Declare implementations and dependencies via standard interfaces for operations (#8)

Adds a new subdirectory `interfaces` where standard interfaces
  are defined. Additional interfaces for a given operation can
  be added with an `AliasOf` type operator. Provides type
  operators that give the return type, full function type, and
  the type of a dependency on, a given operator.

  Resolves #6.

Co-authored-by: Glen Whitney <glen@studioinfinity.org>
Co-authored-by: Jos de Jong <wjosdejong@gmail.com>
Reviewed-on: #8

src/interfaces/arithmetic.ts

@@ -0,0 +1,18 @@
+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

@@ -0,0 +1,10 @@
+// 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

@@ -0,0 +1,9 @@
+// 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

@@ -0,0 +1,75 @@
+/*****
+ * 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.
+ ****/
+
+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> = {
+   [K in keyof AssociatedTypes<T>]:
+   T extends AssociatedTypes<T>[K]['type'] ? AssociatedTypes<T>[K][Name] : never
+}[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]>
+export type Dependencies<Name extends SignatureKey<T>, T> = {[K in Name]: Signature<K, T>}
+
+export type AliasOf<Name extends string, T> = T & {aliasOf?: Name}