7.74/3.52 YES 9.44/4.02 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 9.44/4.02 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 9.44/4.02 9.44/4.02 9.44/4.02 H-Termination with start terms of the given HASKELL could be proven: 9.44/4.02 9.44/4.02 (0) HASKELL 9.44/4.02 (1) BR [EQUIVALENT, 0 ms] 9.44/4.02 (2) HASKELL 9.44/4.02 (3) COR [EQUIVALENT, 0 ms] 9.44/4.02 (4) HASKELL 9.44/4.02 (5) Narrow [SOUND, 0 ms] 9.44/4.02 (6) QDP 9.44/4.02 (7) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.44/4.02 (8) YES 9.44/4.02 9.44/4.02 9.44/4.02 ---------------------------------------- 9.44/4.02 9.44/4.02 (0) 9.44/4.02 Obligation: 9.44/4.02 mainModule Main 9.44/4.02 module Main where { 9.44/4.02 import qualified Prelude; 9.44/4.02 data List a = Cons a (List a) | Nil ; 9.44/4.02 9.44/4.02 scanl :: (b -> a -> b) -> b -> List a -> List b; 9.44/4.02 scanl f q xs = Cons q (scanl0 f q xs); 9.44/4.02 9.44/4.02 scanl0 f q Nil = Nil; 9.44/4.02 scanl0 f q (Cons x xs) = scanl f (f q x) xs; 9.44/4.02 9.44/4.02 scanl1 :: (a -> a -> a) -> List a -> List a; 9.44/4.02 scanl1 vv Nil = Nil; 9.44/4.02 scanl1 f (Cons x xs) = scanl f x xs; 9.44/4.02 9.44/4.02 } 9.44/4.02 9.44/4.02 ---------------------------------------- 9.44/4.02 9.44/4.02 (1) BR (EQUIVALENT) 9.44/4.02 Replaced joker patterns by fresh variables and removed binding patterns. 9.44/4.02 ---------------------------------------- 9.44/4.02 9.44/4.02 (2) 9.44/4.02 Obligation: 9.44/4.02 mainModule Main 9.44/4.02 module Main where { 9.44/4.02 import qualified Prelude; 9.44/4.02 data List a = Cons a (List a) | Nil ; 9.44/4.02 9.44/4.02 scanl :: (b -> a -> b) -> b -> List a -> List b; 9.44/4.02 scanl f q xs = Cons q (scanl0 f q xs); 9.44/4.02 9.44/4.02 scanl0 f q Nil = Nil; 9.44/4.02 scanl0 f q (Cons x xs) = scanl f (f q x) xs; 9.44/4.02 9.44/4.02 scanl1 :: (a -> a -> a) -> List a -> List a; 9.44/4.02 scanl1 vv Nil = Nil; 9.44/4.02 scanl1 f (Cons x xs) = scanl f x xs; 9.44/4.02 9.44/4.02 } 9.44/4.02 9.44/4.02 ---------------------------------------- 9.44/4.02 9.44/4.02 (3) COR (EQUIVALENT) 9.44/4.02 Cond Reductions: 9.44/4.02 The following Function with conditions 9.44/4.02 "undefined |Falseundefined; 9.44/4.02 " 9.44/4.02 is transformed to 9.44/4.02 "undefined = undefined1; 9.44/4.02 " 9.44/4.02 "undefined0 True = undefined; 9.44/4.02 " 9.44/4.02 "undefined1 = undefined0 False; 9.44/4.02 " 9.44/4.02 9.44/4.02 ---------------------------------------- 9.44/4.02 9.44/4.02 (4) 9.44/4.02 Obligation: 9.44/4.02 mainModule Main 9.44/4.02 module Main where { 9.44/4.02 import qualified Prelude; 9.44/4.02 data List a = Cons a (List a) | Nil ; 9.44/4.02 9.44/4.02 scanl :: (a -> b -> a) -> a -> List b -> List a; 9.44/4.02 scanl f q xs = Cons q (scanl0 f q xs); 9.44/4.02 9.44/4.02 scanl0 f q Nil = Nil; 9.44/4.02 scanl0 f q (Cons x xs) = scanl f (f q x) xs; 9.44/4.02 9.44/4.02 scanl1 :: (a -> a -> a) -> List a -> List a; 9.44/4.02 scanl1 vv Nil = Nil; 9.44/4.02 scanl1 f (Cons x xs) = scanl f x xs; 9.44/4.02 9.44/4.02 } 9.44/4.02 9.44/4.02 ---------------------------------------- 9.44/4.02 9.44/4.02 (5) Narrow (SOUND) 9.44/4.02 Haskell To QDPs 9.44/4.02 9.44/4.02 digraph dp_graph { 9.44/4.02 node [outthreshold=100, inthreshold=100];1[label="scanl1",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 9.44/4.02 3[label="scanl1 vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 9.44/4.02 4[label="scanl1 vy3 vy4",fontsize=16,color="burlywood",shape="triangle"];19[label="vy4/Cons vy40 vy41",fontsize=10,color="white",style="solid",shape="box"];4 -> 19[label="",style="solid", color="burlywood", weight=9]; 9.44/4.02 19 -> 5[label="",style="solid", color="burlywood", weight=3]; 9.44/4.02 20[label="vy4/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 20[label="",style="solid", color="burlywood", weight=9]; 9.44/4.02 20 -> 6[label="",style="solid", color="burlywood", weight=3]; 9.44/4.02 5[label="scanl1 vy3 (Cons vy40 vy41)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3]; 9.44/4.02 6[label="scanl1 vy3 Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3]; 9.44/4.02 7[label="scanl vy3 vy40 vy41",fontsize=16,color="black",shape="triangle"];7 -> 9[label="",style="solid", color="black", weight=3]; 9.44/4.02 8[label="Nil",fontsize=16,color="green",shape="box"];9[label="Cons vy40 (scanl0 vy3 vy40 vy41)",fontsize=16,color="green",shape="box"];9 -> 10[label="",style="dashed", color="green", weight=3]; 9.44/4.02 10[label="scanl0 vy3 vy40 vy41",fontsize=16,color="burlywood",shape="box"];21[label="vy41/Cons vy410 vy411",fontsize=10,color="white",style="solid",shape="box"];10 -> 21[label="",style="solid", color="burlywood", weight=9]; 9.44/4.02 21 -> 11[label="",style="solid", color="burlywood", weight=3]; 9.44/4.02 22[label="vy41/Nil",fontsize=10,color="white",style="solid",shape="box"];10 -> 22[label="",style="solid", color="burlywood", weight=9]; 9.44/4.02 22 -> 12[label="",style="solid", color="burlywood", weight=3]; 9.44/4.02 11[label="scanl0 vy3 vy40 (Cons vy410 vy411)",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3]; 9.44/4.02 12[label="scanl0 vy3 vy40 Nil",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3]; 9.44/4.02 13 -> 7[label="",style="dashed", color="red", weight=0]; 9.44/4.02 13[label="scanl vy3 (vy3 vy40 vy410) vy411",fontsize=16,color="magenta"];13 -> 15[label="",style="dashed", color="magenta", weight=3]; 9.44/4.02 13 -> 16[label="",style="dashed", color="magenta", weight=3]; 9.44/4.02 14[label="Nil",fontsize=16,color="green",shape="box"];15[label="vy3 vy40 vy410",fontsize=16,color="green",shape="box"];15 -> 17[label="",style="dashed", color="green", weight=3]; 9.44/4.02 15 -> 18[label="",style="dashed", color="green", weight=3]; 9.44/4.02 16[label="vy411",fontsize=16,color="green",shape="box"];17[label="vy40",fontsize=16,color="green",shape="box"];18[label="vy410",fontsize=16,color="green",shape="box"];} 9.44/4.02 9.44/4.02 ---------------------------------------- 9.44/4.02 9.44/4.02 (6) 9.44/4.02 Obligation: 9.44/4.02 Q DP problem: 9.44/4.02 The TRS P consists of the following rules: 9.44/4.02 9.44/4.02 new_scanl(vy3, Cons(vy410, vy411), h) -> new_scanl(vy3, vy411, h) 9.44/4.02 9.44/4.02 R is empty. 9.44/4.02 Q is empty. 9.44/4.02 We have to consider all minimal (P,Q,R)-chains. 9.44/4.02 ---------------------------------------- 9.44/4.02 9.44/4.02 (7) QDPSizeChangeProof (EQUIVALENT) 9.44/4.02 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. 9.44/4.02 9.44/4.02 From the DPs we obtained the following set of size-change graphs: 9.44/4.02 *new_scanl(vy3, Cons(vy410, vy411), h) -> new_scanl(vy3, vy411, h) 9.44/4.02 The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 9.44/4.02 9.44/4.02 9.44/4.02 ---------------------------------------- 9.44/4.02 9.44/4.02 (8) 9.44/4.02 YES 9.76/4.15 EOF