7.72/3.52 YES 9.43/4.04 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 9.43/4.04 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 9.43/4.04 9.43/4.04 9.43/4.04 H-Termination with start terms of the given HASKELL could be proven: 9.43/4.04 9.43/4.04 (0) HASKELL 9.43/4.04 (1) BR [EQUIVALENT, 0 ms] 9.43/4.04 (2) HASKELL 9.43/4.04 (3) COR [EQUIVALENT, 0 ms] 9.43/4.04 (4) HASKELL 9.43/4.04 (5) Narrow [SOUND, 0 ms] 9.43/4.04 (6) QDP 9.43/4.04 (7) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.43/4.04 (8) YES 9.43/4.04 9.43/4.04 9.43/4.04 ---------------------------------------- 9.43/4.04 9.43/4.04 (0) 9.43/4.04 Obligation: 9.43/4.04 mainModule Main 9.43/4.04 module Main where { 9.43/4.04 import qualified Prelude; 9.43/4.04 data List a = Cons a (List a) | Nil ; 9.43/4.04 9.43/4.04 data MyBool = MyTrue | MyFalse ; 9.43/4.04 9.43/4.04 foldl :: (b -> a -> b) -> b -> List a -> b; 9.43/4.04 foldl f z Nil = z; 9.43/4.04 foldl f z (Cons x xs) = foldl f (f z x) xs; 9.43/4.04 9.43/4.04 foldl1 :: (a -> a -> a) -> List a -> a; 9.43/4.04 foldl1 f (Cons x xs) = foldl f x xs; 9.43/4.04 9.43/4.04 ltEsMyBool :: MyBool -> MyBool -> MyBool; 9.43/4.04 ltEsMyBool MyFalse MyFalse = MyTrue; 9.43/4.04 ltEsMyBool MyFalse MyTrue = MyTrue; 9.43/4.04 ltEsMyBool MyTrue MyFalse = MyFalse; 9.43/4.04 ltEsMyBool MyTrue MyTrue = MyTrue; 9.43/4.04 9.43/4.04 min0 x y MyTrue = y; 9.43/4.04 9.43/4.04 min1 x y MyTrue = x; 9.43/4.04 min1 x y MyFalse = min0 x y otherwise; 9.43/4.04 9.43/4.04 min2 x y = min1 x y (ltEsMyBool x y); 9.43/4.04 9.43/4.04 minMyBool :: MyBool -> MyBool -> MyBool; 9.43/4.04 minMyBool x y = min2 x y; 9.43/4.04 9.43/4.04 minimumMyBool :: List MyBool -> MyBool; 9.43/4.04 minimumMyBool = foldl1 minMyBool; 9.43/4.04 9.43/4.04 otherwise :: MyBool; 9.43/4.04 otherwise = MyTrue; 9.43/4.04 9.43/4.04 } 9.43/4.04 9.43/4.04 ---------------------------------------- 9.43/4.04 9.43/4.04 (1) BR (EQUIVALENT) 9.43/4.04 Replaced joker patterns by fresh variables and removed binding patterns. 9.43/4.04 ---------------------------------------- 9.43/4.04 9.43/4.04 (2) 9.43/4.04 Obligation: 9.43/4.04 mainModule Main 9.43/4.04 module Main where { 9.43/4.04 import qualified Prelude; 9.43/4.04 data List a = Cons a (List a) | Nil ; 9.43/4.04 9.43/4.04 data MyBool = MyTrue | MyFalse ; 9.43/4.04 9.43/4.04 foldl :: (a -> b -> a) -> a -> List b -> a; 9.43/4.04 foldl f z Nil = z; 9.43/4.04 foldl f z (Cons x xs) = foldl f (f z x) xs; 9.43/4.04 9.43/4.04 foldl1 :: (a -> a -> a) -> List a -> a; 9.43/4.04 foldl1 f (Cons x xs) = foldl f x xs; 9.43/4.04 9.43/4.04 ltEsMyBool :: MyBool -> MyBool -> MyBool; 9.43/4.04 ltEsMyBool MyFalse MyFalse = MyTrue; 9.43/4.04 ltEsMyBool MyFalse MyTrue = MyTrue; 9.43/4.04 ltEsMyBool MyTrue MyFalse = MyFalse; 9.43/4.04 ltEsMyBool MyTrue MyTrue = MyTrue; 9.43/4.04 9.43/4.04 min0 x y MyTrue = y; 9.43/4.04 9.43/4.04 min1 x y MyTrue = x; 9.43/4.04 min1 x y MyFalse = min0 x y otherwise; 9.43/4.04 9.43/4.04 min2 x y = min1 x y (ltEsMyBool x y); 9.43/4.04 9.43/4.04 minMyBool :: MyBool -> MyBool -> MyBool; 9.43/4.04 minMyBool x y = min2 x y; 9.43/4.04 9.43/4.04 minimumMyBool :: List MyBool -> MyBool; 9.43/4.04 minimumMyBool = foldl1 minMyBool; 9.43/4.04 9.43/4.04 otherwise :: MyBool; 9.43/4.04 otherwise = MyTrue; 9.43/4.04 9.43/4.04 } 9.43/4.04 9.43/4.04 ---------------------------------------- 9.43/4.04 9.43/4.04 (3) COR (EQUIVALENT) 9.43/4.04 Cond Reductions: 9.43/4.04 The following Function with conditions 9.43/4.04 "undefined |Falseundefined; 9.43/4.04 " 9.43/4.04 is transformed to 9.43/4.04 "undefined = undefined1; 9.43/4.04 " 9.43/4.04 "undefined0 True = undefined; 9.43/4.04 " 9.43/4.04 "undefined1 = undefined0 False; 9.43/4.04 " 9.43/4.04 9.43/4.04 ---------------------------------------- 9.43/4.04 9.43/4.04 (4) 9.43/4.04 Obligation: 9.43/4.04 mainModule Main 9.43/4.04 module Main where { 9.43/4.04 import qualified Prelude; 9.43/4.04 data List a = Cons a (List a) | Nil ; 9.43/4.04 9.43/4.04 data MyBool = MyTrue | MyFalse ; 9.43/4.04 9.43/4.04 foldl :: (b -> a -> b) -> b -> List a -> b; 9.43/4.04 foldl f z Nil = z; 9.43/4.04 foldl f z (Cons x xs) = foldl f (f z x) xs; 9.43/4.04 9.43/4.04 foldl1 :: (a -> a -> a) -> List a -> a; 9.43/4.04 foldl1 f (Cons x xs) = foldl f x xs; 9.43/4.04 9.43/4.04 ltEsMyBool :: MyBool -> MyBool -> MyBool; 9.43/4.04 ltEsMyBool MyFalse MyFalse = MyTrue; 9.43/4.04 ltEsMyBool MyFalse MyTrue = MyTrue; 9.43/4.04 ltEsMyBool MyTrue MyFalse = MyFalse; 9.43/4.04 ltEsMyBool MyTrue MyTrue = MyTrue; 9.43/4.04 9.43/4.04 min0 x y MyTrue = y; 9.43/4.04 9.43/4.04 min1 x y MyTrue = x; 9.43/4.04 min1 x y MyFalse = min0 x y otherwise; 9.43/4.04 9.43/4.04 min2 x y = min1 x y (ltEsMyBool x y); 9.43/4.04 9.43/4.04 minMyBool :: MyBool -> MyBool -> MyBool; 9.43/4.04 minMyBool x y = min2 x y; 9.43/4.04 9.43/4.04 minimumMyBool :: List MyBool -> MyBool; 9.43/4.04 minimumMyBool = foldl1 minMyBool; 9.43/4.04 9.43/4.04 otherwise :: MyBool; 9.43/4.04 otherwise = MyTrue; 9.43/4.04 9.43/4.04 } 9.43/4.04 9.43/4.04 ---------------------------------------- 9.43/4.04 9.43/4.04 (5) Narrow (SOUND) 9.43/4.04 Haskell To QDPs 9.43/4.04 9.43/4.04 digraph dp_graph { 9.43/4.04 node [outthreshold=100, inthreshold=100];1[label="minimumMyBool",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 9.43/4.04 3[label="minimumMyBool vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3]; 9.43/4.04 4[label="foldl1 minMyBool 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]; 9.43/4.04 33 -> 5[label="",style="solid", color="burlywood", weight=3]; 9.43/4.04 34[label="vx3/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 34[label="",style="solid", color="burlywood", weight=9]; 9.43/4.04 34 -> 6[label="",style="solid", color="burlywood", weight=3]; 9.43/4.04 5[label="foldl1 minMyBool (Cons vx30 vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3]; 9.43/4.04 6[label="foldl1 minMyBool Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3]; 9.43/4.04 7[label="foldl minMyBool 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]; 9.43/4.04 35 -> 9[label="",style="solid", color="burlywood", weight=3]; 9.43/4.04 36[label="vx31/Nil",fontsize=10,color="white",style="solid",shape="box"];7 -> 36[label="",style="solid", color="burlywood", weight=9]; 9.43/4.04 36 -> 10[label="",style="solid", color="burlywood", weight=3]; 9.43/4.04 8[label="error []",fontsize=16,color="red",shape="box"];9[label="foldl minMyBool vx30 (Cons vx310 vx311)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3]; 9.43/4.04 10[label="foldl minMyBool vx30 Nil",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3]; 9.43/4.04 11 -> 7[label="",style="dashed", color="red", weight=0]; 9.43/4.04 11[label="foldl minMyBool (minMyBool vx30 vx310) vx311",fontsize=16,color="magenta"];11 -> 13[label="",style="dashed", color="magenta", weight=3]; 9.43/4.04 11 -> 14[label="",style="dashed", color="magenta", weight=3]; 9.43/4.04 12[label="vx30",fontsize=16,color="green",shape="box"];13[label="vx311",fontsize=16,color="green",shape="box"];14[label="minMyBool vx30 vx310",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3]; 9.43/4.04 15[label="min2 vx30 vx310",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3]; 9.43/4.04 16[label="min1 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]; 9.43/4.04 37 -> 17[label="",style="solid", color="burlywood", weight=3]; 9.43/4.04 38[label="vx30/MyFalse",fontsize=10,color="white",style="solid",shape="box"];16 -> 38[label="",style="solid", color="burlywood", weight=9]; 9.43/4.04 38 -> 18[label="",style="solid", color="burlywood", weight=3]; 9.43/4.04 17[label="min1 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]; 9.43/4.04 39 -> 19[label="",style="solid", color="burlywood", weight=3]; 9.43/4.04 40[label="vx310/MyFalse",fontsize=10,color="white",style="solid",shape="box"];17 -> 40[label="",style="solid", color="burlywood", weight=9]; 9.43/4.04 40 -> 20[label="",style="solid", color="burlywood", weight=3]; 9.43/4.04 18[label="min1 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]; 9.43/4.04 41 -> 21[label="",style="solid", color="burlywood", weight=3]; 9.43/4.04 42[label="vx310/MyFalse",fontsize=10,color="white",style="solid",shape="box"];18 -> 42[label="",style="solid", color="burlywood", weight=9]; 9.43/4.04 42 -> 22[label="",style="solid", color="burlywood", weight=3]; 9.43/4.04 19[label="min1 MyTrue MyTrue (ltEsMyBool MyTrue MyTrue)",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3]; 9.43/4.04 20[label="min1 MyTrue MyFalse (ltEsMyBool MyTrue MyFalse)",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3]; 9.43/4.04 21[label="min1 MyFalse MyTrue (ltEsMyBool MyFalse MyTrue)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3]; 9.43/4.04 22[label="min1 MyFalse MyFalse (ltEsMyBool MyFalse MyFalse)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3]; 9.43/4.04 23[label="min1 MyTrue MyTrue MyTrue",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3]; 9.43/4.04 24[label="min1 MyTrue MyFalse MyFalse",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3]; 9.43/4.04 25[label="min1 MyFalse MyTrue MyTrue",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3]; 9.43/4.04 26[label="min1 MyFalse MyFalse MyTrue",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3]; 9.43/4.04 27[label="MyTrue",fontsize=16,color="green",shape="box"];28[label="min0 MyTrue MyFalse otherwise",fontsize=16,color="black",shape="box"];28 -> 31[label="",style="solid", color="black", weight=3]; 9.43/4.04 29[label="MyFalse",fontsize=16,color="green",shape="box"];30[label="MyFalse",fontsize=16,color="green",shape="box"];31[label="min0 MyTrue MyFalse MyTrue",fontsize=16,color="black",shape="box"];31 -> 32[label="",style="solid", color="black", weight=3]; 9.43/4.04 32[label="MyFalse",fontsize=16,color="green",shape="box"];} 9.43/4.04 9.43/4.04 ---------------------------------------- 9.43/4.04 9.43/4.04 (6) 9.43/4.04 Obligation: 9.43/4.04 Q DP problem: 9.43/4.04 The TRS P consists of the following rules: 9.43/4.04 9.43/4.04 new_foldl(vx30, Cons(vx310, vx311)) -> new_foldl(new_min1(vx30, vx310), vx311) 9.43/4.04 9.43/4.04 The TRS R consists of the following rules: 9.43/4.04 9.43/4.04 new_min1(MyTrue, MyFalse) -> MyFalse 9.43/4.04 new_min1(MyFalse, MyTrue) -> MyFalse 9.43/4.04 new_min1(MyFalse, MyFalse) -> MyFalse 9.43/4.04 new_min1(MyTrue, MyTrue) -> MyTrue 9.43/4.04 9.43/4.04 The set Q consists of the following terms: 9.43/4.04 9.43/4.04 new_min1(MyTrue, MyTrue) 9.43/4.04 new_min1(MyFalse, MyFalse) 9.43/4.04 new_min1(MyTrue, MyFalse) 9.43/4.04 new_min1(MyFalse, MyTrue) 9.43/4.04 9.43/4.04 We have to consider all minimal (P,Q,R)-chains. 9.43/4.04 ---------------------------------------- 9.43/4.04 9.43/4.04 (7) QDPSizeChangeProof (EQUIVALENT) 9.43/4.04 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. 9.43/4.04 9.43/4.04 From the DPs we obtained the following set of size-change graphs: 9.43/4.04 *new_foldl(vx30, Cons(vx310, vx311)) -> new_foldl(new_min1(vx30, vx310), vx311) 9.43/4.04 The graph contains the following edges 2 > 2 9.43/4.04 9.43/4.04 9.43/4.04 ---------------------------------------- 9.43/4.04 9.43/4.04 (8) 9.43/4.04 YES 9.71/4.13 EOF