8.42/3.70	YES
10.21/4.21	proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
10.21/4.21	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.21/4.21	
10.21/4.21	
10.21/4.21	H-Termination with start terms of the given HASKELL could be proven:
10.21/4.21	
10.21/4.21	(0) HASKELL
10.21/4.21	(1) BR [EQUIVALENT, 0 ms]
10.21/4.21	(2) HASKELL
10.21/4.21	(3) COR [EQUIVALENT, 0 ms]
10.21/4.21	(4) HASKELL
10.21/4.21	(5) NumRed [SOUND, 0 ms]
10.21/4.21	(6) HASKELL
10.21/4.21	(7) Narrow [SOUND, 0 ms]
10.21/4.21	(8) AND
10.21/4.21	    (9) QDP
10.21/4.21	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.21/4.21	        (11) YES
10.21/4.21	    (12) QDP
10.21/4.21	        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.21/4.21	        (14) YES
10.21/4.21	
10.21/4.21	
10.21/4.21	----------------------------------------
10.21/4.21	
10.21/4.21	(0)
10.21/4.21	Obligation:
10.21/4.21	mainModule Main 
10.21/4.21	module Main where {
10.21/4.21	import qualified Prelude;
10.21/4.21	}
10.21/4.21	
10.21/4.21	----------------------------------------
10.21/4.21	
10.21/4.21	(1) BR (EQUIVALENT)
10.21/4.21	Replaced joker patterns by fresh variables and removed binding patterns.
10.21/4.21	----------------------------------------
10.21/4.21	
10.21/4.21	(2)
10.21/4.21	Obligation:
10.21/4.21	mainModule Main 
10.21/4.21	module Main where {
10.21/4.21	import qualified Prelude;
10.21/4.21	}
10.21/4.21	
10.21/4.21	----------------------------------------
10.21/4.21	
10.21/4.21	(3) COR (EQUIVALENT)
10.21/4.21	Cond Reductions:
10.21/4.21	The following Function with conditions
10.21/4.21	"undefined |Falseundefined;
10.21/4.21	"
10.21/4.21	is transformed to
10.21/4.21	"undefined  = undefined1;
10.21/4.21	"
10.21/4.21	"undefined0 True = undefined;
10.21/4.21	"
10.21/4.21	"undefined1  = undefined0 False;
10.21/4.21	"
10.21/4.21	
10.21/4.21	----------------------------------------
10.21/4.21	
10.21/4.21	(4)
10.21/4.21	Obligation:
10.21/4.21	mainModule Main 
10.21/4.21	module Main where {
10.21/4.21	import qualified Prelude;
10.21/4.21	}
10.21/4.21	
10.21/4.21	----------------------------------------
10.21/4.21	
10.21/4.21	(5) NumRed (SOUND)
10.21/4.21	Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
10.21/4.21	----------------------------------------
10.21/4.21	
10.21/4.21	(6)
10.21/4.21	Obligation:
10.21/4.21	mainModule Main 
10.21/4.21	module Main where {
10.21/4.21	import qualified Prelude;
10.21/4.21	}
10.21/4.21	
10.21/4.21	----------------------------------------
10.21/4.21	
10.21/4.21	(7) Narrow (SOUND)
10.21/4.21	Haskell To QDPs
10.21/4.21	
10.21/4.21	digraph dp_graph {
10.21/4.21	node [outthreshold=100, inthreshold=100];1[label="isOctDigit",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.21/4.21	3[label="isOctDigit vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
10.21/4.21	4 -> 8[label="",style="dashed", color="red", weight=0];
10.21/4.21	4[label="vx3 >= Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))))))))))))))))))))))))))))))))))))) && vx3 <= Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))))))))))))))))))))))))))))))))))))))))))))",fontsize=16,color="magenta"];4 -> 9[label="",style="dashed", color="magenta", weight=3];
10.21/4.21	4 -> 10[label="",style="dashed", color="magenta", weight=3];
10.21/4.21	4 -> 11[label="",style="dashed", color="magenta", weight=3];
10.21/4.21	9[label="vx3",fontsize=16,color="green",shape="box"];10[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))",fontsize=16,color="green",shape="box"];11[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))))))))))))))))))))))))))))))))))))))))))",fontsize=16,color="green",shape="box"];8[label="vx8 >= Char (Succ vx9) && vx8 <= Char (Succ vx10)",fontsize=16,color="black",shape="triangle"];8 -> 15[label="",style="solid", color="black", weight=3];
10.21/4.21	15[label="compare vx8 (Char (Succ vx9)) /= LT && vx8 <= Char (Succ vx10)",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
10.21/4.21	16[label="not (compare vx8 (Char (Succ vx9)) == LT) && vx8 <= Char (Succ vx10)",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
10.21/4.21	17[label="not (primCmpChar vx8 (Char (Succ vx9)) == LT) && vx8 <= Char (Succ vx10)",fontsize=16,color="burlywood",shape="box"];143[label="vx8/Char vx80",fontsize=10,color="white",style="solid",shape="box"];17 -> 143[label="",style="solid", color="burlywood", weight=9];
10.21/4.21	143 -> 18[label="",style="solid", color="burlywood", weight=3];
10.21/4.21	18[label="not (primCmpChar (Char vx80) (Char (Succ vx9)) == LT) && Char vx80 <= Char (Succ vx10)",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
10.21/4.21	19 -> 85[label="",style="dashed", color="red", weight=0];
10.21/4.21	19[label="not (primCmpNat vx80 (Succ vx9) == LT) && Char vx80 <= Char (Succ vx10)",fontsize=16,color="magenta"];19 -> 86[label="",style="dashed", color="magenta", weight=3];
10.21/4.21	19 -> 87[label="",style="dashed", color="magenta", weight=3];
10.21/4.21	19 -> 88[label="",style="dashed", color="magenta", weight=3];
10.21/4.21	86[label="Succ vx9",fontsize=16,color="green",shape="box"];87[label="Char vx80 <= Char (Succ vx10)",fontsize=16,color="black",shape="box"];87 -> 99[label="",style="solid", color="black", weight=3];
10.21/4.21	88[label="vx80",fontsize=16,color="green",shape="box"];85[label="not (primCmpNat vx800000 vx9000 == LT) && vx12",fontsize=16,color="burlywood",shape="triangle"];144[label="vx800000/Succ vx8000000",fontsize=10,color="white",style="solid",shape="box"];85 -> 144[label="",style="solid", color="burlywood", weight=9];
10.21/4.21	144 -> 100[label="",style="solid", color="burlywood", weight=3];
10.21/4.21	145[label="vx800000/Zero",fontsize=10,color="white",style="solid",shape="box"];85 -> 145[label="",style="solid", color="burlywood", weight=9];
10.21/4.21	145 -> 101[label="",style="solid", color="burlywood", weight=3];
10.21/4.21	99[label="compare (Char vx80) (Char (Succ vx10)) /= GT",fontsize=16,color="black",shape="box"];99 -> 102[label="",style="solid", color="black", weight=3];
10.21/4.21	100[label="not (primCmpNat (Succ vx8000000) vx9000 == LT) && vx12",fontsize=16,color="burlywood",shape="box"];146[label="vx9000/Succ vx90000",fontsize=10,color="white",style="solid",shape="box"];100 -> 146[label="",style="solid", color="burlywood", weight=9];
10.21/4.21	146 -> 103[label="",style="solid", color="burlywood", weight=3];
10.21/4.21	147[label="vx9000/Zero",fontsize=10,color="white",style="solid",shape="box"];100 -> 147[label="",style="solid", color="burlywood", weight=9];
10.21/4.21	147 -> 104[label="",style="solid", color="burlywood", weight=3];
10.21/4.21	101[label="not (primCmpNat Zero vx9000 == LT) && vx12",fontsize=16,color="burlywood",shape="box"];148[label="vx9000/Succ vx90000",fontsize=10,color="white",style="solid",shape="box"];101 -> 148[label="",style="solid", color="burlywood", weight=9];
10.21/4.21	148 -> 105[label="",style="solid", color="burlywood", weight=3];
10.21/4.21	149[label="vx9000/Zero",fontsize=10,color="white",style="solid",shape="box"];101 -> 149[label="",style="solid", color="burlywood", weight=9];
10.21/4.21	149 -> 106[label="",style="solid", color="burlywood", weight=3];
10.21/4.21	102[label="not (compare (Char vx80) (Char (Succ vx10)) == GT)",fontsize=16,color="black",shape="box"];102 -> 107[label="",style="solid", color="black", weight=3];
10.21/4.21	103[label="not (primCmpNat (Succ vx8000000) (Succ vx90000) == LT) && vx12",fontsize=16,color="black",shape="box"];103 -> 108[label="",style="solid", color="black", weight=3];
10.21/4.21	104[label="not (primCmpNat (Succ vx8000000) Zero == LT) && vx12",fontsize=16,color="black",shape="box"];104 -> 109[label="",style="solid", color="black", weight=3];
10.21/4.21	105[label="not (primCmpNat Zero (Succ vx90000) == LT) && vx12",fontsize=16,color="black",shape="box"];105 -> 110[label="",style="solid", color="black", weight=3];
10.21/4.21	106[label="not (primCmpNat Zero Zero == LT) && vx12",fontsize=16,color="black",shape="box"];106 -> 111[label="",style="solid", color="black", weight=3];
10.21/4.21	107[label="not (primCmpChar (Char vx80) (Char (Succ vx10)) == GT)",fontsize=16,color="black",shape="box"];107 -> 112[label="",style="solid", color="black", weight=3];
10.21/4.21	108 -> 85[label="",style="dashed", color="red", weight=0];
10.21/4.21	108[label="not (primCmpNat vx8000000 vx90000 == LT) && vx12",fontsize=16,color="magenta"];108 -> 113[label="",style="dashed", color="magenta", weight=3];
10.21/4.21	108 -> 114[label="",style="dashed", color="magenta", weight=3];
10.21/4.21	109[label="not (GT == LT) && vx12",fontsize=16,color="black",shape="box"];109 -> 115[label="",style="solid", color="black", weight=3];
10.21/4.21	110[label="not (LT == LT) && vx12",fontsize=16,color="black",shape="box"];110 -> 116[label="",style="solid", color="black", weight=3];
10.21/4.21	111[label="not (EQ == LT) && vx12",fontsize=16,color="black",shape="box"];111 -> 117[label="",style="solid", color="black", weight=3];
10.21/4.21	112[label="not (primCmpNat vx80 (Succ vx10) == GT)",fontsize=16,color="burlywood",shape="box"];150[label="vx80/Succ vx800",fontsize=10,color="white",style="solid",shape="box"];112 -> 150[label="",style="solid", color="burlywood", weight=9];
10.21/4.21	150 -> 118[label="",style="solid", color="burlywood", weight=3];
10.21/4.21	151[label="vx80/Zero",fontsize=10,color="white",style="solid",shape="box"];112 -> 151[label="",style="solid", color="burlywood", weight=9];
10.21/4.21	151 -> 119[label="",style="solid", color="burlywood", weight=3];
10.21/4.21	113[label="vx90000",fontsize=16,color="green",shape="box"];114[label="vx8000000",fontsize=16,color="green",shape="box"];115[label="not False && vx12",fontsize=16,color="black",shape="triangle"];115 -> 120[label="",style="solid", color="black", weight=3];
10.21/4.21	116[label="not True && vx12",fontsize=16,color="black",shape="box"];116 -> 121[label="",style="solid", color="black", weight=3];
10.21/4.21	117 -> 115[label="",style="dashed", color="red", weight=0];
10.21/4.21	117[label="not False && vx12",fontsize=16,color="magenta"];118[label="not (primCmpNat (Succ vx800) (Succ vx10) == GT)",fontsize=16,color="black",shape="box"];118 -> 122[label="",style="solid", color="black", weight=3];
10.21/4.21	119[label="not (primCmpNat Zero (Succ vx10) == GT)",fontsize=16,color="black",shape="box"];119 -> 123[label="",style="solid", color="black", weight=3];
10.21/4.21	120[label="True && vx12",fontsize=16,color="black",shape="box"];120 -> 124[label="",style="solid", color="black", weight=3];
10.21/4.21	121[label="False && vx12",fontsize=16,color="black",shape="box"];121 -> 125[label="",style="solid", color="black", weight=3];
10.21/4.21	122[label="not (primCmpNat vx800 vx10 == GT)",fontsize=16,color="burlywood",shape="triangle"];152[label="vx800/Succ vx8000",fontsize=10,color="white",style="solid",shape="box"];122 -> 152[label="",style="solid", color="burlywood", weight=9];
10.21/4.21	152 -> 126[label="",style="solid", color="burlywood", weight=3];
10.21/4.21	153[label="vx800/Zero",fontsize=10,color="white",style="solid",shape="box"];122 -> 153[label="",style="solid", color="burlywood", weight=9];
10.21/4.21	153 -> 127[label="",style="solid", color="burlywood", weight=3];
10.21/4.21	123[label="not (LT == GT)",fontsize=16,color="black",shape="triangle"];123 -> 128[label="",style="solid", color="black", weight=3];
10.21/4.21	124[label="vx12",fontsize=16,color="green",shape="box"];125[label="False",fontsize=16,color="green",shape="box"];126[label="not (primCmpNat (Succ vx8000) vx10 == GT)",fontsize=16,color="burlywood",shape="box"];154[label="vx10/Succ vx100",fontsize=10,color="white",style="solid",shape="box"];126 -> 154[label="",style="solid", color="burlywood", weight=9];
10.21/4.21	154 -> 129[label="",style="solid", color="burlywood", weight=3];
10.21/4.21	155[label="vx10/Zero",fontsize=10,color="white",style="solid",shape="box"];126 -> 155[label="",style="solid", color="burlywood", weight=9];
10.21/4.21	155 -> 130[label="",style="solid", color="burlywood", weight=3];
10.21/4.21	127[label="not (primCmpNat Zero vx10 == GT)",fontsize=16,color="burlywood",shape="box"];156[label="vx10/Succ vx100",fontsize=10,color="white",style="solid",shape="box"];127 -> 156[label="",style="solid", color="burlywood", weight=9];
10.21/4.21	156 -> 131[label="",style="solid", color="burlywood", weight=3];
10.21/4.21	157[label="vx10/Zero",fontsize=10,color="white",style="solid",shape="box"];127 -> 157[label="",style="solid", color="burlywood", weight=9];
10.21/4.21	157 -> 132[label="",style="solid", color="burlywood", weight=3];
10.21/4.21	128[label="not False",fontsize=16,color="black",shape="triangle"];128 -> 133[label="",style="solid", color="black", weight=3];
10.21/4.21	129[label="not (primCmpNat (Succ vx8000) (Succ vx100) == GT)",fontsize=16,color="black",shape="box"];129 -> 134[label="",style="solid", color="black", weight=3];
10.21/4.21	130[label="not (primCmpNat (Succ vx8000) Zero == GT)",fontsize=16,color="black",shape="box"];130 -> 135[label="",style="solid", color="black", weight=3];
10.21/4.21	131[label="not (primCmpNat Zero (Succ vx100) == GT)",fontsize=16,color="black",shape="box"];131 -> 136[label="",style="solid", color="black", weight=3];
10.21/4.21	132[label="not (primCmpNat Zero Zero == GT)",fontsize=16,color="black",shape="box"];132 -> 137[label="",style="solid", color="black", weight=3];
10.21/4.21	133[label="True",fontsize=16,color="green",shape="box"];134 -> 122[label="",style="dashed", color="red", weight=0];
10.21/4.21	134[label="not (primCmpNat vx8000 vx100 == GT)",fontsize=16,color="magenta"];134 -> 138[label="",style="dashed", color="magenta", weight=3];
10.21/4.21	134 -> 139[label="",style="dashed", color="magenta", weight=3];
10.21/4.21	135[label="not (GT == GT)",fontsize=16,color="black",shape="box"];135 -> 140[label="",style="solid", color="black", weight=3];
10.21/4.21	136 -> 123[label="",style="dashed", color="red", weight=0];
10.21/4.21	136[label="not (LT == GT)",fontsize=16,color="magenta"];137[label="not (EQ == GT)",fontsize=16,color="black",shape="box"];137 -> 141[label="",style="solid", color="black", weight=3];
10.21/4.21	138[label="vx8000",fontsize=16,color="green",shape="box"];139[label="vx100",fontsize=16,color="green",shape="box"];140[label="not True",fontsize=16,color="black",shape="box"];140 -> 142[label="",style="solid", color="black", weight=3];
10.21/4.21	141 -> 128[label="",style="dashed", color="red", weight=0];
10.21/4.21	141[label="not False",fontsize=16,color="magenta"];142[label="False",fontsize=16,color="green",shape="box"];}
10.21/4.21	
10.21/4.21	----------------------------------------
10.21/4.21	
10.21/4.21	(8)
10.21/4.21	Complex Obligation (AND)
10.21/4.21	
10.21/4.21	----------------------------------------
10.21/4.21	
10.21/4.21	(9)
10.21/4.21	Obligation:
10.21/4.21	Q DP problem:
10.21/4.21	The TRS P consists of the following rules:
10.21/4.21	
10.21/4.21	   new_not(Succ(vx8000), Succ(vx100)) -> new_not(vx8000, vx100)
10.21/4.21	
10.21/4.21	R is empty.
10.21/4.21	Q is empty.
10.21/4.21	We have to consider all minimal (P,Q,R)-chains.
10.21/4.21	----------------------------------------
10.21/4.21	
10.21/4.21	(10) QDPSizeChangeProof (EQUIVALENT)
10.21/4.21	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. 
10.21/4.21	
10.21/4.21	From the DPs we obtained the following set of size-change graphs:
10.21/4.21	*new_not(Succ(vx8000), Succ(vx100)) -> new_not(vx8000, vx100)
10.21/4.21	The graph contains the following edges 1 > 1, 2 > 2
10.21/4.21	
10.21/4.21	
10.21/4.21	----------------------------------------
10.21/4.21	
10.21/4.21	(11)
10.21/4.21	YES
10.21/4.21	
10.21/4.21	----------------------------------------
10.21/4.21	
10.21/4.21	(12)
10.21/4.21	Obligation:
10.21/4.21	Q DP problem:
10.21/4.21	The TRS P consists of the following rules:
10.21/4.21	
10.21/4.21	   new_asAs(Succ(vx8000000), Succ(vx90000), vx12) -> new_asAs(vx8000000, vx90000, vx12)
10.21/4.21	
10.21/4.21	R is empty.
10.21/4.21	Q is empty.
10.21/4.21	We have to consider all minimal (P,Q,R)-chains.
10.21/4.21	----------------------------------------
10.21/4.21	
10.21/4.21	(13) QDPSizeChangeProof (EQUIVALENT)
10.21/4.21	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. 
10.21/4.21	
10.21/4.21	From the DPs we obtained the following set of size-change graphs:
10.21/4.21	*new_asAs(Succ(vx8000000), Succ(vx90000), vx12) -> new_asAs(vx8000000, vx90000, vx12)
10.21/4.21	The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
10.21/4.21	
10.21/4.21	
10.21/4.21	----------------------------------------
10.21/4.21	
10.21/4.21	(14)
10.21/4.21	YES
10.41/4.27	EOF