diff --git a/Jikka.cabal b/Jikka.cabal
index f4d13798..ee1068b8 100644
--- a/Jikka.cabal
+++ b/Jikka.cabal
@@ -84,6 +84,7 @@ library
       Jikka.Core.Convert.RemoveUnusedVars
       Jikka.Core.Convert.SegmentTree
       Jikka.Core.Convert.ShortCutFusion
+      Jikka.Core.Convert.SortAbs
       Jikka.Core.Convert.SpecializeFoldl
       Jikka.Core.Convert.TrivialLetElimination
       Jikka.Core.Convert.TypeInfer
@@ -250,6 +251,7 @@ test-suite jikka-test
       Jikka.Core.Convert.RemoveUnusedVarsSpec
       Jikka.Core.Convert.SegmentTreeSpec
       Jikka.Core.Convert.ShortCutFusionSpec
+      Jikka.Core.Convert.SortAbsSpec
       Jikka.Core.Convert.SpecializeFoldlSpec
       Jikka.Core.Convert.TrivialLetEliminationSpec
       Jikka.Core.Convert.TypeInferSpec
diff --git a/examples/data/METADATA.json b/examples/data/METADATA.json
index 75ff297a..c0d1f9c4 100644
--- a/examples/data/METADATA.json
+++ b/examples/data/METADATA.json
@@ -11,5 +11,6 @@
   "sum_sum_plus_one": "jikka-judge://sum-sum-plus-one",
   "sum_sum_plus_one_lt": "jikka-judge://sum-sum-plus-one-lt",
   "sum_sum_plus_two": "jikka-judge://sum-sum-plus-two",
-  "sum_sum_square": "jikka-judge://sum-sum-square"
+  "sum_sum_square": "jikka-judge://sum-sum-square",
+  "sum_sum_abs_one": "jikka-judge://sum-sum-abs-one"
 }
diff --git a/examples/sum_sum_abs_one.py b/examples/sum_sum_abs_one.py
new file mode 100644
index 00000000..bba1c900
--- /dev/null
+++ b/examples/sum_sum_abs_one.py
@@ -0,0 +1,19 @@
+# https://judge.kimiyuki.net/problem/sum-sum-abs-one
+from typing import *
+
+def solve(a: List[int]) -> int:
+    ans = 0
+    for a_i in a:
+        for a_j in a:
+            ans += abs(a_i - a_j)
+    return ans
+
+def main() -> None:
+    n = int(input())
+    a = list(map(int, input().split()))
+    assert len(a) == n
+    ans = solve(a)
+    print(ans)
+
+if __name__ == "__main__":
+    main()
diff --git a/scripts/integration_tests.py b/scripts/integration_tests.py
index bad67b4b..d6c1c777 100644
--- a/scripts/integration_tests.py
+++ b/scripts/integration_tests.py
@@ -58,7 +58,7 @@ def generate_test_cases_of_library_checker(*, problem_id: str, is_jikka_judge: b
 
     # collect testcases
     testcases: List[Tuple[pathlib.Path, pathlib.Path]] = []
-    for inputcase in (info_toml.parent / 'in').iterdir():
+    for inputcase in sorted((info_toml.parent / 'in').iterdir()):  # sorting makes examples first
         outputcase = info_toml.parent / 'out' / (inputcase.stem + '.out')
         testcases.append((inputcase, outputcase))
     return testcases
@@ -69,7 +69,7 @@ def collect_test_cases(script: pathlib.Path, *, tempdir: pathlib.Path, library_c
     testcases: List[Tuple[pathlib.Path, pathlib.Path]] = []
 
     # text files
-    for path in pathlib.Path('examples', 'data').iterdir():
+    for path in sorted(pathlib.Path('examples', 'data').iterdir()):
         if path.name[:-len(''.join(path.suffixes))] != script.stem:
             continue
         if path.suffix != '.in':
@@ -77,7 +77,7 @@ def collect_test_cases(script: pathlib.Path, *, tempdir: pathlib.Path, library_c
         testcases.append((path, path.with_suffix('.out')))
 
     # using generators
-    for generator_path in pathlib.Path('examples', 'data').glob(glob.escape(script.stem) + '*.generator.py'):
+    for generator_path in sorted(pathlib.Path('examples', 'data').glob(glob.escape(script.stem) + '*.generator.py')):
         _, testset_name, _, _ = generator_path.name.split('.')
 
         for solver_ext in ('.py', '.cpp'):
diff --git a/src/Jikka/Core/Convert.hs b/src/Jikka/Core/Convert.hs
index f576e335..f50c31b9 100644
--- a/src/Jikka/Core/Convert.hs
+++ b/src/Jikka/Core/Convert.hs
@@ -37,6 +37,7 @@ import qualified Jikka.Core.Convert.PropagateMod as PropagateMod
 import qualified Jikka.Core.Convert.RemoveUnusedVars as RemoveUnusedVars
 import qualified Jikka.Core.Convert.SegmentTree as SegmentTree
 import qualified Jikka.Core.Convert.ShortCutFusion as ShortCutFusion
+import qualified Jikka.Core.Convert.SortAbs as SortAbs
 import qualified Jikka.Core.Convert.SpecializeFoldl as SpecializeFoldl
 import qualified Jikka.Core.Convert.TrivialLetElimination as TrivialLetElimination
 import qualified Jikka.Core.Convert.TypeInfer as TypeInfer
@@ -49,6 +50,7 @@ run'' prog = do
   prog <- UnpackTuple.run prog
   prog <- MatrixExponentiation.run prog
   prog <- SpecializeFoldl.run prog
+  prog <- SortAbs.run prog
   prog <- MakeScanl.run prog
   prog <- PropagateMod.run prog
   prog <- ConstantPropagation.run prog
diff --git a/src/Jikka/Core/Convert/Alpha.hs b/src/Jikka/Core/Convert/Alpha.hs
index 9424c22a..9db7457d 100644
--- a/src/Jikka/Core/Convert/Alpha.hs
+++ b/src/Jikka/Core/Convert/Alpha.hs
@@ -10,7 +10,13 @@
 -- Maintainer  : kimiyuki95@gmail.com
 -- Stability   : experimental
 -- Portability : portable
-module Jikka.Core.Convert.Alpha where
+module Jikka.Core.Convert.Alpha
+  ( run,
+    runProgram,
+    runToplevelExpr,
+    runExpr,
+  )
+where
 
 import Jikka.Common.Alpha
 import Jikka.Common.Error
@@ -22,31 +28,35 @@ rename x = do
   i <- nextCounter
   return $ VarName (base ++ "$" ++ show i)
 
-runExpr :: (MonadAlpha m, MonadError Error m) => [(VarName, VarName)] -> Expr -> m Expr
-runExpr env = \case
+runExpr' :: (MonadAlpha m, MonadError Error m) => [(VarName, VarName)] -> Expr -> m Expr
+runExpr' env = \case
   Var x -> case lookup x env of
     Nothing -> throwInternalError $ "undefined variable: " ++ unVarName x
     Just y -> return $ Var y
   Lit lit -> return $ Lit lit
-  App f e -> App <$> runExpr env f <*> runExpr env e
+  App f e -> App <$> runExpr' env f <*> runExpr' env e
   Lam x t body -> do
     y <- rename x
-    body <- runExpr ((x, y) : env) body
+    body <- runExpr' ((x, y) : env) body
     return $ Lam y t body
   Let x t e1 e2 -> do
-    e1 <- runExpr env e1
+    e1 <- runExpr' env e1
     y <- rename x
-    e2 <- runExpr ((x, y) : env) e2
+    e2 <- runExpr' ((x, y) : env) e2
     return $ Let y t e1 e2
-  Assert e1 e2 -> Assert <$> runExpr env e1 <*> runExpr env e2
+  Assert e1 e2 -> Assert <$> runExpr' env e1 <*> runExpr' env e2
 
-runToplevelExpr :: (MonadAlpha m, MonadError Error m) => [(VarName, VarName)] -> ToplevelExpr -> m ToplevelExpr
-runToplevelExpr env = \case
-  ResultExpr e -> ResultExpr <$> runExpr env e
+runExpr :: (MonadAlpha m, MonadError Error m) => [(VarName, Type)] -> Expr -> m Expr
+runExpr env e = wrapError' "Jikka.Core.Convert.Alpha" $ do
+  runExpr' (map (\(x, _) -> (x, x)) env) e
+
+runToplevelExpr' :: (MonadAlpha m, MonadError Error m) => [(VarName, VarName)] -> ToplevelExpr -> m ToplevelExpr
+runToplevelExpr' env = \case
+  ResultExpr e -> ResultExpr <$> runExpr' env e
   ToplevelLet x t e cont -> do
     y <- rename x
-    e <- runExpr env e
-    cont <- runToplevelExpr ((x, y) : env) cont
+    e <- runExpr' env e
+    cont <- runToplevelExpr' ((x, y) : env) cont
     return $ ToplevelLet y t e cont
   ToplevelLetRec f args ret body cont -> do
     g <- rename f
@@ -55,13 +65,18 @@ runToplevelExpr env = \case
       return (x, y, t)
     let args1 = map (\(x, y, _) -> (x, y)) args
     let args2 = map (\(_, y, t) -> (y, t)) args
-    body <- runExpr (args1 ++ (f, g) : env) body
-    cont <- runToplevelExpr ((f, g) : env) cont
+    body <- runExpr' (args1 ++ (f, g) : env) body
+    cont <- runToplevelExpr' ((f, g) : env) cont
     return $ ToplevelLetRec g args2 ret body cont
-  ToplevelAssert e1 e2 -> ToplevelAssert <$> runExpr env e1 <*> runToplevelExpr env e2
+  ToplevelAssert e1 e2 -> ToplevelAssert <$> runExpr' env e1 <*> runToplevelExpr' env e2
+
+runToplevelExpr :: (MonadAlpha m, MonadError Error m) => [(VarName, Type)] -> ToplevelExpr -> m ToplevelExpr
+runToplevelExpr env e = wrapError' "Jikka.Core.Convert.Alpha" $ do
+  runToplevelExpr' (map (\(x, _) -> (x, x)) env) e
 
 runProgram :: (MonadAlpha m, MonadError Error m) => Program -> m Program
-runProgram = runToplevelExpr []
+runProgram prog = wrapError' "Jikka.Core.Convert.Alpha" $ do
+  runToplevelExpr' [] prog
 
 -- | `run` renames variables in exprs to avoid name conflictions, even if the scopes of two variables are distinct.
 --
@@ -81,5 +96,4 @@ runProgram = runToplevelExpr []
 -- > in x2 = x0 + y1
 -- > x2 + y1
 run :: (MonadAlpha m, MonadError Error m) => Program -> m Program
-run prog = wrapError' "Jikka.Core.Convert.Alpha" $ do
-  runToplevelExpr [] prog
+run = runProgram
diff --git a/src/Jikka/Core/Convert/SortAbs.hs b/src/Jikka/Core/Convert/SortAbs.hs
new file mode 100644
index 00000000..f8d7251a
--- /dev/null
+++ b/src/Jikka/Core/Convert/SortAbs.hs
@@ -0,0 +1,135 @@
+{-# LANGUAGE FlexibleContexts #-}
+{-# LANGUAGE LambdaCase #-}
+
+-- |
+-- Module      : Jikka.Core.Convert.SortAbs
+-- Description : remove abs with sorting. / sort によって abs を除去します。
+-- Copyright   : (c) Kimiyuki Onaka, 2021
+-- License     : Apache License 2.0
+-- Maintainer  : kimiyuki95@gmail.com
+-- Stability   : experimental
+-- Portability : portable
+--
+-- \[
+--     \newcommand\int{\mathbf{int}}
+--     \newcommand\bool{\mathbf{bool}}
+--     \newcommand\list{\mathbf{list}}
+-- \]
+module Jikka.Core.Convert.SortAbs
+  ( run,
+
+    -- * internal rules
+    rule,
+  )
+where
+
+import Control.Monad.Trans.Maybe
+import Jikka.Common.Alpha
+import Jikka.Common.Error
+import qualified Jikka.Core.Convert.Alpha as Alpha
+import Jikka.Core.Language.ArithmeticExpr
+import Jikka.Core.Language.Beta
+import Jikka.Core.Language.BuiltinPatterns
+import Jikka.Core.Language.Expr
+import Jikka.Core.Language.Lint
+import Jikka.Core.Language.QuasiRules
+import Jikka.Core.Language.RewriteRules
+import Jikka.Core.Language.Util
+
+-- | @replaceAbsDelta x y z e@ replaces \(\levert x - y \rvert\) in \(e\) with \(z\).
+replaceAbsDelta :: VarName -> VarName -> VarName -> Expr -> Expr
+replaceAbsDelta x y z e = mapSubExpr go [] e
+  where
+    go _ = \case
+      Abs' e | isZeroArithmeticExpr (parseArithmeticExpr (Minus' e (Minus' (Var x) (Var y)))) -> Var z
+      Abs' e | isZeroArithmeticExpr (parseArithmeticExpr (Minus' e (Minus' (Var y) (Var x)))) -> Var z
+      e -> e
+
+swapTwoVars :: MonadAlpha m => VarName -> VarName -> Expr -> m Expr
+swapTwoVars x y e = do
+  x' <- genVarName x
+  y' <- genVarName y
+  e <- substitute x (Var x') e
+  e <- substitute y (Var y') e
+  e <- substitute x' (Var y) e
+  substitute y' (Var x) e
+
+-- | TODO: accept more functions
+isSymmetric :: MonadAlpha m => VarName -> VarName -> Expr -> m Bool
+isSymmetric x y f = do
+  g <- swapTwoVars x y f
+  return $ parseArithmeticExpr g == parseArithmeticExpr f
+
+rule :: (MonadAlpha m, MonadError Error m) => RewriteRule m
+rule = makeRewriteRule "sum/sum/abs/symmetric" $ \env -> \case
+  Sum' (Map' IntTy _ (Lam x _ (Sum' (Map' _ _ (Lam y _ f) xs'))) xs) | xs' == xs -> runMaybeT $ do
+    delta <- lift genVarName'
+    let f' = replaceAbsDelta x y delta f
+    guard $ f' /= f -- f has |x - y|
+    guard =<< lift (isSymmetric x y f') -- symmetric
+    ys <- lift $ genVarName'' xs
+    i <- lift genVarName'
+    j <- lift genVarName'
+    lt <- lift $ substitute delta (Minus' (Var x) (Var y)) f'
+    eq <- lift $ substitute delta (LitInt' 0) f'
+    gt <- lift $ substitute delta (Minus' (Var y) (Var x)) f'
+    let ctx = Let y IntTy (At' IntTy (Var ys) (Var j))
+    let lt' = Sum' (Map' IntTy IntTy (Lam j IntTy (ctx lt)) (Range1' (Var i)))
+    let eq' = Let j IntTy (Var i) (ctx eq)
+    let gt' = Sum' (Map' IntTy IntTy (Lam j IntTy (ctx gt)) (Range2' (Plus' (Var i) (LitInt' 1)) (Len' IntTy (Var ys))))
+    let e =
+          Let ys (ListTy IntTy) (Sorted' IntTy xs) $
+            Sum'
+              ( Map'
+                  IntTy
+                  IntTy
+                  ( Lam
+                      i
+                      IntTy
+                      ( Let
+                          x
+                          IntTy
+                          (At' IntTy (Var ys) (Var i))
+                          (Plus' (Plus' lt' eq') gt')
+                      )
+                  )
+                  (Range1' (Len' IntTy (Var ys)))
+              )
+    lift $ Alpha.runExpr (typeEnv env) e
+  _ -> return Nothing
+
+runProgram :: (MonadAlpha m, MonadError Error m) => Program -> m Program
+runProgram = applyRewriteRuleProgram' rule
+
+-- | `run` reduces \(\lvert \sum _ {a_i \in a} \sum _ {a_j \in a} f(a, a_i, a_j) \rvert\) to \(\mathbf{let}~ b = \mathrm{sort}(a) ~\mathbf{in}~ \sum \sum f'(a, a_i, a_j)\) when \(f\) contains \(\lvert a_i - a_j \rvert\) and \(f(a, a_i, a_j) = f(a, a_j, a_i)\) holds.
+--
+-- == Example
+--
+-- Before:
+--
+-- > sum (map (fun (a_i: int) ->
+-- >     sum (map (fun (a_j: int) ->
+-- >         abs (a_i - a_j)
+-- >     ) a)
+-- > ) a)
+--
+-- After:
+--
+-- > let b = sort a
+-- > in sum (map (fun (i: int) ->
+-- >     (sum (map (fun (b_j: int) ->
+-- >         b_i - b_j
+-- >     ) b[:i])
+-- >     + 0
+-- >     + sum (map (fun (b_j: int) ->
+-- >         b_j - b_i
+-- >     ) b[i + 1:]))
+-- > ) (range (length b)))
+run :: (MonadAlpha m, MonadError Error m) => Program -> m Program
+run prog = wrapError' "Jikka.Core.Convert.SortAbs" $ do
+  precondition $ do
+    ensureWellTyped prog
+  prog <- runProgram prog
+  postcondition $ do
+    ensureWellTyped prog
+  return prog
diff --git a/src/Jikka/Core/Language/Util.hs b/src/Jikka/Core/Language/Util.hs
index 3b5f700e..d343b503 100644
--- a/src/Jikka/Core/Language/Util.hs
+++ b/src/Jikka/Core/Language/Util.hs
@@ -30,6 +30,11 @@ genVarName x = do
 genVarName' :: MonadAlpha m => m VarName
 genVarName' = genVarName (VarName "_")
 
+genVarName'' :: MonadAlpha m => Expr -> m VarName
+genVarName'' = \case
+  Var x -> genVarName x
+  _ -> genVarName'
+
 mapSubTypesM :: Monad m => (Type -> m Type) -> Type -> m Type
 mapSubTypesM f = go
   where
diff --git a/test/Jikka/Core/Convert/SortAbsSpec.hs b/test/Jikka/Core/Convert/SortAbsSpec.hs
new file mode 100644
index 00000000..ccfb73c7
--- /dev/null
+++ b/test/Jikka/Core/Convert/SortAbsSpec.hs
@@ -0,0 +1,51 @@
+{-# LANGUAGE OverloadedStrings #-}
+
+module Jikka.Core.Convert.SortAbsSpec (spec) where
+
+import Jikka.Common.Alpha
+import Jikka.Common.Error
+import Jikka.Core.Convert.SortAbs (run)
+import qualified Jikka.Core.Convert.TypeInfer as TypeInfer
+import Jikka.Core.Format (formatProgram)
+import Jikka.Core.Language.Expr
+import Jikka.Core.Parse (parseProgram)
+import Test.Hspec
+
+run' :: Program -> Either Error Program
+run' = flip evalAlphaT 0 . run
+
+parseProgram' :: [String] -> Program
+parseProgram' = fromSuccess . flip evalAlphaT 100 . (TypeInfer.run <=< parseProgram . unlines)
+
+spec :: Spec
+spec = describe "run" $ do
+  it "works about sum" $ do
+    let prog =
+          parseProgram'
+            [ "fun (a: int list) ->",
+              "    sum (map (fun (a_i: int) ->",
+              "        sum (map (fun (a_j: int) ->",
+              "            abs (a_i - a_j)",
+              "        ) a)",
+              "    ) a)"
+            ]
+    let expected =
+          parseProgram'
+            [ "fun (a: int list) ->",
+              "    let a$6 = sorted a",
+              "    in sum (map (fun ($7: int) ->",
+              "        let a_i$8 = a$6[$7] in",
+              "        sum (map (fun ($9: int) ->",
+              "            let a_j$10 = a$6[$9]",
+              "            in a_i$8 - a_j$10",
+              "        ) (range $7))",
+              "        + (let $11 = $7",
+              "           in let a_j$12 = a$6[$11]",
+              "           in 0)",
+              "        + sum (map (fun ($13: int) ->",
+              "            let a_j$14 = a$6[$13]",
+              "            in a_j$14 - a_i$8",
+              "        ) (range2 ($7 + 1) (len a$6)))",
+              "    ) (range (len a$6)))"
+            ]
+    (formatProgram <$> run' prog) `shouldBe` Right (formatProgram expected)