8.22/3.60	YES
9.84/4.08	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.84/4.08	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.84/4.08	
9.84/4.08	
9.84/4.08	H-Termination with start terms of the given HASKELL could be proven:
9.84/4.08	
9.84/4.08	(0) HASKELL
9.84/4.08	(1) BR [EQUIVALENT, 0 ms]
9.84/4.08	(2) HASKELL
9.84/4.08	(3) COR [EQUIVALENT, 0 ms]
9.84/4.08	(4) HASKELL
9.84/4.08	(5) Narrow [SOUND, 0 ms]
9.84/4.08	(6) QDP
9.84/4.08	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.84/4.08	(8) YES
9.84/4.08	
9.84/4.08	
9.84/4.08	----------------------------------------
9.84/4.08	
9.84/4.08	(0)
9.84/4.08	Obligation:
9.84/4.08	mainModule Main 
9.84/4.08	module Main where {
9.84/4.08	import qualified Prelude;
9.84/4.08	}
9.84/4.08	
9.84/4.08	----------------------------------------
9.84/4.08	
9.84/4.08	(1) BR (EQUIVALENT)
9.84/4.08	Replaced joker patterns by fresh variables and removed binding patterns.
9.84/4.08	----------------------------------------
9.84/4.08	
9.84/4.08	(2)
9.84/4.08	Obligation:
9.84/4.08	mainModule Main 
9.84/4.08	module Main where {
9.84/4.08	import qualified Prelude;
9.84/4.08	}
9.84/4.08	
9.84/4.08	----------------------------------------
9.84/4.08	
9.84/4.08	(3) COR (EQUIVALENT)
9.84/4.08	Cond Reductions:
9.84/4.08	The following Function with conditions
9.84/4.08	"undefined |Falseundefined;
9.84/4.08	"
9.84/4.08	is transformed to
9.84/4.08	"undefined  = undefined1;
9.84/4.08	"
9.84/4.08	"undefined0 True = undefined;
9.84/4.08	"
9.84/4.08	"undefined1  = undefined0 False;
9.84/4.08	"
9.84/4.08	
9.84/4.08	----------------------------------------
9.84/4.08	
9.84/4.08	(4)
9.84/4.08	Obligation:
9.84/4.08	mainModule Main 
9.84/4.08	module Main where {
9.84/4.08	import qualified Prelude;
9.84/4.08	}
9.84/4.08	
9.84/4.08	----------------------------------------
9.84/4.08	
9.84/4.08	(5) Narrow (SOUND)
9.84/4.08	Haskell To QDPs
9.84/4.08	
9.84/4.08	digraph dp_graph {
9.84/4.08	node [outthreshold=100, inthreshold=100];1[label="(<)",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.84/4.08	3[label="(<) vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.84/4.08	4[label="(<) vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.84/4.08	5[label="compare vx3 vx4 == LT",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.84/4.08	6[label="primCmpChar vx3 vx4 == LT",fontsize=16,color="burlywood",shape="box"];25[label="vx3/Char vx30",fontsize=10,color="white",style="solid",shape="box"];6 -> 25[label="",style="solid", color="burlywood", weight=9];
9.84/4.08	25 -> 7[label="",style="solid", color="burlywood", weight=3];
9.84/4.08	7[label="primCmpChar (Char vx30) vx4 == LT",fontsize=16,color="burlywood",shape="box"];26[label="vx4/Char vx40",fontsize=10,color="white",style="solid",shape="box"];7 -> 26[label="",style="solid", color="burlywood", weight=9];
9.84/4.08	26 -> 8[label="",style="solid", color="burlywood", weight=3];
9.84/4.08	8[label="primCmpChar (Char vx30) (Char vx40) == LT",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9.84/4.08	9[label="primCmpNat vx30 vx40 == LT",fontsize=16,color="burlywood",shape="triangle"];27[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];9 -> 27[label="",style="solid", color="burlywood", weight=9];
9.84/4.08	27 -> 10[label="",style="solid", color="burlywood", weight=3];
9.84/4.08	28[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 28[label="",style="solid", color="burlywood", weight=9];
9.84/4.08	28 -> 11[label="",style="solid", color="burlywood", weight=3];
9.84/4.08	10[label="primCmpNat (Succ vx300) vx40 == LT",fontsize=16,color="burlywood",shape="box"];29[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];10 -> 29[label="",style="solid", color="burlywood", weight=9];
9.84/4.08	29 -> 12[label="",style="solid", color="burlywood", weight=3];
9.84/4.08	30[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 30[label="",style="solid", color="burlywood", weight=9];
9.84/4.08	30 -> 13[label="",style="solid", color="burlywood", weight=3];
9.84/4.08	11[label="primCmpNat Zero vx40 == LT",fontsize=16,color="burlywood",shape="box"];31[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];11 -> 31[label="",style="solid", color="burlywood", weight=9];
9.84/4.08	31 -> 14[label="",style="solid", color="burlywood", weight=3];
9.84/4.08	32[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 32[label="",style="solid", color="burlywood", weight=9];
9.84/4.08	32 -> 15[label="",style="solid", color="burlywood", weight=3];
9.84/4.08	12[label="primCmpNat (Succ vx300) (Succ vx400) == LT",fontsize=16,color="black",shape="box"];12 -> 16[label="",style="solid", color="black", weight=3];
9.84/4.08	13[label="primCmpNat (Succ vx300) Zero == LT",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3];
9.84/4.08	14[label="primCmpNat Zero (Succ vx400) == LT",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3];
9.84/4.08	15[label="primCmpNat Zero Zero == LT",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
9.84/4.08	16 -> 9[label="",style="dashed", color="red", weight=0];
9.84/4.08	16[label="primCmpNat vx300 vx400 == LT",fontsize=16,color="magenta"];16 -> 20[label="",style="dashed", color="magenta", weight=3];
9.84/4.08	16 -> 21[label="",style="dashed", color="magenta", weight=3];
9.84/4.08	17[label="GT == LT",fontsize=16,color="black",shape="box"];17 -> 22[label="",style="solid", color="black", weight=3];
9.84/4.08	18[label="LT == LT",fontsize=16,color="black",shape="box"];18 -> 23[label="",style="solid", color="black", weight=3];
9.84/4.08	19[label="EQ == LT",fontsize=16,color="black",shape="box"];19 -> 24[label="",style="solid", color="black", weight=3];
9.84/4.08	20[label="vx400",fontsize=16,color="green",shape="box"];21[label="vx300",fontsize=16,color="green",shape="box"];22[label="False",fontsize=16,color="green",shape="box"];23[label="True",fontsize=16,color="green",shape="box"];24[label="False",fontsize=16,color="green",shape="box"];}
9.84/4.08	
9.84/4.08	----------------------------------------
9.84/4.08	
9.84/4.08	(6)
9.84/4.08	Obligation:
9.84/4.08	Q DP problem:
9.84/4.08	The TRS P consists of the following rules:
9.84/4.08	
9.84/4.08	   new_esEs(Succ(vx300), Succ(vx400)) -> new_esEs(vx300, vx400)
9.84/4.08	
9.84/4.08	R is empty.
9.84/4.08	Q is empty.
9.84/4.08	We have to consider all minimal (P,Q,R)-chains.
9.84/4.08	----------------------------------------
9.84/4.08	
9.84/4.08	(7) QDPSizeChangeProof (EQUIVALENT)
9.84/4.08	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. 
9.84/4.08	
9.84/4.08	From the DPs we obtained the following set of size-change graphs:
9.84/4.08	*new_esEs(Succ(vx300), Succ(vx400)) -> new_esEs(vx300, vx400)
9.84/4.08	The graph contains the following edges 1 > 1, 2 > 2
9.84/4.08	
9.84/4.08	
9.84/4.08	----------------------------------------
9.84/4.08	
9.84/4.08	(8)
9.84/4.08	YES
9.84/4.11	EOF