7.97/3.63 YES 9.98/4.15 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 9.98/4.15 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 9.98/4.15 9.98/4.15 9.98/4.15 H-Termination with start terms of the given HASKELL could be proven: 9.98/4.15 9.98/4.15 (0) HASKELL 9.98/4.15 (1) LR [EQUIVALENT, 0 ms] 9.98/4.15 (2) HASKELL 9.98/4.15 (3) BR [EQUIVALENT, 0 ms] 9.98/4.15 (4) HASKELL 9.98/4.15 (5) COR [EQUIVALENT, 0 ms] 9.98/4.15 (6) HASKELL 9.98/4.15 (7) Narrow [SOUND, 0 ms] 9.98/4.15 (8) QDP 9.98/4.15 (9) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.98/4.15 (10) YES 9.98/4.15 9.98/4.15 9.98/4.15 ---------------------------------------- 9.98/4.15 9.98/4.15 (0) 9.98/4.15 Obligation: 9.98/4.15 mainModule Main 9.98/4.15 module Main where { 9.98/4.15 import qualified Prelude; 9.98/4.15 } 9.98/4.15 9.98/4.15 ---------------------------------------- 9.98/4.15 9.98/4.15 (1) LR (EQUIVALENT) 9.98/4.15 Lambda Reductions: 9.98/4.15 The following Lambda expression 9.98/4.15 "\xs->return (x : xs)" 9.98/4.15 is transformed to 9.98/4.15 "sequence0 x xs = return (x : xs); 9.98/4.15 " 9.98/4.15 The following Lambda expression 9.98/4.15 "\x->sequence cs >>= sequence0 x" 9.98/4.15 is transformed to 9.98/4.15 "sequence1 cs x = sequence cs >>= sequence0 x; 9.98/4.15 " 9.98/4.15 9.98/4.15 ---------------------------------------- 9.98/4.15 9.98/4.15 (2) 9.98/4.15 Obligation: 9.98/4.15 mainModule Main 9.98/4.15 module Main where { 9.98/4.15 import qualified Prelude; 9.98/4.15 } 9.98/4.15 9.98/4.15 ---------------------------------------- 9.98/4.15 9.98/4.15 (3) BR (EQUIVALENT) 9.98/4.15 Replaced joker patterns by fresh variables and removed binding patterns. 9.98/4.15 ---------------------------------------- 9.98/4.15 9.98/4.15 (4) 9.98/4.15 Obligation: 9.98/4.15 mainModule Main 9.98/4.15 module Main where { 9.98/4.15 import qualified Prelude; 9.98/4.15 } 9.98/4.15 9.98/4.15 ---------------------------------------- 9.98/4.15 9.98/4.15 (5) COR (EQUIVALENT) 9.98/4.15 Cond Reductions: 9.98/4.15 The following Function with conditions 9.98/4.15 "undefined |Falseundefined; 9.98/4.15 " 9.98/4.15 is transformed to 9.98/4.15 "undefined = undefined1; 9.98/4.15 " 9.98/4.15 "undefined0 True = undefined; 9.98/4.15 " 9.98/4.15 "undefined1 = undefined0 False; 9.98/4.15 " 9.98/4.15 9.98/4.15 ---------------------------------------- 9.98/4.15 9.98/4.15 (6) 9.98/4.15 Obligation: 9.98/4.15 mainModule Main 9.98/4.15 module Main where { 9.98/4.15 import qualified Prelude; 9.98/4.15 } 9.98/4.15 9.98/4.15 ---------------------------------------- 9.98/4.15 9.98/4.15 (7) Narrow (SOUND) 9.98/4.15 Haskell To QDPs 9.98/4.15 9.98/4.15 digraph dp_graph { 9.98/4.15 node [outthreshold=100, inthreshold=100];1[label="mapM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 9.98/4.15 3[label="mapM vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 9.98/4.15 4[label="mapM vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 9.98/4.15 5[label="sequence . map vx3",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 9.98/4.15 6[label="sequence (map vx3 vx4)",fontsize=16,color="burlywood",shape="triangle"];32[label="vx4/vx40 : vx41",fontsize=10,color="white",style="solid",shape="box"];6 -> 32[label="",style="solid", color="burlywood", weight=9]; 9.98/4.15 32 -> 7[label="",style="solid", color="burlywood", weight=3]; 9.98/4.15 33[label="vx4/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 33[label="",style="solid", color="burlywood", weight=9]; 9.98/4.15 33 -> 8[label="",style="solid", color="burlywood", weight=3]; 9.98/4.15 7[label="sequence (map vx3 (vx40 : vx41))",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3]; 9.98/4.15 8[label="sequence (map vx3 [])",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3]; 9.98/4.15 9[label="sequence (vx3 vx40 : map vx3 vx41)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3]; 9.98/4.15 10[label="sequence []",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3]; 9.98/4.15 11 -> 13[label="",style="dashed", color="red", weight=0]; 9.98/4.15 11[label="vx3 vx40 >>= sequence1 (map vx3 vx41)",fontsize=16,color="magenta"];11 -> 14[label="",style="dashed", color="magenta", weight=3]; 9.98/4.15 12[label="return []",fontsize=16,color="black",shape="box"];12 -> 15[label="",style="solid", color="black", weight=3]; 9.98/4.15 14[label="vx3 vx40",fontsize=16,color="green",shape="box"];14 -> 19[label="",style="dashed", color="green", weight=3]; 9.98/4.15 13[label="vx5 >>= sequence1 (map vx3 vx41)",fontsize=16,color="burlywood",shape="triangle"];34[label="vx5/Nothing",fontsize=10,color="white",style="solid",shape="box"];13 -> 34[label="",style="solid", color="burlywood", weight=9]; 9.98/4.15 34 -> 17[label="",style="solid", color="burlywood", weight=3]; 9.98/4.15 35[label="vx5/Just vx50",fontsize=10,color="white",style="solid",shape="box"];13 -> 35[label="",style="solid", color="burlywood", weight=9]; 9.98/4.15 35 -> 18[label="",style="solid", color="burlywood", weight=3]; 9.98/4.15 15[label="Just []",fontsize=16,color="green",shape="box"];19[label="vx40",fontsize=16,color="green",shape="box"];17[label="Nothing >>= sequence1 (map vx3 vx41)",fontsize=16,color="black",shape="box"];17 -> 20[label="",style="solid", color="black", weight=3]; 9.98/4.15 18[label="Just vx50 >>= sequence1 (map vx3 vx41)",fontsize=16,color="black",shape="box"];18 -> 21[label="",style="solid", color="black", weight=3]; 9.98/4.15 20[label="Nothing",fontsize=16,color="green",shape="box"];21[label="sequence1 (map vx3 vx41) vx50",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3]; 9.98/4.15 22 -> 23[label="",style="dashed", color="red", weight=0]; 9.98/4.15 22[label="sequence (map vx3 vx41) >>= sequence0 vx50",fontsize=16,color="magenta"];22 -> 24[label="",style="dashed", color="magenta", weight=3]; 9.98/4.15 24 -> 6[label="",style="dashed", color="red", weight=0]; 9.98/4.15 24[label="sequence (map vx3 vx41)",fontsize=16,color="magenta"];24 -> 25[label="",style="dashed", color="magenta", weight=3]; 9.98/4.15 23[label="vx6 >>= sequence0 vx50",fontsize=16,color="burlywood",shape="triangle"];36[label="vx6/Nothing",fontsize=10,color="white",style="solid",shape="box"];23 -> 36[label="",style="solid", color="burlywood", weight=9]; 9.98/4.15 36 -> 26[label="",style="solid", color="burlywood", weight=3]; 9.98/4.15 37[label="vx6/Just vx60",fontsize=10,color="white",style="solid",shape="box"];23 -> 37[label="",style="solid", color="burlywood", weight=9]; 9.98/4.15 37 -> 27[label="",style="solid", color="burlywood", weight=3]; 9.98/4.15 25[label="vx41",fontsize=16,color="green",shape="box"];26[label="Nothing >>= sequence0 vx50",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3]; 9.98/4.15 27[label="Just vx60 >>= sequence0 vx50",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3]; 9.98/4.15 28[label="Nothing",fontsize=16,color="green",shape="box"];29[label="sequence0 vx50 vx60",fontsize=16,color="black",shape="box"];29 -> 30[label="",style="solid", color="black", weight=3]; 9.98/4.15 30[label="return (vx50 : vx60)",fontsize=16,color="black",shape="box"];30 -> 31[label="",style="solid", color="black", weight=3]; 9.98/4.15 31[label="Just (vx50 : vx60)",fontsize=16,color="green",shape="box"];} 9.98/4.15 9.98/4.15 ---------------------------------------- 9.98/4.15 9.98/4.15 (8) 9.98/4.15 Obligation: 9.98/4.15 Q DP problem: 9.98/4.15 The TRS P consists of the following rules: 9.98/4.15 9.98/4.15 new_gtGtEs(vx3, vx41, h, ba) -> new_sequence(vx3, vx41, h, ba) 9.98/4.15 new_sequence(vx3, :(vx40, vx41), h, ba) -> new_gtGtEs(vx3, vx41, h, ba) 9.98/4.15 9.98/4.15 R is empty. 9.98/4.15 Q is empty. 9.98/4.15 We have to consider all minimal (P,Q,R)-chains. 9.98/4.15 ---------------------------------------- 9.98/4.15 9.98/4.15 (9) QDPSizeChangeProof (EQUIVALENT) 9.98/4.15 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. 9.98/4.15 9.98/4.15 From the DPs we obtained the following set of size-change graphs: 9.98/4.15 *new_sequence(vx3, :(vx40, vx41), h, ba) -> new_gtGtEs(vx3, vx41, h, ba) 9.98/4.15 The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4 9.98/4.15 9.98/4.15 9.98/4.15 *new_gtGtEs(vx3, vx41, h, ba) -> new_sequence(vx3, vx41, h, ba) 9.98/4.15 The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4 9.98/4.15 9.98/4.15 9.98/4.15 ---------------------------------------- 9.98/4.15 9.98/4.15 (10) 9.98/4.15 YES 10.13/4.19 EOF