8.02/3.93	YES
10.35/4.48	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.35/4.48	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.35/4.48	
10.35/4.48	
10.35/4.48	H-Termination with start terms of the given HASKELL could be proven:
10.35/4.48	
10.35/4.48	(0) HASKELL
10.35/4.48	(1) LR [EQUIVALENT, 0 ms]
10.35/4.48	(2) HASKELL
10.35/4.48	(3) BR [EQUIVALENT, 0 ms]
10.35/4.48	(4) HASKELL
10.35/4.48	(5) COR [EQUIVALENT, 0 ms]
10.35/4.48	(6) HASKELL
10.35/4.48	(7) NumRed [SOUND, 0 ms]
10.35/4.48	(8) HASKELL
10.35/4.48	(9) Narrow [SOUND, 0 ms]
10.35/4.48	(10) QDP
10.35/4.48	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.35/4.48	(12) YES
10.35/4.48	
10.35/4.48	
10.35/4.48	----------------------------------------
10.35/4.48	
10.35/4.48	(0)
10.35/4.48	Obligation:
10.35/4.48	mainModule Main 
10.35/4.48	module Main where {
10.35/4.48	import qualified Prelude;
10.35/4.48	}
10.35/4.48	
10.35/4.48	----------------------------------------
10.35/4.48	
10.35/4.48	(1) LR (EQUIVALENT)
10.35/4.48	Lambda Reductions:
10.35/4.48	The following Lambda expression
10.35/4.48	"\_->q"
10.35/4.48	is transformed to
10.35/4.48	"gtGt0 q _ = q;
10.35/4.48	"
10.35/4.48	
10.35/4.48	----------------------------------------
10.35/4.48	
10.35/4.48	(2)
10.35/4.48	Obligation:
10.35/4.48	mainModule Main 
10.35/4.48	module Main where {
10.35/4.48	import qualified Prelude;
10.35/4.48	}
10.35/4.48	
10.35/4.48	----------------------------------------
10.35/4.48	
10.35/4.48	(3) BR (EQUIVALENT)
10.35/4.48	Replaced joker patterns by fresh variables and removed binding patterns.
10.35/4.48	----------------------------------------
10.35/4.48	
10.35/4.48	(4)
10.35/4.48	Obligation:
10.35/4.48	mainModule Main 
10.35/4.48	module Main where {
10.35/4.48	import qualified Prelude;
10.35/4.48	}
10.35/4.48	
10.35/4.48	----------------------------------------
10.35/4.48	
10.35/4.48	(5) COR (EQUIVALENT)
10.35/4.48	Cond Reductions:
10.35/4.48	The following Function with conditions
10.35/4.48	"randomSelect (x : []) = x;
10.35/4.48	randomSelect (x : xs)|terminatorrandomSelect xs|otherwisex;
10.35/4.48	"
10.35/4.48	is transformed to
10.35/4.48	"randomSelect (x : []) = randomSelect3 (x : []);
10.35/4.48	randomSelect (x : xs) = randomSelect2 (x : xs);
10.35/4.48	"
10.35/4.48	"randomSelect1 x xs True = randomSelect xs;
10.35/4.48	randomSelect1 x xs False = randomSelect0 x xs otherwise;
10.35/4.48	"
10.35/4.48	"randomSelect0 x xs True = x;
10.35/4.48	"
10.35/4.48	"randomSelect2 (x : xs) = randomSelect1 x xs terminator;
10.35/4.48	"
10.35/4.48	"randomSelect3 (x : []) = x;
10.35/4.48	randomSelect3 wv = randomSelect2 wv;
10.35/4.48	"
10.35/4.48	The following Function with conditions
10.35/4.48	"undefined |Falseundefined;
10.35/4.48	"
10.35/4.48	is transformed to
10.35/4.48	"undefined  = undefined1;
10.35/4.48	"
10.35/4.48	"undefined0 True = undefined;
10.35/4.48	"
10.35/4.48	"undefined1  = undefined0 False;
10.35/4.48	"
10.35/4.48	
10.35/4.48	----------------------------------------
10.35/4.48	
10.35/4.48	(6)
10.35/4.48	Obligation:
10.35/4.48	mainModule Main 
10.35/4.48	module Main where {
10.35/4.48	import qualified Prelude;
10.35/4.48	}
10.35/4.48	
10.35/4.48	----------------------------------------
10.35/4.48	
10.35/4.48	(7) NumRed (SOUND)
10.35/4.48	Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
10.35/4.48	----------------------------------------
10.35/4.48	
10.35/4.48	(8)
10.35/4.48	Obligation:
10.35/4.48	mainModule Main 
10.35/4.48	module Main where {
10.35/4.48	import qualified Prelude;
10.35/4.48	}
10.35/4.48	
10.35/4.48	----------------------------------------
10.35/4.48	
10.35/4.48	(9) Narrow (SOUND)
10.35/4.48	Haskell To QDPs
10.35/4.48	
10.35/4.48	digraph dp_graph {
10.35/4.48	node [outthreshold=100, inthreshold=100];1[label="putStrLn",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.35/4.48	3[label="putStrLn ww3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
10.35/4.48	4 -> 5[label="",style="dashed", color="red", weight=0];
10.35/4.48	4[label="putStr ww3 >> putChar (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))",fontsize=16,color="magenta"];4 -> 6[label="",style="dashed", color="magenta", weight=3];
10.35/4.48	4 -> 7[label="",style="dashed", color="magenta", weight=3];
10.35/4.48	6[label="ww3",fontsize=16,color="green",shape="box"];7[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];5[label="putStr ww5 >> putChar (Char (Succ ww6))",fontsize=16,color="black",shape="triangle"];5 -> 8[label="",style="solid", color="black", weight=3];
10.35/4.48	8[label="putStr ww5 >>= gtGt0 (putChar (Char (Succ ww6)))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
10.35/4.48	9 -> 92[label="",style="dashed", color="red", weight=0];
10.35/4.48	9[label="primbindIO (putStr ww5) (gtGt0 (putChar (Char (Succ ww6))))",fontsize=16,color="magenta"];9 -> 93[label="",style="dashed", color="magenta", weight=3];
10.35/4.48	9 -> 94[label="",style="dashed", color="magenta", weight=3];
10.35/4.48	93[label="putChar (Char (Succ ww6))",fontsize=16,color="black",shape="box"];93 -> 195[label="",style="solid", color="black", weight=3];
10.35/4.48	94[label="putStr ww5",fontsize=16,color="burlywood",shape="triangle"];248[label="ww5/ww50 : ww51",fontsize=10,color="white",style="solid",shape="box"];94 -> 248[label="",style="solid", color="burlywood", weight=9];
10.35/4.48	248 -> 196[label="",style="solid", color="burlywood", weight=3];
10.35/4.48	249[label="ww5/[]",fontsize=10,color="white",style="solid",shape="box"];94 -> 249[label="",style="solid", color="burlywood", weight=9];
10.35/4.48	249 -> 197[label="",style="solid", color="burlywood", weight=3];
10.35/4.48	92[label="primbindIO ww9 (gtGt0 ww8)",fontsize=16,color="burlywood",shape="triangle"];250[label="ww9/IO ww90",fontsize=10,color="white",style="solid",shape="box"];92 -> 250[label="",style="solid", color="burlywood", weight=9];
10.35/4.48	250 -> 198[label="",style="solid", color="burlywood", weight=3];
10.35/4.48	251[label="ww9/AProVE_IO ww90",fontsize=10,color="white",style="solid",shape="box"];92 -> 251[label="",style="solid", color="burlywood", weight=9];
10.35/4.48	251 -> 199[label="",style="solid", color="burlywood", weight=3];
10.35/4.48	252[label="ww9/AProVE_Exception ww90",fontsize=10,color="white",style="solid",shape="box"];92 -> 252[label="",style="solid", color="burlywood", weight=9];
10.35/4.48	252 -> 200[label="",style="solid", color="burlywood", weight=3];
10.35/4.48	253[label="ww9/AProVE_Error ww90",fontsize=10,color="white",style="solid",shape="box"];92 -> 253[label="",style="solid", color="burlywood", weight=9];
10.35/4.48	253 -> 201[label="",style="solid", color="burlywood", weight=3];
10.35/4.48	195 -> 225[label="",style="dashed", color="red", weight=0];
10.35/4.48	195[label="(seq Char (Succ ww6) output)",fontsize=16,color="magenta"];195 -> 226[label="",style="dashed", color="magenta", weight=3];
10.35/4.48	195 -> 227[label="",style="dashed", color="magenta", weight=3];
10.35/4.48	196[label="putStr (ww50 : ww51)",fontsize=16,color="black",shape="box"];196 -> 203[label="",style="solid", color="black", weight=3];
10.35/4.48	197[label="putStr []",fontsize=16,color="black",shape="box"];197 -> 204[label="",style="solid", color="black", weight=3];
10.35/4.48	198[label="primbindIO (IO ww90) (gtGt0 ww8)",fontsize=16,color="black",shape="box"];198 -> 205[label="",style="solid", color="black", weight=3];
10.35/4.48	199[label="primbindIO (AProVE_IO ww90) (gtGt0 ww8)",fontsize=16,color="black",shape="box"];199 -> 206[label="",style="solid", color="black", weight=3];
10.35/4.48	200[label="primbindIO (AProVE_Exception ww90) (gtGt0 ww8)",fontsize=16,color="black",shape="box"];200 -> 207[label="",style="solid", color="black", weight=3];
10.35/4.48	201[label="primbindIO (AProVE_Error ww90) (gtGt0 ww8)",fontsize=16,color="black",shape="box"];201 -> 208[label="",style="solid", color="black", weight=3];
10.35/4.48	226 -> 204[label="",style="dashed", color="red", weight=0];
10.35/4.48	226[label="output",fontsize=16,color="magenta"];227[label="Char (Succ ww6)",fontsize=16,color="green",shape="box"];225[label="(seq ww50 ww11)",fontsize=16,color="black",shape="triangle"];225 -> 229[label="",style="solid", color="black", weight=3];
10.35/4.48	203 -> 210[label="",style="dashed", color="red", weight=0];
10.35/4.48	203[label="putChar ww50 >> putStr ww51",fontsize=16,color="magenta"];203 -> 211[label="",style="dashed", color="magenta", weight=3];
10.35/4.48	204[label="output",fontsize=16,color="black",shape="triangle"];204 -> 212[label="",style="solid", color="black", weight=3];
10.35/4.48	205[label="error []",fontsize=16,color="red",shape="box"];206[label="gtGt0 ww8 ww90",fontsize=16,color="black",shape="box"];206 -> 213[label="",style="solid", color="black", weight=3];
10.35/4.48	207[label="AProVE_Exception ww90",fontsize=16,color="green",shape="box"];208[label="AProVE_Error ww90",fontsize=16,color="green",shape="box"];229[label="enforceWHNF (WHNF ww50) ww11",fontsize=16,color="black",shape="box"];229 -> 232[label="",style="solid", color="black", weight=3];
10.35/4.48	211 -> 94[label="",style="dashed", color="red", weight=0];
10.35/4.48	211[label="putStr ww51",fontsize=16,color="magenta"];211 -> 214[label="",style="dashed", color="magenta", weight=3];
10.35/4.48	210[label="putChar ww50 >> ww10",fontsize=16,color="black",shape="triangle"];210 -> 215[label="",style="solid", color="black", weight=3];
10.35/4.48	212[label="randomSelect (aIOE IOError_FullError : aIOE IOError_PermDenied : AProVE_IO () : [])",fontsize=16,color="black",shape="box"];212 -> 216[label="",style="solid", color="black", weight=3];
10.35/4.48	213[label="ww8",fontsize=16,color="green",shape="box"];232[label="ww11",fontsize=16,color="green",shape="box"];214[label="ww51",fontsize=16,color="green",shape="box"];215[label="putChar ww50 >>= gtGt0 ww10",fontsize=16,color="black",shape="box"];215 -> 217[label="",style="solid", color="black", weight=3];
10.35/4.48	216[label="randomSelect2 (aIOE IOError_FullError : aIOE IOError_PermDenied : AProVE_IO () : [])",fontsize=16,color="black",shape="box"];216 -> 218[label="",style="solid", color="black", weight=3];
10.35/4.48	217 -> 92[label="",style="dashed", color="red", weight=0];
10.35/4.48	217[label="primbindIO (putChar ww50) (gtGt0 ww10)",fontsize=16,color="magenta"];217 -> 219[label="",style="dashed", color="magenta", weight=3];
10.35/4.48	217 -> 220[label="",style="dashed", color="magenta", weight=3];
10.35/4.48	218[label="randomSelect1 (aIOE IOError_FullError) (aIOE IOError_PermDenied : AProVE_IO () : []) terminator",fontsize=16,color="black",shape="box"];218 -> 221[label="",style="solid", color="black", weight=3];
10.35/4.48	219[label="ww10",fontsize=16,color="green",shape="box"];220[label="putChar ww50",fontsize=16,color="black",shape="box"];220 -> 222[label="",style="solid", color="black", weight=3];
10.35/4.48	221[label="randomSelect1 (aIOE IOError_FullError) (aIOE IOError_PermDenied : AProVE_IO () : []) ter5m",fontsize=16,color="burlywood",shape="box"];254[label="ter5m/False",fontsize=10,color="white",style="solid",shape="box"];221 -> 254[label="",style="solid", color="burlywood", weight=9];
10.35/4.48	254 -> 223[label="",style="solid", color="burlywood", weight=3];
10.35/4.48	255[label="ter5m/True",fontsize=10,color="white",style="solid",shape="box"];221 -> 255[label="",style="solid", color="burlywood", weight=9];
10.35/4.48	255 -> 224[label="",style="solid", color="burlywood", weight=3];
10.35/4.48	222 -> 225[label="",style="dashed", color="red", weight=0];
10.35/4.48	222[label="(seq ww50 output)",fontsize=16,color="magenta"];222 -> 228[label="",style="dashed", color="magenta", weight=3];
10.35/4.48	223[label="randomSelect1 (aIOE IOError_FullError) (aIOE IOError_PermDenied : AProVE_IO () : []) False",fontsize=16,color="black",shape="box"];223 -> 230[label="",style="solid", color="black", weight=3];
10.35/4.48	224[label="randomSelect1 (aIOE IOError_FullError) (aIOE IOError_PermDenied : AProVE_IO () : []) True",fontsize=16,color="black",shape="box"];224 -> 231[label="",style="solid", color="black", weight=3];
10.35/4.48	228 -> 204[label="",style="dashed", color="red", weight=0];
10.35/4.48	228[label="output",fontsize=16,color="magenta"];230[label="randomSelect0 (aIOE IOError_FullError) (aIOE IOError_PermDenied : AProVE_IO () : []) otherwise",fontsize=16,color="black",shape="box"];230 -> 233[label="",style="solid", color="black", weight=3];
10.35/4.48	231[label="randomSelect (aIOE IOError_PermDenied : AProVE_IO () : [])",fontsize=16,color="black",shape="box"];231 -> 234[label="",style="solid", color="black", weight=3];
10.35/4.48	233[label="randomSelect0 (aIOE IOError_FullError) (aIOE IOError_PermDenied : AProVE_IO () : []) True",fontsize=16,color="black",shape="box"];233 -> 235[label="",style="solid", color="black", weight=3];
10.35/4.48	234[label="randomSelect2 (aIOE IOError_PermDenied : AProVE_IO () : [])",fontsize=16,color="black",shape="box"];234 -> 236[label="",style="solid", color="black", weight=3];
10.35/4.48	235[label="aIOE IOError_FullError",fontsize=16,color="black",shape="box"];235 -> 237[label="",style="solid", color="black", weight=3];
10.35/4.48	236[label="randomSelect1 (aIOE IOError_PermDenied) (AProVE_IO () : []) terminator",fontsize=16,color="black",shape="box"];236 -> 238[label="",style="solid", color="black", weight=3];
10.35/4.48	237[label="AProVE_Exception (AET_IOError (IOError IOError_FullError [] [] Nothing))",fontsize=16,color="green",shape="box"];238[label="randomSelect1 (aIOE IOError_PermDenied) (AProVE_IO () : []) ter6m",fontsize=16,color="burlywood",shape="box"];256[label="ter6m/False",fontsize=10,color="white",style="solid",shape="box"];238 -> 256[label="",style="solid", color="burlywood", weight=9];
10.35/4.48	256 -> 239[label="",style="solid", color="burlywood", weight=3];
10.35/4.48	257[label="ter6m/True",fontsize=10,color="white",style="solid",shape="box"];238 -> 257[label="",style="solid", color="burlywood", weight=9];
10.35/4.48	257 -> 240[label="",style="solid", color="burlywood", weight=3];
10.35/4.48	239[label="randomSelect1 (aIOE IOError_PermDenied) (AProVE_IO () : []) False",fontsize=16,color="black",shape="box"];239 -> 241[label="",style="solid", color="black", weight=3];
10.35/4.48	240[label="randomSelect1 (aIOE IOError_PermDenied) (AProVE_IO () : []) True",fontsize=16,color="black",shape="box"];240 -> 242[label="",style="solid", color="black", weight=3];
10.35/4.48	241[label="randomSelect0 (aIOE IOError_PermDenied) (AProVE_IO () : []) otherwise",fontsize=16,color="black",shape="box"];241 -> 243[label="",style="solid", color="black", weight=3];
10.35/4.48	242[label="randomSelect (AProVE_IO () : [])",fontsize=16,color="black",shape="box"];242 -> 244[label="",style="solid", color="black", weight=3];
10.35/4.48	243[label="randomSelect0 (aIOE IOError_PermDenied) (AProVE_IO () : []) True",fontsize=16,color="black",shape="box"];243 -> 245[label="",style="solid", color="black", weight=3];
10.35/4.48	244[label="randomSelect3 (AProVE_IO () : [])",fontsize=16,color="black",shape="box"];244 -> 246[label="",style="solid", color="black", weight=3];
10.35/4.48	245[label="aIOE IOError_PermDenied",fontsize=16,color="black",shape="box"];245 -> 247[label="",style="solid", color="black", weight=3];
10.35/4.48	246[label="AProVE_IO ()",fontsize=16,color="green",shape="box"];247[label="AProVE_Exception (AET_IOError (IOError IOError_PermDenied [] [] Nothing))",fontsize=16,color="green",shape="box"];}
10.35/4.48	
10.35/4.48	----------------------------------------
10.35/4.48	
10.35/4.48	(10)
10.35/4.48	Obligation:
10.35/4.48	Q DP problem:
10.35/4.48	The TRS P consists of the following rules:
10.35/4.48	
10.35/4.48	   new_putStr(:(ww50, ww51)) -> new_putStr(ww51)
10.35/4.48	
10.35/4.48	R is empty.
10.35/4.48	Q is empty.
10.35/4.48	We have to consider all minimal (P,Q,R)-chains.
10.35/4.48	----------------------------------------
10.35/4.48	
10.35/4.48	(11) QDPSizeChangeProof (EQUIVALENT)
10.35/4.48	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. 
10.35/4.48	
10.35/4.48	From the DPs we obtained the following set of size-change graphs:
10.35/4.48	*new_putStr(:(ww50, ww51)) -> new_putStr(ww51)
10.35/4.48	The graph contains the following edges 1 > 1
10.35/4.48	
10.35/4.48	
10.35/4.48	----------------------------------------
10.35/4.48	
10.35/4.48	(12)
10.35/4.48	YES
10.35/4.52	EOF