9.45/3.92	YES
11.32/4.51	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
11.32/4.51	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.32/4.51	
11.32/4.51	
11.32/4.51	H-Termination with start terms of the given HASKELL could be proven:
11.32/4.51	
11.32/4.51	(0) HASKELL
11.32/4.51	(1) LR [EQUIVALENT, 0 ms]
11.32/4.51	(2) HASKELL
11.32/4.51	(3) BR [EQUIVALENT, 0 ms]
11.32/4.51	(4) HASKELL
11.32/4.51	(5) COR [EQUIVALENT, 0 ms]
11.32/4.51	(6) HASKELL
11.32/4.51	(7) Narrow [SOUND, 0 ms]
11.32/4.51	(8) AND
11.32/4.51	    (9) QDP
11.32/4.51	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.32/4.51	        (11) YES
11.32/4.51	    (12) QDP
11.32/4.51	        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.32/4.51	        (14) YES
11.32/4.51	    (15) QDP
11.32/4.51	        (16) DependencyGraphProof [EQUIVALENT, 0 ms]
11.32/4.51	        (17) AND
11.32/4.51	            (18) QDP
11.32/4.51	                (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.32/4.51	                (20) YES
11.32/4.51	            (21) QDP
11.32/4.51	                (22) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.32/4.51	                (23) YES
11.32/4.51	            (24) QDP
11.32/4.51	                (25) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.32/4.51	                (26) YES
11.32/4.51	
11.32/4.51	
11.32/4.51	----------------------------------------
11.32/4.51	
11.32/4.51	(0)
11.32/4.51	Obligation:
11.32/4.51	mainModule Main 
11.32/4.51	module Maybe where {
11.32/4.51	import qualified Main;
11.32/4.51	import qualified Monad;
11.32/4.51	import qualified Prelude;
11.32/4.51	}
11.32/4.51	module Main where {
11.32/4.51	import qualified Maybe;
11.32/4.51	import qualified Monad;
11.32/4.51	import qualified Prelude;
11.32/4.51	}
11.32/4.51	module Monad where {
11.32/4.51	import qualified Main;
11.32/4.51	import qualified Maybe;
11.32/4.51	import qualified Prelude;
11.32/4.51	zipWithM_ :: Monad c => (d  ->  a  ->  c b)  ->  [d]  ->  [a]  ->  c ();
11.32/4.51	zipWithM_ f xs ys = sequence_ (zipWith f xs ys);
11.32/4.51	
11.32/4.51	}
11.32/4.51	
11.32/4.51	----------------------------------------
11.32/4.51	
11.32/4.51	(1) LR (EQUIVALENT)
11.32/4.51	Lambda Reductions:
11.32/4.51	The following Lambda expression
11.32/4.51	"\_->q"
11.32/4.51	is transformed to
11.32/4.51	"gtGt0 q _ = q;
11.32/4.51	"
11.32/4.51	
11.32/4.51	----------------------------------------
11.32/4.51	
11.32/4.51	(2)
11.32/4.51	Obligation:
11.32/4.51	mainModule Main 
11.32/4.51	module Maybe where {
11.32/4.51	import qualified Main;
11.32/4.51	import qualified Monad;
11.32/4.51	import qualified Prelude;
11.32/4.51	}
11.32/4.51	module Main where {
11.32/4.51	import qualified Maybe;
11.32/4.51	import qualified Monad;
11.32/4.51	import qualified Prelude;
11.32/4.51	}
11.32/4.51	module Monad where {
11.32/4.51	import qualified Main;
11.32/4.51	import qualified Maybe;
11.32/4.51	import qualified Prelude;
11.32/4.51	zipWithM_ :: Monad c => (b  ->  a  ->  c d)  ->  [b]  ->  [a]  ->  c ();
11.32/4.51	zipWithM_ f xs ys = sequence_ (zipWith f xs ys);
11.32/4.51	
11.32/4.51	}
11.32/4.51	
11.32/4.51	----------------------------------------
11.32/4.51	
11.32/4.51	(3) BR (EQUIVALENT)
11.32/4.51	Replaced joker patterns by fresh variables and removed binding patterns.
11.32/4.51	----------------------------------------
11.32/4.51	
11.32/4.51	(4)
11.32/4.51	Obligation:
11.32/4.51	mainModule Main 
11.32/4.51	module Maybe where {
11.32/4.51	import qualified Main;
11.32/4.51	import qualified Monad;
11.32/4.51	import qualified Prelude;
11.32/4.51	}
11.32/4.51	module Main where {
11.32/4.51	import qualified Maybe;
11.32/4.51	import qualified Monad;
11.32/4.51	import qualified Prelude;
11.32/4.51	}
11.32/4.51	module Monad where {
11.32/4.51	import qualified Main;
11.32/4.51	import qualified Maybe;
11.32/4.51	import qualified Prelude;
11.32/4.51	zipWithM_ :: Monad c => (d  ->  b  ->  c a)  ->  [d]  ->  [b]  ->  c ();
11.32/4.51	zipWithM_ f xs ys = sequence_ (zipWith f xs ys);
11.32/4.51	
11.32/4.51	}
11.32/4.51	
11.32/4.51	----------------------------------------
11.32/4.51	
11.32/4.51	(5) COR (EQUIVALENT)
11.32/4.51	Cond Reductions:
11.32/4.51	The following Function with conditions
11.32/4.51	"undefined |Falseundefined;
11.32/4.51	"
11.32/4.51	is transformed to
11.32/4.51	"undefined  = undefined1;
11.32/4.51	"
11.32/4.51	"undefined0 True = undefined;
11.32/4.51	"
11.32/4.51	"undefined1  = undefined0 False;
11.32/4.51	"
11.32/4.51	
11.32/4.51	----------------------------------------
11.32/4.51	
11.32/4.51	(6)
11.32/4.51	Obligation:
11.32/4.51	mainModule Main 
11.32/4.51	module Maybe where {
11.32/4.51	import qualified Main;
11.32/4.51	import qualified Monad;
11.32/4.51	import qualified Prelude;
11.32/4.51	}
11.32/4.51	module Main where {
11.32/4.51	import qualified Maybe;
11.32/4.51	import qualified Monad;
11.32/4.51	import qualified Prelude;
11.32/4.51	}
11.32/4.51	module Monad where {
11.32/4.51	import qualified Main;
11.32/4.51	import qualified Maybe;
11.32/4.51	import qualified Prelude;
11.32/4.51	zipWithM_ :: Monad d => (c  ->  b  ->  d a)  ->  [c]  ->  [b]  ->  d ();
11.32/4.51	zipWithM_ f xs ys = sequence_ (zipWith f xs ys);
11.32/4.51	
11.32/4.51	}
11.32/4.51	
11.32/4.51	----------------------------------------
11.32/4.51	
11.32/4.51	(7) Narrow (SOUND)
11.32/4.51	Haskell To QDPs
11.32/4.51	
11.32/4.51	digraph dp_graph {
11.32/4.51	node [outthreshold=100, inthreshold=100];1[label="Monad.zipWithM_",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.32/4.51	3[label="Monad.zipWithM_ ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
11.32/4.51	4[label="Monad.zipWithM_ ww3 ww4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
11.32/4.51	5[label="Monad.zipWithM_ ww3 ww4 ww5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
11.32/4.51	6[label="sequence_ (zipWith ww3 ww4 ww5)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
11.32/4.51	7[label="foldr (>>) (return ()) (zipWith ww3 ww4 ww5)",fontsize=16,color="burlywood",shape="triangle"];89[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];7 -> 89[label="",style="solid", color="burlywood", weight=9];
11.32/4.51	89 -> 8[label="",style="solid", color="burlywood", weight=3];
11.32/4.51	90[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 90[label="",style="solid", color="burlywood", weight=9];
11.32/4.51	90 -> 9[label="",style="solid", color="burlywood", weight=3];
11.32/4.51	8[label="foldr (>>) (return ()) (zipWith ww3 (ww40 : ww41) ww5)",fontsize=16,color="burlywood",shape="box"];91[label="ww5/ww50 : ww51",fontsize=10,color="white",style="solid",shape="box"];8 -> 91[label="",style="solid", color="burlywood", weight=9];
11.32/4.51	91 -> 10[label="",style="solid", color="burlywood", weight=3];
11.32/4.51	92[label="ww5/[]",fontsize=10,color="white",style="solid",shape="box"];8 -> 92[label="",style="solid", color="burlywood", weight=9];
11.32/4.51	92 -> 11[label="",style="solid", color="burlywood", weight=3];
11.32/4.51	9[label="foldr (>>) (return ()) (zipWith ww3 [] ww5)",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3];
11.32/4.51	10[label="foldr (>>) (return ()) (zipWith ww3 (ww40 : ww41) (ww50 : ww51))",fontsize=16,color="black",shape="box"];10 -> 13[label="",style="solid", color="black", weight=3];
11.32/4.51	11[label="foldr (>>) (return ()) (zipWith ww3 (ww40 : ww41) [])",fontsize=16,color="black",shape="box"];11 -> 14[label="",style="solid", color="black", weight=3];
11.32/4.51	12[label="foldr (>>) (return ()) []",fontsize=16,color="black",shape="triangle"];12 -> 15[label="",style="solid", color="black", weight=3];
11.32/4.51	13[label="foldr (>>) (return ()) (ww3 ww40 ww50 : zipWith ww3 ww41 ww51)",fontsize=16,color="black",shape="box"];13 -> 16[label="",style="solid", color="black", weight=3];
11.32/4.51	14 -> 12[label="",style="dashed", color="red", weight=0];
11.32/4.51	14[label="foldr (>>) (return ()) []",fontsize=16,color="magenta"];15[label="return ()",fontsize=16,color="blue",shape="box"];93[label="return :: () -> Maybe ()",fontsize=10,color="white",style="solid",shape="box"];15 -> 93[label="",style="solid", color="blue", weight=9];
11.32/4.51	93 -> 17[label="",style="solid", color="blue", weight=3];
11.32/4.51	94[label="return :: () -> IO ()",fontsize=10,color="white",style="solid",shape="box"];15 -> 94[label="",style="solid", color="blue", weight=9];
11.32/4.51	94 -> 18[label="",style="solid", color="blue", weight=3];
11.32/4.51	95[label="return :: () -> [] ()",fontsize=10,color="white",style="solid",shape="box"];15 -> 95[label="",style="solid", color="blue", weight=9];
11.32/4.51	95 -> 19[label="",style="solid", color="blue", weight=3];
11.32/4.51	16[label="(>>) ww3 ww40 ww50 foldr (>>) (return ()) (zipWith ww3 ww41 ww51)",fontsize=16,color="blue",shape="box"];96[label=">> :: (Maybe a) -> (Maybe ()) -> Maybe ()",fontsize=10,color="white",style="solid",shape="box"];16 -> 96[label="",style="solid", color="blue", weight=9];
11.32/4.51	96 -> 31[label="",style="solid", color="blue", weight=3];
11.32/4.51	97[label=">> :: (IO a) -> (IO ()) -> IO ()",fontsize=10,color="white",style="solid",shape="box"];16 -> 97[label="",style="solid", color="blue", weight=9];
11.32/4.51	97 -> 32[label="",style="solid", color="blue", weight=3];
11.32/4.51	98[label=">> :: ([] a) -> ([] ()) -> [] ()",fontsize=10,color="white",style="solid",shape="box"];16 -> 98[label="",style="solid", color="blue", weight=9];
11.32/4.51	98 -> 33[label="",style="solid", color="blue", weight=3];
11.32/4.51	17[label="return ()",fontsize=16,color="black",shape="box"];17 -> 22[label="",style="solid", color="black", weight=3];
11.32/4.51	18[label="return ()",fontsize=16,color="black",shape="box"];18 -> 23[label="",style="solid", color="black", weight=3];
11.32/4.51	19[label="return ()",fontsize=16,color="black",shape="box"];19 -> 24[label="",style="solid", color="black", weight=3];
11.32/4.51	31 -> 27[label="",style="dashed", color="red", weight=0];
11.32/4.51	31[label="(>>) ww3 ww40 ww50 foldr (>>) (return ()) (zipWith ww3 ww41 ww51)",fontsize=16,color="magenta"];31 -> 37[label="",style="dashed", color="magenta", weight=3];
11.32/4.51	32 -> 28[label="",style="dashed", color="red", weight=0];
11.32/4.51	32[label="(>>) ww3 ww40 ww50 foldr (>>) (return ()) (zipWith ww3 ww41 ww51)",fontsize=16,color="magenta"];32 -> 38[label="",style="dashed", color="magenta", weight=3];
11.32/4.51	33 -> 29[label="",style="dashed", color="red", weight=0];
11.32/4.51	33[label="(>>) ww3 ww40 ww50 foldr (>>) (return ()) (zipWith ww3 ww41 ww51)",fontsize=16,color="magenta"];33 -> 39[label="",style="dashed", color="magenta", weight=3];
11.32/4.51	22[label="Just ()",fontsize=16,color="green",shape="box"];23[label="primretIO ()",fontsize=16,color="black",shape="box"];23 -> 30[label="",style="solid", color="black", weight=3];
11.32/4.51	24[label="() : []",fontsize=16,color="green",shape="box"];37 -> 7[label="",style="dashed", color="red", weight=0];
11.32/4.51	37[label="foldr (>>) (return ()) (zipWith ww3 ww41 ww51)",fontsize=16,color="magenta"];37 -> 42[label="",style="dashed", color="magenta", weight=3];
11.32/4.51	37 -> 43[label="",style="dashed", color="magenta", weight=3];
11.32/4.51	27[label="(>>) ww3 ww40 ww50 ww6",fontsize=16,color="black",shape="triangle"];27 -> 34[label="",style="solid", color="black", weight=3];
11.32/4.51	38 -> 7[label="",style="dashed", color="red", weight=0];
11.32/4.51	38[label="foldr (>>) (return ()) (zipWith ww3 ww41 ww51)",fontsize=16,color="magenta"];38 -> 44[label="",style="dashed", color="magenta", weight=3];
11.32/4.51	38 -> 45[label="",style="dashed", color="magenta", weight=3];
11.32/4.51	28[label="(>>) ww3 ww40 ww50 ww6",fontsize=16,color="black",shape="triangle"];28 -> 35[label="",style="solid", color="black", weight=3];
11.32/4.51	39 -> 7[label="",style="dashed", color="red", weight=0];
11.32/4.51	39[label="foldr (>>) (return ()) (zipWith ww3 ww41 ww51)",fontsize=16,color="magenta"];39 -> 46[label="",style="dashed", color="magenta", weight=3];
11.32/4.51	39 -> 47[label="",style="dashed", color="magenta", weight=3];
11.32/4.51	29[label="(>>) ww3 ww40 ww50 ww6",fontsize=16,color="black",shape="triangle"];29 -> 36[label="",style="solid", color="black", weight=3];
11.32/4.51	30[label="AProVE_IO ()",fontsize=16,color="green",shape="box"];42[label="ww51",fontsize=16,color="green",shape="box"];43[label="ww41",fontsize=16,color="green",shape="box"];34 -> 40[label="",style="dashed", color="red", weight=0];
11.32/4.51	34[label="ww3 ww40 ww50 >>= gtGt0 ww6",fontsize=16,color="magenta"];34 -> 41[label="",style="dashed", color="magenta", weight=3];
11.32/4.51	44[label="ww51",fontsize=16,color="green",shape="box"];45[label="ww41",fontsize=16,color="green",shape="box"];35[label="ww3 ww40 ww50 >>= gtGt0 ww6",fontsize=16,color="black",shape="box"];35 -> 48[label="",style="solid", color="black", weight=3];
11.32/4.51	46[label="ww51",fontsize=16,color="green",shape="box"];47[label="ww41",fontsize=16,color="green",shape="box"];36 -> 49[label="",style="dashed", color="red", weight=0];
11.32/4.51	36[label="ww3 ww40 ww50 >>= gtGt0 ww6",fontsize=16,color="magenta"];36 -> 50[label="",style="dashed", color="magenta", weight=3];
11.32/4.51	41[label="ww3 ww40 ww50",fontsize=16,color="green",shape="box"];41 -> 51[label="",style="dashed", color="green", weight=3];
11.32/4.51	41 -> 52[label="",style="dashed", color="green", weight=3];
11.32/4.51	40[label="ww7 >>= gtGt0 ww6",fontsize=16,color="burlywood",shape="triangle"];99[label="ww7/Nothing",fontsize=10,color="white",style="solid",shape="box"];40 -> 99[label="",style="solid", color="burlywood", weight=9];
11.32/4.51	99 -> 53[label="",style="solid", color="burlywood", weight=3];
11.32/4.51	100[label="ww7/Just ww70",fontsize=10,color="white",style="solid",shape="box"];40 -> 100[label="",style="solid", color="burlywood", weight=9];
11.32/4.51	100 -> 54[label="",style="solid", color="burlywood", weight=3];
11.32/4.51	48 -> 55[label="",style="dashed", color="red", weight=0];
11.32/4.51	48[label="primbindIO (ww3 ww40 ww50) (gtGt0 ww6)",fontsize=16,color="magenta"];48 -> 56[label="",style="dashed", color="magenta", weight=3];
11.32/4.51	50[label="ww3 ww40 ww50",fontsize=16,color="green",shape="box"];50 -> 57[label="",style="dashed", color="green", weight=3];
11.32/4.51	50 -> 58[label="",style="dashed", color="green", weight=3];
11.32/4.51	49[label="ww8 >>= gtGt0 ww6",fontsize=16,color="burlywood",shape="triangle"];101[label="ww8/ww80 : ww81",fontsize=10,color="white",style="solid",shape="box"];49 -> 101[label="",style="solid", color="burlywood", weight=9];
11.32/4.51	101 -> 59[label="",style="solid", color="burlywood", weight=3];
11.32/4.51	102[label="ww8/[]",fontsize=10,color="white",style="solid",shape="box"];49 -> 102[label="",style="solid", color="burlywood", weight=9];
11.32/4.51	102 -> 60[label="",style="solid", color="burlywood", weight=3];
11.32/4.51	51[label="ww40",fontsize=16,color="green",shape="box"];52[label="ww50",fontsize=16,color="green",shape="box"];53[label="Nothing >>= gtGt0 ww6",fontsize=16,color="black",shape="box"];53 -> 61[label="",style="solid", color="black", weight=3];
11.32/4.51	54[label="Just ww70 >>= gtGt0 ww6",fontsize=16,color="black",shape="box"];54 -> 62[label="",style="solid", color="black", weight=3];
11.32/4.51	56[label="ww3 ww40 ww50",fontsize=16,color="green",shape="box"];56 -> 69[label="",style="dashed", color="green", weight=3];
11.32/4.51	56 -> 70[label="",style="dashed", color="green", weight=3];
11.32/4.51	55[label="primbindIO ww9 (gtGt0 ww6)",fontsize=16,color="burlywood",shape="triangle"];103[label="ww9/IO ww90",fontsize=10,color="white",style="solid",shape="box"];55 -> 103[label="",style="solid", color="burlywood", weight=9];
11.32/4.51	103 -> 65[label="",style="solid", color="burlywood", weight=3];
11.32/4.51	104[label="ww9/AProVE_IO ww90",fontsize=10,color="white",style="solid",shape="box"];55 -> 104[label="",style="solid", color="burlywood", weight=9];
11.32/4.51	104 -> 66[label="",style="solid", color="burlywood", weight=3];
11.32/4.51	105[label="ww9/AProVE_Exception ww90",fontsize=10,color="white",style="solid",shape="box"];55 -> 105[label="",style="solid", color="burlywood", weight=9];
11.32/4.51	105 -> 67[label="",style="solid", color="burlywood", weight=3];
11.32/4.51	106[label="ww9/AProVE_Error ww90",fontsize=10,color="white",style="solid",shape="box"];55 -> 106[label="",style="solid", color="burlywood", weight=9];
11.32/4.51	106 -> 68[label="",style="solid", color="burlywood", weight=3];
11.32/4.51	57[label="ww40",fontsize=16,color="green",shape="box"];58[label="ww50",fontsize=16,color="green",shape="box"];59[label="ww80 : ww81 >>= gtGt0 ww6",fontsize=16,color="black",shape="box"];59 -> 71[label="",style="solid", color="black", weight=3];
11.32/4.51	60[label="[] >>= gtGt0 ww6",fontsize=16,color="black",shape="box"];60 -> 72[label="",style="solid", color="black", weight=3];
11.32/4.51	61[label="Nothing",fontsize=16,color="green",shape="box"];62[label="gtGt0 ww6 ww70",fontsize=16,color="black",shape="box"];62 -> 73[label="",style="solid", color="black", weight=3];
11.32/4.51	69[label="ww40",fontsize=16,color="green",shape="box"];70[label="ww50",fontsize=16,color="green",shape="box"];65[label="primbindIO (IO ww90) (gtGt0 ww6)",fontsize=16,color="black",shape="box"];65 -> 74[label="",style="solid", color="black", weight=3];
11.32/4.51	66[label="primbindIO (AProVE_IO ww90) (gtGt0 ww6)",fontsize=16,color="black",shape="box"];66 -> 75[label="",style="solid", color="black", weight=3];
11.32/4.51	67[label="primbindIO (AProVE_Exception ww90) (gtGt0 ww6)",fontsize=16,color="black",shape="box"];67 -> 76[label="",style="solid", color="black", weight=3];
11.32/4.51	68[label="primbindIO (AProVE_Error ww90) (gtGt0 ww6)",fontsize=16,color="black",shape="box"];68 -> 77[label="",style="solid", color="black", weight=3];
11.32/4.51	71 -> 78[label="",style="dashed", color="red", weight=0];
11.32/4.51	71[label="gtGt0 ww6 ww80 ++ (ww81 >>= gtGt0 ww6)",fontsize=16,color="magenta"];71 -> 79[label="",style="dashed", color="magenta", weight=3];
11.32/4.51	72[label="[]",fontsize=16,color="green",shape="box"];73[label="ww6",fontsize=16,color="green",shape="box"];74[label="error []",fontsize=16,color="red",shape="box"];75[label="gtGt0 ww6 ww90",fontsize=16,color="black",shape="box"];75 -> 80[label="",style="solid", color="black", weight=3];
11.32/4.51	76[label="AProVE_Exception ww90",fontsize=16,color="green",shape="box"];77[label="AProVE_Error ww90",fontsize=16,color="green",shape="box"];79 -> 49[label="",style="dashed", color="red", weight=0];
11.32/4.51	79[label="ww81 >>= gtGt0 ww6",fontsize=16,color="magenta"];79 -> 81[label="",style="dashed", color="magenta", weight=3];
11.32/4.51	78[label="gtGt0 ww6 ww80 ++ ww10",fontsize=16,color="black",shape="triangle"];78 -> 82[label="",style="solid", color="black", weight=3];
11.32/4.51	80[label="ww6",fontsize=16,color="green",shape="box"];81[label="ww81",fontsize=16,color="green",shape="box"];82[label="ww6 ++ ww10",fontsize=16,color="burlywood",shape="triangle"];107[label="ww6/ww60 : ww61",fontsize=10,color="white",style="solid",shape="box"];82 -> 107[label="",style="solid", color="burlywood", weight=9];
11.32/4.51	107 -> 83[label="",style="solid", color="burlywood", weight=3];
11.32/4.51	108[label="ww6/[]",fontsize=10,color="white",style="solid",shape="box"];82 -> 108[label="",style="solid", color="burlywood", weight=9];
11.32/4.51	108 -> 84[label="",style="solid", color="burlywood", weight=3];
11.32/4.51	83[label="(ww60 : ww61) ++ ww10",fontsize=16,color="black",shape="box"];83 -> 85[label="",style="solid", color="black", weight=3];
11.32/4.51	84[label="[] ++ ww10",fontsize=16,color="black",shape="box"];84 -> 86[label="",style="solid", color="black", weight=3];
11.32/4.51	85[label="ww60 : ww61 ++ ww10",fontsize=16,color="green",shape="box"];85 -> 87[label="",style="dashed", color="green", weight=3];
11.32/4.51	86[label="ww10",fontsize=16,color="green",shape="box"];87 -> 82[label="",style="dashed", color="red", weight=0];
11.32/4.51	87[label="ww61 ++ ww10",fontsize=16,color="magenta"];87 -> 88[label="",style="dashed", color="magenta", weight=3];
11.32/4.51	88[label="ww61",fontsize=16,color="green",shape="box"];}
11.32/4.51	
11.32/4.51	----------------------------------------
11.32/4.51	
11.32/4.51	(8)
11.32/4.51	Complex Obligation (AND)
11.32/4.51	
11.32/4.51	----------------------------------------
11.32/4.51	
11.32/4.51	(9)
11.32/4.51	Obligation:
11.32/4.51	Q DP problem:
11.32/4.51	The TRS P consists of the following rules:
11.32/4.51	
11.32/4.51	   new_gtGtEs(:(ww80, ww81), ww6, h) -> new_gtGtEs(ww81, ww6, h)
11.32/4.51	
11.32/4.51	R is empty.
11.32/4.51	Q is empty.
11.32/4.51	We have to consider all minimal (P,Q,R)-chains.
11.32/4.51	----------------------------------------
11.32/4.51	
11.32/4.51	(10) QDPSizeChangeProof (EQUIVALENT)
11.32/4.51	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. 
11.32/4.51	
11.32/4.51	From the DPs we obtained the following set of size-change graphs:
11.32/4.51	*new_gtGtEs(:(ww80, ww81), ww6, h) -> new_gtGtEs(ww81, ww6, h)
11.32/4.51	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
11.32/4.51	
11.32/4.51	
11.32/4.51	----------------------------------------
11.32/4.51	
11.32/4.51	(11)
11.32/4.51	YES
11.32/4.51	
11.32/4.51	----------------------------------------
11.32/4.51	
11.32/4.51	(12)
11.32/4.51	Obligation:
11.32/4.51	Q DP problem:
11.32/4.51	The TRS P consists of the following rules:
11.32/4.51	
11.32/4.51	   new_psPs(:(ww60, ww61), ww10) -> new_psPs(ww61, ww10)
11.32/4.51	
11.32/4.51	R is empty.
11.32/4.51	Q is empty.
11.32/4.51	We have to consider all minimal (P,Q,R)-chains.
11.32/4.51	----------------------------------------
11.32/4.51	
11.32/4.51	(13) QDPSizeChangeProof (EQUIVALENT)
11.32/4.51	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. 
11.32/4.51	
11.32/4.51	From the DPs we obtained the following set of size-change graphs:
11.32/4.51	*new_psPs(:(ww60, ww61), ww10) -> new_psPs(ww61, ww10)
11.32/4.51	The graph contains the following edges 1 > 1, 2 >= 2
11.32/4.51	
11.32/4.51	
11.32/4.51	----------------------------------------
11.32/4.51	
11.32/4.51	(14)
11.32/4.51	YES
11.32/4.51	
11.32/4.51	----------------------------------------
11.32/4.51	
11.32/4.51	(15)
11.32/4.51	Obligation:
11.32/4.51	Q DP problem:
11.32/4.51	The TRS P consists of the following rules:
11.32/4.51	
11.32/4.51	   new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), ty_[], h, ba, bb) -> new_foldr(ww3, ww41, ww51, ty_[], h, ba, bb)
11.32/4.51	   new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), ty_Maybe, h, ba, bb) -> new_foldr(ww3, ww41, ww51, ty_Maybe, h, ba, bb)
11.32/4.51	   new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), ty_IO, h, ba, bb) -> new_foldr(ww3, ww41, ww51, ty_IO, h, ba, bb)
11.32/4.51	
11.32/4.51	R is empty.
11.32/4.51	Q is empty.
11.32/4.51	We have to consider all minimal (P,Q,R)-chains.
11.32/4.51	----------------------------------------
11.32/4.51	
11.32/4.51	(16) DependencyGraphProof (EQUIVALENT)
11.32/4.51	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs.
11.32/4.51	----------------------------------------
11.32/4.51	
11.32/4.51	(17)
11.32/4.51	Complex Obligation (AND)
11.32/4.51	
11.32/4.51	----------------------------------------
11.32/4.51	
11.32/4.51	(18)
11.32/4.51	Obligation:
11.32/4.51	Q DP problem:
11.32/4.51	The TRS P consists of the following rules:
11.32/4.51	
11.32/4.51	   new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), ty_IO, h, ba, bb) -> new_foldr(ww3, ww41, ww51, ty_IO, h, ba, bb)
11.32/4.51	
11.32/4.51	R is empty.
11.32/4.51	Q is empty.
11.32/4.51	We have to consider all minimal (P,Q,R)-chains.
11.32/4.51	----------------------------------------
11.32/4.51	
11.32/4.51	(19) QDPSizeChangeProof (EQUIVALENT)
11.32/4.51	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. 
11.32/4.51	
11.32/4.51	From the DPs we obtained the following set of size-change graphs:
11.32/4.51	*new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), ty_IO, h, ba, bb) -> new_foldr(ww3, ww41, ww51, ty_IO, h, ba, bb)
11.32/4.51	The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7
11.32/4.51	
11.32/4.51	
11.32/4.51	----------------------------------------
11.32/4.51	
11.32/4.51	(20)
11.32/4.51	YES
11.32/4.51	
11.32/4.51	----------------------------------------
11.32/4.51	
11.32/4.51	(21)
11.32/4.51	Obligation:
11.32/4.51	Q DP problem:
11.32/4.51	The TRS P consists of the following rules:
11.32/4.51	
11.32/4.51	   new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), ty_Maybe, h, ba, bb) -> new_foldr(ww3, ww41, ww51, ty_Maybe, h, ba, bb)
11.32/4.51	
11.32/4.51	R is empty.
11.32/4.51	Q is empty.
11.32/4.51	We have to consider all minimal (P,Q,R)-chains.
11.32/4.51	----------------------------------------
11.32/4.51	
11.32/4.51	(22) QDPSizeChangeProof (EQUIVALENT)
11.32/4.51	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. 
11.32/4.51	
11.32/4.51	From the DPs we obtained the following set of size-change graphs:
11.32/4.51	*new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), ty_Maybe, h, ba, bb) -> new_foldr(ww3, ww41, ww51, ty_Maybe, h, ba, bb)
11.32/4.51	The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7
11.32/4.51	
11.32/4.51	
11.32/4.51	----------------------------------------
11.32/4.51	
11.32/4.51	(23)
11.32/4.51	YES
11.32/4.51	
11.32/4.51	----------------------------------------
11.32/4.51	
11.32/4.51	(24)
11.32/4.51	Obligation:
11.32/4.51	Q DP problem:
11.32/4.51	The TRS P consists of the following rules:
11.32/4.51	
11.32/4.51	   new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), ty_[], h, ba, bb) -> new_foldr(ww3, ww41, ww51, ty_[], h, ba, bb)
11.32/4.51	
11.32/4.51	R is empty.
11.32/4.51	Q is empty.
11.32/4.51	We have to consider all minimal (P,Q,R)-chains.
11.32/4.51	----------------------------------------
11.32/4.51	
11.32/4.51	(25) QDPSizeChangeProof (EQUIVALENT)
11.32/4.51	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. 
11.32/4.51	
11.32/4.51	From the DPs we obtained the following set of size-change graphs:
11.32/4.51	*new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), ty_[], h, ba, bb) -> new_foldr(ww3, ww41, ww51, ty_[], h, ba, bb)
11.32/4.51	The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7
11.32/4.51	
11.32/4.51	
11.32/4.51	----------------------------------------
11.32/4.51	
11.32/4.51	(26)
11.32/4.51	YES
11.59/4.56	EOF