fix: traverse

Signed-off-by: Dr. Carsten Leue <carsten.leue@de.ibm.com>

v2/array/array.go

@@ -260,6 +260,8 @@ func Empty[A any]() []A {
 }
 
 // Zero returns an empty array of type A (alias for Empty).
+//
+//go:inline
 func Zero[A any]() []A {
 	return Empty[A]()
 }

v2/array/generic/array.go

@@ -25,8 +25,10 @@ import (
 )
 
 // Of constructs a single element array
+//
+//go:inline
 func Of[GA ~[]A, A any](value A) GA {
-	return GA{value}
+	return array.Of[GA](value)
 }
 
 func Reduce[GA ~[]A, A, B any](f func(B, A) B, initial B) func(GA) B {

v2/array/generic/monoid.go

@@ -0,0 +1,34 @@
+package generic
+
+import (
+	"github.com/IBM/fp-go/v2/internal/array"
+	M "github.com/IBM/fp-go/v2/monoid"
+	S "github.com/IBM/fp-go/v2/semigroup"
+)
+
+// Monoid returns a Monoid instance for arrays.
+// The Monoid combines arrays through concatenation, with an empty array as the identity element.
+//
+// Example:
+//
+//	m := array.Monoid[int]()
+//	result := m.Concat([]int{1, 2}, []int{3, 4}) // [1, 2, 3, 4]
+//	empty := m.Empty() // []
+//
+//go:inline
+func Monoid[GT ~[]T, T any]() M.Monoid[GT] {
+	return M.MakeMonoid(array.Concat[GT], Empty[GT]())
+}
+
+// Semigroup returns a Semigroup instance for arrays.
+// The Semigroup combines arrays through concatenation.
+//
+// Example:
+//
+//	s := array.Semigroup[int]()
+//	result := s.Concat([]int{1, 2}, []int{3, 4}) // [1, 2, 3, 4]
+//
+//go:inline
+func Semigroup[GT ~[]T, T any]() S.Semigroup[GT] {
+	return S.MakeSemigroup(array.Concat[GT])
+}

v2/array/misc_test.go

@@ -18,7 +18,6 @@ package array
 import (
 	"testing"
 
-	O "github.com/IBM/fp-go/v2/option"
 	OR "github.com/IBM/fp-go/v2/ord"
 	"github.com/stretchr/testify/assert"
 )
@@ -103,39 +102,6 @@ func TestSortByKey(t *testing.T) {
 	assert.Equal(t, "Charlie", result[2].Name)
 }
 
-func TestMonadTraverse(t *testing.T) {
-	result := MonadTraverse(
-		O.Of[[]int],
-		O.Map[[]int, func(int) []int],
-		O.Ap[[]int, int],
-		[]int{1, 3, 5},
-		func(n int) O.Option[int] {
-			if n%2 == 1 {
-				return O.Some(n * 2)
-			}
-			return O.None[int]()
-		},
-	)
-
-	assert.Equal(t, O.Some([]int{2, 6, 10}), result)
-
-	// Test with None case
-	result2 := MonadTraverse(
-		O.Of[[]int],
-		O.Map[[]int, func(int) []int],
-		O.Ap[[]int, int],
-		[]int{1, 2, 3},
-		func(n int) O.Option[int] {
-			if n%2 == 1 {
-				return O.Some(n * 2)
-			}
-			return O.None[int]()
-		},
-	)
-
-	assert.Equal(t, O.None[[]int](), result2)
-}
-
 func TestUniqByKey(t *testing.T) {
 	type Person struct {
 		Name string

v2/array/monoid.go

@@ -16,27 +16,12 @@
 package array
 
 import (
+	G "github.com/IBM/fp-go/v2/array/generic"
 	"github.com/IBM/fp-go/v2/internal/array"
 	M "github.com/IBM/fp-go/v2/monoid"
 	S "github.com/IBM/fp-go/v2/semigroup"
 )
 
-func concat[T any](left, right []T) []T {
-	// some performance checks
-	ll := len(left)
-	if ll == 0 {
-		return right
-	}
-	lr := len(right)
-	if lr == 0 {
-		return left
-	}
-	// need to copy
-	buf := make([]T, ll+lr)
-	copy(buf[copy(buf, left):], right)
-	return buf
-}
-
 // Monoid returns a Monoid instance for arrays.
 // The Monoid combines arrays through concatenation, with an empty array as the identity element.
 //
@@ -45,8 +30,10 @@ func concat[T any](left, right []T) []T {
 //	m := array.Monoid[int]()
 //	result := m.Concat([]int{1, 2}, []int{3, 4}) // [1, 2, 3, 4]
 //	empty := m.Empty() // []
+//
+//go:inline
 func Monoid[T any]() M.Monoid[[]T] {
-	return M.MakeMonoid(concat[T], Empty[T]())
+	return G.Monoid[[]T]()
 }
 
 // Semigroup returns a Semigroup instance for arrays.
@@ -56,8 +43,10 @@ func Monoid[T any]() M.Monoid[[]T] {
 //
 //	s := array.Semigroup[int]()
 //	result := s.Concat([]int{1, 2}, []int{3, 4}) // [1, 2, 3, 4]
+//
+//go:inline
 func Semigroup[T any]() S.Semigroup[[]T] {
-	return S.MakeSemigroup(concat[T])
+	return G.Semigroup[[]T]()
 }
 
 func addLen[A any](count int, data []A) int {

v2/array/sequence.go

@@ -16,10 +16,18 @@
 package array
 
 import (
-	F "github.com/IBM/fp-go/v2/function"
+	"github.com/IBM/fp-go/v2/internal/array"
+	M "github.com/IBM/fp-go/v2/monoid"
 	O "github.com/IBM/fp-go/v2/option"
 )
 
+func MonadSequence[HKTA, HKTRA any](
+	fof func(HKTA) HKTRA,
+	m M.Monoid[HKTRA],
+	ma []HKTA) HKTRA {
+	return array.MonadSequence(fof, m.Empty(), m.Concat, ma)
+}
+
 // Sequence takes an array where elements are HKT<A> (higher kinded type) and,
 // using an applicative of that HKT, returns an HKT of []A.
 //
@@ -55,16 +63,11 @@ import (
 //	    option.MonadAp[[]int, int],
 //	)
 //	result := seq(opts) // Some([1, 2, 3])
-func Sequence[A, HKTA, HKTRA, HKTFRA any](
-	_of func([]A) HKTRA,
-	_map func(HKTRA, func([]A) func(A) []A) HKTFRA,
-	_ap func(HKTFRA, HKTA) HKTRA,
+func Sequence[HKTA, HKTRA any](
+	fof func(HKTA) HKTRA,
+	m M.Monoid[HKTRA],
 ) func([]HKTA) HKTRA {
-	ca := F.Curry2(Append[A])
-	empty := _of(Empty[A]())
-	return Reduce(func(fas HKTRA, fa HKTA) HKTRA {
-		return _ap(_map(fas, ca), fa)
-	}, empty)
+	return array.Sequence[[]HKTA](fof, m.Empty(), m.Concat)
 }
 
 // ArrayOption returns a function to convert a sequence of options into an option of a sequence.
@@ -86,10 +89,10 @@ func Sequence[A, HKTA, HKTRA, HKTFRA any](
 //	    option.Some(3),
 //	}
 //	result2 := array.ArrayOption[int]()(opts2) // None
-func ArrayOption[A any]() func([]Option[A]) Option[[]A] {
-	return Sequence(
-		O.Of[[]A],
-		O.MonadMap[[]A, func(A) []A],
-		O.MonadAp[[]A, A],
+func ArrayOption[A any](ma []Option[A]) Option[[]A] {
+	return MonadSequence(
+		O.Map(Of[A]),
+		O.ApplicativeMonoid(Monoid[A]()),
+		ma,
 	)
 }

v2/array/sequence_test.go

@@ -24,8 +24,7 @@ import (
 )
 
 func TestSequenceOption(t *testing.T) {
-	seq := ArrayOption[int]()
 
-	assert.Equal(t, O.Of([]int{1, 3}), seq([]O.Option[int]{O.Of(1), O.Of(3)}))
-	assert.Equal(t, O.None[[]int](), seq([]O.Option[int]{O.Of(1), O.None[int]()}))
+	assert.Equal(t, O.Of([]int{1, 3}), ArrayOption([]O.Option[int]{O.Of(1), O.Of(3)}))
+	assert.Equal(t, O.None[[]int](), ArrayOption([]O.Option[int]{O.Of(1), O.None[int]()}))
 }

v2/array/traverse.go

@@ -80,3 +80,25 @@ func MonadTraverse[A, B, HKTB, HKTAB, HKTRB any](
 
 	return array.MonadTraverse(fof, fmap, fap, ta, f)
 }
+
+//go:inline
+func TraverseWithIndex[A, B, HKTB, HKTAB, HKTRB any](
+	fof func([]B) HKTRB,
+	fmap func(func([]B) func(B) []B) func(HKTRB) HKTAB,
+	fap func(HKTB) func(HKTAB) HKTRB,
+
+	f func(int, A) HKTB) func([]A) HKTRB {
+	return array.TraverseWithIndex[[]A](fof, fmap, fap, f)
+}
+
+//go:inline
+func MonadTraverseWithIndex[A, B, HKTB, HKTAB, HKTRB any](
+	fof func([]B) HKTRB,
+	fmap func(func([]B) func(B) []B) func(HKTRB) HKTAB,
+	fap func(HKTB) func(HKTAB) HKTRB,
+
+	ta []A,
+	f func(int, A) HKTB) HKTRB {
+
+	return array.MonadTraverseWithIndex(fof, fmap, fap, ta, f)
+}

v2/context/readerioresult/reader_test.go

@@ -312,7 +312,7 @@ func TestMonadChainFirstLeft(t *testing.T) {
 			Left[int](originalErr),
 			func(e error) ReaderIOResult[int] {
 				capturedError = e
-				return Right[int](999) // This Right value is ignored
+				return Right(999) // This Right value is ignored
 			},
 		)
 		actualResult := result(ctx)()
@@ -324,7 +324,7 @@ func TestMonadChainFirstLeft(t *testing.T) {
 	t.Run("Right value passes through", func(t *testing.T) {
 		sideEffectCalled := false
 		result := MonadChainFirstLeft(
-			Right[int](42),
+			Right(42),
 			func(e error) ReaderIOResult[int] {
 				sideEffectCalled = true
 				return Left[int](fmt.Errorf("should not be called"))
@@ -343,7 +343,7 @@ func TestMonadChainFirstLeft(t *testing.T) {
 			func(e error) ReaderIOResult[int] {
 				effectCount++
 				// Try to return Right, but original Left should still be returned
-				return Right[int](999)
+				return Right(999)
 			},
 		)
 		actualResult := result(ctx)()
@@ -378,7 +378,7 @@ func TestChainFirstLeft(t *testing.T) {
 		originalErr := fmt.Errorf("test error")
 		chainFn := ChainFirstLeft[int](func(e error) ReaderIOResult[int] {
 			captured = e
-			return Right[int](42) // This Right is ignored
+			return Right(42) // This Right is ignored
 		})
 		result := F.Pipe1(
 			Left[int](originalErr),
@@ -394,10 +394,10 @@ func TestChainFirstLeft(t *testing.T) {
 		called := false
 		chainFn := ChainFirstLeft[int](func(e error) ReaderIOResult[int] {
 			called = true
-			return Right[int](0)
+			return Right(0)
 		})
 		result := F.Pipe1(
-			Right[int](100),
+			Right(100),
 			chainFn,
 		)
 		assert.False(t, called)
@@ -409,7 +409,7 @@ func TestChainFirstLeft(t *testing.T) {
 		originalErr := fmt.Errorf("original")
 		chainFn := ChainFirstLeft[int](func(e error) ReaderIOResult[int] {
 			// Try to return Right, but original Left should still be returned
-			return Right[int](999)
+			return Right(999)
 		})
 
 		result := F.Pipe1(

v2/context/readerioresult/traverse.go

@@ -16,8 +16,8 @@
 package readerioresult
 
 import (
+	"github.com/IBM/fp-go/v2/array"
 	"github.com/IBM/fp-go/v2/function"
-	"github.com/IBM/fp-go/v2/internal/array"
 	"github.com/IBM/fp-go/v2/internal/record"
 )
 
@@ -29,7 +29,7 @@ import (
 //
 // Returns a function that transforms an array into a ReaderIOResult of an array.
 func TraverseArray[A, B any](f Kleisli[A, B]) Kleisli[[]A, []B] {
-	return array.Traverse[[]A](
+	return array.Traverse(
 		Of[[]B],
 		Map[[]B, func(B) []B],
 		Ap[[]B, B],
@@ -46,7 +46,7 @@ func TraverseArray[A, B any](f Kleisli[A, B]) Kleisli[[]A, []B] {
 //
 // Returns a function that transforms an array into a ReaderIOResult of an array.
 func TraverseArrayWithIndex[A, B any](f func(int, A) ReaderIOResult[B]) Kleisli[[]A, []B] {
-	return array.TraverseWithIndex[[]A](
+	return array.TraverseWithIndex(
 		Of[[]B],
 		Map[[]B, func(B) []B],
 		Ap[[]B, B],
@@ -135,22 +135,20 @@ func MonadTraverseArraySeq[A, B any](as []A, f Kleisli[A, B]) ReaderIOResult[[]B
 //
 // Returns a function that transforms an array into a ReaderIOResult of an array.
 func TraverseArraySeq[A, B any](f Kleisli[A, B]) Kleisli[[]A, []B] {
-	return array.Traverse[[]A](
+	return array.Traverse(
 		Of[[]B],
 		Map[[]B, func(B) []B],
 		ApSeq[[]B, B],
-
 		f,
 	)
 }
 
 // TraverseArrayWithIndexSeq uses transforms an array [[]A] into [[]ReaderIOResult[B]] and then resolves that into a [ReaderIOResult[[]B]]
 func TraverseArrayWithIndexSeq[A, B any](f func(int, A) ReaderIOResult[B]) Kleisli[[]A, []B] {
-	return array.TraverseWithIndex[[]A](
+	return array.TraverseWithIndex(
 		Of[[]B],
 		Map[[]B, func(B) []B],
 		ApSeq[[]B, B],
-
 		f,
 	)
 }
@@ -230,22 +228,20 @@ func MonadTraverseArrayPar[A, B any](as []A, f Kleisli[A, B]) ReaderIOResult[[]B
 //
 // Returns a function that transforms an array into a ReaderIOResult of an array.
 func TraverseArrayPar[A, B any](f Kleisli[A, B]) Kleisli[[]A, []B] {
-	return array.Traverse[[]A](
+	return array.Traverse(
 		Of[[]B],
 		Map[[]B, func(B) []B],
 		ApPar[[]B, B],
-
 		f,
 	)
 }
 
 // TraverseArrayWithIndexPar uses transforms an array [[]A] into [[]ReaderIOResult[B]] and then resolves that into a [ReaderIOResult[[]B]]
 func TraverseArrayWithIndexPar[A, B any](f func(int, A) ReaderIOResult[B]) Kleisli[[]A, []B] {
-	return array.TraverseWithIndex[[]A](
+	return array.TraverseWithIndex(
 		Of[[]B],
 		Map[[]B, func(B) []B],
 		ApPar[[]B, B],
-
 		f,
 	)
 }

v2/internal/array/array.go

@@ -15,6 +15,10 @@
 
 package array
 
+func Of[GA ~[]A, A any](a A) GA {
+	return GA{a}
+}
+
 func Slice[GA ~[]A, A any](low, high int) func(as GA) GA {
 	return func(as GA) GA {
 		length := len(as)
@@ -140,7 +144,7 @@ func UpsertAt[GA ~[]A, A any](a A) func(GA) GA {
 func MonadMap[GA ~[]A, GB ~[]B, A, B any](as GA, f func(a A) B) GB {
 	count := len(as)
 	bs := make(GB, count)
-	for i := count - 1; i >= 0; i-- {
+	for i := range count {
 		bs[i] = f(as[i])
 	}
 	return bs
@@ -155,7 +159,7 @@ func Map[GA ~[]A, GB ~[]B, A, B any](f func(a A) B) func(GA) GB {
 func MonadMapWithIndex[GA ~[]A, GB ~[]B, A, B any](as GA, f func(idx int, a A) B) GB {
 	count := len(as)
 	bs := make(GB, count)
-	for i := count - 1; i >= 0; i-- {
+	for i := range count {
 		bs[i] = f(i, as[i])
 	}
 	return bs
@@ -164,3 +168,19 @@ func MonadMapWithIndex[GA ~[]A, GB ~[]B, A, B any](as GA, f func(idx int, a A) B
 func ConstNil[GA ~[]A, A any]() GA {
 	return (GA)(nil)
 }
+
+func Concat[GT ~[]T, T any](left, right GT) GT {
+	// some performance checks
+	ll := len(left)
+	if ll == 0 {
+		return right
+	}
+	lr := len(right)
+	if lr == 0 {
+		return left
+	}
+	// need to copy
+	buf := make(GT, ll+lr)
+	copy(buf[copy(buf, left):], right)
+	return buf
+}

v2/internal/array/traverse.go

@@ -19,6 +19,72 @@ import (
 	F "github.com/IBM/fp-go/v2/function"
 )
 
+func MonadSequenceSegment[HKTB, HKTRB any](
+	fof func(HKTB) HKTRB,
+	empty HKTRB,
+	concat func(HKTRB, HKTRB) HKTRB,
+	fbs []HKTB,
+	start, end int,
+) HKTRB {
+
+	switch end - start {
+	case 0:
+		return empty
+	case 1:
+		return fof(fbs[start])
+	default:
+		mid := (start + end) / 2
+		return concat(
+			MonadSequenceSegment(fof, empty, concat, fbs, start, mid),
+			MonadSequenceSegment(fof, empty, concat, fbs, mid, end),
+		)
+	}
+}
+
+func SequenceSegment[HKTB, HKTRB any](
+	fof func(HKTB) HKTRB,
+	empty HKTRB,
+	concat func(HKTRB, HKTRB) HKTRB,
+) func([]HKTB) HKTRB {
+
+	concat_f := func(left, right func([]HKTB) HKTRB) func([]HKTB) HKTRB {
+		return func(fbs []HKTB) HKTRB {
+			return concat(left(fbs), right(fbs))
+		}
+	}
+	empty_f := F.Constant1[[]HKTB](empty)
+	at := func(idx int) func([]HKTB) HKTRB {
+		return func(fbs []HKTB) HKTRB {
+			return fof(fbs[idx])
+		}
+	}
+
+	var divide func(start, end int) func([]HKTB) HKTRB
+	divide = func(start, end int) func([]HKTB) HKTRB {
+		switch end - start {
+		case 0:
+			return empty_f
+		case 1:
+			return at(start)
+		default:
+			mid := (start + end) / 2
+			left := divide(start, mid)
+			right := divide(mid, end)
+
+			return concat_f(left, right)
+		}
+	}
+
+	// TODO this could be cached by length
+	get_divide := func(len int) func([]HKTB) HKTRB {
+		return divide(0, len)
+	}
+
+	return func(fbs []HKTB) HKTRB {
+		return get_divide(len(fbs))(fbs)
+	}
+}
+
 /*
 *
 We need to pass the members of the applicative explicitly, because golang does neither support higher kinded types nor template methods on structs or interfaces
@@ -79,6 +145,34 @@ func TraverseWithIndex[GA ~[]A, GB ~[]B, A, B, HKTB, HKTAB, HKTRB any](
 	}
 }
 
+/*
+*
+We need to pass the members of the applicative explicitly, because golang does neither support higher kinded types nor template methods on structs or interfaces
+
+HKTRB = HKT<GB>
+HKTB = HKT<B>
+HKTAB = HKT<func(A)B>
+*/
+func MonadSequence[GA ~[]HKTA, HKTA, HKTRA any](
+	fof func(HKTA) HKTRA,
+	empty HKTRA,
+	concat func(HKTRA, HKTRA) HKTRA,
+
+	ta GA) HKTRA {
+	return MonadSequenceSegment(fof, empty, concat, ta, 0, len(ta))
+}
+
+func Sequence[GA ~[]HKTA, HKTA, HKTRA any](
+	fof func(HKTA) HKTRA,
+	empty HKTRA,
+	concat func(HKTRA, HKTRA) HKTRA,
+) func(GA) HKTRA {
+
+	return func(ma GA) HKTRA {
+		return MonadSequence(fof, empty, concat, ma)
+	}
+}
+
 func MonadTraverseReduce[GA ~[]A, GB, A, B, HKTB, HKTAB, HKTRB any](
 	fof func(GB) HKTRB,
 	fmap func(func(GB) func(B) GB) func(HKTRB) HKTAB,

v2/internal/iter/iter.go

@@ -2,6 +2,7 @@ package iter
 
 import (
 	F "github.com/IBM/fp-go/v2/function"
+	M "github.com/IBM/fp-go/v2/monoid"
 )
 
 func MonadReduceWithIndex[GA ~func(yield func(A) bool), A, B any](fa GA, f func(int, B, A) B, initial B) B {
@@ -59,3 +60,41 @@ func Prepend[GA ~func(yield func(A) bool), A any](head A) func(GA) GA {
 func Empty[GA ~func(yield func(A) bool), A any]() GA {
 	return func(_ func(A) bool) {}
 }
+
+func ToArray[GA ~func(yield func(A) bool), GB ~[]A, A any](fa GA) GB {
+	bs := make(GB, 0)
+	for a := range fa {
+		bs = append(bs, a)
+	}
+	return bs
+}
+
+func MonadMapToArray[GA ~func(yield func(A) bool), GB ~[]B, A, B any](fa GA, f func(A) B) GB {
+	bs := make(GB, 0)
+	for a := range fa {
+		bs = append(bs, f(a))
+	}
+	return bs
+}
+
+func MapToArray[GA ~func(yield func(A) bool), GB ~[]B, A, B any](f func(A) B) func(GA) GB {
+	return F.Bind2nd(MonadMapToArray[GA, GB], f)
+}
+
+func MonadMapToArrayWithIndex[GA ~func(yield func(A) bool), GB ~[]B, A, B any](fa GA, f func(int, A) B) GB {
+	bs := make(GB, 0)
+	var i int
+	for a := range fa {
+		bs = append(bs, f(i, a))
+		i += 1
+	}
+	return bs
+}
+
+func MapToArrayWithIndex[GA ~func(yield func(A) bool), GB ~[]B, A, B any](f func(int, A) B) func(GA) GB {
+	return F.Bind2nd(MonadMapToArrayWithIndex[GA, GB], f)
+}
+
+func Monoid[GA ~func(yield func(A) bool), A any]() M.Monoid[GA] {
+	return M.MakeMonoid(Concat[GA], Empty[GA]())
+}

v2/internal/iter/traverse.go

@@ -17,6 +17,8 @@ package iter
 
 import (
 	F "github.com/IBM/fp-go/v2/function"
+	INTA "github.com/IBM/fp-go/v2/internal/array"
+	M "github.com/IBM/fp-go/v2/monoid"
 )
 
 /*
@@ -27,14 +29,65 @@ HKTRB = HKT<GB>
 HKTB = HKT<B>
 HKTAB = HKT<func(A)B>
 */
-func MonadTraverse[GA ~func(yield func(A) bool), GB ~func(yield func(B) bool), A, B, HKTB, HKTAB, HKTRB any](
-	fof func(GB) HKTRB,
-	fmap func(func(GB) func(B) GB) func(HKTRB) HKTAB,
-	fap func(HKTB) func(HKTAB) HKTRB,
+func MonadTraverse[GA ~func(yield func(A) bool), GB ~func(yield func(B) bool), A, B, HKT_B, HKT_GB_GB, HKT_GB any](
+	fmap_b func(HKT_B, func(B) GB) HKT_GB,
+
+	fof_gb func(GB) HKT_GB,
+	fmap_gb func(HKT_GB, func(GB) func(GB) GB) HKT_GB_GB,
+	fap_gb func(HKT_GB_GB, HKT_GB) HKT_GB,
 
 	ta GA,
-	f func(A) HKTB) HKTRB {
-	return MonadTraverseReduce(fof, fmap, fap, ta, f, MonadAppend[GB, B], Empty[GB]())
+	f func(A) HKT_B) HKT_GB {
+
+	fof := F.Bind2nd(fmap_b, Of[GB])
+
+	empty := fof_gb(Empty[GB]())
+
+	cb := F.Curry2(Concat[GB])
+	concat_gb := F.Bind2nd(fmap_gb, cb)
+	concat := func(first HKT_GB, second HKT_GB) HKT_GB {
+		return fap_gb(concat_gb(first), second)
+	}
+
+	// convert to an array
+	hktb := MonadMapToArray[GA, []HKT_B](ta, f)
+	return INTA.MonadSequenceSegment(fof, empty, concat, hktb, 0, len(hktb))
+}
+
+func Traverse[GA ~func(yield func(A) bool), GB ~func(yield func(B) bool), A, B, HKT_B, HKT_GB_GB, HKT_GB any](
+	fmap_b func(func(B) GB) func(HKT_B) HKT_GB,
+
+	fof_gb func(GB) HKT_GB,
+	fmap_gb func(func(GB) func(GB) GB) func(HKT_GB) HKT_GB_GB,
+	fap_gb func(HKT_GB_GB, HKT_GB) HKT_GB,
+
+	f func(A) HKT_B) func(GA) HKT_GB {
+
+	fof := fmap_b(Of[GB])
+	empty := fof_gb(Empty[GB]())
+	cb := F.Curry2(Concat[GB])
+	concat_gb := fmap_gb(cb)
+
+	concat := func(first, second HKT_GB) HKT_GB {
+		return fap_gb(concat_gb(first), second)
+	}
+
+	return func(ma GA) HKT_GB {
+		// return INTA.SequenceSegment(fof, empty, concat)(MapToArray[GA, []HKT_B](f)(ma))
+		hktb := MonadMapToArray[GA, []HKT_B](ma, f)
+		return INTA.MonadSequenceSegment(fof, empty, concat, hktb, 0, len(hktb))
+	}
+}
+
+func MonadSequence[GA ~func(yield func(HKTA) bool), HKTA, HKTRA any](
+	fof func(HKTA) HKTRA,
+	m M.Monoid[HKTRA],
+
+	ta GA) HKTRA {
+
+	// convert to an array
+	hktb := ToArray[GA, []HKTA](ta)
+	return INTA.MonadSequenceSegment(fof, m.Empty(), m.Concat, hktb, 0, len(hktb))
 }
 
 /*
@@ -45,37 +98,35 @@ HKTRB = HKT<GB>
 HKTB = HKT<B>
 HKTAB = HKT<func(A)B>
 */
-func MonadTraverseWithIndex[GA ~func(yield func(A) bool), GB ~func(yield func(B) bool), A, B, HKTB, HKTAB, HKTRB any](
-	fof func(GB) HKTRB,
-	fmap func(func(GB) func(B) GB) func(HKTRB) HKTAB,
-	fap func(HKTB) func(HKTAB) HKTRB,
+func MonadTraverseWithIndex[GA ~func(yield func(A) bool), A, HKTB, HKTRB any](
+	fof func(HKTB) HKTRB,
+	m M.Monoid[HKTRB],
 
 	ta GA,
 	f func(int, A) HKTB) HKTRB {
-	return MonadTraverseReduceWithIndex(fof, fmap, fap, ta, f, MonadAppend[GB, B], Empty[GB]())
+
+	// convert to an array
+	hktb := MonadMapToArrayWithIndex[GA, []HKTB](ta, f)
+	return INTA.MonadSequenceSegment(fof, m.Empty(), m.Concat, hktb, 0, len(hktb))
 }
 
-func Traverse[GA ~func(yield func(A) bool), GB ~func(yield func(B) bool), A, B, HKTB, HKTAB, HKTRB any](
-	fof func(GB) HKTRB,
-	fmap func(func(GB) func(B) GB) func(HKTRB) HKTAB,
-	fap func(HKTB) func(HKTAB) HKTRB,
+func Sequence[GA ~func(yield func(HKTA) bool), HKTA, HKTRA any](
+	fof func(HKTA) HKTRA,
+	m M.Monoid[HKTRA]) func(GA) HKTRA {
 
-	f func(A) HKTB) func(GA) HKTRB {
-
-	return func(ma GA) HKTRB {
-		return MonadTraverse(fof, fmap, fap, ma, f)
+	return func(ma GA) HKTRA {
+		return MonadSequence(fof, m, ma)
 	}
 }
 
-func TraverseWithIndex[GA ~func(yield func(A) bool), GB ~func(yield func(B) bool), A, B, HKTB, HKTAB, HKTRB any](
-	fof func(GB) HKTRB,
-	fmap func(func(GB) func(B) GB) func(HKTRB) HKTAB,
-	fap func(HKTB) func(HKTAB) HKTRB,
+func TraverseWithIndex[GA ~func(yield func(A) bool), A, HKTB, HKTRB any](
+	fof func(HKTB) HKTRB,
+	m M.Monoid[HKTRB],
 
 	f func(int, A) HKTB) func(GA) HKTRB {
 
 	return func(ma GA) HKTRB {
-		return MonadTraverseWithIndex(fof, fmap, fap, ma, f)
+		return MonadTraverseWithIndex(fof, m, ma, f)
 	}
 }
 

v2/io/traverse.go

@@ -63,9 +63,11 @@ func TraverseArray[A, B any](f Kleisli[A, B]) Kleisli[[]A, []B] {
 
 func TraverseIter[A, B any](f Kleisli[A, B]) Kleisli[Seq[A], Seq[B]] {
 	return INTI.Traverse[Seq[A]](
+		Map[B],
+
 		Of[Seq[B]],
-		Map[Seq[B], func(B) Seq[B]],
-		Ap[Seq[B], B],
+		Map[Seq[B]],
+		MonadAp[Seq[B]],
 
 		f,
 	)

v2/iterator/iter/iter.go

@@ -885,3 +885,13 @@ func MonadZip[A, B any](fb Seq[B], fa Seq[A]) Seq2[A, B] {
 func Zip[A, B any](fa Seq[A]) func(Seq[B]) Seq2[A, B] {
 	return F.Bind2nd(MonadZip[A, B], fa)
 }
+
+//go:inline
+func MonadMapToArray[A, B any](fa Seq[A], f func(A) B) []B {
+	return G.MonadMapToArray[Seq[A], []B](fa, f)
+}
+
+//go:inline
+func MapToArray[A, B any](f func(A) B) func(Seq[A]) []B {
+	return G.MapToArray[Seq[A], []B](f)
+}

v2/iterator/iter/monoid.go

@@ -33,5 +33,5 @@ import (
 //
 //go:inline
 func Monoid[T any]() M.Monoid[Seq[T]] {
-	return M.MakeMonoid(G.Concat[Seq[T]], Empty[T]())
+	return G.Monoid[Seq[T]]()
 }

v2/option/apply.go

@@ -42,6 +42,8 @@ func ApplySemigroup[A any](s S.Semigroup[A]) S.Semigroup[Option[A]] {
 //	optMonoid := ApplicativeMonoid(intMonoid)
 //	result := optMonoid.Concat(Some(2), Some(3)) // Some(5)
 //	result := optMonoid.Empty() // Some(0)
+//
+//go:inline
 func ApplicativeMonoid[A any](m M.Monoid[A]) M.Monoid[Option[A]] {
 	return M.ApplicativeMonoid(Of[A], MonadMap[A, func(A) A], MonadAp[A, A], m)
 }

v2/option/iter.go

@@ -55,11 +55,22 @@ import (
 //	result := TraverseIter(parse)(invalidStrings)
 //	// result is None because "invalid" cannot be parsed
 func TraverseIter[A, B any](f Kleisli[A, B]) Kleisli[Seq[A], Seq[B]] {
+
 	return INTI.Traverse[Seq[A]](
+		Map[B],
+
 		Of[Seq[B]],
-		Map[Seq[B], func(B) Seq[B]],
-		Ap[Seq[B]],
+		Map[Seq[B]],
+		MonadAp[Seq[B]],
 
 		f,
 	)
 }
+
+func SequenceIter[A any](as Seq[Option[A]]) Option[Seq[A]] {
+	return INTI.MonadSequence(
+		Map(INTI.Of[Seq[A]]),
+		ApplicativeMonoid(INTI.Monoid[Seq[A]]()),
+		as,
+	)
+}

v2/reader/array.go

@@ -17,7 +17,7 @@ package reader
 
 import (
 	"github.com/IBM/fp-go/v2/function"
-	"github.com/IBM/fp-go/v2/internal/array"
+	G "github.com/IBM/fp-go/v2/reader/generic"
 )
 
 // MonadTraverseArray transforms each element of an array using a function that returns a Reader,
@@ -38,13 +38,7 @@ import (
 //	r := reader.MonadTraverseArray(numbers, addPrefix)
 //	result := r(Config{Prefix: "num"}) // ["num1", "num2", "num3"]
 func MonadTraverseArray[R, A, B any](ma []A, f Kleisli[R, A, B]) Reader[R, []B] {
-	return array.MonadTraverse(
-		Of[R, []B],
-		Map[R, []B, func(B) []B],
-		Ap[[]B, R, B],
-		ma,
-		f,
-	)
+	return G.MonadTraverseArray[Reader[R, B], Reader[R, []B], []A](ma, f)
 }
 
 // TraverseArray transforms each element of an array using a function that returns a Reader,
@@ -63,12 +57,7 @@ func MonadTraverseArray[R, A, B any](ma []A, f Kleisli[R, A, B]) Reader[R, []B]
 //	r := transform([]int{1, 2, 3})
 //	result := r(Config{Multiplier: 10}) // [10, 20, 30]
 func TraverseArray[R, A, B any](f Kleisli[R, A, B]) func([]A) Reader[R, []B] {
-	return array.Traverse[[]A](
-		Of[R, []B],
-		Map[R, []B, func(B) []B],
-		Ap[[]B, R, B],
-		f,
-	)
+	return G.TraverseArray[Reader[R, B], Reader[R, []B], []A](f)
 }
 
 // TraverseArrayWithIndex transforms each element of an array using a function that takes
@@ -89,12 +78,7 @@ func TraverseArray[R, A, B any](f Kleisli[R, A, B]) func([]A) Reader[R, []B] {
 //	r := transform([]string{"a", "b", "c"})
 //	result := r(Config{Prefix: "item"}) // ["item[0]:a", "item[1]:b", "item[2]:c"]
 func TraverseArrayWithIndex[R, A, B any](f func(int, A) Reader[R, B]) func([]A) Reader[R, []B] {
-	return array.TraverseWithIndex[[]A](
-		Of[R, []B],
-		Map[R, []B, func(B) []B],
-		Ap[[]B, R, B],
-		f,
-	)
+	return G.TraverseArrayWithIndex[Reader[R, B], Reader[R, []B], []A](f)
 }
 
 // SequenceArray converts an array of Readers into a single Reader containing an array.

v2/reader/generic/monoid.go

@@ -0,0 +1,15 @@
+package generic
+
+import (
+	M "github.com/IBM/fp-go/v2/monoid"
+)
+
+//go:inline
+func ApplicativeMonoid[GA ~func(R) A, R, A any](m M.Monoid[A]) M.Monoid[GA] {
+	return M.ApplicativeMonoid(
+		Of[GA, R, A],
+		MonadMap[GA, func(R) func(A) A],
+		MonadAp[GA, GA, func(R) func(A) A],
+		m,
+	)
+}