7.97/3.61 YES 9.31/4.08 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 9.31/4.08 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 9.31/4.08 9.31/4.08 9.31/4.08 H-Termination with start terms of the given HASKELL could be proven: 9.31/4.08 9.31/4.08 (0) HASKELL 9.31/4.08 (1) BR [EQUIVALENT, 0 ms] 9.31/4.08 (2) HASKELL 9.31/4.08 (3) COR [EQUIVALENT, 0 ms] 9.31/4.08 (4) HASKELL 9.31/4.08 (5) Narrow [SOUND, 0 ms] 9.31/4.08 (6) AND 9.31/4.08 (7) QDP 9.31/4.08 (8) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.31/4.08 (9) YES 9.31/4.08 (10) QDP 9.31/4.08 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.31/4.08 (12) YES 9.31/4.08 (13) QDP 9.31/4.08 (14) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.31/4.08 (15) YES 9.31/4.08 9.31/4.08 9.31/4.08 ---------------------------------------- 9.31/4.08 9.31/4.08 (0) 9.31/4.08 Obligation: 9.31/4.08 mainModule Main 9.31/4.08 module Main where { 9.31/4.08 import qualified Prelude; 9.31/4.08 data List a = Cons a (List a) | Nil ; 9.31/4.08 9.31/4.08 gtGtEsNil :: List b -> (b -> List a) -> List a; 9.31/4.08 gtGtEsNil (Cons x xs) f = psPs (f x) (gtGtEsNil xs f); 9.31/4.08 gtGtEsNil Nil f = Nil; 9.31/4.08 9.31/4.08 psPs :: List a -> List a -> List a; 9.31/4.08 psPs Nil ys = ys; 9.31/4.08 psPs (Cons x xs) ys = Cons x (psPs xs ys); 9.31/4.08 9.31/4.08 returnNil :: a -> List a; 9.31/4.08 returnNil x = Cons x Nil; 9.31/4.08 9.31/4.08 sequence Nil = returnNil Nil; 9.31/4.08 sequence (Cons c cs) = gtGtEsNil c (sequence1 cs); 9.31/4.08 9.31/4.08 sequence0 x xs = returnNil (Cons x xs); 9.31/4.08 9.31/4.08 sequence1 cs x = gtGtEsNil (sequence cs) (sequence0 x); 9.31/4.08 9.31/4.08 } 9.31/4.08 9.31/4.08 ---------------------------------------- 9.31/4.08 9.31/4.08 (1) BR (EQUIVALENT) 9.31/4.08 Replaced joker patterns by fresh variables and removed binding patterns. 9.31/4.08 ---------------------------------------- 9.31/4.08 9.31/4.08 (2) 9.31/4.08 Obligation: 9.31/4.08 mainModule Main 9.31/4.08 module Main where { 9.31/4.08 import qualified Prelude; 9.31/4.08 data List a = Cons a (List a) | Nil ; 9.31/4.08 9.31/4.08 gtGtEsNil :: List a -> (a -> List b) -> List b; 9.31/4.08 gtGtEsNil (Cons x xs) f = psPs (f x) (gtGtEsNil xs f); 9.31/4.08 gtGtEsNil Nil f = Nil; 9.31/4.08 9.31/4.08 psPs :: List a -> List a -> List a; 9.31/4.08 psPs Nil ys = ys; 9.31/4.08 psPs (Cons x xs) ys = Cons x (psPs xs ys); 9.31/4.08 9.31/4.08 returnNil :: a -> List a; 9.31/4.08 returnNil x = Cons x Nil; 9.31/4.08 9.31/4.08 sequence Nil = returnNil Nil; 9.31/4.08 sequence (Cons c cs) = gtGtEsNil c (sequence1 cs); 9.31/4.08 9.31/4.08 sequence0 x xs = returnNil (Cons x xs); 9.31/4.08 9.31/4.08 sequence1 cs x = gtGtEsNil (sequence cs) (sequence0 x); 9.31/4.08 9.31/4.08 } 9.31/4.08 9.31/4.08 ---------------------------------------- 9.31/4.08 9.31/4.08 (3) COR (EQUIVALENT) 9.31/4.08 Cond Reductions: 9.31/4.08 The following Function with conditions 9.31/4.08 "undefined |Falseundefined; 9.31/4.08 " 9.31/4.08 is transformed to 9.31/4.08 "undefined = undefined1; 9.31/4.08 " 9.31/4.08 "undefined0 True = undefined; 9.31/4.08 " 9.31/4.08 "undefined1 = undefined0 False; 9.31/4.08 " 9.31/4.08 9.31/4.08 ---------------------------------------- 9.31/4.08 9.31/4.08 (4) 9.31/4.08 Obligation: 9.31/4.08 mainModule Main 9.31/4.08 module Main where { 9.31/4.08 import qualified Prelude; 9.31/4.08 data List a = Cons a (List a) | Nil ; 9.31/4.08 9.31/4.08 gtGtEsNil :: List a -> (a -> List b) -> List b; 9.31/4.08 gtGtEsNil (Cons x xs) f = psPs (f x) (gtGtEsNil xs f); 9.31/4.08 gtGtEsNil Nil f = Nil; 9.31/4.08 9.31/4.08 psPs :: List a -> List a -> List a; 9.31/4.08 psPs Nil ys = ys; 9.31/4.08 psPs (Cons x xs) ys = Cons x (psPs xs ys); 9.31/4.08 9.31/4.08 returnNil :: a -> List a; 9.31/4.08 returnNil x = Cons x Nil; 9.31/4.08 9.31/4.08 sequence Nil = returnNil Nil; 9.31/4.08 sequence (Cons c cs) = gtGtEsNil c (sequence1 cs); 9.31/4.08 9.31/4.08 sequence0 x xs = returnNil (Cons x xs); 9.31/4.08 9.31/4.08 sequence1 cs x = gtGtEsNil (sequence cs) (sequence0 x); 9.31/4.08 9.31/4.08 } 9.31/4.08 9.31/4.08 ---------------------------------------- 9.31/4.08 9.31/4.08 (5) Narrow (SOUND) 9.31/4.08 Haskell To QDPs 9.31/4.08 9.31/4.08 digraph dp_graph { 9.31/4.08 node [outthreshold=100, inthreshold=100];1[label="sequence",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 9.31/4.08 3[label="sequence vx3",fontsize=16,color="burlywood",shape="triangle"];48[label="vx3/Cons vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];3 -> 48[label="",style="solid", color="burlywood", weight=9]; 9.31/4.08 48 -> 4[label="",style="solid", color="burlywood", weight=3]; 9.31/4.08 49[label="vx3/Nil",fontsize=10,color="white",style="solid",shape="box"];3 -> 49[label="",style="solid", color="burlywood", weight=9]; 9.31/4.08 49 -> 5[label="",style="solid", color="burlywood", weight=3]; 9.31/4.08 4[label="sequence (Cons vx30 vx31)",fontsize=16,color="black",shape="box"];4 -> 6[label="",style="solid", color="black", weight=3]; 9.31/4.08 5[label="sequence Nil",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3]; 9.31/4.08 6[label="gtGtEsNil vx30 (sequence1 vx31)",fontsize=16,color="burlywood",shape="triangle"];50[label="vx30/Cons vx300 vx301",fontsize=10,color="white",style="solid",shape="box"];6 -> 50[label="",style="solid", color="burlywood", weight=9]; 9.31/4.08 50 -> 8[label="",style="solid", color="burlywood", weight=3]; 9.31/4.08 51[label="vx30/Nil",fontsize=10,color="white",style="solid",shape="box"];6 -> 51[label="",style="solid", color="burlywood", weight=9]; 9.31/4.08 51 -> 9[label="",style="solid", color="burlywood", weight=3]; 9.31/4.08 7[label="returnNil Nil",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3]; 9.31/4.08 8[label="gtGtEsNil (Cons vx300 vx301) (sequence1 vx31)",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3]; 9.31/4.08 9[label="gtGtEsNil Nil (sequence1 vx31)",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3]; 9.31/4.08 10[label="Cons Nil Nil",fontsize=16,color="green",shape="box"];11 -> 13[label="",style="dashed", color="red", weight=0]; 9.31/4.08 11[label="psPs (sequence1 vx31 vx300) (gtGtEsNil vx301 (sequence1 vx31))",fontsize=16,color="magenta"];11 -> 14[label="",style="dashed", color="magenta", weight=3]; 9.31/4.08 12[label="Nil",fontsize=16,color="green",shape="box"];14 -> 6[label="",style="dashed", color="red", weight=0]; 9.31/4.08 14[label="gtGtEsNil vx301 (sequence1 vx31)",fontsize=16,color="magenta"];14 -> 15[label="",style="dashed", color="magenta", weight=3]; 9.31/4.08 13[label="psPs (sequence1 vx31 vx300) vx4",fontsize=16,color="black",shape="triangle"];13 -> 16[label="",style="solid", color="black", weight=3]; 9.31/4.08 15[label="vx301",fontsize=16,color="green",shape="box"];16 -> 17[label="",style="dashed", color="red", weight=0]; 9.31/4.08 16[label="psPs (gtGtEsNil (sequence vx31) (sequence0 vx300)) vx4",fontsize=16,color="magenta"];16 -> 18[label="",style="dashed", color="magenta", weight=3]; 9.31/4.08 18 -> 3[label="",style="dashed", color="red", weight=0]; 9.31/4.08 18[label="sequence vx31",fontsize=16,color="magenta"];18 -> 19[label="",style="dashed", color="magenta", weight=3]; 9.31/4.08 17[label="psPs (gtGtEsNil vx5 (sequence0 vx300)) vx4",fontsize=16,color="burlywood",shape="triangle"];52[label="vx5/Cons vx50 vx51",fontsize=10,color="white",style="solid",shape="box"];17 -> 52[label="",style="solid", color="burlywood", weight=9]; 9.31/4.08 52 -> 20[label="",style="solid", color="burlywood", weight=3]; 9.31/4.08 53[label="vx5/Nil",fontsize=10,color="white",style="solid",shape="box"];17 -> 53[label="",style="solid", color="burlywood", weight=9]; 9.31/4.08 53 -> 21[label="",style="solid", color="burlywood", weight=3]; 9.31/4.08 19[label="vx31",fontsize=16,color="green",shape="box"];20[label="psPs (gtGtEsNil (Cons vx50 vx51) (sequence0 vx300)) vx4",fontsize=16,color="black",shape="box"];20 -> 22[label="",style="solid", color="black", weight=3]; 9.31/4.08 21[label="psPs (gtGtEsNil Nil (sequence0 vx300)) vx4",fontsize=16,color="black",shape="box"];21 -> 23[label="",style="solid", color="black", weight=3]; 9.31/4.08 22[label="psPs (psPs (sequence0 vx300 vx50) (gtGtEsNil vx51 (sequence0 vx300))) vx4",fontsize=16,color="black",shape="box"];22 -> 24[label="",style="solid", color="black", weight=3]; 9.31/4.08 23[label="psPs Nil vx4",fontsize=16,color="black",shape="triangle"];23 -> 25[label="",style="solid", color="black", weight=3]; 9.31/4.08 24[label="psPs (psPs (returnNil (Cons vx300 vx50)) (gtGtEsNil vx51 (sequence0 vx300))) vx4",fontsize=16,color="black",shape="box"];24 -> 26[label="",style="solid", color="black", weight=3]; 9.31/4.08 25[label="vx4",fontsize=16,color="green",shape="box"];26[label="psPs (psPs (Cons (Cons vx300 vx50) Nil) (gtGtEsNil vx51 (sequence0 vx300))) vx4",fontsize=16,color="black",shape="box"];26 -> 27[label="",style="solid", color="black", weight=3]; 9.31/4.08 27 -> 28[label="",style="dashed", color="red", weight=0]; 9.31/4.08 27[label="psPs (Cons (Cons vx300 vx50) (psPs Nil (gtGtEsNil vx51 (sequence0 vx300)))) vx4",fontsize=16,color="magenta"];27 -> 29[label="",style="dashed", color="magenta", weight=3]; 9.31/4.08 29 -> 23[label="",style="dashed", color="red", weight=0]; 9.31/4.08 29[label="psPs Nil (gtGtEsNil vx51 (sequence0 vx300))",fontsize=16,color="magenta"];29 -> 30[label="",style="dashed", color="magenta", weight=3]; 9.31/4.08 28[label="psPs (Cons (Cons vx300 vx50) vx6) vx4",fontsize=16,color="black",shape="triangle"];28 -> 31[label="",style="solid", color="black", weight=3]; 9.31/4.08 30[label="gtGtEsNil vx51 (sequence0 vx300)",fontsize=16,color="burlywood",shape="triangle"];54[label="vx51/Cons vx510 vx511",fontsize=10,color="white",style="solid",shape="box"];30 -> 54[label="",style="solid", color="burlywood", weight=9]; 9.31/4.08 54 -> 32[label="",style="solid", color="burlywood", weight=3]; 9.31/4.08 55[label="vx51/Nil",fontsize=10,color="white",style="solid",shape="box"];30 -> 55[label="",style="solid", color="burlywood", weight=9]; 9.31/4.08 55 -> 33[label="",style="solid", color="burlywood", weight=3]; 9.31/4.08 31[label="Cons (Cons vx300 vx50) (psPs vx6 vx4)",fontsize=16,color="green",shape="box"];31 -> 34[label="",style="dashed", color="green", weight=3]; 9.31/4.08 32[label="gtGtEsNil (Cons vx510 vx511) (sequence0 vx300)",fontsize=16,color="black",shape="box"];32 -> 35[label="",style="solid", color="black", weight=3]; 9.31/4.08 33[label="gtGtEsNil Nil (sequence0 vx300)",fontsize=16,color="black",shape="box"];33 -> 36[label="",style="solid", color="black", weight=3]; 9.31/4.08 34[label="psPs vx6 vx4",fontsize=16,color="burlywood",shape="triangle"];56[label="vx6/Cons vx60 vx61",fontsize=10,color="white",style="solid",shape="box"];34 -> 56[label="",style="solid", color="burlywood", weight=9]; 9.31/4.09 56 -> 37[label="",style="solid", color="burlywood", weight=3]; 9.31/4.09 57[label="vx6/Nil",fontsize=10,color="white",style="solid",shape="box"];34 -> 57[label="",style="solid", color="burlywood", weight=9]; 9.31/4.09 57 -> 38[label="",style="solid", color="burlywood", weight=3]; 9.31/4.09 35 -> 34[label="",style="dashed", color="red", weight=0]; 9.31/4.09 35[label="psPs (sequence0 vx300 vx510) (gtGtEsNil vx511 (sequence0 vx300))",fontsize=16,color="magenta"];35 -> 39[label="",style="dashed", color="magenta", weight=3]; 9.31/4.09 35 -> 40[label="",style="dashed", color="magenta", weight=3]; 9.31/4.09 36[label="Nil",fontsize=16,color="green",shape="box"];37[label="psPs (Cons vx60 vx61) vx4",fontsize=16,color="black",shape="box"];37 -> 41[label="",style="solid", color="black", weight=3]; 9.31/4.09 38[label="psPs Nil vx4",fontsize=16,color="black",shape="box"];38 -> 42[label="",style="solid", color="black", weight=3]; 9.31/4.09 39 -> 30[label="",style="dashed", color="red", weight=0]; 9.31/4.09 39[label="gtGtEsNil vx511 (sequence0 vx300)",fontsize=16,color="magenta"];39 -> 43[label="",style="dashed", color="magenta", weight=3]; 9.31/4.09 40[label="sequence0 vx300 vx510",fontsize=16,color="black",shape="box"];40 -> 44[label="",style="solid", color="black", weight=3]; 9.31/4.09 41[label="Cons vx60 (psPs vx61 vx4)",fontsize=16,color="green",shape="box"];41 -> 45[label="",style="dashed", color="green", weight=3]; 9.31/4.09 42[label="vx4",fontsize=16,color="green",shape="box"];43[label="vx511",fontsize=16,color="green",shape="box"];44[label="returnNil (Cons vx300 vx510)",fontsize=16,color="black",shape="box"];44 -> 46[label="",style="solid", color="black", weight=3]; 9.31/4.09 45 -> 34[label="",style="dashed", color="red", weight=0]; 9.31/4.09 45[label="psPs vx61 vx4",fontsize=16,color="magenta"];45 -> 47[label="",style="dashed", color="magenta", weight=3]; 9.31/4.09 46[label="Cons (Cons vx300 vx510) Nil",fontsize=16,color="green",shape="box"];47[label="vx61",fontsize=16,color="green",shape="box"];} 9.31/4.09 9.31/4.09 ---------------------------------------- 9.31/4.09 9.31/4.09 (6) 9.31/4.09 Complex Obligation (AND) 9.31/4.09 9.31/4.09 ---------------------------------------- 9.31/4.09 9.31/4.09 (7) 9.31/4.09 Obligation: 9.31/4.09 Q DP problem: 9.31/4.09 The TRS P consists of the following rules: 9.31/4.09 9.31/4.09 new_psPs(Cons(vx60, vx61), vx4, h) -> new_psPs(vx61, vx4, h) 9.31/4.09 9.31/4.09 R is empty. 9.31/4.09 Q is empty. 9.31/4.09 We have to consider all minimal (P,Q,R)-chains. 9.31/4.09 ---------------------------------------- 9.31/4.09 9.31/4.09 (8) QDPSizeChangeProof (EQUIVALENT) 9.31/4.09 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. 9.31/4.09 9.31/4.09 From the DPs we obtained the following set of size-change graphs: 9.31/4.09 *new_psPs(Cons(vx60, vx61), vx4, h) -> new_psPs(vx61, vx4, h) 9.31/4.09 The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 9.31/4.09 9.31/4.09 9.31/4.09 ---------------------------------------- 9.31/4.09 9.31/4.09 (9) 9.31/4.09 YES 9.31/4.09 9.31/4.09 ---------------------------------------- 9.31/4.09 9.31/4.09 (10) 9.31/4.09 Obligation: 9.31/4.09 Q DP problem: 9.31/4.09 The TRS P consists of the following rules: 9.31/4.09 9.31/4.09 new_sequence(Cons(Cons(vx300, vx301), vx31), h) -> new_gtGtEsNil0(vx301, vx31, h) 9.31/4.09 new_gtGtEsNil0(Cons(vx300, vx301), vx31, h) -> new_psPs0(vx31, vx300, new_gtGtEsNil1(vx301, vx31, h), h) 9.31/4.09 new_sequence(Cons(Cons(vx300, vx301), vx31), h) -> new_psPs0(vx31, vx300, new_gtGtEsNil1(vx301, vx31, h), h) 9.31/4.09 new_gtGtEsNil0(Cons(vx300, vx301), vx31, h) -> new_gtGtEsNil0(vx301, vx31, h) 9.31/4.09 new_psPs0(vx31, vx300, vx4, h) -> new_sequence(vx31, h) 9.31/4.09 9.31/4.09 The TRS R consists of the following rules: 9.31/4.09 9.31/4.09 new_psPs5(Nil, vx4, h) -> vx4 9.31/4.09 new_gtGtEsNil2(Nil, vx300, h) -> Nil 9.31/4.09 new_sequence0(Cons(vx30, vx31), h) -> new_gtGtEsNil1(vx30, vx31, h) 9.31/4.09 new_gtGtEsNil1(Cons(vx300, vx301), vx31, h) -> new_psPs2(vx31, vx300, new_gtGtEsNil1(vx301, vx31, h), h) 9.31/4.09 new_sequence0(Nil, h) -> Cons(Nil, Nil) 9.31/4.09 new_psPs1(vx4, h) -> vx4 9.31/4.09 new_psPs2(vx31, vx300, vx4, h) -> new_psPs3(new_sequence0(vx31, h), vx300, vx4, h) 9.31/4.09 new_psPs3(Cons(vx50, vx51), vx300, vx4, h) -> new_psPs4(vx300, vx50, new_psPs1(new_gtGtEsNil2(vx51, vx300, h), h), vx4, h) 9.31/4.09 new_psPs4(vx300, vx50, vx6, vx4, h) -> Cons(Cons(vx300, vx50), new_psPs5(vx6, vx4, h)) 9.31/4.09 new_psPs3(Nil, vx300, vx4, h) -> new_psPs1(vx4, h) 9.31/4.09 new_gtGtEsNil1(Nil, vx31, h) -> Nil 9.31/4.09 new_gtGtEsNil2(Cons(vx510, vx511), vx300, h) -> new_psPs5(Cons(Cons(vx300, vx510), Nil), new_gtGtEsNil2(vx511, vx300, h), h) 9.31/4.09 new_psPs5(Cons(vx60, vx61), vx4, h) -> Cons(vx60, new_psPs5(vx61, vx4, h)) 9.31/4.09 9.31/4.09 The set Q consists of the following terms: 9.31/4.09 9.31/4.09 new_gtGtEsNil2(Nil, x0, x1) 9.31/4.09 new_psPs5(Nil, x0, x1) 9.31/4.09 new_psPs1(x0, x1) 9.31/4.09 new_gtGtEsNil1(Cons(x0, x1), x2, x3) 9.31/4.09 new_psPs2(x0, x1, x2, x3) 9.31/4.09 new_sequence0(Cons(x0, x1), x2) 9.31/4.09 new_psPs3(Cons(x0, x1), x2, x3, x4) 9.31/4.09 new_psPs4(x0, x1, x2, x3, x4) 9.31/4.09 new_psPs5(Cons(x0, x1), x2, x3) 9.31/4.09 new_sequence0(Nil, x0) 9.31/4.09 new_psPs3(Nil, x0, x1, x2) 9.31/4.09 new_gtGtEsNil2(Cons(x0, x1), x2, x3) 9.31/4.09 new_gtGtEsNil1(Nil, x0, x1) 9.31/4.09 9.31/4.09 We have to consider all minimal (P,Q,R)-chains. 9.31/4.09 ---------------------------------------- 9.31/4.09 9.31/4.09 (11) QDPSizeChangeProof (EQUIVALENT) 9.31/4.09 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. 9.31/4.09 9.31/4.09 From the DPs we obtained the following set of size-change graphs: 9.31/4.09 *new_psPs0(vx31, vx300, vx4, h) -> new_sequence(vx31, h) 9.31/4.09 The graph contains the following edges 1 >= 1, 4 >= 2 9.31/4.09 9.31/4.09 9.31/4.09 *new_gtGtEsNil0(Cons(vx300, vx301), vx31, h) -> new_gtGtEsNil0(vx301, vx31, h) 9.31/4.09 The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 9.31/4.09 9.31/4.09 9.31/4.09 *new_gtGtEsNil0(Cons(vx300, vx301), vx31, h) -> new_psPs0(vx31, vx300, new_gtGtEsNil1(vx301, vx31, h), h) 9.31/4.09 The graph contains the following edges 2 >= 1, 1 > 2, 3 >= 4 9.31/4.09 9.31/4.09 9.31/4.09 *new_sequence(Cons(Cons(vx300, vx301), vx31), h) -> new_gtGtEsNil0(vx301, vx31, h) 9.31/4.09 The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3 9.31/4.09 9.31/4.09 9.31/4.09 *new_sequence(Cons(Cons(vx300, vx301), vx31), h) -> new_psPs0(vx31, vx300, new_gtGtEsNil1(vx301, vx31, h), h) 9.31/4.09 The graph contains the following edges 1 > 1, 1 > 2, 2 >= 4 9.31/4.09 9.31/4.09 9.31/4.09 ---------------------------------------- 9.31/4.09 9.31/4.09 (12) 9.31/4.09 YES 9.31/4.09 9.31/4.09 ---------------------------------------- 9.31/4.09 9.31/4.09 (13) 9.31/4.09 Obligation: 9.31/4.09 Q DP problem: 9.31/4.09 The TRS P consists of the following rules: 9.31/4.09 9.31/4.09 new_gtGtEsNil(Cons(vx510, vx511), vx300, h) -> new_gtGtEsNil(vx511, vx300, h) 9.31/4.09 9.31/4.09 R is empty. 9.31/4.09 Q is empty. 9.31/4.09 We have to consider all minimal (P,Q,R)-chains. 9.31/4.09 ---------------------------------------- 9.31/4.09 9.31/4.09 (14) QDPSizeChangeProof (EQUIVALENT) 9.31/4.09 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. 9.31/4.09 9.31/4.09 From the DPs we obtained the following set of size-change graphs: 9.31/4.09 *new_gtGtEsNil(Cons(vx510, vx511), vx300, h) -> new_gtGtEsNil(vx511, vx300, h) 9.31/4.09 The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 9.31/4.09 9.31/4.09 9.31/4.09 ---------------------------------------- 9.31/4.09 9.31/4.09 (15) 9.31/4.09 YES 9.66/4.13 EOF