Add more stuff to equality and more logging

Need to get names in there though.

newt/eq.newt

@@ -1,22 +1,29 @@
 module Equality
 
--- we don't have implicits yet, so this won't typecheck
+-- Leibniz equality
 
 Eq : {A : U} -> A -> A -> U
-Eq = \ {A} => \ x => \ y => (P : A -> U) -> P x -> P y
+Eq = \ {A} x y => (P : A -> U) -> P x -> P y
 
 refl : {A : U} {x : A} -> Eq x x
 refl = \ P Px => Px
 
+trans : {A : U} {x y z : A} -> Eq x y -> Eq y z -> Eq x z
+trans = \ Exy Eyz => Eyz (\ w => Eq x w) Exy
+
+sym : {A : U} {x y : A} -> Eq x y -> Eq y x
+sym = \ Exy => Exy (\ z => Eq z x) refl
+
 id : {A} -> A -> A
 id = \ x => x
 
 coerce : {A B : U} -> Eq A B -> A -> B
--- coerce refl a = a
 coerce = \ EqAB a => EqAB id a
 
--- can I write J without pattern matching?
--- J : {A : U} {x y : A} (eq : Eq x y) ->
--- (mot : (x : A) (P : Eq x y) -> U)
--- (b : mot y refl) ->
--- mot x eq
+-- J : {A : U} ->
+--     {C : (x y : A) -> Eq x y -> U} ->
+--     (c : (x : _) -> C x x refl) ->
+--     (x y : A) ->
+--     (p : Eq x y) ->
+--     C x y p
+-- J = \ c x y eq => eq (\ z => C x z _) (c x)