8.15/3.56	YES
9.94/4.05	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.94/4.05	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.94/4.05	
9.94/4.05	
9.94/4.05	H-Termination with start terms of the given HASKELL could be proven:
9.94/4.05	
9.94/4.05	(0) HASKELL
9.94/4.05	(1) BR [EQUIVALENT, 0 ms]
9.94/4.05	(2) HASKELL
9.94/4.05	(3) COR [EQUIVALENT, 0 ms]
9.94/4.05	(4) HASKELL
9.94/4.05	(5) NumRed [SOUND, 0 ms]
9.94/4.05	(6) HASKELL
9.94/4.05	(7) Narrow [SOUND, 0 ms]
9.94/4.05	(8) QDP
9.94/4.05	(9) TransformationProof [EQUIVALENT, 0 ms]
9.94/4.05	(10) QDP
9.94/4.05	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
9.94/4.05	(12) QDP
9.94/4.05	(13) UsableRulesProof [EQUIVALENT, 0 ms]
9.94/4.05	(14) QDP
9.94/4.05	(15) QReductionProof [EQUIVALENT, 0 ms]
9.94/4.05	(16) QDP
9.94/4.05	(17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.94/4.05	(18) YES
9.94/4.05	
9.94/4.05	
9.94/4.05	----------------------------------------
9.94/4.05	
9.94/4.05	(0)
9.94/4.05	Obligation:
9.94/4.05	mainModule Main 
9.94/4.05	module Main where {
9.94/4.05	import qualified Prelude;
9.94/4.05	}
9.94/4.05	
9.94/4.05	----------------------------------------
9.94/4.05	
9.94/4.05	(1) BR (EQUIVALENT)
9.94/4.05	Replaced joker patterns by fresh variables and removed binding patterns.
9.94/4.05	----------------------------------------
9.94/4.05	
9.94/4.05	(2)
9.94/4.05	Obligation:
9.94/4.05	mainModule Main 
9.94/4.05	module Main where {
9.94/4.05	import qualified Prelude;
9.94/4.05	}
9.94/4.05	
9.94/4.05	----------------------------------------
9.94/4.05	
9.94/4.05	(3) COR (EQUIVALENT)
9.94/4.05	Cond Reductions:
9.94/4.05	The following Function with conditions
9.94/4.05	"undefined |Falseundefined;
9.94/4.05	"
9.94/4.05	is transformed to
9.94/4.05	"undefined  = undefined1;
9.94/4.05	"
9.94/4.05	"undefined0 True = undefined;
9.94/4.05	"
9.94/4.05	"undefined1  = undefined0 False;
9.94/4.05	"
9.94/4.05	The following Function with conditions
9.94/4.05	"take n vx|n <= 0[];
9.94/4.05	take vy [] = [];
9.94/4.05	take n (x : xs) = x : take (n - 1) xs;
9.94/4.05	"
9.94/4.05	is transformed to
9.94/4.05	"take n vx = take3 n vx;
9.94/4.05	take vy [] = take1 vy [];
9.94/4.05	take n (x : xs) = take0 n (x : xs);
9.94/4.05	"
9.94/4.05	"take0 n (x : xs) = x : take (n - 1) xs;
9.94/4.05	"
9.94/4.05	"take1 vy [] = [];
9.94/4.05	take1 wv ww = take0 wv ww;
9.94/4.05	"
9.94/4.05	"take2 n vx True = [];
9.94/4.05	take2 n vx False = take1 n vx;
9.94/4.05	"
9.94/4.05	"take3 n vx = take2 n vx (n <= 0);
9.94/4.05	take3 wx wy = take1 wx wy;
9.94/4.05	"
9.94/4.05	
9.94/4.05	----------------------------------------
9.94/4.05	
9.94/4.05	(4)
9.94/4.05	Obligation:
9.94/4.05	mainModule Main 
9.94/4.05	module Main where {
9.94/4.05	import qualified Prelude;
9.94/4.05	}
9.94/4.05	
9.94/4.05	----------------------------------------
9.94/4.05	
9.94/4.05	(5) NumRed (SOUND)
9.94/4.05	Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
9.94/4.05	----------------------------------------
9.94/4.05	
9.94/4.05	(6)
9.94/4.05	Obligation:
9.94/4.05	mainModule Main 
9.94/4.05	module Main where {
9.94/4.05	import qualified Prelude;
9.94/4.05	}
9.94/4.05	
9.94/4.05	----------------------------------------
9.94/4.05	
9.94/4.05	(7) Narrow (SOUND)
9.94/4.05	Haskell To QDPs
9.94/4.05	
9.94/4.05	digraph dp_graph {
9.94/4.05	node [outthreshold=100, inthreshold=100];1[label="take",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.94/4.05	3[label="take wz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.94/4.05	4[label="take wz3 wz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.94/4.05	5[label="take3 wz3 wz4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.94/4.05	6[label="take2 wz3 wz4 (wz3 <= Pos Zero)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
9.94/4.05	7[label="take2 wz3 wz4 (compare wz3 (Pos Zero) /= GT)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
9.94/4.05	8[label="take2 wz3 wz4 (not (compare wz3 (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9.94/4.05	9[label="take2 wz3 wz4 (not (primCmpInt wz3 (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];48[label="wz3/Pos wz30",fontsize=10,color="white",style="solid",shape="box"];9 -> 48[label="",style="solid", color="burlywood", weight=9];
9.94/4.05	48 -> 10[label="",style="solid", color="burlywood", weight=3];
9.94/4.05	49[label="wz3/Neg wz30",fontsize=10,color="white",style="solid",shape="box"];9 -> 49[label="",style="solid", color="burlywood", weight=9];
9.94/4.05	49 -> 11[label="",style="solid", color="burlywood", weight=3];
9.94/4.05	10[label="take2 (Pos wz30) wz4 (not (primCmpInt (Pos wz30) (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];50[label="wz30/Succ wz300",fontsize=10,color="white",style="solid",shape="box"];10 -> 50[label="",style="solid", color="burlywood", weight=9];
9.94/4.05	50 -> 12[label="",style="solid", color="burlywood", weight=3];
9.94/4.05	51[label="wz30/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 51[label="",style="solid", color="burlywood", weight=9];
9.94/4.05	51 -> 13[label="",style="solid", color="burlywood", weight=3];
9.94/4.05	11[label="take2 (Neg wz30) wz4 (not (primCmpInt (Neg wz30) (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];52[label="wz30/Succ wz300",fontsize=10,color="white",style="solid",shape="box"];11 -> 52[label="",style="solid", color="burlywood", weight=9];
9.94/4.05	52 -> 14[label="",style="solid", color="burlywood", weight=3];
9.94/4.05	53[label="wz30/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 53[label="",style="solid", color="burlywood", weight=9];
9.94/4.05	53 -> 15[label="",style="solid", color="burlywood", weight=3];
9.94/4.05	12[label="take2 (Pos (Succ wz300)) wz4 (not (primCmpInt (Pos (Succ wz300)) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];12 -> 16[label="",style="solid", color="black", weight=3];
9.94/4.05	13[label="take2 (Pos Zero) wz4 (not (primCmpInt (Pos Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3];
9.94/4.05	14[label="take2 (Neg (Succ wz300)) wz4 (not (primCmpInt (Neg (Succ wz300)) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3];
9.94/4.05	15[label="take2 (Neg Zero) wz4 (not (primCmpInt (Neg Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
9.94/4.05	16[label="take2 (Pos (Succ wz300)) wz4 (not (primCmpNat (Succ wz300) Zero == GT))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
9.94/4.05	17[label="take2 (Pos Zero) wz4 (not (EQ == GT))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
9.94/4.05	18[label="take2 (Neg (Succ wz300)) wz4 (not (LT == GT))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
9.94/4.05	19[label="take2 (Neg Zero) wz4 (not (EQ == GT))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
9.94/4.05	20[label="take2 (Pos (Succ wz300)) wz4 (not (GT == GT))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
9.94/4.05	21[label="take2 (Pos Zero) wz4 (not False)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
9.94/4.05	22[label="take2 (Neg (Succ wz300)) wz4 (not False)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
9.94/4.05	23[label="take2 (Neg Zero) wz4 (not False)",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
9.94/4.05	24[label="take2 (Pos (Succ wz300)) wz4 (not True)",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
9.94/4.05	25[label="take2 (Pos Zero) wz4 True",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
9.94/4.05	26[label="take2 (Neg (Succ wz300)) wz4 True",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
9.94/4.05	27[label="take2 (Neg Zero) wz4 True",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
9.94/4.05	28[label="take2 (Pos (Succ wz300)) wz4 False",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
9.94/4.05	29[label="[]",fontsize=16,color="green",shape="box"];30[label="[]",fontsize=16,color="green",shape="box"];31[label="[]",fontsize=16,color="green",shape="box"];32[label="take1 (Pos (Succ wz300)) wz4",fontsize=16,color="burlywood",shape="box"];54[label="wz4/wz40 : wz41",fontsize=10,color="white",style="solid",shape="box"];32 -> 54[label="",style="solid", color="burlywood", weight=9];
9.94/4.05	54 -> 33[label="",style="solid", color="burlywood", weight=3];
9.94/4.05	55[label="wz4/[]",fontsize=10,color="white",style="solid",shape="box"];32 -> 55[label="",style="solid", color="burlywood", weight=9];
9.94/4.05	55 -> 34[label="",style="solid", color="burlywood", weight=3];
9.94/4.05	33[label="take1 (Pos (Succ wz300)) (wz40 : wz41)",fontsize=16,color="black",shape="box"];33 -> 35[label="",style="solid", color="black", weight=3];
9.94/4.05	34[label="take1 (Pos (Succ wz300)) []",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3];
9.94/4.05	35[label="take0 (Pos (Succ wz300)) (wz40 : wz41)",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
9.94/4.05	36[label="[]",fontsize=16,color="green",shape="box"];37[label="wz40 : take (Pos (Succ wz300) - Pos (Succ Zero)) wz41",fontsize=16,color="green",shape="box"];37 -> 38[label="",style="dashed", color="green", weight=3];
9.94/4.05	38 -> 4[label="",style="dashed", color="red", weight=0];
9.94/4.05	38[label="take (Pos (Succ wz300) - Pos (Succ Zero)) wz41",fontsize=16,color="magenta"];38 -> 39[label="",style="dashed", color="magenta", weight=3];
9.94/4.05	38 -> 40[label="",style="dashed", color="magenta", weight=3];
9.94/4.05	39[label="wz41",fontsize=16,color="green",shape="box"];40[label="Pos (Succ wz300) - Pos (Succ Zero)",fontsize=16,color="black",shape="box"];40 -> 41[label="",style="solid", color="black", weight=3];
9.94/4.05	41[label="primMinusInt (Pos (Succ wz300)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];41 -> 42[label="",style="solid", color="black", weight=3];
9.94/4.05	42[label="primMinusNat (Succ wz300) (Succ Zero)",fontsize=16,color="black",shape="box"];42 -> 43[label="",style="solid", color="black", weight=3];
9.94/4.05	43[label="primMinusNat wz300 Zero",fontsize=16,color="burlywood",shape="box"];56[label="wz300/Succ wz3000",fontsize=10,color="white",style="solid",shape="box"];43 -> 56[label="",style="solid", color="burlywood", weight=9];
9.94/4.05	56 -> 44[label="",style="solid", color="burlywood", weight=3];
9.94/4.05	57[label="wz300/Zero",fontsize=10,color="white",style="solid",shape="box"];43 -> 57[label="",style="solid", color="burlywood", weight=9];
9.94/4.05	57 -> 45[label="",style="solid", color="burlywood", weight=3];
9.94/4.05	44[label="primMinusNat (Succ wz3000) Zero",fontsize=16,color="black",shape="box"];44 -> 46[label="",style="solid", color="black", weight=3];
9.94/4.05	45[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];45 -> 47[label="",style="solid", color="black", weight=3];
9.94/4.05	46[label="Pos (Succ wz3000)",fontsize=16,color="green",shape="box"];47[label="Pos Zero",fontsize=16,color="green",shape="box"];}
9.94/4.05	
9.94/4.05	----------------------------------------
9.94/4.05	
9.94/4.05	(8)
9.94/4.05	Obligation:
9.94/4.05	Q DP problem:
9.94/4.05	The TRS P consists of the following rules:
9.94/4.05	
9.94/4.05	   new_take(Pos(Succ(wz300)), :(wz40, wz41), h) -> new_take(new_primMinusNat(wz300), wz41, h)
9.94/4.05	
9.94/4.05	The TRS R consists of the following rules:
9.94/4.05	
9.94/4.05	   new_primMinusNat(Succ(wz3000)) -> Pos(Succ(wz3000))
9.94/4.05	   new_primMinusNat(Zero) -> Pos(Zero)
9.94/4.05	
9.94/4.05	The set Q consists of the following terms:
9.94/4.05	
9.94/4.05	   new_primMinusNat(Succ(x0))
9.94/4.05	   new_primMinusNat(Zero)
9.94/4.05	
9.94/4.05	We have to consider all minimal (P,Q,R)-chains.
9.94/4.05	----------------------------------------
9.94/4.05	
9.94/4.05	(9) TransformationProof (EQUIVALENT)
9.94/4.05	By narrowing [LPAR04] the rule new_take(Pos(Succ(wz300)), :(wz40, wz41), h) -> new_take(new_primMinusNat(wz300), wz41, h) at position [0] we obtained the following new rules [LPAR04]:
9.94/4.05	
9.94/4.05	   (new_take(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_take(Pos(Succ(x0)), y2, y3),new_take(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_take(Pos(Succ(x0)), y2, y3))
9.94/4.05	   (new_take(Pos(Succ(Zero)), :(y1, y2), y3) -> new_take(Pos(Zero), y2, y3),new_take(Pos(Succ(Zero)), :(y1, y2), y3) -> new_take(Pos(Zero), y2, y3))
9.94/4.05	
9.94/4.05	
9.94/4.05	----------------------------------------
9.94/4.05	
9.94/4.05	(10)
9.94/4.05	Obligation:
9.94/4.05	Q DP problem:
9.94/4.05	The TRS P consists of the following rules:
9.94/4.05	
9.94/4.05	   new_take(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_take(Pos(Succ(x0)), y2, y3)
9.94/4.05	   new_take(Pos(Succ(Zero)), :(y1, y2), y3) -> new_take(Pos(Zero), y2, y3)
9.94/4.05	
9.94/4.05	The TRS R consists of the following rules:
9.94/4.05	
9.94/4.05	   new_primMinusNat(Succ(wz3000)) -> Pos(Succ(wz3000))
9.94/4.05	   new_primMinusNat(Zero) -> Pos(Zero)
9.94/4.05	
9.94/4.05	The set Q consists of the following terms:
9.94/4.05	
9.94/4.05	   new_primMinusNat(Succ(x0))
9.94/4.05	   new_primMinusNat(Zero)
9.94/4.05	
9.94/4.05	We have to consider all minimal (P,Q,R)-chains.
9.94/4.05	----------------------------------------
9.94/4.05	
9.94/4.05	(11) DependencyGraphProof (EQUIVALENT)
9.94/4.05	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
9.94/4.05	----------------------------------------
9.94/4.05	
9.94/4.05	(12)
9.94/4.05	Obligation:
9.94/4.05	Q DP problem:
9.94/4.05	The TRS P consists of the following rules:
9.94/4.05	
9.94/4.05	   new_take(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_take(Pos(Succ(x0)), y2, y3)
9.94/4.05	
9.94/4.05	The TRS R consists of the following rules:
9.94/4.05	
9.94/4.05	   new_primMinusNat(Succ(wz3000)) -> Pos(Succ(wz3000))
9.94/4.05	   new_primMinusNat(Zero) -> Pos(Zero)
9.94/4.05	
9.94/4.05	The set Q consists of the following terms:
9.94/4.05	
9.94/4.05	   new_primMinusNat(Succ(x0))
9.94/4.05	   new_primMinusNat(Zero)
9.94/4.05	
9.94/4.05	We have to consider all minimal (P,Q,R)-chains.
9.94/4.05	----------------------------------------
9.94/4.05	
9.94/4.05	(13) UsableRulesProof (EQUIVALENT)
9.94/4.05	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
9.94/4.05	----------------------------------------
9.94/4.05	
9.94/4.05	(14)
9.94/4.05	Obligation:
9.94/4.05	Q DP problem:
9.94/4.05	The TRS P consists of the following rules:
9.94/4.05	
9.94/4.05	   new_take(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_take(Pos(Succ(x0)), y2, y3)
9.94/4.05	
9.94/4.05	R is empty.
9.94/4.05	The set Q consists of the following terms:
9.94/4.05	
9.94/4.05	   new_primMinusNat(Succ(x0))
9.94/4.05	   new_primMinusNat(Zero)
9.94/4.05	
9.94/4.05	We have to consider all minimal (P,Q,R)-chains.
9.94/4.05	----------------------------------------
9.94/4.05	
9.94/4.05	(15) QReductionProof (EQUIVALENT)
9.94/4.05	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
9.94/4.05	
9.94/4.05	   new_primMinusNat(Succ(x0))
9.94/4.05	   new_primMinusNat(Zero)
9.94/4.05	
9.94/4.05	
9.94/4.05	----------------------------------------
9.94/4.05	
9.94/4.05	(16)
9.94/4.05	Obligation:
9.94/4.05	Q DP problem:
9.94/4.05	The TRS P consists of the following rules:
9.94/4.05	
9.94/4.05	   new_take(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_take(Pos(Succ(x0)), y2, y3)
9.94/4.05	
9.94/4.05	R is empty.
9.94/4.05	Q is empty.
9.94/4.05	We have to consider all minimal (P,Q,R)-chains.
9.94/4.05	----------------------------------------
9.94/4.05	
9.94/4.05	(17) QDPSizeChangeProof (EQUIVALENT)
9.94/4.05	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. 
9.94/4.05	
9.94/4.05	From the DPs we obtained the following set of size-change graphs:
9.94/4.05	*new_take(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_take(Pos(Succ(x0)), y2, y3)
9.94/4.05	The graph contains the following edges 2 > 2, 3 >= 3
9.94/4.05	
9.94/4.05	
9.94/4.05	----------------------------------------
9.94/4.05	
9.94/4.05	(18)
9.94/4.05	YES
10.05/4.10	EOF