# Preliminaries

(* Switch Coq into implicit argument mode *)

Global Set Implicit Arguments.
Global Unset Strict Implicit.

(* Load basic Coq libraries *)

Require Export Omega List Morphisms.

(* Inversion tactic *)

Ltac inv H := inversion H; subst; clear H.

(* Beteva destuct tactics *)

Tactic Notation "destruct" "_":=
match goal with
| [ |- context[match ?X with _ => _ end] ] => destruct X
| [ H : context[match ?X with _ => _ end] |- _ ] => destruct X
end.

Tactic Notation "destruct" "_" "eqn" ":" ident(E) :=
match goal with
| [ |- context[match ?X with _ => _ end] ] => destruct X eqn:E
| [ H : context[match ?X with _ => _ end] |- _ ] => destruct X eqn:E
end.

Tactic Notation "destruct" "_" "eqn" ":" "_" :=
let E := fresh "E" in
match goal with
| [ |- context[match ?X with _ => _ end] ] => destruct X eqn:E
| [ H : context[match ?X with _ => _ end] |- _ ] => destruct X eqn:E
end.

Tactic Notation "destruct" "*" :=
repeat destruct _.

Tactic Notation "destruct" "*" "eqn" ":" ident(E) :=
repeat (let E := fresh E in destruct _ eqn:E; try progress inv E); try now congruence.

Tactic Notation "destruct" "*" "eqn" ":" "_" := destruct * eqn:E.

(* ** Decidability *)

Definition dec (X : Prop) : Type := {X} + {~ X}.

Notation "'eq_dec' X" := (forall x y : X, dec (x=y)) (at level 70).

(* Register dec as a type class *)

Existing Class dec.

Definition decision (X : Prop) (D : dec X) : dec X := D.
Arguments decision X {D}.

Tactic Notation "decide" constr(p) :=
destruct (decision p).
Tactic Notation "decide" constr(p) "as" simple_intropattern(i) :=
destruct (decision p) as i.

(* Hints for auto concerning dec *)

Hint Unfold dec.
Hint Extern 4 =>
match goal with
| [ |- dec ?p ] => exact (decision p)
end.

(* Improves type class inference *)

Hint Extern 4 =>
match goal with
| [ |- dec ((fun _ => _) _) ] => simpl
end : typeclass_instances.

(* Regiseva instance rules for dec *)

Instance True_dec : dec True :=
left I.

Instance False_dec : dec False :=
right (fun A => A).

Instance impl_dec (X Y : Prop) :
dec X -> dec Y -> dec (X -> Y).
Proof.
unfold dec; tauto.
Defined.

Instance and_dec (X Y : Prop) :
dec X -> dec Y -> dec (X /\ Y).
Proof.
unfold dec; tauto.
Defined.

Instance or_dec (X Y : Prop) :
dec X -> dec Y -> dec (X \/ Y).
Proof.
unfold dec; tauto.
Defined.

(* Coq standard modules make "not" and "iff" opaque for type class inference, can be seen with Print HintDb typeclass_instances. *)

Instance not_dec (X : Prop) :
dec X -> dec (~ X).
Proof.
unfold not. auto.
Defined.

Instance iff_dec (X Y : Prop) :
dec X -> dec Y -> dec (X <-> Y).
Proof.
unfold iff. auto.
Qed.

Lemma dec_DN X :
dec X -> ~~ X -> X.
Proof.
unfold dec; tauto.
Qed.

Lemma dec_DM_and X Y :
dec X -> dec Y -> ~ (X /\ Y) -> ~ X \/ ~ Y.
Proof.
unfold dec; tauto.
Qed.

Lemma dec_DM_impl X Y :
dec X -> dec Y -> ~ (X -> Y) -> X /\ ~ Y.
Proof.
unfold dec; tauto.
Qed.

Lemma dec_prop_iff (X Y : Prop) :
(X <-> Y) -> dec X -> dec Y.
Proof.
unfold dec; tauto.
Defined.

Instance bool_eq_dec :
eq_dec bool.
Proof.
intros x y. hnf. decide equality.
Defined.

Instance nat_eq_dec :
eq_dec nat.
Proof.
intros x y. hnf. decide equality.
Defined.

Instance nat_le_dec (x y : nat) : dec (x <= y) :=
le_dec x y.

(* * Lists *)

(* Notations for lists *)

Definition equi X (A B : list X) : Prop :=
incl A B /\ incl B A.

Hint Unfold equi.

Export ListNotations.
Notation "| A |" := (length A) (at level 65).
Notation "x 'el' A" := (In x A) (at level 70).
Notation "A <<= B" := (incl A B) (at level 70).
Notation "A === B" := (equi A B) (at level 70).

(* The following comments are for coqdoc *)

(* A useful lemma *)

Lemma list_cycle (X : Type) (A : list X) x :
x::A <> A.
Proof.
intros B.
assert (C: |x::A| <> |A|) by (simpl; omega).
apply C. now rewrite B.
Qed.

(* Decidability laws for lists *)

Instance list_eq_dec X :
eq_dec X -> eq_dec (list X).
Proof.
intros D. apply list_eq_dec. exact D.
Defined.

Instance list_in_dec (X : Type) (x : X) (A : list X) :
eq_dec X -> dec (x el A).
Proof.
intros D. apply in_dec. exact D.
Defined.

Lemma list_sigma_forall X A (p : X -> Prop) (p_dec : forall x, dec (p x)) :
{x | x el A /\ p x} + {forall x, x el A -> ~ p x}.
Proof.
induction A as [|x A]; simpl.
- tauto.
- destruct IHA as [[y [D E]]|D].
+ eauto.
+ destruct (p_dec x) as [E|E].
* eauto.
* right. intros y [[]|F]; auto.
Defined.

Arguments list_sigma_forall {X} A p {p_dec}.

Instance list_forall_dec X A (p : X -> Prop) (p_dec : forall x, dec (p x)) :
dec (forall x, x el A -> p x).
Proof.
destruct (list_sigma_forall A (fun x => ~ p x)) as [[x [D E]]|D].
- right. auto.
- left. intros x E. apply dec_DN; auto.
Defined.

Instance list_exists_dec X A (p : X -> Prop) (p_dec : forall x, dec (p x)) :
dec (exists x, x el A /\ p x).
Proof.
destruct (list_sigma_forall A p) as [[x [D E]]|D].
- eauto.
- right. intros [x [E F]]. exact (D x E F).
Defined.

Lemma list_exists_DM X A (p : X -> Prop) :
(forall x, dec (p x)) ->
~ (forall x, x el A -> ~ p x) -> exists x, x el A /\ p x.
Proof.
intros D E.
destruct (list_sigma_forall A p) as [F|F].
+ destruct F as [x F]. eauto.
Qed.

Lemma list_cc X (p : X -> Prop) A :
(forall x, dec (p x)) ->
(exists x, x el A /\ p x) -> {x | x el A /\ p x}.
Proof.
intros D E.
destruct (list_sigma_forall A p) as [[x [F G]]|F].
- eauto.
- exfalso. destruct E as [x [G H]]. apply (F x); auto.
Defined.

(* Membership

We use the following facts from the standard library List.
in_eq : x el x::A
in_nil : ~ x el nil
in_cons : x el A -> x el y::A
in_or_app : x el A \/ x el B -> x el A++B
in_app_iff : x el A++B <-> x el A \/ x el B
in_map_iff : y el map f A <-> exists x, f x = y /\ x el A
*)

Hint Resolve in_eq in_nil in_cons in_or_app.

Lemma in_sing X (x y : X) :
x el [y] -> x = y.

Proof. simpl. intros [[]|[]]. reflexivity. Qed.

Lemma in_cons_neq X (x y : X) A :
x el y::A -> x <> y -> x el A.

Proof. simpl. intros [[]|D] E; congruence. Qed.

(* Inclusion
-
-We use the following facts from the standard library List.
-- A <<= B = forall y, x el A -> x el B
-- incl_refl : A <<= A
-- incl_tl : A <<= B -> A <<= x::B
-- incl_cons : x el B -> A <<= B -> x::A <<= B
-- incl_appl : A <<= B -> A <<= B++C
-- incl_appr : A <<= C -> A <<= B++C
-- incl_app : A <<= C -> B <<= C -> A++B <<= C
-*)

Hint Resolve incl_refl incl_tl incl_cons incl_appl incl_appr incl_app.

Lemma incl_nil X (A : list X) :
nil <<= A.

Proof. intros x []. Qed.

Hint Resolve incl_nil.

Lemma incl_map X Y A B (f : X -> Y) :
A <<= B -> map f A <<= map f B.

Proof.
intros D y E. apply in_map_iff in E as [x [E E']].
subst y. apply in_map_iff. eauto.
Qed.

Section Inclusion.
Variable X : Type.
Implicit Types A B : list X.

Lemma incl_nil_eq A :
A <<= nil -> A=nil.

Proof.
intros D. destruct A as [|x A].
- reflexivity.
- exfalso. apply (D x). auto.
Qed.

Lemma incl_shift x A B :
A <<= B -> x::A <<= x::B.

Proof. intros D y E. destruct E; subst; auto. Qed.

Lemma incl_lcons x A B :
x::A <<= B <-> x el B /\ A <<= B.

Proof.
split.
- intros D. split; hnf; auto.
- intros [D E] z [F|F]; subst; auto.
Qed.

Lemma incl_rcons x A B :
A <<= x::B -> ~ x el A -> A <<= B.

Proof. intros C D y E. destruct (C y E) as [F|F]; congruence. Qed.

Lemma incl_lrcons x A B :
x::A <<= x::B -> ~ x el A -> A <<= B.

Proof.
intros C D y E.
assert (F: y el x::B) by auto.
destruct F as [F|F]; congruence.
Qed.

End Inclusion.

Hint Resolve incl_shift.

Definition inclp (X : Type) (A : list X) (p : X -> Prop) : Prop :=
forall x, x el A -> p x.

(* Setoid rewriting with list inclusion and list equivalence *)

Instance in_equi_proper X :
Proper (eq ==> @equi X ==> iff) (@In X).

Proof. hnf. intros x y []. hnf. firstorder. Qed.

Instance incl_equi_proper X :
Proper (@equi X ==> @equi X ==> iff) (@incl X).

Proof. hnf. intros x y D. hnf. firstorder. Qed.

Instance incl_preorder X : PreOrder (@incl X).

Proof. constructor; hnf; unfold incl; auto. Qed.

Instance equi_Equivalence X : Equivalence (@equi X).

Proof. constructor; hnf; firstorder. Qed.

Instance cons_equi_proper X :
Proper (eq ==> @equi X ==> @equi X) (@cons X).

Proof. hnf. intros x y []. hnf. firstorder. Qed.

Instance app_equi_proper X :
Proper (@equi X ==> @equi X ==> @equi X) (@app X).

Proof.
hnf. intros A B D. hnf. intros A' B' E.
destruct D, E; auto.
Qed.

(* Equivalence *)

Section Equi.
Variable X : Type.
Implicit Types A B : list X.

Lemma equi_push x A :
x el A -> A === x::A.

Proof. auto. Qed.

Lemma equi_dup x A :
x::A === x::x::A.

Proof. auto. Qed.

Lemma equi_swap x y A:
x::y::A === y::x::A.

Proof. split; intros z; simpl; tauto. Qed.

Lemma equi_shift x A B :
x::A++B === A++x::B.

Proof.
split; intros y.
- intros [D|D].
+ subst; auto.
+ apply in_app_iff in D as [D|D]; auto.
- intros D. apply in_app_iff in D as [D|D].
+ auto.
+ destruct D; subst; auto.
Qed.

Lemma equi_rotate x A :
x::A === A++[x].

Proof.
split; intros y; simpl.
- intros [D|D]; subst; auto.
- intros D. apply in_app_iff in D as [D|D].
+ auto.
+ apply in_sing in D. auto.
Qed.
End Equi.

(* * Filter *)

Definition filter (X : Type) (p : X -> Prop) (p_dec : forall x, dec (p x)) : list X -> list X :=
fix f A := match A with
| nil => nil
| x::A' => if decision (p x) then x :: f A' else f A'
end.

Arguments filter {X} p {p_dec} A.

Section FilterLemmas.
Variable X : Type.
Variable p : X -> Prop.
Context {p_dec : forall x, dec (p x)}.

Lemma in_filter_iff x A :
x el filter p A <-> x el A /\ p x.

Proof.
induction A as [|y A]; simpl.
- tauto.
- decide (p y) as [B|B]; simpl;
rewrite IHA; intuition; subst; tauto.
Qed.

Lemma filter_incl A :
filter p A <<= A.

Proof.
intros x D. apply in_filter_iff in D. apply D.
Qed.

Lemma filter_mono A B :
A <<= B -> filter p A <<= filter p B.

Proof.
intros D x E. apply in_filter_iff in E as [E E'].
apply in_filter_iff. auto.
Qed.

Lemma filter_fst x A :
p x -> filter p (x::A) = x::filter p A.
Proof.
simpl. decide (p x); tauto.
Qed.

Lemma filter_app A B :
filter p (A ++ B) = filter p A ++ filter p B.
Proof.
induction A as [|y A]; simpl.
- reflexivity.
- rewrite IHA. decide (p y); reflexivity.
Qed.

Lemma filter_fst' x A :
~ p x -> filter p (x::A) = filter p A.
Proof.
simpl. decide (p x); tauto.
Qed.

End FilterLemmas.

(* * Element removal *)

Section Removal.
Variable X : Type.
Context {eq_X_dec : eq_dec X}.

Definition rem (A : list X) (x : X) : list X :=
filter (fun z => z <> x) A.

Lemma in_rem_iff x A y :
x el rem A y <-> x el A /\ x <> y.
Proof.
apply in_filter_iff.
Qed.

Lemma rem_not_in x y A :
x = y \/ ~ x el A -> ~ x el rem A y.
Proof.
intros D E. apply in_rem_iff in E. tauto.
Qed.

Lemma rem_incl A x :
rem A x <<= A.
Proof.
apply filter_incl.
Qed.

Lemma rem_mono A B x :
A <<= B -> rem A x <<= rem B x.
Proof.
apply filter_mono.
Qed.

Lemma rem_cons A B x :
A <<= B -> rem (x::A) x <<= B.
Proof.
intros E y F. apply E. apply in_rem_iff in F.
destruct F as [[|]]; congruence.
Qed.

Lemma rem_cons' A B x y :
x el B -> rem A y <<= B -> rem (x::A) y <<= B.
Proof.
intros E F u G.
apply in_rem_iff in G as [[[]|G] H]. exact E.
apply F. apply in_rem_iff. auto.
Qed.

Lemma rem_in x y A :
x el rem A y -> x el A.
Proof.
apply rem_incl.
Qed.

Lemma rem_neq x y A :
x <> y -> x el A -> x el rem A y.
Proof.
intros E F. apply in_rem_iff. auto.
Qed.

Lemma rem_app x A B :
x el A -> B <<= A ++ rem B x.
Proof.
intros E y F. decide (x=y) as [[]|]; auto using rem_neq.
Qed.

Lemma rem_equi x A :
x::A === x::rem A x.
Proof.
split; intros y;
intros [[]|E]; decide (x=y) as [[]|D];
eauto using rem_in, rem_neq.
Qed.

Lemma rem_fst x A :
rem (x::A) x = rem A x.
Proof.
unfold rem. rewrite filter_fst'; auto.
Qed.

Lemma rem_fst' x y A :
x <> y -> rem (x::A) y = x::rem A y.
Proof.
intros E. unfold rem. rewrite filter_fst; auto.
Qed.

End Removal.

Hint Resolve rem_not_in rem_incl rem_mono rem_cons rem_cons' rem_app rem_in rem_neq.

(* * Duplicate-free lists *)

Inductive dupfree (X : Type) : list X -> Prop :=
| dupfreeN : dupfree nil
| dupfreeC x A : ~ x el A -> dupfree A -> dupfree (x::A).

Section Dupfree.
Variable X : Type.
Implicit Types A B : list X.

Lemma dupfree_inv x A :
dupfree (x::A) <-> ~ x el A /\ dupfree A.
Proof.
split; intros D.
- inv D; auto.
- apply dupfreeC; tauto.
Qed.

Lemma dupfree_map Y A (f : X -> Y) :
(forall x y, x el A -> y el A -> f x = f y -> x=y) ->
dupfree A -> dupfree (map f A).

Proof.
intros D E. induction E as [|x A E' E]; simpl.
- constructor.
- constructor; [|now auto].
intros F. apply in_map_iff in F as [y [F F']].
rewrite (D y x) in F'; auto.
Qed.

Lemma dupfree_filter p (p_dec : forall x, dec (p x)) A :
dupfree A -> dupfree (filter p A).

Proof.
intros D. induction D as [|x A C D]; simpl.
- left.
- decide (p x) as [E|E]; [|exact IHD].
right; [|exact IHD].
intros F. apply C. apply filter_incl in F. exact F.
Qed.

Lemma dupfree_dec A :
eq_dec X -> dec (dupfree A).

Proof.
intros D. induction A as [|x A].
- left. left.
- decide (x el A) as [E|E].
+ right. intros F. inv F; tauto.
+ destruct (IHA) as [F|F].
* auto using dupfree.
* right. intros G. inv G; tauto.
Qed.

End Dupfree.

Section Undup.
Variable X : Type.
Context {eq_X_dec : eq_dec X}.
Implicit Types A B : list X.

Fixpoint undup (A : list X) : list X :=
match A with
| nil => nil
| x::A' => if decision (x el A') then undup A' else x :: undup A'
end.

Lemma undup_fp_equi A :
undup A === A.
Proof.
induction A as [|x A]; simpl.
- reflexivity.
- decide (x el A) as [E|E]; rewrite IHA; auto.
Qed.

Lemma dupfree_undup A :
dupfree (undup A).
Proof.
induction A as [|x A]; simpl.
- left.
- decide (x el A) as [E|E]; auto.
right; auto. now rewrite undup_fp_equi.
Qed.

Lemma undup_incl A B :
A <<= B <-> undup A <<= undup B.
Proof.
now do 2 rewrite undup_fp_equi.
Qed.

Lemma undup_equi A B :
A === B <-> undup A === undup B.
Proof.
now do 2 rewrite undup_fp_equi.
Qed.

Lemma undup_eq A :
dupfree A -> undup A = A.
Proof.
intros E. induction E as [|x A E F]; simpl.
- reflexivity.
- rewrite IHF. decide (x el A) as [G|G]; tauto.
Qed.

Lemma undup_idempotent A :
undup (undup A) = undup A.

Proof. apply undup_eq, dupfree_undup. Qed.

End Undup.

Section DupfreeLength.
Variable X : Type.
Implicit Types A B : list X.

Lemma dupfree_reorder A x :
dupfree A -> x el A ->
exists A', A === x::A' /\ |A'| < |A| /\ dupfree (x::A').

Proof.
intros E. revert x. induction E as [|y A H]; intros x F.
- destruct F as [F|F].
+ subst y. exists A. auto using dupfree.
+ specialize (IHE x F). destruct IHE as [A' [G [K1 K2]]].
exists (y::A'). split; [|split].
* rewrite G. apply equi_swap.
* simpl. omega.
* { apply dupfree_inv in K2 as [K2 K3]. right.
- intros [M|M]; subst; auto.
- right; [|exact K3].
intros M; apply H. apply G. auto. }
Qed.

Lemma dupfree_le A B :
dupfree A -> dupfree B -> A <<= B -> |A| <= |B|.

Proof.
intros E; revert B.
induction A as [|x A]; simpl; intros B F G.
- omega.
- apply incl_lcons in G as [G H].
destruct (dupfree_reorder F G) as [B' [K [L M]]].
apply dupfree_inv in E as [E1 E2].
apply dupfree_inv in M as [M1 M2].
cut (A <<= B').
{ intros N. specialize (IHA E2 B' M2 N). omega. }
apply incl_rcons with (x:=x); [|exact E1].
rewrite H. apply K.
Qed.

Lemma dupfree_eq A B :
dupfree A -> dupfree B -> A === B -> |A|=|B|.

Proof.
intros D E [F G].
apply (dupfree_le D E) in F.
apply (dupfree_le E D) in G.
omega.
Qed.

Lemma dupfree_lt A B x :
dupfree A -> dupfree B -> A <<= B ->
x el B -> ~ x el A -> |A| < |B|.

Proof.
intros E F G H K.
destruct (dupfree_reorder F H) as [B' [L [M N]]].
rewrite (dupfree_eq F N L).
cut (|A|<=|B'|). { simpl; omega. }
apply dupfree_le.
- exact E.
- now inv N.
- apply incl_rcons with (x:=x).
+ rewrite G. apply L.
+ exact K.
Qed.

Lemma dupfree_ex A B :
eq_dec X -> dupfree A -> dupfree B -> |A| < |B| -> exists x, x el B /\ ~ x el A.

Proof.
intros D E F H.
destruct (list_sigma_forall B (fun x => ~ x el A)) as [[x K]|K].
- exists x; exact K.
- exfalso.
assert (L : B <<= A).
{ intros x L. apply dec_DN; auto. }
apply dupfree_le in L; auto; omega.
Qed.

Lemma dupfree_equi A B :
eq_dec X -> dupfree A -> dupfree B -> A <<= B -> |A|=|B| -> A === B.

Proof.
intros C D E F G. split. exact F.
destruct (list_sigma_forall B (fun x => ~ x el A)) as [[x [H K]]|H].
- exfalso. assert (L:=dupfree_lt D E F H K). omega.
- intros x L. apply dec_DN; auto.
Qed.

End DupfreeLength.

(* * Cardinality *)

Section Cardinality.
Variable X : Type.
Context {eq_X_dec : eq_dec X}.
Implicit Types A B : list X.

Definition card (A : list X) : nat := |undup A|.

Lemma card_le A B :
A <<= B -> card A <= card B.

Proof.
intros E. apply dupfree_le.
- apply dupfree_undup.
- apply dupfree_undup.
- apply undup_incl, E.
Qed.

Lemma card_eq A B :
A === B -> card A = card B.

Proof.
intros [E F]. apply card_le in E. apply card_le in F. omega.
Qed.

Lemma card_equi A B :
A <<= B -> card A = card B -> A === B.
Proof.
intros D E.
apply <- undup_equi. apply -> undup_incl in D.
apply dupfree_equi; auto using dupfree_undup.
Qed.

Lemma card_lt A B x :
A <<= B -> x el B -> ~ x el A -> card A < card B.

Proof.
intros D E F.
apply (dupfree_lt (A:= undup A) (B:= undup B) (x:=x)).
- apply dupfree_undup.
- apply dupfree_undup.
- apply undup_incl, D.
- apply undup_fp_equi, E.
- rewrite undup_fp_equi. exact F.
Qed.

Lemma card_or A B :
A <<= B -> A === B \/ card A < card B.

Proof.
intros D.
decide (card A = card B) as [F|F].
- left. apply card_equi; auto.
- right. apply card_le in D. omega.
Qed.

Lemma card_ex A B :
card A < card B -> exists x, x el B /\ ~ x el A.

Proof.
intros E.
destruct (dupfree_ex (A:=undup A) (B:=undup B)) as [x F].
- exact eq_X_dec.
- apply dupfree_undup.
- apply dupfree_undup.
- exact E.
- exists x. setoid_rewrite undup_fp_equi in F. exact F.
(*Coq bug: Must use setoid_rewrite here *)
Qed.

Lemma card_cons x A :
card (x::A) = if decision (x el A) then card A else 1 + card A.
Proof.
unfold card at 1; simpl. now decide (x el A).
Qed.

Lemma card_cons_rem x A :
card (x::A) = 1 + card (rem A x).
Proof.
rewrite (card_eq (rem_equi x A)).
rewrite card_cons.
decide (x el rem A x) as [D|D].
- apply in_rem_iff in D; tauto.
- reflexivity.
Qed.

Lemma card_0 A :
card A = 0 -> A = nil.
Proof.
destruct A as [|x A]; intros D.
- reflexivity.
- exfalso. rewrite card_cons_rem in D. omega.
Qed.

End Cardinality.

Instance card_equi_proper X (D: eq_dec X) :
Proper (@equi X ==> eq) (@card X D).
Proof.
hnf. apply card_eq.
Qed.

Lemma complete_induction (p : nat -> Prop) (x : nat) :
(forall x, (forall y, y<x -> p y) -> p x) -> p x.

Proof. intros A. apply A. induction x ; intros y B.
exfalso ; omega.
apply A. intros z C. apply IHx. omega. Qed.

Lemma size_induction X (f : X -> nat) (p : X -> Prop) :
(forall x, (forall y, f y < f x -> p y) -> p x) ->
forall x, p x.

Proof.
intros IH x. apply IH.
assert (G: forall n y, f y < n -> p y).
{ intros n. induction n.
- intros y B. exfalso. omega.
- intros y B. apply IH. intros z C. apply IHn. omega. }
apply G.
Qed.

Section pos.

Definition elAt := nth_error.
Notation "A '.[' i ']'" := (elAt A i) (no associativity, at level 50).

Fixpoint pos (X : Type) {e : eq_dec X} (s : X) (A : list X) :=
match A with
| nil => None
| a :: A => if decision (s = a) then Some 0 else match pos s A with None => None | Some n => Some (S n) end
end.

Lemma el_pos X (E : eq_dec X) s A : s el A -> exists m, pos s A = Some m.
Proof.
revert s; induction A; simpl; intros s H.
- decide (s = a) as [D | D]; eauto;
destruct H; try congruence.
destruct (IHA s H) as [n Hn]; eexists; now rewrite Hn.
Qed.

Lemma pos_elAt X (_ : eq_dec X) s A i : pos s A = Some i -> A .[i] = Some s.
Proof.
revert i s. induction A; intros i s.
- destruct i; inversion 1.
- simpl. decide (s = a).
+ inversion 1; subst; reflexivity.
+ destruct i; destruct (pos s A) eqn:B; inversion 1; subst; eauto.
Qed.

Lemma elAt_app X (A : list X) i B s : A .[i] = Some s -> (A ++ B).[i] = Some s.
Proof.
revert s B i. induction A; intros s B i H; destruct i; simpl; intuition; inv H.
Qed.

Lemma elAt_el (X : Type) A (s : X) m : A .[ m ] = Some s -> s el A.
Proof.
revert A. induction m; intros []; inversion 1; eauto.
Qed.

Lemma el_elAt X {_ : eq_dec X} (s : X) A : s el A -> exists m, A .[ m ] = Some s.
Proof.
intros H; destruct (el_pos _ H); eexists; eauto using pos_elAt.
Qed.

Lemma dupfree_elAt X (A : list X) n m s : dupfree A -> A.[n] = Some s -> A.[m] = Some s -> n = m.
Proof with try tauto.
intros H; revert n m; induction A; simpl; intros n m H1 H2.
- destruct n; inv H1.
- destruct n, m; inv H...
+ inv H1. simpl in H2. eapply elAt_el in H2...
+ inv H2. simpl in H1. eapply elAt_el in H1...
+ inv H1. inv H2. rewrite IHA with n m...
Qed.

Lemma nth_error_none A n l : nth_error l n = @None A -> length l <= n.
Proof. revert n;
induction l; intros n.
- simpl; omega.
- simpl. intros. destruct n. inv H. inv H. assert (| l | <= n). eauto. omega.
Qed.

End pos.