8.93/3.88 YES 11.07/4.44 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 11.07/4.44 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 11.07/4.44 11.07/4.44 11.07/4.44 H-Termination with start terms of the given HASKELL could be proven: 11.07/4.44 11.07/4.44 (0) HASKELL 11.07/4.44 (1) LR [EQUIVALENT, 0 ms] 11.07/4.44 (2) HASKELL 11.07/4.44 (3) BR [EQUIVALENT, 0 ms] 11.07/4.44 (4) HASKELL 11.07/4.44 (5) COR [EQUIVALENT, 0 ms] 11.07/4.44 (6) HASKELL 11.07/4.44 (7) Narrow [SOUND, 0 ms] 11.07/4.44 (8) QDP 11.07/4.44 (9) QDPSizeChangeProof [EQUIVALENT, 0 ms] 11.07/4.44 (10) YES 11.07/4.44 11.07/4.44 11.07/4.44 ---------------------------------------- 11.07/4.44 11.07/4.44 (0) 11.07/4.44 Obligation: 11.07/4.44 mainModule Main 11.07/4.44 module Maybe where { 11.07/4.44 import qualified Main; 11.07/4.44 import qualified Monad; 11.07/4.44 import qualified Prelude; 11.07/4.44 } 11.07/4.44 module Main where { 11.07/4.44 import qualified Maybe; 11.07/4.44 import qualified Monad; 11.07/4.44 import qualified Prelude; 11.07/4.44 } 11.07/4.44 module Monad where { 11.07/4.44 import qualified Main; 11.07/4.44 import qualified Maybe; 11.07/4.44 import qualified Prelude; 11.07/4.44 liftM :: Monad a => (c -> b) -> a c -> a b; 11.07/4.44 liftM f m1 = m1 >>= (\x1 ->return (f x1)); 11.07/4.44 11.07/4.44 } 11.07/4.44 11.07/4.44 ---------------------------------------- 11.07/4.44 11.07/4.44 (1) LR (EQUIVALENT) 11.07/4.44 Lambda Reductions: 11.07/4.44 The following Lambda expression 11.07/4.44 "\x1->return (f x1)" 11.07/4.44 is transformed to 11.07/4.44 "liftM0 f x1 = return (f x1); 11.07/4.44 " 11.07/4.44 11.07/4.44 ---------------------------------------- 11.07/4.44 11.07/4.44 (2) 11.07/4.44 Obligation: 11.07/4.44 mainModule Main 11.07/4.44 module Maybe where { 11.07/4.44 import qualified Main; 11.07/4.44 import qualified Monad; 11.07/4.44 import qualified Prelude; 11.07/4.44 } 11.07/4.44 module Main where { 11.07/4.44 import qualified Maybe; 11.07/4.44 import qualified Monad; 11.07/4.44 import qualified Prelude; 11.07/4.44 } 11.07/4.44 module Monad where { 11.07/4.44 import qualified Main; 11.07/4.44 import qualified Maybe; 11.07/4.44 import qualified Prelude; 11.07/4.44 liftM :: Monad c => (b -> a) -> c b -> c a; 11.07/4.44 liftM f m1 = m1 >>= liftM0 f; 11.07/4.44 11.07/4.44 liftM0 f x1 = return (f x1); 11.07/4.44 11.07/4.44 } 11.07/4.44 11.07/4.44 ---------------------------------------- 11.07/4.44 11.07/4.44 (3) BR (EQUIVALENT) 11.07/4.44 Replaced joker patterns by fresh variables and removed binding patterns. 11.07/4.44 ---------------------------------------- 11.07/4.44 11.07/4.44 (4) 11.07/4.44 Obligation: 11.07/4.44 mainModule Main 11.07/4.44 module Maybe where { 11.07/4.44 import qualified Main; 11.07/4.44 import qualified Monad; 11.07/4.44 import qualified Prelude; 11.07/4.44 } 11.07/4.44 module Main where { 11.07/4.44 import qualified Maybe; 11.07/4.44 import qualified Monad; 11.07/4.44 import qualified Prelude; 11.07/4.44 } 11.07/4.44 module Monad where { 11.07/4.44 import qualified Main; 11.07/4.44 import qualified Maybe; 11.07/4.44 import qualified Prelude; 11.07/4.44 liftM :: Monad c => (a -> b) -> c a -> c b; 11.07/4.44 liftM f m1 = m1 >>= liftM0 f; 11.07/4.44 11.07/4.44 liftM0 f x1 = return (f x1); 11.07/4.44 11.07/4.44 } 11.07/4.44 11.07/4.44 ---------------------------------------- 11.07/4.44 11.07/4.44 (5) COR (EQUIVALENT) 11.07/4.44 Cond Reductions: 11.07/4.44 The following Function with conditions 11.07/4.44 "undefined |Falseundefined; 11.07/4.44 " 11.07/4.44 is transformed to 11.07/4.44 "undefined = undefined1; 11.07/4.44 " 11.07/4.44 "undefined0 True = undefined; 11.07/4.44 " 11.07/4.44 "undefined1 = undefined0 False; 11.07/4.44 " 11.07/4.44 11.07/4.44 ---------------------------------------- 11.07/4.44 11.07/4.44 (6) 11.07/4.44 Obligation: 11.07/4.44 mainModule Main 11.07/4.44 module Maybe where { 11.07/4.44 import qualified Main; 11.07/4.44 import qualified Monad; 11.07/4.44 import qualified Prelude; 11.07/4.44 } 11.07/4.44 module Main where { 11.07/4.44 import qualified Maybe; 11.07/4.44 import qualified Monad; 11.07/4.44 import qualified Prelude; 11.07/4.44 } 11.07/4.44 module Monad where { 11.07/4.44 import qualified Main; 11.07/4.44 import qualified Maybe; 11.07/4.44 import qualified Prelude; 11.07/4.44 liftM :: Monad b => (c -> a) -> b c -> b a; 11.07/4.44 liftM f m1 = m1 >>= liftM0 f; 11.07/4.44 11.07/4.44 liftM0 f x1 = return (f x1); 11.07/4.44 11.07/4.44 } 11.07/4.44 11.07/4.44 ---------------------------------------- 11.07/4.44 11.07/4.44 (7) Narrow (SOUND) 11.07/4.44 Haskell To QDPs 11.07/4.44 11.07/4.44 digraph dp_graph { 11.07/4.44 node [outthreshold=100, inthreshold=100];1[label="Monad.liftM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 11.07/4.44 3[label="Monad.liftM vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 11.07/4.44 4[label="Monad.liftM vy3 vy4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 11.07/4.44 5[label="vy4 >>= Monad.liftM0 vy3",fontsize=16,color="burlywood",shape="triangle"];20[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];5 -> 20[label="",style="solid", color="burlywood", weight=9]; 11.07/4.44 20 -> 6[label="",style="solid", color="burlywood", weight=3]; 11.07/4.44 21[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 21[label="",style="solid", color="burlywood", weight=9]; 11.07/4.44 21 -> 7[label="",style="solid", color="burlywood", weight=3]; 11.07/4.44 6[label="vy40 : vy41 >>= Monad.liftM0 vy3",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3]; 11.07/4.44 7[label="[] >>= Monad.liftM0 vy3",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3]; 11.07/4.44 8 -> 10[label="",style="dashed", color="red", weight=0]; 11.07/4.44 8[label="Monad.liftM0 vy3 vy40 ++ (vy41 >>= Monad.liftM0 vy3)",fontsize=16,color="magenta"];8 -> 11[label="",style="dashed", color="magenta", weight=3]; 11.07/4.44 9[label="[]",fontsize=16,color="green",shape="box"];11 -> 5[label="",style="dashed", color="red", weight=0]; 11.07/4.44 11[label="vy41 >>= Monad.liftM0 vy3",fontsize=16,color="magenta"];11 -> 12[label="",style="dashed", color="magenta", weight=3]; 11.07/4.44 10[label="Monad.liftM0 vy3 vy40 ++ vy5",fontsize=16,color="black",shape="triangle"];10 -> 13[label="",style="solid", color="black", weight=3]; 11.07/4.44 12[label="vy41",fontsize=16,color="green",shape="box"];13[label="return (vy3 vy40) ++ vy5",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3]; 11.07/4.44 14[label="(vy3 vy40 : []) ++ vy5",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3]; 11.07/4.44 15[label="vy3 vy40 : [] ++ vy5",fontsize=16,color="green",shape="box"];15 -> 16[label="",style="dashed", color="green", weight=3]; 11.07/4.44 15 -> 17[label="",style="dashed", color="green", weight=3]; 11.07/4.44 16[label="vy3 vy40",fontsize=16,color="green",shape="box"];16 -> 18[label="",style="dashed", color="green", weight=3]; 11.07/4.44 17[label="[] ++ vy5",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3]; 11.07/4.44 18[label="vy40",fontsize=16,color="green",shape="box"];19[label="vy5",fontsize=16,color="green",shape="box"];} 11.07/4.44 11.07/4.44 ---------------------------------------- 11.07/4.44 11.07/4.44 (8) 11.07/4.44 Obligation: 11.07/4.44 Q DP problem: 11.07/4.44 The TRS P consists of the following rules: 11.07/4.44 11.07/4.44 new_gtGtEs(:(vy40, vy41), vy3, h, ba) -> new_gtGtEs(vy41, vy3, h, ba) 11.07/4.44 11.07/4.44 R is empty. 11.07/4.44 Q is empty. 11.07/4.44 We have to consider all minimal (P,Q,R)-chains. 11.07/4.44 ---------------------------------------- 11.07/4.44 11.07/4.44 (9) QDPSizeChangeProof (EQUIVALENT) 11.07/4.44 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. 11.07/4.44 11.07/4.44 From the DPs we obtained the following set of size-change graphs: 11.07/4.44 *new_gtGtEs(:(vy40, vy41), vy3, h, ba) -> new_gtGtEs(vy41, vy3, h, ba) 11.07/4.44 The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4 11.07/4.44 11.07/4.44 11.07/4.44 ---------------------------------------- 11.07/4.44 11.07/4.44 (10) 11.07/4.44 YES 11.11/4.48 EOF