8.24/3.71 YES 10.44/4.28 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 10.44/4.28 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 10.44/4.28 10.44/4.28 10.44/4.28 H-Termination with start terms of the given HASKELL could be proven: 10.44/4.28 10.44/4.28 (0) HASKELL 10.44/4.28 (1) BR [EQUIVALENT, 0 ms] 10.44/4.28 (2) HASKELL 10.44/4.28 (3) COR [EQUIVALENT, 0 ms] 10.44/4.28 (4) HASKELL 10.44/4.28 (5) Narrow [SOUND, 0 ms] 10.44/4.28 (6) QDP 10.44/4.28 (7) QDPSizeChangeProof [EQUIVALENT, 0 ms] 10.44/4.28 (8) YES 10.44/4.28 10.44/4.28 10.44/4.28 ---------------------------------------- 10.44/4.28 10.44/4.28 (0) 10.44/4.28 Obligation: 10.44/4.28 mainModule Main 10.44/4.28 module Main where { 10.44/4.28 import qualified Prelude; 10.44/4.28 data Main.Char = Char MyInt ; 10.44/4.28 10.44/4.28 data MyBool = MyTrue | MyFalse ; 10.44/4.28 10.44/4.28 data MyInt = Pos Main.Nat | Neg Main.Nat ; 10.44/4.28 10.44/4.28 data Main.Nat = Succ Main.Nat | Zero ; 10.44/4.28 10.44/4.28 data Ordering = LT | EQ | GT ; 10.44/4.28 10.44/4.28 compareMyInt :: MyInt -> MyInt -> Ordering; 10.44/4.28 compareMyInt = primCmpInt; 10.44/4.28 10.44/4.28 esEsOrdering :: Ordering -> Ordering -> MyBool; 10.44/4.28 esEsOrdering LT LT = MyTrue; 10.44/4.28 esEsOrdering LT EQ = MyFalse; 10.44/4.28 esEsOrdering LT GT = MyFalse; 10.44/4.28 esEsOrdering EQ LT = MyFalse; 10.44/4.28 esEsOrdering EQ EQ = MyTrue; 10.44/4.28 esEsOrdering EQ GT = MyFalse; 10.44/4.28 esEsOrdering GT LT = MyFalse; 10.44/4.28 esEsOrdering GT EQ = MyFalse; 10.44/4.28 esEsOrdering GT GT = MyTrue; 10.44/4.28 10.44/4.28 fromEnumChar :: Main.Char -> MyInt; 10.44/4.28 fromEnumChar = primCharToInt; 10.44/4.28 10.44/4.28 isAscii :: Main.Char -> MyBool; 10.44/4.28 isAscii c = ltMyInt (fromEnumChar c) (Main.Pos (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ Main.Zero))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))); 10.44/4.28 10.44/4.28 ltMyInt :: MyInt -> MyInt -> MyBool; 10.44/4.28 ltMyInt x y = esEsOrdering (compareMyInt x y) LT; 10.44/4.28 10.44/4.28 primCharToInt :: Main.Char -> MyInt; 10.44/4.28 primCharToInt (Main.Char x) = x; 10.44/4.28 10.44/4.28 primCmpInt :: MyInt -> MyInt -> Ordering; 10.44/4.28 primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ; 10.44/4.28 primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ; 10.44/4.28 primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ; 10.44/4.28 primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ; 10.44/4.28 primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y; 10.44/4.28 primCmpInt (Main.Pos x) (Main.Neg y) = GT; 10.44/4.28 primCmpInt (Main.Neg x) (Main.Pos y) = LT; 10.44/4.28 primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x; 10.44/4.28 10.44/4.28 primCmpNat :: Main.Nat -> Main.Nat -> Ordering; 10.44/4.28 primCmpNat Main.Zero Main.Zero = EQ; 10.44/4.28 primCmpNat Main.Zero (Main.Succ y) = LT; 10.44/4.28 primCmpNat (Main.Succ x) Main.Zero = GT; 10.44/4.28 primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y; 10.44/4.28 10.44/4.28 } 10.44/4.28 10.44/4.28 ---------------------------------------- 10.44/4.28 10.44/4.28 (1) BR (EQUIVALENT) 10.44/4.28 Replaced joker patterns by fresh variables and removed binding patterns. 10.44/4.28 ---------------------------------------- 10.44/4.28 10.44/4.28 (2) 10.44/4.28 Obligation: 10.44/4.28 mainModule Main 10.44/4.28 module Main where { 10.44/4.28 import qualified Prelude; 10.44/4.28 data Main.Char = Char MyInt ; 10.44/4.28 10.44/4.28 data MyBool = MyTrue | MyFalse ; 10.44/4.28 10.44/4.28 data MyInt = Pos Main.Nat | Neg Main.Nat ; 10.44/4.28 10.44/4.28 data Main.Nat = Succ Main.Nat | Zero ; 10.44/4.28 10.44/4.28 data Ordering = LT | EQ | GT ; 10.44/4.28 10.44/4.28 compareMyInt :: MyInt -> MyInt -> Ordering; 10.44/4.28 compareMyInt = primCmpInt; 10.44/4.28 10.44/4.28 esEsOrdering :: Ordering -> Ordering -> MyBool; 10.44/4.28 esEsOrdering LT LT = MyTrue; 10.44/4.28 esEsOrdering LT EQ = MyFalse; 10.44/4.28 esEsOrdering LT GT = MyFalse; 10.44/4.28 esEsOrdering EQ LT = MyFalse; 10.44/4.28 esEsOrdering EQ EQ = MyTrue; 10.44/4.28 esEsOrdering EQ GT = MyFalse; 10.44/4.28 esEsOrdering GT LT = MyFalse; 10.44/4.28 esEsOrdering GT EQ = MyFalse; 10.44/4.28 esEsOrdering GT GT = MyTrue; 10.44/4.28 10.44/4.28 fromEnumChar :: Main.Char -> MyInt; 10.44/4.28 fromEnumChar = primCharToInt; 10.44/4.28 10.44/4.28 isAscii :: Main.Char -> MyBool; 10.44/4.28 isAscii c = ltMyInt (fromEnumChar c) (Main.Pos (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ Main.Zero))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))); 10.44/4.28 10.44/4.28 ltMyInt :: MyInt -> MyInt -> MyBool; 10.44/4.28 ltMyInt x y = esEsOrdering (compareMyInt x y) LT; 10.44/4.28 10.44/4.28 primCharToInt :: Main.Char -> MyInt; 10.44/4.28 primCharToInt (Main.Char x) = x; 10.44/4.28 10.44/4.28 primCmpInt :: MyInt -> MyInt -> Ordering; 10.44/4.28 primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ; 10.44/4.28 primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ; 10.44/4.28 primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ; 10.44/4.28 primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ; 10.44/4.28 primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y; 10.44/4.28 primCmpInt (Main.Pos x) (Main.Neg y) = GT; 10.44/4.28 primCmpInt (Main.Neg x) (Main.Pos y) = LT; 10.44/4.28 primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x; 10.44/4.28 10.44/4.28 primCmpNat :: Main.Nat -> Main.Nat -> Ordering; 10.44/4.28 primCmpNat Main.Zero Main.Zero = EQ; 10.44/4.28 primCmpNat Main.Zero (Main.Succ y) = LT; 10.44/4.28 primCmpNat (Main.Succ x) Main.Zero = GT; 10.44/4.28 primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y; 10.44/4.28 10.44/4.28 } 10.44/4.28 10.44/4.28 ---------------------------------------- 10.44/4.28 10.44/4.28 (3) COR (EQUIVALENT) 10.44/4.28 Cond Reductions: 10.44/4.28 The following Function with conditions 10.44/4.28 "undefined |Falseundefined; 10.44/4.28 " 10.44/4.28 is transformed to 10.44/4.28 "undefined = undefined1; 10.44/4.28 " 10.44/4.28 "undefined0 True = undefined; 10.44/4.28 " 10.44/4.28 "undefined1 = undefined0 False; 10.44/4.28 " 10.44/4.28 10.44/4.28 ---------------------------------------- 10.44/4.28 10.44/4.28 (4) 10.44/4.28 Obligation: 10.44/4.28 mainModule Main 10.44/4.28 module Main where { 10.44/4.28 import qualified Prelude; 10.44/4.28 data Main.Char = Char MyInt ; 10.44/4.28 10.44/4.28 data MyBool = MyTrue | MyFalse ; 10.44/4.28 10.44/4.28 data MyInt = Pos Main.Nat | Neg Main.Nat ; 10.44/4.28 10.44/4.28 data Main.Nat = Succ Main.Nat | Zero ; 10.44/4.28 10.44/4.28 data Ordering = LT | EQ | GT ; 10.44/4.28 10.44/4.28 compareMyInt :: MyInt -> MyInt -> Ordering; 10.44/4.28 compareMyInt = primCmpInt; 10.44/4.28 10.44/4.28 esEsOrdering :: Ordering -> Ordering -> MyBool; 10.44/4.28 esEsOrdering LT LT = MyTrue; 10.44/4.28 esEsOrdering LT EQ = MyFalse; 10.44/4.28 esEsOrdering LT GT = MyFalse; 10.44/4.28 esEsOrdering EQ LT = MyFalse; 10.44/4.28 esEsOrdering EQ EQ = MyTrue; 10.44/4.28 esEsOrdering EQ GT = MyFalse; 10.44/4.28 esEsOrdering GT LT = MyFalse; 10.44/4.28 esEsOrdering GT EQ = MyFalse; 10.44/4.28 esEsOrdering GT GT = MyTrue; 10.44/4.28 10.44/4.28 fromEnumChar :: Main.Char -> MyInt; 10.44/4.28 fromEnumChar = primCharToInt; 10.44/4.28 10.44/4.28 isAscii :: Main.Char -> MyBool; 10.44/4.28 isAscii c = ltMyInt (fromEnumChar c) (Main.Pos (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ Main.Zero))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))); 10.44/4.28 10.44/4.28 ltMyInt :: MyInt -> MyInt -> MyBool; 10.44/4.28 ltMyInt x y = esEsOrdering (compareMyInt x y) LT; 10.44/4.28 10.44/4.28 primCharToInt :: Main.Char -> MyInt; 10.44/4.28 primCharToInt (Main.Char x) = x; 10.44/4.28 10.44/4.28 primCmpInt :: MyInt -> MyInt -> Ordering; 10.44/4.28 primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ; 10.44/4.28 primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ; 10.44/4.28 primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ; 10.44/4.28 primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ; 10.44/4.28 primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y; 10.44/4.28 primCmpInt (Main.Pos x) (Main.Neg y) = GT; 10.44/4.28 primCmpInt (Main.Neg x) (Main.Pos y) = LT; 10.44/4.28 primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x; 10.44/4.28 10.44/4.28 primCmpNat :: Main.Nat -> Main.Nat -> Ordering; 10.44/4.28 primCmpNat Main.Zero Main.Zero = EQ; 10.44/4.28 primCmpNat Main.Zero (Main.Succ y) = LT; 10.44/4.28 primCmpNat (Main.Succ x) Main.Zero = GT; 10.44/4.28 primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y; 10.44/4.28 10.44/4.28 } 10.44/4.28 10.44/4.28 ---------------------------------------- 10.44/4.28 10.44/4.28 (5) Narrow (SOUND) 10.44/4.28 Haskell To QDPs 10.44/4.28 10.44/4.28 digraph dp_graph { 10.44/4.28 node [outthreshold=100, inthreshold=100];1[label="isAscii",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 10.44/4.28 3[label="isAscii vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3]; 10.44/4.28 4 -> 5[label="",style="dashed", color="red", weight=0]; 10.44/4.28 4[label="ltMyInt (fromEnumChar vx3) (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))",fontsize=16,color="magenta"];4 -> 6[label="",style="dashed", color="magenta", weight=3]; 10.44/4.28 4 -> 7[label="",style="dashed", color="magenta", weight=3]; 10.44/4.28 6[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))",fontsize=16,color="green",shape="box"];7[label="vx3",fontsize=16,color="green",shape="box"];5[label="ltMyInt (fromEnumChar vx5) (Pos (Succ vx6))",fontsize=16,color="black",shape="triangle"];5 -> 8[label="",style="solid", color="black", weight=3]; 10.44/4.28 8[label="esEsOrdering (compareMyInt (fromEnumChar vx5) (Pos (Succ vx6))) LT",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3]; 10.44/4.28 9[label="esEsOrdering (primCmpInt (fromEnumChar vx5) (Pos (Succ vx6))) LT",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3]; 10.44/4.28 10[label="esEsOrdering (primCmpInt (primCharToInt vx5) (Pos (Succ vx6))) LT",fontsize=16,color="burlywood",shape="box"];40[label="vx5/Char vx50",fontsize=10,color="white",style="solid",shape="box"];10 -> 40[label="",style="solid", color="burlywood", weight=9]; 10.44/4.28 40 -> 11[label="",style="solid", color="burlywood", weight=3]; 10.44/4.28 11[label="esEsOrdering (primCmpInt (primCharToInt (Char vx50)) (Pos (Succ vx6))) LT",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3]; 10.44/4.28 12[label="esEsOrdering (primCmpInt vx50 (Pos (Succ vx6))) LT",fontsize=16,color="burlywood",shape="box"];41[label="vx50/Pos vx500",fontsize=10,color="white",style="solid",shape="box"];12 -> 41[label="",style="solid", color="burlywood", weight=9]; 10.44/4.28 41 -> 13[label="",style="solid", color="burlywood", weight=3]; 10.44/4.28 42[label="vx50/Neg vx500",fontsize=10,color="white",style="solid",shape="box"];12 -> 42[label="",style="solid", color="burlywood", weight=9]; 10.44/4.28 42 -> 14[label="",style="solid", color="burlywood", weight=3]; 10.44/4.28 13[label="esEsOrdering (primCmpInt (Pos vx500) (Pos (Succ vx6))) LT",fontsize=16,color="burlywood",shape="box"];43[label="vx500/Succ vx5000",fontsize=10,color="white",style="solid",shape="box"];13 -> 43[label="",style="solid", color="burlywood", weight=9]; 10.44/4.28 43 -> 15[label="",style="solid", color="burlywood", weight=3]; 10.44/4.28 44[label="vx500/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 44[label="",style="solid", color="burlywood", weight=9]; 10.44/4.28 44 -> 16[label="",style="solid", color="burlywood", weight=3]; 10.44/4.28 14[label="esEsOrdering (primCmpInt (Neg vx500) (Pos (Succ vx6))) LT",fontsize=16,color="burlywood",shape="box"];45[label="vx500/Succ vx5000",fontsize=10,color="white",style="solid",shape="box"];14 -> 45[label="",style="solid", color="burlywood", weight=9]; 10.44/4.28 45 -> 17[label="",style="solid", color="burlywood", weight=3]; 10.44/4.28 46[label="vx500/Zero",fontsize=10,color="white",style="solid",shape="box"];14 -> 46[label="",style="solid", color="burlywood", weight=9]; 10.44/4.28 46 -> 18[label="",style="solid", color="burlywood", weight=3]; 10.44/4.28 15[label="esEsOrdering (primCmpInt (Pos (Succ vx5000)) (Pos (Succ vx6))) LT",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3]; 10.44/4.28 16[label="esEsOrdering (primCmpInt (Pos Zero) (Pos (Succ vx6))) LT",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3]; 10.44/4.28 17[label="esEsOrdering (primCmpInt (Neg (Succ vx5000)) (Pos (Succ vx6))) LT",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3]; 10.44/4.28 18[label="esEsOrdering (primCmpInt (Neg Zero) (Pos (Succ vx6))) LT",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3]; 10.44/4.28 19[label="esEsOrdering (primCmpNat (Succ vx5000) (Succ vx6)) LT",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3]; 10.44/4.28 20[label="esEsOrdering (primCmpNat Zero (Succ vx6)) LT",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3]; 10.44/4.28 21[label="esEsOrdering LT LT",fontsize=16,color="black",shape="triangle"];21 -> 25[label="",style="solid", color="black", weight=3]; 10.44/4.28 22 -> 21[label="",style="dashed", color="red", weight=0]; 10.44/4.28 22[label="esEsOrdering LT LT",fontsize=16,color="magenta"];23[label="esEsOrdering (primCmpNat vx5000 vx6) LT",fontsize=16,color="burlywood",shape="triangle"];47[label="vx5000/Succ vx50000",fontsize=10,color="white",style="solid",shape="box"];23 -> 47[label="",style="solid", color="burlywood", weight=9]; 10.44/4.28 47 -> 26[label="",style="solid", color="burlywood", weight=3]; 10.44/4.28 48[label="vx5000/Zero",fontsize=10,color="white",style="solid",shape="box"];23 -> 48[label="",style="solid", color="burlywood", weight=9]; 10.44/4.28 48 -> 27[label="",style="solid", color="burlywood", weight=3]; 10.44/4.28 24 -> 21[label="",style="dashed", color="red", weight=0]; 10.44/4.28 24[label="esEsOrdering LT LT",fontsize=16,color="magenta"];25[label="MyTrue",fontsize=16,color="green",shape="box"];26[label="esEsOrdering (primCmpNat (Succ vx50000) vx6) LT",fontsize=16,color="burlywood",shape="box"];49[label="vx6/Succ vx60",fontsize=10,color="white",style="solid",shape="box"];26 -> 49[label="",style="solid", color="burlywood", weight=9]; 10.44/4.28 49 -> 28[label="",style="solid", color="burlywood", weight=3]; 10.44/4.28 50[label="vx6/Zero",fontsize=10,color="white",style="solid",shape="box"];26 -> 50[label="",style="solid", color="burlywood", weight=9]; 10.44/4.28 50 -> 29[label="",style="solid", color="burlywood", weight=3]; 10.44/4.28 27[label="esEsOrdering (primCmpNat Zero vx6) LT",fontsize=16,color="burlywood",shape="box"];51[label="vx6/Succ vx60",fontsize=10,color="white",style="solid",shape="box"];27 -> 51[label="",style="solid", color="burlywood", weight=9]; 10.44/4.28 51 -> 30[label="",style="solid", color="burlywood", weight=3]; 10.44/4.28 52[label="vx6/Zero",fontsize=10,color="white",style="solid",shape="box"];27 -> 52[label="",style="solid", color="burlywood", weight=9]; 10.44/4.28 52 -> 31[label="",style="solid", color="burlywood", weight=3]; 10.44/4.28 28[label="esEsOrdering (primCmpNat (Succ vx50000) (Succ vx60)) LT",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3]; 10.44/4.28 29[label="esEsOrdering (primCmpNat (Succ vx50000) Zero) LT",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3]; 10.44/4.28 30[label="esEsOrdering (primCmpNat Zero (Succ vx60)) LT",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3]; 10.44/4.28 31[label="esEsOrdering (primCmpNat Zero Zero) LT",fontsize=16,color="black",shape="box"];31 -> 35[label="",style="solid", color="black", weight=3]; 10.44/4.28 32 -> 23[label="",style="dashed", color="red", weight=0]; 10.44/4.28 32[label="esEsOrdering (primCmpNat vx50000 vx60) LT",fontsize=16,color="magenta"];32 -> 36[label="",style="dashed", color="magenta", weight=3]; 10.44/4.28 32 -> 37[label="",style="dashed", color="magenta", weight=3]; 10.44/4.28 33[label="esEsOrdering GT LT",fontsize=16,color="black",shape="box"];33 -> 38[label="",style="solid", color="black", weight=3]; 10.44/4.28 34 -> 21[label="",style="dashed", color="red", weight=0]; 10.44/4.28 34[label="esEsOrdering LT LT",fontsize=16,color="magenta"];35[label="esEsOrdering EQ LT",fontsize=16,color="black",shape="box"];35 -> 39[label="",style="solid", color="black", weight=3]; 10.44/4.28 36[label="vx50000",fontsize=16,color="green",shape="box"];37[label="vx60",fontsize=16,color="green",shape="box"];38[label="MyFalse",fontsize=16,color="green",shape="box"];39[label="MyFalse",fontsize=16,color="green",shape="box"];} 10.44/4.28 10.44/4.28 ---------------------------------------- 10.44/4.28 10.44/4.28 (6) 10.44/4.28 Obligation: 10.44/4.28 Q DP problem: 10.44/4.28 The TRS P consists of the following rules: 10.44/4.28 10.44/4.28 new_esEsOrdering(Main.Succ(vx50000), Main.Succ(vx60)) -> new_esEsOrdering(vx50000, vx60) 10.44/4.28 10.44/4.28 R is empty. 10.44/4.28 Q is empty. 10.44/4.28 We have to consider all minimal (P,Q,R)-chains. 10.44/4.28 ---------------------------------------- 10.44/4.28 10.44/4.28 (7) QDPSizeChangeProof (EQUIVALENT) 10.44/4.28 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 10.44/4.28 10.44/4.28 From the DPs we obtained the following set of size-change graphs: 10.44/4.28 *new_esEsOrdering(Main.Succ(vx50000), Main.Succ(vx60)) -> new_esEsOrdering(vx50000, vx60) 10.44/4.28 The graph contains the following edges 1 > 1, 2 > 2 10.44/4.28 10.44/4.28 10.44/4.28 ---------------------------------------- 10.44/4.28 10.44/4.28 (8) 10.44/4.28 YES 10.44/4.34 EOF