diff --git a/src/typechecker/type.hpp b/src/typechecker/type.hpp
index dca778b..282ecde 100644
--- a/src/typechecker/type.hpp
+++ b/src/typechecker/type.hpp
@@ -137,8 +137,8 @@ namespace Solstice {
     };
 
     struct Function {
-        std::unordered_map<std::vector<Type>, Type> signatureToReturnType;
-
+        PossibleType returnTypes;
+        std::vector<PossibleType> argumentTypes;
         Function() = default;
     };
 
diff --git a/src/typechecker/typechecker.cpp b/src/typechecker/typechecker.cpp
index 0bc65a5..56d1d5f 100644
--- a/src/typechecker/typechecker.cpp
+++ b/src/typechecker/typechecker.cpp
@@ -142,12 +142,16 @@ namespace Solstice {
         }
 
         checkNodeType(node.children[1]);
-        auto nodeType = node.children[1].ptype.getOnlyType();
-        if (!nodeType.has_value()) {
-            throw std::runtime_error("cannot assign ambiguous type to name");
+        if (node.children[1].ptype.possiblities.empty() && !node.children[1].ptype.unknownType) {
+            throw std::runtime_error("cannot assign name with no possible types");
         }
-        variables[*id] = {{*nodeType}};
+        variables[*id] = node.children[1].ptype;
         variables[*id].isConstant = true;
+        // if not narrowed down to a single type yet, keep it open so later
+        // usages of this name can continue narrowing it
+        if (!variables[*id].getOnlyType().has_value()) {
+            variables[*id].unknownType = true;
+        }
     }
 
     void TypeChecker::checkFunctionBindNodeType(Node& node) {
@@ -162,11 +166,18 @@ namespace Solstice {
         // create sub-checker 
         TypeChecker typeChecker(*this);
         typeChecker.setNode(node.children[2]);
+        typeChecker.inFunction = true;
+
+        // track parameter names in declaration order so we can pull their
+        // narrowed types back out once the body has been checked
+        std::vector<std::string> paramNames;
 
         for (const auto& arg : node.children[1].children) {
             switch (arg.type) {
                 case NodeType::Identifier: {
-                    typeChecker.setVariableUnknown(*arg.getIdentifier());
+                    auto name = *arg.getIdentifier();
+                    typeChecker.setVariableUnknown(name);
+                    paramNames.push_back(name);
                     break;
                 }
                 case NodeType::SetType: {
@@ -181,7 +192,9 @@ namespace Solstice {
                     if (types.find(*typeId) == types.end()) {
                         throw std::runtime_error("unknown type " + *typeId);
                     }
-                    typeChecker.setVariable(*arg.children[0].getIdentifier(), types[*typeId]);
+                    auto name = *arg.children[0].getIdentifier();
+                    typeChecker.setVariable(name, types[*typeId]);
+                    paramNames.push_back(name);
                     break;
                 }
                 default: {
@@ -191,19 +204,22 @@ namespace Solstice {
         }
 
         typeChecker.checkTypes();
-        
-        auto nodeType = node.children[2].ptype.getOnlyType();
-        if (!nodeType.has_value()) {
-            throw std::runtime_error("cannot assign ambiguous type to name");
+
+        // collect the (possibly still-polymorphic) narrowed argument types,
+        // and the inferred return type from the body's implicit return value
+        Function fn;
+        for (const auto& name : paramNames) {
+            fn.argumentTypes.push_back(typeChecker.variables[name]);
         }
-        variables[*id] = {{*nodeType}};
-        variables[*id].isConstant = true;
+        fn.returnTypes = node.children[2].ptype;
+
+        functions[*id] = fn;
     }
 
     Narrowing TypeChecker::doesNodeChildrenNeedNarrowing(Node& node) {
-        Narrowing ret;
+        Narrowing ret = Narrowing::None;
         if (
-            node.children[0].type == NodeType::Expression &&
+            node.children[0].type == NodeType::Identifier &&
             variables.find(*node.children[0].getIdentifier()) != variables.end() &&
             variables[*node.children[0].getIdentifier()].unknownType
         )  {
@@ -211,7 +227,7 @@ namespace Solstice {
         }
 
         if (
-            node.children[1].type == NodeType::Expression &&
+            node.children[1].type == NodeType::Identifier &&
             variables.find(*node.children[1].getIdentifier()) != variables.end() &&
             variables[*node.children[1].getIdentifier()].unknownType
         )  {
@@ -234,96 +250,73 @@ namespace Solstice {
             case Narrowing::Both: {
                 auto& leftVar = variables[*left.getIdentifier()];
                 auto& rightVar = variables[*right.getIdentifier()];
-                
-                if (leftVar.possiblities.empty() && rightVar.possiblities.empty()) {
-                    for (const auto& [key, value] : overloads) {
-                        leftVar.possiblities.insert(key.left);
-                        rightVar.possiblities.insert(key.right);
+
+                // an empty possibility set means "still fully open" (no prior
+                // narrowing yet), so treat it as no constraint; otherwise only
+                // keep overloads compatible with what's already been narrowed
+                std::unordered_set<Type> newLeft;
+                std::unordered_set<Type> newRight;
+                for (const auto& [key, value] : overloads) {
+                    bool leftOk = leftVar.possiblities.empty() || leftVar.possiblities.find(key.left) != leftVar.possiblities.end();
+                    bool rightOk = rightVar.possiblities.empty() || rightVar.possiblities.find(key.right) != rightVar.possiblities.end();
+                    if (leftOk && rightOk) {
+                        newLeft.insert(key.left);
+                        newRight.insert(key.right);
                         node.ptype.possiblities.insert(value);
                     }
-                } else if (leftVar.possiblities.empty()) {
-                    std::unordered_set<Type> rightMatches;
-                    for (const auto& [key, value] : overloads) {
-                        if (rightVar.possiblities.find(key.right) != rightVar.possiblities.end()) {
-                            rightMatches.insert(key.left);
-                            node.ptype.possiblities.insert(value);
-                        }
-                    }
-                    if (rightMatches.empty()) {
-                        throw std::runtime_error("no valid overload compatible with previous statements");
-                    }
-                    leftVar.possiblities = rightMatches;
-                } else if (rightVar.possiblities.empty()) {
-                    std::unordered_set<Type> leftMatches;
-                    for (const auto& [key, value] : overloads) {
-                        if (leftVar.possiblities.find(key.right) != leftVar.possiblities.end()) {
-                            leftMatches.insert(key.left);
-                            node.ptype.possiblities.insert(value);
-                        }
-                    }
-                    if (leftMatches.empty()) {
-                        throw std::runtime_error("no valid overload compatible with previous statements");
-                    }
-                    rightVar.possiblities = leftMatches;
-                } else {
-                    std::unordered_set<Type> left;
-                    std::unordered_set<Type> right;
-                    for (const auto& [key, value] : overloads) {
-                        if (
-                            leftVar.possiblities.find(key.right) != leftVar.possiblities.end() &&
-                            rightVar.possiblities.find(key.right) != rightVar.possiblities.end()
-                        )  {
-                            left.insert(key.left);
-                            right.insert(key.right);
-                            node.ptype.possiblities.insert(value);
-                        }
-                    }
-                    if (left.empty() || right.empty()) {
-                        throw std::runtime_error("no valid overload compatible with previous statements");
-                    }
                 }
+                if (newLeft.empty() || newRight.empty()) {
+                    throw std::runtime_error("no valid overload compatible with previous statements");
+                }
+                leftVar.possiblities = newLeft;
+                rightVar.possiblities = newRight;
 
                 break;
             }
             case Narrowing::Left: {
                 auto& leftVar = variables[*left.getIdentifier()];
-                std::unordered_set<Type> rightMatches;
+                std::unordered_set<Type> newLeft;
                 for (const auto& [key, value] : overloads) {
-                    if (right.ptype.possiblities.find(key.right) != right.ptype.possiblities.end()) {
-                        rightMatches.insert(key.left);
+                    bool leftOk = leftVar.possiblities.empty() || leftVar.possiblities.find(key.left) != leftVar.possiblities.end();
+                    bool rightOk = right.ptype.possiblities.find(key.right) != right.ptype.possiblities.end();
+                    if (leftOk && rightOk) {
+                        newLeft.insert(key.left);
                         node.ptype.possiblities.insert(value);
                     }
                 }
-                if (rightMatches.empty()) {
+                if (newLeft.empty()) {
                     throw std::runtime_error("no valid overload compatible with previous statements");
                 }
-                leftVar.possiblities = rightMatches;
+                leftVar.possiblities = newLeft;
                 break;
             }
             case Narrowing::Right: {
                 auto& rightVar = variables[*right.getIdentifier()];
-                std::unordered_set<Type> rightMatches;
+                std::unordered_set<Type> newRight;
                 for (const auto& [key, value] : overloads) {
-                    if (left.ptype.possiblities.find(key.left) != left.ptype.possiblities.end()) {
-                        rightMatches.insert(key.left);
+                    bool leftOk = left.ptype.possiblities.find(key.left) != left.ptype.possiblities.end();
+                    bool rightOk = rightVar.possiblities.empty() || rightVar.possiblities.find(key.right) != rightVar.possiblities.end();
+                    if (leftOk && rightOk) {
+                        newRight.insert(key.right);
                         node.ptype.possiblities.insert(value);
                     }
                 }
-                if (rightMatches.empty()) {
+                if (newRight.empty()) {
                     throw std::runtime_error("no valid overload compatible with previous statements");
                 }
-                rightVar.possiblities = rightMatches;
+                rightVar.possiblities = newRight;
                 break;
             }
             case Narrowing::None: {
                 for (const auto& [key, value] : overloads) {
                     if (
-                        left.ptype.possiblities.find(key.right) != left.ptype.possiblities.end() &&
+                        left.ptype.possiblities.find(key.left) != left.ptype.possiblities.end() &&
                         right.ptype.possiblities.find(key.right) != right.ptype.possiblities.end()
                     )  {
                         node.ptype.possiblities.insert(value);
                     }
                 }
+                break;
             }
         }
 
@@ -344,16 +337,27 @@ namespace Solstice {
         }
         
         checkNodeType(node.children[1]);
-        auto nodeType = node.children[1].ptype.getOnlyType();
-        if (!nodeType.has_value()) {
-            throw std::runtime_error("cannot assign ambiguous type to name");
+        if (node.children[1].ptype.possiblities.empty() && !node.children[1].ptype.unknownType) {
+            throw std::runtime_error("cannot assign name with no possible types");
+        }
+        variables[*id] = node.children[1].ptype;
+        // if not narrowed down to a single type yet, keep it open so later
+        // usages of this name can continue narrowing it
+        if (!variables[*id].getOnlyType().has_value()) {
+            variables[*id].unknownType = true;
         }
-        variables[*id] = {{*nodeType}};
     }
 
     void TypeChecker::checkCodeBlockType(Node& node) {
-        
-        return;
+        if (node.children.empty()) {
+            throw std::runtime_error("code block has no statements");
+        }
+        for (auto& child : node.children) {
+            checkNodeType(child);
+        }
+        // implicit return: the value of a code block is the value of its
+        // last statement (there is no explicit 'return' keyword)
+        node.ptype = node.children.back().ptype;
     }
 
     void TypeChecker::checkAddType(Node& node) {
@@ -366,6 +370,55 @@ namespace Solstice {
 
     }
 
+    void TypeChecker::checkSubtractType(Node& node) {
+        checkNodeType(node.children[0]);
+        checkNodeType(node.children[1]);
+
+        narrowBinaryNode(node, subtractOverloads);
+    }
+
+    void TypeChecker::checkMultiplyType(Node& node) {
+        checkNodeType(node.children[0]);
+        checkNodeType(node.children[1]);
+
+        narrowBinaryNode(node, multiplyOverloads);
+    }
+
+    void TypeChecker::checkDivideType(Node& node) {
+        checkNodeType(node.children[0]);
+        checkNodeType(node.children[1]);
+
+        narrowBinaryNode(node, divideOverloads);
+    }
+
+    void TypeChecker::checkEqualType(Node& node) {
+        checkNodeType(node.children[0]);
+        checkNodeType(node.children[1]);
+
+        narrowBinaryNode(node, equalOverloads);
+    }
+
+    void TypeChecker::checkNotEqualType(Node& node) {
+        checkNodeType(node.children[0]);
+        checkNodeType(node.children[1]);
+
+        narrowBinaryNode(node, notEqualOverloads);
+    }
+
+    void TypeChecker::checkGreaterThanType(Node& node) {
+        checkNodeType(node.children[0]);
+        checkNodeType(node.children[1]);
+
+        narrowBinaryNode(node, greaterThanOverloads);
+    }
+
+    void TypeChecker::checkLesserThanType(Node& node) {
+        checkNodeType(node.children[0]);
+        checkNodeType(node.children[1]);
+
+        narrowBinaryNode(node, lesserThanOverloads);
+    }
+
 
     void TypeChecker::checkNodeType(Node& node) {
         switch (node.type) {
@@ -411,7 +464,6 @@ namespace Solstice {
             case NodeType::Add:
                 checkAddType(node);
                 break;
-                /*
             case NodeType::Subtract:
                 checkSubtractType(node);
                 break;
@@ -433,7 +485,6 @@ namespace Solstice {
             case NodeType::LesserThan:
                 checkLesserThanType(node);
                 break;
-                */
 
         }
     }
diff --git a/src/typechecker/typechecker.hpp b/src/typechecker/typechecker.hpp
index 617336a..3f9e759 100644
--- a/src/typechecker/typechecker.hpp
+++ b/src/typechecker/typechecker.hpp
@@ -18,6 +18,7 @@ namespace Solstice {
     class TypeChecker {
         std::unordered_map<std::string, PossibleType> variables;
         std::unordered_map<std::string, Type> types;
+        std::unordered_map<std::string, Function> functions;
 
         std::unordered_map<TypePair, Type> addOverloads;
         std::unordered_map<TypePair, Type> subtractOverloads;
@@ -80,5 +81,13 @@ namespace Solstice {
         }
 
         void checkTypes();
+
+        const Function* getFunction(const std::string& name) const {
+            auto it = functions.find(name);
+            if (it == functions.end()) {
+                return nullptr;
+            }
+            return &it->second;
+        }
     };
 }