7.87/3.55 YES 9.64/4.05 proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs 9.64/4.05 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 9.64/4.05 9.64/4.05 9.64/4.05 H-Termination with start terms of the given HASKELL could be proven: 9.64/4.05 9.64/4.05 (0) HASKELL 9.64/4.05 (1) BR [EQUIVALENT, 0 ms] 9.64/4.05 (2) HASKELL 9.64/4.05 (3) COR [EQUIVALENT, 0 ms] 9.64/4.05 (4) HASKELL 9.64/4.05 (5) Narrow [SOUND, 0 ms] 9.64/4.05 (6) QDP 9.64/4.05 (7) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.64/4.05 (8) YES 9.64/4.05 9.64/4.05 9.64/4.05 ---------------------------------------- 9.64/4.05 9.64/4.05 (0) 9.64/4.05 Obligation: 9.64/4.05 mainModule Main 9.64/4.05 module Main where { 9.64/4.05 import qualified Prelude; 9.64/4.05 data List a = Cons a (List a) | Nil ; 9.64/4.05 9.64/4.05 data MyBool = MyTrue | MyFalse ; 9.64/4.05 9.64/4.05 foldl :: (b -> a -> b) -> b -> List a -> b; 9.64/4.05 foldl f z Nil = z; 9.64/4.05 foldl f z (Cons x xs) = foldl f (f z x) xs; 9.64/4.05 9.64/4.05 foldl1 :: (a -> a -> a) -> List a -> a; 9.64/4.05 foldl1 f (Cons x xs) = foldl f x xs; 9.64/4.05 9.64/4.05 ltEsMyBool :: MyBool -> MyBool -> MyBool; 9.64/4.05 ltEsMyBool MyFalse MyFalse = MyTrue; 9.64/4.05 ltEsMyBool MyFalse MyTrue = MyTrue; 9.64/4.05 ltEsMyBool MyTrue MyFalse = MyFalse; 9.64/4.05 ltEsMyBool MyTrue MyTrue = MyTrue; 9.64/4.05 9.64/4.05 max0 x y MyTrue = x; 9.64/4.05 9.64/4.05 max1 x y MyTrue = y; 9.64/4.05 max1 x y MyFalse = max0 x y otherwise; 9.64/4.05 9.64/4.05 max2 x y = max1 x y (ltEsMyBool x y); 9.64/4.05 9.64/4.05 maxMyBool :: MyBool -> MyBool -> MyBool; 9.64/4.05 maxMyBool x y = max2 x y; 9.64/4.05 9.64/4.05 maximumMyBool :: List MyBool -> MyBool; 9.64/4.05 maximumMyBool = foldl1 maxMyBool; 9.64/4.05 9.64/4.05 otherwise :: MyBool; 9.64/4.05 otherwise = MyTrue; 9.64/4.05 9.64/4.05 } 9.64/4.05 9.64/4.05 ---------------------------------------- 9.64/4.05 9.64/4.05 (1) BR (EQUIVALENT) 9.64/4.05 Replaced joker patterns by fresh variables and removed binding patterns. 9.64/4.05 ---------------------------------------- 9.64/4.05 9.64/4.05 (2) 9.64/4.05 Obligation: 9.64/4.05 mainModule Main 9.64/4.05 module Main where { 9.64/4.05 import qualified Prelude; 9.64/4.05 data List a = Cons a (List a) | Nil ; 9.64/4.05 9.64/4.05 data MyBool = MyTrue | MyFalse ; 9.64/4.05 9.64/4.05 foldl :: (a -> b -> a) -> a -> List b -> a; 9.64/4.05 foldl f z Nil = z; 9.64/4.05 foldl f z (Cons x xs) = foldl f (f z x) xs; 9.64/4.05 9.64/4.05 foldl1 :: (a -> a -> a) -> List a -> a; 9.64/4.05 foldl1 f (Cons x xs) = foldl f x xs; 9.64/4.05 9.64/4.05 ltEsMyBool :: MyBool -> MyBool -> MyBool; 9.64/4.05 ltEsMyBool MyFalse MyFalse = MyTrue; 9.64/4.05 ltEsMyBool MyFalse MyTrue = MyTrue; 9.64/4.05 ltEsMyBool MyTrue MyFalse = MyFalse; 9.64/4.05 ltEsMyBool MyTrue MyTrue = MyTrue; 9.64/4.05 9.64/4.05 max0 x y MyTrue = x; 9.64/4.05 9.64/4.05 max1 x y MyTrue = y; 9.64/4.05 max1 x y MyFalse = max0 x y otherwise; 9.64/4.05 9.64/4.05 max2 x y = max1 x y (ltEsMyBool x y); 9.64/4.05 9.64/4.05 maxMyBool :: MyBool -> MyBool -> MyBool; 9.64/4.05 maxMyBool x y = max2 x y; 9.64/4.05 9.64/4.05 maximumMyBool :: List MyBool -> MyBool; 9.64/4.05 maximumMyBool = foldl1 maxMyBool; 9.64/4.05 9.64/4.05 otherwise :: MyBool; 9.64/4.05 otherwise = MyTrue; 9.64/4.05 9.64/4.05 } 9.64/4.05 9.64/4.05 ---------------------------------------- 9.64/4.05 9.64/4.05 (3) COR (EQUIVALENT) 9.64/4.05 Cond Reductions: 9.64/4.05 The following Function with conditions 9.64/4.05 "undefined |Falseundefined; 9.64/4.05 " 9.64/4.05 is transformed to 9.64/4.05 "undefined = undefined1; 9.64/4.05 " 9.64/4.05 "undefined0 True = undefined; 9.64/4.05 " 9.64/4.05 "undefined1 = undefined0 False; 9.64/4.05 " 9.64/4.05 9.64/4.05 ---------------------------------------- 9.64/4.05 9.64/4.05 (4) 9.64/4.05 Obligation: 9.64/4.05 mainModule Main 9.64/4.05 module Main where { 9.64/4.05 import qualified Prelude; 9.64/4.05 data List a = Cons a (List a) | Nil ; 9.64/4.05 9.64/4.05 data MyBool = MyTrue | MyFalse ; 9.64/4.05 9.64/4.05 foldl :: (b -> a -> b) -> b -> List a -> b; 9.64/4.05 foldl f z Nil = z; 9.64/4.05 foldl f z (Cons x xs) = foldl f (f z x) xs; 9.64/4.05 9.64/4.05 foldl1 :: (a -> a -> a) -> List a -> a; 9.64/4.05 foldl1 f (Cons x xs) = foldl f x xs; 9.64/4.05 9.64/4.05 ltEsMyBool :: MyBool -> MyBool -> MyBool; 9.64/4.05 ltEsMyBool MyFalse MyFalse = MyTrue; 9.64/4.05 ltEsMyBool MyFalse MyTrue = MyTrue; 9.64/4.05 ltEsMyBool MyTrue MyFalse = MyFalse; 9.64/4.05 ltEsMyBool MyTrue MyTrue = MyTrue; 9.64/4.05 9.64/4.05 max0 x y MyTrue = x; 9.64/4.05 9.64/4.05 max1 x y MyTrue = y; 9.64/4.05 max1 x y MyFalse = max0 x y otherwise; 9.64/4.05 9.64/4.05 max2 x y = max1 x y (ltEsMyBool x y); 9.64/4.05 9.64/4.05 maxMyBool :: MyBool -> MyBool -> MyBool; 9.64/4.05 maxMyBool x y = max2 x y; 9.64/4.05 9.64/4.05 maximumMyBool :: List MyBool -> MyBool; 9.64/4.05 maximumMyBool = foldl1 maxMyBool; 9.64/4.05 9.64/4.05 otherwise :: MyBool; 9.64/4.05 otherwise = MyTrue; 9.64/4.05 9.64/4.05 } 9.64/4.05 9.64/4.05 ---------------------------------------- 9.64/4.05 9.64/4.05 (5) Narrow (SOUND) 9.64/4.05 Haskell To QDPs 9.64/4.05 9.64/4.05 digraph dp_graph { 9.64/4.05 node [outthreshold=100, inthreshold=100];1[label="maximumMyBool",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 9.64/4.05 3[label="maximumMyBool vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3]; 9.64/4.05 4[label="foldl1 maxMyBool vx3",fontsize=16,color="burlywood",shape="box"];33[label="vx3/Cons vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 33[label="",style="solid", color="burlywood", weight=9]; 9.64/4.05 33 -> 5[label="",style="solid", color="burlywood", weight=3]; 9.64/4.05 34[label="vx3/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 34[label="",style="solid", color="burlywood", weight=9]; 9.64/4.05 34 -> 6[label="",style="solid", color="burlywood", weight=3]; 9.64/4.05 5[label="foldl1 maxMyBool (Cons vx30 vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3]; 9.64/4.05 6[label="foldl1 maxMyBool Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3]; 9.64/4.05 7[label="foldl maxMyBool vx30 vx31",fontsize=16,color="burlywood",shape="triangle"];35[label="vx31/Cons vx310 vx311",fontsize=10,color="white",style="solid",shape="box"];7 -> 35[label="",style="solid", color="burlywood", weight=9]; 9.64/4.05 35 -> 9[label="",style="solid", color="burlywood", weight=3]; 9.64/4.05 36[label="vx31/Nil",fontsize=10,color="white",style="solid",shape="box"];7 -> 36[label="",style="solid", color="burlywood", weight=9]; 9.64/4.05 36 -> 10[label="",style="solid", color="burlywood", weight=3]; 9.64/4.05 8[label="error []",fontsize=16,color="red",shape="box"];9[label="foldl maxMyBool vx30 (Cons vx310 vx311)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3]; 9.64/4.05 10[label="foldl maxMyBool vx30 Nil",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3]; 9.64/4.05 11 -> 7[label="",style="dashed", color="red", weight=0]; 9.64/4.05 11[label="foldl maxMyBool (maxMyBool vx30 vx310) vx311",fontsize=16,color="magenta"];11 -> 13[label="",style="dashed", color="magenta", weight=3]; 9.64/4.05 11 -> 14[label="",style="dashed", color="magenta", weight=3]; 9.64/4.05 12[label="vx30",fontsize=16,color="green",shape="box"];13[label="vx311",fontsize=16,color="green",shape="box"];14[label="maxMyBool vx30 vx310",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3]; 9.64/4.05 15[label="max2 vx30 vx310",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3]; 9.64/4.05 16[label="max1 vx30 vx310 (ltEsMyBool vx30 vx310)",fontsize=16,color="burlywood",shape="box"];37[label="vx30/MyTrue",fontsize=10,color="white",style="solid",shape="box"];16 -> 37[label="",style="solid", color="burlywood", weight=9]; 9.64/4.05 37 -> 17[label="",style="solid", color="burlywood", weight=3]; 9.64/4.05 38[label="vx30/MyFalse",fontsize=10,color="white",style="solid",shape="box"];16 -> 38[label="",style="solid", color="burlywood", weight=9]; 9.64/4.05 38 -> 18[label="",style="solid", color="burlywood", weight=3]; 9.64/4.05 17[label="max1 MyTrue vx310 (ltEsMyBool MyTrue vx310)",fontsize=16,color="burlywood",shape="box"];39[label="vx310/MyTrue",fontsize=10,color="white",style="solid",shape="box"];17 -> 39[label="",style="solid", color="burlywood", weight=9]; 9.64/4.05 39 -> 19[label="",style="solid", color="burlywood", weight=3]; 9.64/4.05 40[label="vx310/MyFalse",fontsize=10,color="white",style="solid",shape="box"];17 -> 40[label="",style="solid", color="burlywood", weight=9]; 9.64/4.05 40 -> 20[label="",style="solid", color="burlywood", weight=3]; 9.64/4.05 18[label="max1 MyFalse vx310 (ltEsMyBool MyFalse vx310)",fontsize=16,color="burlywood",shape="box"];41[label="vx310/MyTrue",fontsize=10,color="white",style="solid",shape="box"];18 -> 41[label="",style="solid", color="burlywood", weight=9]; 9.64/4.05 41 -> 21[label="",style="solid", color="burlywood", weight=3]; 9.64/4.05 42[label="vx310/MyFalse",fontsize=10,color="white",style="solid",shape="box"];18 -> 42[label="",style="solid", color="burlywood", weight=9]; 9.64/4.05 42 -> 22[label="",style="solid", color="burlywood", weight=3]; 9.64/4.05 19[label="max1 MyTrue MyTrue (ltEsMyBool MyTrue MyTrue)",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3]; 9.64/4.05 20[label="max1 MyTrue MyFalse (ltEsMyBool MyTrue MyFalse)",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3]; 9.64/4.05 21[label="max1 MyFalse MyTrue (ltEsMyBool MyFalse MyTrue)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3]; 9.64/4.05 22[label="max1 MyFalse MyFalse (ltEsMyBool MyFalse MyFalse)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3]; 9.64/4.06 23[label="max1 MyTrue MyTrue MyTrue",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3]; 9.64/4.06 24[label="max1 MyTrue MyFalse MyFalse",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3]; 9.64/4.06 25[label="max1 MyFalse MyTrue MyTrue",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3]; 9.64/4.06 26[label="max1 MyFalse MyFalse MyTrue",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3]; 9.64/4.06 27[label="MyTrue",fontsize=16,color="green",shape="box"];28[label="max0 MyTrue MyFalse otherwise",fontsize=16,color="black",shape="box"];28 -> 31[label="",style="solid", color="black", weight=3]; 9.64/4.06 29[label="MyTrue",fontsize=16,color="green",shape="box"];30[label="MyFalse",fontsize=16,color="green",shape="box"];31[label="max0 MyTrue MyFalse MyTrue",fontsize=16,color="black",shape="box"];31 -> 32[label="",style="solid", color="black", weight=3]; 9.64/4.06 32[label="MyTrue",fontsize=16,color="green",shape="box"];} 9.64/4.06 9.64/4.06 ---------------------------------------- 9.64/4.06 9.64/4.06 (6) 9.64/4.06 Obligation: 9.64/4.06 Q DP problem: 9.64/4.06 The TRS P consists of the following rules: 9.64/4.06 9.64/4.06 new_foldl(vx30, Cons(vx310, vx311)) -> new_foldl(new_max1(vx30, vx310), vx311) 9.64/4.06 9.64/4.06 The TRS R consists of the following rules: 9.64/4.06 9.64/4.06 new_max1(MyFalse, MyFalse) -> MyFalse 9.64/4.06 new_max1(MyTrue, MyTrue) -> MyTrue 9.64/4.06 new_max1(MyTrue, MyFalse) -> MyTrue 9.64/4.06 new_max1(MyFalse, MyTrue) -> MyTrue 9.64/4.06 9.64/4.06 The set Q consists of the following terms: 9.64/4.06 9.64/4.06 new_max1(MyTrue, MyTrue) 9.64/4.06 new_max1(MyFalse, MyFalse) 9.64/4.06 new_max1(MyTrue, MyFalse) 9.64/4.06 new_max1(MyFalse, MyTrue) 9.64/4.06 9.64/4.06 We have to consider all minimal (P,Q,R)-chains. 9.64/4.06 ---------------------------------------- 9.64/4.06 9.64/4.06 (7) QDPSizeChangeProof (EQUIVALENT) 9.64/4.06 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. 9.64/4.06 9.64/4.06 From the DPs we obtained the following set of size-change graphs: 9.64/4.06 *new_foldl(vx30, Cons(vx310, vx311)) -> new_foldl(new_max1(vx30, vx310), vx311) 9.64/4.06 The graph contains the following edges 2 > 2 9.64/4.06 9.64/4.06 9.64/4.06 ---------------------------------------- 9.64/4.06 9.64/4.06 (8) 9.64/4.06 YES 9.85/4.09 EOF