8.53/3.61	YES
10.71/4.17	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.71/4.17	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.71/4.17	
10.71/4.17	
10.71/4.17	H-Termination with start terms of the given HASKELL could be proven:
10.71/4.17	
10.71/4.17	(0) HASKELL
10.71/4.17	(1) LR [EQUIVALENT, 0 ms]
10.71/4.17	(2) HASKELL
10.71/4.17	(3) BR [EQUIVALENT, 0 ms]
10.71/4.17	(4) HASKELL
10.71/4.17	(5) COR [EQUIVALENT, 0 ms]
10.71/4.17	(6) HASKELL
10.71/4.17	(7) NumRed [SOUND, 0 ms]
10.71/4.17	(8) HASKELL
10.71/4.17	(9) Narrow [SOUND, 0 ms]
10.71/4.17	(10) QDP
10.71/4.17	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.71/4.17	(12) YES
10.71/4.17	
10.71/4.17	
10.71/4.17	----------------------------------------
10.71/4.17	
10.71/4.17	(0)
10.71/4.17	Obligation:
10.71/4.17	mainModule Main 
10.71/4.17	module Main where {
10.71/4.17	import qualified Prelude;
10.71/4.17	}
10.71/4.17	
10.71/4.17	----------------------------------------
10.71/4.17	
10.71/4.17	(1) LR (EQUIVALENT)
10.71/4.17	Lambda Reductions:
10.71/4.17	The following Lambda expression
10.71/4.17	"\n_->n + 1"
10.71/4.17	is transformed to
10.71/4.17	"length0 n _ = n + 1;
10.71/4.17	"
10.71/4.17	
10.71/4.17	----------------------------------------
10.71/4.17	
10.71/4.17	(2)
10.71/4.17	Obligation:
10.71/4.17	mainModule Main 
10.71/4.17	module Main where {
10.71/4.17	import qualified Prelude;
10.71/4.17	}
10.71/4.17	
10.71/4.17	----------------------------------------
10.71/4.17	
10.71/4.17	(3) BR (EQUIVALENT)
10.71/4.17	Replaced joker patterns by fresh variables and removed binding patterns.
10.71/4.17	----------------------------------------
10.71/4.17	
10.71/4.17	(4)
10.71/4.17	Obligation:
10.71/4.17	mainModule Main 
10.71/4.17	module Main where {
10.71/4.17	import qualified Prelude;
10.71/4.17	}
10.71/4.17	
10.71/4.17	----------------------------------------
10.71/4.17	
10.71/4.17	(5) COR (EQUIVALENT)
10.71/4.17	Cond Reductions:
10.71/4.17	The following Function with conditions
10.71/4.17	"undefined |Falseundefined;
10.71/4.17	"
10.71/4.17	is transformed to
10.71/4.17	"undefined  = undefined1;
10.71/4.17	"
10.71/4.17	"undefined0 True = undefined;
10.71/4.17	"
10.71/4.17	"undefined1  = undefined0 False;
10.71/4.17	"
10.71/4.17	
10.71/4.17	----------------------------------------
10.71/4.17	
10.71/4.17	(6)
10.71/4.17	Obligation:
10.71/4.17	mainModule Main 
10.71/4.17	module Main where {
10.71/4.17	import qualified Prelude;
10.71/4.17	}
10.71/4.17	
10.71/4.17	----------------------------------------
10.71/4.17	
10.71/4.17	(7) NumRed (SOUND)
10.71/4.17	Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
10.71/4.17	----------------------------------------
10.71/4.17	
10.71/4.17	(8)
10.71/4.17	Obligation:
10.71/4.17	mainModule Main 
10.71/4.17	module Main where {
10.71/4.17	import qualified Prelude;
10.71/4.17	}
10.71/4.17	
10.71/4.17	----------------------------------------
10.71/4.17	
10.71/4.17	(9) Narrow (SOUND)
10.71/4.17	Haskell To QDPs
10.71/4.17	
10.71/4.17	digraph dp_graph {
10.71/4.17	node [outthreshold=100, inthreshold=100];1[label="length",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.71/4.17	3[label="length vy3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
10.71/4.17	4[label="foldl' length0 (Pos Zero) vy3",fontsize=16,color="burlywood",shape="box"];54[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];4 -> 54[label="",style="solid", color="burlywood", weight=9];
10.71/4.17	54 -> 5[label="",style="solid", color="burlywood", weight=3];
10.71/4.17	55[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 55[label="",style="solid", color="burlywood", weight=9];
10.71/4.17	55 -> 6[label="",style="solid", color="burlywood", weight=3];
10.71/4.17	5[label="foldl' length0 (Pos Zero) (vy30 : vy31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
10.71/4.17	6[label="foldl' length0 (Pos Zero) []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
10.71/4.17	7[label="(foldl' length0 $! length0 (Pos Zero) vy30)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
10.71/4.17	8[label="Pos Zero",fontsize=16,color="green",shape="box"];9[label="(length0 (Pos Zero) vy30 `seq` foldl' length0 (length0 (Pos Zero) vy30))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10.71/4.17	10 -> 23[label="",style="dashed", color="red", weight=0];
10.71/4.17	10[label="enforceWHNF (WHNF (length0 (Pos Zero) vy30)) (foldl' length0 (length0 (Pos Zero) vy30)) vy31",fontsize=16,color="magenta"];10 -> 24[label="",style="dashed", color="magenta", weight=3];
10.71/4.17	10 -> 25[label="",style="dashed", color="magenta", weight=3];
10.71/4.17	10 -> 26[label="",style="dashed", color="magenta", weight=3];
10.71/4.17	10 -> 27[label="",style="dashed", color="magenta", weight=3];
10.71/4.17	24[label="vy31",fontsize=16,color="green",shape="box"];25[label="vy30",fontsize=16,color="green",shape="box"];26[label="Zero",fontsize=16,color="green",shape="box"];27[label="Zero",fontsize=16,color="green",shape="box"];23[label="enforceWHNF (WHNF (length0 (Pos vy5) vy310)) (foldl' length0 (length0 (Pos vy4) vy310)) vy311",fontsize=16,color="black",shape="triangle"];23 -> 30[label="",style="solid", color="black", weight=3];
10.71/4.17	30[label="enforceWHNF (WHNF (Pos vy5 + Pos (Succ Zero))) (foldl' length0 (Pos vy5 + Pos (Succ Zero))) vy311",fontsize=16,color="black",shape="box"];30 -> 31[label="",style="solid", color="black", weight=3];
10.71/4.17	31[label="enforceWHNF (WHNF (primPlusInt (Pos vy5) (Pos (Succ Zero)))) (foldl' length0 (primPlusInt (Pos vy5) (Pos (Succ Zero)))) vy311",fontsize=16,color="black",shape="box"];31 -> 32[label="",style="solid", color="black", weight=3];
10.71/4.17	32[label="enforceWHNF (WHNF (Pos (primPlusNat vy5 (Succ Zero)))) (foldl' length0 (Pos (primPlusNat vy5 (Succ Zero)))) vy311",fontsize=16,color="black",shape="box"];32 -> 33[label="",style="solid", color="black", weight=3];
10.71/4.17	33[label="foldl' length0 (Pos (primPlusNat vy5 (Succ Zero))) vy311",fontsize=16,color="burlywood",shape="box"];56[label="vy311/vy3110 : vy3111",fontsize=10,color="white",style="solid",shape="box"];33 -> 56[label="",style="solid", color="burlywood", weight=9];
10.71/4.17	56 -> 34[label="",style="solid", color="burlywood", weight=3];
10.71/4.17	57[label="vy311/[]",fontsize=10,color="white",style="solid",shape="box"];33 -> 57[label="",style="solid", color="burlywood", weight=9];
10.71/4.17	57 -> 35[label="",style="solid", color="burlywood", weight=3];
10.71/4.17	34[label="foldl' length0 (Pos (primPlusNat vy5 (Succ Zero))) (vy3110 : vy3111)",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3];
10.71/4.17	35[label="foldl' length0 (Pos (primPlusNat vy5 (Succ Zero))) []",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
10.71/4.17	36[label="(foldl' length0 $! length0 (Pos (primPlusNat vy5 (Succ Zero))) vy3110)",fontsize=16,color="black",shape="box"];36 -> 38[label="",style="solid", color="black", weight=3];
10.71/4.17	37[label="Pos (primPlusNat vy5 (Succ Zero))",fontsize=16,color="green",shape="box"];37 -> 39[label="",style="dashed", color="green", weight=3];
10.71/4.17	38[label="(length0 (Pos (primPlusNat vy5 (Succ Zero))) vy3110 `seq` foldl' length0 (length0 (Pos (primPlusNat vy5 (Succ Zero))) vy3110))",fontsize=16,color="black",shape="box"];38 -> 40[label="",style="solid", color="black", weight=3];
10.71/4.17	39[label="primPlusNat vy5 (Succ Zero)",fontsize=16,color="burlywood",shape="triangle"];58[label="vy5/Succ vy50",fontsize=10,color="white",style="solid",shape="box"];39 -> 58[label="",style="solid", color="burlywood", weight=9];
10.71/4.17	58 -> 41[label="",style="solid", color="burlywood", weight=3];
10.71/4.17	59[label="vy5/Zero",fontsize=10,color="white",style="solid",shape="box"];39 -> 59[label="",style="solid", color="burlywood", weight=9];
10.71/4.17	59 -> 42[label="",style="solid", color="burlywood", weight=3];
10.71/4.17	40 -> 23[label="",style="dashed", color="red", weight=0];
10.71/4.17	40[label="enforceWHNF (WHNF (length0 (Pos (primPlusNat vy5 (Succ Zero))) vy3110)) (foldl' length0 (length0 (Pos (primPlusNat vy5 (Succ Zero))) vy3110)) vy3111",fontsize=16,color="magenta"];40 -> 43[label="",style="dashed", color="magenta", weight=3];
10.71/4.17	40 -> 44[label="",style="dashed", color="magenta", weight=3];
10.71/4.17	40 -> 45[label="",style="dashed", color="magenta", weight=3];
10.71/4.17	40 -> 46[label="",style="dashed", color="magenta", weight=3];
10.71/4.17	41[label="primPlusNat (Succ vy50) (Succ Zero)",fontsize=16,color="black",shape="box"];41 -> 47[label="",style="solid", color="black", weight=3];
10.71/4.17	42[label="primPlusNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];42 -> 48[label="",style="solid", color="black", weight=3];
10.71/4.17	43[label="vy3111",fontsize=16,color="green",shape="box"];44[label="vy3110",fontsize=16,color="green",shape="box"];45 -> 39[label="",style="dashed", color="red", weight=0];
10.71/4.17	45[label="primPlusNat vy5 (Succ Zero)",fontsize=16,color="magenta"];46 -> 39[label="",style="dashed", color="red", weight=0];
10.71/4.17	46[label="primPlusNat vy5 (Succ Zero)",fontsize=16,color="magenta"];47[label="Succ (Succ (primPlusNat vy50 Zero))",fontsize=16,color="green",shape="box"];47 -> 49[label="",style="dashed", color="green", weight=3];
10.71/4.17	48[label="Succ Zero",fontsize=16,color="green",shape="box"];49[label="primPlusNat vy50 Zero",fontsize=16,color="burlywood",shape="box"];60[label="vy50/Succ vy500",fontsize=10,color="white",style="solid",shape="box"];49 -> 60[label="",style="solid", color="burlywood", weight=9];
10.71/4.17	60 -> 50[label="",style="solid", color="burlywood", weight=3];
10.71/4.17	61[label="vy50/Zero",fontsize=10,color="white",style="solid",shape="box"];49 -> 61[label="",style="solid", color="burlywood", weight=9];
10.71/4.17	61 -> 51[label="",style="solid", color="burlywood", weight=3];
10.71/4.17	50[label="primPlusNat (Succ vy500) Zero",fontsize=16,color="black",shape="box"];50 -> 52[label="",style="solid", color="black", weight=3];
10.71/4.17	51[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];51 -> 53[label="",style="solid", color="black", weight=3];
10.71/4.17	52[label="Succ vy500",fontsize=16,color="green",shape="box"];53[label="Zero",fontsize=16,color="green",shape="box"];}
10.71/4.17	
10.71/4.17	----------------------------------------
10.71/4.17	
10.71/4.17	(10)
10.71/4.17	Obligation:
10.71/4.17	Q DP problem:
10.71/4.17	The TRS P consists of the following rules:
10.71/4.17	
10.71/4.17	   new_enforceWHNF(vy5, vy310, vy4, :(vy3110, vy3111), h) -> new_enforceWHNF(new_primPlusNat(vy5), vy3110, new_primPlusNat(vy5), vy3111, h)
10.71/4.17	
10.71/4.17	The TRS R consists of the following rules:
10.71/4.17	
10.71/4.17	   new_primPlusNat(Succ(vy50)) -> Succ(Succ(new_primPlusNat0(vy50)))
10.71/4.17	   new_primPlusNat(Zero) -> Succ(Zero)
10.71/4.17	   new_primPlusNat0(Succ(vy500)) -> Succ(vy500)
10.71/4.17	   new_primPlusNat0(Zero) -> Zero
10.71/4.17	
10.71/4.17	The set Q consists of the following terms:
10.71/4.17	
10.71/4.17	   new_primPlusNat0(Zero)
10.71/4.17	   new_primPlusNat(Succ(x0))
10.71/4.17	   new_primPlusNat0(Succ(x0))
10.71/4.17	   new_primPlusNat(Zero)
10.71/4.17	
10.71/4.17	We have to consider all minimal (P,Q,R)-chains.
10.71/4.17	----------------------------------------
10.71/4.17	
10.71/4.17	(11) QDPSizeChangeProof (EQUIVALENT)
10.71/4.17	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. 
10.71/4.17	
10.71/4.17	From the DPs we obtained the following set of size-change graphs:
10.71/4.17	*new_enforceWHNF(vy5, vy310, vy4, :(vy3110, vy3111), h) -> new_enforceWHNF(new_primPlusNat(vy5), vy3110, new_primPlusNat(vy5), vy3111, h)
10.71/4.17	The graph contains the following edges 4 > 2, 4 > 4, 5 >= 5
10.71/4.17	
10.71/4.17	
10.71/4.17	----------------------------------------
10.71/4.17	
10.71/4.17	(12)
10.71/4.17	YES
10.78/7.63	EOF