8.13/3.71 YES 9.71/4.14 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 9.71/4.14 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 9.71/4.14 9.71/4.14 9.71/4.14 H-Termination with start terms of the given HASKELL could be proven: 9.71/4.14 9.71/4.14 (0) HASKELL 9.71/4.14 (1) BR [EQUIVALENT, 0 ms] 9.71/4.14 (2) HASKELL 9.71/4.14 (3) COR [EQUIVALENT, 0 ms] 9.71/4.14 (4) HASKELL 9.71/4.14 (5) Narrow [SOUND, 0 ms] 9.71/4.14 (6) QDP 9.71/4.14 (7) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.71/4.14 (8) YES 9.71/4.14 9.71/4.14 9.71/4.14 ---------------------------------------- 9.71/4.14 9.71/4.14 (0) 9.71/4.14 Obligation: 9.71/4.14 mainModule Main 9.71/4.14 module Main where { 9.71/4.14 import qualified Prelude; 9.71/4.14 } 9.71/4.14 9.71/4.14 ---------------------------------------- 9.71/4.14 9.71/4.14 (1) BR (EQUIVALENT) 9.71/4.14 Replaced joker patterns by fresh variables and removed binding patterns. 9.71/4.14 ---------------------------------------- 9.71/4.14 9.71/4.14 (2) 9.71/4.14 Obligation: 9.71/4.14 mainModule Main 9.71/4.14 module Main where { 9.71/4.14 import qualified Prelude; 9.71/4.14 } 9.71/4.14 9.71/4.14 ---------------------------------------- 9.71/4.14 9.71/4.14 (3) COR (EQUIVALENT) 9.71/4.14 Cond Reductions: 9.71/4.14 The following Function with conditions 9.71/4.14 "min x y|x <= yx|otherwisey; 9.71/4.14 " 9.71/4.14 is transformed to 9.71/4.14 "min x y = min2 x y; 9.71/4.14 " 9.71/4.14 "min0 x y True = y; 9.71/4.14 " 9.71/4.14 "min1 x y True = x; 9.71/4.14 min1 x y False = min0 x y otherwise; 9.71/4.14 " 9.71/4.14 "min2 x y = min1 x y (x <= y); 9.71/4.14 " 9.71/4.14 The following Function with conditions 9.71/4.14 "undefined |Falseundefined; 9.71/4.14 " 9.71/4.14 is transformed to 9.71/4.14 "undefined = undefined1; 9.71/4.14 " 9.71/4.14 "undefined0 True = undefined; 9.71/4.14 " 9.71/4.14 "undefined1 = undefined0 False; 9.71/4.14 " 9.71/4.14 9.71/4.14 ---------------------------------------- 9.71/4.14 9.71/4.14 (4) 9.71/4.14 Obligation: 9.71/4.14 mainModule Main 9.71/4.14 module Main where { 9.71/4.14 import qualified Prelude; 9.71/4.14 } 9.71/4.14 9.71/4.14 ---------------------------------------- 9.71/4.14 9.71/4.14 (5) Narrow (SOUND) 9.71/4.14 Haskell To QDPs 9.71/4.14 9.71/4.14 digraph dp_graph { 9.71/4.14 node [outthreshold=100, inthreshold=100];1[label="min",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 9.71/4.14 3[label="min vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 9.71/4.14 4[label="min vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 9.71/4.14 5[label="min2 vx3 vx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 9.71/4.14 6[label="min1 vx3 vx4 (vx3 <= vx4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3]; 9.71/4.14 7[label="min1 vx3 vx4 (compare vx3 vx4 /= GT)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3]; 9.71/4.14 8[label="min1 vx3 vx4 (not (compare vx3 vx4 == GT))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3]; 9.71/4.14 9[label="min1 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]; 9.71/4.14 302 -> 10[label="",style="solid", color="burlywood", weight=3]; 9.71/4.14 10[label="min1 (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]; 9.71/4.14 303 -> 11[label="",style="solid", color="burlywood", weight=3]; 9.71/4.14 11[label="min1 (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]; 9.71/4.14 12[label="min1 (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]; 9.71/4.14 304 -> 13[label="",style="solid", color="burlywood", weight=3]; 9.71/4.14 305[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];12 -> 305[label="",style="solid", color="burlywood", weight=9]; 9.71/4.14 305 -> 14[label="",style="solid", color="burlywood", weight=3]; 9.71/4.14 13[label="min1 (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]; 9.71/4.14 306 -> 15[label="",style="solid", color="burlywood", weight=3]; 9.71/4.14 307[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 307[label="",style="solid", color="burlywood", weight=9]; 9.71/4.14 307 -> 16[label="",style="solid", color="burlywood", weight=3]; 9.71/4.14 14[label="min1 (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]; 9.71/4.14 308 -> 17[label="",style="solid", color="burlywood", weight=3]; 9.71/4.14 309[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];14 -> 309[label="",style="solid", color="burlywood", weight=9]; 9.71/4.14 309 -> 18[label="",style="solid", color="burlywood", weight=3]; 9.71/4.14 15[label="min1 (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]; 9.71/4.14 16[label="min1 (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]; 9.71/4.14 17[label="min1 (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]; 9.71/4.14 18[label="min1 (Char Zero) (Char Zero) (not (primCmpNat Zero Zero == GT))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3]; 9.71/4.14 19 -> 240[label="",style="dashed", color="red", weight=0]; 9.71/4.14 19[label="min1 (Char (Succ vx300)) (Char (Succ vx400)) (not (primCmpNat vx300 vx400 == GT))",fontsize=16,color="magenta"];19 -> 241[label="",style="dashed", color="magenta", weight=3]; 9.71/4.14 19 -> 242[label="",style="dashed", color="magenta", weight=3]; 9.71/4.14 19 -> 243[label="",style="dashed", color="magenta", weight=3]; 9.71/4.14 19 -> 244[label="",style="dashed", color="magenta", weight=3]; 9.71/4.14 20[label="min1 (Char (Succ vx300)) (Char Zero) (not (GT == GT))",fontsize=16,color="black",shape="box"];20 -> 25[label="",style="solid", color="black", weight=3]; 9.71/4.14 21[label="min1 (Char Zero) (Char (Succ vx400)) (not (LT == GT))",fontsize=16,color="black",shape="box"];21 -> 26[label="",style="solid", color="black", weight=3]; 9.71/4.14 22[label="min1 (Char Zero) (Char Zero) (not (EQ == GT))",fontsize=16,color="black",shape="box"];22 -> 27[label="",style="solid", color="black", weight=3]; 9.71/4.14 241[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="min1 (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]; 9.71/4.14 310 -> 281[label="",style="solid", color="burlywood", weight=3]; 9.71/4.14 311[label="vx29/Zero",fontsize=10,color="white",style="solid",shape="box"];240 -> 311[label="",style="solid", color="burlywood", weight=9]; 9.71/4.14 311 -> 282[label="",style="solid", color="burlywood", weight=3]; 9.71/4.14 25[label="min1 (Char (Succ vx300)) (Char Zero) (not True)",fontsize=16,color="black",shape="box"];25 -> 32[label="",style="solid", color="black", weight=3]; 9.71/4.14 26[label="min1 (Char Zero) (Char (Succ vx400)) (not False)",fontsize=16,color="black",shape="box"];26 -> 33[label="",style="solid", color="black", weight=3]; 9.71/4.14 27[label="min1 (Char Zero) (Char Zero) (not False)",fontsize=16,color="black",shape="box"];27 -> 34[label="",style="solid", color="black", weight=3]; 9.71/4.14 281[label="min1 (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]; 9.71/4.14 312 -> 283[label="",style="solid", color="burlywood", weight=3]; 9.71/4.14 313[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];281 -> 313[label="",style="solid", color="burlywood", weight=9]; 9.71/4.14 313 -> 284[label="",style="solid", color="burlywood", weight=3]; 9.71/4.14 282[label="min1 (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]; 9.71/4.14 314 -> 285[label="",style="solid", color="burlywood", weight=3]; 9.71/4.14 315[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];282 -> 315[label="",style="solid", color="burlywood", weight=9]; 9.71/4.14 315 -> 286[label="",style="solid", color="burlywood", weight=3]; 9.71/4.14 32[label="min1 (Char (Succ vx300)) (Char Zero) False",fontsize=16,color="black",shape="box"];32 -> 39[label="",style="solid", color="black", weight=3]; 9.71/4.14 33[label="min1 (Char Zero) (Char (Succ vx400)) True",fontsize=16,color="black",shape="box"];33 -> 40[label="",style="solid", color="black", weight=3]; 9.71/4.14 34[label="min1 (Char Zero) (Char Zero) True",fontsize=16,color="black",shape="box"];34 -> 41[label="",style="solid", color="black", weight=3]; 9.71/4.14 283[label="min1 (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]; 9.71/4.14 284[label="min1 (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]; 9.71/4.14 285[label="min1 (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]; 9.71/4.14 286[label="min1 (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]; 9.71/4.14 39[label="min0 (Char (Succ vx300)) (Char Zero) otherwise",fontsize=16,color="black",shape="box"];39 -> 47[label="",style="solid", color="black", weight=3]; 9.71/4.14 40[label="Char Zero",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]; 9.71/4.14 287[label="min1 (Char (Succ vx27)) (Char (Succ vx28)) (not (primCmpNat vx290 vx300 == GT))",fontsize=16,color="magenta"];287 -> 291[label="",style="dashed", color="magenta", weight=3]; 9.71/4.14 287 -> 292[label="",style="dashed", color="magenta", weight=3]; 9.71/4.14 288[label="min1 (Char (Succ vx27)) (Char (Succ vx28)) (not (GT == GT))",fontsize=16,color="black",shape="box"];288 -> 293[label="",style="solid", color="black", weight=3]; 9.71/4.14 289[label="min1 (Char (Succ vx27)) (Char (Succ vx28)) (not (LT == GT))",fontsize=16,color="black",shape="box"];289 -> 294[label="",style="solid", color="black", weight=3]; 9.71/4.14 290[label="min1 (Char (Succ vx27)) (Char (Succ vx28)) (not (EQ == GT))",fontsize=16,color="black",shape="box"];290 -> 295[label="",style="solid", color="black", weight=3]; 9.71/4.14 47[label="min0 (Char (Succ vx300)) (Char Zero) True",fontsize=16,color="black",shape="box"];47 -> 55[label="",style="solid", color="black", weight=3]; 9.71/4.14 291[label="vx290",fontsize=16,color="green",shape="box"];292[label="vx300",fontsize=16,color="green",shape="box"];293[label="min1 (Char (Succ vx27)) (Char (Succ vx28)) (not True)",fontsize=16,color="black",shape="box"];293 -> 296[label="",style="solid", color="black", weight=3]; 9.71/4.14 294[label="min1 (Char (Succ vx27)) (Char (Succ vx28)) (not False)",fontsize=16,color="black",shape="triangle"];294 -> 297[label="",style="solid", color="black", weight=3]; 9.71/4.14 295 -> 294[label="",style="dashed", color="red", weight=0]; 9.71/4.14 295[label="min1 (Char (Succ vx27)) (Char (Succ vx28)) (not False)",fontsize=16,color="magenta"];55[label="Char Zero",fontsize=16,color="green",shape="box"];296[label="min1 (Char (Succ vx27)) (Char (Succ vx28)) False",fontsize=16,color="black",shape="box"];296 -> 298[label="",style="solid", color="black", weight=3]; 9.71/4.14 297[label="min1 (Char (Succ vx27)) (Char (Succ vx28)) True",fontsize=16,color="black",shape="box"];297 -> 299[label="",style="solid", color="black", weight=3]; 9.71/4.14 298[label="min0 (Char (Succ vx27)) (Char (Succ vx28)) otherwise",fontsize=16,color="black",shape="box"];298 -> 300[label="",style="solid", color="black", weight=3]; 9.71/4.14 299[label="Char (Succ vx27)",fontsize=16,color="green",shape="box"];300[label="min0 (Char (Succ vx27)) (Char (Succ vx28)) True",fontsize=16,color="black",shape="box"];300 -> 301[label="",style="solid", color="black", weight=3]; 9.71/4.14 301[label="Char (Succ vx28)",fontsize=16,color="green",shape="box"];} 9.71/4.14 9.71/4.14 ---------------------------------------- 9.71/4.14 9.71/4.14 (6) 9.71/4.14 Obligation: 9.71/4.14 Q DP problem: 9.71/4.14 The TRS P consists of the following rules: 9.71/4.14 9.71/4.14 new_min1(vx27, vx28, Succ(vx290), Succ(vx300)) -> new_min1(vx27, vx28, vx290, vx300) 9.71/4.14 9.71/4.14 R is empty. 9.71/4.14 Q is empty. 9.71/4.14 We have to consider all minimal (P,Q,R)-chains. 9.71/4.14 ---------------------------------------- 9.71/4.14 9.71/4.14 (7) QDPSizeChangeProof (EQUIVALENT) 9.71/4.14 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. 9.71/4.14 9.71/4.14 From the DPs we obtained the following set of size-change graphs: 9.71/4.14 *new_min1(vx27, vx28, Succ(vx290), Succ(vx300)) -> new_min1(vx27, vx28, vx290, vx300) 9.71/4.14 The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 9.71/4.14 9.71/4.14 9.71/4.14 ---------------------------------------- 9.71/4.14 9.71/4.14 (8) 9.71/4.14 YES 9.87/4.18 EOF