Rel

src/Rel.lidr

@@ -0,0 +1,371 @@
+= Rel : Properties of Relations
+
+> module Rel
+
+\todo[inline]{Add hyperlinks}
+
+This short (and optional) chapter develops some basic definitions and a few
+theorems about binary relations in Idris. The key definitions are repeated where
+they are actually used (in the `Smallstep` chapter), so readers who are already
+comfortable with these ideas can safely skim or skip this chapter. However,
+relations are also a good source of exercises for developing facility with
+Idris's basic reasoning facilities, so it may be useful to look at this material
+just after the `IndProp` chapter.
+
+A binary _relation_ on a set \idr{t} is a family of propositions parameterized
+by two elements of \idr{t} — i.e., a proposition about pairs of elements of
+\idr{t}.
+
+> Relation : Type -> Type
+> Relation t = t -> t -> Type
+
+\todo[inline]{Edit, there's n-relation \idr{Data.Rel} in \idr{contrib}, but no
+\idr{Relation}}
+
+Confusingly, the Idris standard library hijacks the generic term "relation" for
+this specific instance of the idea. To maintain consistency with the library, we
+will do the same. So, henceforth the Idris identifier `relation` will always
+refer to a binary relation between some set and itself, whereas the English word
+"relation" can refer either to the specific Idris 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.
+
+\todo[inline]{There's a similar concept called \idr{LTE} in \idr{Prelude.Nat},
+but it's defined by induction from zero}
+
+An example relation on \idr{Nat} is \idr{Le}, the less-than-or-equal-to
+relation, which we usually write \idr{n1 <= n2}.
+
+\todo[inline]{Copied from `IndProp`}
+
+> data Le : (n : Nat) -> Nat -> Type where
+>   Le_n : Le n n
+>   Le_S : Le n m -> Le n (S m)
+>
+> syntax [m] "<='" [n] = Le m n
+
+```idris
+λΠ> the (Relation Nat) Le
+Le : Nat -> Nat -> Type
+```
+
+\todo[inline]{Edit to show it (probably) doesn't matter in Idris}
+
+(Why did we write it this way instead of starting with \idr{data Le : Relation
+Nat ...}? Because we wanted to put the first \idr{Nat} to the left of the
+\idr{:}, which makes Idris generate a somewhat nicer induction principle for
+reasoning about \idr{<='}.)
+
+
+== Basic Properties
+
+As anyone knows who has taken an undergraduate discrete math course, there is a
+lot to be said about relations in general, including ways of classifying
+relations (as reflexive, transitive, etc.), theorems that can be proved
+generically about certain sorts of relations, constructions that build one
+relation from another, etc. For example...
+
+
+=== Partial Functions
+
+A relation \idr{r} on a set \idr{t} is a _partial function_ if, for every
+\idr{x}, there is at most one \idr{y} such that \idr{r x y} — i.e., \idr{r x y1}
+and \idr{r x y2} together imply \idr{y1 = y2}.
+
+> Partial_function : (r : Relation t) -> Type
+> Partial_function {t} r = (x, y1, y2 : t) -> r x y1 -> r x y2 -> y1 = y2
+
+\todo[inline]{"Earlier" = in \idr{IndProp}, add hyperlink?}
+
+For example, the \idr{Next_nat} relation defined earlier is a partial function.
+
+> data Next_nat : (n : Nat) -> Nat -> Type where
+>   NN : Next_nat n (S n)
+
+```idris
+λΠ> the (Relation Nat) Next_nat
+Next_nat : Nat -> Nat -> Type
+```
+
+> next_nat_partial_function : Partial_function Next_nat
+> next_nat_partial_function x (S x) (S x) NN NN = Refl
+
+However, the \idr{<='} relation on numbers is not a partial function. (Assume,
+for a contradiction, that \idr{<='} is a partial function. But then, since
+\idr{0 <=' 0} and \idr{0 <=' 1}, it follows that \idr{0 = 1}. This is nonsense,
+so our assumption was contradictory.)
+
+> le_not_a_partial_function : Not (Partial_function Le)
+> le_not_a_partial_function f = absurd $ f 0 0 1 Le_n (Le_S Le_n)
+
+
+==== Exercise: 2 stars, optional
+
+\ \todo[inline]{Again, "earlier" = \idr{IndProp}}
+
+Show that the \idr{Total_relation} defined in earlier is not a partial function.
+
+> -- FILL IN HERE
+
+$\square$
+
+
+==== Exercise: 2 stars, optional
+
+Show that the \idr{Empty_relation} that we defined earlier is a partial
+function.
+
+> --FILL IN HERE
+
+$\square$
+
+
+=== Reflexive Relations
+
+A _reflexive_ relation on a set \idr{t} is one for which every element of
+\idr{t} is related to itself.
+
+> Reflexive : (r : Relation t) -> Type
+> Reflexive {t} r = (a : t) -> r a a
+
+> le_reflexive : Reflexive Le
+> le_reflexive n = Le_n {n}
+
+
+=== Transitive Relations
+
+A relation \idr{r} is _transitive_ if \idr{r a c} holds whenever \idr{r a b} and
+\idr{r b c} do.
+
+> Transitive : (r : Relation t) -> Type
+> Transitive {t} r = (a, b, c : t) -> r a b -> r b c -> r a c
+
+> le_trans : Transitive Le
+> le_trans _ _ _ lab Le_n = lab
+> le_trans a b (S c) lab (Le_S lbc) = Le_S $ le_trans a b c lab lbc
+
+\todo[inline]{Copied here}
+
+> Lt : (n, m : Nat) -> Type
+> Lt n m = Le (S n) m
+
+> syntax [m] "<'" [n] = Lt m n
+
+> lt_trans : Transitive Lt
+> lt_trans a b c lab lbc = le_trans (S a) (S b) c (Le_S lab) lbc
+
+
+==== Exercise: 2 stars, optional
+
+We can also prove \idr{lt_trans} more laboriously by induction, without using
+\idr{le_trans}. Do this.
+
+> lt_trans' : Transitive Lt
+> -- Prove this by induction on evidence that a is less than c.
+> lt_trans' a b c lab lbc = ?lt_trans__rhs
+
+$\square$
+
+
+==== Exercise: 2 stars, optional
+
+\ \todo[inline]{Not sure how is this different from \idr{lt_trans'}?}
+
+Prove the same thing again by induction on \idr{c}.
+
+> lt_trans'' : Transitive Lt
+> lt_trans'' a b c lab lbc = ?lt_trans___rhs
+
+$\square$
+
+The transitivity of \idr{Le}, in turn, can be used to prove some facts that will
+be useful later (e.g., for the proof of antisymmetry below)...
+
+> le_Sn_le : ((S n) <=' m) -> (n <=' m)
+> le_Sn_le {n} {m} = le_trans n (S n) m (Le_S Le_n)
+
+
+==== Exercise: 1 star, optional
+
+> le_S_n : ((S n) <=' (S m)) -> (n <=' m)
+> le_S_n less = ?le_S_n_rhs
+
+$\square$
+
+
+==== Exercise: 2 stars, optional (le_Sn_n_inf)
+
+Provide an informal proof of the following theorem:
+
+Theorem: For every \idr{n}, \idr{Not ((S n) <=' n)}
+
+A formal proof of this is an optional exercise below, but try writing an
+informal proof without doing the formal proof first.
+
+Proof:
+
+> -- FILL IN HERE
+
+$\square$
+
+
+==== Exercise: 1 star, optional
+
+> le_Sn_n : Not ((S n) <=' n)
+> le_Sn_n = ?le_Sn_n_rhs
+
+$\square$
+
+Reflexivity and transitivity are the main concepts we'll need for later
+chapters, but, for a bit of additional practice working with relations in Idris,
+let's look at a few other common ones...
+
+
+=== Symmetric and Antisymmetric Relations
+
+A relation \idr{r} is _symmetric_ if \idr{r a b} implies \idr{r b a}.
+
+> Symmetric : (r : Relation t) -> Type
+> Symmetric {t} r = (a, b : t) -> r a b -> r b a
+
+
+==== Exercise: 2 stars, optional
+
+> le_not_symmetric : Not (Symmetric Le)
+> le_not_symmetric = ?le_not_symmetric_rhs
+
+$\square$
+
+A relation \idr{r} is _antisymmetric_ if \idr{r a b} and \idr{r b a} together
+imply \idr{a = b} — that is, if the only "cycles" in \idr{r} are trivial ones.
+
+> Antisymmetric : (r : Relation t) -> Type
+> Antisymmetric {t} r = (a, b : t) -> r a b -> r b a -> a = b
+
+
+==== Exercise: 2 stars, optional
+
+> le_antisymmetric : Antisymmetric Le
+> le_antisymmetric = ?le_antisymmetric_rhs
+
+$\square$
+
+
+==== Exercise: 2 stars, optional
+
+> le_step : (n <' m) -> (m <=' (S p)) -> (n <=' p)
+> le_step ltnm lemsp = ?le_step_rhs
+
+$\square$
+
+
+=== Equivalence Relations
+
+A relation is an _equivalence_ if it's reflexive, symmetric, and transitive.
+
+> Equivalence : (r : Relation t) -> Type
+> Equivalence r = (Reflexive r, Symmetric r, Transitive r)
+
+
+=== Partial Orders and Preorders
+
+\ \todo[inline]{Edit}
+
+A relation is a _partial order_ when it's reflexive, _anti_-symmetric, and
+transitive. In the Idris standard library it's called just "order" for short.
+
+> Order : (r : Relation t) -> Type
+> Order r = (Reflexive r, Antisymmetric r, Transitive r)
+
+A preorder is almost like a partial order, but doesn't have to be antisymmetric.
+
+> Preorder : (r : Relation t) -> Type
+> Preorder r = (Reflexive r, Transitive r)
+
+> le_order : Order Le
+> le_order = (le_reflexive, le_antisymmetric, le_trans)
+
+
+== Reflexive, Transitive Closure
+
+\ \todo[inline]{Edit}
+
+The _reflexive, transitive closure_ of a relation \idr{r} is the smallest
+relation that contains \idr{r} and that is both reflexive and transitive.
+Formally, it is defined like this in the Relations module of the Idris standard
+library:
+
+> data Clos_refl_trans : (r : Relation t) -> Relation t where
+>   Rt_step : r x y -> Clos_refl_trans r x y
+>   Rt_refl : Clos_refl_trans r x x
+>   Rt_trans : 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 \idr{Next_nat} relation
+coincides with the \idr{Le} relation.
+
+\todo[inline]{Copied `<->` for now}
+
+> iff : {p,q : Type} -> Type
+> iff {p} {q} = (p -> q, q -> p)
+>
+> syntax [p] "<->" [q] = iff {p} {q}
+
+> next_nat_closure_is_le : (n <=' m) <-> (Clos_refl_trans Next_nat n m)
+> next_nat_closure_is_le = (to, fro)
+>   where
+>     to : Le n m -> Clos_refl_trans Next_nat n m
+>     to Le_n = Rt_refl
+>     to (Le_S {m} le) = Rt_trans {y=m} (to le) (Rt_step NN)
+>     fro : Clos_refl_trans Next_nat n m -> Le n m
+>     fro (Rt_step NN) = Le_S Le_n
+>     fro Rt_refl = Le_n
+>     fro (Rt_trans {x=n} {y} {z=m} ny ym) =
+>       le_trans n y m (fro ny) (fro ym)
+
+The above definition of reflexive, transitive closure is natural: it says,
+explicitly, that the reflexive and transitive closure of \idr{r} is the least
+relation that includes \idr{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, since the "nondeterminism" of the \idr{Rt_trans} rule can
+sometimes lead to tricky inductions. Here is a more useful definition:
+
+> data Clos_refl_trans_1n : (r : Relation t) -> (x : t) -> t -> Type where
+>   Rt1n_refl : Clos_refl_trans_1n r x x
+>   Rt1n_trans : r x y -> Clos_refl_trans_1n r y z -> Clos_refl_trans_1n r x z
+
+\todo[inline]{Edit}
+
+Our new definition of reflexive, transitive closure "bundles" the \idr{Rt_step}
+and \idr{Rt_trans} rules into the single rule step. The left-hand premise of
+this step is a single use of \idr{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 \idr{Clos_refl_trans_1n} mimics the
+behavior of the two "missing" \idr{Clos_refl_trans} constructors.
+
+> rsc_R : r x y -> Clos_refl_trans_1n r x y
+> rsc_R rxy = Rt1n_trans rxy Rt1n_refl
+
+
+==== Exercise: 2 stars, optional (rsc_trans)
+
+> rsc_trans : Clos_refl_trans_1n r x y -> Clos_refl_trans_1n r y z ->
+>             Clos_refl_trans_1n r x z
+> rsc_trans crxy cryz = ?rsc_trans_rhs
+
+$\square$
+
+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)
+
+> rtc_rsc_coincide : (Clos_refl_trans r x y) <-> (Clos_refl_trans_1n r x y)
+> rtc_rsc_coincide = ?rtc_rsc_coincide_rhs
+
+$\square$