diff --git a/src/core/PocomathInstance.mjs b/src/core/PocomathInstance.mjs
index 22720d0..1fce6b3 100644
--- a/src/core/PocomathInstance.mjs
+++ b/src/core/PocomathInstance.mjs
@@ -1,8 +1,8 @@
 /* Core of pocomath: create an instance */
 import typed from 'typed-function'
-import dependencyExtractor from './dependencyExtractor.mjs'
+import {dependencyExtractor, generateTypeExtractor} from './extractors.mjs'
 import {makeChain} from './Chain.mjs'
-import {subsetOfKeys, typesOfSignature} from './utils.mjs'
+import {typesOfSignature, subsetOfKeys} from './utils.mjs'
 
 const anySpec = {} // fixed dummy specification of 'any' type
 
@@ -461,9 +461,12 @@ export default class PocomathInstance {
                   this._addAffect(depname, name)
                }
                for (const type of typesOfSignature(signature)) {
-                  if (this._templateParam(type)) continue
-                  this._usedTypes.add(type)
-                  this._addAffect(':' + type, name)
+                  for (const word of type.split(/[<>]/)) {
+                     if (word.length == 0) continue
+                     if (this._templateParam(word)) continue
+                     this._usedTypes.add(word)
+                     this._addAffect(':' + word, name)
+                  }
                }
             }
          }
@@ -526,23 +529,38 @@ export default class PocomathInstance {
       if (!imps) {
          throw new SyntaxError(`No implementations for ${name}`)
       }
-      const usableEntries = Object.entries(imps).filter(
-         ([signature]) => subsetOfKeys(typesOfSignature(signature), this.Types))
+      /* Collect the entries we know the types for */
+      const usableEntries = []
+      for (const entry of Object.entries(imps)) {
+         let keep = true
+         for (const type of typesOfSignature(entry[0])) {
+            if (type in this.Types) continue
+            const baseType = type.split('<')[0]
+            if (baseType in this._templateTypes) continue
+            keep = false
+            break
+         }
+         if (keep) usableEntries.push(entry)
+      }
       if (usableEntries.length === 0) {
          throw new SyntaxError(
             `Every implementation for ${name} uses an undefined type;\n`
                + `    signatures: ${Object.keys(imps)}`)
       }
-      // Mark this method as being in the midst of being reassembled
+      /* Initial error checking done; mark this method as being
+       * in the midst of being reassembled
+       */
       Object.defineProperty(this, name, {configurable: true, value: 'limbo'})
       const tf_imps = {}
       for (const [rawSignature, behavior] of usableEntries) {
          /* Check if it's an ordinary non-template signature */
          let explicit = true
          for (const type of typesOfSignature(rawSignature)) {
-            if (this._templateParam(type)) { // template types need better check
-               explicit = false
-               break
+            for (const word of type.split(/[<>]/)) {
+               if (this._templateParam(word)) {
+                  explicit = false
+                  break
+               }
             }
          }
          if (explicit) {
@@ -591,36 +609,31 @@ export default class PocomathInstance {
                      innerRefs[dep] = refs[outerName]
                   }
                }
-               const original = behavior.does(innerRefs)
                return behavior.does(innerRefs)
             }
             this._addTFimplementation(tf_imps, signature, {uses, does: patch})
          }
          /* Now add the catchall signature */
+         let templateCall = `<${theTemplateParam}>`
+         /* Relying here that the base of 'Foo<T>' is 'Foo': */
+         let baseSignature = substituteInSig(trimSignature, templateCall, '')
+         /* Any remaining template params are top-level */
          const signature = substituteInSig(
-            trimSignature, theTemplateParam, 'any')
+            baseSignature, theTemplateParam, 'any')
          /* The catchall signature has to detect the actual type of the call
-          * and add the new instantiations. We should really be using the
-          * typed-function parser to do the manipulations below, but we don't
-          * have access. The firs section prepares the type inference data:
+          * and add the new instantiations.
+          * First, prepare the type inference data:
           */
          const parTypes = trimSignature.split(',')
-         const inferences = []
-         const typer = entity => this.typeOf(entity)
-         let ambiguous = true
-         for (let parType of parTypes) {
-            parType = parType.trim()
-            if (parType.slice(0,3) === '...') {
-               parType = parType.slice(3).trim()
-            }
-            if (parType === theTemplateParam) {
-               inferences.push(typer)
-               ambiguous = false
-            } else {
-               inferences.push(false)
-            }
-         }
-         if (ambiguous) {
+         const topTyper = entity => this.typeOf(entity)
+         const inferences = parTypes.map(
+            type => generateTypeExtractor(
+               type,
+               theTemplateParam,
+               topTyper,
+               this.joinTypes.bind(this),
+               this._templateTypes))
+         if (inferences.every(x => !x)) { // all false
             throw new SyntaxError(
                `Cannot find template parameter in ${rawSignature}`)
          }
@@ -654,9 +667,24 @@ export default class PocomathInstance {
                usedConversions = true
                instantiateFor = self.joinTypes(argTypes, usedConversions)
                if (instantiateFor === 'any') {
+                  // Need a more informative error message here
                   throw TypeError('No common type for arguments to ' + name)
                }
             }
+            /* Generate the list of actual wanted types */
+            const wantTypes = parTypes.map(type => substituteInSig(
+               type, theTemplateParam, instantiateFor))
+            /* Now we have to add any actual types that are relevant
+             * to this invocation. Namely, that would be every formal parameter
+             * type in the invocation, with the parameter template instantiated
+             * by instantiateFor, and for all of instantiateFor's "prior types"
+             */
+            for (j = 0; j < parTypes.length; ++j) {
+               if (wantTypes[i] !== parTypes[i] && wantTypes.includes('<')) {
+                  // actually used the param and is a template
+                  self._ensureTemplateTypes(parTypes[i], instantiateFor)
+               }
+            }
             /* Transform the arguments if we used any conversions: */
             if (usedConversions) {
                i = - 1
@@ -774,4 +802,12 @@ export default class PocomathInstance {
       }
       imps[signature] = does(refs)
    }
+
+   /* HERE!! This function needs to analyze the template and make sure the
+    * instantiations of it for type and all prior types of type are present
+    * in the instance
+    */
+   _ensureTemplateTypes(template, type) {
+   }
+
 }
diff --git a/src/core/dependencyExtractor.mjs b/src/core/dependencyExtractor.mjs
deleted file mode 100644
index 1b1091c..0000000
--- a/src/core/dependencyExtractor.mjs
+++ /dev/null
@@ -1,12 +0,0 @@
-/* Call this with an empty Set object S, and it returns an entity E
- * from which properties can be extracted, and at any time S will
- * contain all of the property names that have been extracted from E.
- */
-export default function dependencyExtractor(destinationSet) {
-   return new Proxy({}, {
-      get: (target, property) => {
-         destinationSet.add(property)
-         return {}
-      }
-   })
-}
diff --git a/src/core/extractors.mjs b/src/core/extractors.mjs
new file mode 100644
index 0000000..ff63d9d
--- /dev/null
+++ b/src/core/extractors.mjs
@@ -0,0 +1,41 @@
+/* Call this with an empty Set object S, and it returns an entity E
+ * from which properties can be extracted, and at any time S will
+ * contain all of the property names that have been extracted from E.
+ */
+export function dependencyExtractor(destinationSet) {
+   return new Proxy({}, {
+      get: (target, property) => {
+         destinationSet.add(property)
+         return {}
+      }
+   })
+}
+
+/* Given a (template) type name, what the template parameter is,
+ * a top level typer, and a library of templates,
+ * produces a function that will extract the instantantion type from an
+ * instance. Currently relies heavily on there being only unary templates.
+ *
+ * We should really be using the typed-function parser to do the
+ * manipulations below, but at the moment we don't have access.
+ */
+export function generateTypeExtractor(
+   type, param, topTyper, typeJoiner, templates)
+{
+   type = type.trim()
+   if (type.slice(0,3) === '...') {
+      type = type.slice(3).trim()
+   }
+   if (type === param) return topTyper
+   if (!(type.includes('<'))) return false // no template type to extract
+   const base = type.split('<',1)[0]
+   if (!(base in templates)) return false // unknown template
+   const arg = type.slice(base.length+1, -1)
+   const argExtractor = generateTypeExtractor(
+      arg, param, topTyper, typeJointer, templates)
+   if (!argExtractor) return false
+   return templates[base].infer({
+      typeOf: argExtractor,
+      joinTypes: typeJoiner
+   })
+}
diff --git a/test/core/_dependencyExtractor.mjs b/test/core/dependencyExtractor.mjs
similarity index 89%
rename from test/core/_dependencyExtractor.mjs
rename to test/core/dependencyExtractor.mjs
index 91e0e40..bc1683f 100644
--- a/test/core/_dependencyExtractor.mjs
+++ b/test/core/dependencyExtractor.mjs
@@ -1,5 +1,5 @@
 import assert from 'assert'
-import dependencyExtractor from '../../src/core/dependencyExtractor.mjs'
+import {dependencyExtractor} from '../../src/core/extractors.mjs'
 
 describe('dependencyExtractor', () => {
    it('will record the keys of a destructuring function', () => {