x (list_nat_max xs))).
by apply Nat.max_le_compat_l.
rewrite Max.max_assoc.
rewrite Max.max_comm with a x.
by rewrite <- Max.max_assoc.
Qed.
Lemma list_nat_max_le_app :
forall xs x,
x <= list_nat_max (x :: xs).
Proof.
induction xs; intro x; simpls.
by rewrite Max.max_0_r.
apply Nat.le_trans with (list_nat_max (x :: xs)); simpls.
apply Nat.max_le_compat_l.
rewrite <- list_nat_max_app.
apply list_nat_max_tail.
Qed.
Lemma list_nat_max_In :
forall xs x,
In x xs -> x <= list_nat_max xs.
Proof.
induction xs; intros x H; simpls.
destruct H as [H | H]. clarify.
rewrite <- list_nat_max_app.
apply list_nat_max_le_app.
apply Nat.le_trans with (list_nat_max xs); auto.
rewrite <- list_nat_max_app.
apply list_nat_max_tail.
Qed.
Lemma list_nat_max_In_neg :
forall xs,
~ In ((list_nat_max xs) + 1) xs.
Proof.
intros xs H.
apply list_nat_max_In in H.
apply Nat.nle_succ_diag_l with (n := list_nat_max xs).
omega.
Qed.
(** Given any list of natural numbers, one can always find a natural number
that is not in this list. *)
Lemma list_nat_max_notin :
forall xs : list nat,
exists x : nat, ~ In x xs.
Proof.
intro xs. exists ((list_nat_max xs) + 1).
intro H. by apply list_nat_max_In_neg in H.
Qed.
Close Scope nat_scope.
(** *** Lists of integers *)
Open Scope Z_scope.
Fixpoint list_Z_max (xs : list Z) : Z :=
match xs with
| nil => 0
| x :: xs' => Z.max x (list_Z_max xs')
end.
Lemma list_Z_max_app :
forall xs x,
list_Z_max (x :: xs) = Z.max x (list_Z_max xs).
Proof.
induction xs; intro x; simpls.
Qed.
Lemma list_Z_max_tail :
forall xs x,
list_Z_max xs <= list_Z_max (x :: xs).
Proof.
induction xs; intro x; simpls.
apply Z.le_max_r.
apply Z.le_trans with (Z.max a (Z.max x (list_Z_max xs))).
by apply Z.max_le_compat_l.
rewrite Z.max_assoc.
rewrite Z.max_comm with a x.
rewrite <- Z.max_assoc.
reflexivity.
Qed.
Lemma list_Z_max_le_app :
forall xs x,
x <= list_Z_max (x :: xs).
Proof.
induction xs; intro x; simpls.
apply Z.le_max_l.
apply Z.le_trans with (list_Z_max (x :: xs)); simpls.
apply Z.max_le_compat_l.
rewrite <- list_Z_max_app.
apply list_Z_max_tail.
Qed.
Lemma list_Z_max_In :
forall xs x,
In x xs -> x <= list_Z_max xs.
Proof.
induction xs; intros x H; simpls.
destruct H as [H | H]. clarify.
rewrite <- list_Z_max_app.
apply list_Z_max_le_app.
apply Z.le_trans with (list_Z_max xs); auto.
rewrite <- list_Z_max_app.
apply list_Z_max_tail.
Qed.
Lemma list_Z_max_In_neg :
forall xs,
~ In ((list_Z_max xs) + 1) xs.
Proof.
intros xs H.
apply list_Z_max_In in H.
apply Z.nle_succ_diag_l with (n := list_Z_max xs).
omega.
Qed.
(** Given any (finite) list of integers, one can always find an
integer that is not in this list. *)
Lemma list_Z_max_notin :
forall xs : list Z,
exists x : Z, ~ In x xs.
Proof.
intro xs. exists ((list_Z_max xs) + 1).
intro H. by apply list_Z_max_In_neg in H.
Qed.
Close Scope Z_scope.
(** *** List removal *)
(** The [removeFirst] operation removes only the _first_ occurrence of a
given element [x] in a given list [xs]. *)
Section ListRemoval.
Variable T : Type.
Hypothesis eq_dec : forall x y : T, { x = y } + { x <> y }.
Fixpoint removeFirst (x : T)(xs : list T) : list T :=
match xs with
| [] => []
| y :: xs' => if eq_dec x y then xs' else y :: removeFirst x xs'
end.
Lemma Permutation_moveleft :
forall (xs : list T)(x : T),
In x xs -> Permutation xs (x :: removeFirst x xs).
Proof.
induction xs; intros x H; simpls.
destruct H as [H | H]. clarify.
apply perm_skip. destruct (eq_dec x x); vauto.
destruct (eq_dec x a); vauto.
transitivity (a :: x :: removeFirst x xs).
apply perm_skip. by apply IHxs.
apply perm_swap.
Qed.
End ListRemoval.
(** *** Permutations *)
(** Different results on permutations on certain substructures and operations on lists. *)
Lemma map_permut {T U} :
forall (xs ys : list T)(f : T -> U),
Permutation xs ys ->
Permutation (map f xs) (map f ys).
Proof.
intros xs ys f PERM.
induction PERM; simpls.
by apply perm_skip.
by apply perm_swap.
by transitivity (map f l').
Qed.
Add Parametric Morphism {T U} : (@map T U)
with signature eq ==> @Permutation T ==> @Permutation U
as map_permut_mor.
Proof.
intros f xs ys H.
by apply map_permut.
Qed.
(** ** Finite sets *)
(** *** Auxiliary results *)
(** Some results that are not present in the ListSet library. *)
Lemma set_In_permutation {T}:
forall (xs ys : set T)(e : T),
Permutation xs ys -> set_In e xs -> set_In e ys.
Proof.
intros xs ys e H.
induction H; simpls; desf; intuition vauto.
Qed.
Add Parametric Morphism {T} : (@set_In T)
with signature eq ==> (@Permutation T) ==> iff
as set_In_permut_mor.
Proof.
intros e xs ys H1. split; intro H2.
by apply set_In_permutation with xs.
apply set_In_permutation with ys; auto.
Qed.
Lemma set_remove_permutation {T} (eq_dec : forall x y : T, { x = y } + { x <> y }) :
forall (xs ys : set T)(e : T),
Permutation xs ys ->
Permutation (set_remove eq_dec e xs) (set_remove eq_dec e ys).
Proof.
intros xs ys e H.
induction H; simpls; desf; intuition vauto.
Qed.
Hint Resolve set_remove_permutation : core.
Add Parametric Morphism {T} : (@set_remove T)
with signature eq ==> eq ==> (@Permutation T) ==> (@Permutation T)
as set_remove_permut_mor.
Proof.
intros eq_dec e xs ys H.
by apply set_remove_permutation.
Qed.
Lemma set_In_remove {T} (eq_dec : forall x y : T, { x = y } + { x <> y }) :
forall xs x y,
x <> y -> set_In x (set_remove eq_dec y xs) <-> set_In x xs.
Proof.
induction xs; ins; desf; simpls; intuition vauto.
right. by apply IHxs with y.
right. by apply IHxs.
Qed.
Lemma set_remove_swap {T} (eq_dec : forall x y : T, { x = y } + { x <> y }):
forall xs x y,
set_remove eq_dec x (set_remove eq_dec y xs) =
set_remove eq_dec y (set_remove eq_dec x xs).
Proof.
induction xs; ins; simpls; desf; simpls; desf; intuition vauto.
by rewrite IHxs.
Qed.
Lemma set_remove_cons {T} (eq_dec : forall x y : T, { x = y } + { x <> y }) :
forall (xs : set T)(x y : T),
x <> y -> y :: set_remove eq_dec x xs = set_remove eq_dec x (y :: xs).
Proof.
induction xs; ins; desf.
Qed.
(** *** Subset *)
Definition subset {T} (X Y : set T) : Prop :=
forall e : T, set_In e X -> set_In e Y.
Global Instance subset_refl {T} :
Reflexive (@subset T).
Proof.
by repeat red.
Qed.
Hint Resolve subset_refl : core.
Global Instance subset_trans {T} :
Transitive (@subset T).
Proof.
red. intros ???????.
by apply H0, H.
Qed.
Section Subset.
Variable T : Type.
Hypothesis eq_dec : forall x y : T, { x = y } + { x <> y }.
Lemma subset_add :
forall (e : T)(X : set T),
subset [e] (set_add eq_dec e X).
Proof.
red. red. ins. desf.
by apply set_add_intro2.
Qed.
Lemma subset_add_mono :
forall (e : T)(X : set T),
subset X (set_add eq_dec e X).
Proof.
red. ins. by apply set_add_intro1.
Qed.
Lemma subset_union_mono_l :
forall (X Y : set T),
subset X (set_union eq_dec X Y).
Proof.
red. red. ins.
by apply set_union_intro1.
Qed.
Lemma subset_union_mono_r :
forall (X Y : set T),
subset X (set_union eq_dec Y X).
Proof.
red. red. ins.
by apply set_union_intro2.
Qed.
Lemma subset_union_l :
forall (X1 X2 Y : set T),
subset X1 X2 -> subset (set_union eq_dec X1 Y) (set_union eq_dec X2 Y).
Proof.
intros X1 X2 Y H1 e H2.
apply set_union_intro.
apply set_union_elim in H2. des.
left. by apply H1.
by right.
Qed.
Lemma subset_union_r :
forall (X Y1 Y2 : set T),
subset Y1 Y2 -> subset (set_union eq_dec X Y1) (set_union eq_dec X Y2).
Proof.
intros X1 X2 Y H1 e H2.
apply set_union_intro.
apply set_union_elim in H2. des.
by left.
right. by apply H1.
Qed.
End Subset.
Add Parametric Morphism {T} (eq_dec : forall x y : T, { x = y } + { x <> y }) : (set_union eq_dec)
with signature (@subset T) ==> (@subset T) ==> (@subset T)
as set_union_subset_mor.
Proof.
intros X1 Y1 H1 X2 Y2 H2.
transitivity (set_union eq_dec Y1 X2).
by apply subset_union_l.
by apply subset_union_r.
Qed.
(** *** Sublists *)
(** A slightly different definition than [subset]; here elements are allowed to
occur multiple times in the specified lists. *)
(** TODO: prove antisymmetry. *)
Section Sublist.
Variable T : Type.
Hypothesis eq_dec : forall x y : T, { x = y } + { x <> y }.
Fixpoint sublist (xs ys : set T) : Prop :=
match xs with
| nil => True
| x :: xs' => set_In x ys /\ sublist xs' (set_remove eq_dec x ys)
end.
Lemma sublist_In :
forall (xs ys : set T)(x : T),
sublist xs ys -> set_In x xs -> set_In x ys.
Proof.
induction xs; ins; simpls; desf; intuition vauto.
assert (x = a \/ x <> a). tauto. desf.
apply IHxs with (x := x) in H1; vauto.
by rewrite set_In_remove in H1.
Qed.
Lemma sublist_cons :
forall (xs ys : set T)(x : T),
sublist xs (x :: ys) <-> sublist (set_remove eq_dec x xs) ys.
Proof.
induction xs; ins; simpls; desf; simpls;
desf; intuition vauto; by apply IHxs.
Qed.
Lemma sublist_nil :
forall xs : set T,
sublist xs nil -> xs = nil.
Proof.
induction xs; ins; simpls; desf.
Qed.
Lemma sublist_remove_In :
forall (ys xs : set T)(x : T),
set_In x ys ->
sublist xs (set_remove eq_dec x ys) <-> sublist (x :: xs) ys.
Proof.
induction ys; ins; simpls; desf; intuition vauto.
Qed.
Lemma sublist_remove :
forall (xs ys : set T)(x : T),
sublist xs ys -> sublist (set_remove eq_dec x xs) (set_remove eq_dec x ys).
Proof.
induction xs, ys; ins; simpls; desf; simpls; desf; intuition vauto;
try by apply IHxs.
- by rewrite <- sublist_cons.
- right. rewrite set_In_remove; auto.
- repeat rewrite set_remove_cons; auto.
rewrite set_remove_swap.
apply IHxs. by rewrite <- set_remove_cons.
Qed.
Global Instance sublist_refl :
Reflexive sublist.
Proof.
red. induction x; simpls; intuition vauto; desf.
Qed.
Hint Resolve sublist_refl : core.
Global Instance sublist_trans :
Transitive sublist.
Proof.
red. induction x, y; simpls; desf; intuition vauto.
- by apply IHx with y.
- by apply IHx with y.
- rewrite <- set_In_remove; vauto.
apply sublist_In with y; vauto.
- apply IHx with (t :: set_remove eq_dec a y); auto.
rewrite set_remove_cons; vauto.
apply sublist_remove.
by apply sublist_remove_In.
Qed.
Lemma sublist_permutation_r :
forall xs ys1 ys2 : set T,
Permutation ys1 ys2 -> sublist xs ys1 -> sublist xs ys2.
Proof.
induction xs; ins; intuition vauto.
by rewrite <- H.
apply IHxs with (set_remove eq_dec a ys1); vauto.
by apply set_remove_permutation.
Qed.
Lemma sublist_permutation_l :
forall xs1 xs2 : set T,
Permutation xs1 xs2 ->
forall ys, sublist xs1 ys -> sublist xs2 ys.
Proof.
intros xs1 xs2 H.
induction H; ins; simpls; desf; intuition vauto.
- by apply IHPermutation.
- assert (x = y \/ x <> y). tauto. desf.
by rewrite set_In_remove in H0.
- assert (x = y \/ x <> y). tauto. desf.
rewrite set_In_remove. done. intro. by apply H2.
- by rewrite set_remove_swap.
- by apply IHPermutation2, IHPermutation1.
Qed.
Add Parametric Morphism : sublist
with signature (@Permutation T) ==> (@Permutation T) ==> iff
as sublist_permut_mor.
Proof.
intros xs1 ys1 H1 xs2 ys2 H2. split; intro H3.
apply sublist_permutation_l with xs1; vauto.
apply sublist_permutation_r with xs2; vauto.
apply sublist_permutation_l with ys1. by symmetry.
apply sublist_permutation_r with ys2; auto.
Qed.
Lemma sublist_permutation :
forall xs ys,
Permutation xs ys -> sublist xs ys.
Proof.
intros xs ys H.
destruct H; simpls; desf; simpls; desf; intuition vauto.
by rewrite H.
by rewrite H, H0.
Qed.
Hint Resolve sublist_permutation : core.
End Sublist.
(** ** Finite maps *)
(** A list of pairs is a _finite map_ if there are no duplicates among the left elements *)
Definition fmap_dom {T U} (xs : list (T * U)) : list T :=
fst (split xs).
Definition fmap {T U} (xs : list (T * U)) : Prop :=
NoDup (fmap_dom xs).
Lemma fmap_dom_cons {T U} :
forall (xs : list (T * U))(x : T * U),
fmap_dom (x :: xs) = fst x :: fmap_dom xs.
Proof.
induction xs; intro x; simpls;
unfold fmap_dom; simpls; desf.
Qed.
Lemma fmap_dom_In {T U} :
forall (xs : list (T * U))(x : T * U),
In x xs -> In (fst x) (fmap_dom xs).
Proof.
induction xs; intros x H; simpls.
destruct H as [H | H]. clarify.
rewrite fmap_dom_cons.
by apply in_eq.
rewrite fmap_dom_cons.
by apply in_cons, IHxs.
Qed.
(** The following definition determines whether a list of pairs is
_projected_ onto a given mapping. *)
Definition projected {T U} (xs : list (T * U))(f : T -> U) : Prop :=
forall x : T * U, In x xs -> f (fst x) = snd x.