8.28/3.56	YES
10.08/4.06	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.08/4.06	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.08/4.06	
10.08/4.06	
10.08/4.06	H-Termination with start terms of the given HASKELL could be proven:
10.08/4.06	
10.08/4.06	(0) HASKELL
10.08/4.06	(1) BR [EQUIVALENT, 0 ms]
10.08/4.06	(2) HASKELL
10.08/4.06	(3) COR [EQUIVALENT, 0 ms]
10.08/4.06	(4) HASKELL
10.08/4.06	(5) NumRed [SOUND, 0 ms]
10.08/4.06	(6) HASKELL
10.08/4.06	(7) Narrow [SOUND, 0 ms]
10.08/4.06	(8) QDP
10.08/4.06	(9) TransformationProof [EQUIVALENT, 0 ms]
10.08/4.06	(10) QDP
10.08/4.06	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
10.08/4.06	(12) QDP
10.08/4.06	(13) UsableRulesProof [EQUIVALENT, 0 ms]
10.08/4.06	(14) QDP
10.08/4.06	(15) QReductionProof [EQUIVALENT, 0 ms]
10.08/4.06	(16) QDP
10.08/4.06	(17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.08/4.06	(18) YES
10.08/4.06	
10.08/4.06	
10.08/4.06	----------------------------------------
10.08/4.06	
10.08/4.06	(0)
10.08/4.06	Obligation:
10.08/4.06	mainModule Main 
10.08/4.06	module Main where {
10.08/4.06	import qualified Prelude;
10.08/4.06	}
10.08/4.06	
10.08/4.06	----------------------------------------
10.08/4.06	
10.08/4.06	(1) BR (EQUIVALENT)
10.08/4.06	Replaced joker patterns by fresh variables and removed binding patterns.
10.08/4.06	----------------------------------------
10.08/4.06	
10.08/4.06	(2)
10.08/4.06	Obligation:
10.08/4.06	mainModule Main 
10.08/4.06	module Main where {
10.08/4.06	import qualified Prelude;
10.08/4.06	}
10.08/4.06	
10.08/4.06	----------------------------------------
10.08/4.06	
10.08/4.06	(3) COR (EQUIVALENT)
10.08/4.06	Cond Reductions:
10.08/4.06	The following Function with conditions
10.08/4.06	"drop n xs|n <= 0xs;
10.08/4.06	drop vw [] = [];
10.08/4.06	drop n (vx : xs) = drop (n - 1) xs;
10.08/4.06	"
10.08/4.06	is transformed to
10.08/4.06	"drop n xs = drop3 n xs;
10.08/4.06	drop vw [] = drop1 vw [];
10.08/4.06	drop n (vx : xs) = drop0 n (vx : xs);
10.08/4.06	"
10.08/4.06	"drop0 n (vx : xs) = drop (n - 1) xs;
10.08/4.06	"
10.08/4.06	"drop1 vw [] = [];
10.08/4.06	drop1 wv ww = drop0 wv ww;
10.08/4.06	"
10.08/4.06	"drop2 n xs True = xs;
10.08/4.06	drop2 n xs False = drop1 n xs;
10.08/4.06	"
10.08/4.06	"drop3 n xs = drop2 n xs (n <= 0);
10.08/4.06	drop3 wx wy = drop1 wx wy;
10.08/4.06	"
10.08/4.06	The following Function with conditions
10.08/4.06	"undefined |Falseundefined;
10.08/4.06	"
10.08/4.06	is transformed to
10.08/4.06	"undefined  = undefined1;
10.08/4.06	"
10.08/4.06	"undefined0 True = undefined;
10.08/4.06	"
10.08/4.06	"undefined1  = undefined0 False;
10.08/4.06	"
10.08/4.06	
10.08/4.06	----------------------------------------
10.08/4.06	
10.08/4.06	(4)
10.08/4.06	Obligation:
10.08/4.06	mainModule Main 
10.08/4.06	module Main where {
10.08/4.06	import qualified Prelude;
10.08/4.06	}
10.08/4.06	
10.08/4.06	----------------------------------------
10.08/4.06	
10.08/4.06	(5) NumRed (SOUND)
10.08/4.06	Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
10.08/4.06	----------------------------------------
10.08/4.06	
10.08/4.06	(6)
10.08/4.06	Obligation:
10.08/4.06	mainModule Main 
10.08/4.06	module Main where {
10.08/4.06	import qualified Prelude;
10.08/4.06	}
10.08/4.06	
10.08/4.06	----------------------------------------
10.08/4.06	
10.08/4.06	(7) Narrow (SOUND)
10.08/4.06	Haskell To QDPs
10.08/4.06	
10.08/4.06	digraph dp_graph {
10.08/4.06	node [outthreshold=100, inthreshold=100];1[label="drop",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.08/4.06	3[label="drop wz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.08/4.06	4[label="drop wz3 wz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
10.08/4.06	5[label="drop3 wz3 wz4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
10.08/4.06	6[label="drop2 wz3 wz4 (wz3 <= Pos Zero)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
10.08/4.06	7[label="drop2 wz3 wz4 (compare wz3 (Pos Zero) /= GT)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
10.08/4.06	8[label="drop2 wz3 wz4 (not (compare wz3 (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
10.08/4.06	9[label="drop2 wz3 wz4 (not (primCmpInt wz3 (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];47[label="wz3/Pos wz30",fontsize=10,color="white",style="solid",shape="box"];9 -> 47[label="",style="solid", color="burlywood", weight=9];
10.08/4.06	47 -> 10[label="",style="solid", color="burlywood", weight=3];
10.08/4.06	48[label="wz3/Neg wz30",fontsize=10,color="white",style="solid",shape="box"];9 -> 48[label="",style="solid", color="burlywood", weight=9];
10.08/4.06	48 -> 11[label="",style="solid", color="burlywood", weight=3];
10.08/4.06	10[label="drop2 (Pos wz30) wz4 (not (primCmpInt (Pos wz30) (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];49[label="wz30/Succ wz300",fontsize=10,color="white",style="solid",shape="box"];10 -> 49[label="",style="solid", color="burlywood", weight=9];
10.08/4.06	49 -> 12[label="",style="solid", color="burlywood", weight=3];
10.08/4.06	50[label="wz30/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 50[label="",style="solid", color="burlywood", weight=9];
10.08/4.06	50 -> 13[label="",style="solid", color="burlywood", weight=3];
10.08/4.06	11[label="drop2 (Neg wz30) wz4 (not (primCmpInt (Neg wz30) (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];51[label="wz30/Succ wz300",fontsize=10,color="white",style="solid",shape="box"];11 -> 51[label="",style="solid", color="burlywood", weight=9];
10.08/4.06	51 -> 14[label="",style="solid", color="burlywood", weight=3];
10.08/4.06	52[label="wz30/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 52[label="",style="solid", color="burlywood", weight=9];
10.08/4.06	52 -> 15[label="",style="solid", color="burlywood", weight=3];
10.08/4.06	12[label="drop2 (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];
10.08/4.06	13[label="drop2 (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];
10.08/4.06	14[label="drop2 (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];
10.08/4.06	15[label="drop2 (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];
10.08/4.06	16[label="drop2 (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];
10.08/4.06	17[label="drop2 (Pos Zero) wz4 (not (EQ == GT))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
10.08/4.06	18[label="drop2 (Neg (Succ wz300)) wz4 (not (LT == GT))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
10.08/4.06	19[label="drop2 (Neg Zero) wz4 (not (EQ == GT))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
10.08/4.06	20[label="drop2 (Pos (Succ wz300)) wz4 (not (GT == GT))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
10.08/4.06	21[label="drop2 (Pos Zero) wz4 (not False)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
10.08/4.06	22[label="drop2 (Neg (Succ wz300)) wz4 (not False)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
10.08/4.06	23[label="drop2 (Neg Zero) wz4 (not False)",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
10.08/4.06	24[label="drop2 (Pos (Succ wz300)) wz4 (not True)",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
10.08/4.06	25[label="drop2 (Pos Zero) wz4 True",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
10.08/4.06	26[label="drop2 (Neg (Succ wz300)) wz4 True",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
10.08/4.06	27[label="drop2 (Neg Zero) wz4 True",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
10.08/4.06	28[label="drop2 (Pos (Succ wz300)) wz4 False",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
10.08/4.06	29[label="wz4",fontsize=16,color="green",shape="box"];30[label="wz4",fontsize=16,color="green",shape="box"];31[label="wz4",fontsize=16,color="green",shape="box"];32[label="drop1 (Pos (Succ wz300)) wz4",fontsize=16,color="burlywood",shape="box"];53[label="wz4/wz40 : wz41",fontsize=10,color="white",style="solid",shape="box"];32 -> 53[label="",style="solid", color="burlywood", weight=9];
10.08/4.06	53 -> 33[label="",style="solid", color="burlywood", weight=3];
10.08/4.06	54[label="wz4/[]",fontsize=10,color="white",style="solid",shape="box"];32 -> 54[label="",style="solid", color="burlywood", weight=9];
10.08/4.06	54 -> 34[label="",style="solid", color="burlywood", weight=3];
10.08/4.06	33[label="drop1 (Pos (Succ wz300)) (wz40 : wz41)",fontsize=16,color="black",shape="box"];33 -> 35[label="",style="solid", color="black", weight=3];
10.08/4.06	34[label="drop1 (Pos (Succ wz300)) []",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3];
10.08/4.06	35[label="drop0 (Pos (Succ wz300)) (wz40 : wz41)",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
10.08/4.06	36[label="[]",fontsize=16,color="green",shape="box"];37 -> 4[label="",style="dashed", color="red", weight=0];
10.08/4.06	37[label="drop (Pos (Succ wz300) - Pos (Succ Zero)) wz41",fontsize=16,color="magenta"];37 -> 38[label="",style="dashed", color="magenta", weight=3];
10.08/4.06	37 -> 39[label="",style="dashed", color="magenta", weight=3];
10.08/4.06	38[label="wz41",fontsize=16,color="green",shape="box"];39[label="Pos (Succ wz300) - Pos (Succ Zero)",fontsize=16,color="black",shape="box"];39 -> 40[label="",style="solid", color="black", weight=3];
10.08/4.06	40[label="primMinusInt (Pos (Succ wz300)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];40 -> 41[label="",style="solid", color="black", weight=3];
10.08/4.06	41[label="primMinusNat (Succ wz300) (Succ Zero)",fontsize=16,color="black",shape="box"];41 -> 42[label="",style="solid", color="black", weight=3];
10.08/4.06	42[label="primMinusNat wz300 Zero",fontsize=16,color="burlywood",shape="box"];55[label="wz300/Succ wz3000",fontsize=10,color="white",style="solid",shape="box"];42 -> 55[label="",style="solid", color="burlywood", weight=9];
10.08/4.06	55 -> 43[label="",style="solid", color="burlywood", weight=3];
10.08/4.06	56[label="wz300/Zero",fontsize=10,color="white",style="solid",shape="box"];42 -> 56[label="",style="solid", color="burlywood", weight=9];
10.08/4.06	56 -> 44[label="",style="solid", color="burlywood", weight=3];
10.08/4.06	43[label="primMinusNat (Succ wz3000) Zero",fontsize=16,color="black",shape="box"];43 -> 45[label="",style="solid", color="black", weight=3];
10.08/4.06	44[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];44 -> 46[label="",style="solid", color="black", weight=3];
10.08/4.06	45[label="Pos (Succ wz3000)",fontsize=16,color="green",shape="box"];46[label="Pos Zero",fontsize=16,color="green",shape="box"];}
10.08/4.06	
10.08/4.06	----------------------------------------
10.08/4.06	
10.08/4.06	(8)
10.08/4.06	Obligation:
10.08/4.06	Q DP problem:
10.08/4.06	The TRS P consists of the following rules:
10.08/4.06	
10.08/4.06	   new_drop(Pos(Succ(wz300)), :(wz40, wz41), h) -> new_drop(new_primMinusNat(wz300), wz41, h)
10.08/4.06	
10.08/4.06	The TRS R consists of the following rules:
10.08/4.06	
10.08/4.06	   new_primMinusNat(Succ(wz3000)) -> Pos(Succ(wz3000))
10.08/4.06	   new_primMinusNat(Zero) -> Pos(Zero)
10.08/4.06	
10.08/4.06	The set Q consists of the following terms:
10.08/4.06	
10.08/4.06	   new_primMinusNat(Succ(x0))
10.08/4.06	   new_primMinusNat(Zero)
10.08/4.06	
10.08/4.06	We have to consider all minimal (P,Q,R)-chains.
10.08/4.06	----------------------------------------
10.08/4.06	
10.08/4.06	(9) TransformationProof (EQUIVALENT)
10.08/4.06	By narrowing [LPAR04] the rule new_drop(Pos(Succ(wz300)), :(wz40, wz41), h) -> new_drop(new_primMinusNat(wz300), wz41, h) at position [0] we obtained the following new rules [LPAR04]:
10.08/4.06	
10.08/4.06	   (new_drop(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_drop(Pos(Succ(x0)), y2, y3),new_drop(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_drop(Pos(Succ(x0)), y2, y3))
10.08/4.06	   (new_drop(Pos(Succ(Zero)), :(y1, y2), y3) -> new_drop(Pos(Zero), y2, y3),new_drop(Pos(Succ(Zero)), :(y1, y2), y3) -> new_drop(Pos(Zero), y2, y3))
10.08/4.06	
10.08/4.06	
10.08/4.06	----------------------------------------
10.08/4.06	
10.08/4.06	(10)
10.08/4.06	Obligation:
10.08/4.06	Q DP problem:
10.08/4.06	The TRS P consists of the following rules:
10.08/4.06	
10.08/4.06	   new_drop(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_drop(Pos(Succ(x0)), y2, y3)
10.08/4.06	   new_drop(Pos(Succ(Zero)), :(y1, y2), y3) -> new_drop(Pos(Zero), y2, y3)
10.08/4.06	
10.08/4.06	The TRS R consists of the following rules:
10.08/4.06	
10.08/4.06	   new_primMinusNat(Succ(wz3000)) -> Pos(Succ(wz3000))
10.08/4.06	   new_primMinusNat(Zero) -> Pos(Zero)
10.08/4.06	
10.08/4.06	The set Q consists of the following terms:
10.08/4.06	
10.08/4.06	   new_primMinusNat(Succ(x0))
10.08/4.06	   new_primMinusNat(Zero)
10.08/4.06	
10.08/4.06	We have to consider all minimal (P,Q,R)-chains.
10.08/4.06	----------------------------------------
10.08/4.06	
10.08/4.06	(11) DependencyGraphProof (EQUIVALENT)
10.08/4.06	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
10.08/4.06	----------------------------------------
10.08/4.06	
10.08/4.06	(12)
10.08/4.06	Obligation:
10.08/4.06	Q DP problem:
10.08/4.06	The TRS P consists of the following rules:
10.08/4.06	
10.08/4.06	   new_drop(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_drop(Pos(Succ(x0)), y2, y3)
10.08/4.06	
10.08/4.06	The TRS R consists of the following rules:
10.08/4.06	
10.08/4.06	   new_primMinusNat(Succ(wz3000)) -> Pos(Succ(wz3000))
10.08/4.06	   new_primMinusNat(Zero) -> Pos(Zero)
10.08/4.06	
10.08/4.06	The set Q consists of the following terms:
10.08/4.06	
10.08/4.06	   new_primMinusNat(Succ(x0))
10.08/4.06	   new_primMinusNat(Zero)
10.08/4.06	
10.08/4.06	We have to consider all minimal (P,Q,R)-chains.
10.08/4.06	----------------------------------------
10.08/4.06	
10.08/4.06	(13) UsableRulesProof (EQUIVALENT)
10.08/4.06	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.
10.08/4.06	----------------------------------------
10.08/4.06	
10.08/4.06	(14)
10.08/4.06	Obligation:
10.08/4.06	Q DP problem:
10.08/4.06	The TRS P consists of the following rules:
10.08/4.06	
10.08/4.06	   new_drop(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_drop(Pos(Succ(x0)), y2, y3)
10.08/4.06	
10.08/4.06	R is empty.
10.08/4.06	The set Q consists of the following terms:
10.08/4.06	
10.08/4.06	   new_primMinusNat(Succ(x0))
10.08/4.06	   new_primMinusNat(Zero)
10.08/4.06	
10.08/4.06	We have to consider all minimal (P,Q,R)-chains.
10.08/4.06	----------------------------------------
10.08/4.06	
10.08/4.06	(15) QReductionProof (EQUIVALENT)
10.08/4.06	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
10.08/4.06	
10.08/4.06	   new_primMinusNat(Succ(x0))
10.08/4.06	   new_primMinusNat(Zero)
10.08/4.06	
10.08/4.06	
10.08/4.06	----------------------------------------
10.08/4.06	
10.08/4.06	(16)
10.08/4.06	Obligation:
10.08/4.06	Q DP problem:
10.08/4.06	The TRS P consists of the following rules:
10.08/4.06	
10.08/4.06	   new_drop(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_drop(Pos(Succ(x0)), y2, y3)
10.08/4.06	
10.08/4.06	R is empty.
10.08/4.06	Q is empty.
10.08/4.06	We have to consider all minimal (P,Q,R)-chains.
10.08/4.06	----------------------------------------
10.08/4.06	
10.08/4.06	(17) QDPSizeChangeProof (EQUIVALENT)
10.08/4.06	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. 
10.08/4.06	
10.08/4.06	From the DPs we obtained the following set of size-change graphs:
10.08/4.06	*new_drop(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_drop(Pos(Succ(x0)), y2, y3)
10.08/4.06	The graph contains the following edges 2 > 2, 3 >= 3
10.08/4.06	
10.08/4.06	
10.08/4.06	----------------------------------------
10.08/4.06	
10.08/4.06	(18)
10.08/4.06	YES
10.08/4.11	EOF