8.14/3.63 YES 10.05/4.20 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 10.05/4.20 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 10.05/4.20 10.05/4.20 10.05/4.20 H-Termination with start terms of the given HASKELL could be proven: 10.05/4.20 10.05/4.20 (0) HASKELL 10.05/4.20 (1) BR [EQUIVALENT, 0 ms] 10.05/4.20 (2) HASKELL 10.05/4.20 (3) COR [EQUIVALENT, 0 ms] 10.05/4.20 (4) HASKELL 10.05/4.20 (5) Narrow [SOUND, 0 ms] 10.05/4.20 (6) AND 10.05/4.20 (7) QDP 10.05/4.20 (8) QDPSizeChangeProof [EQUIVALENT, 0 ms] 10.05/4.20 (9) YES 10.05/4.20 (10) QDP 10.05/4.20 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 10.05/4.20 (12) YES 10.05/4.20 10.05/4.20 10.05/4.20 ---------------------------------------- 10.05/4.20 10.05/4.20 (0) 10.05/4.20 Obligation: 10.05/4.20 mainModule Main 10.05/4.20 module Main where { 10.05/4.20 import qualified Prelude; 10.05/4.20 data Main.Char = Char MyInt ; 10.05/4.20 10.05/4.20 data MyBool = MyTrue | MyFalse ; 10.05/4.20 10.05/4.20 data MyInt = Pos Main.Nat | Neg Main.Nat ; 10.05/4.20 10.05/4.20 data Main.Nat = Succ Main.Nat | Zero ; 10.05/4.20 10.05/4.20 data Ordering = LT | EQ | GT ; 10.05/4.20 10.05/4.20 asAs :: MyBool -> MyBool -> MyBool; 10.05/4.20 asAs MyFalse x = MyFalse; 10.05/4.20 asAs MyTrue x = x; 10.05/4.20 10.05/4.20 compareChar :: Main.Char -> Main.Char -> Ordering; 10.05/4.20 compareChar = primCmpChar; 10.05/4.20 10.05/4.20 esEsOrdering :: Ordering -> Ordering -> MyBool; 10.05/4.20 esEsOrdering LT LT = MyTrue; 10.05/4.20 esEsOrdering LT EQ = MyFalse; 10.05/4.20 esEsOrdering LT GT = MyFalse; 10.05/4.20 esEsOrdering EQ LT = MyFalse; 10.05/4.20 esEsOrdering EQ EQ = MyTrue; 10.05/4.20 esEsOrdering EQ GT = MyFalse; 10.05/4.20 esEsOrdering GT LT = MyFalse; 10.05/4.20 esEsOrdering GT EQ = MyFalse; 10.05/4.20 esEsOrdering GT GT = MyTrue; 10.05/4.20 10.05/4.20 fsEsOrdering :: Ordering -> Ordering -> MyBool; 10.05/4.20 fsEsOrdering x y = not (esEsOrdering x y); 10.05/4.20 10.05/4.20 gtEsChar :: Main.Char -> Main.Char -> MyBool; 10.05/4.20 gtEsChar x y = fsEsOrdering (compareChar x y) LT; 10.05/4.20 10.05/4.20 isDigit :: Main.Char -> MyBool; 10.05/4.20 isDigit c = asAs (gtEsChar c (Main.Char (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.Zero))))))))))))))))))))))))))))))))))))))))))))))))))) (ltEsChar c (Main.Char (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.Zero)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))); 10.05/4.20 10.05/4.20 ltEsChar :: Main.Char -> Main.Char -> MyBool; 10.05/4.20 ltEsChar x y = fsEsOrdering (compareChar x y) GT; 10.05/4.20 10.05/4.20 not :: MyBool -> MyBool; 10.05/4.20 not MyTrue = MyFalse; 10.05/4.20 not MyFalse = MyTrue; 10.05/4.20 10.05/4.20 primCmpChar :: Main.Char -> Main.Char -> Ordering; 10.05/4.20 primCmpChar (Main.Char x) (Main.Char y) = primCmpInt x y; 10.05/4.20 10.05/4.20 primCmpInt :: MyInt -> MyInt -> Ordering; 10.05/4.20 primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ; 10.05/4.20 primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ; 10.05/4.20 primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ; 10.05/4.20 primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ; 10.05/4.20 primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y; 10.05/4.20 primCmpInt (Main.Pos x) (Main.Neg y) = GT; 10.05/4.20 primCmpInt (Main.Neg x) (Main.Pos y) = LT; 10.05/4.20 primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x; 10.05/4.20 10.05/4.20 primCmpNat :: Main.Nat -> Main.Nat -> Ordering; 10.05/4.20 primCmpNat Main.Zero Main.Zero = EQ; 10.05/4.20 primCmpNat Main.Zero (Main.Succ y) = LT; 10.05/4.20 primCmpNat (Main.Succ x) Main.Zero = GT; 10.05/4.20 primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y; 10.05/4.20 10.05/4.20 } 10.05/4.20 10.05/4.20 ---------------------------------------- 10.05/4.20 10.05/4.20 (1) BR (EQUIVALENT) 10.05/4.20 Replaced joker patterns by fresh variables and removed binding patterns. 10.05/4.20 ---------------------------------------- 10.05/4.20 10.05/4.20 (2) 10.05/4.20 Obligation: 10.05/4.20 mainModule Main 10.05/4.20 module Main where { 10.05/4.20 import qualified Prelude; 10.05/4.20 data Main.Char = Char MyInt ; 10.05/4.20 10.05/4.20 data MyBool = MyTrue | MyFalse ; 10.05/4.20 10.05/4.20 data MyInt = Pos Main.Nat | Neg Main.Nat ; 10.05/4.20 10.05/4.20 data Main.Nat = Succ Main.Nat | Zero ; 10.05/4.20 10.05/4.20 data Ordering = LT | EQ | GT ; 10.05/4.20 10.05/4.20 asAs :: MyBool -> MyBool -> MyBool; 10.05/4.20 asAs MyFalse x = MyFalse; 10.05/4.20 asAs MyTrue x = x; 10.05/4.20 10.05/4.20 compareChar :: Main.Char -> Main.Char -> Ordering; 10.05/4.20 compareChar = primCmpChar; 10.05/4.20 10.05/4.20 esEsOrdering :: Ordering -> Ordering -> MyBool; 10.05/4.20 esEsOrdering LT LT = MyTrue; 10.05/4.20 esEsOrdering LT EQ = MyFalse; 10.05/4.20 esEsOrdering LT GT = MyFalse; 10.05/4.20 esEsOrdering EQ LT = MyFalse; 10.05/4.20 esEsOrdering EQ EQ = MyTrue; 10.05/4.20 esEsOrdering EQ GT = MyFalse; 10.05/4.20 esEsOrdering GT LT = MyFalse; 10.05/4.20 esEsOrdering GT EQ = MyFalse; 10.05/4.20 esEsOrdering GT GT = MyTrue; 10.05/4.20 10.05/4.20 fsEsOrdering :: Ordering -> Ordering -> MyBool; 10.05/4.20 fsEsOrdering x y = not (esEsOrdering x y); 10.05/4.20 10.05/4.20 gtEsChar :: Main.Char -> Main.Char -> MyBool; 10.05/4.20 gtEsChar x y = fsEsOrdering (compareChar x y) LT; 10.05/4.20 10.05/4.20 isDigit :: Main.Char -> MyBool; 10.05/4.20 isDigit c = asAs (gtEsChar c (Main.Char (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.Zero))))))))))))))))))))))))))))))))))))))))))))))))))) (ltEsChar c (Main.Char (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.Zero)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))); 10.05/4.20 10.05/4.20 ltEsChar :: Main.Char -> Main.Char -> MyBool; 10.05/4.20 ltEsChar x y = fsEsOrdering (compareChar x y) GT; 10.05/4.20 10.05/4.20 not :: MyBool -> MyBool; 10.05/4.20 not MyTrue = MyFalse; 10.05/4.20 not MyFalse = MyTrue; 10.05/4.20 10.05/4.20 primCmpChar :: Main.Char -> Main.Char -> Ordering; 10.05/4.20 primCmpChar (Main.Char x) (Main.Char y) = primCmpInt x y; 10.05/4.20 10.05/4.20 primCmpInt :: MyInt -> MyInt -> Ordering; 10.05/4.20 primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ; 10.05/4.20 primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ; 10.05/4.20 primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ; 10.05/4.20 primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ; 10.05/4.20 primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y; 10.05/4.20 primCmpInt (Main.Pos x) (Main.Neg y) = GT; 10.05/4.20 primCmpInt (Main.Neg x) (Main.Pos y) = LT; 10.05/4.20 primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x; 10.05/4.20 10.05/4.20 primCmpNat :: Main.Nat -> Main.Nat -> Ordering; 10.05/4.20 primCmpNat Main.Zero Main.Zero = EQ; 10.05/4.20 primCmpNat Main.Zero (Main.Succ y) = LT; 10.05/4.20 primCmpNat (Main.Succ x) Main.Zero = GT; 10.05/4.20 primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y; 10.05/4.20 10.05/4.20 } 10.05/4.20 10.05/4.20 ---------------------------------------- 10.05/4.20 10.05/4.20 (3) COR (EQUIVALENT) 10.05/4.20 Cond Reductions: 10.05/4.20 The following Function with conditions 10.05/4.20 "undefined |Falseundefined; 10.05/4.20 " 10.05/4.20 is transformed to 10.05/4.20 "undefined = undefined1; 10.05/4.20 " 10.05/4.20 "undefined0 True = undefined; 10.05/4.20 " 10.05/4.20 "undefined1 = undefined0 False; 10.05/4.20 " 10.05/4.20 10.05/4.20 ---------------------------------------- 10.05/4.20 10.05/4.20 (4) 10.05/4.20 Obligation: 10.05/4.20 mainModule Main 10.05/4.20 module Main where { 10.05/4.20 import qualified Prelude; 10.05/4.20 data Main.Char = Char MyInt ; 10.05/4.20 10.05/4.20 data MyBool = MyTrue | MyFalse ; 10.05/4.20 10.05/4.20 data MyInt = Pos Main.Nat | Neg Main.Nat ; 10.05/4.20 10.05/4.20 data Main.Nat = Succ Main.Nat | Zero ; 10.05/4.20 10.05/4.20 data Ordering = LT | EQ | GT ; 10.05/4.20 10.05/4.20 asAs :: MyBool -> MyBool -> MyBool; 10.05/4.20 asAs MyFalse x = MyFalse; 10.05/4.20 asAs MyTrue x = x; 10.05/4.20 10.05/4.20 compareChar :: Main.Char -> Main.Char -> Ordering; 10.05/4.20 compareChar = primCmpChar; 10.05/4.20 10.05/4.20 esEsOrdering :: Ordering -> Ordering -> MyBool; 10.05/4.20 esEsOrdering LT LT = MyTrue; 10.05/4.20 esEsOrdering LT EQ = MyFalse; 10.05/4.20 esEsOrdering LT GT = MyFalse; 10.05/4.20 esEsOrdering EQ LT = MyFalse; 10.05/4.20 esEsOrdering EQ EQ = MyTrue; 10.05/4.20 esEsOrdering EQ GT = MyFalse; 10.05/4.20 esEsOrdering GT LT = MyFalse; 10.05/4.20 esEsOrdering GT EQ = MyFalse; 10.05/4.20 esEsOrdering GT GT = MyTrue; 10.05/4.20 10.05/4.20 fsEsOrdering :: Ordering -> Ordering -> MyBool; 10.05/4.20 fsEsOrdering x y = not (esEsOrdering x y); 10.05/4.20 10.05/4.20 gtEsChar :: Main.Char -> Main.Char -> MyBool; 10.05/4.20 gtEsChar x y = fsEsOrdering (compareChar x y) LT; 10.05/4.20 10.05/4.20 isDigit :: Main.Char -> MyBool; 10.05/4.20 isDigit c = asAs (gtEsChar c (Main.Char (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.Zero))))))))))))))))))))))))))))))))))))))))))))))))))) (ltEsChar c (Main.Char (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.Zero)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))); 10.05/4.20 10.05/4.20 ltEsChar :: Main.Char -> Main.Char -> MyBool; 10.05/4.20 ltEsChar x y = fsEsOrdering (compareChar x y) GT; 10.05/4.20 10.05/4.20 not :: MyBool -> MyBool; 10.05/4.20 not MyTrue = MyFalse; 10.05/4.20 not MyFalse = MyTrue; 10.05/4.20 10.05/4.20 primCmpChar :: Main.Char -> Main.Char -> Ordering; 10.05/4.20 primCmpChar (Main.Char x) (Main.Char y) = primCmpInt x y; 10.05/4.20 10.05/4.20 primCmpInt :: MyInt -> MyInt -> Ordering; 10.05/4.20 primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ; 10.05/4.20 primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ; 10.05/4.20 primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ; 10.05/4.20 primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ; 10.05/4.20 primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y; 10.05/4.20 primCmpInt (Main.Pos x) (Main.Neg y) = GT; 10.05/4.20 primCmpInt (Main.Neg x) (Main.Pos y) = LT; 10.05/4.20 primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x; 10.05/4.20 10.05/4.20 primCmpNat :: Main.Nat -> Main.Nat -> Ordering; 10.05/4.20 primCmpNat Main.Zero Main.Zero = EQ; 10.05/4.20 primCmpNat Main.Zero (Main.Succ y) = LT; 10.05/4.20 primCmpNat (Main.Succ x) Main.Zero = GT; 10.05/4.20 primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y; 10.05/4.20 10.05/4.20 } 10.05/4.20 10.05/4.20 ---------------------------------------- 10.05/4.20 10.05/4.20 (5) Narrow (SOUND) 10.05/4.20 Haskell To QDPs 10.05/4.20 10.05/4.20 digraph dp_graph { 10.05/4.20 node [outthreshold=100, inthreshold=100];1[label="isDigit",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 10.05/4.20 3[label="isDigit vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3]; 10.05/4.20 4 -> 8[label="",style="dashed", color="red", weight=0]; 10.05/4.20 4[label="asAs (gtEsChar vx3 (Char (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 Zero))))))))))))))))))))))))))))))))))))))))))))))))))) (ltEsChar vx3 (Char (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 Zero))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))",fontsize=16,color="magenta"];4 -> 9[label="",style="dashed", color="magenta", weight=3]; 10.05/4.20 4 -> 10[label="",style="dashed", color="magenta", weight=3]; 10.05/4.20 4 -> 11[label="",style="dashed", color="magenta", weight=3]; 10.05/4.20 9[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 Zero)))))))))))))))))))))))))))))))))))))))))))))))))))))))",fontsize=16,color="green",shape="box"];10[label="vx3",fontsize=16,color="green",shape="box"];11[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 Zero))))))))))))))))))))))))))))))))))))))))))))))",fontsize=16,color="green",shape="box"];8[label="asAs (gtEsChar vx8 (Char (Pos (Succ vx9)))) (ltEsChar vx8 (Char (Pos (Succ vx10))))",fontsize=16,color="black",shape="triangle"];8 -> 15[label="",style="solid", color="black", weight=3]; 10.05/4.20 15[label="asAs (fsEsOrdering (compareChar vx8 (Char (Pos (Succ vx9)))) LT) (ltEsChar vx8 (Char (Pos (Succ vx10))))",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3]; 10.05/4.20 16[label="asAs (not (esEsOrdering (compareChar vx8 (Char (Pos (Succ vx9)))) LT)) (ltEsChar vx8 (Char (Pos (Succ vx10))))",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3]; 10.05/4.20 17[label="asAs (not (esEsOrdering (primCmpChar vx8 (Char (Pos (Succ vx9)))) LT)) (ltEsChar vx8 (Char (Pos (Succ vx10))))",fontsize=16,color="burlywood",shape="box"];175[label="vx8/Char vx80",fontsize=10,color="white",style="solid",shape="box"];17 -> 175[label="",style="solid", color="burlywood", weight=9]; 10.05/4.20 175 -> 18[label="",style="solid", color="burlywood", weight=3]; 10.05/4.20 18[label="asAs (not (esEsOrdering (primCmpChar (Char vx80) (Char (Pos (Succ vx9)))) LT)) (ltEsChar (Char vx80) (Char (Pos (Succ vx10))))",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3]; 10.05/4.20 19[label="asAs (not (esEsOrdering (primCmpInt vx80 (Pos (Succ vx9))) LT)) (ltEsChar (Char vx80) (Char (Pos (Succ vx10))))",fontsize=16,color="burlywood",shape="box"];176[label="vx80/Pos vx800",fontsize=10,color="white",style="solid",shape="box"];19 -> 176[label="",style="solid", color="burlywood", weight=9]; 10.05/4.20 176 -> 20[label="",style="solid", color="burlywood", weight=3]; 10.05/4.20 177[label="vx80/Neg vx800",fontsize=10,color="white",style="solid",shape="box"];19 -> 177[label="",style="solid", color="burlywood", weight=9]; 10.05/4.20 177 -> 21[label="",style="solid", color="burlywood", weight=3]; 10.05/4.20 20[label="asAs (not (esEsOrdering (primCmpInt (Pos vx800) (Pos (Succ vx9))) LT)) (ltEsChar (Char (Pos vx800)) (Char (Pos (Succ vx10))))",fontsize=16,color="burlywood",shape="box"];178[label="vx800/Succ vx8000",fontsize=10,color="white",style="solid",shape="box"];20 -> 178[label="",style="solid", color="burlywood", weight=9]; 10.05/4.20 178 -> 22[label="",style="solid", color="burlywood", weight=3]; 10.05/4.20 179[label="vx800/Zero",fontsize=10,color="white",style="solid",shape="box"];20 -> 179[label="",style="solid", color="burlywood", weight=9]; 10.05/4.20 179 -> 23[label="",style="solid", color="burlywood", weight=3]; 10.05/4.20 21[label="asAs (not (esEsOrdering (primCmpInt (Neg vx800) (Pos (Succ vx9))) LT)) (ltEsChar (Char (Neg vx800)) (Char (Pos (Succ vx10))))",fontsize=16,color="burlywood",shape="box"];180[label="vx800/Succ vx8000",fontsize=10,color="white",style="solid",shape="box"];21 -> 180[label="",style="solid", color="burlywood", weight=9]; 10.05/4.20 180 -> 24[label="",style="solid", color="burlywood", weight=3]; 10.05/4.20 181[label="vx800/Zero",fontsize=10,color="white",style="solid",shape="box"];21 -> 181[label="",style="solid", color="burlywood", weight=9]; 10.05/4.20 181 -> 25[label="",style="solid", color="burlywood", weight=3]; 10.05/4.20 22[label="asAs (not (esEsOrdering (primCmpInt (Pos (Succ vx8000)) (Pos (Succ vx9))) LT)) (ltEsChar (Char (Pos (Succ vx8000))) (Char (Pos (Succ vx10))))",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3]; 10.05/4.20 23[label="asAs (not (esEsOrdering (primCmpInt (Pos Zero) (Pos (Succ vx9))) LT)) (ltEsChar (Char (Pos Zero)) (Char (Pos (Succ vx10))))",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3]; 10.05/4.20 24[label="asAs (not (esEsOrdering (primCmpInt (Neg (Succ vx8000)) (Pos (Succ vx9))) LT)) (ltEsChar (Char (Neg (Succ vx8000))) (Char (Pos (Succ vx10))))",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3]; 10.05/4.20 25[label="asAs (not (esEsOrdering (primCmpInt (Neg Zero) (Pos (Succ vx9))) LT)) (ltEsChar (Char (Neg Zero)) (Char (Pos (Succ vx10))))",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3]; 10.05/4.20 26 -> 105[label="",style="dashed", color="red", weight=0]; 10.05/4.20 26[label="asAs (not (esEsOrdering (primCmpNat (Succ vx8000) (Succ vx9)) LT)) (ltEsChar (Char (Pos (Succ vx8000))) (Char (Pos (Succ vx10))))",fontsize=16,color="magenta"];26 -> 106[label="",style="dashed", color="magenta", weight=3]; 10.05/4.20 26 -> 107[label="",style="dashed", color="magenta", weight=3]; 10.05/4.20 26 -> 108[label="",style="dashed", color="magenta", weight=3]; 10.05/4.20 27 -> 105[label="",style="dashed", color="red", weight=0]; 10.05/4.20 27[label="asAs (not (esEsOrdering (primCmpNat Zero (Succ vx9)) LT)) (ltEsChar (Char (Pos Zero)) (Char (Pos (Succ vx10))))",fontsize=16,color="magenta"];27 -> 109[label="",style="dashed", color="magenta", weight=3]; 10.05/4.20 27 -> 110[label="",style="dashed", color="magenta", weight=3]; 10.05/4.20 27 -> 111[label="",style="dashed", color="magenta", weight=3]; 10.05/4.20 28[label="asAs (not (esEsOrdering LT LT)) (ltEsChar (Char (Neg (Succ vx8000))) (Char (Pos (Succ vx10))))",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3]; 10.05/4.20 29[label="asAs (not (esEsOrdering LT LT)) (ltEsChar (Char (Neg Zero)) (Char (Pos (Succ vx10))))",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3]; 10.05/4.20 106[label="ltEsChar (Char (Pos (Succ vx8000))) (Char (Pos (Succ vx10)))",fontsize=16,color="black",shape="box"];106 -> 122[label="",style="solid", color="black", weight=3]; 10.05/4.20 107[label="Succ vx8000",fontsize=16,color="green",shape="box"];108[label="Succ vx9",fontsize=16,color="green",shape="box"];105[label="asAs (not (esEsOrdering (primCmpNat vx8000000 vx9000) LT)) vx12",fontsize=16,color="burlywood",shape="triangle"];182[label="vx8000000/Succ vx80000000",fontsize=10,color="white",style="solid",shape="box"];105 -> 182[label="",style="solid", color="burlywood", weight=9]; 10.05/4.20 182 -> 123[label="",style="solid", color="burlywood", weight=3]; 10.05/4.20 183[label="vx8000000/Zero",fontsize=10,color="white",style="solid",shape="box"];105 -> 183[label="",style="solid", color="burlywood", weight=9]; 10.05/4.20 183 -> 124[label="",style="solid", color="burlywood", weight=3]; 10.05/4.20 109[label="ltEsChar (Char (Pos Zero)) (Char (Pos (Succ vx10)))",fontsize=16,color="black",shape="box"];109 -> 125[label="",style="solid", color="black", weight=3]; 10.05/4.20 110[label="Zero",fontsize=16,color="green",shape="box"];111[label="Succ vx9",fontsize=16,color="green",shape="box"];32 -> 81[label="",style="dashed", color="red", weight=0]; 10.05/4.20 32[label="asAs (not MyTrue) (ltEsChar (Char (Neg (Succ vx8000))) (Char (Pos (Succ vx10))))",fontsize=16,color="magenta"];32 -> 82[label="",style="dashed", color="magenta", weight=3]; 10.05/4.20 33 -> 81[label="",style="dashed", color="red", weight=0]; 10.05/4.20 33[label="asAs (not MyTrue) (ltEsChar (Char (Neg Zero)) (Char (Pos (Succ vx10))))",fontsize=16,color="magenta"];33 -> 83[label="",style="dashed", color="magenta", weight=3]; 10.05/4.20 122[label="fsEsOrdering (compareChar (Char (Pos (Succ vx8000))) (Char (Pos (Succ vx10)))) GT",fontsize=16,color="black",shape="box"];122 -> 128[label="",style="solid", color="black", weight=3]; 10.05/4.20 123[label="asAs (not (esEsOrdering (primCmpNat (Succ vx80000000) vx9000) LT)) vx12",fontsize=16,color="burlywood",shape="box"];184[label="vx9000/Succ vx90000",fontsize=10,color="white",style="solid",shape="box"];123 -> 184[label="",style="solid", color="burlywood", weight=9]; 10.05/4.20 184 -> 129[label="",style="solid", color="burlywood", weight=3]; 10.05/4.20 185[label="vx9000/Zero",fontsize=10,color="white",style="solid",shape="box"];123 -> 185[label="",style="solid", color="burlywood", weight=9]; 10.05/4.20 185 -> 130[label="",style="solid", color="burlywood", weight=3]; 10.05/4.20 124[label="asAs (not (esEsOrdering (primCmpNat Zero vx9000) LT)) vx12",fontsize=16,color="burlywood",shape="box"];186[label="vx9000/Succ vx90000",fontsize=10,color="white",style="solid",shape="box"];124 -> 186[label="",style="solid", color="burlywood", weight=9]; 10.05/4.20 186 -> 131[label="",style="solid", color="burlywood", weight=3]; 10.05/4.20 187[label="vx9000/Zero",fontsize=10,color="white",style="solid",shape="box"];124 -> 187[label="",style="solid", color="burlywood", weight=9]; 10.05/4.20 187 -> 132[label="",style="solid", color="burlywood", weight=3]; 10.05/4.20 125[label="fsEsOrdering (compareChar (Char (Pos Zero)) (Char (Pos (Succ vx10)))) GT",fontsize=16,color="black",shape="box"];125 -> 133[label="",style="solid", color="black", weight=3]; 10.05/4.20 82[label="ltEsChar (Char (Neg (Succ vx8000))) (Char (Pos (Succ vx10)))",fontsize=16,color="black",shape="box"];82 -> 87[label="",style="solid", color="black", weight=3]; 10.05/4.20 81[label="asAs (not MyTrue) vx11",fontsize=16,color="black",shape="triangle"];81 -> 88[label="",style="solid", color="black", weight=3]; 10.05/4.20 83[label="ltEsChar (Char (Neg Zero)) (Char (Pos (Succ vx10)))",fontsize=16,color="black",shape="box"];83 -> 89[label="",style="solid", color="black", weight=3]; 10.05/4.20 128[label="not (esEsOrdering (compareChar (Char (Pos (Succ vx8000))) (Char (Pos (Succ vx10)))) GT)",fontsize=16,color="black",shape="box"];128 -> 136[label="",style="solid", color="black", weight=3]; 10.05/4.20 129[label="asAs (not (esEsOrdering (primCmpNat (Succ vx80000000) (Succ vx90000)) LT)) vx12",fontsize=16,color="black",shape="box"];129 -> 137[label="",style="solid", color="black", weight=3]; 10.05/4.20 130[label="asAs (not (esEsOrdering (primCmpNat (Succ vx80000000) Zero) LT)) vx12",fontsize=16,color="black",shape="box"];130 -> 138[label="",style="solid", color="black", weight=3]; 10.05/4.20 131[label="asAs (not (esEsOrdering (primCmpNat Zero (Succ vx90000)) LT)) vx12",fontsize=16,color="black",shape="box"];131 -> 139[label="",style="solid", color="black", weight=3]; 10.05/4.20 132[label="asAs (not (esEsOrdering (primCmpNat Zero Zero) LT)) vx12",fontsize=16,color="black",shape="box"];132 -> 140[label="",style="solid", color="black", weight=3]; 10.05/4.20 133[label="not (esEsOrdering (compareChar (Char (Pos Zero)) (Char (Pos (Succ vx10)))) GT)",fontsize=16,color="black",shape="box"];133 -> 141[label="",style="solid", color="black", weight=3]; 10.05/4.20 87[label="fsEsOrdering (compareChar (Char (Neg (Succ vx8000))) (Char (Pos (Succ vx10)))) GT",fontsize=16,color="black",shape="box"];87 -> 99[label="",style="solid", color="black", weight=3]; 10.05/4.20 88[label="asAs MyFalse vx11",fontsize=16,color="black",shape="box"];88 -> 100[label="",style="solid", color="black", weight=3]; 10.05/4.20 89[label="fsEsOrdering (compareChar (Char (Neg Zero)) (Char (Pos (Succ vx10)))) GT",fontsize=16,color="black",shape="box"];89 -> 101[label="",style="solid", color="black", weight=3]; 10.05/4.20 136[label="not (esEsOrdering (primCmpChar (Char (Pos (Succ vx8000))) (Char (Pos (Succ vx10)))) GT)",fontsize=16,color="black",shape="box"];136 -> 144[label="",style="solid", color="black", weight=3]; 10.05/4.20 137 -> 105[label="",style="dashed", color="red", weight=0]; 10.05/4.20 137[label="asAs (not (esEsOrdering (primCmpNat vx80000000 vx90000) LT)) vx12",fontsize=16,color="magenta"];137 -> 145[label="",style="dashed", color="magenta", weight=3]; 10.05/4.20 137 -> 146[label="",style="dashed", color="magenta", weight=3]; 10.05/4.20 138[label="asAs (not (esEsOrdering GT LT)) vx12",fontsize=16,color="black",shape="box"];138 -> 147[label="",style="solid", color="black", weight=3]; 10.05/4.20 139[label="asAs (not (esEsOrdering LT LT)) vx12",fontsize=16,color="black",shape="box"];139 -> 148[label="",style="solid", color="black", weight=3]; 10.05/4.20 140[label="asAs (not (esEsOrdering EQ LT)) vx12",fontsize=16,color="black",shape="box"];140 -> 149[label="",style="solid", color="black", weight=3]; 10.05/4.20 141[label="not (esEsOrdering (primCmpChar (Char (Pos Zero)) (Char (Pos (Succ vx10)))) GT)",fontsize=16,color="black",shape="box"];141 -> 150[label="",style="solid", color="black", weight=3]; 10.05/4.20 99[label="not (esEsOrdering (compareChar (Char (Neg (Succ vx8000))) (Char (Pos (Succ vx10)))) GT)",fontsize=16,color="black",shape="box"];99 -> 126[label="",style="solid", color="black", weight=3]; 10.05/4.20 100[label="MyFalse",fontsize=16,color="green",shape="box"];101[label="not (esEsOrdering (compareChar (Char (Neg Zero)) (Char (Pos (Succ vx10)))) GT)",fontsize=16,color="black",shape="box"];101 -> 127[label="",style="solid", color="black", weight=3]; 10.05/4.20 144[label="not (esEsOrdering (primCmpInt (Pos (Succ vx8000)) (Pos (Succ vx10))) GT)",fontsize=16,color="black",shape="box"];144 -> 152[label="",style="solid", color="black", weight=3]; 10.05/4.20 145[label="vx80000000",fontsize=16,color="green",shape="box"];146[label="vx90000",fontsize=16,color="green",shape="box"];147[label="asAs (not MyFalse) vx12",fontsize=16,color="black",shape="triangle"];147 -> 153[label="",style="solid", color="black", weight=3]; 10.05/4.20 148 -> 81[label="",style="dashed", color="red", weight=0]; 10.05/4.20 148[label="asAs (not MyTrue) vx12",fontsize=16,color="magenta"];148 -> 154[label="",style="dashed", color="magenta", weight=3]; 10.05/4.20 149 -> 147[label="",style="dashed", color="red", weight=0]; 10.05/4.20 149[label="asAs (not MyFalse) vx12",fontsize=16,color="magenta"];150[label="not (esEsOrdering (primCmpInt (Pos Zero) (Pos (Succ vx10))) GT)",fontsize=16,color="black",shape="box"];150 -> 155[label="",style="solid", color="black", weight=3]; 10.05/4.20 126[label="not (esEsOrdering (primCmpChar (Char (Neg (Succ vx8000))) (Char (Pos (Succ vx10)))) GT)",fontsize=16,color="black",shape="box"];126 -> 134[label="",style="solid", color="black", weight=3]; 10.05/4.20 127[label="not (esEsOrdering (primCmpChar (Char (Neg Zero)) (Char (Pos (Succ vx10)))) GT)",fontsize=16,color="black",shape="box"];127 -> 135[label="",style="solid", color="black", weight=3]; 10.05/4.20 152[label="not (esEsOrdering (primCmpNat (Succ vx8000) (Succ vx10)) GT)",fontsize=16,color="black",shape="box"];152 -> 157[label="",style="solid", color="black", weight=3]; 10.05/4.20 153[label="asAs MyTrue vx12",fontsize=16,color="black",shape="box"];153 -> 158[label="",style="solid", color="black", weight=3]; 10.05/4.20 154[label="vx12",fontsize=16,color="green",shape="box"];155[label="not (esEsOrdering (primCmpNat Zero (Succ vx10)) GT)",fontsize=16,color="black",shape="box"];155 -> 159[label="",style="solid", color="black", weight=3]; 10.05/4.20 134[label="not (esEsOrdering (primCmpInt (Neg (Succ vx8000)) (Pos (Succ vx10))) GT)",fontsize=16,color="black",shape="box"];134 -> 142[label="",style="solid", color="black", weight=3]; 10.05/4.20 135[label="not (esEsOrdering (primCmpInt (Neg Zero) (Pos (Succ vx10))) GT)",fontsize=16,color="black",shape="box"];135 -> 143[label="",style="solid", color="black", weight=3]; 10.05/4.20 157[label="not (esEsOrdering (primCmpNat vx8000 vx10) GT)",fontsize=16,color="burlywood",shape="triangle"];188[label="vx8000/Succ vx80000",fontsize=10,color="white",style="solid",shape="box"];157 -> 188[label="",style="solid", color="burlywood", weight=9]; 10.05/4.20 188 -> 160[label="",style="solid", color="burlywood", weight=3]; 10.05/4.20 189[label="vx8000/Zero",fontsize=10,color="white",style="solid",shape="box"];157 -> 189[label="",style="solid", color="burlywood", weight=9]; 10.05/4.20 189 -> 161[label="",style="solid", color="burlywood", weight=3]; 10.05/4.20 158[label="vx12",fontsize=16,color="green",shape="box"];159 -> 142[label="",style="dashed", color="red", weight=0]; 10.05/4.20 159[label="not (esEsOrdering LT GT)",fontsize=16,color="magenta"];142[label="not (esEsOrdering LT GT)",fontsize=16,color="black",shape="triangle"];142 -> 151[label="",style="solid", color="black", weight=3]; 10.05/4.20 143 -> 142[label="",style="dashed", color="red", weight=0]; 10.05/4.20 143[label="not (esEsOrdering LT GT)",fontsize=16,color="magenta"];160[label="not (esEsOrdering (primCmpNat (Succ vx80000) vx10) GT)",fontsize=16,color="burlywood",shape="box"];190[label="vx10/Succ vx100",fontsize=10,color="white",style="solid",shape="box"];160 -> 190[label="",style="solid", color="burlywood", weight=9]; 10.05/4.20 190 -> 162[label="",style="solid", color="burlywood", weight=3]; 10.05/4.20 191[label="vx10/Zero",fontsize=10,color="white",style="solid",shape="box"];160 -> 191[label="",style="solid", color="burlywood", weight=9]; 10.05/4.20 191 -> 163[label="",style="solid", color="burlywood", weight=3]; 10.05/4.20 161[label="not (esEsOrdering (primCmpNat Zero vx10) GT)",fontsize=16,color="burlywood",shape="box"];192[label="vx10/Succ vx100",fontsize=10,color="white",style="solid",shape="box"];161 -> 192[label="",style="solid", color="burlywood", weight=9]; 10.05/4.20 192 -> 164[label="",style="solid", color="burlywood", weight=3]; 10.05/4.20 193[label="vx10/Zero",fontsize=10,color="white",style="solid",shape="box"];161 -> 193[label="",style="solid", color="burlywood", weight=9]; 10.05/4.20 193 -> 165[label="",style="solid", color="burlywood", weight=3]; 10.05/4.20 151[label="not MyFalse",fontsize=16,color="black",shape="triangle"];151 -> 156[label="",style="solid", color="black", weight=3]; 10.05/4.20 162[label="not (esEsOrdering (primCmpNat (Succ vx80000) (Succ vx100)) GT)",fontsize=16,color="black",shape="box"];162 -> 166[label="",style="solid", color="black", weight=3]; 10.05/4.20 163[label="not (esEsOrdering (primCmpNat (Succ vx80000) Zero) GT)",fontsize=16,color="black",shape="box"];163 -> 167[label="",style="solid", color="black", weight=3]; 10.05/4.20 164[label="not (esEsOrdering (primCmpNat Zero (Succ vx100)) GT)",fontsize=16,color="black",shape="box"];164 -> 168[label="",style="solid", color="black", weight=3]; 10.05/4.20 165[label="not (esEsOrdering (primCmpNat Zero Zero) GT)",fontsize=16,color="black",shape="box"];165 -> 169[label="",style="solid", color="black", weight=3]; 10.05/4.20 156[label="MyTrue",fontsize=16,color="green",shape="box"];166 -> 157[label="",style="dashed", color="red", weight=0]; 10.05/4.20 166[label="not (esEsOrdering (primCmpNat vx80000 vx100) GT)",fontsize=16,color="magenta"];166 -> 170[label="",style="dashed", color="magenta", weight=3]; 10.05/4.20 166 -> 171[label="",style="dashed", color="magenta", weight=3]; 10.05/4.20 167[label="not (esEsOrdering GT GT)",fontsize=16,color="black",shape="box"];167 -> 172[label="",style="solid", color="black", weight=3]; 10.05/4.20 168 -> 142[label="",style="dashed", color="red", weight=0]; 10.05/4.20 168[label="not (esEsOrdering LT GT)",fontsize=16,color="magenta"];169[label="not (esEsOrdering EQ GT)",fontsize=16,color="black",shape="box"];169 -> 173[label="",style="solid", color="black", weight=3]; 10.05/4.20 170[label="vx100",fontsize=16,color="green",shape="box"];171[label="vx80000",fontsize=16,color="green",shape="box"];172[label="not MyTrue",fontsize=16,color="black",shape="box"];172 -> 174[label="",style="solid", color="black", weight=3]; 10.05/4.20 173 -> 151[label="",style="dashed", color="red", weight=0]; 10.05/4.20 173[label="not MyFalse",fontsize=16,color="magenta"];174[label="MyFalse",fontsize=16,color="green",shape="box"];} 10.05/4.20 10.05/4.20 ---------------------------------------- 10.05/4.20 10.05/4.20 (6) 10.05/4.20 Complex Obligation (AND) 10.05/4.20 10.05/4.20 ---------------------------------------- 10.05/4.20 10.05/4.20 (7) 10.05/4.20 Obligation: 10.05/4.20 Q DP problem: 10.05/4.20 The TRS P consists of the following rules: 10.05/4.20 10.05/4.20 new_asAs(Main.Succ(vx80000000), Main.Succ(vx90000), vx12) -> new_asAs(vx80000000, vx90000, vx12) 10.05/4.20 10.05/4.20 R is empty. 10.05/4.20 Q is empty. 10.05/4.20 We have to consider all minimal (P,Q,R)-chains. 10.05/4.20 ---------------------------------------- 10.05/4.20 10.05/4.20 (8) QDPSizeChangeProof (EQUIVALENT) 10.05/4.20 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. 10.05/4.20 10.05/4.20 From the DPs we obtained the following set of size-change graphs: 10.05/4.20 *new_asAs(Main.Succ(vx80000000), Main.Succ(vx90000), vx12) -> new_asAs(vx80000000, vx90000, vx12) 10.05/4.20 The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3 10.05/4.20 10.05/4.20 10.05/4.20 ---------------------------------------- 10.05/4.20 10.05/4.20 (9) 10.05/4.20 YES 10.05/4.20 10.05/4.20 ---------------------------------------- 10.05/4.20 10.05/4.20 (10) 10.05/4.20 Obligation: 10.05/4.20 Q DP problem: 10.05/4.20 The TRS P consists of the following rules: 10.05/4.20 10.05/4.20 new_not(Main.Succ(vx80000), Main.Succ(vx100)) -> new_not(vx80000, vx100) 10.05/4.20 10.05/4.20 R is empty. 10.05/4.20 Q is empty. 10.05/4.20 We have to consider all minimal (P,Q,R)-chains. 10.05/4.20 ---------------------------------------- 10.05/4.20 10.05/4.20 (11) QDPSizeChangeProof (EQUIVALENT) 10.05/4.20 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. 10.05/4.20 10.05/4.20 From the DPs we obtained the following set of size-change graphs: 10.05/4.20 *new_not(Main.Succ(vx80000), Main.Succ(vx100)) -> new_not(vx80000, vx100) 10.05/4.20 The graph contains the following edges 1 > 1, 2 > 2 10.05/4.20 10.05/4.20 10.05/4.20 ---------------------------------------- 10.05/4.20 10.05/4.20 (12) 10.05/4.20 YES 10.15/4.24 EOF