RelProperties of Relations

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

Require Export SfLib.

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) := XXProp.

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.
  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).
  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.

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.
  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.
  intros n m o Hnm Hmo.
  induction Hmo.
  Case "le_n". apply Hnm.
  Case "le_S". apply le_S. apply IHHmo. Qed.

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).

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.

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).

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,
  (nm) ((clos_refl_trans next_nat) n m).
  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.)
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.
  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.
  (* 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.
  (* FILL IN HERE *) Admitted.