diff --git a/compiler/erg_common/dict.rs b/compiler/erg_common/dict.rs
index 4aaddcc2..233da03f 100644
--- a/compiler/erg_common/dict.rs
+++ b/compiler/erg_common/dict.rs
@@ -175,8 +175,8 @@ impl<K: Hash + Eq, V> Dict<K, V> {
     }
 
     #[inline]
-    pub fn insert(&mut self, k: K, v: V) {
-        self.dict.insert(k, v);
+    pub fn insert(&mut self, k: K, v: V) -> Option<V> {
+        self.dict.insert(k, v)
     }
 
     #[inline]
diff --git a/compiler/erg_compiler/context/initialize/mod.rs b/compiler/erg_compiler/context/initialize/mod.rs
index ebfd0a88..c61477b9 100644
--- a/compiler/erg_compiler/context/initialize/mod.rs
+++ b/compiler/erg_compiler/context/initialize/mod.rs
@@ -7,6 +7,7 @@ pub mod py_mods;
 use std::path::PathBuf;
 
 use erg_common::config::ErgConfig;
+use erg_common::dict;
 // use erg_common::error::Location;
 use erg_common::vis::Visibility;
 use erg_common::Str;
@@ -470,7 +471,12 @@ impl Context {
         obj.register_builtin_impl("__sizeof__", fn0_met(Obj, Nat), Const, Public);
         obj.register_builtin_impl("__repr__", fn0_met(Obj, Str), Immutable, Public);
         obj.register_builtin_impl("__str__", fn0_met(Obj, Str), Immutable, Public);
-        obj.register_builtin_impl("__dict__", fn0_met(Obj, dict(Str, Obj)), Immutable, Public);
+        obj.register_builtin_impl(
+            "__dict__",
+            fn0_met(Obj, dict! {Str => Obj}.into()),
+            Immutable,
+            Public,
+        );
         obj.register_builtin_impl(
             "__bytes__",
             fn0_met(Obj, builtin_mono("Bytes")),
@@ -907,7 +913,7 @@ impl Context {
             Self::builtin_poly_class("Set", vec![PS::t_nd("T"), PS::named_nd("N", Nat)], 10);
         let n = mono_q_tp("N");
         let m = mono_q_tp("M");
-        let set_t = set(mono_q("T"), n.clone());
+        let set_t = set_t(mono_q("T"), n.clone());
         set_.register_superclass(Obj, &obj);
         set_.register_marker_trait(builtin_poly("Output", vec![ty_tp(mono_q("T"))]));
         let t = fn_met(
diff --git a/compiler/erg_compiler/hir.rs b/compiler/erg_compiler/hir.rs
index d83c85fe..99195096 100644
--- a/compiler/erg_compiler/hir.rs
+++ b/compiler/erg_compiler/hir.rs
@@ -1,6 +1,7 @@
 /// defines High-level Intermediate Representation
 use std::fmt;
 
+use erg_common::dict::Dict as HashMap;
 use erg_common::error::Location;
 use erg_common::traits::{Locational, NestedDisplay, Stream};
 use erg_common::vis::{Field, Visibility};
@@ -14,7 +15,7 @@ use erg_common::{
 use erg_parser::ast::{fmt_lines, DefId, DefKind, Params, TypeSpec, VarName};
 use erg_parser::token::{Token, TokenKind};
 
-use erg_type::constructors::{array, set, tuple};
+use erg_type::constructors::{array, dict_t, set_t, tuple};
 use erg_type::typaram::TyParam;
 use erg_type::value::{TypeKind, ValueObj};
 use erg_type::{impl_t, impl_t_for_enum, HasType, Type};
@@ -734,11 +735,16 @@ impl_display_from_nested!(NormalDict);
 impl_locational!(NormalDict, l_brace, r_brace);
 
 impl NormalDict {
-    pub const fn new(l_brace: Token, r_brace: Token, t: Type, kvs: Vec<KeyValue>) -> Self {
+    pub fn new(
+        l_brace: Token,
+        r_brace: Token,
+        kv_ts: HashMap<TyParam, TyParam>,
+        kvs: Vec<KeyValue>,
+    ) -> Self {
         Self {
             l_brace,
             r_brace,
-            t,
+            t: dict_t(TyParam::Dict(kv_ts)),
             kvs,
         }
     }
@@ -801,7 +807,7 @@ impl_t!(NormalSet);
 
 impl NormalSet {
     pub fn new(l_brace: Token, r_brace: Token, elem_t: Type, elems: Args) -> Self {
-        let t = set(elem_t, TyParam::value(elems.len()));
+        let t = set_t(elem_t, TyParam::value(elems.len()));
         Self {
             l_brace,
             r_brace,
diff --git a/compiler/erg_compiler/lower.rs b/compiler/erg_compiler/lower.rs
index 0ca580cf..1946c607 100644
--- a/compiler/erg_compiler/lower.rs
+++ b/compiler/erg_compiler/lower.rs
@@ -4,6 +4,7 @@
 
 use erg_common::astr::AtomicStr;
 use erg_common::config::ErgConfig;
+use erg_common::dict;
 use erg_common::error::{Location, MultiErrorDisplay};
 use erg_common::set;
 use erg_common::set::Set;
@@ -18,7 +19,8 @@ use erg_parser::token::{Token, TokenKind};
 use erg_parser::Parser;
 
 use erg_type::constructors::{
-    array, array_mut, builtin_mono, builtin_poly, free_var, func, mono, proc, quant, set, set_mut,
+    array, array_mut, builtin_mono, builtin_poly, free_var, func, mono, proc, quant, set_mut,
+    set_t, ty_tp,
 };
 use erg_type::free::Constraint;
 use erg_type::typaram::TyParam;
@@ -362,7 +364,6 @@ impl ASTLowerer {
         let mut union = Type::Never;
         let mut new_set = vec![];
         for elem in elems {
-            // TODO: Check if the object's type implements Eq
             let elem = self.lower_expr(elem.expr)?;
             union = self.ctx.union(&union, elem.ref_t());
             if union.is_intersection_type() {
@@ -448,7 +449,7 @@ impl ASTLowerer {
                 } else if self.ctx.subtype_of(&elem.t(), &Type::Type) {
                     builtin_poly("SetType", vec![TyParam::t(elem.t()), TyParam::Value(v)])
                 } else {
-                    set(elem.t(), TyParam::Value(v))
+                    set_t(elem.t(), TyParam::Value(v))
                 }
             }
             Ok(v @ ValueObj::Mut(_)) if v.class() == builtin_mono("Nat!") => {
@@ -469,7 +470,7 @@ impl ASTLowerer {
                         vec![TyParam::t(elem.t()), TyParam::erased(Type::Nat)],
                     )
                 } else {
-                    set(elem.t(), TyParam::erased(Type::Nat))
+                    set_t(elem.t(), TyParam::erased(Type::Nat))
                 }
             }
         }
@@ -484,8 +485,81 @@ impl ASTLowerer {
         }
     }
 
-    fn lower_normal_dict(&mut self, _dict: ast::NormalDict) -> LowerResult<hir::NormalDict> {
-        todo!()
+    fn lower_normal_dict(&mut self, dict: ast::NormalDict) -> LowerResult<hir::NormalDict> {
+        log!(info "enter {}({dict})", fn_name!());
+        let mut union = dict! {};
+        let mut new_kvs = vec![];
+        for kv in dict.kvs {
+            let loc = kv.loc();
+            let key = self.lower_expr(kv.key)?;
+            let value = self.lower_expr(kv.value)?;
+            if union.insert(key.t(), value.t()).is_none() {
+                return Err(LowerErrors::from(LowerError::syntax_error(
+                    self.cfg.input.clone(),
+                    line!() as usize,
+                    loc,
+                    AtomicStr::arc(&self.ctx.name[..]),
+                    switch_lang!(
+                        "japanese" => "集合の要素は全て同じ型である必要があります",
+                        "simplified_chinese" => "集合元素必须全部是相同类型",
+                        "traditional_chinese" => "集合元素必須全部是相同類型",
+                        "english" => "all elements of a set must be of the same type",
+                    ),
+                    Some(
+                        switch_lang!(
+                            "japanese" => "Int or Strなど明示的に型を指定してください",
+                            "simplified_chinese" => "明确指定类型，例如：Int or Str",
+                            "traditional_chinese" => "明確指定類型，例如：Int or Str",
+                            "english" => "please specify the type explicitly, e.g. Int or Str",
+                        )
+                        .into(),
+                    ),
+                )));
+            }
+            new_kvs.push(hir::KeyValue::new(key, value));
+        }
+        /*let sup = builtin_poly("Eq", vec![TyParam::t(elem_t.clone())]);
+        let loc = Location::concat(&dict.l_brace, &dict.r_brace);
+        // check if elem_t is Eq
+        if let Err(errs) = self.ctx.sub_unify(&elem_t, &sup, loc, None) {
+            self.errs.extend(errs.into_iter());
+        }*/
+        let kv_ts = if union.is_empty() {
+            dict! {
+                ty_tp(free_var(self.ctx.level, Constraint::new_type_of(Type::Type))) =>
+                    ty_tp(free_var(self.ctx.level, Constraint::new_type_of(Type::Type)))
+            }
+        } else {
+            union
+                .into_iter()
+                .map(|(k, v)| (TyParam::t(k), TyParam::t(v)))
+                .collect()
+        };
+        // TODO: lint
+        /*
+        if is_duplicated {
+            self.warns.push(LowerWarning::syntax_error(
+                self.cfg.input.clone(),
+                line!() as usize,
+                normal_set.loc(),
+                AtomicStr::arc(&self.ctx.name[..]),
+                switch_lang!(
+                    "japanese" => "要素が重複しています",
+                    "simplified_chinese" => "元素重复",
+                    "traditional_chinese" => "元素重複",
+                    "english" => "Elements are duplicated",
+                ),
+                None,
+            ));
+        }
+        Ok(normal_set)
+        */
+        Ok(hir::NormalDict::new(
+            dict.l_brace,
+            dict.r_brace,
+            kv_ts,
+            new_kvs,
+        ))
     }
 
     fn lower_acc(&mut self, acc: ast::Accessor) -> LowerResult<hir::Accessor> {
diff --git a/compiler/erg_parser/ast.rs b/compiler/erg_parser/ast.rs
index d73e2489..e3222509 100644
--- a/compiler/erg_parser/ast.rs
+++ b/compiler/erg_parser/ast.rs
@@ -613,8 +613,8 @@ impl KeyValue {
 
 #[derive(Clone, Debug, PartialEq, Eq, Hash)]
 pub struct NormalDict {
-    pub(crate) l_brace: Token,
-    pub(crate) r_brace: Token,
+    pub l_brace: Token,
+    pub r_brace: Token,
     pub kvs: Vec<KeyValue>,
 }
 
diff --git a/compiler/erg_type/constructors.rs b/compiler/erg_type/constructors.rs
index 32da9479..31ce967e 100644
--- a/compiler/erg_type/constructors.rs
+++ b/compiler/erg_type/constructors.rs
@@ -35,16 +35,13 @@ pub fn array_mut(elem_t: Type, len: TyParam) -> Type {
     builtin_poly("Array!", vec![TyParam::t(elem_t), len])
 }
 
-pub fn dict(k_t: Type, v_t: Type) -> Type {
-    builtin_poly("Dict", vec![TyParam::t(k_t), TyParam::t(v_t)])
-}
-
+// FIXME
 pub fn tuple(args: Vec<Type>) -> Type {
     let name = format!("Tuple{}", args.len());
     builtin_poly(name, args.into_iter().map(TyParam::t).collect())
 }
 
-pub fn set(elem_t: Type, len: TyParam) -> Type {
+pub fn set_t(elem_t: Type, len: TyParam) -> Type {
     builtin_poly("Set", vec![TyParam::t(elem_t), len])
 }
 
@@ -52,6 +49,10 @@ pub fn set_mut(elem_t: Type, len: TyParam) -> Type {
     builtin_poly("Set!", vec![TyParam::t(elem_t), len])
 }
 
+pub fn dict_t(dict: TyParam) -> Type {
+    builtin_poly("Dict", vec![dict])
+}
+
 #[inline]
 pub fn range(t: Type) -> Type {
     builtin_poly("Range", vec![TyParam::t(t)])
diff --git a/compiler/erg_type/deserialize.rs b/compiler/erg_type/deserialize.rs
index 54c45ec3..2d89077a 100644
--- a/compiler/erg_type/deserialize.rs
+++ b/compiler/erg_type/deserialize.rs
@@ -5,6 +5,7 @@ use std::string::FromUtf8Error;
 use erg_common::astr::AtomicStr;
 use erg_common::cache::CacheSet;
 use erg_common::config::{ErgConfig, Input};
+use erg_common::dict::Dict;
 use erg_common::error::{ErrorCore, ErrorKind, Location};
 use erg_common::serialize::DataTypePrefix;
 use erg_common::{fn_name, switch_lang};
@@ -105,7 +106,7 @@ pub type DeserializeResult<T> = Result<T, DeserializeError>;
 pub struct Deserializer {
     str_cache: CacheSet<str>,
     arr_cache: CacheSet<[ValueObj]>,
-    dict_cache: CacheSet<[(ValueObj, ValueObj)]>,
+    _dict_cache: CacheSet<Dict<ValueObj, ValueObj>>,
 }
 
 impl Deserializer {
@@ -113,7 +114,7 @@ impl Deserializer {
         Self {
             str_cache: CacheSet::new(),
             arr_cache: CacheSet::new(),
-            dict_cache: CacheSet::new(),
+            _dict_cache: CacheSet::new(),
         }
     }
 
@@ -146,11 +147,6 @@ impl Deserializer {
         ValueObj::Array(self.arr_cache.get(arr))
     }
 
-    /// TODO: 使わない？
-    pub fn get_cached_dict(&mut self, dict: &[(ValueObj, ValueObj)]) -> ValueObj {
-        ValueObj::Dict(self.dict_cache.get(dict))
-    }
-
     pub fn vec_to_bytes<const LEN: usize>(vector: Vec<u8>) -> [u8; LEN] {
         let mut arr = [0u8; LEN];
         for (arr_elem, vec_elem) in arr.iter_mut().zip(vector.iter()) {
diff --git a/compiler/erg_type/lib.rs b/compiler/erg_type/lib.rs
index 65dbd745..d343bd88 100644
--- a/compiler/erg_type/lib.rs
+++ b/compiler/erg_type/lib.rs
@@ -14,6 +14,7 @@ use std::fmt;
 use std::ops::{Range, RangeInclusive};
 use std::path::PathBuf;
 
+use constructors::dict_t;
 use erg_common::dict::Dict;
 use erg_common::set::Set;
 use erg_common::traits::LimitedDisplay;
@@ -1446,6 +1447,16 @@ impl From<RangeInclusive<&TyParam>> for Type {
     }
 }
 
+impl From<Dict<Type, Type>> for Type {
+    fn from(d: Dict<Type, Type>) -> Self {
+        let d = d
+            .into_iter()
+            .map(|(k, v)| (TyParam::t(k), TyParam::t(v)))
+            .collect();
+        dict_t(TyParam::Dict(d))
+    }
+}
+
 fn get_t_from_tp(tp: &TyParam) -> Option<Type> {
     match tp {
         TyParam::FreeVar(fv) if fv.is_linked() => get_t_from_tp(&fv.crack()),
diff --git a/compiler/erg_type/typaram.rs b/compiler/erg_type/typaram.rs
index c3bb7470..6859a124 100644
--- a/compiler/erg_type/typaram.rs
+++ b/compiler/erg_type/typaram.rs
@@ -2,6 +2,7 @@ use std::cmp::Ordering;
 use std::fmt;
 use std::ops::{Add, Div, Mul, Neg, Range, RangeInclusive, Sub};
 
+use erg_common::dict::Dict;
 use erg_common::traits::LimitedDisplay;
 
 use crate::constructors::int_interval;
@@ -127,6 +128,7 @@ pub enum TyParam {
     Array(Vec<TyParam>),
     Set(Vec<TyParam>),
     Tuple(Vec<TyParam>),
+    Dict(Dict<TyParam, TyParam>),
     Mono(Str),
     MonoProj {
         obj: Box<TyParam>,
@@ -300,6 +302,7 @@ impl LimitedDisplay for TyParam {
                 }
                 write!(f, "}}")
             }
+            Self::Dict(dict) => write!(f, "{dict}"),
             Self::Tuple(tuple) => {
                 write!(f, "(")?;
                 for (i, t) in tuple.iter().enumerate() {
diff --git a/compiler/erg_type/value.rs b/compiler/erg_type/value.rs
index 4328d4ef..10d8cc18 100644
--- a/compiler/erg_type/value.rs
+++ b/compiler/erg_type/value.rs
@@ -12,13 +12,14 @@ use erg_common::dict::Dict;
 use erg_common::error::ErrorCore;
 use erg_common::serialize::*;
 use erg_common::set;
+use erg_common::set::Set;
 use erg_common::shared::Shared;
 use erg_common::vis::Field;
 use erg_common::{dict, fmt_iter, impl_display_from_debug, switch_lang};
 use erg_common::{RcArray, Str};
 
 use crate::codeobj::CodeObj;
-use crate::constructors::{array, builtin_mono, builtin_poly, refinement, set as const_set, tuple};
+use crate::constructors::{array, builtin_mono, builtin_poly, refinement, set_t, tuple};
 use crate::free::fresh_varname;
 use crate::typaram::TyParam;
 use crate::{ConstSubr, HasType, Predicate, Type};
@@ -124,8 +125,8 @@ pub enum ValueObj {
     Str(Str),
     Bool(bool),
     Array(Rc<[ValueObj]>),
-    Set(Rc<[ValueObj]>),
-    Dict(Rc<[(ValueObj, ValueObj)]>),
+    Set(Set<ValueObj>),
+    Dict(Dict<ValueObj, ValueObj>),
     Tuple(Rc<[ValueObj]>),
     Record(Dict<Field, ValueObj>),
     Code(Box<CodeObj>),
@@ -508,7 +509,7 @@ impl ValueObj {
             ),
             Self::Dict(_dict) => todo!(),
             Self::Tuple(tup) => tuple(tup.iter().map(|v| v.class()).collect()),
-            Self::Set(st) => const_set(st.iter().next().unwrap().class(), TyParam::value(st.len())),
+            Self::Set(st) => set_t(st.iter().next().unwrap().class(), TyParam::value(st.len())),
             Self::Code(_) => Type::Code,
             Self::Record(rec) => {
                 Type::Record(rec.iter().map(|(k, v)| (k.clone(), v.class())).collect())