RelProperties of Relations

(* $Date: 2013-04-01 09:15:45 -0400 (Mon, 01 Apr 2013) $ *)

Require Export SfLib.

A (binary) relation is just a parameterized proposition. As you know from your undergraduate discrete math course, there are a lot of ways of discussing and describing relations in general — ways of classifying relations (are they reflexive, transitive, etc.), theorems that can be proved generically about classes of relations, constructions that build one relation from another, etc. Let us pause here to review a few that will be useful in what follows.

A (binary) relation on a set X is a proposition parameterized by two Xs — i.e., it is a logical assertion involving two values from the set X.

Definition relation (X: Type) := X→X→Prop.

Somewhat confusingly, the Coq standard library hijacks the generic term "relation" for this specific instance. To maintain consistency with the library, we will do the same. So, henceforth the Coq identifier relation will always refer to a binary relation between some set and itself, while the English word "relation" can refer either to the specific Coq concept or the more general concept of a relation between any number of possibly different sets. The context of the discussion should always make clear which is meant.

An example relation on nat is le, the less-that-or-equal-to relation which we usually write like this n1 <= n2.

Print le.
(* ====>
Inductive le (n : nat) : nat -> Prop :=
le_n : n <= n
| le_S : forall m : nat, n <= m -> n <= S m
*)
Check le : nat → nat → Prop.
Check le : relation nat.

Basic Properties of Relations

A relation R on a set X is a partial function if, for every x, there is at most one y such that R x y — i.e., if R x y1 and R x y2 together imply y1 = y2.

Definition partial_function {X: Type} (R: relation X) :=
∀x y1 y2 : X, R x y1 → R x y2 → y1 = y2.

For example, the next_nat relation defined in Logic.v is a partial function.

(* Print next_nat.
(* ====>
Inductive next_nat (n : nat) : nat -> Prop :=
  nn : next_nat n (S n)
*)
Check next_nat : relation nat.

Theorem next_nat_partial_function :
   partial_function next_nat.
Proof.
  unfold partial_function.
  intros x y1 y2 H1 H2.
  inversion H1. inversion H2.
  reflexivity.  Qed. *)

However, the <= relation on numbers is not a partial function.

This can be shown by contradiction. In short: Assume, for a contradiction, that <= is a partial function. But then, since 0 <= 0 and 0 <= 1, it follows that 0 = 1. This is nonsense, so our assumption was contradictory.

Theorem le_not_a_partial_function :
  ~ (partial_function le).
Proof.
  unfold not. unfold partial_function. intros Hc.
  assert (0 = 1) as Nonsense.
   Case "Proof of assertion".
   apply Hc with (x := 0).
     apply le_n.
     apply le_S. apply le_n.
  inversion Nonsense. Qed.

Exercise: 2 stars, optional

Show that the total_relation defined in Logic.v is not a partial function.

(* FILL IN HERE *)

☐

Exercise: 2 stars, optional

Show that the empty_relation defined in Logic.v is a partial function.

(* FILL IN HERE *)

☐

A reflexive relation on a set X is one for which every element of X is related to itself.

Definition reflexive {X: Type} (R: relation X) :=
  ∀a : X, R a a.

Theorem le_reflexive :
  reflexive le.
Proof.
  unfold reflexive. intros n. apply le_n. Qed.

A relation R is transitive if R a c holds whenever R a b and R b c do.

Definition transitive {X: Type} (R: relation X) :=
  ∀a b c : X, (R a b) → (R b c) → (R a c).

Theorem le_trans :
  transitive le.
Proof.
  intros n m o Hnm Hmo.
  induction Hmo.
  Case "le_n". apply Hnm.
  Case "le_S". apply le_S. apply IHHmo. Qed.

Theorem lt_trans:
  transitive lt.
Proof.
  unfold lt. unfold transitive.
  intros n m o Hnm Hmo.
  apply le_S in Hnm.
  apply le_trans with (a := (S n)) (b := (S m)) (c := o).
  apply Hnm.
  apply Hmo. Qed.

Exercise: 2 stars, optional

We can also prove lt_trans more laboriously by induction, without using le_trans. Do this.

Theorem lt_trans' :
  transitive lt.
Proof.
  (* Prove this by induction on evidence that m is less than o. *)
  unfold lt. unfold transitive.
  intros n m o Hnm Hmo.
  induction Hmo as [| m' Hm'o].
    (* FILL IN HERE *) Admitted.

☐

Exercise: 2 stars, optional

Prove the same thing again by induction on o.

Theorem lt_trans'' :
  transitive lt.
Proof.
  unfold lt. unfold transitive.
  intros n m o Hnm Hmo.
  induction o as [| o'].
  (* FILL IN HERE *) Admitted.

☐

The transitivity of le, in turn, can be used to prove some facts that will be useful later (e.g., for the proof of antisymmetry below)...

Theorem le_Sn_le : ∀n m, S n <= m → n <= m.
Proof.
  intros n m H. apply le_trans with (S n).
    apply le_S. apply le_n.
    apply H. Qed.

Exercise: 1 star, optional

Theorem le_S_n : ∀n m,
(S n <= S m) → (n <= m).
Proof.
(* FILL IN HERE *) Admitted.

☐

Exercise: 2 stars, optional (le_Sn_n_inf)

Provide an informal proof of the following theorem:

Theorem: For every n, ~(S n <= n)

A formal proof of this is an optional exercise below, but try the informal proof without doing the formal proof first.

Proof: (* FILL IN HERE *)
☐

Exercise: 1 star, optional

Theorem le_Sn_n : ∀n,
~ (S n <= n).
Proof.
(* FILL IN HERE *) Admitted.

☐

Reflexivity and transitivity are the main concepts we'll need for later chapters, but, for a bit of additional practice working with relations in Coq, here are a few more common ones.

A relation R is symmetric if R a b implies R b a.

Definition symmetric {X: Type} (R: relation X) :=
∀a b : X, (R a b) → (R b a).

Exercise: 2 stars, optional

Theorem le_not_symmetric :
~ (symmetric le).
Proof.
(* FILL IN HERE *) Admitted.

☐

A relation R is antisymmetric if R a b and R b a together imply a = b — that is, if the only "cycles" in R are trivial ones.

Definition antisymmetric {X: Type} (R: relation X) :=
∀a b : X, (R a b) → (R b a) → a = b.

Exercise: 2 stars, optional

Theorem le_antisymmetric :
antisymmetric le.
Proof.
(* FILL IN HERE *) Admitted.

☐

Exercise: 2 stars, optional

Theorem le_step : ∀n m p,
  n < m →
  m <= S p →
  n <= p.
Proof.
  (* FILL IN HERE *) Admitted.

☐

A relation is an equivalence if it's reflexive, symmetric, and transitive.

Definition equivalence {X:Type} (R: relation X) :=
(reflexive R) ∧ (symmetric R) ∧ (transitive R).

A relation is a partial order when it's reflexive, anti-symmetric, and transitive. In the Coq standard library it's called just "order" for short.

Definition order {X:Type} (R: relation X) :=
(reflexive R) ∧ (antisymmetric R) ∧ (transitive R).

A preorder is almost like a partial order, but doesn't have to be antisymmetric.

Definition preorder {X:Type} (R: relation X) :=
  (reflexive R) ∧ (transitive R).

Theorem le_order :
  order le.
Proof.
  unfold order. split.
    Case "refl". apply le_reflexive.
    split.
      Case "antisym". apply le_antisymmetric.
      Case "transitive.". apply le_trans. Qed.

Reflexive, Transitive Closure

The reflexive, transitive closure of a relation R is the smallest relation that contains R and that is both reflexive and transitive. Formally, it is defined like this in the Relations module of the Coq standard library:

Inductive clos_refl_trans {A: Type} (R: relation A) : relation A :=
    | rt_step : ∀x y, R x y → clos_refl_trans R x y
    | rt_refl : ∀x, clos_refl_trans R x x
    | rt_trans : ∀x y z,
          clos_refl_trans R x y →
          clos_refl_trans R y z →
          clos_refl_trans R x z.

For example, the reflexive and transitive closure of the next_nat relation coincides with the le relation.

Theorem next_nat_closure_is_le : ∀n m,
  (n <= m) ↔ ((clos_refl_trans next_nat) n m).
Proof.
  intros n m. split.
    Case "→".
      intro H. induction H.
      SCase "le_n". apply rt_refl.
      SCase "le_S".
        apply rt_trans with m. apply IHle. apply rt_step. apply nn.
    Case "←".
      intro H. induction H.
      SCase "rt_step". inversion H. apply le_S. apply le_n.
      SCase "rt_refl". apply le_n.
      SCase "rt_trans".
        apply le_trans with y.
        apply IHclos_refl_trans1.
        apply IHclos_refl_trans2. Qed.

The above definition of reflexive, transitive closure is natural — it says, explicitly, that the reflexive and transitive closure of R is the least relation that includes R and that is closed under rules of reflexivity and transitivity. But it turns out that this definition is not very convenient for doing proofs — the "nondeterminism" of the rt_trans rule can sometimes lead to tricky inductions.

Here is a more useful definition...

Inductive refl_step_closure {X:Type} (R: relation X) : relation X :=
  | rsc_refl : ∀(x : X), refl_step_closure R x x
  | rsc_step : ∀(x y z : X),
                    R x y →
                    refl_step_closure R y z →
                    refl_step_closure R x z.

(Note that, aside from the naming of the constructors, this definition is the same as the multi step relation used in many other chapters.)

(The following Tactic Notation definitions are explained in Imp.v. You can ignore them if you haven't read that chapter yet.)

Tactic Notation "rt_cases" tactic(first) ident(c) :=
  first;
  [ Case_aux c "rt_step" | Case_aux c "rt_refl"
  | Case_aux c "rt_trans" ].

Tactic Notation "rsc_cases" tactic(first) ident(c) :=
  first;
  [ Case_aux c "rsc_refl" | Case_aux c "rsc_step" ].

Our new definition of reflexive, transitive closure "bundles" the rt_step and rt_trans rules into the single rule step. The left-hand premise of this step is a single use of R, leading to a much simpler induction principle.

Before we go on, we should check that the two definitions do indeed define the same relation...

First, we prove two lemmas showing that refl_step_closure mimics the behavior of the two "missing" clos_refl_trans constructors.

Theorem rsc_R : ∀(X:Type) (R:relation X) (x y : X),
       R x y → refl_step_closure R x y.
Proof.
  intros X R x y H.
  apply rsc_step with y. apply H. apply rsc_refl. Qed.

Exercise: 2 stars, optional (rsc_trans)

Theorem rsc_trans :
  ∀(X:Type) (R: relation X) (x y z : X),
      refl_step_closure R x y →
      refl_step_closure R y z →
      refl_step_closure R x z.
Proof.
  (* FILL IN HERE *) Admitted.

☐

Then we use these facts to prove that the two definitions of reflexive, transitive closure do indeed define the same relation.

Exercise: 3 stars, optional (rtc_rsc_coincide)

Theorem rtc_rsc_coincide :
         ∀(X:Type) (R: relation X) (x y : X),
  clos_refl_trans R x y ↔ refl_step_closure R x y.
Proof.
  (* FILL IN HERE *) Admitted.

☐