9.02/3.84 YES 10.49/4.31 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 10.49/4.31 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 10.49/4.31 10.49/4.31 10.49/4.31 H-Termination with start terms of the given HASKELL could be proven: 10.49/4.31 10.49/4.31 (0) HASKELL 10.49/4.31 (1) BR [EQUIVALENT, 0 ms] 10.49/4.31 (2) HASKELL 10.49/4.31 (3) COR [EQUIVALENT, 0 ms] 10.49/4.31 (4) HASKELL 10.49/4.31 (5) Narrow [SOUND, 0 ms] 10.49/4.31 (6) AND 10.49/4.31 (7) QDP 10.49/4.31 (8) QDPSizeChangeProof [EQUIVALENT, 0 ms] 10.49/4.31 (9) YES 10.49/4.31 (10) QDP 10.49/4.31 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 10.49/4.31 (12) YES 10.49/4.31 10.49/4.31 10.49/4.31 ---------------------------------------- 10.49/4.31 10.49/4.31 (0) 10.49/4.31 Obligation: 10.49/4.31 mainModule Main 10.49/4.31 module Main where { 10.49/4.31 import qualified Prelude; 10.49/4.31 } 10.49/4.31 10.49/4.31 ---------------------------------------- 10.49/4.31 10.49/4.31 (1) BR (EQUIVALENT) 10.49/4.31 Replaced joker patterns by fresh variables and removed binding patterns. 10.49/4.31 ---------------------------------------- 10.49/4.31 10.49/4.31 (2) 10.49/4.31 Obligation: 10.49/4.31 mainModule Main 10.49/4.31 module Main where { 10.49/4.31 import qualified Prelude; 10.49/4.31 } 10.49/4.31 10.49/4.31 ---------------------------------------- 10.49/4.31 10.49/4.31 (3) COR (EQUIVALENT) 10.49/4.31 Cond Reductions: 10.49/4.31 The following Function with conditions 10.49/4.31 "min x y|x <= yx|otherwisey; 10.49/4.31 " 10.49/4.31 is transformed to 10.49/4.31 "min x y = min2 x y; 10.49/4.31 " 10.49/4.31 "min0 x y True = y; 10.49/4.31 " 10.49/4.31 "min1 x y True = x; 10.49/4.31 min1 x y False = min0 x y otherwise; 10.49/4.31 " 10.49/4.31 "min2 x y = min1 x y (x <= y); 10.49/4.31 " 10.49/4.31 The following Function with conditions 10.49/4.31 "undefined |Falseundefined; 10.49/4.31 " 10.49/4.31 is transformed to 10.49/4.31 "undefined = undefined1; 10.49/4.31 " 10.49/4.31 "undefined0 True = undefined; 10.49/4.31 " 10.49/4.31 "undefined1 = undefined0 False; 10.49/4.31 " 10.49/4.31 10.49/4.31 ---------------------------------------- 10.49/4.31 10.49/4.31 (4) 10.49/4.31 Obligation: 10.49/4.31 mainModule Main 10.49/4.31 module Main where { 10.49/4.31 import qualified Prelude; 10.49/4.31 } 10.49/4.31 10.49/4.31 ---------------------------------------- 10.49/4.31 10.49/4.31 (5) Narrow (SOUND) 10.49/4.31 Haskell To QDPs 10.49/4.31 10.49/4.31 digraph dp_graph { 10.49/4.31 node [outthreshold=100, inthreshold=100];1[label="min",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 10.49/4.31 3[label="min vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 10.49/4.31 4[label="min vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 10.49/4.31 5[label="min2 vx3 vx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 10.49/4.31 6[label="min1 vx3 vx4 (vx3 <= vx4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3]; 10.49/4.31 7[label="min1 vx3 vx4 (compare vx3 vx4 /= GT)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3]; 10.49/4.31 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]; 10.49/4.31 9[label="min1 vx3 vx4 (not (primCmpInt vx3 vx4 == GT))",fontsize=16,color="burlywood",shape="box"];591[label="vx3/Pos vx30",fontsize=10,color="white",style="solid",shape="box"];9 -> 591[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 591 -> 10[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 592[label="vx3/Neg vx30",fontsize=10,color="white",style="solid",shape="box"];9 -> 592[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 592 -> 11[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 10[label="min1 (Pos vx30) vx4 (not (primCmpInt (Pos vx30) vx4 == GT))",fontsize=16,color="burlywood",shape="box"];593[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];10 -> 593[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 593 -> 12[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 594[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 594[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 594 -> 13[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 11[label="min1 (Neg vx30) vx4 (not (primCmpInt (Neg vx30) vx4 == GT))",fontsize=16,color="burlywood",shape="box"];595[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];11 -> 595[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 595 -> 14[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 596[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 596[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 596 -> 15[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 12[label="min1 (Pos (Succ vx300)) vx4 (not (primCmpInt (Pos (Succ vx300)) vx4 == GT))",fontsize=16,color="burlywood",shape="box"];597[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];12 -> 597[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 597 -> 16[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 598[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];12 -> 598[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 598 -> 17[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 13[label="min1 (Pos Zero) vx4 (not (primCmpInt (Pos Zero) vx4 == GT))",fontsize=16,color="burlywood",shape="box"];599[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];13 -> 599[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 599 -> 18[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 600[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];13 -> 600[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 600 -> 19[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 14[label="min1 (Neg (Succ vx300)) vx4 (not (primCmpInt (Neg (Succ vx300)) vx4 == GT))",fontsize=16,color="burlywood",shape="box"];601[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];14 -> 601[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 601 -> 20[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 602[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];14 -> 602[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 602 -> 21[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 15[label="min1 (Neg Zero) vx4 (not (primCmpInt (Neg Zero) vx4 == GT))",fontsize=16,color="burlywood",shape="box"];603[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];15 -> 603[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 603 -> 22[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 604[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];15 -> 604[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 604 -> 23[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 16[label="min1 (Pos (Succ vx300)) (Pos vx40) (not (primCmpInt (Pos (Succ vx300)) (Pos vx40) == GT))",fontsize=16,color="black",shape="box"];16 -> 24[label="",style="solid", color="black", weight=3]; 10.49/4.31 17[label="min1 (Pos (Succ vx300)) (Neg vx40) (not (primCmpInt (Pos (Succ vx300)) (Neg vx40) == GT))",fontsize=16,color="black",shape="box"];17 -> 25[label="",style="solid", color="black", weight=3]; 10.49/4.31 18[label="min1 (Pos Zero) (Pos vx40) (not (primCmpInt (Pos Zero) (Pos vx40) == GT))",fontsize=16,color="burlywood",shape="box"];605[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];18 -> 605[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 605 -> 26[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 606[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];18 -> 606[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 606 -> 27[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 19[label="min1 (Pos Zero) (Neg vx40) (not (primCmpInt (Pos Zero) (Neg vx40) == GT))",fontsize=16,color="burlywood",shape="box"];607[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];19 -> 607[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 607 -> 28[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 608[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];19 -> 608[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 608 -> 29[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 20[label="min1 (Neg (Succ vx300)) (Pos vx40) (not (primCmpInt (Neg (Succ vx300)) (Pos vx40) == GT))",fontsize=16,color="black",shape="box"];20 -> 30[label="",style="solid", color="black", weight=3]; 10.49/4.31 21[label="min1 (Neg (Succ vx300)) (Neg vx40) (not (primCmpInt (Neg (Succ vx300)) (Neg vx40) == GT))",fontsize=16,color="black",shape="box"];21 -> 31[label="",style="solid", color="black", weight=3]; 10.49/4.31 22[label="min1 (Neg Zero) (Pos vx40) (not (primCmpInt (Neg Zero) (Pos vx40) == GT))",fontsize=16,color="burlywood",shape="box"];609[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];22 -> 609[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 609 -> 32[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 610[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];22 -> 610[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 610 -> 33[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 23[label="min1 (Neg Zero) (Neg vx40) (not (primCmpInt (Neg Zero) (Neg vx40) == GT))",fontsize=16,color="burlywood",shape="box"];611[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];23 -> 611[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 611 -> 34[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 612[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];23 -> 612[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 612 -> 35[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 24[label="min1 (Pos (Succ vx300)) (Pos vx40) (not (primCmpNat (Succ vx300) vx40 == GT))",fontsize=16,color="burlywood",shape="box"];613[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];24 -> 613[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 613 -> 36[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 614[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];24 -> 614[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 614 -> 37[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 25[label="min1 (Pos (Succ vx300)) (Neg vx40) (not (GT == GT))",fontsize=16,color="black",shape="box"];25 -> 38[label="",style="solid", color="black", weight=3]; 10.49/4.31 26[label="min1 (Pos Zero) (Pos (Succ vx400)) (not (primCmpInt (Pos Zero) (Pos (Succ vx400)) == GT))",fontsize=16,color="black",shape="box"];26 -> 39[label="",style="solid", color="black", weight=3]; 10.49/4.31 27[label="min1 (Pos Zero) (Pos Zero) (not (primCmpInt (Pos Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];27 -> 40[label="",style="solid", color="black", weight=3]; 10.49/4.31 28[label="min1 (Pos Zero) (Neg (Succ vx400)) (not (primCmpInt (Pos Zero) (Neg (Succ vx400)) == GT))",fontsize=16,color="black",shape="box"];28 -> 41[label="",style="solid", color="black", weight=3]; 10.49/4.31 29[label="min1 (Pos Zero) (Neg Zero) (not (primCmpInt (Pos Zero) (Neg Zero) == GT))",fontsize=16,color="black",shape="box"];29 -> 42[label="",style="solid", color="black", weight=3]; 10.49/4.31 30[label="min1 (Neg (Succ vx300)) (Pos vx40) (not (LT == GT))",fontsize=16,color="black",shape="box"];30 -> 43[label="",style="solid", color="black", weight=3]; 10.49/4.31 31[label="min1 (Neg (Succ vx300)) (Neg vx40) (not (primCmpNat vx40 (Succ vx300) == GT))",fontsize=16,color="burlywood",shape="box"];615[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];31 -> 615[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 615 -> 44[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 616[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];31 -> 616[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 616 -> 45[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 32[label="min1 (Neg Zero) (Pos (Succ vx400)) (not (primCmpInt (Neg Zero) (Pos (Succ vx400)) == GT))",fontsize=16,color="black",shape="box"];32 -> 46[label="",style="solid", color="black", weight=3]; 10.49/4.31 33[label="min1 (Neg Zero) (Pos Zero) (not (primCmpInt (Neg Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];33 -> 47[label="",style="solid", color="black", weight=3]; 10.49/4.31 34[label="min1 (Neg Zero) (Neg (Succ vx400)) (not (primCmpInt (Neg Zero) (Neg (Succ vx400)) == GT))",fontsize=16,color="black",shape="box"];34 -> 48[label="",style="solid", color="black", weight=3]; 10.49/4.31 35[label="min1 (Neg Zero) (Neg Zero) (not (primCmpInt (Neg Zero) (Neg Zero) == GT))",fontsize=16,color="black",shape="box"];35 -> 49[label="",style="solid", color="black", weight=3]; 10.49/4.31 36[label="min1 (Pos (Succ vx300)) (Pos (Succ vx400)) (not (primCmpNat (Succ vx300) (Succ vx400) == GT))",fontsize=16,color="black",shape="box"];36 -> 50[label="",style="solid", color="black", weight=3]; 10.49/4.31 37[label="min1 (Pos (Succ vx300)) (Pos Zero) (not (primCmpNat (Succ vx300) Zero == GT))",fontsize=16,color="black",shape="box"];37 -> 51[label="",style="solid", color="black", weight=3]; 10.49/4.31 38[label="min1 (Pos (Succ vx300)) (Neg vx40) (not True)",fontsize=16,color="black",shape="box"];38 -> 52[label="",style="solid", color="black", weight=3]; 10.49/4.31 39[label="min1 (Pos Zero) (Pos (Succ vx400)) (not (primCmpNat Zero (Succ vx400) == GT))",fontsize=16,color="black",shape="box"];39 -> 53[label="",style="solid", color="black", weight=3]; 10.49/4.31 40[label="min1 (Pos Zero) (Pos Zero) (not (EQ == GT))",fontsize=16,color="black",shape="box"];40 -> 54[label="",style="solid", color="black", weight=3]; 10.49/4.31 41[label="min1 (Pos Zero) (Neg (Succ vx400)) (not (GT == GT))",fontsize=16,color="black",shape="box"];41 -> 55[label="",style="solid", color="black", weight=3]; 10.49/4.31 42[label="min1 (Pos Zero) (Neg Zero) (not (EQ == GT))",fontsize=16,color="black",shape="box"];42 -> 56[label="",style="solid", color="black", weight=3]; 10.49/4.31 43[label="min1 (Neg (Succ vx300)) (Pos vx40) (not False)",fontsize=16,color="black",shape="box"];43 -> 57[label="",style="solid", color="black", weight=3]; 10.49/4.31 44[label="min1 (Neg (Succ vx300)) (Neg (Succ vx400)) (not (primCmpNat (Succ vx400) (Succ vx300) == GT))",fontsize=16,color="black",shape="box"];44 -> 58[label="",style="solid", color="black", weight=3]; 10.49/4.31 45[label="min1 (Neg (Succ vx300)) (Neg Zero) (not (primCmpNat Zero (Succ vx300) == GT))",fontsize=16,color="black",shape="box"];45 -> 59[label="",style="solid", color="black", weight=3]; 10.49/4.31 46[label="min1 (Neg Zero) (Pos (Succ vx400)) (not (LT == GT))",fontsize=16,color="black",shape="box"];46 -> 60[label="",style="solid", color="black", weight=3]; 10.49/4.31 47[label="min1 (Neg Zero) (Pos Zero) (not (EQ == GT))",fontsize=16,color="black",shape="box"];47 -> 61[label="",style="solid", color="black", weight=3]; 10.49/4.31 48[label="min1 (Neg Zero) (Neg (Succ vx400)) (not (primCmpNat (Succ vx400) Zero == GT))",fontsize=16,color="black",shape="box"];48 -> 62[label="",style="solid", color="black", weight=3]; 10.49/4.31 49[label="min1 (Neg Zero) (Neg Zero) (not (EQ == GT))",fontsize=16,color="black",shape="box"];49 -> 63[label="",style="solid", color="black", weight=3]; 10.49/4.31 50 -> 453[label="",style="dashed", color="red", weight=0]; 10.49/4.31 50[label="min1 (Pos (Succ vx300)) (Pos (Succ vx400)) (not (primCmpNat vx300 vx400 == GT))",fontsize=16,color="magenta"];50 -> 454[label="",style="dashed", color="magenta", weight=3]; 10.49/4.31 50 -> 455[label="",style="dashed", color="magenta", weight=3]; 10.49/4.31 50 -> 456[label="",style="dashed", color="magenta", weight=3]; 10.49/4.31 50 -> 457[label="",style="dashed", color="magenta", weight=3]; 10.49/4.31 51[label="min1 (Pos (Succ vx300)) (Pos Zero) (not (GT == GT))",fontsize=16,color="black",shape="box"];51 -> 66[label="",style="solid", color="black", weight=3]; 10.49/4.31 52[label="min1 (Pos (Succ vx300)) (Neg vx40) False",fontsize=16,color="black",shape="box"];52 -> 67[label="",style="solid", color="black", weight=3]; 10.49/4.31 53[label="min1 (Pos Zero) (Pos (Succ vx400)) (not (LT == GT))",fontsize=16,color="black",shape="box"];53 -> 68[label="",style="solid", color="black", weight=3]; 10.49/4.31 54[label="min1 (Pos Zero) (Pos Zero) (not False)",fontsize=16,color="black",shape="box"];54 -> 69[label="",style="solid", color="black", weight=3]; 10.49/4.31 55[label="min1 (Pos Zero) (Neg (Succ vx400)) (not True)",fontsize=16,color="black",shape="box"];55 -> 70[label="",style="solid", color="black", weight=3]; 10.49/4.31 56[label="min1 (Pos Zero) (Neg Zero) (not False)",fontsize=16,color="black",shape="box"];56 -> 71[label="",style="solid", color="black", weight=3]; 10.49/4.31 57[label="min1 (Neg (Succ vx300)) (Pos vx40) True",fontsize=16,color="black",shape="box"];57 -> 72[label="",style="solid", color="black", weight=3]; 10.49/4.31 58 -> 523[label="",style="dashed", color="red", weight=0]; 10.49/4.31 58[label="min1 (Neg (Succ vx300)) (Neg (Succ vx400)) (not (primCmpNat vx400 vx300 == GT))",fontsize=16,color="magenta"];58 -> 524[label="",style="dashed", color="magenta", weight=3]; 10.49/4.31 58 -> 525[label="",style="dashed", color="magenta", weight=3]; 10.49/4.31 58 -> 526[label="",style="dashed", color="magenta", weight=3]; 10.49/4.31 58 -> 527[label="",style="dashed", color="magenta", weight=3]; 10.49/4.31 59[label="min1 (Neg (Succ vx300)) (Neg Zero) (not (LT == GT))",fontsize=16,color="black",shape="box"];59 -> 75[label="",style="solid", color="black", weight=3]; 10.49/4.31 60[label="min1 (Neg Zero) (Pos (Succ vx400)) (not False)",fontsize=16,color="black",shape="box"];60 -> 76[label="",style="solid", color="black", weight=3]; 10.49/4.31 61[label="min1 (Neg Zero) (Pos Zero) (not False)",fontsize=16,color="black",shape="box"];61 -> 77[label="",style="solid", color="black", weight=3]; 10.49/4.31 62[label="min1 (Neg Zero) (Neg (Succ vx400)) (not (GT == GT))",fontsize=16,color="black",shape="box"];62 -> 78[label="",style="solid", color="black", weight=3]; 10.49/4.31 63[label="min1 (Neg Zero) (Neg Zero) (not False)",fontsize=16,color="black",shape="box"];63 -> 79[label="",style="solid", color="black", weight=3]; 10.49/4.31 454[label="vx300",fontsize=16,color="green",shape="box"];455[label="vx400",fontsize=16,color="green",shape="box"];456[label="vx300",fontsize=16,color="green",shape="box"];457[label="vx400",fontsize=16,color="green",shape="box"];453[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not (primCmpNat vx43 vx44 == GT))",fontsize=16,color="burlywood",shape="triangle"];617[label="vx43/Succ vx430",fontsize=10,color="white",style="solid",shape="box"];453 -> 617[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 617 -> 490[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 618[label="vx43/Zero",fontsize=10,color="white",style="solid",shape="box"];453 -> 618[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 618 -> 491[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 66[label="min1 (Pos (Succ vx300)) (Pos Zero) (not True)",fontsize=16,color="black",shape="box"];66 -> 84[label="",style="solid", color="black", weight=3]; 10.49/4.31 67[label="min0 (Pos (Succ vx300)) (Neg vx40) otherwise",fontsize=16,color="black",shape="box"];67 -> 85[label="",style="solid", color="black", weight=3]; 10.49/4.31 68[label="min1 (Pos Zero) (Pos (Succ vx400)) (not False)",fontsize=16,color="black",shape="box"];68 -> 86[label="",style="solid", color="black", weight=3]; 10.49/4.31 69[label="min1 (Pos Zero) (Pos Zero) True",fontsize=16,color="black",shape="box"];69 -> 87[label="",style="solid", color="black", weight=3]; 10.49/4.31 70[label="min1 (Pos Zero) (Neg (Succ vx400)) False",fontsize=16,color="black",shape="box"];70 -> 88[label="",style="solid", color="black", weight=3]; 10.49/4.31 71[label="min1 (Pos Zero) (Neg Zero) True",fontsize=16,color="black",shape="box"];71 -> 89[label="",style="solid", color="black", weight=3]; 10.49/4.31 72[label="Neg (Succ vx300)",fontsize=16,color="green",shape="box"];524[label="vx300",fontsize=16,color="green",shape="box"];525[label="vx400",fontsize=16,color="green",shape="box"];526[label="vx300",fontsize=16,color="green",shape="box"];527[label="vx400",fontsize=16,color="green",shape="box"];523[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not (primCmpNat vx52 vx53 == GT))",fontsize=16,color="burlywood",shape="triangle"];619[label="vx52/Succ vx520",fontsize=10,color="white",style="solid",shape="box"];523 -> 619[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 619 -> 564[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 620[label="vx52/Zero",fontsize=10,color="white",style="solid",shape="box"];523 -> 620[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 620 -> 565[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 75[label="min1 (Neg (Succ vx300)) (Neg Zero) (not False)",fontsize=16,color="black",shape="box"];75 -> 94[label="",style="solid", color="black", weight=3]; 10.49/4.31 76[label="min1 (Neg Zero) (Pos (Succ vx400)) True",fontsize=16,color="black",shape="box"];76 -> 95[label="",style="solid", color="black", weight=3]; 10.49/4.31 77[label="min1 (Neg Zero) (Pos Zero) True",fontsize=16,color="black",shape="box"];77 -> 96[label="",style="solid", color="black", weight=3]; 10.49/4.31 78[label="min1 (Neg Zero) (Neg (Succ vx400)) (not True)",fontsize=16,color="black",shape="box"];78 -> 97[label="",style="solid", color="black", weight=3]; 10.49/4.31 79[label="min1 (Neg Zero) (Neg Zero) True",fontsize=16,color="black",shape="box"];79 -> 98[label="",style="solid", color="black", weight=3]; 10.49/4.31 490[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not (primCmpNat (Succ vx430) vx44 == GT))",fontsize=16,color="burlywood",shape="box"];621[label="vx44/Succ vx440",fontsize=10,color="white",style="solid",shape="box"];490 -> 621[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 621 -> 496[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 622[label="vx44/Zero",fontsize=10,color="white",style="solid",shape="box"];490 -> 622[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 622 -> 497[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 491[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not (primCmpNat Zero vx44 == GT))",fontsize=16,color="burlywood",shape="box"];623[label="vx44/Succ vx440",fontsize=10,color="white",style="solid",shape="box"];491 -> 623[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 623 -> 498[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 624[label="vx44/Zero",fontsize=10,color="white",style="solid",shape="box"];491 -> 624[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 624 -> 499[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 84[label="min1 (Pos (Succ vx300)) (Pos Zero) False",fontsize=16,color="black",shape="box"];84 -> 103[label="",style="solid", color="black", weight=3]; 10.49/4.31 85[label="min0 (Pos (Succ vx300)) (Neg vx40) True",fontsize=16,color="black",shape="box"];85 -> 104[label="",style="solid", color="black", weight=3]; 10.49/4.31 86[label="min1 (Pos Zero) (Pos (Succ vx400)) True",fontsize=16,color="black",shape="box"];86 -> 105[label="",style="solid", color="black", weight=3]; 10.49/4.31 87[label="Pos Zero",fontsize=16,color="green",shape="box"];88[label="min0 (Pos Zero) (Neg (Succ vx400)) otherwise",fontsize=16,color="black",shape="box"];88 -> 106[label="",style="solid", color="black", weight=3]; 10.49/4.31 89[label="Pos Zero",fontsize=16,color="green",shape="box"];564[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not (primCmpNat (Succ vx520) vx53 == GT))",fontsize=16,color="burlywood",shape="box"];625[label="vx53/Succ vx530",fontsize=10,color="white",style="solid",shape="box"];564 -> 625[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 625 -> 568[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 626[label="vx53/Zero",fontsize=10,color="white",style="solid",shape="box"];564 -> 626[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 626 -> 569[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 565[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not (primCmpNat Zero vx53 == GT))",fontsize=16,color="burlywood",shape="box"];627[label="vx53/Succ vx530",fontsize=10,color="white",style="solid",shape="box"];565 -> 627[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 627 -> 570[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 628[label="vx53/Zero",fontsize=10,color="white",style="solid",shape="box"];565 -> 628[label="",style="solid", color="burlywood", weight=9]; 10.49/4.31 628 -> 571[label="",style="solid", color="burlywood", weight=3]; 10.49/4.31 94[label="min1 (Neg (Succ vx300)) (Neg Zero) True",fontsize=16,color="black",shape="box"];94 -> 111[label="",style="solid", color="black", weight=3]; 10.49/4.31 95[label="Neg Zero",fontsize=16,color="green",shape="box"];96[label="Neg Zero",fontsize=16,color="green",shape="box"];97[label="min1 (Neg Zero) (Neg (Succ vx400)) False",fontsize=16,color="black",shape="box"];97 -> 112[label="",style="solid", color="black", weight=3]; 10.49/4.31 98[label="Neg Zero",fontsize=16,color="green",shape="box"];496[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not (primCmpNat (Succ vx430) (Succ vx440) == GT))",fontsize=16,color="black",shape="box"];496 -> 504[label="",style="solid", color="black", weight=3]; 10.49/4.31 497[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not (primCmpNat (Succ vx430) Zero == GT))",fontsize=16,color="black",shape="box"];497 -> 505[label="",style="solid", color="black", weight=3]; 10.49/4.31 498[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not (primCmpNat Zero (Succ vx440) == GT))",fontsize=16,color="black",shape="box"];498 -> 506[label="",style="solid", color="black", weight=3]; 10.49/4.31 499[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not (primCmpNat Zero Zero == GT))",fontsize=16,color="black",shape="box"];499 -> 507[label="",style="solid", color="black", weight=3]; 10.49/4.31 103[label="min0 (Pos (Succ vx300)) (Pos Zero) otherwise",fontsize=16,color="black",shape="box"];103 -> 118[label="",style="solid", color="black", weight=3]; 10.49/4.31 104[label="Neg vx40",fontsize=16,color="green",shape="box"];105[label="Pos Zero",fontsize=16,color="green",shape="box"];106[label="min0 (Pos Zero) (Neg (Succ vx400)) True",fontsize=16,color="black",shape="box"];106 -> 119[label="",style="solid", color="black", weight=3]; 10.49/4.31 568[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not (primCmpNat (Succ vx520) (Succ vx530) == GT))",fontsize=16,color="black",shape="box"];568 -> 574[label="",style="solid", color="black", weight=3]; 10.49/4.31 569[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not (primCmpNat (Succ vx520) Zero == GT))",fontsize=16,color="black",shape="box"];569 -> 575[label="",style="solid", color="black", weight=3]; 10.49/4.31 570[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not (primCmpNat Zero (Succ vx530) == GT))",fontsize=16,color="black",shape="box"];570 -> 576[label="",style="solid", color="black", weight=3]; 10.49/4.31 571[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not (primCmpNat Zero Zero == GT))",fontsize=16,color="black",shape="box"];571 -> 577[label="",style="solid", color="black", weight=3]; 10.49/4.31 111[label="Neg (Succ vx300)",fontsize=16,color="green",shape="box"];112[label="min0 (Neg Zero) (Neg (Succ vx400)) otherwise",fontsize=16,color="black",shape="box"];112 -> 125[label="",style="solid", color="black", weight=3]; 10.49/4.31 504 -> 453[label="",style="dashed", color="red", weight=0]; 10.49/4.31 504[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not (primCmpNat vx430 vx440 == GT))",fontsize=16,color="magenta"];504 -> 512[label="",style="dashed", color="magenta", weight=3]; 10.49/4.31 504 -> 513[label="",style="dashed", color="magenta", weight=3]; 10.49/4.31 505[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not (GT == GT))",fontsize=16,color="black",shape="box"];505 -> 514[label="",style="solid", color="black", weight=3]; 10.49/4.31 506[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not (LT == GT))",fontsize=16,color="black",shape="box"];506 -> 515[label="",style="solid", color="black", weight=3]; 10.49/4.31 507[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not (EQ == GT))",fontsize=16,color="black",shape="box"];507 -> 516[label="",style="solid", color="black", weight=3]; 10.49/4.31 118[label="min0 (Pos (Succ vx300)) (Pos Zero) True",fontsize=16,color="black",shape="box"];118 -> 133[label="",style="solid", color="black", weight=3]; 10.49/4.31 119[label="Neg (Succ vx400)",fontsize=16,color="green",shape="box"];574 -> 523[label="",style="dashed", color="red", weight=0]; 10.49/4.31 574[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not (primCmpNat vx520 vx530 == GT))",fontsize=16,color="magenta"];574 -> 579[label="",style="dashed", color="magenta", weight=3]; 10.49/4.31 574 -> 580[label="",style="dashed", color="magenta", weight=3]; 10.49/4.31 575[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not (GT == GT))",fontsize=16,color="black",shape="box"];575 -> 581[label="",style="solid", color="black", weight=3]; 10.49/4.31 576[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not (LT == GT))",fontsize=16,color="black",shape="box"];576 -> 582[label="",style="solid", color="black", weight=3]; 10.49/4.31 577[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not (EQ == GT))",fontsize=16,color="black",shape="box"];577 -> 583[label="",style="solid", color="black", weight=3]; 10.49/4.31 125[label="min0 (Neg Zero) (Neg (Succ vx400)) True",fontsize=16,color="black",shape="box"];125 -> 141[label="",style="solid", color="black", weight=3]; 10.49/4.31 512[label="vx430",fontsize=16,color="green",shape="box"];513[label="vx440",fontsize=16,color="green",shape="box"];514[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not True)",fontsize=16,color="black",shape="box"];514 -> 566[label="",style="solid", color="black", weight=3]; 10.49/4.31 515[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not False)",fontsize=16,color="black",shape="triangle"];515 -> 567[label="",style="solid", color="black", weight=3]; 10.49/4.31 516 -> 515[label="",style="dashed", color="red", weight=0]; 10.49/4.31 516[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) (not False)",fontsize=16,color="magenta"];133[label="Pos Zero",fontsize=16,color="green",shape="box"];579[label="vx530",fontsize=16,color="green",shape="box"];580[label="vx520",fontsize=16,color="green",shape="box"];581[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not True)",fontsize=16,color="black",shape="box"];581 -> 585[label="",style="solid", color="black", weight=3]; 10.49/4.31 582[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not False)",fontsize=16,color="black",shape="triangle"];582 -> 586[label="",style="solid", color="black", weight=3]; 10.49/4.31 583 -> 582[label="",style="dashed", color="red", weight=0]; 10.49/4.31 583[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) (not False)",fontsize=16,color="magenta"];141[label="Neg (Succ vx400)",fontsize=16,color="green",shape="box"];566[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) False",fontsize=16,color="black",shape="box"];566 -> 572[label="",style="solid", color="black", weight=3]; 10.49/4.31 567[label="min1 (Pos (Succ vx41)) (Pos (Succ vx42)) True",fontsize=16,color="black",shape="box"];567 -> 573[label="",style="solid", color="black", weight=3]; 10.49/4.31 585[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) False",fontsize=16,color="black",shape="box"];585 -> 587[label="",style="solid", color="black", weight=3]; 10.49/4.31 586[label="min1 (Neg (Succ vx50)) (Neg (Succ vx51)) True",fontsize=16,color="black",shape="box"];586 -> 588[label="",style="solid", color="black", weight=3]; 10.49/4.31 572[label="min0 (Pos (Succ vx41)) (Pos (Succ vx42)) otherwise",fontsize=16,color="black",shape="box"];572 -> 578[label="",style="solid", color="black", weight=3]; 10.49/4.31 573[label="Pos (Succ vx41)",fontsize=16,color="green",shape="box"];587[label="min0 (Neg (Succ vx50)) (Neg (Succ vx51)) otherwise",fontsize=16,color="black",shape="box"];587 -> 589[label="",style="solid", color="black", weight=3]; 10.49/4.31 588[label="Neg (Succ vx50)",fontsize=16,color="green",shape="box"];578[label="min0 (Pos (Succ vx41)) (Pos (Succ vx42)) True",fontsize=16,color="black",shape="box"];578 -> 584[label="",style="solid", color="black", weight=3]; 10.49/4.31 589[label="min0 (Neg (Succ vx50)) (Neg (Succ vx51)) True",fontsize=16,color="black",shape="box"];589 -> 590[label="",style="solid", color="black", weight=3]; 10.49/4.31 584[label="Pos (Succ vx42)",fontsize=16,color="green",shape="box"];590[label="Neg (Succ vx51)",fontsize=16,color="green",shape="box"];} 10.49/4.31 10.49/4.31 ---------------------------------------- 10.49/4.31 10.49/4.31 (6) 10.49/4.31 Complex Obligation (AND) 10.49/4.31 10.49/4.31 ---------------------------------------- 10.49/4.31 10.49/4.31 (7) 10.49/4.31 Obligation: 10.49/4.31 Q DP problem: 10.49/4.31 The TRS P consists of the following rules: 10.49/4.31 10.49/4.31 new_min1(vx50, vx51, Succ(vx520), Succ(vx530)) -> new_min1(vx50, vx51, vx520, vx530) 10.49/4.31 10.49/4.31 R is empty. 10.49/4.31 Q is empty. 10.49/4.31 We have to consider all minimal (P,Q,R)-chains. 10.49/4.31 ---------------------------------------- 10.49/4.31 10.49/4.31 (8) QDPSizeChangeProof (EQUIVALENT) 10.49/4.31 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. 10.49/4.31 10.49/4.31 From the DPs we obtained the following set of size-change graphs: 10.49/4.31 *new_min1(vx50, vx51, Succ(vx520), Succ(vx530)) -> new_min1(vx50, vx51, vx520, vx530) 10.49/4.31 The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 10.49/4.31 10.49/4.31 10.49/4.31 ---------------------------------------- 10.49/4.31 10.49/4.31 (9) 10.49/4.31 YES 10.49/4.31 10.49/4.31 ---------------------------------------- 10.49/4.31 10.49/4.31 (10) 10.49/4.31 Obligation: 10.49/4.31 Q DP problem: 10.49/4.31 The TRS P consists of the following rules: 10.49/4.31 10.49/4.31 new_min10(vx41, vx42, Succ(vx430), Succ(vx440)) -> new_min10(vx41, vx42, vx430, vx440) 10.49/4.31 10.49/4.31 R is empty. 10.49/4.31 Q is empty. 10.49/4.31 We have to consider all minimal (P,Q,R)-chains. 10.49/4.31 ---------------------------------------- 10.49/4.31 10.49/4.31 (11) QDPSizeChangeProof (EQUIVALENT) 10.49/4.31 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. 10.49/4.31 10.49/4.31 From the DPs we obtained the following set of size-change graphs: 10.49/4.31 *new_min10(vx41, vx42, Succ(vx430), Succ(vx440)) -> new_min10(vx41, vx42, vx430, vx440) 10.49/4.31 The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 10.49/4.31 10.49/4.31 10.49/4.31 ---------------------------------------- 10.49/4.31 10.49/4.31 (12) 10.49/4.31 YES 10.78/4.35 EOF