+
+// We will only build permutations with 0 or 1 elements in each of the left and
+// right sets. This covers all the reduced form permutations. Any longer
+// permutations will be similar to one of the reduced forms.
+func permuteExistingNumbers(generation int, universe surrealUniverse) []surrealNumber {
+ var numbers []surrealNumber
+
+ // First build permutations with 1 element in each set.
+ for i := 0; i < universe.cardinality(); i++ {
+ for j := 0; j < universe.cardinality(); j++ {
+ var leftSet, rightSet surrealSet
+ leftSet.insert(universe.numbers[i], universe)
+ rightSet.insert(universe.numbers[j], universe)
+ newSurrealNumber := surrealNumber{leftSet,rightSet,0,generation}
+ numbers = append(numbers, newSurrealNumber)
+ }
+ }
+ // Now build permutations with one empty set and one 1-element set.
+ for i := 0; i < universe.cardinality(); i++ {
+ var tempSet surrealSet
+ tempSet.insert(universe.numbers[i], universe)
+ newSurrealNumber := surrealNumber{tempSet,surrealSet{},0,generation}
+ numbers = append(numbers, newSurrealNumber)
+ newSurrealNumber = surrealNumber{surrealSet{},tempSet,0,generation}
+ numbers = append(numbers, newSurrealNumber)
+ }
+
+ return numbers
+}
+
+func pruneInvalidNumbers(candidates []surrealNumber, universe surrealUniverse) []surrealNumber {
+ var numbers []surrealNumber
+ for i := 0; i < len(candidates); i++ {
+ if isValidSurrealNumber(candidates[i], universe) {
+ numbers = append(numbers, candidates[i])
+ }
+ }
+ return numbers
+}
+
+// Although a non-reduced-form number is technically valid, for the purposes of
+// this program we're declaring it invalid.
+func isValidSurrealNumber(candidate surrealNumber, universe surrealUniverse) bool {
+ // Is the number in reduced form?
+ if candidate.leftSet.cardinality() > 1 || candidate.rightSet.cardinality() > 1 {
+ return false
+ }
+
+ // Is the number valid per Axiom 2?
+ if candidate.leftSet.cardinality() != 0 && candidate.rightSet.cardinality() != 0 {
+ for i := 0; i < candidate.leftSet.cardinality(); i++ {
+ for j := 0; j < candidate.leftSet.cardinality(); j++ {
+ // Since candidates are drawn from the universe, this comparison is allowed.
+ if universe.greaterThanOrEqual(candidate.leftSet.members[i], candidate.rightSet.members[j]) {
+ return false
+ }
+ }
+ }
+ }
+
+ return true
+}
+
+func addNumbersToUniverse(numbers []surrealNumber, universe *surrealUniverse) {
+ for i := 0; i < len(numbers); i++ {
+ universe.insert(numbers[i])
+ }
+}
+
+// Pretty print a number, indenting to indicate generation.
+func printlnNumber(num surrealNumber) {
+ for i := 0; i < num.generation; i++ {
+ fmt.Printf(".\t\t")
+ }
+ fmt.Printf("%s = ", toBase26(num.identifier))
+ fmt.Printf("<")
+ if num.leftSet.cardinality() == 0 {
+ fmt.Printf("---")
+ } else {
+ fmt.Printf("%s", toBase26(num.leftSet.members[0].identifier))
+ }
+ fmt.Printf("|")
+ if num.rightSet.cardinality() == 0 {
+ fmt.Printf("---")
+ } else {
+ fmt.Printf("%s", toBase26(num.rightSet.members[0].identifier))
+ }
+ fmt.Printf(">")
+ fmt.Printf("\n")
+}
+
+// This function is a total hack, intended only to avoid printing identifiers
+// as actual numbers since we don't want to imply any particular association to
+// specific 'real' numbers yet. Instead, print them with [a-z] as base-26
+// integers.
+func toBase26(num uint) (base26String []byte) {
+ // For alignment reasons, we never print more than three digits. Thus, any
+ // number greater than (26^3)-1 is out of range.
+ if num >= 26*26*26 {
+ base26String = []byte("xxx")
+ } else {
+ var firstByte, secondByte, thirdByte byte
+ thirdByte = byte((num % 26) + uint('a'));
+ num = num / 26;
+ secondByte = byte((num % 26) + uint('a'));
+ num = num / 26;
+ firstByte = byte((num % 26) + uint('a'));
+ num = num / 26;
+ base26String = append(base26String, firstByte)
+ base26String = append(base26String, secondByte)
+ base26String = append(base26String, thirdByte)
+ }
+ return base26String
+}