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selectionSort20172_sorts.v
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(** 116297 - Tópicos Avançados em Computadores - 2017/2 **)
(** Provas Formais: Uma Introdução à Teoria de Tipos - Turma B **)
(** Prof. Flávio L. C. de Moura **)
(** Email: [email protected] **)
(** Homepage: http://flaviomoura.mat.br **)
(** Aluno: **)
(** Matrícula: **)
Require Import Arith List.
Require Import Recdef.
Fixpoint select_min (x: nat) (l: list nat) : nat * list nat :=
match l with
| nil => (x,l)
| h :: tl => if (le_lt_dec x h) then
let (m,l') := select_min x tl in
(m, h::l')
else
let (m,l') := select_min h tl in
(m, x::l')
end.
Lemma select_min_length: forall l l' x y, select_min x l = (y, l') -> length l = length l'.
Proof.
induction l.
- simpl.
intros l x y H.
inversion H; subst.
reflexivity.
- intros l' x y.
assert (H: select_min x l = (y, l') -> length l = length l').
{
apply IHl.
}
generalize dependent y.
generalize dependent x.
induction l'.
+ intros x y IH H.
simpl in H.
destruct (le_lt_dec x a).
* destruct (select_min x l).
inversion H.
* destruct (select_min a l).
inversion H.
+ intros x y IH H.
simpl in H.
destruct (le_lt_dec x a).
* simpl. apply f_equal.
apply IHl with x y.
destruct (select_min x l).
inversion H; subst.
reflexivity.
* simpl. apply f_equal.
apply IHl with a y.
destruct (select_min a l).
inversion H; subst.
reflexivity.
Qed.
Function select (l: list nat) {measure length} : list nat :=
match l with
| nil => l
| h :: tl =>
let (m,l') := select_min h tl in
(m :: (select l'))
end.
Proof.
intros.
apply select_min_length in teq0.
rewrite <- teq0.
simpl.
apply lt_n_Sn.
Qed.
Inductive ordenada : list nat -> Prop :=
| lista_vazia : ordenada nil
| lista_1: forall n : nat, ordenada (cons n nil)
| lista_nv : forall (x y : nat) (l : list nat), ordenada (cons y l) -> x <= y -> ordenada (cons x (cons y l)).
Lemma ordenada_sub: forall l n, ordenada (n :: l) -> ordenada l.
Proof.
induction l.
- intros n H.
apply lista_vazia.
- intros n' Hcons.
inversion Hcons. subst.
assumption.
Qed.
Fixpoint num_oc n l :=
match l with
| nil => 0
| cons h tl =>
match eq_nat_dec n h with
| left _ => S(num_oc n tl)
| right _ => num_oc n tl
end
end.
Definition equiv l l' := forall n:nat, num_oc n l = num_oc n l'.
Lemma equiv_nil: forall l, equiv nil l -> l = nil.
Proof.
induction l.
- intro H.
reflexivity.
- intro H.
unfold equiv in H.
assert (H': num_oc a nil = num_oc a (a :: l)).
{
apply H.
}
simpl in H'.
destruct (Nat.eq_dec a a).
+ inversion H'.
+ apply False_ind.
apply n; reflexivity.
Qed.
Lemma equiv_trans: forall l l' l'', equiv l l' -> equiv l' l'' -> equiv l l''.
Proof.
intros l l' l'' H H'.
unfold equiv in *.
intro n.
apply eq_trans with (num_oc n l').
apply H.
apply H'.
Qed.
Lemma equiv_cons: forall l l' a, equiv l l' -> equiv (a::l) (a::l').
Proof.
intros l l' n H.
unfold equiv in *.
intros n'.
simpl.
destruct (Nat.eq_dec n' n).
- apply f_equal.
apply H.
- apply H.
Qed.
Lemma equiv_cons_comm: forall l l' x z, equiv l l' -> equiv (z :: x :: l) (x :: z :: l').
Proof.
intros l l' x z H.
unfold equiv in *.
intro n. simpl.
destruct (Nat.eq_dec n z).
- destruct (Nat.eq_dec n x).
+ apply f_equal. apply f_equal.
apply H.
+ apply f_equal.
apply H.
- destruct (Nat.eq_dec n x).
+ apply f_equal.
apply H.
+ apply H.
Qed.
Lemma equiv_cons_cons: forall l l' x y z, equiv (x :: l) (y :: l') -> equiv (x :: z :: l) (y :: z :: l').
Proof.
intros l l' x y z H.
assert (H': equiv (z :: x :: l) (z :: y :: l')).
{
apply equiv_cons; assumption.
}
apply equiv_trans with (z :: x :: l).
- apply equiv_cons_comm.
unfold equiv; reflexivity.
- apply equiv_trans with (z :: y :: l').
+ assumption.
+ apply equiv_cons_comm.
unfold equiv; reflexivity.
Qed.
Lemma select_min_cons_le: forall l l' x y a, select_min x (a::l) = (y,a::l') -> x <= a -> select_min x l = (y,l').
Proof.
intros l l' x y a H H'.
simpl in H.
destruct (le_lt_dec x a).
- destruct (select_min x l).
inversion H; subst.
reflexivity.
- destruct (select_min a l).
apply le_not_lt in H'.
contradiction.
Qed.
Lemma select_min_cons_lt: forall l l' x y a, select_min x (a::l) = (y,x::l') -> a < x -> select_min a l = (y,l').
Proof.
intros l l' x y a H H'.
simpl in H.
destruct (le_lt_dec x a).
- destruct (select_min x l).
apply le_not_lt in l0.
contradiction.
- destruct (select_min a l).
inversion H; subst.
reflexivity.
Qed.
Lemma select_min_equiv: forall l l' x y, select_min x l = (y, l') -> equiv (x::l) (y::l').
Proof.
induction l.
- intros l x y H.
simpl in H.
inversion H; subst.
unfold equiv; reflexivity.
- intros l' x y. case l'.
+ simpl. intro H.
destruct (le_lt_dec x a).
* destruct (select_min x l).
inversion H.
* destruct (select_min a l).
inversion H.
+ intros n l'' H.
assert (H' := H).
simpl in H.
destruct (le_lt_dec x a).
* destruct (select_min x l).
inversion H; subst.
apply equiv_cons_cons.
clear H.
apply IHl.
apply select_min_cons_le in H'.
** assumption.
** assumption.
* destruct (select_min a l).
inversion H; subst.
apply equiv_trans with (a :: n :: l).
** apply equiv_cons_comm.
unfold equiv; reflexivity.
** apply equiv_cons_cons.
apply IHl.
apply select_min_cons_lt with n; assumption.
Qed.
Lemma selectionSort_equiv: forall l, equiv l (select l).
Proof.
intro l.
functional induction (select l).
- unfold equiv.
reflexivity.
- unfold equiv in *.
intro n.
simpl.
destruct (Nat.eq_dec n h).
+ destruct (Nat.eq_dec n m).
* apply f_equal.
apply select_min_equiv in e0.
subst.
unfold equiv in e0.
apply eq_trans with (num_oc m l').
assert (H: num_oc m (m :: tl) = num_oc m (m :: l')).
{
apply e0.
}
simpl in H.
destruct (Nat.eq_dec m m).
** inversion H; subst.
reflexivity.
** apply False_ind.
apply n; reflexivity.
** apply IHl0.
* apply select_min_equiv in e0.
unfold equiv in e0.
assert (H: num_oc n (h :: tl) = num_oc n (m :: l')).
{ apply e0. }
clear e0.
simpl in H.
subst.
destruct (Nat.eq_dec h h).
** destruct (Nat.eq_dec h m).
*** contradiction.
*** rewrite H.
apply IHl0.
** destruct (Nat.eq_dec h m).
*** contradiction.
*** apply False_ind.
apply n; reflexivity.
+ destruct (Nat.eq_dec n m).
* subst.
apply select_min_equiv in e0.
unfold equiv in e0.
assert (H := e0 m).
simpl in H.
destruct (Nat.eq_dec m h).
** destruct (Nat.eq_dec m m).
*** contradiction.
*** apply False_ind.
apply n; reflexivity.
** destruct (Nat.eq_dec m m).
*** rewrite H.
apply f_equal.
apply IHl0.
*** apply False_ind.
apply n1; reflexivity.
* apply select_min_equiv in e0.
unfold equiv in e0.
assert (H := e0 n).
simpl in H.
destruct (Nat.eq_dec n h).
** contradiction.
** destruct (Nat.eq_dec n m).
*** contradiction.
*** rewrite H.
apply IHl0.
Qed.
(** Selection Sort sorts. *)
Lemma forall_leq_head: forall y h l, Forall (fun z => y <= z) (h::l) -> y <= h.
Proof.
intros y h l H.
inversion H; subst.
assumption.
Qed.
Inductive Permutation : list nat -> list nat -> Prop :=
| perm_nil: Permutation nil nil
| perm_skip x l l' : Permutation l l' -> Permutation (x::l) (x::l')
| perm_swap x y l : Permutation (y::x::l) (x::y::l)
| perm_trans l l' l'' :
Permutation l l' -> Permutation l' l'' -> Permutation l l''.
Lemma forall_permutation: forall y l l', Permutation l l' -> Forall (fun z => y <= z) l -> Forall (fun z => y <= z) l'.
Proof.
intros y l l' H H'.
induction H.
- apply Forall_nil.
- inversion H'; subst.
apply Forall_cons.
+ assumption.
+ apply IHPermutation.
assumption.
- inversion H'; subst.
inversion H2; subst.
apply Forall_cons.
+ assumption.
+ apply Forall_cons; assumption.
- apply IHPermutation2.
apply IHPermutation1.
assumption.
Qed.
Lemma Permutation_implies_equiv: forall l l', Permutation l l' -> equiv l l'.
Proof.
intros l l' H.
induction H.
- unfold equiv; reflexivity.
- unfold equiv in *.
intro n.
simpl num_oc.
destruct (Nat.eq_dec n x).
+ apply f_equal.
apply IHPermutation.
+ apply IHPermutation.
- unfold equiv in *.
intro n.
simpl num_oc at 1.
destruct (Nat.eq_dec n y).
+ destruct (Nat.eq_dec n x).
* simpl num_oc.
destruct (Nat.eq_dec n x).
** destruct (Nat.eq_dec n y).
*** reflexivity.
*** contradiction.
** destruct (Nat.eq_dec n y).
*** contradiction.
*** contradiction.
* simpl num_oc.
destruct (Nat.eq_dec n x).
** destruct (Nat.eq_dec n y).
*** contradiction.
*** contradiction.
** destruct (Nat.eq_dec n y).
*** reflexivity.
*** contradiction.
+ destruct (Nat.eq_dec n x).
* simpl num_oc.
destruct (Nat.eq_dec n x).
** destruct (Nat.eq_dec n y).
*** contradiction.
*** reflexivity.
** destruct (Nat.eq_dec n y).
*** reflexivity.
*** contradiction.
* simpl num_oc.
destruct (Nat.eq_dec n x).
** destruct (Nat.eq_dec n y).
*** contradiction.
*** contradiction.
** destruct (Nat.eq_dec n y).
*** contradiction.
*** reflexivity.
- unfold equiv in *.
intro n.
rewrite <- IHPermutation2.
apply IHPermutation1.
Qed.
Lemma list_length_ind: forall (A : Type) (P : list A -> Prop),
P nil ->
(forall (l : list A), (forall (l' : list A), length l' < length l -> P l') -> P l) ->
forall l : list A, P l.
Proof.
intros A P BI PI.
assert (H: forall (l1 l2: list A), length l2 <= length l1 -> P l2).
{
induction l1.
- intros l H.
inversion H.
apply length_zero_iff_nil in H1.
subst. assumption.
- intros l2 H.
apply PI.
simpl in H.
intros l' H'.
apply IHl1.
assert (H'': length l' < S (length l1)).
{
apply Nat.lt_le_trans with (length l2); assumption.
}
apply lt_n_Sm_le; assumption.
}
intro l.
apply H with l.
auto.
Qed.
Lemma equiv_implies_Permutation: forall l l', equiv l l' -> Permutation l l'.
Proof.
intros l. induction l using list_length_ind.
- intros l' H.
apply equiv_nil in H. subst.
apply perm_nil.
- intros l' H'.
generalize dependent l'. intro l'. case l'.
+ admit.
+ intros.
Admitted.
Lemma Permutation_equiv: forall l l', Permutation l l' <-> equiv l l'.
Proof.
intros l l'.
split.
- apply Permutation_implies_equiv.
- apply equiv_implies_Permutation.
Qed.
Lemma forall_equiv: forall y l l', Forall (fun z => y <= z) l -> equiv l l' -> Forall (fun z => y <= z) l'.
Proof.
intros y l l' H H'.
apply equiv_implies_Permutation in H'.
apply forall_permutation with l; assumption.
Qed.
Lemma select_min_leq: forall h l m l', select_min h l = (m, l') -> m <= h.
Proof.
intros h l; revert h; induction l using list_length_ind.
- intros h m l' H.
inversion H.
auto.
- intros h m l' H'.
generalize dependent l.
intro l.
case l.
+ intros H H1.
inversion H1.
apply le_refl.
+ intros n l0 H H1.
simpl in H1.
destruct (le_lt_dec h n).
* destruct (select_min h l0) eqn: H2.
inversion H1; subst.
apply H with l0 l2.
** simpl.
apply lt_n_Sn.
** assumption.
* destruct (select_min n l0) eqn: H2.
inversion H1; subst.
apply le_trans with n.
** apply H with l0 l2.
*** simpl.
apply lt_n_Sn.
*** assumption.
** apply Nat.lt_le_incl in l1.
assumption.
Qed.
Lemma select_min_smallest:
forall x l y l', select_min x l = (y, l') ->
Forall (fun z => y <= z) l'.
Proof.
intros x l; revert x; induction l using list_length_ind.
- intros x y l' HI.
inversion HI; subst.
apply Forall_nil.
- intros x y l' H'.
generalize dependent l'. intro l'.
case l'.
+ intros H H1.
inversion H1; subst.
apply Forall_nil.
+ intros n l0 H H1.
simpl in H1.
destruct (select_min x l0) eqn: H2.
inversion H1. subst.
apply forall_leq_head with l0.
apply Forall_cons.
Admitted.
Lemma select_forall: forall m l, ordenada l ->
Forall (fun z => m <= z) l ->
ordenada (m :: l).
Proof.
intros m l.
case l.
- intros H H'.
apply lista_1.
- intros n l' H H'.
apply lista_nv.
+ assumption.
+ inversion H'; subst.
assumption.
Qed.
Theorem selectionSort_sorts: forall l, ordenada (select l).
Proof.
intro l.
functional induction (select l).
- apply lista_vazia.
- generalize dependent l'.
intro l'.
induction l' using list_length_ind.
+ intros H H1.
rewrite select_equation.
apply lista_1.
+ intros H0 H1.
case l'.
* rewrite select_equation.
apply lista_1.
* intros n l.
rewrite select_equation.
destruct (select_min n l) as [m' l''] eqn: H2.
apply lista_nv.
** apply H.
**
case l'.
+ intros H H1.
rewrite select_equation.
apply lista_1.
+ intros n l Hl Hord.
rewrite select_equation in *.
destruct (select_min n l) as [m' l''] eqn: H.
assert (H': ordenada (m' :: select l'') ->
Forall (fun z => m <= z) (m' :: select l'') ->
ordenada (m :: (m' :: select l''))).
{ apply select_forall. }
apply lista_nv.
* assumption.
* inversion H.
inversion Hl.
apply select_min_leq in H.
apply select_min_smallest in H2.
apply forall_leq_head in H2.
admit.
Admitted.
(** Exercício extra. *)
Lemma select_min_min: forall h1 l1 m1 m2 h2 l2 l3, select_min h1 l1 = (m1, h2::l2) -> select_min h2 l2 = (m2, l3) -> m1 <= m2.
Proof.
Admitted.