9.16/4.30 YES 10.91/4.85 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 10.91/4.85 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 10.91/4.85 10.91/4.85 10.91/4.85 H-Termination with start terms of the given HASKELL could be proven: 10.91/4.85 10.91/4.85 (0) HASKELL 10.91/4.85 (1) LR [EQUIVALENT, 0 ms] 10.91/4.85 (2) HASKELL 10.91/4.85 (3) BR [EQUIVALENT, 0 ms] 10.91/4.85 (4) HASKELL 10.91/4.85 (5) COR [EQUIVALENT, 0 ms] 10.91/4.85 (6) HASKELL 10.91/4.85 (7) Narrow [SOUND, 0 ms] 10.91/4.85 (8) QDP 10.91/4.85 (9) QDPSizeChangeProof [EQUIVALENT, 0 ms] 10.91/4.85 (10) YES 10.91/4.85 10.91/4.85 10.91/4.85 ---------------------------------------- 10.91/4.85 10.91/4.85 (0) 10.91/4.85 Obligation: 10.91/4.85 mainModule Main 10.91/4.85 module Maybe where { 10.91/4.85 import qualified Main; 10.91/4.85 import qualified Monad; 10.91/4.85 import qualified Prelude; 10.91/4.85 } 10.91/4.85 module Main where { 10.91/4.85 import qualified Maybe; 10.91/4.85 import qualified Monad; 10.91/4.85 import qualified Prelude; 10.91/4.85 } 10.91/4.85 module Monad where { 10.91/4.85 import qualified Main; 10.91/4.85 import qualified Maybe; 10.91/4.85 import qualified Prelude; 10.91/4.85 zipWithM :: Monad d => (a -> b -> d c) -> [a] -> [b] -> d [c]; 10.91/4.85 zipWithM f xs ys = sequence (zipWith f xs ys); 10.91/4.85 10.91/4.85 } 10.91/4.85 10.91/4.85 ---------------------------------------- 10.91/4.85 10.91/4.85 (1) LR (EQUIVALENT) 10.91/4.85 Lambda Reductions: 10.91/4.85 The following Lambda expression 10.91/4.86 "\xs->return (x : xs)" 10.91/4.86 is transformed to 10.91/4.86 "sequence0 x xs = return (x : xs); 10.91/4.86 " 10.91/4.86 The following Lambda expression 10.91/4.86 "\x->sequence cs >>= sequence0 x" 10.91/4.86 is transformed to 10.91/4.86 "sequence1 cs x = sequence cs >>= sequence0 x; 10.91/4.86 " 10.91/4.86 10.91/4.86 ---------------------------------------- 10.91/4.86 10.91/4.86 (2) 10.91/4.86 Obligation: 10.91/4.86 mainModule Main 10.91/4.86 module Maybe where { 10.91/4.86 import qualified Main; 10.91/4.86 import qualified Monad; 10.91/4.86 import qualified Prelude; 10.91/4.86 } 10.91/4.86 module Main where { 10.91/4.86 import qualified Maybe; 10.91/4.86 import qualified Monad; 10.91/4.86 import qualified Prelude; 10.91/4.86 } 10.91/4.86 module Monad where { 10.91/4.86 import qualified Main; 10.91/4.86 import qualified Maybe; 10.91/4.86 import qualified Prelude; 10.91/4.86 zipWithM :: Monad c => (a -> b -> c d) -> [a] -> [b] -> c [d]; 10.91/4.86 zipWithM f xs ys = sequence (zipWith f xs ys); 10.91/4.86 10.91/4.86 } 10.91/4.86 10.91/4.86 ---------------------------------------- 10.91/4.86 10.91/4.86 (3) BR (EQUIVALENT) 10.91/4.86 Replaced joker patterns by fresh variables and removed binding patterns. 10.91/4.86 ---------------------------------------- 10.91/4.86 10.91/4.86 (4) 10.91/4.86 Obligation: 10.91/4.86 mainModule Main 10.91/4.86 module Maybe where { 10.91/4.86 import qualified Main; 10.91/4.86 import qualified Monad; 10.91/4.86 import qualified Prelude; 10.91/4.86 } 10.91/4.86 module Main where { 10.91/4.86 import qualified Maybe; 10.91/4.86 import qualified Monad; 10.91/4.86 import qualified Prelude; 10.91/4.86 } 10.91/4.86 module Monad where { 10.91/4.86 import qualified Main; 10.91/4.86 import qualified Maybe; 10.91/4.86 import qualified Prelude; 10.91/4.86 zipWithM :: Monad a => (b -> c -> a d) -> [b] -> [c] -> a [d]; 10.91/4.86 zipWithM f xs ys = sequence (zipWith f xs ys); 10.91/4.86 10.91/4.86 } 10.91/4.86 10.91/4.86 ---------------------------------------- 10.91/4.86 10.91/4.86 (5) COR (EQUIVALENT) 10.91/4.86 Cond Reductions: 10.91/4.86 The following Function with conditions 10.91/4.86 "undefined |Falseundefined; 10.91/4.86 " 10.91/4.86 is transformed to 10.91/4.86 "undefined = undefined1; 10.91/4.86 " 10.91/4.86 "undefined0 True = undefined; 10.91/4.86 " 10.91/4.86 "undefined1 = undefined0 False; 10.91/4.86 " 10.91/4.86 10.91/4.86 ---------------------------------------- 10.91/4.86 10.91/4.86 (6) 10.91/4.86 Obligation: 10.91/4.86 mainModule Main 10.91/4.86 module Maybe where { 10.91/4.86 import qualified Main; 10.91/4.86 import qualified Monad; 10.91/4.86 import qualified Prelude; 10.91/4.86 } 10.91/4.86 module Main where { 10.91/4.86 import qualified Maybe; 10.91/4.86 import qualified Monad; 10.91/4.86 import qualified Prelude; 10.91/4.86 } 10.91/4.86 module Monad where { 10.91/4.86 import qualified Main; 10.91/4.86 import qualified Maybe; 10.91/4.86 import qualified Prelude; 10.91/4.86 zipWithM :: Monad b => (d -> c -> b a) -> [d] -> [c] -> b [a]; 10.91/4.86 zipWithM f xs ys = sequence (zipWith f xs ys); 10.91/4.86 10.91/4.86 } 10.91/4.86 10.91/4.86 ---------------------------------------- 10.91/4.86 10.91/4.86 (7) Narrow (SOUND) 10.91/4.86 Haskell To QDPs 10.91/4.86 10.91/4.86 digraph dp_graph { 10.91/4.86 node [outthreshold=100, inthreshold=100];1[label="Monad.zipWithM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 10.91/4.86 3[label="Monad.zipWithM wv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 10.91/4.86 4[label="Monad.zipWithM wv3 wv4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3]; 10.91/4.86 5[label="Monad.zipWithM wv3 wv4 wv5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3]; 10.91/4.86 6[label="sequence (zipWith wv3 wv4 wv5)",fontsize=16,color="burlywood",shape="triangle"];38[label="wv4/wv40 : wv41",fontsize=10,color="white",style="solid",shape="box"];6 -> 38[label="",style="solid", color="burlywood", weight=9]; 10.91/4.86 38 -> 7[label="",style="solid", color="burlywood", weight=3]; 10.91/4.86 39[label="wv4/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 39[label="",style="solid", color="burlywood", weight=9]; 10.91/4.86 39 -> 8[label="",style="solid", color="burlywood", weight=3]; 10.91/4.86 7[label="sequence (zipWith wv3 (wv40 : wv41) wv5)",fontsize=16,color="burlywood",shape="box"];40[label="wv5/wv50 : wv51",fontsize=10,color="white",style="solid",shape="box"];7 -> 40[label="",style="solid", color="burlywood", weight=9]; 10.91/4.86 40 -> 9[label="",style="solid", color="burlywood", weight=3]; 10.91/4.86 41[label="wv5/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 41[label="",style="solid", color="burlywood", weight=9]; 10.91/4.86 41 -> 10[label="",style="solid", color="burlywood", weight=3]; 10.91/4.86 8[label="sequence (zipWith wv3 [] wv5)",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3]; 10.91/4.86 9[label="sequence (zipWith wv3 (wv40 : wv41) (wv50 : wv51))",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3]; 10.91/4.86 10[label="sequence (zipWith wv3 (wv40 : wv41) [])",fontsize=16,color="black",shape="box"];10 -> 13[label="",style="solid", color="black", weight=3]; 10.91/4.86 11[label="sequence []",fontsize=16,color="black",shape="triangle"];11 -> 14[label="",style="solid", color="black", weight=3]; 10.91/4.86 12[label="sequence (wv3 wv40 wv50 : zipWith wv3 wv41 wv51)",fontsize=16,color="black",shape="box"];12 -> 15[label="",style="solid", color="black", weight=3]; 10.91/4.86 13 -> 11[label="",style="dashed", color="red", weight=0]; 10.91/4.86 13[label="sequence []",fontsize=16,color="magenta"];14[label="return []",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3]; 10.91/4.86 15 -> 17[label="",style="dashed", color="red", weight=0]; 10.91/4.86 15[label="wv3 wv40 wv50 >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="magenta"];15 -> 18[label="",style="dashed", color="magenta", weight=3]; 10.91/4.86 16[label="Just []",fontsize=16,color="green",shape="box"];18[label="wv3 wv40 wv50",fontsize=16,color="green",shape="box"];18 -> 23[label="",style="dashed", color="green", weight=3]; 10.91/4.86 18 -> 24[label="",style="dashed", color="green", weight=3]; 10.91/4.86 17[label="wv6 >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="burlywood",shape="triangle"];42[label="wv6/Nothing",fontsize=10,color="white",style="solid",shape="box"];17 -> 42[label="",style="solid", color="burlywood", weight=9]; 10.91/4.86 42 -> 21[label="",style="solid", color="burlywood", weight=3]; 10.91/4.86 43[label="wv6/Just wv60",fontsize=10,color="white",style="solid",shape="box"];17 -> 43[label="",style="solid", color="burlywood", weight=9]; 10.91/4.86 43 -> 22[label="",style="solid", color="burlywood", weight=3]; 10.91/4.86 23[label="wv40",fontsize=16,color="green",shape="box"];24[label="wv50",fontsize=16,color="green",shape="box"];21[label="Nothing >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3]; 10.91/4.86 22[label="Just wv60 >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3]; 10.91/4.86 25[label="Nothing",fontsize=16,color="green",shape="box"];26[label="sequence1 (zipWith wv3 wv41 wv51) wv60",fontsize=16,color="black",shape="box"];26 -> 27[label="",style="solid", color="black", weight=3]; 10.91/4.86 27 -> 28[label="",style="dashed", color="red", weight=0]; 10.91/4.86 27[label="sequence (zipWith wv3 wv41 wv51) >>= sequence0 wv60",fontsize=16,color="magenta"];27 -> 29[label="",style="dashed", color="magenta", weight=3]; 10.91/4.86 29 -> 6[label="",style="dashed", color="red", weight=0]; 10.91/4.86 29[label="sequence (zipWith wv3 wv41 wv51)",fontsize=16,color="magenta"];29 -> 30[label="",style="dashed", color="magenta", weight=3]; 10.91/4.86 29 -> 31[label="",style="dashed", color="magenta", weight=3]; 10.91/4.86 28[label="wv7 >>= sequence0 wv60",fontsize=16,color="burlywood",shape="triangle"];44[label="wv7/Nothing",fontsize=10,color="white",style="solid",shape="box"];28 -> 44[label="",style="solid", color="burlywood", weight=9]; 10.91/4.86 44 -> 32[label="",style="solid", color="burlywood", weight=3]; 10.91/4.86 45[label="wv7/Just wv70",fontsize=10,color="white",style="solid",shape="box"];28 -> 45[label="",style="solid", color="burlywood", weight=9]; 10.91/4.86 45 -> 33[label="",style="solid", color="burlywood", weight=3]; 10.91/4.86 30[label="wv41",fontsize=16,color="green",shape="box"];31[label="wv51",fontsize=16,color="green",shape="box"];32[label="Nothing >>= sequence0 wv60",fontsize=16,color="black",shape="box"];32 -> 34[label="",style="solid", color="black", weight=3]; 10.91/4.86 33[label="Just wv70 >>= sequence0 wv60",fontsize=16,color="black",shape="box"];33 -> 35[label="",style="solid", color="black", weight=3]; 10.91/4.86 34[label="Nothing",fontsize=16,color="green",shape="box"];35[label="sequence0 wv60 wv70",fontsize=16,color="black",shape="box"];35 -> 36[label="",style="solid", color="black", weight=3]; 10.91/4.86 36[label="return (wv60 : wv70)",fontsize=16,color="black",shape="box"];36 -> 37[label="",style="solid", color="black", weight=3]; 10.91/4.86 37[label="Just (wv60 : wv70)",fontsize=16,color="green",shape="box"];} 10.91/4.86 10.91/4.86 ---------------------------------------- 10.91/4.86 10.91/4.86 (8) 10.91/4.86 Obligation: 10.91/4.86 Q DP problem: 10.91/4.86 The TRS P consists of the following rules: 10.91/4.86 10.91/4.86 new_gtGtEs(wv3, wv41, wv51, h, ba, bb) -> new_sequence(wv3, wv41, wv51, h, ba, bb) 10.91/4.86 new_sequence(wv3, :(wv40, wv41), :(wv50, wv51), h, ba, bb) -> new_gtGtEs(wv3, wv41, wv51, h, ba, bb) 10.91/4.86 10.91/4.86 R is empty. 10.91/4.86 Q is empty. 10.91/4.86 We have to consider all minimal (P,Q,R)-chains. 10.91/4.86 ---------------------------------------- 10.91/4.86 10.91/4.86 (9) QDPSizeChangeProof (EQUIVALENT) 10.91/4.86 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. 10.91/4.86 10.91/4.86 From the DPs we obtained the following set of size-change graphs: 10.91/4.86 *new_sequence(wv3, :(wv40, wv41), :(wv50, wv51), h, ba, bb) -> new_gtGtEs(wv3, wv41, wv51, h, ba, bb) 10.91/4.86 The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6 10.91/4.86 10.91/4.86 10.91/4.86 *new_gtGtEs(wv3, wv41, wv51, h, ba, bb) -> new_sequence(wv3, wv41, wv51, h, ba, bb) 10.91/4.86 The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6 10.91/4.86 10.91/4.86 10.91/4.86 ---------------------------------------- 10.91/4.86 10.91/4.86 (10) 10.91/4.86 YES 11.28/7.01 EOF