feat: Add generic operation square and numeric unequal (#4)

Co-authored-by: Jos de Jong <wjosdejong@gmail.com>
Co-authored-by: Glen Whitney <glen@studioinfinity.org>
Reviewed-on: #4

.gitignore

@@ -2,7 +2,7 @@
 *~
 # Typescript
 # emitted code
-build
+obj
 
 # ---> Node
 # Logs
@@ -129,9 +129,6 @@ dist
 # Stores VSCode versions used for testing VSCode extensions
 .vscode-test
 
-# Webstorm
-.idea
-
 # yarn v2
 .yarn/cache
 .yarn/unplugged

.npmrc

@@ -1 +0,0 @@
-shell-emulator=true

README.md

@@ -1,14 +1,3 @@
 # typocomath
 
-A final (?) prototype for a refactor of mathjs, culminating the picomath, pocomath, typomath series. Provides an extensible core with "fuzzy" types for its operations, that can at any time generate exact .d.ts file for its current state.
+A final (?) prototype for a refactor of mathjs, culminating the picomath, pocomath, typomath series. Provides an extensible core with "fuzzy" types for its operations, that can at any time generate exact .d.ts file for its current state.
-
-
-Convenience scripts:
-
-* `pnpm build` -- compile the package
-* `pnpm exec` -- run the compiled code produced by `pnpm build`
-* `pnpm go` -- both of the above in sequence.
-
-Important installation note:
-
-after `pnpm install`, you must execute `npx ts-patch install` to activate the ts-macros compiler plugin.

etc/package.json

@@ -1 +0,0 @@
-{"type": "module"}

package.json5

@@ -1,16 +1,11 @@
 {
    name: 'typocomath',
-   version: '0.0.2',
+   version: '0.0.1',
    description: 'A hopeful final typescipt-pragmatic mathjs proof-of-concept',
    main: 'index.ts',
    scripts: {
       test: 'echo "Error: no test specified" && exit 1',
-      build: 'mkdirp build && cpy etc/package.json build --flat && tsc',
-      start: 'node build',
-      go: 'pnpm clean && pnpm build && pnpm start',
-      clean: 'del-cli build',
    },
-   packageManager: 'pnpm',
    keywords: [
       'math',
       'algebra',
@@ -23,13 +18,6 @@
       url: 'https://code.studioinfinity.org/glen/typocomath.git',
    },
    devDependencies: {
-      '@types/node': '20.8.4',
+      typescript: '^4.9.3',
-      'cpy-cli': '5.0.0',
-      'del-cli': '5.1.0',
-      mkdirp: '3.0.1',
-      'source-map': '0.7.4',
-      'ts-macros': '2.6.0',
-      'ts-patch': '3.0.2',
-      typescript: '5.1.6',
    },
 }

pnpm-lock.yaml

@@ -1,920 +1,15 @@
-lockfileVersion: '6.0'
+lockfileVersion: 5.4
 
-settings:
+specifiers:
-  autoInstallPeers: true
+  typescript: ^4.9.3
-  excludeLinksFromLockfile: false
 
 devDependencies:
-  '@types/node':
+  typescript: 4.9.3
-    specifier: 20.8.4
-    version: 20.8.4
-  cpy-cli:
-    specifier: 5.0.0
-    version: 5.0.0
-  del-cli:
-    specifier: 5.1.0
-    version: 5.1.0
-  mkdirp:
-    specifier: 3.0.1
-    version: 3.0.1
-  source-map:
-    specifier: 0.7.4
-    version: 0.7.4
-  ts-macros:
-    specifier: 2.6.0
-    version: 2.6.0(typescript@5.1.6)
-  ts-patch:
-    specifier: 3.0.2
-    version: 3.0.2
-  typescript:
-    specifier: 5.1.6
-    version: 5.1.6
 
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src/Complex/all.ts

@@ -1,6 +1,10 @@
+import {ForType} from '../core/Dispatcher.js'
+import {ComplexReturn} from './type.js'
 import * as Complex from './native.js'
-import * as complex from './arithmetic.js'
-
-export { complex }
 
 export {Complex}
+
+declare module "../core/Dispatcher" {
+   interface ReturnTypes<Params>
+      extends ForType<'Complex', ComplexReturn<Params>> {}
+}

src/Complex/arithmetic.ts

@@ -1,104 +1,139 @@
-import {Complex} from './type.js'
+import {Complex, UnderlyingReal, complex_binary} from './type.js'
-import type {
+import {
-   Dependencies, Signature, Returns, RealType, AliasOf
+   BBinary, Dependency, ConservativeUnary, ConservativeBinary, ImpType
-} from '../interfaces/type.js'
+} from '../core/Dispatcher.js'
-import {$reflect} from '../interfaces/type.js'
 
-declare module "../interfaces/type" {
+declare module "./type" {
-   interface Signatures<T> {
+   interface ComplexReturn<Params> {
-      addReal: AliasOf<'add', (a: T, b: RealType<T>) => T>
+      add: ConservativeBinary<Params, Complex<any>>
-      divideReal: AliasOf<'divide', (a: T, b: RealType<T>) => T>
+      addReal: Params extends [infer Z, infer R]
+         ? [R] extends [UnderlyingReal<Z>] ? Z : never
+         : never
+      unaryMinus: ConservativeUnary<Params, Complex<any>>
+      conj: ConservativeUnary<Params, Complex<any>>
+      subtract: ConservativeBinary<Params, Complex<any>>
+      multiply: ConservativeBinary<Params, Complex<any>>
+      absquare: Params extends [infer Z]
+         ? Z extends Complex<any> ? UnderlyingReal<Z> : never
+         : never
+      reciprocal: ConservativeUnary<Params, Complex<any>>
+      divide: ConservativeBinary<Params, Complex<any>>
+      divideByReal: Params extends [infer Z, infer R]
+         ? [R] extends [UnderlyingReal<Z>] ? Z : never
+         : never
+      // square root that remains the same type
+      conservativeSqrt: ConservativeUnary<Params, Complex<any>>
+      // Same as conservativeSqrt for complex numbers:
+      sqrt: ConservativeUnary<Params, Complex<any>>
+
+      // complex square root of the real type of a complex:
+      complexSqrt: Params extends [infer T] ? Complex<T> : never
    }
 }
 
 export const add =
-   <T>(dep: Dependencies<'add' | 'complex', T>): Signature<'add', Complex<T>> =>
+   <T>(dep: Dependency<'add', [T,T]>):
-   (w, z) => dep.complex(dep.add(w.re, z.re), dep.add(w.im, z.im))
+      ImpType<'add', [Complex<T>, Complex<T>]> =>
+   (w, z) => complex_binary(dep.add(w.re, z.re), dep.add(w.im, z.im))
 
 export const addReal =
-   <T>(dep: Dependencies<'addReal' | 'complex', T>):
+   <T>(dep: Dependency<'addReal', [T, UnderlyingReal<T>]>):
-      Signature<'addReal', Complex<T>> =>
+      ImpType<'addReal', [Complex<T>, UnderlyingReal<T>]> =>
-   (z, r) => dep.complex(dep.addReal(z.re, r), z.im)
+   (z, r) => complex_binary(dep.addReal(z.re, r), z.im)
 
 export const unaryMinus =
-   <T>(dep: Dependencies<'unaryMinus' | 'complex', T>):
+   <T>(dep: Dependency<'unaryMinus', [T]>):
-      Signature<'unaryMinus', Complex<T>> =>
+      ImpType<'unaryMinus', [Complex<T>]> =>
-   z => dep.complex(dep.unaryMinus(z.re), dep.unaryMinus(z.im))
+   z => complex_binary(dep.unaryMinus(z.re), dep.unaryMinus(z.im))
 
 export const conj =
-   <T>(dep: Dependencies<'unaryMinus' | 'conj' | 'complex', T>):
+   <T>(dep: Dependency<'unaryMinus'|'conj', [T]>):
-      Signature<'conj', Complex<T>> =>
+      ImpType<'conj', [Complex<T>]> =>
-   z => dep.complex(dep.conj(z.re), dep.unaryMinus(z.im))
+   z => complex_binary(dep.conj(z.re), dep.unaryMinus(z.im))
 
 export const subtract =
-   <T>(dep: Dependencies<'subtract' | 'complex', T>):
+   <T>(dep: Dependency<'subtract', [T,T]>):
-      Signature<'subtract', Complex<T>> =>
+      ImpType<'subtract', [Complex<T>, Complex<T>]> =>
-   (w, z) => dep.complex(dep.subtract(w.re, z.re), dep.subtract(w.im, z.im))
+   (w, z) => complex_binary(dep.subtract(w.re, z.re), dep.subtract(w.im, z.im))
 
 export const multiply =
-   <T>(dep: Dependencies<
+   <T>(dep: Dependency<'add', [T,T]>
-       'add' | 'subtract' | 'multiply' | 'conj' | 'complex', T>):
+       & Dependency<'subtract', [T,T]>
-      Signature<'multiply', Complex<T>> =>
+       & Dependency<'multiply', [T,T]>
+       & Dependency<'conj', [T]>):
+      ImpType<'multiply', [Complex<T>, Complex<T>]> =>
    (w, z) => {
       const mult = dep.multiply
       const realpart = dep.subtract(
          mult(         w.re,  z.re), mult(dep.conj(w.im), z.im))
       const imagpart = dep.add(
          mult(dep.conj(w.re), z.im), mult(         w.im,  z.re))
-      return dep.complex(realpart, imagpart)
+      return complex_binary(realpart, imagpart)
    }
 
 export const absquare =
-   <T>(dep: Dependencies<'absquare', T>
+   <T>(dep: Dependency<'absquare', [T]>
-       & Dependencies<'add', Returns<'absquare', T>>):
+       & Dependency<'add', BBinary<UnderlyingReal<T>>>):
-      Signature<'absquare', Complex<T>> =>
+      ImpType<'absquare', [Complex<T>]> =>
    z => dep.add(dep.absquare(z.re), dep.absquare(z.im))
 
-export const divideReal =
+export const divideByReal =
-   <T>(dep: Dependencies<'divideReal' | 'complex', T>):
+   <T>(dep: Dependency<'divideByReal', [T, UnderlyingReal<T>]>):
-      Signature<'divideReal', Complex<T>> =>
+      ImpType<'divideByReal', [Complex<T>, UnderlyingReal<T>]> =>
-   (z, r) => dep.complex(dep.divideReal(z.re, r), dep.divideReal(z.im, r))
+   (z, r) => complex_binary(
+      dep.divideByReal(z.re, r), dep.divideByReal(z.im, r))
 
 export const reciprocal =
-   <T>(dep: Dependencies<'conj' | 'absquare' | 'divideReal', Complex<T>>):
+   <T>(dep: Dependency<'conj', [Complex<T>]>
-      Signature<'reciprocal', Complex<T>> =>
+       & Dependency<'absquare', [Complex<T>]>
-   z => dep.divideReal(dep.conj(z), dep.absquare(z))
+       & Dependency<'divideByReal', [Complex<T>, UnderlyingReal<T>]>):
+      ImpType<'reciprocal', [Complex<T>]> =>
+   z => dep.divideByReal(dep.conj(z), dep.absquare(z))
 
 export const divide =
-   <T>(dep: Dependencies<'multiply' | 'reciprocal', Complex<T>>):
+   <T>(dep: Dependency<'multiply', [Complex<T>, Complex<T>]>
-      Signature<'divide', Complex<T>> =>
+       & Dependency<'reciprocal', [Complex<T>]>):
+      ImpType<'divide', [Complex<T>, Complex<T>]> =>
    (w, z) => dep.multiply(w, dep.reciprocal(z))
 
-// The dependencies are slightly tricky here, because there are three types
+export const complexSqrt =
-// involved: Complex<T>, T, and RealType<T>, all of which might be different,
+   <T>(dep: Dependency<'conservativeSqrt', [T]>
-// and we have to get it straight which operations we need on each type, and
+       & Dependency<'isSquare', [T]>
-// in fact, we need `addReal` on both T and Complex<T>, hence the dependency
+       & Dependency<'complex', [T]>
-// with a custom name, not generated via Dependencies<...>
+       & Dependency<'unaryMinus', [T]>
-export const sqrt =
+       & Dependency<'zero', [T]>
-   <T>(dep: Dependencies<'equal' | 'conservativeSqrt' | 'unaryMinus', RealType<T>>
+       & Dependency<'nan', [Complex<T>]>): ImpType<'complexSqrt', [T]> =>
-       & Dependencies<'zero' | 'complex', T>
+   r => {
-       & Dependencies<'absquare' | 're' | 'divideReal', Complex<T>>
+      if (dep.isSquare(r)) return dep.complex(dep.conservativeSqrt(r))
-       & {
+      const negative = dep.unaryMinus(r)
-         addTR: Signature<'addReal', T>,
+      if (dep.isSquare(negative)) {
-         addRR: Signature<'add', RealType<T>>,
+         return complex_binary(
-         addCR: Signature<'addReal', Complex<T>>
+            dep.zero(r), dep.conservativeSqrt(negative))
-       }): Signature<'sqrt', Complex<T>> =>
-   z => {
-      const myabs = dep.conservativeSqrt(dep.absquare(z))
-      const r = dep.re(z)
-      const negr = dep.unaryMinus(r)
-      if (dep.equal(myabs, negr)) {
-         // pure imaginary square root; z.im already zero
-         return dep.complex(
-            dep.zero(z.re), dep.addTR(z.im, dep.conservativeSqrt(negr)))
       }
-      const num = dep.addCR(z, myabs)
+      // neither the real number or its negative is a square; could happen
-      const denomsq = dep.addRR(dep.addRR(myabs, myabs), dep.addRR(r, r))
+      // for example with bigint. So there is no square root. So we have to
-      const denom = dep.conservativeSqrt(denomsq)
+      // return the NaN of the type.
-      return dep.divideReal(num, denom)
+      return dep.nan(dep.complex(r))
    }
 
-$reflect!([
+export const sqrt =
-   add, addReal, unaryMinus, conj, subtract, multiply, absquare, divideReal,
+   <T>(dep: Dependency<'isReal', [Complex<T>]>
-   reciprocal, divide, sqrt
+       & Dependency<'complexSqrt', [T]>
-])
+       & Dependency<'absquare', [Complex<T>]>
+       & Dependency<'conservativeSqrt', [UnderlyingReal<T>]>
+       & Dependency<'addReal', [Complex<T>,UnderlyingReal<T>]>
+       & Dependency<'re', [Complex<T>]>
+       & Dependency<'add', [UnderlyingReal<T>,UnderlyingReal<T>]>
+       & Dependency<'divideByReal', [Complex<T>,UnderlyingReal<T>]>
+      ): ImpType<'sqrt', [Complex<T>]> =>
+   z => {
+      if (dep.isReal(z)) return dep.complexSqrt(z.re)
+      const myabs = dep.conservativeSqrt(dep.absquare(z))
+      const num = dep.addReal(z, myabs)
+      const r = dep.re(z)
+      const denomsq = dep.add(dep.add(myabs, myabs), dep.add(r, r))
+      const denom = dep.conservativeSqrt(denomsq)
+      return dep.divideByReal(num, denom)
+   }
+
+export const conservativeSqrt = sqrt

src/Complex/predicate.ts

@@ -1,13 +1,19 @@
 import {Complex} from './type.js'
-import type {Dependencies, Signature} from '../interfaces/type.js'
+import {Signature, Dependency, ImpType} from '../core/Dispatcher.js'
-import {$reflect} from '../interfaces/type.js'
+
+declare module "./type" {
+   interface ComplexReturn<Params> {
+      isReal: Signature<Params, [Complex<any>], boolean>
+      isSquare: Signature<Params, [Complex<any>], boolean>
+   }
+}
 
 export const isReal =
-   <T>(dep: Dependencies<'add' | 'equal' | 'isReal', T>):
+   <T>(dep: Dependency<'equal', [T,T]>
-      Signature<'isReal', Complex<T>> =>
+       & Dependency<'add', [T,T]>
+       & Dependency<'isReal', [T]>
+      ): ImpType<'isReal', [Complex<T>]> =>
    z => dep.isReal(z.re) && dep.equal(z.re, dep.add(z.re, z.im))
 
-export const isSquare = (): Signature<'isSquare', Complex<unknown>> =>
+export const isSquare: ImpType<'isSquare', [Complex<any>]> =
    z => true // FIXME: not correct for Complex<bigint> once we get there
-
-$reflect!([isReal, isSquare])

src/Complex/relational.ts

@@ -1,8 +1,15 @@
 import {Complex} from './type.js'
-import {Dependencies, Signature} from '../interfaces/type.js'
+import {BBinary, ImpType, Dependency} from '../core/Dispatcher.js'
-import {$reflect} from '../interfaces/type.js'
+
+declare module "./type" {
+   interface ComplexReturn<Params> {
+      equal: Params extends BBinary<infer B>
+         ? B extends Complex<any> ? boolean : never
+         : never
+   }
+}
 
 export const equal =
-   <T>(dep: Dependencies<'equal', T>): Signature<'equal', Complex<T>> =>
+   <T>(dep: Dependency<'equal', [T,T]>):
+      ImpType<'equal', [Complex<T>, Complex<T>]> =>
    (w, z) => dep.equal(w.re, z.re) && dep.equal(w.im, z.im)
-$reflect!([equal])

src/Complex/type.ts

@@ -1,67 +1,87 @@
-import {joinTypes, typeOfDependency} from '../core/Dispatcher.js'
+import {
-import type {
+   joinTypes, typeOfDependency, Dependency, BBinary, ImpType, ImpReturns
-   ZeroType, OneType, NaNType, Dependencies, Signature, Returns
+} from '../core/Dispatcher.js'
-} from '../interfaces/type.js'
-import {$reflect} from '../interfaces/type.js'
 
-export type Complex<T> = { re: T; im: T; }
+export type Complex<T> = {re: T; im: T;}
+
+export type UnderlyingReal<T> =
+   T extends Complex<infer U> ? UnderlyingReal<U> : T
 
 export const Complex_type = {
-   name: 'Complex', // just until we have reflection to tell us
+   test: <T>(dep: {testT: (z: unknown) => z is T}) =>
-   test: <T>(dep: { testT: (z: unknown) => z is T }) =>
       (z: unknown): z is Complex<T> =>
-         typeof z === 'object' && z != null && 're' in z && 'im' in z
+      typeof z === 'object' && 're' in z && 'im' in z
-         && dep.testT(z['re']) && dep.testT(z['im']),
+      && dep.testT(z.re) && dep.testT(z.im),
    infer: (dep: typeOfDependency) =>
       (z: Complex<unknown>) => joinTypes(dep.typeOf(z.re), dep.typeOf(z.im)),
    from: {
-      T: <T>(dep: Dependencies<'zero', T>) => (t: T) =>
+      T: <T>(dep: Dependency<'zero', [T]>) => (t: T) =>
-         ({ re: t, im: dep.zero(t) }),
+         ({re: t, im: dep.zero(t)}),
-      Complex: <U, T>(dep: { convert: (from: U) => T }) =>
+      Complex: <U,T>(dep: {convert: (from: U) => T}) =>
-         (z: Complex<U>) => ({ re: dep.convert(z.re), im: dep.convert(z.im) })
+         (z: Complex<U>) => ({re: dep.convert(z.re), im: dep.convert(z.im)})
    }
 }
 
-declare module "../interfaces/type" {
+export interface ComplexReturn<Params> {
-   interface AssociatedTypes<T> {
+   // Sadly, I can't think of a way to make some nice abbreviation operators
-      Complex: T extends Complex<infer R> ? {
+   // for these generic type specifications because TypeScript generics
-         type: Complex<R>
+   // can't take and use generic parameters, only fully instantiated types.
-         zero: Complex<ZeroType<R>>
+   complex: Params extends [infer U] ? Complex<U> // unary case
-         one:  Complex<OneType<R> | ZeroType<R>>
+      : Params extends BBinary<infer B> ? Complex<B> // binary case
-         nan:  Complex<NaNType<R>>
+      : never
-         real: RealType<R>
-      } : never
-   }
 
-   interface Signatures<T> {
+   // alternatively if it seems better; each definition is simpler, but at
-      complex: ((re: T) => Complex<T>) | ((re: T, im: T) => Complex<T>)
+   // the cost of having two keys here:
-   }
+   // complex_unary: Params extends [infer R] ? Complex<R> : never
+   // complex_binary: Params extends BBinary<infer R> ? Complex<R> : never
+
+   // There is actually a subtlety here that complex_unary really only works
+   // on real types that include their own zero value, so it should really be
+   // complex_unary: Params extends [infer R]
+   //    ? ImpReturns<'zero', [R]> extends R ? Complex<R> : never
+   //    : never
+   // and that might actually simplify some of the typings of other operations,
+   // but we'll leave such fine tuning til later, if we adopt this scheme
+
+   zero: Params extends [infer Z] // unary
+      ? Z extends Complex<infer T> // of a Complex parameter
+         ? ImpReturns<'zero', T> extends T ? Z : never // that has its real 0
+         : never
+      : never
+   one: Params extends [infer Z] // unary
+      ? Z extends Complex<infer T> // of a Complex parameter
+         ? ImpReturns<'one'|'zero', T> extends T ? Z : never // has real 1, 0
+         : never
+      : never
+   nan: Params extends [infer Z] // unary
+      ? Z extends Complex<infer T> // of a Complex parameter
+         ? ImpReturns<'nan', T> extends T ? Z : never // has real NaN
+         : never
+      : never
+   re: Params extends [infer Z]
+      ? Z extends Complex<infer T> ? UnderlyingReal<T> : never
+      : never
 }
 
-export const complex =
+export const complex_unary =
-   <T>(dep: Dependencies<'zero', T>): Signature<'complex', T> =>
+   <T>(dep: Dependency<'zero', [T]>): ImpType<'complex', [T]> =>
-   (a, b) => ({re: a, im: b || dep.zero(a)})
+   t => ({re: t, im: dep.zero(t)})
+export const complex_binary = <T>(t: T, u: T): ImpReturns<'complex', [T,T]> =>
+   ({re: t, im: u})
 
 export const zero =
-   <T>(dep: Dependencies<'zero', T>
+   <T>(dep: Dependency<'zero', [T]>): ImpType<'zero', [Complex<T>]> =>
-       & Dependencies<'complex', Returns<'zero', T>>):
+   z => complex_binary(dep.zero(z.re), dep.zero(z.im))
-      Signature<'zero', Complex<T>> =>
-   z => dep.complex(dep.zero(z.re), dep.zero(z.im))
 
 export const one =
-   <T>(dep: Dependencies<'one' | 'zero', T>
+   <T>(dep: Dependency<'zero' | 'one', [T]>): ImpType<'one', [Complex<T>]> =>
-       & Dependencies<'complex', Returns<'one' | 'zero', T>>):
+   z => // Must provide parameter T, else TS narrows to return type of dep.one
-      Signature<'one', Complex<T>> =>
+   complex_binary<T>(dep.one(z.re), dep.zero(z.im))
-   z => dep.complex(dep.one(z.re), dep.zero(z.im))
 
 export const nan =
-   <T>(dep: Dependencies<'nan', T>
+   <T>(dep: Dependency<'nan', [T]>): ImpType<'nan', [Complex<T>]> =>
-       & Dependencies<'complex', Returns<'nan', T>>):
+   z => complex_binary(dep.nan(z.re), dep.nan(z.im))
-      Signature<'nan', Complex<T>> =>
-   z => dep.complex(dep.nan(z.re), dep.nan(z.im))
 
 export const re =
-   <T>(dep: Dependencies<'re', T>): Signature<'re', Complex<T>> =>
+   <T>(dep: Dependency<'re', [T]>): ImpType<'re', [Complex<T>]> =>
    z => dep.re(z.re)
-
-$reflect!([complex, zero, one, nan, re])

src/core/Dispatcher.ts

@@ -5,29 +5,122 @@
  * for specific types (including their own).
  */
 
-import {parseReflectedType, ImplementationDef} from './parseReflectedType.js'
-
 // First helper types and functions for the Dispatcher
 
 type TypeName = string
 type Parameter = TypeName
-type Signature = Parameter[]
+type InputSignature = Parameter[]
 type DependenciesType = Record<string, Function>
 
-// A "canned" dependency for a builtin function:
 export type typeOfDependency = {typeOf: (x: unknown) => TypeName}
 
-// Utility needed in type definitions
+// All of the implementations must publish descriptions of their
+// return types into the following interface, using the format
+// described just below:
+export interface ReturnTypes<Params> {}
+
+/*****
+   To describe one implementation for a hypothetical operation `foo`, there
+   should be a property of the interface whose name starts with `foo` and whose
+   next character, if any, is an underscore. The type of this property
+   must be the return type of that implementation when Params matches the
+   parameter types of the implementation, and `never` otherwise.
+   Thus to describe an implementation that takes a number and a string and
+   returns a boolean, for example, you could write
+   ```
+   declare module "Dispatcher" {
+      interface ReturnTypes<Params> {
+         foo_example: Params extends [number, string] ? boolean : never
+      }
+   }
+   ```
+   If there is another, generic implementation that takes one argument
+   of any type and returns a Vector of that type, you can say
+   ```
+   ...
+      foo_generic: Params extends [infer T] ? Vector<T> : never
+   ...
+   ```
+   In practice, each subdirectory corresponding to a type, like Complex,
+   defines an interface, like `ComplexReturn<Params>` for the implementations
+   in that subdirectory, which can mostly be defined without suffixes because
+   there's typically just a single implementation within that domain.
+   Then the module responsible for collating all of the implementations for
+   that type inserts all of the properties of that interface into `ReturnTypes`
+   suitably suffixed to avoid collisions.
+
+   One might think that simply defining an implementation for `foo`
+   of type `(n: number, s: string) => boolean` would provide all of the same
+   information as the type of the key `foo_example` in the ReturnTypes
+   interface above, but in practice TypeScript has challenges in extracting
+   types relating to functions. (In particular, there is no
+   way to get the specialized return type of a generic function when it is
+   called on aguments whose specific types match the generic parameters.)
+   Hence the need for this additional mechanism to specify return types, in
+   a way readily suited for TypeScript type computations.
+*****/
+
+// Helpers for specifying signatures
+
+// A basic signature with concrete types
+export type Signature<CandidateParams, ActualParams, Returns> =
+   CandidateParams extends ActualParams ? Returns : never
+
+// A homogeneous binary parameter tuple (comes up a lot, needs a better name?)
+// Typical usage: `foo_impl: Params extends BBinary<infer B> ? B : never`
+// says that this implementation takes two arguments, both of type B, and
+// returns the same type.
+export type BBinary<B> = [B, B]
+
+// A unary signature that preserves the type of its argument, which must
+// extend the given Bound:
+export type ConservativeUnary<CandidateParams, Bound> =
+   CandidateParams extends [infer T] ? T extends Bound ? T : never : never
+
+// A homogeneous binary signature that preserves the common type of its
+// arguments, which must extend the given Bound:
+export type ConservativeBinary<CandidateParams, Bound> =
+   CandidateParams extends BBinary<infer B>
+      ? B extends Bound ? B : never
+      : never
+
+// Helper for collecting return types
+// (Really just adds the literal string Suffix onto the keys of interface IFace)
+export type ForType<Suffix extends string, IFace> = keyof IFace extends string
+   ? {[K in keyof IFace as `${K}_${Suffix}`]: IFace[K]}
+   : never
+
 //dummy implementation for now
 export function joinTypes(a: TypeName, b: TypeName) {
    if (a === b) return a
    return 'any'
 }
 
+// Used to filter keys that match a given operation name
+type BeginsWith<Name extends string> = Name | `${Name}_${string}`
+
+// Look up the return type of an implementation based on its name
+// and the parameters it takes
+export type ImpReturns<Name extends string, Params> =
+   {[K in keyof ReturnTypes<Params>]: K extends BeginsWith<Name>
+    ? ReturnTypes<Params>[K] : never}[keyof ReturnTypes<Params>]
+
+// The type of an implementation (with dependencies satisfied,
+// based on its name and the parameters it takes
+export type ImpType<Name extends string, Params extends unknown[]> =
+   (...args: Params) => ImpReturns<Name, Params>
+
+// The type of a dependency on an implementation based on its name
+// and the parameters it takes (just a simple object with one property
+// named the same as the operation, of value type equal to the type of
+// that implementation. These can be `&`ed together in case of multiple
+// dependencies:
+export type Dependency<Name extends string, Params extends unknown[]> =
+   {[N in Name]: ImpType<N, Params>}
+
 // Now types used in the Dispatcher class itself
 
 type TypeSpecification = {
-   name: string, // just until we get reflection, then we can remove this property
    before?: TypeName[],
    test: ((x: unknown) => boolean)
       | ((d: DependenciesType) => (x: unknown) => boolean),
@@ -41,44 +134,38 @@ type SpecificationsGroup = Record<string, SpecObject>
 export class Dispatcher {
    installSpecification(
       name: string,
-      defn: ImplementationDef,
+      signature: InputSignature,
+      returns: TypeName,
+      dependencies: Record<string, InputSignature>,
       behavior: Function // possible todo: constrain this type based
       // on the signature, return type, and dependencies. Not sure if
       // that's really possible, though.
    ) {
-      console.log('Pretending to install', name, 'with signatures')
+      console.log('Pretending to install', name, signature, '=>', returns)
-      for (const signature of defn.fn.signatures) {
-         console.log('   ', signature.args, '=>', signature.returns)
-      }
-      if (defn.fn.aliasOf) {
-         console.log('   As an alias of', defn.fn.aliasOf)
-      }
       //TODO: implement me
    }
    installType(name: TypeName, typespec: TypeSpecification) {
       console.log('Pretending to install type', name, typespec)
       //TODO: implement me
    }
-   constructor(collection: SpecificationsGroup) {
+   constructor(collection: SpecificationsGroup) { 
-      const implementations = []
       for (const key in collection) {
          console.log('Working on', key)
          for (const identifier in collection[key]) {
-            const item = collection[key][identifier]
+            console.log('Handling', key, ':', identifier)
-            if (typeof item === 'function') {
+            const parts = identifier.split('_')
-               implementations.push([key, identifier, item])
+            if (parts[parts.length - 1] === 'type') {
-            } else {
+               parts.pop()
-               console.log('Handling type', key, ':', identifier)
+               const name = parts.join('_')
                this.installType(
-                  item.name, collection[key][identifier] as TypeSpecification)
+                  name, collection[key][identifier] as TypeSpecification)
+            } else {
+               const name = parts[0]
+               this.installSpecification(
+                  name, ['dunno'], 'unsure', {},
+                  collection[key][identifier] as Function)
             }
          }
       }
-      for (const trio of implementations) {
-         const [k, id, imp] = trio
-         console.log('Handling implementation', id, 'from', k)
-         this.installSpecification(
-            id, parseReflectedType(id, imp.reflectedType), imp)
-      }
    }
 }

src/core/parseReflectedType.ts

@@ -1,279 +0,0 @@
-export type FunctionDef = {
-  name: string,
-  aliasOf?: string,
-  signatures: Array<{
-    args: Array<{ name: string, type: string }>
-    returns: string
-  }>
-}
-
-
-export type ImplementationDef = {
-  fn: FunctionDef,
-  dependencies: Record<string, FunctionDef>
-  genericParameter: string | null
-}
-
-/**
- * Parse a reflected type coming out of TypeScript into a structured object, for example:
- *
- *   '<T>(dep: configDependency & { complex: ((re: number) => Complex<number>) | ((re: number, im: number) => Complex<number>); }) => (a: number) => number | Complex<number>'
- */
-export function parseReflectedType(name: string, reflectedType: string): ImplementationDef {
-  console.log('For', name, 'parsing', reflectedType)
-  const [factoryArgs, fnsClause] = split(reflectedType, '=>', 2).map(trim)
-  const fn = parseAlias(name, fnsClause)
-
-  // extract the generic parameter like '<T>' at the start of the type
-  const genericParameter = factoryArgs.trim().startsWith('<')
-    ? findBlockContents(factoryArgs, '<', '>')?.innerText || null
-    : null
-
-  const factoryArgsInner = findBlockContents(factoryArgs, '(', ')')
-  const depArg = split(factoryArgsInner.innerText, ':').map(trim)[1]
-  const depArgBlocks: string[] = depArg ? split(depArg, '&').map(trim) : []
-
-  const deps = depArgBlocks
-    .filter(depArgBlock => {
-      if (depArgBlock.startsWith('{') || depArgBlock === 'configDependency') {
-        return true
-      } else {
-        throw new SyntaxError(`Cannot parse dependency "${depArgBlock}"`)
-      }
-    })
-    .flatMap(parseDependencies)
-
-  const dependencies: Record<string, FunctionDef> = groupBy(deps, 'name')
-
-  return {fn, dependencies, genericParameter }
-}
-
-function parseDependencies(deps: string): FunctionDef[] {
-  if (deps === 'configDependency') {
-    return [{name: 'config', signatures: [{args: [], returns: 'Config'}]}]
-  }
-  const inner = findBlockContents(deps, '{', '}').innerText
-  return split(inner, ';')
-    .map(trim)
-    .filter(notEmpty)
-    .map(parseDependency)
-}
-
-// parse a dependency like "complex: ((re: number) => Complex<number>) | ((re: number, im: number) => Complex<number>)"
-function parseDependency(dep: string): FunctionDef {
-  const [name, def] = split(dep, ':').map(trim)
-
-  return parseAlias(name, def)
-}
-
-// Parse a possibly aliased function
-function parseAlias(name: string, alias: string): FunctionDef {
-  const { aliasOf, innerSignature } = parseAliasOf(alias)
-
-  return { name, signatures: parseSignatures(innerSignature), aliasOf }
-}
-
-// parse function signatures like ((re: number) => Complex<number>) | ((re: number, im: number) => Complex<number>)
-// But also have to succeed on (a: number) => number | Complex<number>
-// That's why we only split on an alternation bar `|` that's followed by
-// a parenthesis; that way we avoid splitting a union return type. Note
-// this is not necessarily foolproof, as there could be a return type that
-// is a union with a complicated piece that has to be enclosed in parens;
-// but so far it seems to work in practice.
-function parseSignatures(sigs: string) {
-  return split(sigs, /[|]\s*(?=[(])/)
-    .map(trim)
-    .map(stripParenthesis)
-    .map(signature => {
-      const [argsBlock, returns] = split(signature, '=>').map(trim)
-
-      if (!returns) {
-        throw new SyntaxError(`Failed to find return type in '${signature}'`)
-      }
-
-      return {
-        args: parseArgs(argsBlock),
-        returns
-      }
-    })
-}
-
-// parse args like "(re: number, im: number)"
-function parseArgs(argsBlock: string) : Array<{name: string, type: string}> {
-  const args = findBlockContents(argsBlock, '(', ')').innerText
-
-  return split(args, ',')
-    .map(trim)
-    .map(arg => {
-      const [name, type] = split(arg, ':').map(trim)
-      return { name, type}
-    })
-}
-
-// parse "AliasOf<"divide", (a: Complex<T>, b: RealType<Complex<T>>) => Complex<T>>"
-function parseAliasOf(signature: string) : { innerSignature: string, aliasOf: string | undefined } {
-  if (!signature.startsWith('AliasOf')) {
-    return {
-      innerSignature: signature,
-      aliasOf: undefined
-    }
-  }
-
-  const inner = findBlockContents(signature, '<', '>').innerText.trim()
-  const [aliasOfWithQuotes, innerSignature] = split(inner, ',').map(trim)
-  return {
-    innerSignature,
-    aliasOf: aliasOfWithQuotes.substring(1, aliasOfWithQuotes.length - 1) // remove double quotes
-  }
-}
-
-// remove the outer parenthesis, for example "((re: number) => Complex<number>)" returns "(re: number) => Complex<number>"
-function stripParenthesis(text: string) : string {
-  return text.startsWith('(') && text.endsWith(')')
-    ? text.substring(1, text.length - 1)
-    : text
-}
-
-function findBlockContents(text: string, blockStart: string, blockEnd: string, startIndex = 0) : { start: number, end: number, innerText: string } | undefined {
-  let i = startIndex
-
-  while (!matchSubString(text, blockStart, i) && i < text.length) {
-    i++
-  }
-
-  if (i >= text.length) {
-    return undefined
-  }
-
-  i++
-  const start = i
-
-  while (!matchSubString(text, blockEnd, i) || matchSubString(text, '=>', i - 1)) {
-    i = skipBrackets(text, i)
-
-    i++
-  }
-
-  if (i >= text.length) {
-    return undefined
-  }
-  const end = i
-
-  return {
-    start,
-    end,
-    innerText: text.substring(start, end)
-  }
-}
-
-/**
- * Given a string, generate a string source for a regexp that will match
- * exactly the given string.
- * Uses the fact that the only characters that need to be escaped in
- * a character class are \, ], and ^
- */
-function regexpQuote(s: string) {
-  const special = '\\]^'
-  let re = ''
-  for (const char of s) {
-    if (special.includes(char)) re += `\\${char}`
-    else re += `[${char}]`
-  }
-  return re
-}
-
-/**
- * Split a string by a delimiter, but ignore all occurrences of the delimiter
- * that are inside bracket pairs <> () [] {}
- */
-export function split(
-     text: string, delimiter: string | RegExp, pieces = 0): string[] {
-  const delim: RegExp = typeof delimiter === 'string'
-     ? new RegExp(regexpQuote(delimiter), 'y')
-     : new RegExp(delimiter.source, 'y')
-  const parts: string[] = []
-
-  let i = 0
-  let n = 1
-  let start = 0
-  while (i < text.length && (pieces === 0 || n < pieces)) {
-    i = skipBrackets(text, i)
-
-    delim.lastIndex = i
-    const result = delim.exec(text)
-    if (result) {
-      parts.push(text.substring(start, i))
-      n += 1
-      i += result[0].length
-      start = i
-    }
-
-    i++
-  }
-
-  parts.push(text.substring(start))
-
-  return parts
-}
-
-function skipBrackets(text: string, startIndex: number) : number {
-  let level = 0
-  let i = startIndex
-
-  do {
-    if (isBracketOpen(text, i)) {
-      level++
-    }
-
-    if (isBracketClose(text, i) && level > 0) {
-      level--
-    }
-
-    if (level === 0) {
-      break
-    }
-
-    i++
-  } while(i < text.length)
-
-  return i
-}
-
-function isBracketOpen(text: string, index: number) {
-  const char = text[index]
-  return char === '(' || char === '<' || char === '[' || char === '{'
-}
-
-function isBracketClose(text: string, index: number) {
-  const char = text[index]
-  // we need to take care of not matching the ">" of the operator "=>"
-  return char === ')' || (char === '>' && text[index - 1] !== '=') || char === ']' || char === '}'
-}
-
-function matchSubString(text: string, search: string, index: number) : boolean {
-  for (let i = 0; i < search.length; i++) {
-    if (text[i + index] !== search[i]) {
-      return false
-    }
-  }
-
-  return true
-}
-
-function trim(text: string) : string {
-  return text.trim()
-}
-
-function notEmpty(text: string) : boolean {
-  return text.length > 0
-}
-
-function groupBy<T>(items: T[], key: string) : Record<string, T> {
-  const obj: Record<string, T> = {}
-
-  items.forEach((item) => {
-    obj[item[key]] = item
-  })
-
-  return obj
-}

src/generic/all.ts

@@ -1 +1,10 @@
-export * as generic from './native.js'
+import { ForType } from '../core/Dispatcher.js'
+import { GenericReturn } from './type.js'
+import * as generic from './arithmetic.js'
+
+export { generic }
+
+declare module "../core/Dispatcher" {
+   interface ReturnTypes<Params>
+      extends ForType<'generic', GenericReturn<Params>> { }
+}

src/generic/arithmetic.ts

@@ -1,9 +1,39 @@
-import type {Dependencies, Signature} from '../interfaces/type.js'
+import {Dependency, ImpType, ImpReturns} from "../core/Dispatcher";
-import {$reflect} from '../interfaces/type.js'
+
+declare module "./type" {
+  interface GenericReturn<Params> {
+    // Jos: not sure how to define this or why it is needed
+    // square: Signature<Params, [T], T>
+    // square: ConservativeUnary<Params, T>
+    // square: Params extends [infer R]
+    //   ? R extends number ? UnderlyingReal<R> : never
+    //    : never
+
+    // The type of `square` in this interface, instantiated with the type
+    // Params of a parameter list, needs to be the return type of the
+    // operation `square` on those parameters. In other words, `square` gives
+    // a type transformer from the tuple type of its parameters to its return
+    // type.
+    // That's how Dispatcher knows what the return type will be in
+    // `Dependency<'square', [bigint]>`, for example: it instantiates
+    // GenericReturn with Params equal to [bigint] and then grabs the
+    // type of the `square` property. Hence we write:
+
+    square: Params extends [infer T] // square only takes 1 arbitrary parameter
+        ? ImpReturns<'multiply', [T, T]> // and returns whatever multiply does
+        : never; // otherwise if not a single argument, this implementation
+                 // doesn't handle it
+
+    // If square had more than one implementation in this collection, we could
+    // either add more conditional clauses to the above type transformer
+    // as I did in Complex/type.ts for `complex`, or we could have two
+    // different keys that both start with `square_` and Dispatcher will
+    // check both (as I have now done in comments in Complex/type.ts and
+    // verified that also works).
+  }
+}
 
 export const square =
-   <T>(dep: Dependencies<'multiply', T>): Signature<'square', T> =>
+  <T>(dep: Dependency<'multiply', [T, T]>):
-   z => dep.multiply(z, z)
+    ImpType<'square', [T]> =>
-   // z => dep.fooBar(z, z) // fails as desired
+    z => dep.multiply(z, z)
-   // z => dep.multiply(z, 'foo') // still fails as desired
-$reflect!([square])

src/generic/native.ts

@@ -1,2 +0,0 @@
-export * from './arithmetic.js'
-export * from './relational.js'

src/generic/relational.ts

@@ -1,7 +0,0 @@
-import {Dependencies, Signature} from '../interfaces/type.js'
-import {$reflect} from '../interfaces/type.js'
-
-export const unequal =
-   <T>(dep: Dependencies<'equal', T>): Signature<'unequal', T> =>
-   (x, y) => !dep.equal(x, y)
-$reflect!([unequal])

src/generic/type.ts

@@ -0,0 +1,3 @@
+export interface GenericReturn<Params> {
+
+}

src/index.ts

@@ -1,53 +1,4 @@
-import {inspect} from 'node:util'
 import {Dispatcher} from './core/Dispatcher.js'
 import * as Specifications from './all.js'
 
 export default new Dispatcher(Specifications)
-
-import {Complex} from './Complex/type.js'
-import {absquare as absquare_complex} from './Complex/arithmetic.js'
-import {parseReflectedType} from './core/parseReflectedType.js'
-
-const mockRealAdd = (a: number, b: number) => a+b
-const mockComplexAbsquare = (z: Complex<number>) => z.re*z.re + z.im*z.im
-const mockComplex = (re: number, im: number) => ({ re, im })
-
-const config = {
-  predictable: false,
-  epsilon: 1e-14
-}
-
-const quatAbsquare = absquare_complex({
-  add: mockRealAdd,
-  absquare: mockComplexAbsquare
-})
-
-const myabs = quatAbsquare({re: {re: 0, im: 1}, im: {re:2, im: 3}})
-const typeTest: typeof myabs = 7 // check myabs is just a number
-console.log('Result is myabs=', myabs)
-
-const sqrt = Specifications.numbers.sqrt({
-  config,
-  complex: mockComplex
-})
-console.log('Result of sqrt(16)=', sqrt(16))
-console.log('Result of sqrt(-4)=', sqrt(-4))
-
-console.log()
-console.log('1) NUMBER SQRT')
-console.log(`1.1) REFLECTED TYPE: "${Specifications.numbers.sqrt.reflectedType}"`)
-console.log(
-  '1.2) PARSED TYPE:',
-  inspect(
-    parseReflectedType('sqrt', Specifications.numbers.sqrt.reflectedType),
-    { depth: null, colors: true }))
-
-console.log()
-console.log('2) GENERIC SQUARE')
-console.log(`1.1) REFLECTED TYPE: "${Specifications.generic.square.reflectedType}"`)
-console.log('2.2) PARSED TYPE:', inspect(parseReflectedType('square', Specifications.generic.square.reflectedType), { depth: null, colors: true }))
-
-console.log()
-console.log('3) COMPLEX SQRT')
-console.log(`1.1) REFLECTED TYPE: "${Specifications.complex.sqrt.reflectedType}"`)
-console.log('3.2) PARSED TYPE:', inspect(parseReflectedType('sqrt', Specifications.complex.sqrt.reflectedType), { depth: null, colors: true }))

src/interfaces/arithmetic.ts

@@ -1,18 +0,0 @@
-import type {Complex} from '../Complex/type.js'
-import type {RealType} from './type.js'
-
-declare module "./type" {
-   interface Signatures<T> {
-      add:              (a: T, b: T) => T
-      unaryMinus:       (a: T) => T
-      conj:             (a: T) => T
-      subtract:         (a: T, b: T) => T
-      multiply:         (a: T, b: T) => T
-      square:           (a: T) => T
-      absquare:         (a: T) => RealType<T>
-      reciprocal:       (a: T) => T
-      divide:           (a: T, b: T) => T
-      conservativeSqrt: (a: T) => T
-      sqrt: (a: T)=> T extends Complex<unknown> ? T : T | Complex<T>
-   }
-}

src/interfaces/predicate.ts

@@ -1,10 +0,0 @@
-// Warning: a module must have something besides just a "declare module"
-// section; otherwise it is ignored.
-export type UnaryPredicate<T> = (a: T) => boolean
-
-declare module "./type" {
-   interface Signatures<T> {
-      isReal:   (a: T) => boolean
-      isSquare: (a: T) => boolean
-   }
-}

src/interfaces/relational.ts

@@ -1,9 +0,0 @@
-// Warning: a module must have something besides just a "declare module"
-// section; otherwise it is ignored.
-export type BinaryPredicate<T> = (a: T, b: T) => T
-declare module "./type" {
-   interface Signatures<T> {
-      equal:   (a: T, b: T) => boolean
-      unequal: (a: T, b: T) => boolean
-   }
-}

src/interfaces/type.ts

@@ -1,90 +0,0 @@
-import {$$typeToString} from 'ts-macros'
-
-/*****
- * Every typocomath type has some associated types; they need
- * to be published in the following interface. The key is the
- * name of the type, and within the subinterface for that key,
- * the type of the 'type' property is the actual TypeScript type
- * we are associating the other properties to. Note the interface
- * is generic with one parameter, corresponding to the fact that
- * typocomath currently only allows generic types with a single
- * generic parameter. This way, AssociatedTypes<SubType> can give the
- * associated types for a generic type instantiated with SubType.
- * That's not necessary for the 'undefined' type (or if you look in the
- * `numbers` subdirectory, the 'number' type) or any concrete type,
- * but that's OK, the generic parameter doesn't hurt in those cases.
- ****/
-
-type ValueIntersectionByKeyUnion<T,  TKey extends keyof T> = {
-  [P in TKey]: (k: T[P])=>void
-} [TKey] extends ((k: infer I)=>void) ? I : never
-
-export interface AssociatedTypes<T> {
-   undefined: {
-      type: undefined
-      zero: undefined
-      one: undefined
-      nan: undefined
-      real: undefined
-   }
-}
-
-type AssociatedTypeNames = keyof AssociatedTypes<unknown>['undefined']
-type ALookup<T, Name extends AssociatedTypeNames> = ValueIntersectionByKeyUnion<{
-   [K in keyof AssociatedTypes<T>]:
-   T extends AssociatedTypes<T>[K]['type'] ? AssociatedTypes<T>[K][Name] : unknown},
-   keyof AssociatedTypes<T>>
-
-// For everything to compile, zero and one must be subtypes of T:
-export type ZeroType<T> = ALookup<T, 'zero'> & T
-export type OneType<T> = ALookup<T, 'one'> & T
-// But I believe 'nan' really might not be, like I think we will have to use
-// 'undefined' for the nan of 'bigint', as it has nothing at all like NaN,
-// so don't force it:
-export type NaNType<T> = ALookup<T, 'nan'>
-export type RealType<T> = ALookup<T, 'real'>
-
-/*****
- * The global signature patterns for all operations need to be published in the
- * following interface. Each key is the name of an operation (but note that
- * the Dispatcher will automatically merge operations that have the same
- * name when the first underscore `_` and everything thereafter is stripped).
- * The type of each key should be an interface with two properties: 'params'
- * whose type is the type of the parameter list for the operation, and
- * 'returns' whose type is the return type of the operation on those
- * parameters. These types are generic in a parameter type T which should
- * be interpreted as the type that the operation is supposed to "primarily"
- * operate on, although note that some of the parameters and/or return types
- * may depend on T rather than be exactly T.
- * So note that the example 're' below provides essentially the same
- * information that e.g.
- * `type ReOp<T> = (t: T) => RealType<T>`
- * would, but in a way that is much easier to manipulate in TypeScript,
- * and it records the name of the operation as 're' also by virtue of the
- * key 're' in the interface.
- ****/
-export interface Signatures<T> {
-   zero: (a: T) => ZeroType<T>
-   one:  (a: T) => OneType<T>
-   // nan needs to be able to operate on its own output for everything
-   // else to compile. That's why its parameter type is widened:
-   nan:  (a: T | NaNType<T>) => NaNType<T>
-   re:   (a: T) => RealType<T>
-}
-
-type SignatureKey<T> = keyof Signatures<T>
-
-export type Signature<Name extends SignatureKey<T>, T> = Signatures<T>[Name]
-export type Returns<Name extends SignatureKey<T>, T> =
-   ReturnType<Signatures<T>[Name]>
-type Deps<T> =  T extends unknown ? { [K in keyof T]: T[K] } : never;
-export type Dependencies<Name extends SignatureKey<T>, T> =
-   Deps<{[K in Name]: Signature<K, T>}>
-
-export type AliasOf<Name extends string, T> = T & {aliasOf?: Name}
-
-// For defining implementations with type reflection
-export function $reflect<ImplTuple>(tup: ImplTuple) {
-   +[[tup], <T>(elt: T) =>
-     elt.reflectedType = $$typeToString!<T>(true, false, true)]
-}

src/numbers/all.ts

@@ -1,3 +1,10 @@
+import {ForType} from '../core/Dispatcher.js'
+import {NumbersReturn} from './type.js'
 import * as numbers from './native.js'
 
 export {numbers}
+
+declare module "../core/Dispatcher" {
+   interface ReturnTypes<Params>
+      extends ForType<'numbers', NumbersReturn<Params>> {}
+}

src/numbers/arithmetic.ts

@@ -1,35 +1,72 @@
-import type {configDependency} from '../core/Config.js'
+import {configDependency} from '../core/Config.js'
-import type {Dependencies, Signature} from '../interfaces/type.js'
+import {
-import {$reflect} from '../interfaces/type.js'
+   Signature, ConservativeBinary, ConservativeUnary, Dependency, ImpType
+} from '../core/Dispatcher.js'
+import type {Complex, UnderlyingReal} from '../Complex/type.js'
 
-export const add =        (): Signature<'add',        number> => (a, b) => a + b
+declare module "./type" {
-const unaMinus = (a: number) => -a
+   interface NumbersReturn<Params> {
-export const unaryMinus = (): Signature<'unaryMinus', number> => unaMinus
+      // This description loses information: some subtypes like NumInt or
-export const conj =       (): Signature<'conj',       number> => a => a
+      // Positive are closed under addition, but this says that the result
-export const subtract =   (): Signature<'subtract',   number> => (a, b) => a - b
+      // of add is just a number, not still of the reduced type
-export const multiply =   (): Signature<'multiply',   number> => (a, b) => a * b
+      // add: Signature<Params, [number, number], number>
-export const absquare =   (): Signature<'absquare',   number> => a => a * a
-export const reciprocal = (): Signature<'reciprocal', number> => a => 1 / a
-export const divide =     (): Signature<'divide',     number> => (a, b) => a / b
 
-const basicSqrt = (a: number) => isNaN(a) ? NaN : Math.sqrt(a)
+      // Whereas this one preserves information, but lies
-export const conservativeSqrt = (): Signature<'conservativeSqrt', number> =>
+      // because it claims all subtypes of number are closed under addition,
-  basicSqrt
+      // which is not true for `1 | 2 | 3`, for example. But because in
+      // generics that use add we often need to assign the result of add
+      // to something of the exact generic type, generics using add won't
+      // compile unless we lie in this way and assert that add returns
+      // the subtype.
+      add: ConservativeBinary<Params, number>
+      // Not sure how this will need to go when we introduce NumInt.
+
+      addReal: Params extends [infer R, infer S]
+         ? R extends number ? S extends R ? R : never : never
+         : never
+      unaryMinus: ConservativeUnary<Params, number>
+      conj: ConservativeUnary<Params, number>
+      subtract: ConservativeBinary<Params, number>
+      multiply: ConservativeBinary<Params, number>
+      absquare: Params extends [infer R]
+         ? R extends number ? UnderlyingReal<R> : never
+         : never
+      reciprocal: ConservativeUnary<Params, number>
+      divide: ConservativeBinary<Params, number>
+      divideByReal: Params extends [infer R, infer S]
+         ? R extends number ? S extends R ? R : never : never
+         : never
+      // best square root that remains the same type
+      conservativeSqrt: ConservativeUnary<Params, number>
+      // Best we can do for sqrt at compile time, since actual return
+      // type depends on config. Not sure how this will play out
+      // when we make a number-only bundle, but at least the import type
+      // above for Complex<> does not lead to any emitted JavaScript.
+      sqrt: Signature<Params, [number], number | Complex<number>>
+   }
+}
+
+export const add: ImpType<'add', [number, number]> = (a, b) => a + b
+export const addReal = add
+export const unaryMinus: ImpType<'unaryMinus', [number]> = a => -a
+export const conj: ImpType<'conj', [number]> = a => a
+export const subtract: ImpType<'subtract', [number, number]> = (a, b) => a - b
+export const multiply: ImpType<'multiply', [number, number]> = (a, b) => a * b
+export const absquare: ImpType<'absquare', [number]> = a => a*a
+export const reciprocal: ImpType<'reciprocal', [number]> = a => 1/a
+export const divide: ImpType<'divide', [number, number]> = (a, b) => a / b
+export const divideByReal: ImpType<'divideByReal', [number, number]> = divide
+
+export const conservativeSqrt: ImpType<'conservativeSqrt', [number]> =
+   a => isNaN(a) ? NaN : Math.sqrt(a)
 
 export const sqrt =
-  (dep: configDependency
+   (dep: configDependency
-   & Dependencies<'complex', number>): Signature<'sqrt', number> => {
+    & Dependency<'complex', [number, number]>): ImpType<'sqrt', [number]> => {
-    if (dep.config.predictable || !dep.complex) {
+       if (dep.config.predictable || !dep.complex) return conservativeSqrt
-      return basicSqrt
+       return a => {
+          if (isNaN(a)) return NaN
+          if (a >= 0) return Math.sqrt(a)
+          return dep.complex(0, Math.sqrt(unaryMinus(a)))
+       }
     }
-    return a => {
-      if (isNaN(a)) return NaN
-      if (a >= 0) return Math.sqrt(a)
-      return dep.complex(0, Math.sqrt(unaMinus(a)))
-    }
-   }
-
-$reflect!([
-  add, unaryMinus, conj, subtract, multiply, absquare, reciprocal, divide,
-  conservativeSqrt, sqrt
-])

src/numbers/native.ts

@@ -1,4 +1,2 @@
 export * from './type.js'
 export * from './arithmetic.js'
-export * from './predicate.js'
-export * from './relational.js'

src/numbers/predicate.ts

@@ -1,7 +1,11 @@
-import type {Signature} from '../interfaces/type.js'
+import {Signature, ImpType} from '../core/Dispatcher.js'
-import {$reflect} from '../interfaces/type.js'
 
-export const isReal =   (): Signature<'isReal',   number> => (a) => true
+declare module "./type" {
-export const isSquare = (): Signature<'isSquare', number> => (a) => a >= 0
+   interface NumbersReturn<Params> {
+      isReal: Signature<Params, [number], true>
+      isSquare: Signature<Params, [number], boolean>
+   }
+}
 
-$reflect!([isReal, isSquare])
+export const isReal: ImpType<'isReal', [number]> = a => true
+export const isSquare: ImpType<'isSquare', [number]> = a => a >= 0

src/numbers/relational.ts

@@ -1,23 +1,34 @@
 import {configDependency} from '../core/Config.js'
-import {Signature} from '../interfaces/type.js'
+import {Signature, ImpType, Dependency} from '../core/Dispatcher.js'
-import {$reflect} from '../interfaces/type.js'
 
 const DBL_EPSILON = Number.EPSILON || 2.2204460492503130808472633361816E-16
 
-export const equal =
+declare module "./type" {
-   (dep: configDependency): Signature<'equal', number> =>
+   interface NumbersReturn<Params> {
-   (x, y) => {
+      equal: Signature<Params, [number, number], boolean>
-      const eps = dep.config.epsilon
+      unequal: Signature<Params, [number, number], boolean>
-      if (eps === null || eps === undefined) return x === y
+   }
-      if (x === y) return true
+}
-      if (isNaN(x) || isNaN(y)) return false
 
-      if (isFinite(x) && isFinite(y)) {
+export const equal =
-         const diff = Math.abs(x - y)
+   (dep: configDependency): ImpType<'equal', [number, number]> =>
-         if (diff < DBL_EPSILON) return true
+      (x, y) => {
-         return diff <= Math.max(Math.abs(x), Math.abs(y)) * eps
+         const eps = dep.config.epsilon
+         if (eps === null || eps === undefined) return x === y
+         if (x === y) return true
+         if (isNaN(x) || isNaN(y)) return false
+
+         if (isFinite(x) && isFinite(y)) {
+            const diff = Math.abs(x - y)
+            if (diff < DBL_EPSILON) return true
+            return diff <= Math.max(Math.abs(x), Math.abs(y)) * eps
+         }
+
+         return false
       }
 
-      return false
+export const unequal = (dep: Dependency<'equal', [number, number]>):
+   ImpType<'unequal', [number, number]> =>
+   (x, y) => {
+      return !dep.equal(x, y)
    }
-$reflect!([equal])

src/numbers/type.ts

@@ -1,29 +1,44 @@
-import type { Signature } from '../interfaces/type.js'
+import {ImpType} from '../core/Dispatcher.js'
-import {$reflect} from '../interfaces/type.js'
+import type {UnderlyingReal} from '../Complex/type.js'
 
 export const number_type = {
-   name: 'number', // just until we have reflection to tell us
    before: ['Complex'],
    test: (n: unknown): n is number => typeof n === 'number',
-   from: { string: (s: string) => +s }
+   from: {string: s => +s}
 }
 
-declare module "../interfaces/type" {
+
-   interface AssociatedTypes<T> {
+export interface NumbersReturn<Params> {
-      number: {
+   // The following description of the return type of `zero` on a single
-         type: number
+   // number argument has ended up unfortunately rather complicated. However,
-         zero: 0
+   // it illustrates the typing is really working: Suppose we have a
-         one: 1
+   // `type Small = 1 | 2 | 3`. Then Small indeed extends number, but we
-         nan: typeof NaN
+   // can't use the operation `zero(s: Small)` because zero is supposed to
-         real: number
+   // return something of the same type as its argument, but there is no
-      }
+   // zero in Small. Anyhow, in plain language the below says that given
-   }
+   // one parameter of a subtype of number, as long as that subtype includes 0,
+   // the zero operation returns a member of the type `0` (so we know even
+   // at compile time that its value will be 0).
+   zero: Params extends [infer T]
+      ? T extends number ? 0 extends T ? 0 : never : never
+      : never
+   // Note that in any case the simple
+   // zero: Signature<Params, [number], 0>
+   // makes complex fail to compile, because it worries that you might be
+   // making `Complex<Small>` where zero would not return the right type.
+
+   one: Params extends [infer T]
+      ? T extends number ? 1 extends T ? 1 : never : never
+      : never
+   nan: Params extends [infer T]
+      ? T extends number ? typeof NaN extends T ? typeof NaN : never : never
+      : never
+   re: Params extends [infer T]
+      ? T extends number ? UnderlyingReal<T> : never
+      : never
 }
 
-// I don't like the redundancy of repeating 'zero'; any way to eliminate that?
+export const zero: ImpType<'zero', [number]> = a => 0
-export const zero = (): Signature<'zero', number> => (a) => 0
+export const one:  ImpType<'one',  [number]> = a => 1
-export const one =  (): Signature<'one',  number> => (a) => 1
+export const nan:  ImpType<'nan',  [number]> = a => NaN
-export const nan =  (): Signature<'nan',  number> => (a) => NaN
+export const re:   ImpType<'re',   [number]> = a => a
-export const re =   (): Signature<'re',   number> => (a) => a
-
-$reflect!([zero, one, nan, re])

tsconfig.json

@@ -1,12 +1,7 @@
 {
     "compilerOptions": {
-        "target": "esnext",
+        "target": "ES2022",
         "rootDir": "./src",
-        "outDir": "./build",
+        "outDir": "./obj"
-        "moduleResolution": "nodenext",
-        "plugins": [ {
-            "transform": "ts-macros/dist/type-resolve",
-            "transformProgram": true
-        } ]
     }
 }