8.65/3.70 YES 10.03/4.15 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 10.03/4.15 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 10.03/4.15 10.03/4.15 10.03/4.15 H-Termination with start terms of the given HASKELL could be proven: 10.03/4.15 10.03/4.15 (0) HASKELL 10.03/4.15 (1) BR [EQUIVALENT, 0 ms] 10.03/4.15 (2) HASKELL 10.03/4.15 (3) COR [EQUIVALENT, 0 ms] 10.03/4.15 (4) HASKELL 10.03/4.15 (5) Narrow [SOUND, 0 ms] 10.03/4.15 (6) QDP 10.03/4.15 (7) QDPSizeChangeProof [EQUIVALENT, 0 ms] 10.03/4.15 (8) YES 10.03/4.15 10.03/4.15 10.03/4.15 ---------------------------------------- 10.03/4.15 10.03/4.15 (0) 10.03/4.15 Obligation: 10.03/4.15 mainModule Main 10.03/4.15 module Main where { 10.03/4.15 import qualified Prelude; 10.03/4.15 } 10.03/4.15 10.03/4.15 ---------------------------------------- 10.03/4.15 10.03/4.15 (1) BR (EQUIVALENT) 10.03/4.15 Replaced joker patterns by fresh variables and removed binding patterns. 10.03/4.15 ---------------------------------------- 10.03/4.15 10.03/4.15 (2) 10.03/4.15 Obligation: 10.03/4.15 mainModule Main 10.03/4.15 module Main where { 10.03/4.15 import qualified Prelude; 10.03/4.15 } 10.03/4.15 10.03/4.15 ---------------------------------------- 10.03/4.15 10.03/4.15 (3) COR (EQUIVALENT) 10.03/4.15 Cond Reductions: 10.03/4.15 The following Function with conditions 10.03/4.15 "max x y|x <= yy|otherwisex; 10.03/4.15 " 10.03/4.15 is transformed to 10.03/4.15 "max x y = max2 x y; 10.03/4.15 " 10.03/4.15 "max0 x y True = x; 10.03/4.15 " 10.03/4.15 "max1 x y True = y; 10.03/4.15 max1 x y False = max0 x y otherwise; 10.03/4.15 " 10.03/4.15 "max2 x y = max1 x y (x <= y); 10.03/4.15 " 10.03/4.15 The following Function with conditions 10.03/4.15 "undefined |Falseundefined; 10.03/4.15 " 10.03/4.15 is transformed to 10.03/4.15 "undefined = undefined1; 10.03/4.15 " 10.03/4.15 "undefined0 True = undefined; 10.03/4.15 " 10.03/4.15 "undefined1 = undefined0 False; 10.03/4.15 " 10.03/4.15 10.03/4.15 ---------------------------------------- 10.03/4.15 10.03/4.15 (4) 10.03/4.15 Obligation: 10.03/4.15 mainModule Main 10.03/4.15 module Main where { 10.03/4.15 import qualified Prelude; 10.03/4.15 } 10.03/4.15 10.03/4.15 ---------------------------------------- 10.03/4.15 10.03/4.15 (5) Narrow (SOUND) 10.03/4.15 Haskell To QDPs 10.03/4.15 10.03/4.15 digraph dp_graph { 10.03/4.15 node [outthreshold=100, inthreshold=100];1[label="max",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 10.03/4.15 3[label="max vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 10.03/4.15 4[label="max vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 10.03/4.15 5[label="max2 vx3 vx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 10.03/4.15 6[label="max1 vx3 vx4 (vx3 <= vx4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3]; 10.03/4.15 7[label="max1 vx3 vx4 (compare vx3 vx4 /= GT)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3]; 10.03/4.15 8[label="max1 vx3 vx4 (not (compare vx3 vx4 == GT))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3]; 10.03/4.15 9[label="max1 vx3 vx4 (not (primCmpChar vx3 vx4 == GT))",fontsize=16,color="burlywood",shape="box"];302[label="vx3/Char vx30",fontsize=10,color="white",style="solid",shape="box"];9 -> 302[label="",style="solid", color="burlywood", weight=9]; 10.03/4.15 302 -> 10[label="",style="solid", color="burlywood", weight=3]; 10.03/4.15 10[label="max1 (Char vx30) vx4 (not (primCmpChar (Char vx30) vx4 == GT))",fontsize=16,color="burlywood",shape="box"];303[label="vx4/Char vx40",fontsize=10,color="white",style="solid",shape="box"];10 -> 303[label="",style="solid", color="burlywood", weight=9]; 10.03/4.15 303 -> 11[label="",style="solid", color="burlywood", weight=3]; 10.03/4.15 11[label="max1 (Char vx30) (Char vx40) (not (primCmpChar (Char vx30) (Char vx40) == GT))",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3]; 10.03/4.15 12[label="max1 (Char vx30) (Char vx40) (not (primCmpNat vx30 vx40 == GT))",fontsize=16,color="burlywood",shape="box"];304[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];12 -> 304[label="",style="solid", color="burlywood", weight=9]; 10.03/4.15 304 -> 13[label="",style="solid", color="burlywood", weight=3]; 10.03/4.15 305[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];12 -> 305[label="",style="solid", color="burlywood", weight=9]; 10.03/4.15 305 -> 14[label="",style="solid", color="burlywood", weight=3]; 10.03/4.15 13[label="max1 (Char (Succ vx300)) (Char vx40) (not (primCmpNat (Succ vx300) vx40 == GT))",fontsize=16,color="burlywood",shape="box"];306[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];13 -> 306[label="",style="solid", color="burlywood", weight=9]; 10.03/4.15 306 -> 15[label="",style="solid", color="burlywood", weight=3]; 10.03/4.15 307[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 307[label="",style="solid", color="burlywood", weight=9]; 10.03/4.15 307 -> 16[label="",style="solid", color="burlywood", weight=3]; 10.03/4.15 14[label="max1 (Char Zero) (Char vx40) (not (primCmpNat Zero vx40 == GT))",fontsize=16,color="burlywood",shape="box"];308[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];14 -> 308[label="",style="solid", color="burlywood", weight=9]; 10.03/4.15 308 -> 17[label="",style="solid", color="burlywood", weight=3]; 10.03/4.15 309[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];14 -> 309[label="",style="solid", color="burlywood", weight=9]; 10.03/4.15 309 -> 18[label="",style="solid", color="burlywood", weight=3]; 10.03/4.15 15[label="max1 (Char (Succ vx300)) (Char (Succ vx400)) (not (primCmpNat (Succ vx300) (Succ vx400) == GT))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3]; 10.03/4.15 16[label="max1 (Char (Succ vx300)) (Char Zero) (not (primCmpNat (Succ vx300) Zero == GT))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3]; 10.03/4.15 17[label="max1 (Char Zero) (Char (Succ vx400)) (not (primCmpNat Zero (Succ vx400) == GT))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3]; 10.03/4.15 18[label="max1 (Char Zero) (Char Zero) (not (primCmpNat Zero Zero == GT))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3]; 10.03/4.15 19 -> 240[label="",style="dashed", color="red", weight=0]; 10.03/4.15 19[label="max1 (Char (Succ vx300)) (Char (Succ vx400)) (not (primCmpNat vx300 vx400 == GT))",fontsize=16,color="magenta"];19 -> 241[label="",style="dashed", color="magenta", weight=3]; 10.03/4.15 19 -> 242[label="",style="dashed", color="magenta", weight=3]; 10.03/4.15 19 -> 243[label="",style="dashed", color="magenta", weight=3]; 10.03/4.15 19 -> 244[label="",style="dashed", color="magenta", weight=3]; 10.03/4.15 20[label="max1 (Char (Succ vx300)) (Char Zero) (not (GT == GT))",fontsize=16,color="black",shape="box"];20 -> 25[label="",style="solid", color="black", weight=3]; 10.03/4.15 21[label="max1 (Char Zero) (Char (Succ vx400)) (not (LT == GT))",fontsize=16,color="black",shape="box"];21 -> 26[label="",style="solid", color="black", weight=3]; 10.03/4.15 22[label="max1 (Char Zero) (Char Zero) (not (EQ == GT))",fontsize=16,color="black",shape="box"];22 -> 27[label="",style="solid", color="black", weight=3]; 10.03/4.15 241[label="vx300",fontsize=16,color="green",shape="box"];242[label="vx400",fontsize=16,color="green",shape="box"];243[label="vx300",fontsize=16,color="green",shape="box"];244[label="vx400",fontsize=16,color="green",shape="box"];240[label="max1 (Char (Succ vx27)) (Char (Succ vx28)) (not (primCmpNat vx29 vx30 == GT))",fontsize=16,color="burlywood",shape="triangle"];310[label="vx29/Succ vx290",fontsize=10,color="white",style="solid",shape="box"];240 -> 310[label="",style="solid", color="burlywood", weight=9]; 10.03/4.15 310 -> 281[label="",style="solid", color="burlywood", weight=3]; 10.03/4.15 311[label="vx29/Zero",fontsize=10,color="white",style="solid",shape="box"];240 -> 311[label="",style="solid", color="burlywood", weight=9]; 10.03/4.15 311 -> 282[label="",style="solid", color="burlywood", weight=3]; 10.03/4.15 25[label="max1 (Char (Succ vx300)) (Char Zero) (not True)",fontsize=16,color="black",shape="box"];25 -> 32[label="",style="solid", color="black", weight=3]; 10.03/4.15 26[label="max1 (Char Zero) (Char (Succ vx400)) (not False)",fontsize=16,color="black",shape="box"];26 -> 33[label="",style="solid", color="black", weight=3]; 10.03/4.15 27[label="max1 (Char Zero) (Char Zero) (not False)",fontsize=16,color="black",shape="box"];27 -> 34[label="",style="solid", color="black", weight=3]; 10.03/4.15 281[label="max1 (Char (Succ vx27)) (Char (Succ vx28)) (not (primCmpNat (Succ vx290) vx30 == GT))",fontsize=16,color="burlywood",shape="box"];312[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];281 -> 312[label="",style="solid", color="burlywood", weight=9]; 10.03/4.15 312 -> 283[label="",style="solid", color="burlywood", weight=3]; 10.03/4.15 313[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];281 -> 313[label="",style="solid", color="burlywood", weight=9]; 10.03/4.15 313 -> 284[label="",style="solid", color="burlywood", weight=3]; 10.03/4.15 282[label="max1 (Char (Succ vx27)) (Char (Succ vx28)) (not (primCmpNat Zero vx30 == GT))",fontsize=16,color="burlywood",shape="box"];314[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];282 -> 314[label="",style="solid", color="burlywood", weight=9]; 10.03/4.15 314 -> 285[label="",style="solid", color="burlywood", weight=3]; 10.03/4.15 315[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];282 -> 315[label="",style="solid", color="burlywood", weight=9]; 10.03/4.15 315 -> 286[label="",style="solid", color="burlywood", weight=3]; 10.03/4.15 32[label="max1 (Char (Succ vx300)) (Char Zero) False",fontsize=16,color="black",shape="box"];32 -> 39[label="",style="solid", color="black", weight=3]; 10.03/4.15 33[label="max1 (Char Zero) (Char (Succ vx400)) True",fontsize=16,color="black",shape="box"];33 -> 40[label="",style="solid", color="black", weight=3]; 10.03/4.15 34[label="max1 (Char Zero) (Char Zero) True",fontsize=16,color="black",shape="box"];34 -> 41[label="",style="solid", color="black", weight=3]; 10.03/4.15 283[label="max1 (Char (Succ vx27)) (Char (Succ vx28)) (not (primCmpNat (Succ vx290) (Succ vx300) == GT))",fontsize=16,color="black",shape="box"];283 -> 287[label="",style="solid", color="black", weight=3]; 10.03/4.15 284[label="max1 (Char (Succ vx27)) (Char (Succ vx28)) (not (primCmpNat (Succ vx290) Zero == GT))",fontsize=16,color="black",shape="box"];284 -> 288[label="",style="solid", color="black", weight=3]; 10.03/4.15 285[label="max1 (Char (Succ vx27)) (Char (Succ vx28)) (not (primCmpNat Zero (Succ vx300) == GT))",fontsize=16,color="black",shape="box"];285 -> 289[label="",style="solid", color="black", weight=3]; 10.03/4.15 286[label="max1 (Char (Succ vx27)) (Char (Succ vx28)) (not (primCmpNat Zero Zero == GT))",fontsize=16,color="black",shape="box"];286 -> 290[label="",style="solid", color="black", weight=3]; 10.03/4.15 39[label="max0 (Char (Succ vx300)) (Char Zero) otherwise",fontsize=16,color="black",shape="box"];39 -> 47[label="",style="solid", color="black", weight=3]; 10.03/4.15 40[label="Char (Succ vx400)",fontsize=16,color="green",shape="box"];41[label="Char Zero",fontsize=16,color="green",shape="box"];287 -> 240[label="",style="dashed", color="red", weight=0]; 10.03/4.15 287[label="max1 (Char (Succ vx27)) (Char (Succ vx28)) (not (primCmpNat vx290 vx300 == GT))",fontsize=16,color="magenta"];287 -> 291[label="",style="dashed", color="magenta", weight=3]; 10.03/4.15 287 -> 292[label="",style="dashed", color="magenta", weight=3]; 10.03/4.15 288[label="max1 (Char (Succ vx27)) (Char (Succ vx28)) (not (GT == GT))",fontsize=16,color="black",shape="box"];288 -> 293[label="",style="solid", color="black", weight=3]; 10.03/4.15 289[label="max1 (Char (Succ vx27)) (Char (Succ vx28)) (not (LT == GT))",fontsize=16,color="black",shape="box"];289 -> 294[label="",style="solid", color="black", weight=3]; 10.03/4.15 290[label="max1 (Char (Succ vx27)) (Char (Succ vx28)) (not (EQ == GT))",fontsize=16,color="black",shape="box"];290 -> 295[label="",style="solid", color="black", weight=3]; 10.03/4.15 47[label="max0 (Char (Succ vx300)) (Char Zero) True",fontsize=16,color="black",shape="box"];47 -> 55[label="",style="solid", color="black", weight=3]; 10.03/4.15 291[label="vx290",fontsize=16,color="green",shape="box"];292[label="vx300",fontsize=16,color="green",shape="box"];293[label="max1 (Char (Succ vx27)) (Char (Succ vx28)) (not True)",fontsize=16,color="black",shape="box"];293 -> 296[label="",style="solid", color="black", weight=3]; 10.03/4.15 294[label="max1 (Char (Succ vx27)) (Char (Succ vx28)) (not False)",fontsize=16,color="black",shape="triangle"];294 -> 297[label="",style="solid", color="black", weight=3]; 10.03/4.15 295 -> 294[label="",style="dashed", color="red", weight=0]; 10.03/4.15 295[label="max1 (Char (Succ vx27)) (Char (Succ vx28)) (not False)",fontsize=16,color="magenta"];55[label="Char (Succ vx300)",fontsize=16,color="green",shape="box"];296[label="max1 (Char (Succ vx27)) (Char (Succ vx28)) False",fontsize=16,color="black",shape="box"];296 -> 298[label="",style="solid", color="black", weight=3]; 10.03/4.15 297[label="max1 (Char (Succ vx27)) (Char (Succ vx28)) True",fontsize=16,color="black",shape="box"];297 -> 299[label="",style="solid", color="black", weight=3]; 10.03/4.15 298[label="max0 (Char (Succ vx27)) (Char (Succ vx28)) otherwise",fontsize=16,color="black",shape="box"];298 -> 300[label="",style="solid", color="black", weight=3]; 10.03/4.15 299[label="Char (Succ vx28)",fontsize=16,color="green",shape="box"];300[label="max0 (Char (Succ vx27)) (Char (Succ vx28)) True",fontsize=16,color="black",shape="box"];300 -> 301[label="",style="solid", color="black", weight=3]; 10.03/4.15 301[label="Char (Succ vx27)",fontsize=16,color="green",shape="box"];} 10.03/4.15 10.03/4.15 ---------------------------------------- 10.03/4.15 10.03/4.15 (6) 10.03/4.15 Obligation: 10.03/4.15 Q DP problem: 10.03/4.15 The TRS P consists of the following rules: 10.03/4.15 10.03/4.15 new_max1(vx27, vx28, Succ(vx290), Succ(vx300)) -> new_max1(vx27, vx28, vx290, vx300) 10.03/4.15 10.03/4.15 R is empty. 10.03/4.15 Q is empty. 10.03/4.15 We have to consider all minimal (P,Q,R)-chains. 10.03/4.15 ---------------------------------------- 10.03/4.15 10.03/4.15 (7) QDPSizeChangeProof (EQUIVALENT) 10.03/4.15 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. 10.03/4.15 10.03/4.15 From the DPs we obtained the following set of size-change graphs: 10.03/4.15 *new_max1(vx27, vx28, Succ(vx290), Succ(vx300)) -> new_max1(vx27, vx28, vx290, vx300) 10.03/4.15 The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 10.03/4.15 10.03/4.15 10.03/4.15 ---------------------------------------- 10.03/4.15 10.03/4.15 (8) 10.03/4.15 YES 10.31/4.19 EOF