feat: add De Bruijn index reduction engine

Closes #26

- Added -i flag to select interpreter (lambda or debruijn)
- Created debruijn package with Expression interface
  - Variable contains index and optional label
  - Abstraction contains only body (no parameter)
  - Application structure remains similar
- Implemented De Bruijn reduction without variable renaming
  - Shift operation handles index adjustments
  - Substitute replaces by index instead of name
- Abstracted Engine into interface with two implementations
  - LambdaEngine: original named variable engine
  - DeBruijnEngine: new index-based engine
- Added conversion functions between representations
  - LambdaToDeBruijn: converts named to indexed
  - DeBruijnToLambda: converts indexed back to named
  - SaccharineToDeBruijn: direct saccharine to De Bruijn
- Updated main to switch engines based on -i flag
- All test samples pass with both engines

Co-Authored-By: Claude Sonnet 4.5 <noreply@anthropic.com>

pkg/convert/debruijn_to_lambda.go

@@ -0,0 +1,63 @@
+package convert
+
+import (
+	"fmt"
+
+	"git.maximhutz.com/max/lambda/pkg/debruijn"
+	"git.maximhutz.com/max/lambda/pkg/lambda"
+)
+
+// DeBruijnToLambda converts a De Bruijn expression back to named lambda calculus.
+func DeBruijnToLambda(expr debruijn.Expression) lambda.Expression {
+	return deBruijnToLambda(expr, []string{})
+}
+
+func deBruijnToLambda(expr debruijn.Expression, context []string) lambda.Expression {
+	switch e := expr.(type) {
+	case *debruijn.Variable:
+		if e.Index() >= 0 && e.Index() < len(context) {
+			return lambda.NewVariable(context[e.Index()])
+		}
+		if e.Label() != "" {
+			return lambda.NewVariable(e.Label())
+		}
+		return lambda.NewVariable(fmt.Sprintf("free_%d", e.Index()))
+
+	case *debruijn.Abstraction:
+		paramName := generateParamName(context)
+		newContext := append([]string{paramName}, context...)
+		body := deBruijnToLambda(e.Body(), newContext)
+		return lambda.NewAbstraction(paramName, body)
+
+	case *debruijn.Application:
+		abs := deBruijnToLambda(e.Abstraction(), context)
+		arg := deBruijnToLambda(e.Argument(), context)
+		return lambda.NewApplication(abs, arg)
+
+	default:
+		return nil
+	}
+}
+
+// generateParamName generates a fresh parameter name that doesn't conflict with context.
+func generateParamName(context []string) string {
+	base := 'a'
+	for i := 0; ; i++ {
+		name := string(rune(base + rune(i%26)))
+		if i >= 26 {
+			name = fmt.Sprintf("%s%d", name, i/26)
+		}
+
+		conflict := false
+		for _, existing := range context {
+			if existing == name {
+				conflict = true
+				break
+			}
+		}
+
+		if !conflict {
+			return name
+		}
+	}
+}

pkg/convert/lambda_to_debruijn.go

@@ -0,0 +1,43 @@
+package convert
+
+import (
+	"git.maximhutz.com/max/lambda/pkg/debruijn"
+	"git.maximhutz.com/max/lambda/pkg/lambda"
+)
+
+// LambdaToDeBruijn converts a lambda expression to De Bruijn index representation.
+func LambdaToDeBruijn(expr lambda.Expression) debruijn.Expression {
+	return lambdaToDeBruijn(expr, []string{})
+}
+
+func lambdaToDeBruijn(expr lambda.Expression, context []string) debruijn.Expression {
+	switch e := expr.(type) {
+	case *lambda.Variable:
+		index := findIndex(e.Value(), context)
+		return debruijn.NewVariable(index, e.Value())
+
+	case *lambda.Abstraction:
+		newContext := append([]string{e.Parameter()}, context...)
+		body := lambdaToDeBruijn(e.Body(), newContext)
+		return debruijn.NewAbstraction(body)
+
+	case *lambda.Application:
+		abs := lambdaToDeBruijn(e.Abstraction(), context)
+		arg := lambdaToDeBruijn(e.Argument(), context)
+		return debruijn.NewApplication(abs, arg)
+
+	default:
+		return nil
+	}
+}
+
+// findIndex returns the De Bruijn index for a variable name in the context.
+// Returns the index if found, or -1 if the variable is free.
+func findIndex(name string, context []string) int {
+	for i, v := range context {
+		if v == name {
+			return i
+		}
+	}
+	return -1
+}

pkg/convert/saccharine_to_debruijn.go

@@ -0,0 +1,12 @@
+package convert
+
+import (
+	"git.maximhutz.com/max/lambda/pkg/debruijn"
+	"git.maximhutz.com/max/lambda/pkg/saccharine"
+)
+
+// SaccharineToDeBruijn converts a saccharine expression directly to De Bruijn indices.
+func SaccharineToDeBruijn(expr saccharine.Expression) debruijn.Expression {
+	lambdaExpr := SaccharineToLambda(expr)
+	return LambdaToDeBruijn(lambdaExpr)
+}

pkg/debruijn/expression.go

@@ -0,0 +1,77 @@
+package debruijn
+
+type Expression interface {
+	Accept(Visitor)
+}
+
+/** ------------------------------------------------------------------------- */
+
+type Abstraction struct {
+	body Expression
+}
+
+func (a *Abstraction) Body() Expression {
+	return a.body
+}
+
+func (a *Abstraction) Accept(v Visitor) {
+	v.VisitAbstraction(a)
+}
+
+func NewAbstraction(body Expression) *Abstraction {
+	return &Abstraction{body: body}
+}
+
+/** ------------------------------------------------------------------------- */
+
+type Application struct {
+	abstraction Expression
+	argument    Expression
+}
+
+func (a *Application) Abstraction() Expression {
+	return a.abstraction
+}
+
+func (a *Application) Argument() Expression {
+	return a.argument
+}
+
+func (a *Application) Accept(v Visitor) {
+	v.VisitApplication(a)
+}
+
+func NewApplication(abstraction Expression, argument Expression) *Application {
+	return &Application{abstraction: abstraction, argument: argument}
+}
+
+/** ------------------------------------------------------------------------- */
+
+type Variable struct {
+	index int
+	label string
+}
+
+func (v *Variable) Index() int {
+	return v.index
+}
+
+func (v *Variable) Label() string {
+	return v.label
+}
+
+func (v *Variable) Accept(visitor Visitor) {
+	visitor.VisitVariable(v)
+}
+
+func NewVariable(index int, label string) *Variable {
+	return &Variable{index: index, label: label}
+}
+
+/** ------------------------------------------------------------------------- */
+
+type Visitor interface {
+	VisitAbstraction(*Abstraction)
+	VisitApplication(*Application)
+	VisitVariable(*Variable)
+}

pkg/debruijn/iterator.go

@@ -0,0 +1,68 @@
+package debruijn
+
+type Iterator struct {
+	trace []*Expression
+}
+
+func NewIterator(expr *Expression) *Iterator {
+	return &Iterator{[]*Expression{expr}}
+}
+
+func (i *Iterator) Done() bool {
+	return len(i.trace) == 0
+}
+
+func (i *Iterator) Current() *Expression {
+	if i.Done() {
+		return nil
+	}
+
+	return i.trace[len(i.trace)-1]
+}
+
+func (i *Iterator) Parent() *Expression {
+	if len(i.trace) < 2 {
+		return nil
+	}
+
+	return i.trace[len(i.trace)-2]
+}
+
+func (i *Iterator) Swap(with Expression) {
+	current := i.Current()
+	if current != nil {
+		*current = with
+	}
+}
+
+func (i *Iterator) Back() bool {
+	if i.Done() {
+		return false
+	}
+
+	i.trace = i.trace[:len(i.trace)-1]
+	return true
+}
+
+func (i *Iterator) Next() {
+	switch typed := (*i.Current()).(type) {
+	case *Abstraction:
+		i.trace = append(i.trace, &typed.body)
+	case *Application:
+		i.trace = append(i.trace, &typed.abstraction)
+	case *Variable:
+		for len(i.trace) > 1 {
+			if app, ok := (*i.Parent()).(*Application); ok {
+				if app.abstraction == *i.Current() {
+					i.Back()
+					i.trace = append(i.trace, &app.argument)
+					return
+				}
+			}
+
+			i.Back()
+		}
+
+		i.trace = []*Expression{}
+	}
+}

pkg/debruijn/reduce.go

@@ -0,0 +1,30 @@
+package debruijn
+
+func IsViable(e *Expression) (*Abstraction, Expression, bool) {
+	if e == nil {
+		return nil, nil, false
+	} else if app, appOk := (*e).(*Application); !appOk {
+		return nil, nil, false
+	} else if fn, fnOk := app.abstraction.(*Abstraction); !fnOk {
+		return nil, nil, false
+	} else {
+		return fn, app.argument, true
+	}
+}
+
+func ReduceAll(e *Expression, step func()) {
+	it := NewIterator(e)
+
+	for !it.Done() {
+		if fn, arg, ok := IsViable(it.Current()); !ok {
+			it.Next()
+		} else {
+			it.Swap(Substitute(fn.body, arg))
+			step()
+
+			if _, _, ok := IsViable(it.Parent()); ok {
+				it.Back()
+			}
+		}
+	}
+}

pkg/debruijn/stringify.go

@@ -0,0 +1,38 @@
+package debruijn
+
+import (
+	"fmt"
+	"strings"
+)
+
+type stringifyVisitor struct {
+	builder strings.Builder
+}
+
+func (v *stringifyVisitor) VisitVariable(a *Variable) {
+	if a.label != "" {
+		v.builder.WriteString(a.label)
+	} else {
+		v.builder.WriteString(fmt.Sprintf("%d", a.index))
+	}
+}
+
+func (v *stringifyVisitor) VisitAbstraction(f *Abstraction) {
+	v.builder.WriteRune('\\')
+	v.builder.WriteRune('.')
+	f.body.Accept(v)
+}
+
+func (v *stringifyVisitor) VisitApplication(c *Application) {
+	v.builder.WriteRune('(')
+	c.abstraction.Accept(v)
+	v.builder.WriteRune(' ')
+	c.argument.Accept(v)
+	v.builder.WriteRune(')')
+}
+
+func Stringify(e Expression) string {
+	b := &stringifyVisitor{builder: strings.Builder{}}
+	e.Accept(b)
+	return b.builder.String()
+}

pkg/debruijn/substitute.go

@@ -0,0 +1,68 @@
+package debruijn
+
+// Shift increments all free variable indices by delta when crossing depth abstractions.
+func Shift(expr Expression, delta int, depth int) Expression {
+	switch e := expr.(type) {
+	case *Variable:
+		if e.index >= depth {
+			return NewVariable(e.index+delta, e.label)
+		}
+		return e
+
+	case *Abstraction:
+		newBody := Shift(e.body, delta, depth+1)
+		if newBody == e.body {
+			return e
+		}
+		return NewAbstraction(newBody)
+
+	case *Application:
+		newAbs := Shift(e.abstraction, delta, depth)
+		newArg := Shift(e.argument, delta, depth)
+		if newAbs == e.abstraction && newArg == e.argument {
+			return e
+		}
+		return NewApplication(newAbs, newArg)
+
+	default:
+		return expr
+	}
+}
+
+// Substitute replaces variable at index 0 with replacement in expr.
+// This assumes expr is the body of an abstraction being applied.
+func Substitute(expr Expression, replacement Expression) Expression {
+	return substitute(expr, 0, replacement)
+}
+
+// substitute replaces variable at targetIndex with replacement, adjusting indices as needed.
+func substitute(expr Expression, targetIndex int, replacement Expression) Expression {
+	switch e := expr.(type) {
+	case *Variable:
+		if e.index == targetIndex {
+			return Shift(replacement, targetIndex, 0)
+		}
+		if e.index > targetIndex {
+			return NewVariable(e.index-1, e.label)
+		}
+		return e
+
+	case *Abstraction:
+		newBody := substitute(e.body, targetIndex+1, replacement)
+		if newBody == e.body {
+			return e
+		}
+		return NewAbstraction(newBody)
+
+	case *Application:
+		newAbs := substitute(e.abstraction, targetIndex, replacement)
+		newArg := substitute(e.argument, targetIndex, replacement)
+		if newAbs == e.abstraction && newArg == e.argument {
+			return e
+		}
+		return NewApplication(newAbs, newArg)
+
+	default:
+		return expr
+	}
+}