7.99/3.64 YES 10.19/4.23 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 10.19/4.23 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 10.19/4.23 10.19/4.23 10.19/4.23 H-Termination with start terms of the given HASKELL could be proven: 10.19/4.23 10.19/4.23 (0) HASKELL 10.19/4.23 (1) LR [EQUIVALENT, 0 ms] 10.19/4.23 (2) HASKELL 10.19/4.23 (3) BR [EQUIVALENT, 0 ms] 10.19/4.23 (4) HASKELL 10.19/4.23 (5) COR [EQUIVALENT, 0 ms] 10.19/4.23 (6) HASKELL 10.19/4.23 (7) LetRed [EQUIVALENT, 0 ms] 10.19/4.23 (8) HASKELL 10.19/4.23 (9) Narrow [SOUND, 0 ms] 10.19/4.23 (10) QDP 10.19/4.23 (11) MRRProof [EQUIVALENT, 28 ms] 10.19/4.23 (12) QDP 10.19/4.23 (13) DependencyGraphProof [EQUIVALENT, 0 ms] 10.19/4.23 (14) TRUE 10.19/4.23 10.19/4.23 10.19/4.23 ---------------------------------------- 10.19/4.23 10.19/4.23 (0) 10.19/4.23 Obligation: 10.19/4.23 mainModule Main 10.19/4.23 module Main where { 10.19/4.23 import qualified Prelude; 10.19/4.23 } 10.19/4.23 10.19/4.23 ---------------------------------------- 10.19/4.23 10.19/4.23 (1) LR (EQUIVALENT) 10.19/4.23 Lambda Reductions: 10.19/4.23 The following Lambda expression 10.19/4.23 "\(q : _)->q" 10.19/4.23 is transformed to 10.19/4.23 "q1 (q : _) = q; 10.19/4.23 " 10.19/4.23 The following Lambda expression 10.19/4.23 "\qs->qs" 10.19/4.23 is transformed to 10.19/4.23 "qs0 qs = qs; 10.19/4.23 " 10.19/4.23 10.19/4.23 ---------------------------------------- 10.19/4.23 10.19/4.23 (2) 10.19/4.23 Obligation: 10.19/4.23 mainModule Main 10.19/4.23 module Main where { 10.19/4.23 import qualified Prelude; 10.19/4.23 } 10.19/4.23 10.19/4.23 ---------------------------------------- 10.19/4.23 10.19/4.23 (3) BR (EQUIVALENT) 10.19/4.23 Replaced joker patterns by fresh variables and removed binding patterns. 10.19/4.23 ---------------------------------------- 10.19/4.23 10.19/4.23 (4) 10.19/4.23 Obligation: 10.19/4.23 mainModule Main 10.19/4.23 module Main where { 10.19/4.23 import qualified Prelude; 10.19/4.23 } 10.19/4.23 10.19/4.23 ---------------------------------------- 10.19/4.23 10.19/4.23 (5) COR (EQUIVALENT) 10.19/4.23 Cond Reductions: 10.19/4.23 The following Function with conditions 10.19/4.23 "undefined |Falseundefined; 10.19/4.23 " 10.19/4.23 is transformed to 10.19/4.23 "undefined = undefined1; 10.19/4.23 " 10.19/4.23 "undefined0 True = undefined; 10.19/4.23 " 10.19/4.23 "undefined1 = undefined0 False; 10.19/4.23 " 10.19/4.23 10.19/4.23 ---------------------------------------- 10.19/4.23 10.19/4.23 (6) 10.19/4.23 Obligation: 10.19/4.23 mainModule Main 10.19/4.23 module Main where { 10.19/4.23 import qualified Prelude; 10.19/4.23 } 10.19/4.23 10.19/4.23 ---------------------------------------- 10.19/4.23 10.19/4.23 (7) LetRed (EQUIVALENT) 10.19/4.23 Let/Where Reductions: 10.19/4.23 The bindings of the following Let/Where expression 10.19/4.23 "f x q : qs where { 10.19/4.23 q = q1 vu41; 10.19/4.23 ; 10.19/4.23 q1 (q : vw) = q; 10.19/4.23 ; 10.19/4.23 qs = qs0 vu41; 10.19/4.23 ; 10.19/4.23 qs0 qs = qs; 10.19/4.23 ; 10.19/4.23 vu41 = scanr1 f xs; 10.19/4.23 } 10.19/4.23 " 10.19/4.23 are unpacked to the following functions on top level 10.19/4.23 "scanr1Q vy vz = scanr1Q1 vy vz (scanr1Vu41 vy vz); 10.19/4.23 " 10.19/4.23 "scanr1Qs0 vy vz qs = qs; 10.19/4.23 " 10.19/4.23 "scanr1Q1 vy vz (q : vw) = q; 10.19/4.23 " 10.19/4.23 "scanr1Qs vy vz = scanr1Qs0 vy vz (scanr1Vu41 vy vz); 10.19/4.23 " 10.19/4.23 "scanr1Vu41 vy vz = scanr1 vy vz; 10.19/4.23 " 10.19/4.23 10.19/4.23 ---------------------------------------- 10.19/4.23 10.19/4.23 (8) 10.19/4.23 Obligation: 10.19/4.23 mainModule Main 10.19/4.23 module Main where { 10.19/4.23 import qualified Prelude; 10.19/4.23 } 10.19/4.23 10.19/4.23 ---------------------------------------- 10.19/4.23 10.19/4.23 (9) Narrow (SOUND) 10.19/4.23 Haskell To QDPs 10.19/4.23 10.19/4.23 digraph dp_graph { 10.19/4.23 node [outthreshold=100, inthreshold=100];1[label="scanr1",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 10.19/4.23 3[label="scanr1 wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 10.19/4.23 4[label="scanr1 wu3 wu4",fontsize=16,color="burlywood",shape="triangle"];29[label="wu4/wu40 : wu41",fontsize=10,color="white",style="solid",shape="box"];4 -> 29[label="",style="solid", color="burlywood", weight=9]; 10.19/4.23 29 -> 5[label="",style="solid", color="burlywood", weight=3]; 10.19/4.23 30[label="wu4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 30[label="",style="solid", color="burlywood", weight=9]; 10.19/4.23 30 -> 6[label="",style="solid", color="burlywood", weight=3]; 10.19/4.23 5[label="scanr1 wu3 (wu40 : wu41)",fontsize=16,color="burlywood",shape="box"];31[label="wu41/wu410 : wu411",fontsize=10,color="white",style="solid",shape="box"];5 -> 31[label="",style="solid", color="burlywood", weight=9]; 10.19/4.23 31 -> 7[label="",style="solid", color="burlywood", weight=3]; 10.19/4.23 32[label="wu41/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 32[label="",style="solid", color="burlywood", weight=9]; 10.19/4.23 32 -> 8[label="",style="solid", color="burlywood", weight=3]; 10.19/4.23 6[label="scanr1 wu3 []",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3]; 10.19/4.23 7[label="scanr1 wu3 (wu40 : wu410 : wu411)",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3]; 10.19/4.23 8[label="scanr1 wu3 (wu40 : [])",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3]; 10.19/4.23 9[label="[]",fontsize=16,color="green",shape="box"];10[label="wu3 wu40 (scanr1Q wu3 (wu410 : wu411)) : scanr1Qs wu3 (wu410 : wu411)",fontsize=16,color="green",shape="box"];10 -> 12[label="",style="dashed", color="green", weight=3]; 10.19/4.23 10 -> 13[label="",style="dashed", color="green", weight=3]; 10.19/4.23 11[label="wu40 : []",fontsize=16,color="green",shape="box"];12[label="wu3 wu40 (scanr1Q wu3 (wu410 : wu411))",fontsize=16,color="green",shape="box"];12 -> 14[label="",style="dashed", color="green", weight=3]; 10.19/4.23 12 -> 15[label="",style="dashed", color="green", weight=3]; 10.19/4.23 13[label="scanr1Qs wu3 (wu410 : wu411)",fontsize=16,color="black",shape="box"];13 -> 16[label="",style="solid", color="black", weight=3]; 10.19/4.23 14[label="wu40",fontsize=16,color="green",shape="box"];15[label="scanr1Q wu3 (wu410 : wu411)",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3]; 10.19/4.23 16[label="scanr1Qs0 wu3 (wu410 : wu411) (scanr1Vu41 wu3 (wu410 : wu411))",fontsize=16,color="black",shape="box"];16 -> 18[label="",style="solid", color="black", weight=3]; 10.19/4.23 17 -> 21[label="",style="dashed", color="red", weight=0]; 10.19/4.23 17[label="scanr1Q1 wu3 (wu410 : wu411) (scanr1Vu41 wu3 (wu410 : wu411))",fontsize=16,color="magenta"];17 -> 22[label="",style="dashed", color="magenta", weight=3]; 10.19/4.23 18[label="scanr1Vu41 wu3 (wu410 : wu411)",fontsize=16,color="black",shape="triangle"];18 -> 20[label="",style="solid", color="black", weight=3]; 10.19/4.23 22 -> 18[label="",style="dashed", color="red", weight=0]; 10.19/4.23 22[label="scanr1Vu41 wu3 (wu410 : wu411)",fontsize=16,color="magenta"];21[label="scanr1Q1 wu3 (wu410 : wu411) wu5",fontsize=16,color="burlywood",shape="triangle"];33[label="wu5/wu50 : wu51",fontsize=10,color="white",style="solid",shape="box"];21 -> 33[label="",style="solid", color="burlywood", weight=9]; 10.19/4.23 33 -> 24[label="",style="solid", color="burlywood", weight=3]; 10.19/4.23 34[label="wu5/[]",fontsize=10,color="white",style="solid",shape="box"];21 -> 34[label="",style="solid", color="burlywood", weight=9]; 10.19/4.23 34 -> 25[label="",style="solid", color="burlywood", weight=3]; 10.19/4.23 20 -> 4[label="",style="dashed", color="red", weight=0]; 10.19/4.23 20[label="scanr1 wu3 (wu410 : wu411)",fontsize=16,color="magenta"];20 -> 26[label="",style="dashed", color="magenta", weight=3]; 10.19/4.23 24[label="scanr1Q1 wu3 (wu410 : wu411) (wu50 : wu51)",fontsize=16,color="black",shape="box"];24 -> 27[label="",style="solid", color="black", weight=3]; 10.19/4.23 25[label="scanr1Q1 wu3 (wu410 : wu411) []",fontsize=16,color="black",shape="box"];25 -> 28[label="",style="solid", color="black", weight=3]; 10.19/4.23 26[label="wu410 : wu411",fontsize=16,color="green",shape="box"];27[label="wu50",fontsize=16,color="green",shape="box"];28[label="error []",fontsize=16,color="red",shape="box"];} 10.19/4.23 10.19/4.23 ---------------------------------------- 10.19/4.23 10.19/4.23 (10) 10.19/4.23 Obligation: 10.19/4.23 Q DP problem: 10.19/4.23 The TRS P consists of the following rules: 10.19/4.23 10.19/4.23 new_scanr1(wu3, :(wu40, :(wu410, wu411)), h) -> new_scanr1Vu41(wu3, wu410, wu411, h) 10.19/4.23 new_scanr1Vu41(wu3, wu410, wu411, h) -> new_scanr1(wu3, :(wu410, wu411), h) 10.19/4.23 new_scanr1(wu3, :(wu40, :(wu410, wu411)), h) -> new_scanr1(wu3, :(wu410, wu411), h) 10.19/4.23 10.19/4.23 R is empty. 10.19/4.23 Q is empty. 10.19/4.23 We have to consider all minimal (P,Q,R)-chains. 10.19/4.23 ---------------------------------------- 10.19/4.23 10.19/4.23 (11) MRRProof (EQUIVALENT) 10.19/4.23 By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. 10.19/4.23 10.19/4.23 Strictly oriented dependency pairs: 10.19/4.23 10.19/4.23 new_scanr1(wu3, :(wu40, :(wu410, wu411)), h) -> new_scanr1Vu41(wu3, wu410, wu411, h) 10.19/4.23 new_scanr1(wu3, :(wu40, :(wu410, wu411)), h) -> new_scanr1(wu3, :(wu410, wu411), h) 10.19/4.23 10.19/4.23 10.19/4.23 Used ordering: Polynomial interpretation [POLO]: 10.19/4.23 10.19/4.23 POL(:(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 10.19/4.23 POL(new_scanr1(x_1, x_2, x_3)) = x_1 + x_2 + x_3 10.19/4.23 POL(new_scanr1Vu41(x_1, x_2, x_3, x_4)) = 1 + x_1 + 2*x_2 + 2*x_3 + x_4 10.19/4.23 10.19/4.23 10.19/4.23 ---------------------------------------- 10.19/4.23 10.19/4.23 (12) 10.19/4.23 Obligation: 10.19/4.23 Q DP problem: 10.19/4.23 The TRS P consists of the following rules: 10.19/4.23 10.19/4.23 new_scanr1Vu41(wu3, wu410, wu411, h) -> new_scanr1(wu3, :(wu410, wu411), h) 10.19/4.23 10.19/4.23 R is empty. 10.19/4.23 Q is empty. 10.19/4.23 We have to consider all minimal (P,Q,R)-chains. 10.19/4.23 ---------------------------------------- 10.19/4.23 10.19/4.23 (13) DependencyGraphProof (EQUIVALENT) 10.19/4.23 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 10.19/4.23 ---------------------------------------- 10.19/4.23 10.19/4.23 (14) 10.19/4.23 TRUE 10.19/4.27 EOF