diff --git a/src/bigint/native.mjs b/src/bigint/native.mjs
index dc12d9e..6458912 100644
--- a/src/bigint/native.mjs
+++ b/src/bigint/native.mjs
@@ -1,9 +1,11 @@
 import gcdType from '../generic/gcdType.mjs'
+import {identity} from '../generic/identity.mjs'
 
 export * from './Types/bigint.mjs'
 
 export {add} from './add.mjs'
 export {compare} from './compare.mjs'
+export const conjugate = {bigint: () => identity}
 export {divide} from './divide.mjs'
 export const gcd = gcdType('bigint')
 export {isZero} from './isZero.mjs'
diff --git a/src/complex/add.mjs b/src/complex/add.mjs
index 1afd22f..0c178d6 100644
--- a/src/complex/add.mjs
+++ b/src/complex/add.mjs
@@ -7,16 +7,16 @@ export const add = {
     */
    'Complex,number': ({
       'self(number,number)': addNum,
-      'complex(any,any)': cplx
+      'complex(number,number)': cplx
    }) => (z,x) => cplx(addNum(z.re, x), z.im),
 
    'Complex,bigint': ({
       'self(bigint,bigint)': addBigInt,
-      'complex(any,any)': cplx
+      'complex(bigint,bigint)': cplx
    }) => (z,x) => cplx(addBigInt(z.re, x), z.im),
 
    'Complex,Complex': ({
       self,
-      'complex(any,any)': cplx
-   }) => (w,z) => cplx(self(w.re, z.re), self(w.im, z.im))
+      complex
+   }) => (w,z) => complex(self(w.re, z.re), self(w.im, z.im))
 }
diff --git a/src/complex/associate.mjs b/src/complex/associate.mjs
index c6fab9e..f3106b4 100644
--- a/src/complex/associate.mjs
+++ b/src/complex/associate.mjs
@@ -4,7 +4,7 @@ export * from './Types/Complex.mjs'
 export const associate = {
     'Complex,Complex': ({
         'multiply(Complex,Complex)': times,
-        'equal(Complex,Complex)': eq,
+        'equalTT(Complex,Complex)': eq,
         zero,
         one,
         complex,
diff --git a/src/complex/complex.mjs b/src/complex/complex.mjs
index 58d3e2e..c34434c 100644
--- a/src/complex/complex.mjs
+++ b/src/complex/complex.mjs
@@ -9,7 +9,8 @@ export const complex = {
    'undefined': () => u => u,
    'undefined,any': () => (u, y) => u,
    'any,undefined': () => (x, u) => u,
-   'any,any': () => (x, y) => ({re: x, im: y}),
+   'undefined,undefined': () => (u, v) => u,
+   'T,T': () => (x, y) => ({re: x, im: y}),
    /* Take advantage of conversions in typed-function */
    Complex: () => z => z
 }
diff --git a/src/complex/equal.mjs b/src/complex/equalTT.mjs
similarity index 95%
rename from src/complex/equal.mjs
rename to src/complex/equalTT.mjs
index 4eca63f..43fe0d1 100644
--- a/src/complex/equal.mjs
+++ b/src/complex/equalTT.mjs
@@ -1,6 +1,6 @@
 export * from './Types/Complex.mjs'
 
-export const equal = {
+export const equalTT = {
    'Complex,number': ({
       'isZero(number)': isZ,
       'self(number,number)': eqNum
diff --git a/src/complex/native.mjs b/src/complex/native.mjs
index 1dfb45e..420ce88 100644
--- a/src/complex/native.mjs
+++ b/src/complex/native.mjs
@@ -6,7 +6,7 @@ export {add} from './add.mjs'
 export {associate} from './associate.mjs'
 export {complex} from './complex.mjs'
 export {conjugate} from './conjugate.mjs'
-export {equal} from './equal.mjs'
+export {equalTT} from './equalTT.mjs'
 export {gcd} from './gcd.mjs'
 export {invert} from './invert.mjs'
 export {isZero} from './isZero.mjs'
diff --git a/src/core/PocomathInstance.mjs b/src/core/PocomathInstance.mjs
index cc999c1..05636d4 100644
--- a/src/core/PocomathInstance.mjs
+++ b/src/core/PocomathInstance.mjs
@@ -2,7 +2,7 @@
 import typed from 'typed-function'
 import {dependencyExtractor, generateTypeExtractor} from './extractors.mjs'
 import {makeChain} from './Chain.mjs'
-import {typesOfSignature, subsetOfKeys} from './utils.mjs'
+import {typeListOfSignature, typesOfSignature, subsetOfKeys} from './utils.mjs'
 
 const anySpec = {} // fixed dummy specification of 'any' type
 
@@ -30,6 +30,8 @@ export default class PocomathInstance {
       'installType',
       'joinTypes',
       'name',
+      'self',
+      'Templates',
       'typeOf',
       'Types',
       'undefinedTypes'
@@ -41,13 +43,17 @@ export default class PocomathInstance {
       this._affects = {}
       this._typed = typed.create()
       this._typed.clear()
+      this._typed.addTypes([{name: 'ground', test: () => true}])
       /* List of types installed in the instance. We start with just dummies
        * for the 'any' type and for type parameters:
        */
       this.Types = {any: anySpec}
       this.Types[theTemplateParam] = anySpec
+      this.Types.ground = anySpec
       // All the template types that have been defined
-      this._templateTypes = {}
+      this.Templates = {}
+      // The actual type testing functions
+      this._typeTests = {}
       this._subtypes = {} // For each type, gives all of its (in)direct subtypes
       /* The following gives for each type, a set of all types that could
        * match in typed-function's dispatch algorithm before the given type.
@@ -56,8 +62,8 @@ export default class PocomathInstance {
        * might match.
        */
       this._priorTypes = {}
-      this._usedTypes = new Set() // all types that have occurred in a signature
-      this._doomed = new Set() // for detecting circular reference
+      this._seenTypes = new Set() // all types that have occurred in a signature
+      this._invalid = new Set() // methods that are currently invalid
       this._config = {predictable: false, epsilon: 1e-12}
       const self = this
       this.config = new Proxy(this._config, {
@@ -72,6 +78,12 @@ export default class PocomathInstance {
       })
       this._plainFunctions = new Set() // the names of the plain functions
       this._chainRepository = {} // place to store chainified functions
+
+      this._installFunctions({
+         typeOf: {ground: {uses: new Set(), does: () => () => 'any'}}
+      })
+
+      this.joinTypes = this.joinTypes.bind(this)
    }
 
    /**
@@ -181,10 +193,10 @@ export default class PocomathInstance {
 
    _installInstance(other) {
       for (const [type, spec] of Object.entries(other.Types)) {
-         if (type === 'any' || this._templateParam(type)) continue
+         if (spec === anySpec) continue
          this.installType(type, spec)
       }
-      for (const [base, info] of Object.entries(other._templateTypes)) {
+      for (const [base, info] of Object.entries(other.Templates)) {
          this._installTemplateType(info.type, info.spec)
       }
       const migrateImps = {}
@@ -290,7 +302,7 @@ export default class PocomathInstance {
       }
       let beforeType = spec.refines
       if (!beforeType) {
-         beforeType = 'any'
+         beforeType = 'ground'
          for (const other of spec.before || []) {
             if (other in this.Types) {
                beforeType = other
@@ -303,17 +315,14 @@ export default class PocomathInstance {
          const supertypeTest = this.Types[spec.refines].test
          testFn = entity => supertypeTest(entity) && spec.test(entity)
       }
+      this._typeTests[type] = testFn
       this._typed.addTypes([{name: type, test: testFn}], beforeType)
       this.Types[type] = spec
       this._subtypes[type] = new Set()
       this._priorTypes[type] = new Set()
-      // Update all the subtype sets of supertypes up the chain, and
-      // while we are at it add trivial conversions from subtypes to supertypes
-      // to help typed-function match signatures properly:
+      // Update all the subtype sets of supertypes up the chain
       let nextSuper = spec.refines
       while (nextSuper) {
-         this._typed.addConversion(
-            {from: type, to: nextSuper, convert: x => x})
          this._invalidateDependents(':' + nextSuper)
          this._priorTypes[nextSuper].add(type)
          this._subtypes[nextSuper].add(type)
@@ -325,6 +334,7 @@ export default class PocomathInstance {
             // add conversions from "from" to this one and all its supertypes:
             let nextSuper = type
             while (nextSuper) {
+               if (this._priorTypes[nextSuper].has(from)) break
                this._typed.addConversion(
                   {from, to: nextSuper, convert: spec.from[from]})
                this._invalidateDependents(':' + nextSuper)
@@ -384,6 +394,7 @@ export default class PocomathInstance {
       if (!typeA) return typeB
       if (!typeB) return typeA
       if (typeA === 'any' || typeB === 'any') return 'any'
+      if (typeA === 'ground' || typeB === 'ground') return 'ground'
       if (typeA === typeB) return typeA
       const subber = convert ? this._priorTypes : this._subtypes
       if (subber[typeB].has(typeA)) return typeB
@@ -409,7 +420,7 @@ export default class PocomathInstance {
     * signatures of operations, but which have not actually been installed:
     */
    undefinedTypes() {
-      return Array.from(this._usedTypes).filter(t => !(t in this.Types))
+      return Array.from(this._seenTypes).filter(t => !(t in this.Types))
    }
 
    /* Used internally to install a template type */
@@ -418,15 +429,15 @@ export default class PocomathInstance {
       /* For now, just allow a single template per base type; that
        * might need to change later:
        */
-      if (base in this._templateTypes) {
-         if (spec !== this._templateTypes[base].spec) {
+      if (base in this.Templates) {
+         if (spec !== this.Templates[base].spec) {
             throw new SyntaxError(
                `Conflicting definitions of template type ${type}`)
          }
          return
       }
       // Nothing actually happens until we match a template parameter
-      this._templateTypes[base] = {type, spec}
+      this.Templates[base] = {type, spec}
    }
 
    /* Used internally by install, see the documentation there */
@@ -469,7 +480,7 @@ export default class PocomathInstance {
                   for (const word of type.split(/[<>]/)) {
                      if (word.length == 0) continue
                      if (this._templateParam(word)) continue
-                     this._usedTypes.add(word)
+                     this._seenTypes.add(word)
                      this._addAffect(':' + word, name)
                   }
                }
@@ -497,20 +508,20 @@ export default class PocomathInstance {
     * and if it has no implementations so far, set them up.
     */
    _invalidate(name) {
-      if (this._doomed.has(name)) {
-         /* In the midst of a circular invalidation, so do nothing */
-         return
-      }
+      if (this._invalid.has(name)) return
       if (!(name in this._imps)) {
          this._imps[name] = {}
       }
-      this._doomed.add(name)
+      this._invalid.add(name)
       this._invalidateDependents(name)
-      this._doomed.delete(name)
       const self = this
       Object.defineProperty(this, name, {
          configurable: true,
-         get: () => self._bundle(name)
+         get: () => {
+            const result = self._bundle(name)
+            self._invalid.delete(name)
+            return result
+         }
       })
    }
 
@@ -541,7 +552,7 @@ export default class PocomathInstance {
          for (const type of typesOfSignature(entry[0])) {
             if (type in this.Types) continue
             const baseType = type.split('<')[0]
-            if (baseType in this._templateTypes) continue
+            if (baseType in this.Templates) continue
             keep = false
             break
          }
@@ -578,16 +589,10 @@ export default class PocomathInstance {
             behavior.instantiations = new Set()
          }
          let instantiationSet = new Set()
-         let trimSignature = rawSignature
-         if (rawSignature.charAt(0) === '!') {
-            trimSignature = trimSignature.slice(1)
-            instantiationSet = this._usedTypes
-         } else {
-            for (const instType of behavior.instantiations) {
-               instantiationSet.add(instType)
-               for (const other of this._priorTypes[instType]) {
-                  instantiationSet.add(other)
-               }
+         for (const instType of behavior.instantiations) {
+            instantiationSet.add(instType)
+            for (const other of this._priorTypes[instType]) {
+               instantiationSet.add(other)
             }
          }
 
@@ -595,7 +600,7 @@ export default class PocomathInstance {
             if (!(instType in this.Types)) continue
             if (this.Types[instType] === anySpec) continue
             const signature =
-                  substituteInSig(trimSignature, theTemplateParam, instType)
+                  substituteInSig(rawSignature, theTemplateParam, instType)
             /* Don't override an explicit implementation: */
             if (signature in imps) continue
             const uses = new Set()
@@ -622,15 +627,15 @@ export default class PocomathInstance {
          /* Now add the catchall signature */
          let templateCall = `<${theTemplateParam}>`
          /* Relying here that the base of 'Foo<T>' is 'Foo': */
-         let baseSignature = trimSignature.replaceAll(templateCall, '')
+         let baseSignature = rawSignature.replaceAll(templateCall, '')
          /* Any remaining template params are top-level */
          const signature = substituteInSig(
-            baseSignature, theTemplateParam, 'any')
+            baseSignature, theTemplateParam, 'ground')
          /* The catchall signature has to detect the actual type of the call
           * and add the new instantiations.
           * First, prepare the type inference data:
           */
-         const parTypes = trimSignature.split(',')
+         const parTypes = rawSignature.split(',')
          const restParam = (parTypes[parTypes.length-1].slice(0,3) === '...')
          const topTyper = entity => this.typeOf(entity)
          const inferences = parTypes.map(
@@ -639,7 +644,7 @@ export default class PocomathInstance {
                theTemplateParam,
                topTyper,
                this.joinTypes.bind(this),
-               this._templateTypes))
+               this.Templates))
          if (inferences.every(x => !x)) { // all false
             throw new SyntaxError(
                `Cannot find template parameter in ${rawSignature}`)
@@ -700,7 +705,6 @@ export default class PocomathInstance {
                usedConversions = true
                instantiateFor = self.joinTypes(argTypes, usedConversions)
                if (instantiateFor === 'any') {
-                  // Need a more informative error message here
                   throw TypeError(
                      `In call to ${name}, no type unifies arguments `
                         + args.toString() + '; of types ' + argTypes.toString()
@@ -717,7 +721,7 @@ export default class PocomathInstance {
              * by instantiateFor, and for all of instantiateFor's "prior types"
              */
             for (j = 0; j < parTypes.length; ++j) {
-               if (wantTypes[j] !== parTypes[j] && wantTypes[j].includes('<')) {
+               if (wantTypes[j] !== parTypes[j] && parTypes[j].includes('<')) {
                   // actually used the param and is a template
                   self._ensureTemplateTypes(parTypes[j], instantiateFor)
                }
@@ -762,8 +766,9 @@ export default class PocomathInstance {
                   if (self._typed.isTypedFunction(refs[simplifiedDep])) {
                      const subsig = substituteInSig(
                         needsig, theTemplateParam, instantiateFor)
-                     innerRefs[dep] = self._typed.find(
-                        refs[simplifiedDep], subsig)
+                     let resname = simplifiedDep
+                     if (resname === 'self') resname = name
+                     innerRefs[dep] = self._pocoresolve(resname, subsig)
                   } else {
                      innerRefs[dep] = refs[simplifiedDep]
                   }
@@ -841,7 +846,7 @@ export default class PocomathInstance {
                // can bundle up func, and grab its signature if need be
                let destination = this[func]
                if (needsig) {
-                  destination = this._typed.find(destination, needsig)
+                  destination = this._pocoresolve(func, needsig)
                }
                refs[dep] = destination
             }
@@ -886,27 +891,10 @@ export default class PocomathInstance {
                corrected_self_references.push(neededSig)
                continue
             }
-            // No exact match, have to try to get one that matches with
+            // No exact match, try to get one that matches with
             // subtypes since the whole conversion thing in typed-function
             // is too complicated to reproduce
-            let foundSig = false
-            const typeList = typesOfSignature(neededSig)
-            for (const otherSig in imps) {
-               const otherTypeList = typesOfSignature(otherSig)
-               if (typeList.length !== otherTypeList.length) continue
-               const allMatch = true
-               for (let k = 0; k < typeList.length; ++k) {
-                  if (this._subtypes[otherTypeList[k]].has(typeList[k])) {
-                     continue
-                  }
-                  allMatch = false
-                  break
-               }
-               if (allMatch) {
-                  foundSig = otherSig
-                  break
-               }
-            }
+            const foundSig = this._findSubtypeImpl(imps, neededSig)
             if (foundSig) {
                corrected_self_references.push(foundSig)
             } else {
@@ -963,18 +951,14 @@ export default class PocomathInstance {
       if (wantsType in this.Types) return false
       // OK, need to generate the type from the template
       // Set up refines, before, test, and from
-      const newTypeSpec = {}
-      const template = this._templateTypes[base].spec
+      const newTypeSpec = {refines: base}
+      const maybeFrom = {}
+      const template = this.Templates[base].spec
       if (!template) {
          throw new Error(
             `Implementor error in _maybeAddTemplateType ${base} ${instantiator}`)
       }
       const instantiatorSpec = this.Types[instantiator]
-      if (instantiatorSpec.refines) {
-         // Assuming all templates are covariant, for now
-         this._maybeAddTemplateType(base, instantiatorSpec.refines)
-         newTypeSpec.refines = `${base}<${instantiatorSpec.refines}>`
-      }
       let beforeTypes = []
       if (instantiatorSpec.before) {
          beforeTypes = instantiatorSpec.before.map(type => `${base}<${type}>`)
@@ -988,9 +972,8 @@ export default class PocomathInstance {
       if (beforeTypes.length > 0) {
          newTypeSpec.before = beforeTypes
       }
-      newTypeSpec.test = template.test(instantiatorSpec.test)
+      newTypeSpec.test = template.test(this._typeTests[instantiator])
       if (template.from) {
-         newTypeSpec.from = {}
          for (let source in template.from) {
             const instSource = substituteInSig(
                source, theTemplateParam, instantiator)
@@ -1005,16 +988,87 @@ export default class PocomathInstance {
                for (const iFrom in instantiatorSpec.from) {
                   const finalSource = substituteInSig(
                      instSource, templateFromParam, iFrom)
-                  newTypeSpec.from[finalSource] = template.from[source](
+                  maybeFrom[finalSource] = template.from[source](
                      instantiatorSpec.from[iFrom])
                }
+               // Assuming all templates are covariant here, I guess...
+               for (const subType of this._subtypes[instantiator]) {
+                  const finalSource = substituteInSig(
+                     instSource, templateFromParam, subType)
+                  maybeFrom[finalSource] = template.from[source](x => x)
+               }
             } else {
-               newTypeSpec.from[instSource] = template.from[source]
+               maybeFrom[instSource] = template.from[source]
             }
          }
       }
+
+      if (Object.keys(maybeFrom).length > 0) {
+         newTypeSpec.from = maybeFrom
+      }
       this.installType(wantsType, newTypeSpec)
       return wantsType
    }
 
+   _findSubtypeImpl(imps, neededSig) {
+      if (neededSig in imps) return neededSig
+      let foundSig = false
+      const typeList = typeListOfSignature(neededSig)
+      for (const otherSig in imps) {
+         const otherTypeList = typeListOfSignature(otherSig)
+         if (typeList.length !== otherTypeList.length) continue
+         let allMatch = true
+         for (let k = 0; k < typeList.length; ++k) {
+            let myType = typeList[k]
+            let otherType = otherTypeList[k]
+            if (otherType === theTemplateParam) {
+               otherTypeList[k] = 'ground'
+               otherType = 'ground'
+            }
+            if (otherType === '...T') {
+               otherTypeList[k] = '...ground'
+               otherType = 'ground'
+            }
+            const adjustedOtherType = otherType.replaceAll(
+               `<${theTemplateParam}>`, '')
+            if (adjustedOtherType !== otherType) {
+               otherTypeList[k] = adjustedOtherType
+               otherType = adjustedOtherType
+            }
+            if (myType.slice(0,3) === '...') myType = myType.slice(3)
+            if (otherType.slice(0,3) === '...') otherType = otherType.slice(3)
+            if (otherType === 'any') continue
+            if (otherType === 'ground') continue
+            if (!(otherType in this.Types)) {
+               allMatch = false
+               break
+            }
+            if (myType === otherType
+                || this._subtypes[otherType].has(myType)) {
+               continue
+            }
+            allMatch = false
+            break
+         }
+         if (allMatch) {
+            foundSig = otherTypeList.join(',')
+            break
+         }
+      }
+      return foundSig
+   }
+
+   _pocoresolve(name, sig) {
+      const typedfunc = this[name]
+      let result = undefined
+      try {
+         result = this._typed.find(typedfunc, sig, {exact: true})
+      } catch {
+      }
+      if (result) return result
+      const foundsig = this._findSubtypeImpl(this._imps[name], sig)
+      if (foundsig) return this._typed.find(typedfunc, foundsig)
+      return this._typed.find(typedfunc, sig)
+   }
+
 }
diff --git a/src/core/utils.mjs b/src/core/utils.mjs
index 11a879f..db164dd 100644
--- a/src/core/utils.mjs
+++ b/src/core/utils.mjs
@@ -6,6 +6,11 @@ export function subsetOfKeys(set, obj) {
    return true
 }
 
+/* Returns a list of the types mentioned in a typed-function signature */
+export function typeListOfSignature(signature) {
+   return signature.split(',').map(s => s.trim())
+}
+
 /* Returns a set of all of the types mentioned in a typed-function signature */
 export function typesOfSignature(signature) {
    return new Set(signature.split(/[^\w\d]/).filter(s => s.length))
diff --git a/src/generic/arithmetic.mjs b/src/generic/arithmetic.mjs
index 04948df..00faddb 100644
--- a/src/generic/arithmetic.mjs
+++ b/src/generic/arithmetic.mjs
@@ -4,6 +4,7 @@ export * from './Types/generic.mjs'
 
 export const add = reducingOperation
 export const gcd = reducingOperation
+export {identity} from './identity.mjs'
 export {lcm} from './lcm.mjs'
 export {mean} from './mean.mjs'
 export {mod} from './mod.mjs'
diff --git a/src/generic/identity.mjs b/src/generic/identity.mjs
new file mode 100644
index 0000000..2422d2f
--- /dev/null
+++ b/src/generic/identity.mjs
@@ -0,0 +1,3 @@
+export function identity(x) {
+   return x
+}
diff --git a/src/generic/relational.mjs b/src/generic/relational.mjs
index 368f394..939ae19 100644
--- a/src/generic/relational.mjs
+++ b/src/generic/relational.mjs
@@ -7,14 +7,27 @@ export const isZero = {
 }
 
 export const equal = {
-   '!T,T': ({
+   'any,any': ({equalTT, joinTypes, Templates, typeOf}) => (x,y) => {
+      const resultant = joinTypes([typeOf(x), typeOf(y)], 'convert')
+      if (resultant === 'any' || resultant in Templates) {
+         return false
+      }
+      return equalTT(x,y)
+   }
+}
+
+export const equalTT = {
+   'T,T': ({
       'compare(T,T)': cmp,
       'isZero(T)': isZ
-   }) => (x,y) => isZ(cmp(x,y))
+   }) => (x,y) => isZ(cmp(x,y)),
+   // If templates were native to typed-function, we should be able to
+   // do something like:
+   // 'any,any': () => () => false // should only be hit for different types
 }
 
 export const unequal = {
-   'T,T': ({'equal(T.T)': eq}) => (x,y) => !(eq(x,y))
+   'any,any': ({equal}) => (x,y) => !(equal(x,y))
 }
 
 export const larger = {
diff --git a/src/number/native.mjs b/src/number/native.mjs
index 1404ab4..d095574 100644
--- a/src/number/native.mjs
+++ b/src/number/native.mjs
@@ -1,10 +1,12 @@
 import gcdType from '../generic/gcdType.mjs'
+import {identity} from '../generic/identity.mjs'
 
 export * from './Types/number.mjs'
 
 export {abs} from './abs.mjs'
 export {add} from './add.mjs'
 export {compare} from './compare.mjs'
+export const conjugate = {number: () => identity}
 export const gcd = gcdType('NumInt')
 export {invert} from './invert.mjs'
 export {isZero} from './isZero.mjs'
diff --git a/src/ops/floor.mjs b/src/ops/floor.mjs
index 98d524a..e8897e8 100644
--- a/src/ops/floor.mjs
+++ b/src/ops/floor.mjs
@@ -12,7 +12,7 @@ export const floor = {
    // entry with type `bigint|NumInt|GaussianInteger` because they couldn't
    // be separately activated then
 
-   number: ({'equal(number,number)': eq}) => n => {
+   number: ({'equalTT(number,number)': eq}) => n => {
       if (eq(n, Math.round(n))) return Math.round(n)
       return Math.floor(n)
    },
diff --git a/src/pocomath.mjs b/src/pocomath.mjs
index 53e646f..d8df045 100644
--- a/src/pocomath.mjs
+++ b/src/pocomath.mjs
@@ -4,18 +4,10 @@ import * as numbers from './number/native.mjs'
 import * as bigints from './bigint/native.mjs'
 import * as complex from './complex/native.mjs'
 import * as tuple from './tuple/native.mjs'
-// Most of tuple is not ready yet:
-const tupleReady = {
-    Tuple: tuple.Tuple,
-    equal: tuple.equal,
-    isZero: tuple.isZero,
-    length: tuple.length,
-    tuple: tuple.tuple
-}
 import * as generic from './generic/all.mjs'
 import * as ops from './ops/all.mjs'
 
 const math = PocomathInstance.merge(
-    'math', numbers, bigints, complex, tupleReady, generic, ops)
+    'math', numbers, bigints, complex, tuple, generic, ops)
 
 export default math
diff --git a/src/tuple/Types/Tuple.mjs b/src/tuple/Types/Tuple.mjs
index 7b4ed35..0c6c0ae 100644
--- a/src/tuple/Types/Tuple.mjs
+++ b/src/tuple/Types/Tuple.mjs
@@ -23,7 +23,7 @@ Tuple.installType('Tuple<T>', {
    from: {
       'Tuple<U>': convert => tu => ({elts: tu.elts.map(convert)}),
       // Here since there is no U it's a straight conversion:
-      T: t => ({elts: [u]}), // singleton promotion
+      T: t => ({elts: [t]}), // singleton promotion
       // Whereas the following will let you go directly from an element
       // convertible to T to a singleton Tuple<T>. Not sure if we really
       // want that, but we'll try it just for kicks.
@@ -32,48 +32,48 @@ Tuple.installType('Tuple<T>', {
 })
 
 Tuple.promoteUnary = {
-   'Tuple<T>': ({'self(T)': me}) => t => ({elts: t.elts.map(me)})
+   'Tuple<T>': ({'self(T)': me, tuple}) => t => tuple(...(t.elts.map(me)))
 }
 
 Tuple.promoteBinaryUnary = {
-   'Tuple<T>,Tuple<T>': ({'self(T,T)': meB, 'self(T)': meU}) => (s,t) => {
+   'Tuple<T>,Tuple<T>': ({'self(T,T)': meB, 'self(T)': meU, tuple}) => (s,t) => {
       let i = -1
       let result = []
       while (true) {
          i += 1
          if (i < s.elts.length) {
-            if (i < t.elts.length) result.append(meB(s.elts[i], t.elts[i]))
-            else results.append(meU(s.elts[i]))
+            if (i < t.elts.length) result.push(meB(s.elts[i], t.elts[i]))
+            else result.push(meU(s.elts[i]))
             continue
          }
-         if (i < t.elts.length) result.append(meU(t.elts[i]))
+         if (i < t.elts.length) result.push(meU(t.elts[i]))
          else break
       }
-      return {elts: result}
+      return tuple(...result)
    }
 }
 
 Tuple.promoteBinary = {
-   'Tuple<T>,Tuple<T>': ({'self(T,T)': meB}) => (s,t) => {
+   'Tuple<T>,Tuple<T>': ({'self(T,T)': meB, tuple}) => (s,t) => {
       const lim = Math.max(s.elts.length, t.elts.length)
       const result = []
       for (let i = 0; i < lim; ++i) {
-         result.append(meB(s.elts[i], t.elts[i]))
+         result.push(meB(s.elts[i], t.elts[i]))
       }
-      return {elts: result}
+      return tuple(...result)
    }
 }
 
 Tuple.promoteBinaryStrict = {
-   'Tuple<T>,Tuple<T>': ({'self(T,T)': meB}) => (s,t) => {
+   'Tuple<T>,Tuple<T>': ({'self(T,T)': meB, tuple}) => (s,t) => {
       if (s.elts.length !== t.elts.length) {
          throw new RangeError('Tuple length mismatch') // get name of self ??
       }
       const result = []
       for (let i = 0; i < s.elts.length; ++i) {
-         result.append(meB(s.elts[i], t.elts[i]))
+         result.push(meB(s.elts[i], t.elts[i]))
       }
-      return {elts: result}
+      return tuple(...result)
    }
 }
 
diff --git a/src/tuple/equal.mjs b/src/tuple/equal.mjs
deleted file mode 100644
index 8186624..0000000
--- a/src/tuple/equal.mjs
+++ /dev/null
@@ -1,14 +0,0 @@
-export * from './Types/Tuple.mjs'
-
-export const equal = {
-   // Change this to a template implementation (or maybe add template
-   // implementation to handle matching types, and have mixed template-base
-   // method returning false to catch test of two tuples of different types.
-   'Tuple,Tuple': ({self, length}) => (s,t) => {
-      if (length(s) !== length(t)) return false
-      for (let i = 0; i < length(s); ++i) {
-         if (!self(s.elts[i], t.elts[i])) return false
-      }
-      return true
-   }
-}
diff --git a/src/tuple/equalTT.mjs b/src/tuple/equalTT.mjs
new file mode 100644
index 0000000..1606410
--- /dev/null
+++ b/src/tuple/equalTT.mjs
@@ -0,0 +1,11 @@
+export * from './Types/Tuple.mjs'
+
+export const equalTT = {
+   'Tuple<T>,Tuple<T>': ({'self(T,T)': me, 'length(Tuple)': len}) => (s,t) => {
+      if (len(s) !== len(t)) return false
+      for (let i = 0; i < len(s); ++i) {
+         if (!me(s.elts[i], t.elts[i])) return false
+      }
+      return true
+   }
+}
diff --git a/src/tuple/native.mjs b/src/tuple/native.mjs
index fcf6f0d..66ba8f0 100644
--- a/src/tuple/native.mjs
+++ b/src/tuple/native.mjs
@@ -3,8 +3,8 @@ import {Tuple} from './Types/Tuple.mjs'
 export const add = Tuple.promoteBinaryUnary
 export const complex = Tuple.promoteBinaryStrict
 export const conjugate = Tuple.promoteUnary
-// May want to replace equal with compare based on lexicographic ordering?
-export {equal} from './equal.mjs'
+export const divide = Tuple.promoteBinaryStrict
+export {equalTT} from './equalTT.mjs'
 export const invert = Tuple.promoteUnary
 export {isZero} from './isZero.mjs'
 export {length} from './length.mjs'
@@ -14,6 +14,7 @@ export const one = Tuple.promoteUnary
 export const quotient = Tuple.promoteBinaryStrict
 export const roundquotient = Tuple.promoteBinaryStrict
 export const sqrt = Tuple.promoteUnary
+export const subtract = Tuple.promoteBinaryStrict
 export {tuple} from './tuple.mjs'
 export const zero = Tuple.promoteUnary
 
diff --git a/test/custom.mjs b/test/custom.mjs
index d232dd5..6a399b0 100644
--- a/test/custom.mjs
+++ b/test/custom.mjs
@@ -18,9 +18,10 @@ describe('A custom instance', () => {
    it("works when partially assembled", () => {
       bw.install(complex)
       // Not much we can call without any number types:
-      const i3 = {re: 0, im: 3}
-      assert.deepStrictEqual(bw.complex(0, 3), i3)
-      assert.deepStrictEqual(bw.chain(0).complex(3).value, i3)
+      assert.deepStrictEqual(bw.complex(undefined, undefined), undefined)
+      assert.deepStrictEqual(
+         bw.chain(undefined).complex(undefined).value,
+         undefined)
       // Don't have a way to negate things, for example:
       assert.throws(() => bw.negate(2), TypeError)
    })
@@ -34,7 +35,7 @@ describe('A custom instance', () => {
       assert.deepStrictEqual(
          bw.subtract(16, bw.add(3, bw.complex(0,4), 2)),
          math.complex(11, -4)) // note both instances coexist
-      assert.deepStrictEqual(bw.negate(math.complex(3, '8')).im, -8)
+      assert.deepStrictEqual(bw.negate(bw.complex(3, '8')).im, -8)
    })
 
    it("can be assembled piecemeal", () => {
@@ -130,12 +131,9 @@ describe('A custom instance', () => {
          inst.typeMerge(3, inst.complex(4.5,2.1)),
          'Merge to Complex')
       // The following is the current behavior, since 3 converts to 3+0i
-      // and 3n converts to 3n+0ni, both of which are technically Complex.
-      // This will remain the case even with templated Complex, because
-      // both Complex<bigint> and Complex<NumInt> will refine Complex (for the
-      // sake of catching new specializations). Not sure whether that will be
-      // OK or a problem that will have to be dealt with.
-      assert.strictEqual(inst.typeMerge(3, 3n), 'Merge to Complex')
+      // which is technically the same Complex type as 3n+0ni.
+      // This should clear up when Complex is templatized
+      assert.strictEqual(inst.typeMerge(3, inst.complex(3n)), 'Merge to Complex')
       // But types that truly cannot be merged should throw a TypeError
       // Should add a variation of this with a more usual type once there is
       // one not interconvertible with others...
diff --git a/test/tuple/_native.mjs b/test/tuple/_native.mjs
index 7727ac9..2cf56d1 100644
--- a/test/tuple/_native.mjs
+++ b/test/tuple/_native.mjs
@@ -18,11 +18,97 @@ describe('tuple', () => {
          {elts: [math.complex(3), math.complex(2n), math.complex(5.2)]})
    })
 
-   it('can be tested for zero', () => {
+   it('can be tested for zero and equality', () => {
       assert.strictEqual(math.isZero(math.tuple(0,1)), false)
       assert.strictEqual(math.isZero(math.tuple(0n,0n,0n,0n)), true)
       assert.strictEqual(math.isZero(math.tuple(0,0.001,0)), false)
+      assert.deepStrictEqual(math.complex(0,0), {re: 0, im:0})
       assert.strictEqual(math.isZero(math.tuple(0,math.complex(0,0))), true)
+      assert.strictEqual(
+         math.equal(
+            math.tuple(0,math.complex(0,0.1)),
+            math.complex(math.tuple(0,0), math.tuple(0,0.1))),
+         true)
+      assert.strictEqual(
+         math.equal(math.tuple(3n,2n), math.tuple(3,2)),
+         false)
+   })
+
+   it('supports addition', () => {
+      assert.deepStrictEqual(
+         math.add(math.tuple(3,4,5), math.tuple(2,1,0)),
+         math.tuple(5,5,5))
+      assert.deepStrictEqual(
+         math.add(math.tuple(3.25,4.5,5), math.tuple(3,3)),
+         math.tuple(6.25,7.5,5))
+      assert.deepStrictEqual(
+         math.add(math.tuple(math.complex(2,3), 7), math.tuple(4, 5, 6)),
+         math.tuple(math.complex(6,3), math.complex(12), math.complex(6)))
+      assert.deepStrictEqual(
+         math.add(math.tuple(5,6), 7),
+         math.tuple(12,6))
+      assert.deepStrictEqual(
+         math.add(math.tuple(math.complex(5,4),6), 7),
+         math.tuple(math.complex(12,4),math.complex(6)))
+   })
+
+   it('supports subtraction', () => {
+      assert.deepStrictEqual(
+         math.subtract(math.tuple(3n,4n,5n), math.tuple(2n,1n,0n)),
+         math.tuple(1n,3n,5n))
+      assert.throws(
+         () => math.subtract(math.tuple(5,6), math.tuple(7)),
+         /RangeError/)
+   })
+
+   it('makes a tuple of complex and conjugates it', () => {
+      const complexTuple = math.tuple(
+         math.complex(3,1), math.complex(4,2.2), math.complex(5,3))
+      assert.deepStrictEqual(
+         math.complex(math.tuple(3,4,5), math.tuple(1,2.2,3)),
+         complexTuple)
+      assert.deepStrictEqual(
+         math.conjugate(complexTuple),
+         math.tuple(math.complex(3,-1), math.complex(4,-2.2), math.complex(5,-3)))
+   })
+
+   it('supports division', () => {
+      assert.deepStrictEqual(
+         math.divide(math.tuple(3,4,5),math.tuple(1,2,2)),
+         math.tuple(3,2,2.5))
+   })
+
+   it('supports multiplication', () => {
+      assert.deepStrictEqual(
+         math.multiply(math.tuple(3,4,5), math.tuple(1,2,2)),
+         math.tuple(3,8,10))
+   })
+
+   it('supports one and zero', () => {
+      assert.deepStrictEqual(
+         math.one(math.tuple(2n,3n,0n)),
+         math.tuple(1n,1n,1n))
+      assert.deepStrictEqual(
+         math.zero(math.tuple(math.complex(5,2), 3.4)),
+         math.tuple(math.complex(0), math.complex(0)))
+   })
+
+   it('supports quotient and roundquotient', () => {
+      const bigTuple = math.tuple(1n,2n,3n,4n,5n)
+      const bigOnes = math.one(bigTuple)
+      const threes = math.add(bigOnes, bigOnes, bigOnes)
+      assert.deepStrictEqual(
+         math.quotient(bigTuple, threes),
+         math.tuple(0n, 0n, 1n, 1n, 1n))
+      assert.deepStrictEqual(
+         math.roundquotient(bigTuple, threes),
+         math.tuple(0n, 1n, 1n, 1n, 2n))
+   })
+
+   it('supports sqrt', () => {
+      assert.deepStrictEqual(
+         math.sqrt(math.tuple(4,-4,2.25)),
+         math.tuple(2, math.complex(0,2), 1.5))
    })
 
 })