7.81/3.45 YES 9.69/3.99 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 9.69/3.99 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 9.69/3.99 9.69/3.99 9.69/3.99 H-Termination with start terms of the given HASKELL could be proven: 9.69/3.99 9.69/3.99 (0) HASKELL 9.69/3.99 (1) LR [EQUIVALENT, 0 ms] 9.69/3.99 (2) HASKELL 9.69/3.99 (3) BR [EQUIVALENT, 0 ms] 9.69/3.99 (4) HASKELL 9.69/3.99 (5) COR [EQUIVALENT, 0 ms] 9.69/3.99 (6) HASKELL 9.69/3.99 (7) Narrow [SOUND, 0 ms] 9.69/3.99 (8) AND 9.69/3.99 (9) QDP 9.69/3.99 (10) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.69/3.99 (11) YES 9.69/3.99 (12) QDP 9.69/3.99 (13) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.69/3.99 (14) YES 9.69/3.99 (15) QDP 9.69/3.99 (16) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.69/3.99 (17) YES 9.69/3.99 9.69/3.99 9.69/3.99 ---------------------------------------- 9.69/3.99 9.69/3.99 (0) 9.69/3.99 Obligation: 9.69/3.99 mainModule Main 9.69/3.99 module Main where { 9.69/3.99 import qualified Prelude; 9.69/3.99 } 9.69/3.99 9.69/3.99 ---------------------------------------- 9.69/3.99 9.69/3.99 (1) LR (EQUIVALENT) 9.69/3.99 Lambda Reductions: 9.69/3.99 The following Lambda expression 9.69/3.99 "\_->q" 9.69/3.99 is transformed to 9.69/3.99 "gtGt0 q _ = q; 9.69/3.99 " 9.69/3.99 9.69/3.99 ---------------------------------------- 9.69/3.99 9.69/3.99 (2) 9.69/3.99 Obligation: 9.69/3.99 mainModule Main 9.69/3.99 module Main where { 9.69/3.99 import qualified Prelude; 9.69/3.99 } 9.69/3.99 9.69/3.99 ---------------------------------------- 9.69/3.99 9.69/3.99 (3) BR (EQUIVALENT) 9.69/3.99 Replaced joker patterns by fresh variables and removed binding patterns. 9.69/3.99 ---------------------------------------- 9.69/3.99 9.69/3.99 (4) 9.69/3.99 Obligation: 9.69/3.99 mainModule Main 9.69/3.99 module Main where { 9.69/3.99 import qualified Prelude; 9.69/3.99 } 9.69/3.99 9.69/3.99 ---------------------------------------- 9.69/3.99 9.69/3.99 (5) COR (EQUIVALENT) 9.69/3.99 Cond Reductions: 9.69/3.99 The following Function with conditions 9.69/3.99 "undefined |Falseundefined; 9.69/3.99 " 9.69/3.99 is transformed to 9.69/3.99 "undefined = undefined1; 9.69/3.99 " 9.69/3.99 "undefined0 True = undefined; 9.69/3.99 " 9.69/3.99 "undefined1 = undefined0 False; 9.69/3.99 " 9.69/3.99 9.69/3.99 ---------------------------------------- 9.69/3.99 9.69/3.99 (6) 9.69/3.99 Obligation: 9.69/3.99 mainModule Main 9.69/3.99 module Main where { 9.69/3.99 import qualified Prelude; 9.69/3.99 } 9.69/3.99 9.69/3.99 ---------------------------------------- 9.69/3.99 9.69/3.99 (7) Narrow (SOUND) 9.69/3.99 Haskell To QDPs 9.69/3.99 9.69/3.99 digraph dp_graph { 9.69/3.99 node [outthreshold=100, inthreshold=100];1[label="sequence_",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 9.69/3.99 3[label="sequence_ vy3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3]; 9.69/3.99 4[label="foldr (>>) (return ()) vy3",fontsize=16,color="burlywood",shape="triangle"];28[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];4 -> 28[label="",style="solid", color="burlywood", weight=9]; 9.69/3.99 28 -> 5[label="",style="solid", color="burlywood", weight=3]; 9.69/3.99 29[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 29[label="",style="solid", color="burlywood", weight=9]; 9.69/3.99 29 -> 6[label="",style="solid", color="burlywood", weight=3]; 9.69/3.99 5[label="foldr (>>) (return ()) (vy30 : vy31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3]; 9.69/3.99 6[label="foldr (>>) (return ()) []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3]; 9.69/3.99 7 -> 9[label="",style="dashed", color="red", weight=0]; 9.69/3.99 7[label="(>>) vy30 foldr (>>) (return ()) vy31",fontsize=16,color="magenta"];7 -> 10[label="",style="dashed", color="magenta", weight=3]; 9.69/3.99 8[label="return ()",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3]; 9.69/3.99 10 -> 4[label="",style="dashed", color="red", weight=0]; 9.69/3.99 10[label="foldr (>>) (return ()) vy31",fontsize=16,color="magenta"];10 -> 12[label="",style="dashed", color="magenta", weight=3]; 9.69/3.99 9[label="(>>) vy30 vy4",fontsize=16,color="black",shape="triangle"];9 -> 13[label="",style="solid", color="black", weight=3]; 9.69/3.99 11[label="() : []",fontsize=16,color="green",shape="box"];12[label="vy31",fontsize=16,color="green",shape="box"];13[label="vy30 >>= gtGt0 vy4",fontsize=16,color="burlywood",shape="triangle"];30[label="vy30/vy300 : vy301",fontsize=10,color="white",style="solid",shape="box"];13 -> 30[label="",style="solid", color="burlywood", weight=9]; 9.69/3.99 30 -> 14[label="",style="solid", color="burlywood", weight=3]; 9.69/3.99 31[label="vy30/[]",fontsize=10,color="white",style="solid",shape="box"];13 -> 31[label="",style="solid", color="burlywood", weight=9]; 9.69/3.99 31 -> 15[label="",style="solid", color="burlywood", weight=3]; 9.69/3.99 14[label="vy300 : vy301 >>= gtGt0 vy4",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3]; 9.69/3.99 15[label="[] >>= gtGt0 vy4",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3]; 9.69/3.99 16 -> 18[label="",style="dashed", color="red", weight=0]; 9.69/3.99 16[label="gtGt0 vy4 vy300 ++ (vy301 >>= gtGt0 vy4)",fontsize=16,color="magenta"];16 -> 19[label="",style="dashed", color="magenta", weight=3]; 9.69/3.99 17[label="[]",fontsize=16,color="green",shape="box"];19 -> 13[label="",style="dashed", color="red", weight=0]; 9.69/3.99 19[label="vy301 >>= gtGt0 vy4",fontsize=16,color="magenta"];19 -> 20[label="",style="dashed", color="magenta", weight=3]; 9.69/3.99 18[label="gtGt0 vy4 vy300 ++ vy5",fontsize=16,color="black",shape="triangle"];18 -> 21[label="",style="solid", color="black", weight=3]; 9.69/3.99 20[label="vy301",fontsize=16,color="green",shape="box"];21[label="vy4 ++ vy5",fontsize=16,color="burlywood",shape="triangle"];32[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];21 -> 32[label="",style="solid", color="burlywood", weight=9]; 9.69/3.99 32 -> 22[label="",style="solid", color="burlywood", weight=3]; 9.69/3.99 33[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];21 -> 33[label="",style="solid", color="burlywood", weight=9]; 9.69/3.99 33 -> 23[label="",style="solid", color="burlywood", weight=3]; 9.69/3.99 22[label="(vy40 : vy41) ++ vy5",fontsize=16,color="black",shape="box"];22 -> 24[label="",style="solid", color="black", weight=3]; 9.69/3.99 23[label="[] ++ vy5",fontsize=16,color="black",shape="box"];23 -> 25[label="",style="solid", color="black", weight=3]; 9.69/3.99 24[label="vy40 : vy41 ++ vy5",fontsize=16,color="green",shape="box"];24 -> 26[label="",style="dashed", color="green", weight=3]; 9.69/3.99 25[label="vy5",fontsize=16,color="green",shape="box"];26 -> 21[label="",style="dashed", color="red", weight=0]; 9.69/3.99 26[label="vy41 ++ vy5",fontsize=16,color="magenta"];26 -> 27[label="",style="dashed", color="magenta", weight=3]; 9.69/3.99 27[label="vy41",fontsize=16,color="green",shape="box"];} 9.69/3.99 9.69/3.99 ---------------------------------------- 9.69/3.99 9.69/3.99 (8) 9.69/3.99 Complex Obligation (AND) 9.69/3.99 9.69/3.99 ---------------------------------------- 9.69/3.99 9.69/3.99 (9) 9.69/3.99 Obligation: 9.69/3.99 Q DP problem: 9.69/3.99 The TRS P consists of the following rules: 9.69/3.99 9.69/3.99 new_gtGtEs(:(vy300, vy301), vy4, h) -> new_gtGtEs(vy301, vy4, h) 9.69/3.99 9.69/3.99 R is empty. 9.69/3.99 Q is empty. 9.69/3.99 We have to consider all minimal (P,Q,R)-chains. 9.69/3.99 ---------------------------------------- 9.69/3.99 9.69/3.99 (10) QDPSizeChangeProof (EQUIVALENT) 9.69/3.99 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. 9.69/3.99 9.69/3.99 From the DPs we obtained the following set of size-change graphs: 9.69/3.99 *new_gtGtEs(:(vy300, vy301), vy4, h) -> new_gtGtEs(vy301, vy4, h) 9.69/3.99 The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 9.69/3.99 9.69/3.99 9.69/3.99 ---------------------------------------- 9.69/3.99 9.69/3.99 (11) 9.69/3.99 YES 9.69/3.99 9.69/3.99 ---------------------------------------- 9.69/3.99 9.69/3.99 (12) 9.69/3.99 Obligation: 9.69/3.99 Q DP problem: 9.69/3.99 The TRS P consists of the following rules: 9.69/3.99 9.69/3.99 new_psPs(:(vy40, vy41), vy5) -> new_psPs(vy41, vy5) 9.69/3.99 9.69/3.99 R is empty. 9.69/3.99 Q is empty. 9.69/3.99 We have to consider all minimal (P,Q,R)-chains. 9.69/3.99 ---------------------------------------- 9.69/3.99 9.69/3.99 (13) QDPSizeChangeProof (EQUIVALENT) 9.69/3.99 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. 9.69/3.99 9.69/3.99 From the DPs we obtained the following set of size-change graphs: 9.69/3.99 *new_psPs(:(vy40, vy41), vy5) -> new_psPs(vy41, vy5) 9.69/3.99 The graph contains the following edges 1 > 1, 2 >= 2 9.69/3.99 9.69/3.99 9.69/3.99 ---------------------------------------- 9.69/3.99 9.69/3.99 (14) 9.69/3.99 YES 9.69/3.99 9.69/3.99 ---------------------------------------- 9.69/3.99 9.69/3.99 (15) 9.69/3.99 Obligation: 9.69/3.99 Q DP problem: 9.69/3.99 The TRS P consists of the following rules: 9.69/3.99 9.69/3.99 new_foldr(:(vy30, vy31), h) -> new_foldr(vy31, h) 9.69/3.99 9.69/3.99 R is empty. 9.69/3.99 Q is empty. 9.69/3.99 We have to consider all minimal (P,Q,R)-chains. 9.69/3.99 ---------------------------------------- 9.69/3.99 9.69/3.99 (16) QDPSizeChangeProof (EQUIVALENT) 9.69/3.99 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. 9.69/3.99 9.69/3.99 From the DPs we obtained the following set of size-change graphs: 9.69/3.99 *new_foldr(:(vy30, vy31), h) -> new_foldr(vy31, h) 9.69/3.99 The graph contains the following edges 1 > 1, 2 >= 2 9.69/3.99 9.69/3.99 9.69/3.99 ---------------------------------------- 9.69/3.99 9.69/3.99 (17) 9.69/3.99 YES 9.97/4.03 EOF