diff --git a/compiler/erg_compiler/context/eval.rs b/compiler/erg_compiler/context/eval.rs
index baa158d4..e251521d 100644
--- a/compiler/erg_compiler/context/eval.rs
+++ b/compiler/erg_compiler/context/eval.rs
@@ -16,10 +16,12 @@ use erg_parser::token::{Token, TokenKind};
 use erg_type::constructors::{enum_t, mono, mono_proj, poly, ref_, ref_mut, refinement, subr_t};
 use erg_type::typaram::{OpKind, TyParam};
 use erg_type::value::ValueObj;
-use erg_type::{ConstSubr, HasType, Predicate, TyBound, Type, UserConstSubr, ValueArgs};
+use erg_type::{
+    ConstSubr, HasType, ParamTy, Predicate, SubrKind, TyBound, Type, UserConstSubr, ValueArgs,
+};
 
 use crate::context::instantiate::TyVarContext;
-use crate::context::{ClassDefType, Context};
+use crate::context::{ClassDefType, Context, RegistrationMode};
 use crate::error::{EvalError, EvalResult, TyCheckResult};
 
 #[inline]
@@ -343,7 +345,7 @@ impl Context {
     fn eval_const_normal_record(&self, record: &NormalRecord) -> EvalResult<ValueObj> {
         let mut attrs = vec![];
         // HACK: should avoid cloning
-        let mut record_ctx = Context::instant(Str::ever("<unnamed record>"), 2, self.clone());
+        let mut record_ctx = Context::instant(Str::ever("<unnamed record>"), 0, self.clone());
         for attr in record.attrs.iter() {
             let name = attr.sig.ident().map(|i| i.inspect());
             let elem = record_ctx.eval_const_block(&attr.body.block, name)?;
@@ -360,11 +362,63 @@ impl Context {
     }
 
     fn eval_const_lambda(&self, lambda: &Lambda) -> EvalResult<ValueObj> {
+        let mut non_default_params = Vec::with_capacity(lambda.sig.params.non_defaults.len());
+        for sig in lambda.sig.params.non_defaults.iter() {
+            let t = self.instantiate_param_sig_t(sig, None, RegistrationMode::Normal)?;
+            let pt = if let Some(name) = sig.inspect() {
+                ParamTy::kw(name.clone(), t)
+            } else {
+                ParamTy::anonymous(t)
+            };
+            non_default_params.push(pt);
+        }
+        let var_params = if let Some(p) = lambda.sig.params.var_args.as_ref() {
+            let t = self.instantiate_param_sig_t(p, None, RegistrationMode::Normal)?;
+            let pt = if let Some(name) = p.inspect() {
+                ParamTy::kw(name.clone(), t)
+            } else {
+                ParamTy::anonymous(t)
+            };
+            Some(pt)
+        } else {
+            None
+        };
+        let mut default_params = Vec::with_capacity(lambda.sig.params.defaults.len());
+        for sig in lambda.sig.params.defaults.iter() {
+            let t = self.instantiate_param_sig_t(sig, None, RegistrationMode::Normal)?;
+            let pt = if let Some(name) = sig.inspect() {
+                ParamTy::kw(name.clone(), t)
+            } else {
+                ParamTy::anonymous(t)
+            };
+            default_params.push(pt);
+        }
+        // HACK: should avoid cloning
+        let mut lambda_ctx = Context::instant(Str::ever("<lambda>"), 0, self.clone());
+        let return_t = lambda_ctx.eval_const_block(&lambda.body, None)?;
+        // FIXME: lambda: i: Int -> Int
+        // => sig_t: (i: Type) -> Type
+        // => as_type: (i: Int) -> Int
+        let sig_t = subr_t(
+            SubrKind::from(lambda.op.kind),
+            non_default_params.clone(),
+            var_params.clone(),
+            default_params.clone(),
+            enum_t(set![return_t.clone()]),
+        );
+        let as_type = subr_t(
+            SubrKind::from(lambda.op.kind),
+            non_default_params,
+            var_params,
+            default_params,
+            return_t.as_type().ok_or_else(|| todo!())?.into_typ(),
+        );
         let subr = ConstSubr::User(UserConstSubr::new(
             Str::ever("<lambda>"),
             lambda.sig.params.clone(),
             lambda.body.clone(),
-            Type::Uninited,
+            sig_t,
+            Some(as_type),
         ));
         Ok(ValueObj::Subr(subr))
     }
diff --git a/compiler/erg_compiler/context/initialize/const_func.rs b/compiler/erg_compiler/context/initialize/const_func.rs
index 6313b8b8..ff1610b4 100644
--- a/compiler/erg_compiler/context/initialize/const_func.rs
+++ b/compiler/erg_compiler/context/initialize/const_func.rs
@@ -7,24 +7,8 @@ use erg_common::color::{RED, RESET};
 use erg_common::error::{ErrorCore, ErrorKind, Location};
 use erg_type::constructors::{and, mono};
 use erg_type::value::{EvalValueResult, TypeKind, TypeObj, ValueObj};
-use erg_type::Type;
 use erg_type::ValueArgs;
 
-fn value_obj_to_t(value: ValueObj) -> TypeObj {
-    match value {
-        ValueObj::Type(t) => t,
-        ValueObj::Record(rec) => TypeObj::Builtin(Type::Record(
-            rec.into_iter()
-                .map(|(k, v)| (k, value_obj_to_t(v).typ().clone()))
-                .collect(),
-        )),
-        ValueObj::Subr(subr) => {
-            todo!("{subr}")
-        }
-        other => todo!("{other}"),
-    }
-}
-
 /// Requirement: Type, Impl := Type -> ClassType
 pub fn class_func(mut args: ValueArgs, __name__: Option<Str>) -> EvalValueResult<ValueObj> {
     let require = args.remove_left_or_key("Requirement").ok_or_else(|| {
@@ -36,9 +20,9 @@ pub fn class_func(mut args: ValueArgs, __name__: Option<Str>) -> EvalValueResult
             None,
         )
     })?;
-    let require = value_obj_to_t(require);
+    let require = require.as_type().unwrap();
     let impls = args.remove_left_or_key("Impl");
-    let impls = impls.map(value_obj_to_t);
+    let impls = impls.map(|v| v.as_type().unwrap());
     let t = mono(__name__.unwrap_or(Str::ever("<Lambda>")));
     Ok(ValueObj::gen_t(TypeKind::Class, t, require, impls, None))
 }
@@ -54,11 +38,11 @@ pub fn inherit_func(mut args: ValueArgs, __name__: Option<Str>) -> EvalValueResu
             None,
         )
     })?;
-    let sup = value_obj_to_t(sup);
+    let sup = sup.as_type().unwrap();
     let impls = args.remove_left_or_key("Impl");
-    let impls = impls.map(value_obj_to_t);
+    let impls = impls.map(|v| v.as_type().unwrap());
     let additional = args.remove_left_or_key("Additional");
-    let additional = additional.map(value_obj_to_t);
+    let additional = additional.map(|v| v.as_type().unwrap());
     let t = mono(__name__.unwrap_or(Str::ever("<Lambda>")));
     Ok(ValueObj::gen_t(
         TypeKind::Subclass,
@@ -112,9 +96,9 @@ pub fn trait_func(mut args: ValueArgs, __name__: Option<Str>) -> EvalValueResult
             None,
         )
     })?;
-    let require = value_obj_to_t(require);
+    let require = require.as_type().unwrap();
     let impls = args.remove_left_or_key("Impl");
-    let impls = impls.map(value_obj_to_t);
+    let impls = impls.map(|v| v.as_type().unwrap());
     let t = mono(__name__.unwrap_or(Str::ever("<Lambda>")));
     Ok(ValueObj::gen_t(TypeKind::Trait, t, require, impls, None))
 }
@@ -130,11 +114,11 @@ pub fn subsume_func(mut args: ValueArgs, __name__: Option<Str>) -> EvalValueResu
             None,
         )
     })?;
-    let sup = value_obj_to_t(sup);
+    let sup = sup.as_type().unwrap();
     let impls = args.remove_left_or_key("Impl");
-    let impls = impls.map(value_obj_to_t);
+    let impls = impls.map(|v| v.as_type().unwrap());
     let additional = args.remove_left_or_key("Additional");
-    let additional = additional.map(value_obj_to_t);
+    let additional = additional.map(|v| v.as_type().unwrap());
     let t = mono(__name__.unwrap_or(Str::ever("<Lambda>")));
     Ok(ValueObj::gen_t(
         TypeKind::Subtrait,
diff --git a/compiler/erg_compiler/context/initialize/mod.rs b/compiler/erg_compiler/context/initialize/mod.rs
index 8091bcfc..44d542e5 100644
--- a/compiler/erg_compiler/context/initialize/mod.rs
+++ b/compiler/erg_compiler/context/initialize/mod.rs
@@ -1486,7 +1486,7 @@ impl Context {
             vec![param_t("Impl", Type)],
             Class,
         );
-        let class = ConstSubr::Builtin(BuiltinConstSubr::new("Class", class_func, class_t));
+        let class = ConstSubr::Builtin(BuiltinConstSubr::new("Class", class_func, class_t, None));
         self.register_builtin_const("Class", ValueObj::Subr(class));
         let inherit_t = func(
             vec![param_t("Super", Class)],
@@ -1494,7 +1494,12 @@ impl Context {
             vec![param_t("Impl", Type), param_t("Additional", Type)],
             Class,
         );
-        let inherit = ConstSubr::Builtin(BuiltinConstSubr::new("Inherit", inherit_func, inherit_t));
+        let inherit = ConstSubr::Builtin(BuiltinConstSubr::new(
+            "Inherit",
+            inherit_func,
+            inherit_t,
+            None,
+        ));
         self.register_builtin_const("Inherit", ValueObj::Subr(inherit));
         let trait_t = func(
             vec![param_t("Requirement", Type)],
@@ -1502,7 +1507,7 @@ impl Context {
             vec![param_t("Impl", Type)],
             Trait,
         );
-        let trait_ = ConstSubr::Builtin(BuiltinConstSubr::new("Trait", trait_func, trait_t));
+        let trait_ = ConstSubr::Builtin(BuiltinConstSubr::new("Trait", trait_func, trait_t, None));
         self.register_builtin_const("Trait", ValueObj::Subr(trait_));
         let subsume_t = func(
             vec![param_t("Super", Trait)],
@@ -1510,7 +1515,12 @@ impl Context {
             vec![param_t("Impl", Type), param_t("Additional", Type)],
             Trait,
         );
-        let subsume = ConstSubr::Builtin(BuiltinConstSubr::new("Subsume", subsume_func, subsume_t));
+        let subsume = ConstSubr::Builtin(BuiltinConstSubr::new(
+            "Subsume",
+            subsume_func,
+            subsume_t,
+            None,
+        ));
         self.register_builtin_const("Subsume", ValueObj::Subr(subsume));
         // decorators
         let inheritable_t = func1(Class, Class);
@@ -1518,6 +1528,7 @@ impl Context {
             "Inheritable",
             inheritable_func,
             inheritable_t,
+            None,
         ));
         self.register_builtin_const("Inheritable", ValueObj::Subr(inheritable));
     }
diff --git a/compiler/erg_compiler/context/inquire.rs b/compiler/erg_compiler/context/inquire.rs
index b4c4fcc0..4729ecc4 100644
--- a/compiler/erg_compiler/context/inquire.rs
+++ b/compiler/erg_compiler/context/inquire.rs
@@ -1259,6 +1259,7 @@ impl Context {
         }
     }
 
+    /// FIXME: if trait, returns a freevar
     pub(crate) fn rec_get_self_t(&self) -> Option<Type> {
         if self.kind.is_method_def() || self.kind.is_type() {
             // TODO: poly type
diff --git a/compiler/erg_compiler/context/instantiate.rs b/compiler/erg_compiler/context/instantiate.rs
index 7f9b42a5..f9541704 100644
--- a/compiler/erg_compiler/context/instantiate.rs
+++ b/compiler/erg_compiler/context/instantiate.rs
@@ -628,11 +628,7 @@ impl Context {
                 .collect();
                 let return_t = self.instantiate_typespec(&subr.return_t, mode)?;
                 Ok(subr_t(
-                    if subr.arrow.is(TokenKind::FuncArrow) {
-                        SubrKind::Func
-                    } else {
-                        SubrKind::Proc
-                    },
+                    SubrKind::from(subr.arrow.kind),
                     non_defaults,
                     var_args,
                     defaults,
diff --git a/compiler/erg_compiler/context/register.rs b/compiler/erg_compiler/context/register.rs
index 56d70658..93194b1b 100644
--- a/compiler/erg_compiler/context/register.rs
+++ b/compiler/erg_compiler/context/register.rs
@@ -649,6 +649,28 @@ impl Context {
                     todo!()
                 }
             }
+            TypeKind::Trait => {
+                if gen.t.is_monomorphic() {
+                    let ctx = Self::mono_trait(gen.t.name(), self.level);
+                    self.register_gen_mono_type(ident, gen, ctx, Const);
+                } else {
+                    todo!()
+                }
+            }
+            TypeKind::Subtrait => {
+                if gen.t.is_monomorphic() {
+                    let super_classes = vec![gen.require_or_sup.typ().clone()];
+                    // let super_traits = gen.impls.iter().map(|to| to.typ().clone()).collect();
+                    let mut ctx = Self::mono_trait(gen.t.name(), self.level);
+                    for sup in super_classes.into_iter() {
+                        let (_, sup_ctx) = self.get_nominal_type_ctx(&sup).unwrap();
+                        ctx.register_superclass(sup, sup_ctx);
+                    }
+                    self.register_gen_mono_type(ident, gen, ctx, Const);
+                } else {
+                    todo!()
+                }
+            }
             other => todo!("{other:?}"),
         }
     }
diff --git a/compiler/erg_compiler/context/tyvar.rs b/compiler/erg_compiler/context/tyvar.rs
index 18dcae83..a58a82f2 100644
--- a/compiler/erg_compiler/context/tyvar.rs
+++ b/compiler/erg_compiler/context/tyvar.rs
@@ -1027,6 +1027,9 @@ impl Context {
             (Type::MonoProj { .. }, _) => todo!(),
             (_, Type::MonoProj { .. }) => todo!(),
             (Refinement(_), Refinement(_)) => todo!(),
+            (Type::Subr(_) | Type::Record(_), Type) => Ok(()),
+            // TODO Tuple2, ...
+            (Type::Poly{ name, .. }, Type) if &name[..] == "Array" || &name[..] == "Tuple" => Ok(()),
             _ => todo!("{maybe_sub} can be a subtype of {maybe_sup}, but failed to semi-unify (or existential types are not supported)"),
         }
     }
diff --git a/compiler/erg_compiler/lower.rs b/compiler/erg_compiler/lower.rs
index 0719e743..bec72111 100644
--- a/compiler/erg_compiler/lower.rs
+++ b/compiler/erg_compiler/lower.rs
@@ -562,7 +562,14 @@ impl ASTLowerer {
                     name,
                 ));
             }
-            self.ctx.grow(name, ContextKind::Instant, def.sig.vis())?;
+            let kind = if def.is_class_def() {
+                ContextKind::Class
+            } else if def.is_trait_def() {
+                ContextKind::Trait
+            } else {
+                ContextKind::Instant
+            };
+            self.ctx.grow(name, kind, def.sig.vis())?;
             let res = match def.sig {
                 ast::Signature::Subr(sig) => self.lower_subr_def(sig, def.body),
                 ast::Signature::Var(sig) => self.lower_var_def(sig, def.body),
diff --git a/compiler/erg_parser/ast.rs b/compiler/erg_parser/ast.rs
index b03d2b97..8a79e214 100644
--- a/compiler/erg_parser/ast.rs
+++ b/compiler/erg_parser/ast.rs
@@ -225,6 +225,20 @@ impl Args {
     pub fn push_kw(&mut self, arg: KwArg) {
         self.kw_args.push(arg);
     }
+
+    pub fn get_left_or_key(&self, key: &str) -> Option<&Expr> {
+        if !self.pos_args.is_empty() {
+            self.pos_args.get(0).map(|a| &a.expr)
+        } else {
+            self.kw_args.iter().find_map(|a| {
+                if &a.keyword.content[..] == key {
+                    Some(&a.expr)
+                } else {
+                    None
+                }
+            })
+        }
+    }
 }
 
 #[derive(Clone, Debug, PartialEq, Eq, Hash)]
@@ -2862,6 +2876,34 @@ impl Def {
     pub const fn is_subr(&self) -> bool {
         matches!(&self.sig, Signature::Subr(_))
     }
+
+    pub fn is_class_def(&self) -> bool {
+        match self.body.block.first().unwrap() {
+            Expr::Call(call)
+                if call.obj.get_name().map(|n| &n[..]) == Some("Class")
+                    || call.obj.get_name().map(|n| &n[..]) == Some("Inherit") =>
+            {
+                true
+            }
+            Expr::Call(call) if call.obj.get_name().map(|n| &n[..]) == Some("Inheritable") => {
+                if let Some(Expr::Call(inner)) = call.args.get_left_or_key("Class") {
+                    inner.obj.get_name().map(|n| &n[..]) == Some("Class")
+                        || inner.obj.get_name().map(|n| &n[..]) == Some("Inherit")
+                } else {
+                    false
+                }
+            }
+            _ => false,
+        }
+    }
+
+    pub fn is_trait_def(&self) -> bool {
+        matches!(
+            self.body.block.first().unwrap(),
+            Expr::Call(call) if call.obj.get_name().map(|n| &n[..]) == Some("Trait")
+            || call.obj.get_name().map(|n| &n[..]) == Some("Subsume")
+        )
+    }
 }
 
 /// e.g.
diff --git a/compiler/erg_type/lib.rs b/compiler/erg_type/lib.rs
index 77852850..67c0e121 100644
--- a/compiler/erg_type/lib.rs
+++ b/compiler/erg_type/lib.rs
@@ -20,6 +20,7 @@ use erg_common::vis::Field;
 use erg_common::{enum_unwrap, fmt_option, fmt_set_split_with, set, Str};
 
 use erg_parser::ast::{Block, Params};
+use erg_parser::token::TokenKind;
 
 use crate::constructors::{int_interval, mono, mono_q};
 use crate::free::{
@@ -135,16 +136,24 @@ pub struct UserConstSubr {
     name: Str,
     params: Params,
     block: Block,
-    t: Type,
+    sig_t: Type,
+    as_type: Option<Type>,
 }
 
 impl UserConstSubr {
-    pub const fn new(name: Str, params: Params, block: Block, t: Type) -> Self {
+    pub const fn new(
+        name: Str,
+        params: Params,
+        block: Block,
+        sig_t: Type,
+        as_type: Option<Type>,
+    ) -> Self {
         Self {
             name,
             params,
             block,
-            t,
+            sig_t,
+            as_type,
         }
     }
 }
@@ -173,7 +182,8 @@ impl ValueArgs {
 pub struct BuiltinConstSubr {
     name: &'static str,
     subr: fn(ValueArgs, Option<Str>) -> EvalValueResult<ValueObj>,
-    t: Type,
+    sig_t: Type,
+    as_type: Option<Type>,
 }
 
 impl fmt::Display for BuiltinConstSubr {
@@ -186,9 +196,15 @@ impl BuiltinConstSubr {
     pub const fn new(
         name: &'static str,
         subr: fn(ValueArgs, Option<Str>) -> EvalValueResult<ValueObj>,
-        t: Type,
+        sig_t: Type,
+        as_type: Option<Type>,
     ) -> Self {
-        Self { name, subr, t }
+        Self {
+            name,
+            subr,
+            sig_t,
+            as_type,
+        }
     }
 
     pub fn call(&self, args: ValueArgs, __name__: Option<Str>) -> EvalValueResult<ValueObj> {
@@ -214,10 +230,17 @@ impl fmt::Display for ConstSubr {
 }
 
 impl ConstSubr {
-    pub fn class(&self) -> Type {
+    pub fn sig_t(&self) -> &Type {
         match self {
-            ConstSubr::User(user) => user.t.clone(),
-            ConstSubr::Builtin(builtin) => builtin.t.clone(),
+            ConstSubr::User(user) => &user.sig_t,
+            ConstSubr::Builtin(builtin) => &builtin.sig_t,
+        }
+    }
+
+    pub fn as_type(&self) -> Option<&Type> {
+        match self {
+            ConstSubr::User(user) => user.as_type.as_ref(),
+            ConstSubr::Builtin(builtin) => builtin.as_type.as_ref(),
         }
     }
 }
@@ -976,6 +999,16 @@ pub enum SubrKind {
     Proc,
 }
 
+impl From<TokenKind> for SubrKind {
+    fn from(op_kind: TokenKind) -> Self {
+        match op_kind {
+            TokenKind::FuncArrow => Self::Func,
+            TokenKind::ProcArrow => Self::Proc,
+            _ => panic!("invalid token kind for subr kind"),
+        }
+    }
+}
+
 impl SubrKind {
     pub const fn arrow(&self) -> Str {
         match self {
diff --git a/compiler/erg_type/value.rs b/compiler/erg_type/value.rs
index 9b0fe66e..ea68b3c2 100644
--- a/compiler/erg_type/value.rs
+++ b/compiler/erg_type/value.rs
@@ -14,7 +14,7 @@ use erg_common::rccell::RcCell;
 use erg_common::serialize::*;
 use erg_common::set;
 use erg_common::vis::Field;
-use erg_common::{fmt_iter, impl_display_from_debug, switch_lang};
+use erg_common::{dict, fmt_iter, impl_display_from_debug, switch_lang};
 use erg_common::{RcArray, Str};
 
 use crate::codeobj::CodeObj;
@@ -99,6 +99,13 @@ impl TypeObj {
         }
     }
 
+    pub fn into_typ(self) -> Type {
+        match self {
+            TypeObj::Builtin(t) => t,
+            TypeObj::Generated(t) => t.t,
+        }
+    }
+
     pub fn contains_intersec(&self, other: &Type) -> bool {
         match self {
             TypeObj::Builtin(t) => t.contains_intersec(other),
@@ -482,7 +489,7 @@ impl ValueObj {
             Self::Record(rec) => {
                 Type::Record(rec.iter().map(|(k, v)| (k.clone(), v.class())).collect())
             }
-            Self::Subr(subr) => subr.class(),
+            Self::Subr(subr) => subr.sig_t().clone(),
             Self::Type(t_obj) => match t_obj {
                 // TODO: builtin
                 TypeObj::Builtin(_t) => Type::Type,
@@ -771,6 +778,22 @@ impl ValueObj {
             _ => None,
         }
     }
+
+    pub fn as_type(&self) -> Option<TypeObj> {
+        match self {
+            Self::Type(t) => Some(t.clone()),
+            Self::Record(rec) => {
+                let mut attr_ts = dict! {};
+                for (k, v) in rec.iter() {
+                    attr_ts.insert(k.clone(), v.as_type()?.typ().clone());
+                }
+                Some(TypeObj::Builtin(Type::Record(attr_ts)))
+            }
+            Self::Subr(subr) => Some(TypeObj::Builtin(subr.as_type().unwrap().clone())),
+            Self::Array(_) | Self::Tuple(_) | Self::Dict(_) => todo!(),
+            _other => None,
+        }
+    }
 }
 
 pub mod value_set {