diff --git a/nothing-helper.js b/nothing-helper.js
index d54700c..618a959 100644
--- a/nothing-helper.js
+++ b/nothing-helper.js
@@ -1,4 +1,53 @@
-module.exports = {
-  to_int: (f) => f(n => n+1)(0),
-  to_boolean: (f) => f(true)(false),
+const nothing = require('./nothing.js');
+
+const charset = '0123456789BFiuz';
+const from_int = (n) => {
+  let f = nothing.ZERO;
+  for (let _ = 0; _ < n; _++) f = nothing.INC(f);
+  return f;
+};
+const to_int = (f) => nothing.TIMES(f)(i => i+1)(0);
+const from_bool = (b) => b ? nothing.TRUE : nothing.FALSE;
+const to_bool = (f) => f(true)(false);
+const to_array = (l) => {
+  let a = [];
+
+  while (!to_bool(nothing.IS_EMPTY(l))) {
+    a.push(nothing.FIRST(l));
+    l = nothing.REST(l);
+  }
+
+  return a;
+};
+const from_array = (a) => {
+  let l = nothing.EMPTY;
+
+  while (a.length) l = nothing.UNSHIFT(l)(a.pop());
+
+  return l;
+};
+const to_char = (c) => charset[to_int(c)];
+const from_char = (c) => from_int(charset.indexOf(c));
+const to_string = (s) => to_array(s).map(to_char).join('');
+const from_string = (s) => from_array(s.split('').map(from_char));
+const representation_of = (v) => {
+  if (typeof v == "number") return from_int(v);
+  if (typeof v == "boolean") return from_bool(v);
+  if (typeof v == "string") return from_string(v);
+  if (Array.isArray(v)) return from_array(v.map(representation_of));
+  throw Error(`Oops, I don’t know how represent this: ${v}`);
+};
+const zip = (a, b) => a.map((e, i) => [e, b[i]]);
+const check_represent = function(r, v) {
+  if (typeof v == "number") return to_int(r) == v;
+  if (typeof v == "boolean") return to_bool(r) == v;
+  if (typeof v == "string") { console.log("checking ", to_array(r).map(to_int), " against ", v); return to_string(r) == v; }
+  if (Array.isArray(v)) {
+    let a = to_array(r);
+    return a.length == v.length && zip(a, v).every(([r, v]) => check_represent(r,v));
+  }
+  throw Error(`Oops, I don’t know how check this: ${v}`);
+};
+module.exports = {
+  to_string, to_array, representation_of, check_represent, to_int, to_char
 };
diff --git a/nothing.js b/nothing.js
index da1b579..e595508 100644
--- a/nothing.js
+++ b/nothing.js
@@ -18,37 +18,55 @@ const NOT = b => IF(b)(FALSE)(TRUE);
 const AND = a => b => IF(a)(IF(b)(TRUE)(FALSE))(FALSE);
 const OR = a => b => IF(a)(TRUE)(IF(b)(TRUE)(FALSE));
 const IS_ZERO = n => n(x => FALSE)(TRUE);
-const IS_LEQ = n => m => IS_ZERO(SUBTRACT(n)(m)); // we only have positive ints
+const IS_LEQ = n => m => IS_ZERO(SUB(n)(m)); // we only have positive ints
 const IS_EQ = n => m => AND(IS_LEQ(n)(m))(IS_LEQ(m)(n)); // this is stupid
 // time to get recursive!
 const Y = f => (x => f(x(x)))(x => f(x(x))); // lazy recursion
 const Z = f => (x => f(_ => x(x)(_)))(x => f(_ => x(x)(_))); // eager recursion
 const FACT = Z(f => n => IF(IS_ZERO(n))(ONE)(_ => MULT(n)(f(DEC(n)))(_)));
-const DIV = null;
-const MOD = null;
-const PAIR = null;
-const LEFT = null;
-const RIGHT = null;
-const EMPTY = null;
-const UNSHIFT = null;
-const IS_EMPTY = null;
-const FIRST = null;
-const REST = null;
-const INJECT = null;
-const FOLD = null;
-const MAP = null;
-const RANGE = null;
-const SUM = null;
-const PROD = null;
-const CONCAT = null;
-const PUSH = null;
-const REVERSE = null;
-const INC_ALL = null;
-const DOUBLE_ALL = null;
-const TO_DIGITS = null;
-const TO_STRING = null;
-const FIZZBUZZ = null;
+const DIV = Z(f => n => m => IF(IS_LEQ(m)(n))(_ => INC(f(SUB(n)(m))(m))(_))(ZERO));
+const MOD = Z(f => n => m => IF(IS_LEQ(m)(n))(_ => f(SUB(n)(m))(m)(_))(m));
+const PAIR = x => y => f => f(x)(y);
+const LEFT = p => p(x => y => x);
+const RIGHT = p => p (x => y => y);
+const EMPTY = PAIR(TRUE)(TRUE);
+const UNSHIFT = l => x => PAIR(FALSE)(PAIR(x)(l));
+const IS_EMPTY = LEFT;
+const FIRST = l => LEFT(RIGHT(l));
+const REST = l => RIGHT(RIGHT(l));
+const INJECT = Z(f => l => x => g => IF(IS_EMPTY(l))(x)(_ => f(REST(l))(g(x)(FIRST(l)))(g)(_)));
+const FOLD = Z(f => l => x => g => IF(IS_EMPTY(l))(x)(_ => g(f(REST(l))(x)(g))(FIRST(l))(_)));
+const MAP = k => f => FOLD(k)(EMPTY)(l => x => UNSHIFT(l)(f(x)));
+const RANGE = Z(f => n => m => IF(IS_LEQ(n)(m))(_ => UNSHIFT(f(INC(n))(m))(n)(_))(EMPTY));
+const SUM = l => INJECT(l)(ZERO)(ADD);
+const PROD = l => INJECT(l)(ONE)(MULT);
+const CONCAT = k => l => FOLD(k)(l)(UNSHIFT);
+const PUSH = l => x => CONCAT(l)(UNSHIFT(EMPTY)(x));
+const REVERSE = l => FOLD(l)(EMPTY)(PUSH);
+const INC_ALL = l => MAP(l)(INC);
+const DOUBLE_ALL = l => MAP(l)(MULT(TWO));
+const TEN = INC(MULT(THREE)(THREE));
+const RADIX = TEN;
+const FOUR = INC(THREE);
+const FIVE = INC(INC(THREE));
+const FIFTEEN = MULT(THREE)(FIVE);
+const FIZZ = MAP(UNSHIFT(UNSHIFT(UNSHIFT(UNSHIFT(EMPTY)(FOUR))(FOUR))(TWO))(ONE))(ADD(RADIX));
+const BUZZ = MAP(UNSHIFT(UNSHIFT(UNSHIFT(UNSHIFT(EMPTY)(FOUR))(FOUR))(THREE))(ZERO))(ADD(RADIX));
+const FIZZBUZZ = n => MAP(RANGE(ONE)(n))(m =>
+  IF(IS_ZERO(MOD(m)(FIFTEEN)))(
+    CONCAT(FIZZ)(BUZZ)
+  )(IF(IS_ZERO(MOD(m)(THREE)))(
+    FIZZ
+  )(IF(IS_ZERO(MOD(m)(FIVE)))(
+    BUZZ
+  )(
+    m
+  )))
+);
 module.exports = {
   ZERO, ONE, TWO, THREE, TIMES, INC, ADD, MULT, POWER, DEC, SUB, TRUE,
-  FALSE, IF, NOT, AND, OR, IS_ZERO, IS_LEQ, IS_EQ, IS_LEQ, Y, Z, FACT
+  FALSE, IF, NOT, AND, OR, IS_ZERO, IS_LEQ, IS_EQ, IS_LEQ, Y, Z, FACT,
+  DIV, MOD, PAIR, LEFT, RIGHT, EMPTY, UNSHIFT, IS_EMPTY, FIRST, REST, INJECT,
+  FOLD, MAP, RANGE, SUM, PROD, CONCAT, PUSH, REVERSE, INC_ALL, DOUBLE_ALL,
+  TEN, RADIX, FOUR, FIVE, FIFTEEN, FIZZ, BUZZ, FIZZBUZZ
 };