All as a monoid

all_monoid.js

'use strict';

// All (||) operator as a monoid

function All(v) {
  return {
    value: v
  };
}

function binary(v1, v2) {
  return v1 && v2;
}

function empty() {
  return All(true);
}

function append(m1, m2) {
  return All(binary(m1.value, m2.value));
}

function concat(ms) {
  return ms.reduce(append, empty());
}


function assertSameValue(m1, m2) {
  if(m1.value !== m2.value) {
    throw new Error('Expected ' + JSON.stringify(m1) + ' to have the same value as ' + JSON.stringify(m2));
  }
}

// `empty` must act as the identity with respect to `append`

var appendEmptyLeft = append(empty(), All(true));

assertSameValue(appendEmptyLeft, All(true));

var appendEmptyRight = append(All(false), empty());

assertSameValue(appendEmptyRight, All(false));


// `append` must be associative

var generalAppend = append(All(false), All(true));

assertSameValue(
  append(generalAppend, All(false)),
  append(All(false), generalAppend)
);


// `concat` can take a list of monoid values and reduce them to a single value

assertSameValue(
  concat([All(true), All(true), All(true)]),
  All(true)
);

assertSameValue(
  concat([All(true), All(true), All(false)]),
  All(false)
);

function all(bools) {
  return concat(bools.map(All)).value;
}

console.log(all([true, true, true]));
console.log(all([true, false, true]));