diff --git a/src/ast.ts b/src/ast.ts
index 805c323..750f7e0 100644
--- a/src/ast.ts
+++ b/src/ast.ts
@@ -15,6 +15,7 @@ export type ItemKind = {
 
 export type Item = ItemKind & {
   span: Span;
+  id: number;
 };
 
 export type FunctionDef = {
@@ -94,6 +95,7 @@ export type ExprKind =
 
 export type Expr = ExprKind & {
   span: Span;
+  ty?: Ty;
 };
 
 export type Literal =
@@ -128,6 +130,7 @@ export const COMPARISON_KINDS: BinaryKind[] = [
   ">=",
   "!=",
 ];
+export const EQUALITY_KINDS: BinaryKind[] = ["==", "!="];
 export const LOGICAL_KINDS: BinaryKind[] = ["&", "|"];
 export const ARITH_TERM_KINDS: BinaryKind[] = ["+", "-"];
 export const ARITH_FACTOR_KINDS: BinaryKind[] = ["*", "/"];
@@ -174,6 +177,7 @@ export type TypeKind =
 
 export type Type = TypeKind & {
   span: Span;
+  ty?: Ty;
 };
 
 export type Resolution =
@@ -202,8 +206,53 @@ export type Resolution =
     }
   | {
       kind: "builtin";
+      name: string;
     };
 
+export type TyString = {
+  kind: "string";
+};
+
+export type TyInt = {
+  kind: "int";
+};
+
+export type TyBool = {
+  kind: "bool";
+};
+
+export type TyList = {
+  kind: "list";
+  elem: Ty;
+};
+
+export type TyTuple = {
+  kind: "tuple";
+  elems: Ty[];
+};
+
+export type TyUnit = {
+  kind: "tuple";
+  elems: [];
+};
+
+export type TyFn = {
+  kind: "fn";
+  params: Ty[];
+  returnTy: Ty;
+};
+
+export type TyVar = {
+  kind: "var";
+  index: number;
+};
+
+export type Ty = TyString | TyInt | TyBool | TyList | TyTuple | TyFn | TyVar;
+
+export function tyIsUnit(ty: Ty): ty is TyUnit {
+  return ty.kind === "tuple" && ty.elems.length === 0;
+}
+
 // folders
 
 export type FoldFn<T> = (value: T) => T;
@@ -252,6 +301,7 @@ export function super_fold_item(item: Item, folder: Folder): Item {
           body: folder.expr(item.node.body),
           returnType: item.node.returnType && folder.type(item.node.returnType),
         },
+        id: item.id,
       };
     }
   }
diff --git a/src/index.ts b/src/index.ts
index 9698fe6..10d9796 100644
--- a/src/index.ts
+++ b/src/index.ts
@@ -3,19 +3,15 @@ import { tokenize } from "./lexer";
 import { parse } from "./parser";
 import { printAst } from "./printer";
 import { resolve } from "./resolve";
+import { typeck } from "./typeck";
 
 const input = `
-function main(argv: [String]): () = (
-  print(argv);
-  if 1 then (
-    print("AAAAAAAAAAAAAAAAAAAA");
-    let a = 0 in
-    a;
-  ) else (
-    print("AAAAAAAAAAAAAAAAAAAAAA");
-    let b = 0 in
-    b;
-  )
+function main() = (
+  let a = 0 in
+  let b = a in
+  let c = b in
+  let d = c in
+  d;
 );
 `;
 
@@ -38,6 +34,11 @@ function main() {
     console.log("-----AST resolved------");
     const resolvedPrinted = printAst(resolved);
     console.log(resolvedPrinted);
+
+    console.log("-----AST typecked------");
+
+    const typecked = typeck(resolved);
+    console.dir(typecked, { depth: 10 });
   });
 }
 
diff --git a/src/parser.ts b/src/parser.ts
index f5cfdad..40e4c6b 100644
--- a/src/parser.ts
+++ b/src/parser.ts
@@ -27,7 +27,9 @@ export function parse(t: Token[]): Ast {
     items.push(item);
   }
 
-  return items;
+  const withIds = items.map((item, i) => ({ ...item, id: i }));
+
+  return withIds;
 }
 
 function parseItem(t: Token[]): [Token[], Item] {
@@ -79,7 +81,16 @@ function parseItem(t: Token[]): [Token[], Item] {
       body,
     };
 
-    return [t, { kind: "function", node: def, span: tok.span }];
+    return [
+      t,
+      {
+        kind: "function",
+        node: def,
+        span: tok.span,
+        // Assigned later.
+        id: 0,
+      },
+    ];
   } else {
     unexpectedToken(tok);
   }
@@ -114,7 +125,7 @@ function parseExpr(t: Token[]): [Token[], Expr] {
   if (peak.kind === "let") {
     [t] = expectNext(t, "let");
     let name;
-    [t, name] = expectNext<TokenIdent>(t, "identifier");
+    [t, name] = expectNext<TokenIdent & Token>(t, "identifier");
 
     let type = undefined;
     let colon;
@@ -132,7 +143,7 @@ function parseExpr(t: Token[]): [Token[], Expr] {
 
     return [
       t,
-      { kind: "let", name: name.ident, type, rhs, after, span: t[0].span },
+      { kind: "let", name: name.ident, type, rhs, after, span: name.span },
     ];
   }
 
diff --git a/src/printer.ts b/src/printer.ts
index ec330dd..0bb8860 100644
--- a/src/printer.ts
+++ b/src/printer.ts
@@ -5,7 +5,9 @@ import {
   Identifier,
   Item,
   Resolution,
+  Ty,
   Type,
+  tyIsUnit,
 } from "./ast";
 
 export function printAst(ast: Ast): string {
@@ -139,6 +141,33 @@ function printIdent(ident: Identifier): string {
   return `${ident.name}${res}`;
 }
 
+export function printTy(ty: Ty): string {
+  switch (ty.kind) {
+    case "string": {
+      return "String";
+    }
+    case "int": {
+      return "Int";
+    }
+    case "bool": {
+      return "Bool";
+    }
+    case "list": {
+      return `[${printTy(ty.elem)}]`;
+    }
+    case "tuple": {
+      return `(${ty.elems.map(printTy).join(", ")})`;
+    }
+    case "fn": {
+      const ret = tyIsUnit(ty.returnTy) ? "" : `: ${printTy(ty.returnTy)}`;
+      return `fn(${ty.params.map(printTy).join(", ")})${ret}`;
+    }
+    case "var": {
+      return `?${ty.index}`;
+    }
+  }
+}
+
 function linebreak(indent: number): string {
   return `\n${ind(indent)}`;
 }
diff --git a/src/resolve.ts b/src/resolve.ts
index a351a62..4a4ec15 100644
--- a/src/resolve.ts
+++ b/src/resolve.ts
@@ -10,7 +10,14 @@ import {
 } from "./ast";
 import { CompilerError } from "./error";
 
-const BUILTINS = new Set<string>(["print", "String"]);
+const BUILTINS = new Set<string>([
+  "print",
+  "String",
+  "Int",
+  "Bool",
+  "true",
+  "false",
+]);
 
 export function resolve(ast: Ast): Ast {
   const items = new Map<string, number>();
@@ -60,9 +67,9 @@ export function resolve(ast: Ast): Ast {
     }
 
     if (BUILTINS.has(ident.name)) {
-      return { kind: "builtin" };
+      return { kind: "builtin", name: ident.name };
     }
-    
+
     throw new CompilerError(`cannot find ${ident.name}`, ident.span);
   };
 
@@ -92,6 +99,7 @@ export function resolve(ast: Ast): Ast {
               returnType,
               body,
             },
+            id: item.id,
           };
         }
       }
diff --git a/src/typeck.ts b/src/typeck.ts
new file mode 100644
index 0000000..bce1afd
--- /dev/null
+++ b/src/typeck.ts
@@ -0,0 +1,553 @@
+import { check } from "prettier";
+import {
+  Ast,
+  COMPARISON_KINDS,
+  DEFAULT_FOLDER,
+  EQUALITY_KINDS,
+  Expr,
+  ExprBinary,
+  ExprCall,
+  ExprUnary,
+  Folder,
+  Identifier,
+  LOGICAL_KINDS,
+  Resolution,
+  Ty,
+  TyFn,
+  TyVar,
+  Type,
+  binaryExprPrecedenceClass,
+  fold_ast,
+  super_fold_expr,
+} from "./ast";
+import { CompilerError, Span } from "./error";
+import { printTy } from "./printer";
+
+const TY_UNIT: Ty = { kind: "tuple", elems: [] };
+const TY_STRING: Ty = { kind: "string" };
+const TY_BOOL: Ty = { kind: "bool" };
+const TY_INT: Ty = { kind: "int" };
+
+function builtinAsTy(name: string, span: Span): Ty {
+  switch (name) {
+    case "String": {
+      return TY_STRING;
+    }
+    case "Int": {
+      return TY_INT;
+    }
+    case "Bool": {
+      return TY_BOOL;
+    }
+    default: {
+      throw new CompilerError(`\`${name}\` is not a type`, span);
+    }
+  }
+}
+
+function typeOfBuiltinValue(name: string, span: Span): Ty {
+  switch (name) {
+    case "false":
+    case "true":
+      return TY_BOOL;
+    case "print":
+      return { kind: "fn", params: [TY_STRING], returnTy: TY_UNIT };
+    default: {
+      throw new CompilerError(`\`${name}\` cannot be used as a value`, span);
+    }
+  }
+}
+
+function lowerAstTyBase(
+  type: Type,
+  lowerIdentTy: (ident: Identifier) => Ty,
+  typeOfItem: (index: number) => Ty
+): Ty {
+  switch (type.kind) {
+    case "ident": {
+      const res = type.value.res!;
+      switch (res.kind) {
+        case "local": {
+          throw new Error("Cannot resolve local here");
+        }
+        case "item": {
+          return typeOfItem(res.index);
+        }
+        case "builtin": {
+          return builtinAsTy(res.name, type.value.span);
+        }
+      }
+    }
+    case "list": {
+      return {
+        kind: "list",
+        elem: lowerAstTyBase(type.elem, lowerIdentTy, typeOfItem),
+      };
+    }
+    case "tuple": {
+      return {
+        kind: "tuple",
+        elems: type.elems.map((type) =>
+          lowerAstTyBase(type, lowerIdentTy, typeOfItem)
+        ),
+      };
+    }
+  }
+}
+
+export function typeck(ast: Ast): Ast {
+  const itemTys = new Map<number, Ty | null>();
+  function typeOfItem(index: number): Ty {
+    const ty = itemTys.get(index);
+    if (ty) {
+      return ty;
+    }
+    if (ty === null) {
+      throw Error(`cycle computing type of #G${index}`);
+    }
+    itemTys.set(index, null);
+    const item = ast[index];
+    switch (item.kind) {
+      case "function": {
+        const args = item.node.args.map((arg) => lowerAstTy(arg.type));
+        const returnTy: Ty = item.node.returnType
+          ? lowerAstTy(item.node.returnType)
+          : TY_UNIT;
+
+        return { kind: "fn", params: args, returnTy };
+      }
+    }
+  }
+
+  function lowerAstTy(type: Type): Ty {
+    return lowerAstTyBase(
+      type,
+      (ident) => {
+        const res = ident.res!;
+        switch (res.kind) {
+          case "local": {
+            throw new Error("Cannot resolve local here");
+          }
+          case "item": {
+            return typeOfItem(res.index);
+          }
+          case "builtin": {
+            return builtinAsTy(res.name, ident.span);
+          }
+        }
+      },
+      typeOfItem
+    );
+  }
+
+  const checker: Folder = {
+    ...DEFAULT_FOLDER,
+    item(item) {
+      switch (item.kind) {
+        case "function": {
+          const fnTy = typeOfItem(item.id) as TyFn;
+          const body = checkBody(item.node.body, fnTy, typeOfItem);
+
+          const returnType = item.node.returnType && {
+            ...item.node.returnType,
+            ty: fnTy.returnTy,
+          };
+
+          return {
+            kind: "function",
+            node: {
+              name: item.node.name,
+              args: item.node.args.map((arg, i) => ({
+                ...arg,
+                type: { ...arg.type, ty: fnTy.params[i] },
+              })),
+              body,
+              returnType,
+            },
+            span: item.span,
+            id: item.id,
+          };
+        }
+      }
+    },
+  };
+
+  const withTypes = fold_ast(ast, checker);
+
+  return withTypes;
+}
+
+type TyVarRes =
+  | {
+      kind: "final";
+      ty: Ty;
+    }
+  | {
+      kind: "unified";
+      index: number;
+    }
+  | {
+      kind: "unknown";
+    };
+
+export function checkBody(
+  body: Expr,
+  fnTy: TyFn,
+  typeOfItem: (index: number) => Ty
+): Expr {
+  const localTys = [...fnTy.params];
+  const tyVars: TyVarRes[] = [];
+
+  function newVar(): Ty {
+    const index = tyVars.length;
+    tyVars.push({ kind: "unknown" });
+    return { kind: "var", index };
+  }
+
+  function typeOf(res: Resolution, span: Span): Ty {
+    switch (res.kind) {
+      case "local": {
+        const idx = localTys.length - 1 - res.index;
+        return localTys[idx];
+      }
+      case "item": {
+        return typeOfItem(res.index);
+      }
+      case "builtin":
+        return typeOfBuiltinValue(res.name, span);
+    }
+  }
+
+  function lowerAstTy(type: Type): Ty {
+    return lowerAstTyBase(
+      type,
+      (ident) => {
+        const res = ident.res!;
+        return typeOf(res, ident.span);
+      },
+      typeOfItem
+    );
+  }
+
+  function tryResolveVar(variable: number): Ty | undefined {
+    const varRes = tyVars[variable];
+    switch (varRes.kind) {
+      case "final": {
+        return varRes.ty;
+      }
+      case "unified": {
+        const ty = tryResolveVar(varRes.index);
+        if (ty) {
+          tyVars[variable] = { kind: "final", ty };
+          return ty;
+        } else {
+          return undefined;
+        }
+      }
+      case "unknown": {
+        return undefined;
+      }
+    }
+  }
+
+  /**
+   * Try to constrain a type variable to be of a specific type.
+   * INVARIANT: Both sides must not be of res "final", use resolveIfPossible
+   * before calling this.
+   */
+  function constrainVar(variable: number, ty: Ty) {
+    if (ty.kind === "var") {
+      // Point the lhs to the rhs.
+      tyVars[variable] = { kind: "unified", index: ty.index };
+    }
+
+    let idx = variable;
+    let nextVar;
+    while ((nextVar = tyVars[idx]).kind === "unified") {
+      idx = nextVar.index;
+    }
+
+    const root = idx;
+    tyVars[root] = { kind: "final", ty };
+  }
+
+  function resolveIfPossible(ty: Ty): Ty {
+    if (ty.kind === "var") {
+      return tryResolveVar(ty.index) ?? ty;
+    } else {
+      return ty;
+    }
+  }
+
+  function assign(lhs_: Ty, rhs_: Ty, span: Span) {
+    const lhs = resolveIfPossible(lhs_);
+    const rhs = resolveIfPossible(rhs_);
+
+    if (lhs.kind === "var") {
+      constrainVar(lhs.index, rhs);
+      return;
+    }
+    if (rhs.kind === "var") {
+      constrainVar(rhs.index, lhs);
+      return;
+    }
+    // type variable handling here
+
+    switch (lhs.kind) {
+      case "string": {
+        if (rhs.kind === "string") return;
+        break;
+      }
+      case "int": {
+        if (rhs.kind === "int") return;
+        break;
+      }
+      case "bool": {
+        if (rhs.kind === "bool") return;
+        break;
+      }
+      case "list": {
+        if (rhs.kind === "list") {
+          assign(lhs.elem, rhs.elem, span);
+          return;
+        }
+        break;
+      }
+      case "tuple": {
+        if (rhs.kind === "tuple" && lhs.elems.length === rhs.elems.length) {
+          lhs.elems.forEach((lhs, i) => assign(lhs, rhs.elems[i], span));
+          return;
+        }
+        break;
+      }
+      case "fn": {
+        if (rhs.kind === "fn" && lhs.params.length === rhs.params.length) {
+          // swapping because of contravariance in the future maybe
+          lhs.params.forEach((lhs, i) => assign(rhs.params[i], lhs, span));
+
+          assign(lhs.returnTy, rhs.returnTy, span);
+
+          return;
+        }
+        break;
+      }
+    }
+
+    throw new CompilerError(
+      `cannot assign ${printTy(rhs)} to ${printTy(lhs)}`,
+      span
+    );
+  }
+
+  const checker: Folder = {
+    ...DEFAULT_FOLDER,
+    expr(expr) {
+      switch (expr.kind) {
+        case "empty": {
+          return { ...expr, ty: TY_UNIT };
+        }
+        case "let": {
+          const loweredBindingTy = expr.type && lowerAstTy(expr.type);
+          let bindingTy = loweredBindingTy ? loweredBindingTy : newVar();
+
+          const rhs = this.expr(expr.rhs);
+          assign(bindingTy, rhs.ty!, expr.span);
+
+          localTys.push(bindingTy);
+          const after = this.expr(expr.after);
+          localTys.pop();
+
+          const type: Type | undefined = loweredBindingTy && {
+            ...expr.type!,
+            ty: loweredBindingTy!,
+          };
+
+          return {
+            kind: "let",
+            name: expr.name,
+            type,
+            rhs,
+            after,
+            ty: after.ty!,
+            span: expr.span,
+          };
+        }
+        case "block": {
+          const exprs = expr.exprs.map((expr) => this.expr(expr));
+          const ty = exprs.length > 0 ? exprs[exprs.length - 1].ty! : TY_UNIT;
+
+          return {
+            kind: "block",
+            exprs,
+            ty,
+            span: expr.span,
+          };
+        }
+        case "literal": {
+          let ty;
+          switch (expr.value.kind) {
+            case "str": {
+              ty = TY_STRING;
+              break;
+            }
+            case "int": {
+              ty = TY_INT;
+              break;
+            }
+          }
+
+          return { ...expr, ty };
+        }
+        case "ident": {
+          const ty = typeOf(expr.value.res!, expr.value.span);
+
+          return { ...expr, ty };
+        }
+        case "binary": {
+          const lhs = this.expr(expr.lhs);
+          const rhs = this.expr(expr.rhs);
+
+          lhs.ty = resolveIfPossible(lhs.ty!);
+          rhs.ty = resolveIfPossible(rhs.ty!);
+
+          return checkBinary({ ...expr, lhs, rhs });
+        }
+        case "unary": {
+          const rhs = this.expr(expr.rhs);
+          rhs.ty = resolveIfPossible(rhs.ty!);
+          return checkUnary({ ...expr, rhs });
+        }
+        case "call": {
+          const lhs = this.expr(expr.lhs);
+          lhs.ty = resolveIfPossible(lhs.ty!);
+          const lhsTy = lhs.ty!;
+          if (lhsTy.kind !== "fn") {
+            throw new CompilerError(
+              `expression of type ${printTy(lhsTy)} is not callable`,
+              lhs.span
+            );
+          }
+
+          const args = expr.args.map((arg) => this.expr(arg));
+
+          lhsTy.params.forEach((param, i) => {
+            if (!args[i]) {
+              throw new CompilerError(
+                `missing argument of type ${printTy(param)}`,
+                expr.span
+              );
+            }
+            const arg = checker.expr(args[i]);
+
+            assign(param, arg.ty!, args[i].span);
+          });
+
+          if (args.length > lhsTy.params.length) {
+            throw new CompilerError(
+              `too many arguments passed, expected ${lhsTy.params.length}, found ${args.length}`,
+              expr.span
+            );
+          }
+
+          return { ...expr, lhs, args, ty: lhsTy.returnTy };
+        }
+        case "if": {
+          const cond = this.expr(expr.cond);
+          const then = this.expr(expr.then);
+          const elsePart = expr.else && this.expr(expr.else);
+
+          assign(TY_BOOL, cond.ty!, cond.span);
+
+          let ty;
+          if (elsePart) {
+            assign(then.ty!, elsePart.ty!, elsePart.span);
+            ty = then.ty!;
+          } else {
+            assign(TY_UNIT, then.ty!, then.span);
+          }
+
+          return { ...expr, cond, then, else: elsePart, ty };
+        }
+      }
+    },
+  };
+
+  const checked = checker.expr(body);
+
+  assign(fnTy.returnTy, checked.ty!, body.span);
+
+  return checked;
+}
+
+function checkBinary(expr: Expr & ExprBinary): Expr {
+  const checkPrecedence = (inner: Expr, side: string) => {
+    if (inner.kind === "binary") {
+      const ourClass = binaryExprPrecedenceClass(expr.binaryKind);
+      const innerClass = binaryExprPrecedenceClass(inner.binaryKind);
+
+      if (ourClass !== innerClass) {
+        throw new CompilerError(
+          `mixing operators without parentheses is not allowed. ${side} is ${inner.binaryKind}, which is different from ${expr.binaryKind}`,
+          expr.span
+        );
+      }
+    }
+  };
+
+  checkPrecedence(expr.lhs, "left");
+  checkPrecedence(expr.rhs, "right");
+
+  let lhsTy = expr.lhs.ty!;
+  let rhsTy = expr.rhs.ty!;
+
+  if (lhsTy.kind === "int" && rhsTy.kind === "int") {
+    return { ...expr, ty: TY_INT };
+  }
+
+  if (COMPARISON_KINDS.includes(expr.binaryKind)) {
+    if (lhsTy.kind === "string" && rhsTy.kind === "string") {
+      return { ...expr, ty: TY_BOOL };
+    }
+
+    if (EQUALITY_KINDS.includes(expr.binaryKind)) {
+      if (lhsTy.kind === "bool" && rhsTy.kind === "bool") {
+        return { ...expr, ty: TY_BOOL };
+      }
+    }
+  }
+
+  if (LOGICAL_KINDS.includes(expr.binaryKind)) {
+    if (lhsTy.kind === "bool" && rhsTy.kind === "bool") {
+      return { ...expr, ty: TY_BOOL };
+    }
+  }
+
+  throw new CompilerError(
+    `invalid types for binary operation: ${printTy(expr.lhs.ty!)} ${
+      expr.binaryKind
+    } ${printTy(expr.rhs.ty!)}`,
+    expr.span
+  );
+}
+
+function checkUnary(expr: Expr & ExprUnary): Expr {
+  let rhsTy = expr.rhs.ty!;
+
+  if (
+    expr.unaryKind === "!" &&
+    (rhsTy.kind === "int" || rhsTy.kind === "bool")
+  ) {
+    return { ...expr, ty: rhsTy };
+  }
+
+  if (expr.unaryKind === "-" && rhsTy.kind == "int") {
+    return { ...expr, ty: rhsTy };
+  }
+
+  throw new CompilerError(
+    `invalid types for unary operation: ${expr.unaryKind} ${printTy(
+      expr.rhs.ty!
+    )}`,
+    expr.span
+  );
+}