7.90/3.57 YES 9.35/4.02 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 9.35/4.02 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 9.35/4.02 9.35/4.02 9.35/4.02 H-Termination with start terms of the given HASKELL could be proven: 9.35/4.02 9.35/4.02 (0) HASKELL 9.35/4.02 (1) BR [EQUIVALENT, 0 ms] 9.35/4.02 (2) HASKELL 9.35/4.02 (3) COR [EQUIVALENT, 0 ms] 9.35/4.02 (4) HASKELL 9.35/4.02 (5) Narrow [SOUND, 0 ms] 9.35/4.02 (6) QDP 9.35/4.02 (7) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.35/4.02 (8) YES 9.35/4.02 9.35/4.02 9.35/4.02 ---------------------------------------- 9.35/4.02 9.35/4.02 (0) 9.35/4.02 Obligation: 9.35/4.02 mainModule Main 9.35/4.02 module Main where { 9.35/4.02 import qualified Prelude; 9.35/4.02 data List a = Cons a (List a) | Nil ; 9.35/4.02 9.35/4.02 zipWith :: (c -> a -> b) -> List c -> List a -> List b; 9.35/4.02 zipWith z (Cons a as) (Cons b bs) = Cons (z a b) (zipWith z as bs); 9.35/4.02 zipWith vv vw vx = Nil; 9.35/4.02 9.35/4.02 } 9.35/4.02 9.35/4.02 ---------------------------------------- 9.35/4.02 9.35/4.02 (1) BR (EQUIVALENT) 9.35/4.02 Replaced joker patterns by fresh variables and removed binding patterns. 9.35/4.02 ---------------------------------------- 9.35/4.02 9.35/4.02 (2) 9.35/4.02 Obligation: 9.35/4.02 mainModule Main 9.35/4.02 module Main where { 9.35/4.02 import qualified Prelude; 9.35/4.02 data List a = Cons a (List a) | Nil ; 9.35/4.02 9.35/4.02 zipWith :: (b -> c -> a) -> List b -> List c -> List a; 9.35/4.02 zipWith z (Cons a as) (Cons b bs) = Cons (z a b) (zipWith z as bs); 9.35/4.02 zipWith vv vw vx = Nil; 9.35/4.02 9.35/4.02 } 9.35/4.02 9.35/4.02 ---------------------------------------- 9.35/4.02 9.35/4.02 (3) COR (EQUIVALENT) 9.35/4.02 Cond Reductions: 9.35/4.02 The following Function with conditions 9.35/4.02 "undefined |Falseundefined; 9.35/4.02 " 9.35/4.02 is transformed to 9.35/4.02 "undefined = undefined1; 9.35/4.02 " 9.35/4.02 "undefined0 True = undefined; 9.35/4.02 " 9.35/4.02 "undefined1 = undefined0 False; 9.35/4.02 " 9.35/4.02 9.35/4.02 ---------------------------------------- 9.35/4.02 9.35/4.02 (4) 9.35/4.02 Obligation: 9.35/4.02 mainModule Main 9.35/4.02 module Main where { 9.35/4.02 import qualified Prelude; 9.35/4.02 data List a = Cons a (List a) | Nil ; 9.35/4.02 9.35/4.02 zipWith :: (c -> b -> a) -> List c -> List b -> List a; 9.35/4.02 zipWith z (Cons a as) (Cons b bs) = Cons (z a b) (zipWith z as bs); 9.35/4.02 zipWith vv vw vx = Nil; 9.35/4.02 9.35/4.02 } 9.35/4.02 9.35/4.02 ---------------------------------------- 9.35/4.02 9.35/4.02 (5) Narrow (SOUND) 9.35/4.02 Haskell To QDPs 9.35/4.02 9.35/4.02 digraph dp_graph { 9.35/4.02 node [outthreshold=100, inthreshold=100];1[label="zipWith",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 9.35/4.02 3[label="zipWith wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 9.35/4.02 4[label="zipWith wu3 wu4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3]; 9.35/4.02 5[label="zipWith wu3 wu4 wu5",fontsize=16,color="burlywood",shape="triangle"];19[label="wu4/Cons wu40 wu41",fontsize=10,color="white",style="solid",shape="box"];5 -> 19[label="",style="solid", color="burlywood", weight=9]; 9.35/4.02 19 -> 6[label="",style="solid", color="burlywood", weight=3]; 9.35/4.02 20[label="wu4/Nil",fontsize=10,color="white",style="solid",shape="box"];5 -> 20[label="",style="solid", color="burlywood", weight=9]; 9.35/4.02 20 -> 7[label="",style="solid", color="burlywood", weight=3]; 9.35/4.02 6[label="zipWith wu3 (Cons wu40 wu41) wu5",fontsize=16,color="burlywood",shape="box"];21[label="wu5/Cons wu50 wu51",fontsize=10,color="white",style="solid",shape="box"];6 -> 21[label="",style="solid", color="burlywood", weight=9]; 9.35/4.02 21 -> 8[label="",style="solid", color="burlywood", weight=3]; 9.35/4.02 22[label="wu5/Nil",fontsize=10,color="white",style="solid",shape="box"];6 -> 22[label="",style="solid", color="burlywood", weight=9]; 9.35/4.02 22 -> 9[label="",style="solid", color="burlywood", weight=3]; 9.35/4.02 7[label="zipWith wu3 Nil wu5",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3]; 9.35/4.02 8[label="zipWith wu3 (Cons wu40 wu41) (Cons wu50 wu51)",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3]; 9.35/4.02 9[label="zipWith wu3 (Cons wu40 wu41) Nil",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3]; 9.35/4.02 10[label="Nil",fontsize=16,color="green",shape="box"];11[label="Cons (wu3 wu40 wu50) (zipWith wu3 wu41 wu51)",fontsize=16,color="green",shape="box"];11 -> 13[label="",style="dashed", color="green", weight=3]; 9.35/4.02 11 -> 14[label="",style="dashed", color="green", weight=3]; 9.35/4.02 12[label="Nil",fontsize=16,color="green",shape="box"];13[label="wu3 wu40 wu50",fontsize=16,color="green",shape="box"];13 -> 15[label="",style="dashed", color="green", weight=3]; 9.35/4.02 13 -> 16[label="",style="dashed", color="green", weight=3]; 9.35/4.02 14 -> 5[label="",style="dashed", color="red", weight=0]; 9.35/4.02 14[label="zipWith wu3 wu41 wu51",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3]; 9.35/4.02 14 -> 18[label="",style="dashed", color="magenta", weight=3]; 9.35/4.02 15[label="wu40",fontsize=16,color="green",shape="box"];16[label="wu50",fontsize=16,color="green",shape="box"];17[label="wu51",fontsize=16,color="green",shape="box"];18[label="wu41",fontsize=16,color="green",shape="box"];} 9.35/4.02 9.35/4.02 ---------------------------------------- 9.35/4.02 9.35/4.02 (6) 9.35/4.02 Obligation: 9.35/4.02 Q DP problem: 9.35/4.02 The TRS P consists of the following rules: 9.35/4.02 9.35/4.02 new_zipWith(wu3, Cons(wu40, wu41), Cons(wu50, wu51), h, ba, bb) -> new_zipWith(wu3, wu41, wu51, h, ba, bb) 9.35/4.02 9.35/4.02 R is empty. 9.35/4.02 Q is empty. 9.35/4.02 We have to consider all minimal (P,Q,R)-chains. 9.35/4.02 ---------------------------------------- 9.35/4.02 9.35/4.02 (7) QDPSizeChangeProof (EQUIVALENT) 9.35/4.02 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. 9.35/4.02 9.35/4.02 From the DPs we obtained the following set of size-change graphs: 9.35/4.02 *new_zipWith(wu3, Cons(wu40, wu41), Cons(wu50, wu51), h, ba, bb) -> new_zipWith(wu3, wu41, wu51, h, ba, bb) 9.35/4.02 The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6 9.35/4.02 9.35/4.02 9.35/4.02 ---------------------------------------- 9.35/4.02 9.35/4.02 (8) 9.35/4.02 YES 9.40/7.00 EOF