7.95/3.55 MAYBE 9.57/4.07 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 9.57/4.07 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 9.57/4.07 9.57/4.07 9.57/4.07 H-Termination with start terms of the given HASKELL could not be shown: 9.57/4.07 9.57/4.07 (0) HASKELL 9.57/4.07 (1) BR [EQUIVALENT, 0 ms] 9.57/4.07 (2) HASKELL 9.57/4.07 (3) COR [EQUIVALENT, 0 ms] 9.57/4.07 (4) HASKELL 9.57/4.07 (5) LetRed [EQUIVALENT, 0 ms] 9.57/4.07 (6) HASKELL 9.57/4.07 (7) Narrow [SOUND, 0 ms] 9.57/4.07 (8) AND 9.57/4.07 (9) QDP 9.57/4.07 (10) NonTerminationLoopProof [COMPLETE, 0 ms] 9.57/4.07 (11) NO 9.57/4.07 (12) QDP 9.57/4.07 (13) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.57/4.07 (14) YES 9.57/4.07 (15) Narrow [COMPLETE, 0 ms] 9.57/4.07 (16) TRUE 9.57/4.07 9.57/4.07 9.57/4.07 ---------------------------------------- 9.57/4.07 9.57/4.07 (0) 9.57/4.07 Obligation: 9.57/4.07 mainModule Main 9.57/4.07 module Main where { 9.57/4.07 import qualified Prelude; 9.57/4.07 } 9.57/4.07 9.57/4.07 ---------------------------------------- 9.57/4.07 9.57/4.07 (1) BR (EQUIVALENT) 9.57/4.07 Replaced joker patterns by fresh variables and removed binding patterns. 9.57/4.07 ---------------------------------------- 9.57/4.07 9.57/4.07 (2) 9.57/4.07 Obligation: 9.57/4.07 mainModule Main 9.57/4.07 module Main where { 9.57/4.07 import qualified Prelude; 9.57/4.07 } 9.57/4.07 9.57/4.07 ---------------------------------------- 9.57/4.07 9.57/4.07 (3) COR (EQUIVALENT) 9.57/4.07 Cond Reductions: 9.57/4.07 The following Function with conditions 9.57/4.07 "undefined |Falseundefined; 9.57/4.07 " 9.57/4.07 is transformed to 9.57/4.07 "undefined = undefined1; 9.57/4.07 " 9.57/4.07 "undefined0 True = undefined; 9.57/4.07 " 9.57/4.07 "undefined1 = undefined0 False; 9.57/4.07 " 9.57/4.07 9.57/4.07 ---------------------------------------- 9.57/4.07 9.57/4.07 (4) 9.57/4.07 Obligation: 9.57/4.07 mainModule Main 9.57/4.07 module Main where { 9.57/4.07 import qualified Prelude; 9.57/4.07 } 9.57/4.07 9.57/4.07 ---------------------------------------- 9.57/4.07 9.57/4.07 (5) LetRed (EQUIVALENT) 9.57/4.07 Let/Where Reductions: 9.57/4.07 The bindings of the following Let/Where expression 9.57/4.07 "xs' where { 9.57/4.07 xs' = xs ++ xs'; 9.57/4.07 } 9.57/4.07 " 9.57/4.07 are unpacked to the following functions on top level 9.57/4.07 "cycleXs' vx = vx ++ cycleXs' vx; 9.57/4.07 " 9.57/4.07 9.57/4.07 ---------------------------------------- 9.57/4.07 9.57/4.07 (6) 9.57/4.07 Obligation: 9.57/4.07 mainModule Main 9.57/4.07 module Main where { 9.57/4.07 import qualified Prelude; 9.57/4.07 } 9.57/4.07 9.57/4.07 ---------------------------------------- 9.57/4.07 9.57/4.07 (7) Narrow (SOUND) 9.57/4.07 Haskell To QDPs 9.57/4.07 9.57/4.07 digraph dp_graph { 9.57/4.07 node [outthreshold=100, inthreshold=100];1[label="cycle",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 9.57/4.07 3[label="cycle vy3",fontsize=16,color="burlywood",shape="triangle"];20[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];3 -> 20[label="",style="solid", color="burlywood", weight=9]; 9.57/4.07 20 -> 4[label="",style="solid", color="burlywood", weight=3]; 9.57/4.07 21[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];3 -> 21[label="",style="solid", color="burlywood", weight=9]; 9.57/4.07 21 -> 5[label="",style="solid", color="burlywood", weight=3]; 9.57/4.07 4[label="cycle (vy30 : vy31)",fontsize=16,color="black",shape="box"];4 -> 6[label="",style="solid", color="black", weight=3]; 9.57/4.07 5[label="cycle []",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3]; 9.57/4.07 6[label="cycleXs' (vy30 : vy31)",fontsize=16,color="black",shape="triangle"];6 -> 8[label="",style="solid", color="black", weight=3]; 9.57/4.07 7[label="error []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3]; 9.57/4.07 8 -> 10[label="",style="dashed", color="red", weight=0]; 9.57/4.07 8[label="(vy30 : vy31) ++ cycleXs' (vy30 : vy31)",fontsize=16,color="magenta"];8 -> 11[label="",style="dashed", color="magenta", weight=3]; 9.57/4.07 9[label="error []",fontsize=16,color="red",shape="box"];11 -> 6[label="",style="dashed", color="red", weight=0]; 9.57/4.07 11[label="cycleXs' (vy30 : vy31)",fontsize=16,color="magenta"];10[label="(vy30 : vy31) ++ vy4",fontsize=16,color="black",shape="triangle"];10 -> 12[label="",style="solid", color="black", weight=3]; 9.57/4.07 12[label="vy30 : vy31 ++ vy4",fontsize=16,color="green",shape="box"];12 -> 13[label="",style="dashed", color="green", weight=3]; 9.57/4.07 13[label="vy31 ++ vy4",fontsize=16,color="burlywood",shape="triangle"];22[label="vy31/vy310 : vy311",fontsize=10,color="white",style="solid",shape="box"];13 -> 22[label="",style="solid", color="burlywood", weight=9]; 9.57/4.07 22 -> 14[label="",style="solid", color="burlywood", weight=3]; 9.57/4.07 23[label="vy31/[]",fontsize=10,color="white",style="solid",shape="box"];13 -> 23[label="",style="solid", color="burlywood", weight=9]; 9.57/4.07 23 -> 15[label="",style="solid", color="burlywood", weight=3]; 9.57/4.07 14[label="(vy310 : vy311) ++ vy4",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3]; 9.57/4.07 15[label="[] ++ vy4",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3]; 9.57/4.07 16[label="vy310 : vy311 ++ vy4",fontsize=16,color="green",shape="box"];16 -> 18[label="",style="dashed", color="green", weight=3]; 9.57/4.07 17[label="vy4",fontsize=16,color="green",shape="box"];18 -> 13[label="",style="dashed", color="red", weight=0]; 9.57/4.07 18[label="vy311 ++ vy4",fontsize=16,color="magenta"];18 -> 19[label="",style="dashed", color="magenta", weight=3]; 9.57/4.07 19[label="vy311",fontsize=16,color="green",shape="box"];} 9.57/4.07 9.57/4.07 ---------------------------------------- 9.57/4.07 9.57/4.07 (8) 9.57/4.07 Complex Obligation (AND) 9.57/4.07 9.57/4.07 ---------------------------------------- 9.57/4.07 9.57/4.07 (9) 9.57/4.07 Obligation: 9.57/4.07 Q DP problem: 9.57/4.07 The TRS P consists of the following rules: 9.57/4.07 9.57/4.07 new_cycleXs'(vy30, vy31, h) -> new_cycleXs'(vy30, vy31, h) 9.57/4.07 9.57/4.07 R is empty. 9.57/4.07 Q is empty. 9.57/4.07 We have to consider all minimal (P,Q,R)-chains. 9.57/4.07 ---------------------------------------- 9.57/4.07 9.57/4.07 (10) NonTerminationLoopProof (COMPLETE) 9.57/4.07 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 9.57/4.07 Found a loop by semiunifying a rule from P directly. 9.57/4.07 9.57/4.07 s = new_cycleXs'(vy30, vy31, h) evaluates to t =new_cycleXs'(vy30, vy31, h) 9.57/4.07 9.57/4.07 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 9.57/4.07 * Matcher: [ ] 9.57/4.07 * Semiunifier: [ ] 9.57/4.07 9.57/4.07 -------------------------------------------------------------------------------- 9.57/4.07 Rewriting sequence 9.57/4.07 9.57/4.07 The DP semiunifies directly so there is only one rewrite step from new_cycleXs'(vy30, vy31, h) to new_cycleXs'(vy30, vy31, h). 9.57/4.07 9.57/4.07 9.57/4.07 9.57/4.07 9.57/4.07 ---------------------------------------- 9.57/4.07 9.57/4.07 (11) 9.57/4.07 NO 9.57/4.07 9.57/4.07 ---------------------------------------- 9.57/4.07 9.57/4.07 (12) 9.57/4.07 Obligation: 9.57/4.07 Q DP problem: 9.57/4.07 The TRS P consists of the following rules: 9.57/4.07 9.57/4.07 new_psPs(:(vy310, vy311), vy4, h) -> new_psPs(vy311, vy4, h) 9.57/4.07 9.57/4.07 R is empty. 9.57/4.07 Q is empty. 9.57/4.07 We have to consider all minimal (P,Q,R)-chains. 9.57/4.07 ---------------------------------------- 9.57/4.07 9.57/4.07 (13) QDPSizeChangeProof (EQUIVALENT) 9.57/4.07 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. 9.57/4.07 9.57/4.07 From the DPs we obtained the following set of size-change graphs: 9.57/4.07 *new_psPs(:(vy310, vy311), vy4, h) -> new_psPs(vy311, vy4, h) 9.57/4.07 The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 9.57/4.07 9.57/4.07 9.57/4.07 ---------------------------------------- 9.57/4.07 9.57/4.07 (14) 9.57/4.07 YES 9.57/4.07 9.57/4.07 ---------------------------------------- 9.57/4.07 9.57/4.07 (15) Narrow (COMPLETE) 9.57/4.07 Haskell To QDPs 9.57/4.07 9.57/4.07 digraph dp_graph { 9.57/4.07 node [outthreshold=100, inthreshold=100];1[label="cycle",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 9.57/4.07 3[label="cycle vy3",fontsize=16,color="burlywood",shape="triangle"];20[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];3 -> 20[label="",style="solid", color="burlywood", weight=9]; 9.57/4.07 20 -> 4[label="",style="solid", color="burlywood", weight=3]; 9.57/4.07 21[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];3 -> 21[label="",style="solid", color="burlywood", weight=9]; 9.57/4.07 21 -> 5[label="",style="solid", color="burlywood", weight=3]; 9.57/4.07 4[label="cycle (vy30 : vy31)",fontsize=16,color="black",shape="box"];4 -> 6[label="",style="solid", color="black", weight=3]; 9.57/4.07 5[label="cycle []",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3]; 9.57/4.07 6[label="cycleXs' (vy30 : vy31)",fontsize=16,color="black",shape="triangle"];6 -> 8[label="",style="solid", color="black", weight=3]; 9.57/4.07 7[label="error []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3]; 9.57/4.07 8 -> 10[label="",style="dashed", color="red", weight=0]; 9.57/4.07 8[label="(vy30 : vy31) ++ cycleXs' (vy30 : vy31)",fontsize=16,color="magenta"];8 -> 11[label="",style="dashed", color="magenta", weight=3]; 9.57/4.07 9[label="error []",fontsize=16,color="red",shape="box"];11 -> 6[label="",style="dashed", color="red", weight=0]; 9.57/4.07 11[label="cycleXs' (vy30 : vy31)",fontsize=16,color="magenta"];10[label="(vy30 : vy31) ++ vy4",fontsize=16,color="black",shape="triangle"];10 -> 12[label="",style="solid", color="black", weight=3]; 9.57/4.07 12[label="vy30 : vy31 ++ vy4",fontsize=16,color="green",shape="box"];12 -> 13[label="",style="dashed", color="green", weight=3]; 9.57/4.07 13[label="vy31 ++ vy4",fontsize=16,color="burlywood",shape="triangle"];22[label="vy31/vy310 : vy311",fontsize=10,color="white",style="solid",shape="box"];13 -> 22[label="",style="solid", color="burlywood", weight=9]; 9.57/4.07 22 -> 14[label="",style="solid", color="burlywood", weight=3]; 9.57/4.07 23[label="vy31/[]",fontsize=10,color="white",style="solid",shape="box"];13 -> 23[label="",style="solid", color="burlywood", weight=9]; 9.57/4.07 23 -> 15[label="",style="solid", color="burlywood", weight=3]; 9.57/4.07 14[label="(vy310 : vy311) ++ vy4",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3]; 9.57/4.07 15[label="[] ++ vy4",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3]; 9.57/4.07 16[label="vy310 : vy311 ++ vy4",fontsize=16,color="green",shape="box"];16 -> 18[label="",style="dashed", color="green", weight=3]; 9.57/4.07 17[label="vy4",fontsize=16,color="green",shape="box"];18 -> 13[label="",style="dashed", color="red", weight=0]; 9.57/4.07 18[label="vy311 ++ vy4",fontsize=16,color="magenta"];18 -> 19[label="",style="dashed", color="magenta", weight=3]; 9.57/4.07 19[label="vy311",fontsize=16,color="green",shape="box"];} 9.57/4.07 9.57/4.07 ---------------------------------------- 9.57/4.07 9.57/4.07 (16) 9.57/4.07 TRUE 9.97/4.11 EOF