10.55/4.48	YES
12.32/4.96	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.32/4.96	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.32/4.96	
12.32/4.96	
12.32/4.96	H-Termination with start terms of the given HASKELL could be proven:
12.32/4.96	
12.32/4.96	(0) HASKELL
12.32/4.96	(1) BR [EQUIVALENT, 0 ms]
12.32/4.96	(2) HASKELL
12.32/4.96	(3) COR [EQUIVALENT, 21 ms]
12.32/4.96	(4) HASKELL
12.32/4.96	(5) Narrow [SOUND, 0 ms]
12.32/4.96	(6) AND
12.32/4.96	    (7) QDP
12.32/4.96	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.32/4.96	        (9) YES
12.32/4.96	    (10) QDP
12.32/4.96	        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.32/4.96	        (12) YES
12.32/4.96	
12.32/4.96	
12.32/4.96	----------------------------------------
12.32/4.96	
12.32/4.96	(0)
12.32/4.96	Obligation:
12.32/4.96	mainModule Main 
12.32/4.96	module Maybe where {
12.32/4.96	import qualified List;
12.32/4.96	import qualified Main;
12.32/4.96	import qualified Prelude;
12.32/4.96	}
12.32/4.96	module List where {
12.32/4.96	import qualified Main;
12.32/4.96	import qualified Maybe;
12.32/4.96	import qualified Prelude;
12.32/4.96	isPrefixOf :: Eq a => [a]  ->  [a]  ->  Bool;
12.32/4.96	isPrefixOf [] _ = True;
12.32/4.96	isPrefixOf _ [] = False;
12.32/4.96	isPrefixOf (x : xs) (y : ys) = x == y && isPrefixOf xs ys;
12.32/4.96	
12.32/4.96	}
12.32/4.96	module Main where {
12.32/4.96	import qualified List;
12.32/4.96	import qualified Maybe;
12.32/4.96	import qualified Prelude;
12.32/4.96	}
12.32/4.96	
12.32/4.96	----------------------------------------
12.32/4.96	
12.32/4.96	(1) BR (EQUIVALENT)
12.32/4.96	Replaced joker patterns by fresh variables and removed binding patterns.
12.32/4.96	----------------------------------------
12.32/4.96	
12.32/4.96	(2)
12.32/4.96	Obligation:
12.32/4.96	mainModule Main 
12.32/4.96	module Maybe where {
12.32/4.96	import qualified List;
12.32/4.96	import qualified Main;
12.32/4.96	import qualified Prelude;
12.32/4.96	}
12.32/4.96	module List where {
12.32/4.96	import qualified Main;
12.32/4.96	import qualified Maybe;
12.32/4.96	import qualified Prelude;
12.32/4.96	isPrefixOf :: Eq a => [a]  ->  [a]  ->  Bool;
12.32/4.96	isPrefixOf [] wu = True;
12.32/4.96	isPrefixOf wv [] = False;
12.32/4.96	isPrefixOf (x : xs) (y : ys) = x == y && isPrefixOf xs ys;
12.32/4.96	
12.32/4.96	}
12.32/4.96	module Main where {
12.32/4.96	import qualified List;
12.32/4.96	import qualified Maybe;
12.32/4.96	import qualified Prelude;
12.32/4.96	}
12.32/4.96	
12.32/4.96	----------------------------------------
12.32/4.96	
12.32/4.96	(3) COR (EQUIVALENT)
12.32/4.96	Cond Reductions:
12.32/4.96	The following Function with conditions
12.32/4.96	"undefined |Falseundefined;
12.32/4.96	"
12.32/4.96	is transformed to
12.32/4.96	"undefined  = undefined1;
12.32/4.96	"
12.32/4.96	"undefined0 True = undefined;
12.32/4.96	"
12.32/4.96	"undefined1  = undefined0 False;
12.32/4.96	"
12.32/4.96	
12.32/4.96	----------------------------------------
12.32/4.96	
12.32/4.96	(4)
12.32/4.96	Obligation:
12.32/4.96	mainModule Main 
12.32/4.96	module Maybe where {
12.32/4.96	import qualified List;
12.32/4.96	import qualified Main;
12.32/4.96	import qualified Prelude;
12.32/4.96	}
12.32/4.96	module List where {
12.32/4.96	import qualified Main;
12.32/4.96	import qualified Maybe;
12.32/4.96	import qualified Prelude;
12.32/4.96	isPrefixOf :: Eq a => [a]  ->  [a]  ->  Bool;
12.32/4.96	isPrefixOf [] wu = True;
12.32/4.96	isPrefixOf wv [] = False;
12.32/4.96	isPrefixOf (x : xs) (y : ys) = x == y && isPrefixOf xs ys;
12.32/4.96	
12.32/4.96	}
12.32/4.96	module Main where {
12.32/4.96	import qualified List;
12.32/4.96	import qualified Maybe;
12.32/4.96	import qualified Prelude;
12.32/4.96	}
12.32/4.96	
12.32/4.96	----------------------------------------
12.32/4.96	
12.32/4.96	(5) Narrow (SOUND)
12.32/4.96	Haskell To QDPs
12.32/4.96	
12.32/4.96	digraph dp_graph {
12.32/4.96	node [outthreshold=100, inthreshold=100];1[label="List.isPrefixOf",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.32/4.96	3[label="List.isPrefixOf ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.32/4.96	4[label="List.isPrefixOf ww3 ww4",fontsize=16,color="burlywood",shape="triangle"];73[label="ww3/ww30 : ww31",fontsize=10,color="white",style="solid",shape="box"];4 -> 73[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	73 -> 5[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	74[label="ww3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 74[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	74 -> 6[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	5[label="List.isPrefixOf (ww30 : ww31) ww4",fontsize=16,color="burlywood",shape="box"];75[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];5 -> 75[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	75 -> 7[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	76[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 76[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	76 -> 8[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	6[label="List.isPrefixOf [] ww4",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
12.32/4.96	7[label="List.isPrefixOf (ww30 : ww31) (ww40 : ww41)",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
12.32/4.96	8[label="List.isPrefixOf (ww30 : ww31) []",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
12.32/4.96	9[label="True",fontsize=16,color="green",shape="box"];10 -> 12[label="",style="dashed", color="red", weight=0];
12.32/4.96	10[label="ww30 == ww40 && List.isPrefixOf ww31 ww41",fontsize=16,color="magenta"];10 -> 13[label="",style="dashed", color="magenta", weight=3];
12.32/4.96	11[label="False",fontsize=16,color="green",shape="box"];13 -> 4[label="",style="dashed", color="red", weight=0];
12.32/4.96	13[label="List.isPrefixOf ww31 ww41",fontsize=16,color="magenta"];13 -> 14[label="",style="dashed", color="magenta", weight=3];
12.32/4.96	13 -> 15[label="",style="dashed", color="magenta", weight=3];
12.32/4.96	12[label="ww30 == ww40 && ww5",fontsize=16,color="black",shape="triangle"];12 -> 16[label="",style="solid", color="black", weight=3];
12.32/4.96	14[label="ww31",fontsize=16,color="green",shape="box"];15[label="ww41",fontsize=16,color="green",shape="box"];16[label="primEqInt ww30 ww40 && ww5",fontsize=16,color="burlywood",shape="box"];77[label="ww30/Pos ww300",fontsize=10,color="white",style="solid",shape="box"];16 -> 77[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	77 -> 17[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	78[label="ww30/Neg ww300",fontsize=10,color="white",style="solid",shape="box"];16 -> 78[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	78 -> 18[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	17[label="primEqInt (Pos ww300) ww40 && ww5",fontsize=16,color="burlywood",shape="box"];79[label="ww300/Succ ww3000",fontsize=10,color="white",style="solid",shape="box"];17 -> 79[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	79 -> 19[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	80[label="ww300/Zero",fontsize=10,color="white",style="solid",shape="box"];17 -> 80[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	80 -> 20[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	18[label="primEqInt (Neg ww300) ww40 && ww5",fontsize=16,color="burlywood",shape="box"];81[label="ww300/Succ ww3000",fontsize=10,color="white",style="solid",shape="box"];18 -> 81[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	81 -> 21[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	82[label="ww300/Zero",fontsize=10,color="white",style="solid",shape="box"];18 -> 82[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	82 -> 22[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	19[label="primEqInt (Pos (Succ ww3000)) ww40 && ww5",fontsize=16,color="burlywood",shape="box"];83[label="ww40/Pos ww400",fontsize=10,color="white",style="solid",shape="box"];19 -> 83[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	83 -> 23[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	84[label="ww40/Neg ww400",fontsize=10,color="white",style="solid",shape="box"];19 -> 84[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	84 -> 24[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	20[label="primEqInt (Pos Zero) ww40 && ww5",fontsize=16,color="burlywood",shape="box"];85[label="ww40/Pos ww400",fontsize=10,color="white",style="solid",shape="box"];20 -> 85[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	85 -> 25[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	86[label="ww40/Neg ww400",fontsize=10,color="white",style="solid",shape="box"];20 -> 86[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	86 -> 26[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	21[label="primEqInt (Neg (Succ ww3000)) ww40 && ww5",fontsize=16,color="burlywood",shape="box"];87[label="ww40/Pos ww400",fontsize=10,color="white",style="solid",shape="box"];21 -> 87[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	87 -> 27[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	88[label="ww40/Neg ww400",fontsize=10,color="white",style="solid",shape="box"];21 -> 88[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	88 -> 28[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	22[label="primEqInt (Neg Zero) ww40 && ww5",fontsize=16,color="burlywood",shape="box"];89[label="ww40/Pos ww400",fontsize=10,color="white",style="solid",shape="box"];22 -> 89[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	89 -> 29[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	90[label="ww40/Neg ww400",fontsize=10,color="white",style="solid",shape="box"];22 -> 90[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	90 -> 30[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	23[label="primEqInt (Pos (Succ ww3000)) (Pos ww400) && ww5",fontsize=16,color="burlywood",shape="box"];91[label="ww400/Succ ww4000",fontsize=10,color="white",style="solid",shape="box"];23 -> 91[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	91 -> 31[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	92[label="ww400/Zero",fontsize=10,color="white",style="solid",shape="box"];23 -> 92[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	92 -> 32[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	24[label="primEqInt (Pos (Succ ww3000)) (Neg ww400) && ww5",fontsize=16,color="black",shape="box"];24 -> 33[label="",style="solid", color="black", weight=3];
12.32/4.96	25[label="primEqInt (Pos Zero) (Pos ww400) && ww5",fontsize=16,color="burlywood",shape="box"];93[label="ww400/Succ ww4000",fontsize=10,color="white",style="solid",shape="box"];25 -> 93[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	93 -> 34[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	94[label="ww400/Zero",fontsize=10,color="white",style="solid",shape="box"];25 -> 94[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	94 -> 35[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	26[label="primEqInt (Pos Zero) (Neg ww400) && ww5",fontsize=16,color="burlywood",shape="box"];95[label="ww400/Succ ww4000",fontsize=10,color="white",style="solid",shape="box"];26 -> 95[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	95 -> 36[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	96[label="ww400/Zero",fontsize=10,color="white",style="solid",shape="box"];26 -> 96[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	96 -> 37[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	27[label="primEqInt (Neg (Succ ww3000)) (Pos ww400) && ww5",fontsize=16,color="black",shape="box"];27 -> 38[label="",style="solid", color="black", weight=3];
12.32/4.96	28[label="primEqInt (Neg (Succ ww3000)) (Neg ww400) && ww5",fontsize=16,color="burlywood",shape="box"];97[label="ww400/Succ ww4000",fontsize=10,color="white",style="solid",shape="box"];28 -> 97[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	97 -> 39[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	98[label="ww400/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 98[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	98 -> 40[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	29[label="primEqInt (Neg Zero) (Pos ww400) && ww5",fontsize=16,color="burlywood",shape="box"];99[label="ww400/Succ ww4000",fontsize=10,color="white",style="solid",shape="box"];29 -> 99[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	99 -> 41[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	100[label="ww400/Zero",fontsize=10,color="white",style="solid",shape="box"];29 -> 100[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	100 -> 42[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	30[label="primEqInt (Neg Zero) (Neg ww400) && ww5",fontsize=16,color="burlywood",shape="box"];101[label="ww400/Succ ww4000",fontsize=10,color="white",style="solid",shape="box"];30 -> 101[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	101 -> 43[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	102[label="ww400/Zero",fontsize=10,color="white",style="solid",shape="box"];30 -> 102[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	102 -> 44[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	31[label="primEqInt (Pos (Succ ww3000)) (Pos (Succ ww4000)) && ww5",fontsize=16,color="black",shape="box"];31 -> 45[label="",style="solid", color="black", weight=3];
12.32/4.96	32[label="primEqInt (Pos (Succ ww3000)) (Pos Zero) && ww5",fontsize=16,color="black",shape="box"];32 -> 46[label="",style="solid", color="black", weight=3];
12.32/4.96	33[label="False && ww5",fontsize=16,color="black",shape="triangle"];33 -> 47[label="",style="solid", color="black", weight=3];
12.32/4.96	34[label="primEqInt (Pos Zero) (Pos (Succ ww4000)) && ww5",fontsize=16,color="black",shape="box"];34 -> 48[label="",style="solid", color="black", weight=3];
12.32/4.96	35[label="primEqInt (Pos Zero) (Pos Zero) && ww5",fontsize=16,color="black",shape="box"];35 -> 49[label="",style="solid", color="black", weight=3];
12.32/4.96	36[label="primEqInt (Pos Zero) (Neg (Succ ww4000)) && ww5",fontsize=16,color="black",shape="box"];36 -> 50[label="",style="solid", color="black", weight=3];
12.32/4.96	37[label="primEqInt (Pos Zero) (Neg Zero) && ww5",fontsize=16,color="black",shape="box"];37 -> 51[label="",style="solid", color="black", weight=3];
12.32/4.96	38 -> 33[label="",style="dashed", color="red", weight=0];
12.32/4.96	38[label="False && ww5",fontsize=16,color="magenta"];39[label="primEqInt (Neg (Succ ww3000)) (Neg (Succ ww4000)) && ww5",fontsize=16,color="black",shape="box"];39 -> 52[label="",style="solid", color="black", weight=3];
12.32/4.96	40[label="primEqInt (Neg (Succ ww3000)) (Neg Zero) && ww5",fontsize=16,color="black",shape="box"];40 -> 53[label="",style="solid", color="black", weight=3];
12.32/4.96	41[label="primEqInt (Neg Zero) (Pos (Succ ww4000)) && ww5",fontsize=16,color="black",shape="box"];41 -> 54[label="",style="solid", color="black", weight=3];
12.32/4.96	42[label="primEqInt (Neg Zero) (Pos Zero) && ww5",fontsize=16,color="black",shape="box"];42 -> 55[label="",style="solid", color="black", weight=3];
12.32/4.96	43[label="primEqInt (Neg Zero) (Neg (Succ ww4000)) && ww5",fontsize=16,color="black",shape="box"];43 -> 56[label="",style="solid", color="black", weight=3];
12.32/4.96	44[label="primEqInt (Neg Zero) (Neg Zero) && ww5",fontsize=16,color="black",shape="box"];44 -> 57[label="",style="solid", color="black", weight=3];
12.32/4.96	45[label="primEqNat ww3000 ww4000 && ww5",fontsize=16,color="burlywood",shape="triangle"];103[label="ww3000/Succ ww30000",fontsize=10,color="white",style="solid",shape="box"];45 -> 103[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	103 -> 58[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	104[label="ww3000/Zero",fontsize=10,color="white",style="solid",shape="box"];45 -> 104[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	104 -> 59[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	46 -> 33[label="",style="dashed", color="red", weight=0];
12.32/4.96	46[label="False && ww5",fontsize=16,color="magenta"];47[label="False",fontsize=16,color="green",shape="box"];48 -> 33[label="",style="dashed", color="red", weight=0];
12.32/4.96	48[label="False && ww5",fontsize=16,color="magenta"];49[label="True && ww5",fontsize=16,color="black",shape="triangle"];49 -> 60[label="",style="solid", color="black", weight=3];
12.32/4.96	50 -> 33[label="",style="dashed", color="red", weight=0];
12.32/4.96	50[label="False && ww5",fontsize=16,color="magenta"];51 -> 49[label="",style="dashed", color="red", weight=0];
12.32/4.96	51[label="True && ww5",fontsize=16,color="magenta"];52 -> 45[label="",style="dashed", color="red", weight=0];
12.32/4.96	52[label="primEqNat ww3000 ww4000 && ww5",fontsize=16,color="magenta"];52 -> 61[label="",style="dashed", color="magenta", weight=3];
12.32/4.96	52 -> 62[label="",style="dashed", color="magenta", weight=3];
12.32/4.96	53 -> 33[label="",style="dashed", color="red", weight=0];
12.32/4.96	53[label="False && ww5",fontsize=16,color="magenta"];54 -> 33[label="",style="dashed", color="red", weight=0];
12.32/4.96	54[label="False && ww5",fontsize=16,color="magenta"];55 -> 49[label="",style="dashed", color="red", weight=0];
12.32/4.96	55[label="True && ww5",fontsize=16,color="magenta"];56 -> 33[label="",style="dashed", color="red", weight=0];
12.32/4.96	56[label="False && ww5",fontsize=16,color="magenta"];57 -> 49[label="",style="dashed", color="red", weight=0];
12.32/4.96	57[label="True && ww5",fontsize=16,color="magenta"];58[label="primEqNat (Succ ww30000) ww4000 && ww5",fontsize=16,color="burlywood",shape="box"];105[label="ww4000/Succ ww40000",fontsize=10,color="white",style="solid",shape="box"];58 -> 105[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	105 -> 63[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	106[label="ww4000/Zero",fontsize=10,color="white",style="solid",shape="box"];58 -> 106[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	106 -> 64[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	59[label="primEqNat Zero ww4000 && ww5",fontsize=16,color="burlywood",shape="box"];107[label="ww4000/Succ ww40000",fontsize=10,color="white",style="solid",shape="box"];59 -> 107[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	107 -> 65[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	108[label="ww4000/Zero",fontsize=10,color="white",style="solid",shape="box"];59 -> 108[label="",style="solid", color="burlywood", weight=9];
12.32/4.96	108 -> 66[label="",style="solid", color="burlywood", weight=3];
12.32/4.96	60[label="ww5",fontsize=16,color="green",shape="box"];61[label="ww4000",fontsize=16,color="green",shape="box"];62[label="ww3000",fontsize=16,color="green",shape="box"];63[label="primEqNat (Succ ww30000) (Succ ww40000) && ww5",fontsize=16,color="black",shape="box"];63 -> 67[label="",style="solid", color="black", weight=3];
12.32/4.96	64[label="primEqNat (Succ ww30000) Zero && ww5",fontsize=16,color="black",shape="box"];64 -> 68[label="",style="solid", color="black", weight=3];
12.32/4.96	65[label="primEqNat Zero (Succ ww40000) && ww5",fontsize=16,color="black",shape="box"];65 -> 69[label="",style="solid", color="black", weight=3];
12.32/4.96	66[label="primEqNat Zero Zero && ww5",fontsize=16,color="black",shape="box"];66 -> 70[label="",style="solid", color="black", weight=3];
12.32/4.96	67 -> 45[label="",style="dashed", color="red", weight=0];
12.32/4.96	67[label="primEqNat ww30000 ww40000 && ww5",fontsize=16,color="magenta"];67 -> 71[label="",style="dashed", color="magenta", weight=3];
12.32/4.96	67 -> 72[label="",style="dashed", color="magenta", weight=3];
12.32/4.96	68 -> 33[label="",style="dashed", color="red", weight=0];
12.32/4.96	68[label="False && ww5",fontsize=16,color="magenta"];69 -> 33[label="",style="dashed", color="red", weight=0];
12.32/4.96	69[label="False && ww5",fontsize=16,color="magenta"];70 -> 49[label="",style="dashed", color="red", weight=0];
12.32/4.96	70[label="True && ww5",fontsize=16,color="magenta"];71[label="ww40000",fontsize=16,color="green",shape="box"];72[label="ww30000",fontsize=16,color="green",shape="box"];}
12.32/4.96	
12.32/4.96	----------------------------------------
12.32/4.96	
12.32/4.96	(6)
12.32/4.96	Complex Obligation (AND)
12.32/4.96	
12.32/4.96	----------------------------------------
12.32/4.96	
12.32/4.96	(7)
12.32/4.96	Obligation:
12.32/4.96	Q DP problem:
12.32/4.96	The TRS P consists of the following rules:
12.32/4.96	
12.32/4.96	   new_asAs(Succ(ww30000), Succ(ww40000), ww5) -> new_asAs(ww30000, ww40000, ww5)
12.32/4.96	
12.32/4.96	R is empty.
12.32/4.96	Q is empty.
12.32/4.96	We have to consider all minimal (P,Q,R)-chains.
12.32/4.96	----------------------------------------
12.32/4.96	
12.32/4.96	(8) QDPSizeChangeProof (EQUIVALENT)
12.32/4.96	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. 
12.32/4.96	
12.32/4.96	From the DPs we obtained the following set of size-change graphs:
12.32/4.96	*new_asAs(Succ(ww30000), Succ(ww40000), ww5) -> new_asAs(ww30000, ww40000, ww5)
12.32/4.96	The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
12.32/4.96	
12.32/4.96	
12.32/4.96	----------------------------------------
12.32/4.96	
12.32/4.96	(9)
12.32/4.96	YES
12.32/4.96	
12.32/4.96	----------------------------------------
12.32/4.96	
12.32/4.96	(10)
12.32/4.96	Obligation:
12.32/4.96	Q DP problem:
12.32/4.96	The TRS P consists of the following rules:
12.32/4.96	
12.32/4.96	   new_isPrefixOf(:(ww30, ww31), :(ww40, ww41)) -> new_isPrefixOf(ww31, ww41)
12.32/4.96	
12.32/4.96	R is empty.
12.32/4.96	Q is empty.
12.32/4.96	We have to consider all minimal (P,Q,R)-chains.
12.32/4.96	----------------------------------------
12.32/4.96	
12.32/4.96	(11) QDPSizeChangeProof (EQUIVALENT)
12.32/4.96	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. 
12.32/4.96	
12.32/4.96	From the DPs we obtained the following set of size-change graphs:
12.32/4.96	*new_isPrefixOf(:(ww30, ww31), :(ww40, ww41)) -> new_isPrefixOf(ww31, ww41)
12.32/4.96	The graph contains the following edges 1 > 1, 2 > 2
12.32/4.96	
12.32/4.96	
12.32/4.96	----------------------------------------
12.32/4.96	
12.32/4.96	(12)
12.32/4.96	YES
12.32/4.99	EOF