8.26/3.53	YES
10.33/4.10	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.33/4.10	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.33/4.10	
10.33/4.10	
10.33/4.10	H-Termination with start terms of the given HASKELL could be proven:
10.33/4.10	
10.33/4.10	(0) HASKELL
10.33/4.10	(1) LR [EQUIVALENT, 0 ms]
10.33/4.10	(2) HASKELL
10.33/4.10	(3) BR [EQUIVALENT, 0 ms]
10.33/4.10	(4) HASKELL
10.33/4.10	(5) COR [EQUIVALENT, 0 ms]
10.33/4.10	(6) HASKELL
10.33/4.10	(7) LetRed [EQUIVALENT, 0 ms]
10.33/4.10	(8) HASKELL
10.33/4.10	(9) Narrow [SOUND, 0 ms]
10.33/4.10	(10) QDP
10.33/4.10	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.33/4.10	(12) YES
10.33/4.10	
10.33/4.10	
10.33/4.10	----------------------------------------
10.33/4.10	
10.33/4.10	(0)
10.33/4.10	Obligation:
10.33/4.10	mainModule Main 
10.33/4.10	module Main where {
10.33/4.10	import qualified Prelude;
10.33/4.10	}
10.33/4.10	
10.33/4.10	----------------------------------------
10.33/4.10	
10.33/4.10	(1) LR (EQUIVALENT)
10.33/4.10	Lambda Reductions:
10.33/4.10	The following Lambda expression
10.33/4.10	"\(q : _)->q"
10.33/4.10	is transformed to
10.33/4.10	"q1 (q : _) = q;
10.33/4.10	"
10.33/4.10	The following Lambda expression
10.33/4.10	"\qs->qs"
10.33/4.10	is transformed to
10.33/4.10	"qs0 qs = qs;
10.33/4.10	"
10.33/4.10	
10.33/4.10	----------------------------------------
10.33/4.10	
10.33/4.10	(2)
10.33/4.10	Obligation:
10.33/4.10	mainModule Main 
10.33/4.10	module Main where {
10.33/4.10	import qualified Prelude;
10.33/4.10	}
10.33/4.10	
10.33/4.10	----------------------------------------
10.33/4.10	
10.33/4.10	(3) BR (EQUIVALENT)
10.33/4.10	Replaced joker patterns by fresh variables and removed binding patterns.
10.33/4.10	----------------------------------------
10.33/4.10	
10.33/4.10	(4)
10.33/4.10	Obligation:
10.33/4.10	mainModule Main 
10.33/4.10	module Main where {
10.33/4.10	import qualified Prelude;
10.33/4.10	}
10.33/4.10	
10.33/4.10	----------------------------------------
10.33/4.10	
10.33/4.10	(5) COR (EQUIVALENT)
10.33/4.10	Cond Reductions:
10.33/4.10	The following Function with conditions
10.33/4.10	"undefined |Falseundefined;
10.33/4.10	"
10.33/4.10	is transformed to
10.33/4.10	"undefined  = undefined1;
10.33/4.10	"
10.33/4.10	"undefined0 True = undefined;
10.33/4.10	"
10.33/4.10	"undefined1  = undefined0 False;
10.33/4.10	"
10.33/4.10	
10.33/4.10	----------------------------------------
10.33/4.10	
10.33/4.10	(6)
10.33/4.10	Obligation:
10.33/4.10	mainModule Main 
10.33/4.10	module Main where {
10.33/4.10	import qualified Prelude;
10.33/4.10	}
10.33/4.10	
10.33/4.10	----------------------------------------
10.33/4.10	
10.33/4.10	(7) LetRed (EQUIVALENT)
10.33/4.10	Let/Where Reductions:
10.33/4.10	The bindings of the following Let/Where expression
10.33/4.10	"f x q : qs where {
10.33/4.10	q  = q1 vu40;
10.33/4.10	;
10.33/4.10	q1 (q : vw) = q;
10.33/4.10	;
10.33/4.10	qs  = qs0 vu40;
10.33/4.10	;
10.33/4.10	qs0 qs = qs;
10.33/4.10	;
10.33/4.10	vu40  = scanr f q0 xs;
10.33/4.10	} 
10.33/4.10	"
10.33/4.10	are unpacked to the following functions on top level
10.33/4.10	"scanrQs0 vy vz wu qs = qs;
10.33/4.10	"
10.33/4.10	"scanrQs vy vz wu = scanrQs0 vy vz wu (scanrVu40 vy vz wu);
10.33/4.10	"
10.33/4.10	"scanrQ1 vy vz wu (q : vw) = q;
10.33/4.10	"
10.33/4.10	"scanrVu40 vy vz wu = scanr vy vz wu;
10.33/4.10	"
10.33/4.10	"scanrQ vy vz wu = scanrQ1 vy vz wu (scanrVu40 vy vz wu);
10.33/4.10	"
10.33/4.10	
10.33/4.10	----------------------------------------
10.33/4.10	
10.33/4.10	(8)
10.33/4.10	Obligation:
10.33/4.10	mainModule Main 
10.33/4.10	module Main where {
10.33/4.10	import qualified Prelude;
10.33/4.10	}
10.33/4.10	
10.33/4.10	----------------------------------------
10.33/4.10	
10.33/4.10	(9) Narrow (SOUND)
10.33/4.10	Haskell To QDPs
10.33/4.10	
10.33/4.10	digraph dp_graph {
10.33/4.10	node [outthreshold=100, inthreshold=100];1[label="scanr",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.33/4.10	3[label="scanr wv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.33/4.10	4[label="scanr wv3 wv4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
10.33/4.10	5[label="scanr wv3 wv4 wv5",fontsize=16,color="burlywood",shape="triangle"];27[label="wv5/wv50 : wv51",fontsize=10,color="white",style="solid",shape="box"];5 -> 27[label="",style="solid", color="burlywood", weight=9];
10.33/4.10	27 -> 6[label="",style="solid", color="burlywood", weight=3];
10.33/4.10	28[label="wv5/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 28[label="",style="solid", color="burlywood", weight=9];
10.33/4.10	28 -> 7[label="",style="solid", color="burlywood", weight=3];
10.33/4.10	6[label="scanr wv3 wv4 (wv50 : wv51)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
10.33/4.10	7[label="scanr wv3 wv4 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
10.33/4.10	8[label="wv3 wv50 (scanrQ wv3 wv4 wv51) : scanrQs wv3 wv4 wv51",fontsize=16,color="green",shape="box"];8 -> 10[label="",style="dashed", color="green", weight=3];
10.33/4.10	8 -> 11[label="",style="dashed", color="green", weight=3];
10.33/4.10	9[label="wv4 : []",fontsize=16,color="green",shape="box"];10[label="wv3 wv50 (scanrQ wv3 wv4 wv51)",fontsize=16,color="green",shape="box"];10 -> 12[label="",style="dashed", color="green", weight=3];
10.33/4.10	10 -> 13[label="",style="dashed", color="green", weight=3];
10.33/4.10	11[label="scanrQs wv3 wv4 wv51",fontsize=16,color="black",shape="box"];11 -> 14[label="",style="solid", color="black", weight=3];
10.33/4.10	12[label="wv50",fontsize=16,color="green",shape="box"];13[label="scanrQ wv3 wv4 wv51",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
10.33/4.10	14[label="scanrQs0 wv3 wv4 wv51 (scanrVu40 wv3 wv4 wv51)",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
10.33/4.10	15 -> 19[label="",style="dashed", color="red", weight=0];
10.33/4.10	15[label="scanrQ1 wv3 wv4 wv51 (scanrVu40 wv3 wv4 wv51)",fontsize=16,color="magenta"];15 -> 20[label="",style="dashed", color="magenta", weight=3];
10.33/4.10	16[label="scanrVu40 wv3 wv4 wv51",fontsize=16,color="black",shape="triangle"];16 -> 18[label="",style="solid", color="black", weight=3];
10.33/4.10	20 -> 16[label="",style="dashed", color="red", weight=0];
10.33/4.10	20[label="scanrVu40 wv3 wv4 wv51",fontsize=16,color="magenta"];19[label="scanrQ1 wv3 wv4 wv51 wv6",fontsize=16,color="burlywood",shape="triangle"];29[label="wv6/wv60 : wv61",fontsize=10,color="white",style="solid",shape="box"];19 -> 29[label="",style="solid", color="burlywood", weight=9];
10.33/4.10	29 -> 22[label="",style="solid", color="burlywood", weight=3];
10.33/4.10	30[label="wv6/[]",fontsize=10,color="white",style="solid",shape="box"];19 -> 30[label="",style="solid", color="burlywood", weight=9];
10.33/4.10	30 -> 23[label="",style="solid", color="burlywood", weight=3];
10.33/4.10	18 -> 5[label="",style="dashed", color="red", weight=0];
10.33/4.10	18[label="scanr wv3 wv4 wv51",fontsize=16,color="magenta"];18 -> 24[label="",style="dashed", color="magenta", weight=3];
10.33/4.10	22[label="scanrQ1 wv3 wv4 wv51 (wv60 : wv61)",fontsize=16,color="black",shape="box"];22 -> 25[label="",style="solid", color="black", weight=3];
10.33/4.10	23[label="scanrQ1 wv3 wv4 wv51 []",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
10.33/4.10	24[label="wv51",fontsize=16,color="green",shape="box"];25[label="wv60",fontsize=16,color="green",shape="box"];26[label="error []",fontsize=16,color="red",shape="box"];}
10.33/4.10	
10.33/4.10	----------------------------------------
10.33/4.10	
10.33/4.10	(10)
10.33/4.10	Obligation:
10.33/4.10	Q DP problem:
10.33/4.10	The TRS P consists of the following rules:
10.33/4.10	
10.33/4.10	   new_scanr(wv3, wv4, :(wv50, wv51), h, ba) -> new_scanrVu40(wv3, wv4, wv51, h, ba)
10.33/4.10	   new_scanrVu40(wv3, wv4, wv51, h, ba) -> new_scanr(wv3, wv4, wv51, h, ba)
10.33/4.10	   new_scanr(wv3, wv4, :(wv50, wv51), h, ba) -> new_scanr(wv3, wv4, wv51, h, ba)
10.33/4.10	
10.33/4.10	R is empty.
10.33/4.10	Q is empty.
10.33/4.10	We have to consider all minimal (P,Q,R)-chains.
10.33/4.10	----------------------------------------
10.33/4.10	
10.33/4.10	(11) QDPSizeChangeProof (EQUIVALENT)
10.33/4.10	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. 
10.33/4.10	
10.33/4.10	From the DPs we obtained the following set of size-change graphs:
10.33/4.10	*new_scanrVu40(wv3, wv4, wv51, h, ba) -> new_scanr(wv3, wv4, wv51, h, ba)
10.33/4.10	The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5
10.33/4.10	
10.33/4.10	
10.33/4.10	*new_scanr(wv3, wv4, :(wv50, wv51), h, ba) -> new_scanr(wv3, wv4, wv51, h, ba)
10.33/4.10	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5
10.33/4.10	
10.33/4.10	
10.33/4.10	*new_scanr(wv3, wv4, :(wv50, wv51), h, ba) -> new_scanrVu40(wv3, wv4, wv51, h, ba)
10.33/4.10	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5
10.33/4.10	
10.33/4.10	
10.33/4.10	----------------------------------------
10.33/4.10	
10.33/4.10	(12)
10.33/4.10	YES
10.57/4.15	EOF