diff --git a/src/expr.rs b/src/expr.rs
index 592d938..e75811d 100644
--- a/src/expr.rs
+++ b/src/expr.rs
@@ -13,6 +13,7 @@ pub enum Statement {
         val_expr: LTExpr
     },
     LetAssign {
+        typ: Option<TypeTag>,
         var_id: String,
         val_expr: LTExpr,
     },
@@ -24,22 +25,34 @@ pub enum Statement {
     Expr(LTExpr)
 }
 
+#[derive(Clone, Debug)]
+pub enum TypeError {
+    ParseError(laddertypes::parser::ParseError),
+    Mismatch {
+        expected: laddertypes::TypeTerm,
+        received: laddertypes::TypeTerm
+    }
+}
+
+pub type TypeTag = Result< laddertypes::TypeTerm, TypeError >;
+
 #[derive(Clone, Debug)]
 pub enum LTExpr {
     Literal {
-        typ: Option< laddertypes::TypeTerm >,
+        typ: Option<TypeTag>,
         val: tisc::VM_Word
     },
     Symbol {
-        typ: Option< laddertypes::TypeTerm >,
+        typ: Option<TypeTag>,
         symbol: String,
     },
     Application {
+        typ: Option<TypeTag>,
         head: Box<LTExpr>,
         body: Vec<LTExpr>
     },
     Abstraction {
-        args: Vec<(String, Option<laddertypes::TypeTerm>)>,
+        args: Vec<(String, Option<TypeTag>)>,
         body: Box<LTExpr>
     },
     Branch {
@@ -67,18 +80,19 @@ impl LTExpr {
         }
     }
 
-    pub fn abstraction(args: Vec<(&str, &str)>, val_expr: LTExpr) -> LTExpr {
+    pub fn abstraction(args: Vec<(&str, &str)>, body: LTExpr) -> LTExpr {
         LTExpr::Abstraction {
             args: args.into_iter().map(|(arg_name, arg_type)|
                 ( arg_name.into(), None )    
                 //typectx.write().unwrap().parse(t).expect("parse typeterm")
             ).collect(),
-            body: Box::new(val_expr)
+            body: Box::new(body)
         }
     }
 
     pub fn application(head: LTExpr, body: Vec<LTExpr>) -> Self {
         LTExpr::Application {
+            typ: None,
             head: Box::new( head ),
             body: body
         }
diff --git a/src/lexer.rs b/src/lexer.rs
index c8cfa86..07e1699 100644
--- a/src/lexer.rs
+++ b/src/lexer.rs
@@ -10,7 +10,8 @@ pub enum LTIRToken {
     // TripleQuote(String),
 
     Lambda,
-    AssignType,
+    LambdaBody,
+    AssignType( String ),
     AssignValue,
 
     ExprOpen,
@@ -30,6 +31,7 @@ pub enum LexError {
 #[derive(PartialEq, Eq, Clone, Debug)]
 pub enum LexerState {
     Any,
+    TypeTerm( String ),
     Sym( String ),
     Num( i64 ),
     Char( Option<char> )
@@ -39,6 +41,7 @@ impl LexerState {
     fn into_token(self) -> Option< LTIRToken > {
         match self {
             LexerState::Any => None,
+            LexerState::TypeTerm(s) => Some(LTIRToken::AssignType(s)),
             LexerState::Sym(s) => Some(LTIRToken::Symbol(s)),
             LexerState::Num(n) => Some(LTIRToken::Num(n)),
             LexerState::Char(c) => Some(LTIRToken::Char(c?))
@@ -86,11 +89,15 @@ where It: Iterator<Item = char>
                 LexerState::Any => {
                     match c {
                         'λ' => { self.chars.next(); return Some(Ok(LTIRToken::Lambda)); },
+                        '.' => { self.chars.next(); return Some(Ok(LTIRToken::LambdaBody)); },
                         '(' => { self.chars.next(); return Some(Ok(LTIRToken::ExprOpen)); },
                         ')' => { self.chars.next(); return Some(Ok(LTIRToken::ExprClose)); },
                         '{' => { self.chars.next(); return Some(Ok(LTIRToken::BlockOpen)); },
                         '}' => { self.chars.next(); return Some(Ok(LTIRToken::BlockClose)); },
-                        ':' => { self.chars.next(); return Some(Ok(LTIRToken::AssignType)); },
+                        ':' => {
+                            self.chars.next();
+                            state = LexerState::TypeTerm(String::new());
+                        },
                         '=' => { self.chars.next(); return Some(Ok(LTIRToken::AssignValue)); },
                         ';' => { self.chars.next(); return Some(Ok(LTIRToken::StatementSep)); },
                         '\'' => { self.chars.next(); state = LexerState::Char(None); },
@@ -138,12 +145,25 @@ where It: Iterator<Item = char>
                     }
                 }
 
+                LexerState::TypeTerm(s) => {
+                    if *c == '=' || *c == '.' {
+                        if let Some(token) = state.clone().into_token() {
+                            return Some(Ok(token))
+                        }
+                    } else {
+                        if let Some(c) = self.chars.next() {
+                            s.push(c);
+                        }
+                    }
+                }
+
                 _ => {
 
                     if c.is_whitespace()
                         || *c == '(' || *c == ')'
                         || *c == '{' || *c == '}'
-                        || *c == ';' || *c == '=' || *c == ':'
+                        || *c == ';' || *c == '='
+                        || *c == ':' || *c == '.'
                     {
                         // finish the current token
 
@@ -183,4 +203,25 @@ where It: Iterator<Item = char>
     }
 }
 
+mod tests {
+    #[test]
+    fn test_lexer() {
+        let mut lexer = crate::lexer::LTIRLexer::from(
+            "let var1:ℕ=123;
+                let square =λx.* x x;
+
+                let sqrt = λx:ℝ~machine::Float64~machine::Word.(f64-sqrt x);
+                let magnitude =
+                     λx:ℝ
+                    .λy:ℝ
+                    .sqrt (+ (* x x) (* y y));
+                ".chars()
+        );
+
+        for token in lexer {
+            eprintln!("token = {:?}", token);
+        }
+    }
+}
+
 
diff --git a/src/main.rs b/src/main.rs
index 07b62a9..12c3fc9 100644
--- a/src/main.rs
+++ b/src/main.rs
@@ -21,6 +21,7 @@ fn compile(scope: &Arc<RwLock<Scope>>, name: &str, source: &str) -> Vec< tisc::a
     ProcedureCompiler::new(scope)
         .compile(
             &parser::parse_expr(
+                &scope.read().unwrap().typectx,
                 &mut lexer::LTIRLexer::from(
                     source.chars().peekable()
                 ).peekable()
@@ -63,40 +64,81 @@ fn main() {
     linker.add_procedure("main", compile(&main_scope,
         "main",
         "{
-            let print-nullterm = λstr {
-                while (@ str) {
-                    emit (@ str);
-                    ! str (i+ str 1);
-                }
-            };
+             let print-nullterm =
+                 λ str : <Ref <Seq Char~Ascii~machine::Word>>
+                       ~ <Ref <NullTerminatedArray machine::Word>>
+                       ~ machine::Address
+                       ~ machine::Word
+                       .
+                 {
+                     while (@ str) {
+                         emit (@ str);
+                         ! str (i+ str 1);
+                     }
+                 };
 
-            let print-lenprefix = λstr {
+            let print-len =
+                λ len : ℤ_2^64
+                      ~ machine::UInt64
+                      ~ machine::Word
+                      .
+                λ str : <Ref <Seq Char~Ascii~machine::Word>>
+                      ~ <Ref <Array machine::Word>>
+                      ~ machine::Address
+                      ~ machine::Word
+                      .
+                {
+                    let end = (i+ str len);
+                    while (i- str end) {
+                        emit (@ str);
+                        ! str (i+ str 1);
+                    }
+                };
+
+            let print-lenprefix =
+                λ str : <Ref <Seq Char~Ascii~machine::Word>>
+                      ~ <Ref <LenPrefixArray machine::Word>>
+                      ~ <Ref <Struct
+                           <len ℤ_2^64
+                               ~machine::UInt64
+                               ~machine::Word>
+                           <data <Array machine::Word>>
+                         >>
+                      ~ machine::Address
+                      ~ machine::Word
+                      .
+            {
                 let len = (@ str);
                 ! str (i+ str 1);
-                let end = (i+ str len);
-                while (i- str end) {
-                    emit (@ str);
-                    ! str (i+ str 1);
-                }
+                print-len len str;
             };
 
-            let hello = λ{
+            let hello = λ.{
                 print-nullterm hello-string;
                 print-lenprefix pfxstr;
+
+                let isquare = λx:ℤ. i* x x;
+                let imagnitude2 = λx:ℤ.λy:ℤ. i+ (isquare x) (isquare y);
+                let factorial = λn:ℤ.
+                    if( n ){ i* n (factorial (i- n 1)); }
+                    else   { 1; };
+
+                factorial 20;
+
+                if ( i- (imagnitude2 10 20) 500 ) {
+                    emit '?';
+                } else {
+                    emit '!';
+                };
+
+                emit '\n';
+                emit (i+ '0' (isquare 3));
+                emit '\n';
             };
 
             hello;
-
-            let isquare = λx (i* x x);
-            let magnitude2 = λx y {
-                i+ (isquare x) (isquare y);
-            };
-            magnitude2 8 16;
-
-            emit '\n';
-            emit (i+ '0' (isquare 3));
-            emit '\n';
-        }"));
+        }"
+    ));
 
     linker.add_static("hello-string",
         "Hallo Welt!\n\0"
diff --git a/src/parser.rs b/src/parser.rs
index 6a4f48b..921a745 100644
--- a/src/parser.rs
+++ b/src/parser.rs
@@ -1,8 +1,11 @@
 use {
-    std::iter::Peekable,
+    std::{
+        iter::Peekable,
+        sync::{Arc, RwLock}
+    },
     crate::{
         lexer::{LTIRLexer, LTIRToken, LexError},
-        expr::{LTExpr, Statement}
+        expr::{LTExpr, Statement, TypeTag, TypeError}
     }
 };
 
@@ -46,7 +49,34 @@ where It: Iterator<Item = char>
     }
 }
 
+pub fn parse_type_tag<It>(
+    typectx: &Arc<RwLock<laddertypes::dict::TypeDict>>,
+    tokens: &mut Peekable<LTIRLexer<It>>
+) -> Option<TypeTag>
+where It: Iterator<Item = char>
+{
+    if let Some(peektok) = tokens.peek().clone() {
+        match peektok.clone() {
+            Ok(LTIRToken::AssignType(typeterm_str)) => {
+                tokens.next();
+                match typectx.write().unwrap().parse(typeterm_str.as_str()) {
+                    Ok(typeterm) => {
+                        Some(Ok(typeterm))
+                    }
+                    Err(parse_error) => {
+                        Some(Err(TypeError::ParseError(parse_error)))
+                    }
+                }
+            }
+            _ => None
+        }
+    } else {
+        None
+    }
+}
+
 pub fn parse_statement<It>(
+    typectx: &Arc<RwLock<laddertypes::dict::TypeDict>>,
     tokens: &mut Peekable<LTIRLexer<It>>
 ) -> Result< crate::expr::Statement, ParseError >
 where It: Iterator<Item = char>
@@ -59,7 +89,7 @@ where It: Iterator<Item = char>
                         tokens.next();
                         // todo accept address-expression instead of symbol
                         let name = parse_symbol(tokens)?;
-                        let val_expr = parse_expr(tokens)?;
+                        let val_expr = parse_expr(typectx, tokens)?;
                         let _ = parse_expect(tokens, LTIRToken::StatementSep)?;
 
                         Ok(Statement::Assignment {
@@ -70,11 +100,13 @@ where It: Iterator<Item = char>
                     "let" => {
                         tokens.next();
                         let name = parse_symbol(tokens)?;
+                        let typ = parse_type_tag(typectx, tokens);
                         let _ = parse_expect(tokens, LTIRToken::AssignValue);
-                        let val_expr = parse_expr(tokens)?;
+                        let val_expr = parse_expr(typectx, tokens)?;
                         let _ = parse_expect(tokens, LTIRToken::StatementSep)?;
-
+                       
                         Ok(Statement::LetAssign {
+                            typ,
                             var_id: name,
                             val_expr
                         })
@@ -82,28 +114,28 @@ where It: Iterator<Item = char>
                     "while" => {
                         tokens.next();
                         let _ = parse_expect(tokens, LTIRToken::ExprOpen)?;
-                        let cond = parse_expr(tokens)?;
+                        let cond = parse_expr(typectx, tokens)?;
                         let _ = parse_expect(tokens, LTIRToken::ExprClose)?;
                         Ok(Statement::WhileLoop {
                             condition: cond,
-                            body: parse_block(tokens)?
+                            body: parse_block(typectx, tokens)?
                         })
                     }
                     "return" => {
                         tokens.next();
-                        let expr = parse_expr(tokens)?;
+                        let expr = parse_expr(typectx, tokens)?;
                         let _ = parse_expect(tokens, LTIRToken::StatementSep)?;
-                        Ok(Statement::Return(parse_expr(tokens)?))
+                        Ok(Statement::Return(parse_expr(typectx, tokens)?))
                     }
                     _ => {
-                        let expr = parse_expr(tokens)?;
+                        let expr = parse_expr(typectx, tokens)?;
                         let _ = parse_expect(tokens, LTIRToken::StatementSep)?;
                         Ok(Statement::Expr(expr))
                     }
                 }
             }
             Ok(_) => {
-                let expr = parse_expr(tokens)?;
+                let expr = parse_expr(typectx, tokens)?;
                 let _ = parse_expect(tokens, LTIRToken::StatementSep)?;
                 Ok(Statement::Expr(expr))
             },
@@ -115,6 +147,7 @@ where It: Iterator<Item = char>
 }
 
 pub fn parse_block<It>(
+    typectx: &Arc<RwLock<laddertypes::dict::TypeDict>>,
     tokens: &mut Peekable<LTIRLexer<It>>
 ) -> Result< Vec<Statement>, ParseError >
 where It: Iterator<Item = char>
@@ -128,7 +161,7 @@ where It: Iterator<Item = char>
                 tokens.next();
                 return Ok(statements)
             }
-            Ok(_) => { statements.push( parse_statement(tokens)? ); }
+            Ok(_) => { statements.push( parse_statement(typectx, tokens)? ); }
             Err(err) => { return Err(ParseError::LexError(err.clone())); }
         }
     }
@@ -164,6 +197,7 @@ where It: Iterator<Item = char>
 }
 
 pub fn parse_expr<It>(
+    typectx: &Arc<RwLock<laddertypes::dict::TypeDict>>,
     tokens: &mut Peekable<LTIRLexer<It>>
 ) -> Result< crate::expr::LTExpr, ParseError >
 where It: Iterator<Item = char>
@@ -178,9 +212,14 @@ where It: Iterator<Item = char>
 
                     let mut args = Vec::new();
                     while let Some(Ok(LTIRToken::Symbol(_))) = tokens.peek() {
-                        args.push((parse_symbol(tokens)?, None));
+                        args.push((
+                            parse_symbol(tokens)?,
+                            parse_type_tag(typectx, tokens)
+                        ));
                     }
-                    let body = parse_expr(tokens)?;
+
+                    let _ = parse_expect(tokens, LTIRToken::LambdaBody);
+                    let body = parse_expr(typectx, tokens)?;
 
                     return Ok(LTExpr::Abstraction{
                         args,
@@ -200,11 +239,13 @@ where It: Iterator<Item = char>
                         }
                         _ => {}
                     }
-                    children.push(parse_expr(tokens)?);
+                    children.push(parse_expr(typectx, tokens)?);
                 }
             },
             Ok(LTIRToken::ExprClose) => { break; }
-            Ok(LTIRToken::BlockOpen) => { children.push( LTExpr::block(parse_block(tokens)?)); }
+            Ok(LTIRToken::BlockOpen) => {
+                children.push( LTExpr::block(parse_block(typectx, tokens)?));
+            }
             Ok(LTIRToken::BlockClose) => { break; }
             Ok(LTIRToken::StatementSep) => { break; }
             Ok(LTIRToken::Symbol(name)) => {
@@ -212,16 +253,16 @@ where It: Iterator<Item = char>
                     "if" => {
                         tokens.next();
                         let _ = parse_expect(tokens, LTIRToken::ExprOpen)?;
-                        let cond = parse_expr(tokens)?;
+                        let cond = parse_expr(typectx, tokens)?;
                         let _ = parse_expect(tokens, LTIRToken::ExprClose)?;
-                        let if_expr = LTExpr::block(parse_block(tokens)?);
+                        let if_expr = LTExpr::block(parse_block(typectx, tokens)?);
                         let mut else_expr = LTExpr::block(vec![]);
 
                         if let Some(peektok) = tokens.peek() {
                             if let Ok(LTIRToken::Symbol(name)) = peektok {
                                 if name == "else" {
                                     tokens.next();
-                                    else_expr = parse_expr(tokens)?;
+                                    else_expr = parse_expr(typectx, tokens)?;
                                 }
                             }
                         }
@@ -245,6 +286,7 @@ where It: Iterator<Item = char>
     if children.len() > 0 {
         let head = children.remove(0);
         Ok(LTExpr::Application {
+            typ: None,
             head: Box::new(head),
             body: children
         })
diff --git a/src/procedure_compiler.rs b/src/procedure_compiler.rs
index a361146..00f44f4 100644
--- a/src/procedure_compiler.rs
+++ b/src/procedure_compiler.rs
@@ -98,7 +98,7 @@ impl ProcedureCompiler {
                     }
                 }
             }
-            Statement::LetAssign{ var_id, val_expr } => { 
+            Statement::LetAssign{ typ, var_id, val_expr } => { 
                 match val_expr {
                     LTExpr::Abstraction { args:_, body:_ } => {
                         self.symbols.write().unwrap()
@@ -180,7 +180,7 @@ impl ProcedureCompiler {
             LTExpr::Literal { typ, val } => {
                 self.asm = self.asm.lit( *val );
             }
-            LTExpr::Application { head, body } => {
+            LTExpr::Application { typ, head, body } => {
                 for arg in body.iter().rev() {
                     self = self.compile(arg);
                 }
@@ -188,16 +188,19 @@ impl ProcedureCompiler {
             }
             LTExpr::Abstraction { args, body } => {
                 for (arg_name, arg_type) in args.iter() {
-                    let id = self.symbols
-                       .write().unwrap()
-                       .declare_var(
-                            arg_name.clone(),
-                            arg_type.clone().unwrap_or(
-                            laddertypes::TypeTerm::unit())
-                        );
-                    self.asm = self.asm
-                        .lit( id )
-                        .call("data-frame-set");
+                    if let Some(Ok(typeterm)) = arg_type {
+                        let id = self.symbols
+                           .write().unwrap()
+                           .declare_var(
+                                arg_name.clone(),
+                                typeterm.clone()
+                            );
+                        self.asm = self.asm
+                            .lit( id )
+                            .call("data-frame-set");
+                    } else {
+                        eprintln!("invalid type {:?} for argument {}", arg_type, arg_name);
+                    }
                 }
                 self = self.compile(body);
             }
diff --git a/src/runtime.rs b/src/runtime.rs
index 0bf2cce..3554ecd 100644
--- a/src/runtime.rs
+++ b/src/runtime.rs
@@ -89,6 +89,7 @@ pub fn init_runtime(linker: &mut Linker) -> Arc<RwLock<Scope>> {
     linker.add_procedure(
         "i-",
         tisc::Assembler::new()
+            .inst(tisc::VM_Instruction::Swap)
             .inst(tisc::VM_Instruction::BitwiseNot)
             .lit(1)
             .inst(tisc::VM_Instruction::Add)
@@ -174,6 +175,7 @@ pub fn init_runtime(linker: &mut Linker) -> Arc<RwLock<Scope>> {
                     .lit( 1 )
 
                     // [ a sum b -- a sum (b-1) ]
+                    .inst( tisc::VM_Instruction::Swap )
                     .call( "i-" )
 
                     // [ a sum b -- a b sum ]
@@ -245,7 +247,6 @@ pub fn init_runtime(linker: &mut Linker) -> Arc<RwLock<Scope>> {
         tisc::Assembler::new()
             .static_ref("data-frame-ptr")
             .inst( tisc::VM_Instruction::Fetch )
-            .inst( tisc::VM_Instruction::Swap )
             .call("i-")
             .static_ref("data-frame-ptr")
             .inst( tisc::VM_Instruction::Store )