src/complex/__test__/type.spec.js

@@ -1,7 +1,7 @@
 import assert from 'assert'
 import math from '#nanomath'
 import {Complex} from '../Complex.js'
-import {OneOf, ReturnTyping} from '#core/Type.js'
+import {OneOf, ReturnType, ReturnTyping} from '#core/Type.js'
 import {NumberT} from '#number/NumberT.js'
 
 
@@ -38,13 +38,13 @@ describe('complex type operations', () => {
    })
    it('computes cis of an angle', () => {
       const cis = math.cis.resolve(NumberT)
-      assert.strictEqual(cis.returns, OneOf(Complex(NumberT), NumberT))
+         assert.strictEqual(ReturnType(cis), OneOf(Complex(NumberT), NumberT))
       assert.strictEqual(cis(0), 1)
       assert.strictEqual(cis(Math.PI), -1)
       assert(math.equal(cis(Math.PI/3), cplx(0.5, Math.sqrt(3)/2)))
       math.config.returnTyping = ReturnTyping.full
       const ccis = math.cis.resolve(NumberT)
-      assert.strictEqual(ccis.returns, Complex(NumberT))
+      assert.strictEqual(ReturnType(ccis), Complex(NumberT))
       const one = ccis(0)
       assert(one !== 1)
       assert(math.equal(one, 1))

src/complex/arithmetic.js

@@ -1,7 +1,7 @@
 import {Complex} from './Complex.js'
 import {maybeComplex, promoteBinary, promoteUnary} from './helpers.js'
 import {ResolutionError} from '#core/helpers.js'
-import {Returns, ReturnTyping} from '#core/Type.js'
+import {Returns, ReturnType, ReturnTyping} from '#core/Type.js'
 import {match} from '#core/TypePatterns.js'
 import {ReturnsAs} from '#generic/helpers.js'
 
@@ -9,7 +9,7 @@ const {conservative, full, free} = ReturnTyping
 
 export const normsq = match(Complex, (math, C, strategy) => {
    const compNormsq = math.normsq.resolve(C.Component, full)
-   const R = compNormsq.returns
+   const R = ReturnType(compNormsq)
    const add = math.add.resolve([R,R], strategy)
    return ReturnsAs(add, z => add(compNormsq(z.re), compNormsq(z.im)))
 })
@@ -19,19 +19,21 @@ export const add = promoteBinary('add')
 export const conj = match(Complex, (math, C, strategy) => {
    const neg = math.negate.resolve(C.Component, full)
    const compConj = math.conj.resolve(C.Component, full)
-   const cplx = maybeComplex(math, strategy, compConj.returns, neg.returns)
+   const cplx = maybeComplex(
+      math, strategy, ReturnType(compConj), ReturnType(neg))
    return ReturnsAs(cplx, z => cplx(compConj(z.re), neg(z.im)))
 })
 
 export const divide = [
    match([Complex, T => !T.complex], (math, [C, R], strategy) => {
       const div = math.divide.resolve([C.Component, R], full)
-      const cplx = maybeComplex(math, strategy, div.returns, div.returns)
+      const NewComp = ReturnType(div)
+      const cplx = maybeComplex(math, strategy, NewComp, NewComp)
       return ReturnsAs(cplx, (z, r) => cplx(div(z.re, r), div(z.im, r)))
    }),
    match([Complex, Complex], (math, [W, Z], strategy) => {
       const inv = math.invert.resolve(Z, full)
-      const mult = math.multiply.resolve([W, inv.returns], strategy)
+      const mult = math.multiply.resolve([W, ReturnType(inv)], strategy)
       return ReturnsAs(mult, (w, z) => mult(w, inv(z)))
    })
 ]
@@ -39,8 +41,9 @@ export const divide = [
 export const invert = match(Complex, (math, C, strategy) => {
    const conj = math.conj.resolve(C, full)
    const normsq = math.normsq.resolve(C, full)
-   const div = math.divide.resolve([C.Component, normsq.returns], full)
+   const div = math.divide.resolve([C.Component, ReturnType(normsq)], full)
-   const cplx = maybeComplex(math, strategy, div.returns, div.returns)
+   const NewComp = ReturnType(div)
+   const cplx = maybeComplex(math, strategy, NewComp, NewComp)
    return ReturnsAs(cplx, z => {
       const c = conj(z)
       const d = normsq(z)
@@ -53,7 +56,8 @@ export const invert = match(Complex, (math, C, strategy) => {
 export const multiply = [
    match([T => !T.complex, Complex], (math, [R, C], strategy) => {
       const mult = math.multiply.resolve([R, C.Component], full)
-      const cplx = maybeComplex(math, strategy, mult.returns, mult.returns)
+      const NewComp = ReturnType(mult)
+      const cplx = maybeComplex(math, strategy, NewComp, NewComp)
       return ReturnsAs(cplx, (r, z) => cplx(mult(r, z.re), mult(r, z.im)))
    }),
    match([Complex, T => !T.complex], (math, [C, R], strategy) => {
@@ -62,15 +66,19 @@ export const multiply = [
    }),
    match([Complex, Complex], (math, [W, Z], strategy) => {
       const conj = math.conj.resolve(W.Component, full)
-      if (conj.returns !== W.Component) {
+      if (ReturnType(conj) !== W.Component) {
          throw new ResolutionError(
-            `conjugation on ${W.Component} returns type (${conj.returns})`)
+            `conjugation on ${W.Component} returns different type `
+            + `(${ReturnType(conj)})`)
       }
       const mWZ = math.multiply.resolve([W.Component, Z.Component], full)
       const mZW = math.multiply.resolve([Z.Component, W.Component], full)
-      const sub = math.subtract.resolve([mWZ.returns, mZW.returns], full)
+      const TWZ = ReturnType(mWZ)
-      const add = math.add.resolve([mWZ.returns, mZW.returns], full)
+      const TZW = ReturnType(mZW)
-      const cplx = maybeComplex(math, strategy, sub.returns, add.returns)
+      const sub = math.subtract.resolve([TWZ, TZW], full)
+      const add = math.add.resolve([TWZ, TZW], full)
+      const cplx = maybeComplex(
+         math, strategy, ReturnType(sub), ReturnType(add))
       return ReturnsAs(cplx, (w, z) => {
          const real = sub(mWZ(     w.re,  z.re), mZW(z.im, conj(w.im)))
          const imag = add(mWZ(conj(w.re), z.im), mZW(z.re,      w.im))
@@ -85,7 +93,7 @@ export const negate = promoteUnary('negate')
 // integer coordinates.
 export const sqrt = match(Complex, (math, C, strategy) => {
    const re = math.re.resolve(C)
-   const R = re.returns
+   const R = ReturnType(re)
    const isReal = math.isReal.resolve(C)
    // dependencies for the real case:
    const zComp = math.zero(C.Component)
@@ -98,8 +106,8 @@ export const sqrt = match(Complex, (math, C, strategy) => {
    const cplx = math.complex.resolve([C.Component, C.Component], full)
    // additional dependencies for the complex case
    const abs = math.abs.resolve(C, full)
-   if (abs.returns !== R) {
+   if (ReturnType(abs) !== R) {
-      throw new TypeError(`abs on ${C} returns ${abs.returns}, not ${R}`)
+      throw new TypeError(`abs on ${C} returns ${ReturnType(abs)}, not ${R}`)
    }
    const addRR = math.add.resolve([R, R], conservative)
    const twoR = addRR(oneR, oneR)

src/complex/helpers.js

@@ -1,5 +1,5 @@
 import {Complex} from './Complex.js'
-import {ReturnTyping} from '#core/Type.js'
+import {ReturnType, ReturnTyping} from '#core/Type.js'
 import {match} from '#core/TypePatterns.js'
 import {ReturnsAs} from '#generic/helpers.js'
 
@@ -8,7 +8,7 @@ const {free, full} = ReturnTyping
 export const maybeComplex = (math, strategy, Real, Imag) => {
    if (strategy !== free) return math.complex.resolve([Real, Imag], strategy)
    const cplx = math.complex.resolve([Real, Imag], full)
-   const prune = math.pruneImaginary.resolve(cplx.returns, full)
+   const prune = math.pruneImaginary.resolve(ReturnType(cplx), full)
    return ReturnsAs(prune, (r, m) => prune(cplx(r, m)))
 }
 
@@ -17,7 +17,8 @@ export const promoteUnary = (name, overrideStrategy) => match(
    (math, C, strategy) => {
       const compOp = math.resolve(name, C.Component, full)
       if (overrideStrategy) strategy = overrideStrategy
-      const cplx = maybeComplex(math, strategy, compOp.returns, compOp.returns)
+      const NewComp = ReturnType(compOp)
+      const cplx = maybeComplex(math, strategy, NewComp, NewComp)
       return ReturnsAs(cplx, z => cplx(compOp(z.re), compOp(z.im)))
    })
 
@@ -32,7 +33,8 @@ export const promoteBinary = name => match(
    [Complex, Complex],
    (math, [W, Z], strategy) => {
       const compOp = math.resolve(name, [W.Component, Z.Component], full)
-      const cplx = maybeComplex(math, strategy, compOp.returns, compOp.returns)
+      const NewComp = ReturnType(compOp)
+      const cplx = maybeComplex(math, strategy, NewComp, NewComp)
       return ReturnsAs(
          cplx, (w, z) => cplx(compOp(w.re, z.re), compOp(w.im, z.im)))
    })

src/complex/type.js

@@ -1,5 +1,7 @@
 import {Complex} from './Complex.js'
-import {OneOf, Returns, ReturnTyping, TypeOfTypes} from "#core/Type.js"
+import {
+   OneOf, Returns, ReturnType, ReturnTyping, TypeOfTypes
+} from "#core/Type.js"
 import {Any, match} from "#core/TypePatterns.js"
 import {BooleanT} from '#boolean/BooleanT.js'
 import {NumberT} from '#number/NumberT.js'
@@ -32,7 +34,7 @@ export const complex = [
 export const arg = // [  // enable when we have atan2 in mathjs
 //   match(Complex, (math, C) => {
 //      const re = math.re.resolve(C)
-//      const R = re.returns
+//      const R = ReturnType(re)
 //      const im = math.im.resolve(C)
 //      const abs = math.abs.resolve(C)
 //      const atan2 = math.atan2.resolve([R, R], conservative)
@@ -54,11 +56,11 @@ export const associate = match(
       }
       const eq = math.equal.resolve([W, Z], full)
       const neg = math.negate.resolve(Z, full)
-      const eqN = math.equal.resolve([W, neg.returns], full)
+      const eqN = math.equal.resolve([W, ReturnType(neg)], full)
       const mult = math.multiply.resolve([Z, Z], full)
-      const eqM = math.equal.resolve([W, mult.returns], full)
+      const eqM = math.equal.resolve([W, ReturnType(mult)], full)
-      const negM = math.negate.resolve(mult.returns, full)
+      const negM = math.negate.resolve(ReturnType(mult), full)
-      const eqNM = math.equal.resolve([W, negM.returns], full)
+      const eqNM = math.equal.resolve([W, ReturnType(negM)], full)
       const iZ = math.complex(math.zero(Z.Component), math.one(Z.Component))
       return Returns(BooleanT, (w, z) => {
          if (eq(w, z) || eqN(w, neg(z))) return true

src/core/README.md

@@ -13,12 +13,16 @@ As of this writing, the only two types required to be in a TypeDispatcher are
 Undefined (the type inhabited only by `undefined`) and TypeOfTypes (the type
 inhabited exactly by Type objects).
 
-There is also a constant NotAType which is the type-world analogue of NaN for
+There is also a constant Unknown which is the type that is used when it is
-numbers. It is occasionally used for the rare behavior that truly does not
+not possible in advance to determine what the type of an entity is or will
+be. It is, for example, used for the occasional behavior that truly does not
 return any particular type, such as the method `zero` that takes a Type and
-returns its zero element. However, it does not really work as a Type, and in
+returns its zero element. It is also used for the component type of instances
-particular, do _not_ merge it into any TypeDispatcher -- it will disrupt the
+of containers, like Vector, that have inhomogeneous contents -- and therefore,
-type and method resolution process.
+as the return type of `sum` on a `Vector(Unknown)`. However, it lacks much of
+the machinery of other Type entities. In particular, do _not_ attempt to merge
+Unknown as a type into any TypeDispatcher -- it will disrupt the type and
+method resolution process.
 
 ## Core methods
 

src/core/Type.js

@@ -66,8 +66,8 @@ export const Undefined = new Type(
    t => typeof t === 'undefined',
    {zero: undefined, one: undefined, nan: undefined})
 export const TypeOfTypes = new Type(t => t instanceof Type)
-export const NotAType = new Type(() => true) // Danger, do not merge!
+export const Unknown = new Type(() => true) // Danger, do not merge!
-NotAType._doNotMerge = true
+Unknown._doNotMerge = true
 
 const unionDirectory = new ArrayKeyedMap() // make sure only one of each union
 export const OneOf = (...types) => {
@@ -101,7 +101,8 @@ export const OneOf = (...types) => {
    return unionDirectory.get(typeList)
 }
 
-export const Returns = (type, f) => (f.returns = type, f)
+export const Returns = (type, f) => (f.returns = type, f.isBehavior = true, f)
+export const ReturnType = f => f.returns ?? Unknown
 
 export const whichType = typs => Returns(TypeOfTypes, item => {
    for (const type of Object.values(typs)) {

src/core/TypeDispatcher.js

@@ -3,7 +3,9 @@ import ArrayKeyedMap from 'array-keyed-map'
 import {ResolutionError, isPlainFunction, isPlainObject} from './helpers.js'
 import {Implementations, ImplementationsGenerator} from './Implementations.js'
 import {bootstrapTypes} from './type.js'
-import {Returns, ReturnTyping, whichType, Type} from './Type.js'
+import {
+   Returns, ReturnType, ReturnTyping, Unknown, Type, whichType
+} from './Type.js'
 import {
    matched, needsCollection, Passthru, Matcher, match
 } from './TypePatterns.js'
@@ -253,7 +255,7 @@ export class TypeDispatcher {
          if ('actual' in template) { // incorporate conversion
             let convert = template.convertor
             // Check if it's a factory:
-            if (!convert.returns) convert = convert(this, template.actual)
+            if (!convert.isBehavior) convert = convert(this, template.actual)
             extractor = args => convert(args[from])
          }
          return state ? extractor : args => [extractor(args)]
@@ -273,6 +275,14 @@ export class TypeDispatcher {
          throw new ReferenceError(`no method or value for key '${key}'`)
       }
       if (!Array.isArray(types)) types = [types]
+      if (types.some(T => T === Unknown)) {
+         if (!strategy) return this[key]
+         const thisTypeOf = whichType(this.types)
+         return (...args) => {
+            const types = args.map(thisTypeOf)
+            return this.resolve(key, types, strategy)(...args)
+         }
+      }
       if (!strategy) {
          // Avoid recursing on obtaining config
          if (key === 'config') strategy = ReturnTyping.free
@@ -380,12 +390,7 @@ export class TypeDispatcher {
 
          finalBehavior = theBehavior
          if (typeof theBehavior === 'function') {
-            const returning = theBehavior.returns
+            const returning = ReturnType(theBehavior)
-            if (!returning) {
-               throw new TypeError(
-                  `No return type specified for ${key} on ${types} with`
-                  + ` return typing ${ReturnTyping.name(strategy)}`)
-            }
             if (needsCollection(template)) {
                // have to wrap the behavior to collect the actual arguments
                // in the way corresponding to the template. Generating that

src/core/TypePatterns.js

@@ -1,4 +1,4 @@
-import {Type, Undefined} from './Type.js'
+import {ReturnType, Type, Undefined, Unknown} from './Type.js'
 import {isPlainFunction} from './helpers.js'
 
 export class TypePattern {
@@ -189,8 +189,10 @@ export const matched = (template, math) => {
    }
    if (template.matched) {
       let convert = template.convertor
-      if (!convert.returns) convert = convert(math, template.actual)
+      if (!convert.isBehavior) convert = convert(math, template.actual)
-      return convert.returns || template.matched
+      const ConvertsTo = ReturnType(convert)
+      if (ConvertsTo !== Unknown) return ConvertsTo
+      return template.matched
    }
    return template
 }

src/core/__test__/Type.spec.js

@@ -2,7 +2,7 @@ import assert from 'assert'
 import math from '#nanomath'
 import {NumberT} from '#number/NumberT.js'
 
-import {Returns, ReturnTyping} from '../Type.js'
+import {Returns, ReturnType, ReturnTyping} from '../Type.js'
 import {isPlainFunction} from '../helpers.js'
 
 describe('Core types', () => {
@@ -36,7 +36,7 @@ describe('Core types', () => {
    it('provides a return-value labeling', () => {
       const labeledF = Returns(math.types.Undefined, () => undefined)
       assert.strictEqual(typeof labeledF, 'function')
-      assert.strictEqual(labeledF.returns, math.types.Undefined)
+      assert.strictEqual(ReturnType(labeledF), math.types.Undefined)
       assert(isPlainFunction(labeledF))
    })
 

src/core/__test__/TypeDispatcher.spec.js

@@ -6,7 +6,7 @@ import * as numbers from '#number/all.js'
 import {NumberT} from '#number/NumberT.js'
 import {ResolutionError} from "#core/helpers.js"
 import {match, Any} from "#core/TypePatterns.js"
-import {NotAType, Returns, ReturnTyping} from "#core/Type.js"
+import {Returns, ReturnType, ReturnTyping, Unknown} from "#core/Type.js"
 import {plain} from "#number/helpers.js"
 
 describe('TypeDispatcher', () => {
@@ -18,7 +18,7 @@ describe('TypeDispatcher', () => {
       const {NumberT, TypeOfTypes, Undefined} = incremental.types
       assert(NumberT.test(7))
       assert.strictEqual(incremental.add(-1.5, 0.5), -1)
-      assert.throws(() => incremental.add(7, undefined), ResolutionError)
+      assert.throws(() => incremental.add(7, NumberT), ResolutionError)
       // Make Undefined act like zero:
       incremental.merge({add: [
          match([Undefined, Any], (_m, [_U, T]) => Returns(T, (_a, b) => b)),
@@ -26,11 +26,11 @@ describe('TypeDispatcher', () => {
       ]})
       assert.strictEqual(incremental.add(7, undefined), 7)
       assert.strictEqual(
-         incremental.resolve('add', [TypeOfTypes, Undefined]).returns,
+         ReturnType(incremental.resolve('add', [TypeOfTypes, Undefined])),
          TypeOfTypes)
       assert.strictEqual(incremental.add(undefined, -3.25), -3.25)
       assert.strictEqual(
-         incremental.add.resolve([Undefined, NumberT]).returns,
+         ReturnType(incremental.add.resolve([Undefined, NumberT])),
          NumberT)
       // Oops, changed my mind ;-), make it work like NaN with numbers:
       const alwaysNaN = Returns(NumberT, () => NaN)
@@ -40,7 +40,7 @@ describe('TypeDispatcher', () => {
       ]})
       assert(isNaN(incremental.add(undefined, -3.25)))
       assert.strictEqual(
-         incremental.add.resolve([Undefined, NumberT]).returns,
+         ReturnType(incremental.add.resolve([Undefined, NumberT])),
          NumberT)
       assert.strictEqual(incremental.isnan(NaN), 1)
       incremental.merge(booleans)
@@ -68,9 +68,9 @@ describe('TypeDispatcher', () => {
       assert(!bgn._behaviors.negate.has([ReturnTyping.free, BooleanT]))
       assert.strictEqual(bgn.negate(true), -2)
    })
-   it('disallows merging NotAType', () => {
+   it('disallows merging Unknown', () => {
       const doomed = new TypeDispatcher()
-      assert.throws(() => doomed.merge({NaT: NotAType}), TypeError)
+      assert.throws(() => doomed.merge({Unknown}), TypeError)
    })
    it('supports generic types', () => {
       assert.throws(() => NumberT(0), TypeError)

src/coretypes/utils.js

@@ -1,15 +1,23 @@
-import {NotAType, Returns, TypeOfTypes} from '#core/Type.js'
+import {Returns, TypeOfTypes, Unknown} from '#core/Type.js'
 import {match,Any} from "#core/TypePatterns.js"
 import {boolnum} from "#number/helpers.js"
 
+export const haszero = [
+   match(Any, (math, T) => {
+      const answer = math.haszero(T)
+      return Returns(math.typeOf(answer), () => answer)
+   }),
+   match(TypeOfTypes, boolnum(T => 'zero' in T))
+]
+
 export const zero = [
    match(Any, (math, T) => {
       const z = math.zero(T)
       return Returns(T, () => z)
    }),
-   match(TypeOfTypes, Returns(NotAType, t => {
+   match(TypeOfTypes, Returns(Unknown, T => {
-      if ('zero' in t) return t.zero
+      if ('zero' in T) return T.zero
-      throw new RangeError(`type '${t}' has no zero element`)
+      throw new RangeError(`type '${T}' has no zero element`)
    }))
 ]
 
@@ -18,10 +26,10 @@ export const one = [
       const unit = math.one(T)
       return Returns(T, () => unit)
    }),
-   match(TypeOfTypes, Returns(NotAType, t => {
+   match(TypeOfTypes, Returns(Unknown, T => {
-      if ('one' in t) return t.one
+      if ('one' in T) return T.one
       throw new RangeError(
-         `type '${t}' has no unit element designated as "one"`)
+         `type '${T}' has no unit element designated as "one"`)
    }))
 ]
 
@@ -30,7 +38,7 @@ export const hasnan = [
       const answer = math.hasnan(T)
       return Returns(math.typeOf(answer), () => answer)
    }),
-   match(TypeOfTypes, boolnum(t => 'nan' in t))
+   match(TypeOfTypes, boolnum(T => 'nan' in T))
 ]
 
 export const nan = [
@@ -38,9 +46,9 @@ export const nan = [
       const notanum = math.nan(T)
       return Returns(T, () => notanum)
    }),
-   match(TypeOfTypes, Returns(NotAType, t => {
+   match(TypeOfTypes, Returns(Unknown, T => {
-      if ('nan' in t) return t.nan
+      if ('nan' in T) return T.nan
       throw new RangeError(
-         `type '${t}' has no "not a number" element`)
+         `type '${T}' has no "not a number" element`)
    }))
 ]

src/generic/__test__/arithmetic.spec.js

@@ -1,6 +1,6 @@
 import assert from 'assert'
 import math from '#nanomath'
-import {ReturnTyping} from '#core/Type.js'
+import {ReturnType, ReturnTyping} from '#core/Type.js'
 
 const {Complex, NumberT} = math.types
 
@@ -8,7 +8,7 @@ describe('generic arithmetic', () => {
    it('squares anything', () => {
       const sq = math.square
       assert.strictEqual(sq(7), 49)
-      assert.strictEqual(math.square.resolve([NumberT]).returns, NumberT)
+      assert.strictEqual(ReturnType(math.square.resolve([NumberT])), NumberT)
       assert.deepStrictEqual(sq(math.complex(3, 4)), math.complex(-7, 24))
       const eyes = math.complex(0, 2)
       assert.strictEqual(sq(eyes), -4)

src/generic/arithmetic.js

@@ -1,18 +1,20 @@
 import {ReturnsAs} from './helpers.js'
 
-import {Returns, ReturnTyping} from '#core/Type.js'
+import {Returns, ReturnType, ReturnTyping} from '#core/Type.js'
 import {match, Any} from '#core/TypePatterns.js'
 
 export const norm = match(Any, (math, T) => {
    const normsq = math.normsq.resolve(T)
-   const sqrt = math.sqrt.resolve(normsq.returns, ReturnTyping.conservative)
+   const sqrt = math.sqrt.resolve(
+      ReturnType(normsq), ReturnTyping.conservative)
    return ReturnsAs(sqrt, t => sqrt(normsq(t)))
 })
 
 export const abs = norm // coincide for most types (scalars)
 
 export const conj = match(Any, (_math, T) => Returns(T, t => t))
+
 export const square = match(Any, (math, T, strategy) => {
    const mult = math.multiply.resolve([T, T], strategy)
-   return Returns(mult.returns, t => mult(t, t))
+   return ReturnsAs(mult, t => mult(t, t))
 })

src/generic/helpers.js

@@ -1 +1,3 @@
-export const ReturnsAs = (g, f) => (f.returns = g.returns, f)
+import {Returns, ReturnType} from '#core/Type.js'
+
+export const ReturnsAs = (g, f) => Returns(ReturnType(g), f)

src/generic/logical.js

@@ -1,6 +1,6 @@
 import {ReturnsAs} from './helpers.js'
 
-import {OneOf, Returns} from '#core/Type.js'
+import {OneOf, Returns, ReturnType} from '#core/Type.js'
 import {Any, Multiple, match} from '#core/TypePatterns.js'
 import {boolnum} from '#number/helpers.js'
 
@@ -18,7 +18,7 @@ export const and = [
    }),
    match([Any, Any, Any, Multiple(Any)], (math, [T, U, V, Rest]) => {
       const andRest = math.and.resolve([U, V, ...Rest])
-      const andFirst = math.and.resolve([T, andRest.returns])
+      const andFirst = math.and.resolve([T, ReturnType(andRest)])
       return ReturnsAs(
          andFirst, (t, u, v, rest) => andFirst(t, andRest(u, v, ...rest)))
    })
@@ -33,7 +33,7 @@ export const or = [
    }),
    match([Any, Any, Any, Multiple(Any)], (math, [T, U, V, Rest]) => {
       const orRest = math.or.resolve([U, V, ...Rest])
-      const orFirst = math.and.resolve([T, orRest.returns])
+      const orFirst = math.and.resolve([T, ReturnType(orRest)])
       return ReturnsAs(
          orFirst, (t, u, v, rest) => orFirst(t, orRest(u, v, ...rest)))
    })

src/generic/relational.js

@@ -1,5 +1,5 @@
 import {ReturnsAs} from './helpers.js'
-import {Returns, ReturnTyping} from '#core/Type.js'
+import {Returns, ReturnType, ReturnTyping} from '#core/Type.js'
 import {Any, Passthru, match, matched} from '#core/TypePatterns.js'
 import {boolnum} from '#number/helpers.js'
 
@@ -82,15 +82,15 @@ export const sign = match(Any, (math, T) => {
 
 export const unequal = match(Passthru, (math, types) => {
    const eq = math.equal.resolve(types)
-   const not = math.not.resolve(eq.returns)
+   const not = math.not.resolve(ReturnType(eq))
    return ReturnsAs(not, (...args) => not(eq(...args)))
 })
 
 export const larger = match([Any, Any], (math, [T, U]) => {
    const eq = math.equal.resolve([T, U])
    const bigger = math.exceeds.resolve([T, U])
-   const not = math.not.resolve(eq.returns)
+   const not = math.not.resolve(ReturnType(eq))
-   const and = math.and.resolve([not.returns, bigger.returns])
+   const and = math.and.resolve([ReturnType(not), ReturnType(bigger)])
    return ReturnsAs(and, (t, u) => and(not(eq(t, u)), bigger(t, u)))
 })
 
@@ -103,21 +103,21 @@ export const isPositive = match(Any, (math, T) => {
 export const largerEq = match([Any, Any], (math, [T, U]) => {
    const eq = math.equal.resolve([T, U])
    const bigger = math.exceeds.resolve([T, U])
-   const or = math.or.resolve([eq.returns, bigger.returns])
+   const or = math.or.resolve([ReturnType(eq), ReturnType(bigger)])
    return ReturnsAs(or, (t, u) => or(eq(t, u), bigger(t, u)))
 })
 
 export const smaller = match([Any, Any], (math, [T, U]) => {
    const eq = math.equal.resolve([T, U])
    const bigger = math.exceeds.resolve([U, T])
-   const not = math.not.resolve(eq.returns)
+   const not = math.not.resolve(ReturnType(eq))
-   const and = math.and.resolve([not.returns, bigger.returns])
+   const and = math.and.resolve([ReturnType(not), ReturnType(bigger)])
    return ReturnsAs(and, (t, u) => and(not(eq(t, u)), bigger(u, t)))
 })
 
 export const smallerEq = match([Any, Any], (math, [T, U]) => {
    const eq = math.equal.resolve([T, U])
    const bigger = math.exceeds.resolve([U, T])
-   const or = math.or.resolve([eq.returns, bigger.returns])
+   const or = math.or.resolve([ReturnType(eq), ReturnType(bigger)])
    return ReturnsAs(or, (t, u) => or(eq(t, u), bigger(u, t)))
 })

src/number/arithmetic.js

@@ -1,6 +1,6 @@
 import {plain} from './helpers.js'
 import {NumberT} from './NumberT.js'
-import {OneOf, Returns, ReturnTyping} from '#core/Type.js'
+import {OneOf, Returns, ReturnTyping, Undefined} from '#core/Type.js'
 import {match} from '#core/TypePatterns.js'
 import {Complex} from '#complex/Complex.js'
 
@@ -9,7 +9,11 @@ const {conservative, full} = ReturnTyping
 export const abs = plain(Math.abs)
 export const norm = abs
 export const normsq = plain(a => a*a)
-export const add = plain((a, b) => a + b)
+export const add = [
+   plain((a, b) => a + b),
+   match([Undefined, NumberT], Returns(NumberT, () => NaN)),
+   match([NumberT, Undefined], Returns(NumberT, () => NaN))
+]
 export const divide = plain((a, b) => a / b)
 export const cbrt = plain(a => {
    if (a === 0) return a

src/vector/Vector.js

@@ -1,4 +1,4 @@
-import {NotAType, Type} from '#core/Type.js'
+import {Type, Unknown} from '#core/Type.js'
 
 const isVector = v => Array.isArray(v)
 
@@ -12,7 +12,7 @@ export const Vector = new Type(isVector, {
          if ('zero' in CompType) typeOptions.zero = [CompType.zero]
          const vectorCompType = new Type(specTest, typeOptions)
          vectorCompType.Component = CompType
-         vectorCompType.vector = true
+         vectorCompType.vectorDepth = (CompType.vectorDepth ?? 0) + 1
          return vectorCompType
       }
       // Wrapping a generic type in Vector
@@ -37,12 +37,12 @@ export const Vector = new Type(isVector, {
          }
       })
    },
-   specializesTo: VT => VT.vector,
+   specializesTo: VT => 'vectorDepth' in VT && VT.vectorDepth > 0,
    refine(v, typer) {
-      if (!v.length) return this.specialize(NotAType) // what else could we do?
+      if (!v.length) return this.specialize(Unknown) // what else could we do?
       const eltTypes = v.map(elt => typer(elt))
       let compType = eltTypes[0]
-      if (eltTypes.some(T => T !== compType)) compType = NotAType
+      if (eltTypes.some(T => T !== compType)) compType = Unknown
       return this.specialize(compType)
    }
 })

src/vector/__test__/Vector.spec.js

@@ -1,6 +1,6 @@
 import assert from 'assert'
 import {Vector} from '../Vector.js'
-import {NotAType} from '#core/Type.js'
+import {Unknown} from '#core/Type.js'
 import math from '#nanomath'
 
 describe('Vector Type', () => {
@@ -14,7 +14,7 @@ describe('Vector Type', () => {
       const cplx = math.complex
       assert.strictEqual(typ([3, 4]), Vector(NumberT))
       assert.strictEqual(typ([true, false]), Vector(BooleanT))
-      assert.strictEqual(typ([3, false]), Vector(NotAType))
+      assert.strictEqual(typ([3, false]), Vector(Unknown))
       assert.strictEqual(typ([cplx(3, 4), cplx(5)]), Vector(Complex(NumberT)))
       assert.strictEqual(typ([[3, 4], [5]]), Vector(Vector(NumberT)))
    })

src/vector/__test__/arithmetic.spec.js

@@ -0,0 +1,33 @@
+import assert from 'assert'
+import math from '#nanomath'
+
+describe('Vector arithmetic functions', () => {
+   const cplx = math.complex
+   const cV = [cplx(3, 4), cplx(4, -3), cplx(-3, -4)]
+   it('distributes abs elementwise', () => {
+      const abs = math.abs
+      assert.deepStrictEqual(abs([-3, 4, -5]), [3, 4, 5])
+      assert.deepStrictEqual(abs(cV), [5, 5, 5])
+      assert.deepStrictEqual(abs([true, -4, cplx(4, 3)]), [1, 4, 5])
+   })
+   it('computes the norm of a vector or matrix', () => {
+      const norm = math.norm
+      assert.strictEqual(norm([-3, 4, -5]), Math.sqrt(50))
+      assert.strictEqual(norm([[1, 2], [4, 2]]), 5)
+      assert.strictEqual(norm(cV), Math.sqrt(75))
+   })
+   it('adds vectors and matrices', () => {
+      const add = math.add
+      assert.deepStrictEqual(add([-3, 4, -5], [8, 0, 2]), [5, 4, -3])
+      assert.deepStrictEqual(
+         add([[1, 2], [4, 2]], [[0, -1], [3, -4]]), [[1, 1], [7, -2]])
+      assert.deepStrictEqual(
+         add([[1, 2], [4, 2]], [0, -1]), [[1, 1], [4, 1]])
+   })
+   it('computes the sum of a vector', () => {
+      const sum = math.sum
+      assert.strictEqual(sum([-3, 4, -5]), -4)
+      assert.deepStrictEqual(sum(cV), cplx(4, -3))
+      assert.strictEqual(sum([4, true, -2]), 3)
+   })
+})

src/vector/__test__/type.spec.js

@@ -1,5 +1,5 @@
 import assert from 'assert'
-import {NotAType} from '#core/Type.js'
+import {ReturnType, Unknown} from '#core/Type.js'
 import math from '#nanomath'
 
 describe('Vector type functions', () => {
@@ -8,9 +8,11 @@ describe('Vector type functions', () => {
       const {BooleanT, NumberT, Vector} = math.types
       assert.deepStrictEqual(vec(3, 4, 5), [3, 4, 5])
       assert.strictEqual(
-         vec.resolve([NumberT, NumberT, NumberT]).returns, Vector(NumberT))
+         ReturnType(vec.resolve([NumberT, NumberT, NumberT])),
+         Vector(NumberT))
       assert.deepStrictEqual(vec(3, true), [3, true])
       assert.strictEqual(
-         vec.resolve([NumberT, BooleanT]).returns, Vector(NotAType))
+         ReturnType(vec.resolve([NumberT, BooleanT])),
+         Vector(Unknown))
    })
 })

src/vector/all.js

@@ -1,4 +1,5 @@
 export * as typeDefinition from './Vector.js'
+export * as arithmetic from './arithmetic.js'
 export * as logical from './logical.js'
 export * as relational from './relational.js'
 export * as type from './type.js'

src/vector/arithmetic.js

@@ -0,0 +1,33 @@
+import {promoteBinary, promoteUnary} from './helpers.js'
+import {Vector} from './Vector.js'
+
+import {ReturnType} from '#core/Type.js'
+import {match} from '#core/TypePatterns.js'
+import {ReturnsAs} from '#generic/helpers.js'
+
+export const normsq = match(Vector, (math, V) => {
+   const compNormsq = math.normsq.resolve(V.Component)
+   const sum = math.sum.resolve(Vector(ReturnType(compNormsq)))
+   return ReturnsAs(sum, v => sum(v.map(compNormsq)))
+})
+// abs and norm differ only on Vector (and perhaps other collections) --
+// norm computes overall by the generic formula, whereas abs distributes
+// elementwise:
+export const abs = promoteUnary('abs')
+export const add = promoteBinary('add')
+
+export const sum = match(Vector, (math, V) => {
+   const add = math.add.resolve([V.Component, V.Component])
+   const haszero = math.haszero(V.Component)
+   const zero = haszero ? math.zero(V.Component) : undefined
+   return ReturnsAs(add, v => {
+      if (v.length === 0) {
+         if (haszero) return zero
+         throw new TypeError(`Can't sum empty ${V}: no zero element`)
+      }
+      let ix = 0
+      let retval = v[ix]
+      while (++ix < v.length) retval = add(retval, v[ix])
+      return retval
+   })
+})

src/vector/helpers.js

@@ -1,61 +1,44 @@
 import {Vector} from './Vector.js'
-import {NotAType, Returns, Undefined} from '#core/Type.js'
+import {Returns, ReturnType, Undefined} from '#core/Type.js'
 import {Any, match} from '#core/TypePatterns.js'
 
 export const promoteUnary = name => match(Vector, (math, V, strategy) => {
-   if (V.Component === NotAType) {
-      // have to resolve element by element :-(
-      return Returns(V, v => v.map(
-         elt => math.resolve(name, math.typeOf(elt), strategy)(elt)))
-   }
    const compOp = math.resolve(name, V.Component, strategy)
-   return Returns(Vector(compOp.returns), v => v.map(elt => compOp(elt)))
+   return Returns(Vector(ReturnType(compOp)), v => v.map(elt => compOp(elt)))
 })
 
 export const promoteBinary = name => [
    match([Vector, Any], (math, [V, E], strategy) => {
-      const VComp = V.Component
+      const compOp = math.resolve(name, [V.Component, E], strategy)
-      if (VComp === NotAType) {
-         return Returns(V, (v, e) => v.map(
-            f => math.resolve(name, [math.typeOf(f), E], strategy)(f, e)))
-      }
-      const compOp = math.resolve(name, [VComp, E], strategy)
       return Returns(
-         Vector(compOp.returns), (v, e) => v.map(f => compOp(f, e)))
+         Vector(ReturnType(compOp)), (v, e) => v.map(f => compOp(f, e)))
    }),
    match([Any, Vector], (math, [E, V], strategy) => {
-      const VComp = V.Component
+      const compOp = math.resolve(name, [E, V.Component], strategy)
-      if (VComp === NotAType) {
-         return Returns(V, (e, v) => v.map(
-            f => math.resolve(name, [E, math.typeOf(f)], strategy)(e, f)))
-      }
-      const compOp = math.resolve(name, [E, VComp], strategy)
       return Returns(
-         Vector(compOp.returns, (e, v) => v.map(f => compOp(e, f))))
+         Vector(ReturnType(compOp)), (e, v) => v.map(f => compOp(e, f)))
    }),
    match([Vector, Vector], (math, [V, W], strategy) => {
       const VComp = V.Component
       const WComp = W.Component
-      let compOp
+      // special case: if the vector nesting depths do not match,
-      let opNoV
+      // we operate between the elements of the deeper one and the entire
-      let opNoW
+      // more shallow one:
-      let ReturnType
+      if (V.vectorDepth > W.vectorDepth) {
-      if (VComp === NotAType || WComp === NotAType) {
+         const compOp = math.resolve(name, [VComp, W], strategy)
-         const typ = math.typeOf
+         return Returns(
-         compOp = (v, w) => {
+            Vector(ReturnType(compOp)), (v, w) => v.map(f => compOp(f, w)))
-            return math.resolve(name, [typ(v), typ(w)], strategy)(v, w)
-         }
-         opNoV = compOp
-         opNoW = compOp
-         ReturnType = Vector(NotAType)
-      } else {
-         compOp = math.resolve(name, [VComp, WComp], strategy)
-         opNoV = math.resolve(name, [Undefined, WComp], strategy)
-         opNoW = math.resolve(name, [VComp, Undefined], strategy)
-         ReturnType = Vector(compOp.returns)
       }
+      if (V.vectorDepth < W.vectorDepth) {
+         const compOp = math.resolve(name, [V, WComp], strategy)
+         return Returns(
+            Vector(ReturnType(compOp)), (v, w) => w.map(f => compOp(v, f)))
+      }
+      const compOp = math.resolve(name, [VComp, WComp], strategy)
+      const opNoV = math.resolve(name, [Undefined, WComp], strategy)
+      const opNoW = math.resolve(name, [VComp, Undefined], strategy)
       return Returns(
-         ReturnType,
+         Vector(ReturnType(compOp)),
          (v, w) => {
             const vInc = Number(v.length > 1)
             const wInc = Number(w.length >= v.length || w.length > 1)

src/vector/relational.js

@@ -1,7 +1,7 @@
 import {Vector} from './Vector.js'
 import {promoteBinary} from './helpers.js'
 
-import {NotAType, Returns, Undefined} from '#core/Type.js'
+import {Unknown, Returns, ReturnType, Undefined} from '#core/Type.js'
 import {Any, Optional, match} from '#core/TypePatterns.js'
 import {BooleanT} from '#boolean/BooleanT.js'
 
@@ -9,11 +9,6 @@ export const deepEqual = [
    match([Vector, Any], Returns(BooleanT, () => false)),
    match([Any, Vector], Returns(BooleanT, () => false)),
    match([Vector, Vector], (math, [V, W]) => {
-      if (V.Component === NotAType || W.Component === NotAType) {
-         return Returns(BooleanT, (v, w) => v === w
-            || (v.length === w.length
-                && v.every((e, i) => math.deepEqual(e, w[i]))))
-      }
       const compDeep = math.deepEqual.resolve([V.Component, W.Component])
       return Returns(BooleanT, (v,w) => v === w
          || (v.length === w.length
@@ -25,24 +20,14 @@ export const indistinguishable = [
    match([Vector, Any, Optional([Any, Any])], (math, [V, E, T]) => {
       const VComp = V.Component
       if (T.length === 0) { // no tolerances
-         if (VComp === NotAType) {
-            return Returns(V, (v, e) => v.map(f => {
-               return math.indistinguishable.resolve([math.typeOf(f), E])(f, e)
-            }))
-         }
          const same = math.indistinguishable.resolve([VComp, E])
          return Returns(
-            Vector(same.returns), (v, e) => v.map(f => same(f, e)))
+            Vector(ReturnType(same)), (v, e) => v.map(f => same(f, e)))
       }
       const [[RT, AT]] = T
-      if (VComp === NotAType) {
-         return Returns(V, (v, e, [[rT, aT]]) => v.map(f => {
-            return math.indistinguishable(f, e, rT, aT)
-         }))
-      }
       const same = math.indistinguishable.resolve([VComp, E, RT, AT])
       return Returns(
-         Vector(same.returns),
+         Vector(ReturnType(same)),
          (v, e, [[rT, aT]]) => v.map(f => same(f, e, rT, aT)))
    }),
    match([Any, Vector, Optional([Any, Any])], (math, [E, V, T]) => {
@@ -50,30 +35,20 @@ export const indistinguishable = [
       // same is symmetric, even though it probably is
       const VComp = V.Component
       if (T.length === 0) { // no tolerances
-         if (VComp === NotAType) {
-            return Returns(V, (e, v) => v.map(f => {
-               return math.indistinguishable.resolve([E, math.typeOf(f)])(e, f)
-            }))
-         }
          const same = math.indistinguishable.resolve([E, VComp])
          return Returns(
-            Vector(same.returns), (e, v) => v.map(f => same(e, f)))
+            Vector(ReturnType(same)), (e, v) => v.map(f => same(e, f)))
       }
       const [[RT, AT]] = T
-      if (VComp === NotAType) {
-         return Returns(V, (e, v, [[rT, aT]]) => v.map(f => {
-            return math.indistiguishable(e, f, rT, aT)
-         }))
-      }
       const same = math.indistinguishable.resolve([E, VComp, RT, AT])
       return Returns(
-         Vector(same.returns),
+         Vector(ReturnType(same)),
          (e, v, [[rT, aT]]) => v.map(f => same(e, f, rT, aT)))
    }),
    match([Vector, Vector, Optional([Any, Any])], (math, [V, W, T]) => {
       const VComp = V.Component
       const WComp = W.Component
-      const inhomogeneous = VComp === NotAType || WComp === NotAType
+      const inhomogeneous = VComp === Unknown || WComp === Unknown
       let same
       let sameNoV
       let sameNoW
@@ -92,7 +67,7 @@ export const indistinguishable = [
          sameNoW = math.indistinguishable.resolve([VComp, Undefined, RT, AT])
       }
       return Returns(
-         inhomogeneous ? Vector(NotAType) : Vector(same.returns),
+         inhomogeneous ? Vector(Unknown) : Vector(ReturnType(same)),
          (v, w, [tol = [0, 0]]) => {
             const [rT, aT] = tol
             const vInc = Number(v.length > 1)

src/vector/type.js

@@ -1,11 +1,11 @@
 import {Vector} from './Vector.js'
-import {NotAType, Returns} from '#core/Type.js'
+import {Returns, Unknown} from '#core/Type.js'
 import {Any, Multiple, match} from '#core/TypePatterns.js'
 
 export const vector = match(Multiple(Any), (math, [TV]) => {
-   if (!TV.length) return Returns(Vector(NotAType), () => [])
+   if (!TV.length) return Returns(Vector(Unknown), () => [])
    let CompType = TV[0]
-   if (TV.some(T => T !== CompType)) CompType = NotAType
+   if (TV.some(T => T !== CompType)) CompType = Unknown
    return Returns(Vector(CompType), v => v)
 })