8.72/3.89	YES
10.95/4.44	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.95/4.44	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.95/4.44	
10.95/4.44	
10.95/4.44	H-Termination with start terms of the given HASKELL could be proven:
10.95/4.44	
10.95/4.44	(0) HASKELL
10.95/4.44	(1) LR [EQUIVALENT, 0 ms]
10.95/4.44	(2) HASKELL
10.95/4.44	(3) BR [EQUIVALENT, 0 ms]
10.95/4.44	(4) HASKELL
10.95/4.44	(5) COR [EQUIVALENT, 0 ms]
10.95/4.44	(6) HASKELL
10.95/4.44	(7) Narrow [SOUND, 0 ms]
10.95/4.44	(8) AND
10.95/4.44	    (9) QDP
10.95/4.44	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.95/4.44	        (11) YES
10.95/4.44	    (12) QDP
10.95/4.44	        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.95/4.44	        (14) YES
10.95/4.44	    (15) QDP
10.95/4.44	        (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.95/4.44	        (17) YES
10.95/4.44	
10.95/4.44	
10.95/4.44	----------------------------------------
10.95/4.44	
10.95/4.44	(0)
10.95/4.44	Obligation:
10.95/4.44	mainModule Main 
10.95/4.44	module Maybe where {
10.95/4.44	import qualified Main;
10.95/4.44	import qualified Monad;
10.95/4.44	import qualified Prelude;
10.95/4.44	}
10.95/4.44	module Main where {
10.95/4.44	import qualified Maybe;
10.95/4.44	import qualified Monad;
10.95/4.44	import qualified Prelude;
10.95/4.44	}
10.95/4.44	module Monad where {
10.95/4.44	import qualified Main;
10.95/4.44	import qualified Maybe;
10.95/4.44	import qualified Prelude;
10.95/4.44	zipWithM_ :: Monad b => (d  ->  c  ->  b a)  ->  [d]  ->  [c]  ->  b ();
10.95/4.44	zipWithM_ f xs ys = sequence_ (zipWith f xs ys);
10.95/4.44	
10.95/4.44	}
10.95/4.44	
10.95/4.44	----------------------------------------
10.95/4.44	
10.95/4.44	(1) LR (EQUIVALENT)
10.95/4.44	Lambda Reductions:
10.95/4.44	The following Lambda expression
10.95/4.44	"\_->q"
10.95/4.44	is transformed to
10.95/4.44	"gtGt0 q _ = q;
10.95/4.44	"
10.95/4.44	
10.95/4.44	----------------------------------------
10.95/4.44	
10.95/4.44	(2)
10.95/4.44	Obligation:
10.95/4.44	mainModule Main 
10.95/4.44	module Maybe where {
10.95/4.44	import qualified Main;
10.95/4.44	import qualified Monad;
10.95/4.44	import qualified Prelude;
10.95/4.44	}
10.95/4.44	module Main where {
10.95/4.44	import qualified Maybe;
10.95/4.44	import qualified Monad;
10.95/4.44	import qualified Prelude;
10.95/4.44	}
10.95/4.44	module Monad where {
10.95/4.44	import qualified Main;
10.95/4.44	import qualified Maybe;
10.95/4.44	import qualified Prelude;
10.95/4.44	zipWithM_ :: Monad a => (d  ->  c  ->  a b)  ->  [d]  ->  [c]  ->  a ();
10.95/4.44	zipWithM_ f xs ys = sequence_ (zipWith f xs ys);
10.95/4.44	
10.95/4.44	}
10.95/4.44	
10.95/4.44	----------------------------------------
10.95/4.44	
10.95/4.44	(3) BR (EQUIVALENT)
10.95/4.44	Replaced joker patterns by fresh variables and removed binding patterns.
10.95/4.44	----------------------------------------
10.95/4.44	
10.95/4.44	(4)
10.95/4.44	Obligation:
10.95/4.44	mainModule Main 
10.95/4.44	module Maybe where {
10.95/4.44	import qualified Main;
10.95/4.44	import qualified Monad;
10.95/4.44	import qualified Prelude;
10.95/4.44	}
10.95/4.44	module Main where {
10.95/4.44	import qualified Maybe;
10.95/4.44	import qualified Monad;
10.95/4.44	import qualified Prelude;
10.95/4.44	}
10.95/4.44	module Monad where {
10.95/4.44	import qualified Main;
10.95/4.44	import qualified Maybe;
10.95/4.44	import qualified Prelude;
10.95/4.44	zipWithM_ :: Monad d => (b  ->  c  ->  d a)  ->  [b]  ->  [c]  ->  d ();
10.95/4.44	zipWithM_ f xs ys = sequence_ (zipWith f xs ys);
10.95/4.44	
10.95/4.44	}
10.95/4.44	
10.95/4.44	----------------------------------------
10.95/4.44	
10.95/4.44	(5) COR (EQUIVALENT)
10.95/4.44	Cond Reductions:
10.95/4.44	The following Function with conditions
10.95/4.44	"undefined |Falseundefined;
10.95/4.44	"
10.95/4.44	is transformed to
10.95/4.44	"undefined  = undefined1;
10.95/4.44	"
10.95/4.44	"undefined0 True = undefined;
10.95/4.44	"
10.95/4.44	"undefined1  = undefined0 False;
10.95/4.44	"
10.95/4.44	
10.95/4.44	----------------------------------------
10.95/4.44	
10.95/4.44	(6)
10.95/4.44	Obligation:
10.95/4.44	mainModule Main 
10.95/4.44	module Maybe where {
10.95/4.44	import qualified Main;
10.95/4.44	import qualified Monad;
10.95/4.44	import qualified Prelude;
10.95/4.44	}
10.95/4.44	module Main where {
10.95/4.44	import qualified Maybe;
10.95/4.44	import qualified Monad;
10.95/4.44	import qualified Prelude;
10.95/4.44	}
10.95/4.44	module Monad where {
10.95/4.44	import qualified Main;
10.95/4.44	import qualified Maybe;
10.95/4.44	import qualified Prelude;
10.95/4.44	zipWithM_ :: Monad c => (a  ->  b  ->  c d)  ->  [a]  ->  [b]  ->  c ();
10.95/4.44	zipWithM_ f xs ys = sequence_ (zipWith f xs ys);
10.95/4.44	
10.95/4.44	}
10.95/4.44	
10.95/4.44	----------------------------------------
10.95/4.44	
10.95/4.44	(7) Narrow (SOUND)
10.95/4.44	Haskell To QDPs
10.95/4.44	
10.95/4.44	digraph dp_graph {
10.95/4.44	node [outthreshold=100, inthreshold=100];1[label="Monad.zipWithM_",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.95/4.44	3[label="Monad.zipWithM_ ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.95/4.44	4[label="Monad.zipWithM_ ww3 ww4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
10.95/4.44	5[label="Monad.zipWithM_ ww3 ww4 ww5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
10.95/4.44	6[label="sequence_ (zipWith ww3 ww4 ww5)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
10.95/4.44	7[label="foldr (>>) (return ()) (zipWith ww3 ww4 ww5)",fontsize=16,color="burlywood",shape="triangle"];43[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];7 -> 43[label="",style="solid", color="burlywood", weight=9];
10.95/4.44	43 -> 8[label="",style="solid", color="burlywood", weight=3];
10.95/4.44	44[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 44[label="",style="solid", color="burlywood", weight=9];
10.95/4.44	44 -> 9[label="",style="solid", color="burlywood", weight=3];
10.95/4.44	8[label="foldr (>>) (return ()) (zipWith ww3 (ww40 : ww41) ww5)",fontsize=16,color="burlywood",shape="box"];45[label="ww5/ww50 : ww51",fontsize=10,color="white",style="solid",shape="box"];8 -> 45[label="",style="solid", color="burlywood", weight=9];
10.95/4.44	45 -> 10[label="",style="solid", color="burlywood", weight=3];
10.95/4.44	46[label="ww5/[]",fontsize=10,color="white",style="solid",shape="box"];8 -> 46[label="",style="solid", color="burlywood", weight=9];
10.95/4.44	46 -> 11[label="",style="solid", color="burlywood", weight=3];
10.95/4.44	9[label="foldr (>>) (return ()) (zipWith ww3 [] ww5)",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3];
10.95/4.44	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];
10.95/4.44	11[label="foldr (>>) (return ()) (zipWith ww3 (ww40 : ww41) [])",fontsize=16,color="black",shape="box"];11 -> 14[label="",style="solid", color="black", weight=3];
10.95/4.44	12[label="foldr (>>) (return ()) []",fontsize=16,color="black",shape="triangle"];12 -> 15[label="",style="solid", color="black", weight=3];
10.95/4.44	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];
10.95/4.44	14 -> 12[label="",style="dashed", color="red", weight=0];
10.95/4.44	14[label="foldr (>>) (return ()) []",fontsize=16,color="magenta"];15[label="return ()",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
10.95/4.44	16 -> 18[label="",style="dashed", color="red", weight=0];
10.95/4.44	16[label="(>>) ww3 ww40 ww50 foldr (>>) (return ()) (zipWith ww3 ww41 ww51)",fontsize=16,color="magenta"];16 -> 19[label="",style="dashed", color="magenta", weight=3];
10.95/4.44	17[label="() : []",fontsize=16,color="green",shape="box"];19 -> 7[label="",style="dashed", color="red", weight=0];
10.95/4.44	19[label="foldr (>>) (return ()) (zipWith ww3 ww41 ww51)",fontsize=16,color="magenta"];19 -> 20[label="",style="dashed", color="magenta", weight=3];
10.95/4.44	19 -> 21[label="",style="dashed", color="magenta", weight=3];
10.95/4.44	18[label="(>>) ww3 ww40 ww50 ww6",fontsize=16,color="black",shape="triangle"];18 -> 22[label="",style="solid", color="black", weight=3];
10.95/4.44	20[label="ww51",fontsize=16,color="green",shape="box"];21[label="ww41",fontsize=16,color="green",shape="box"];22 -> 23[label="",style="dashed", color="red", weight=0];
10.95/4.44	22[label="ww3 ww40 ww50 >>= gtGt0 ww6",fontsize=16,color="magenta"];22 -> 24[label="",style="dashed", color="magenta", weight=3];
10.95/4.44	24[label="ww3 ww40 ww50",fontsize=16,color="green",shape="box"];24 -> 29[label="",style="dashed", color="green", weight=3];
10.95/4.44	24 -> 30[label="",style="dashed", color="green", weight=3];
10.95/4.44	23[label="ww8 >>= gtGt0 ww6",fontsize=16,color="burlywood",shape="triangle"];47[label="ww8/ww80 : ww81",fontsize=10,color="white",style="solid",shape="box"];23 -> 47[label="",style="solid", color="burlywood", weight=9];
10.95/4.44	47 -> 27[label="",style="solid", color="burlywood", weight=3];
10.95/4.44	48[label="ww8/[]",fontsize=10,color="white",style="solid",shape="box"];23 -> 48[label="",style="solid", color="burlywood", weight=9];
10.95/4.44	48 -> 28[label="",style="solid", color="burlywood", weight=3];
10.95/4.44	29[label="ww40",fontsize=16,color="green",shape="box"];30[label="ww50",fontsize=16,color="green",shape="box"];27[label="ww80 : ww81 >>= gtGt0 ww6",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
10.95/4.44	28[label="[] >>= gtGt0 ww6",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
10.95/4.44	31 -> 33[label="",style="dashed", color="red", weight=0];
10.95/4.44	31[label="gtGt0 ww6 ww80 ++ (ww81 >>= gtGt0 ww6)",fontsize=16,color="magenta"];31 -> 34[label="",style="dashed", color="magenta", weight=3];
10.95/4.44	32[label="[]",fontsize=16,color="green",shape="box"];34 -> 23[label="",style="dashed", color="red", weight=0];
10.95/4.44	34[label="ww81 >>= gtGt0 ww6",fontsize=16,color="magenta"];34 -> 35[label="",style="dashed", color="magenta", weight=3];
10.95/4.44	33[label="gtGt0 ww6 ww80 ++ ww9",fontsize=16,color="black",shape="triangle"];33 -> 36[label="",style="solid", color="black", weight=3];
10.95/4.44	35[label="ww81",fontsize=16,color="green",shape="box"];36[label="ww6 ++ ww9",fontsize=16,color="burlywood",shape="triangle"];49[label="ww6/ww60 : ww61",fontsize=10,color="white",style="solid",shape="box"];36 -> 49[label="",style="solid", color="burlywood", weight=9];
10.95/4.44	49 -> 37[label="",style="solid", color="burlywood", weight=3];
10.95/4.44	50[label="ww6/[]",fontsize=10,color="white",style="solid",shape="box"];36 -> 50[label="",style="solid", color="burlywood", weight=9];
10.95/4.44	50 -> 38[label="",style="solid", color="burlywood", weight=3];
10.95/4.44	37[label="(ww60 : ww61) ++ ww9",fontsize=16,color="black",shape="box"];37 -> 39[label="",style="solid", color="black", weight=3];
10.95/4.44	38[label="[] ++ ww9",fontsize=16,color="black",shape="box"];38 -> 40[label="",style="solid", color="black", weight=3];
10.95/4.44	39[label="ww60 : ww61 ++ ww9",fontsize=16,color="green",shape="box"];39 -> 41[label="",style="dashed", color="green", weight=3];
10.95/4.44	40[label="ww9",fontsize=16,color="green",shape="box"];41 -> 36[label="",style="dashed", color="red", weight=0];
10.95/4.44	41[label="ww61 ++ ww9",fontsize=16,color="magenta"];41 -> 42[label="",style="dashed", color="magenta", weight=3];
10.95/4.44	42[label="ww61",fontsize=16,color="green",shape="box"];}
10.95/4.44	
10.95/4.44	----------------------------------------
10.95/4.44	
10.95/4.44	(8)
10.95/4.44	Complex Obligation (AND)
10.95/4.44	
10.95/4.44	----------------------------------------
10.95/4.44	
10.95/4.44	(9)
10.95/4.44	Obligation:
10.95/4.44	Q DP problem:
10.95/4.44	The TRS P consists of the following rules:
10.95/4.44	
10.95/4.44	   new_gtGtEs(:(ww80, ww81), ww6, h) -> new_gtGtEs(ww81, ww6, h)
10.95/4.44	
10.95/4.44	R is empty.
10.95/4.44	Q is empty.
10.95/4.44	We have to consider all minimal (P,Q,R)-chains.
10.95/4.44	----------------------------------------
10.95/4.44	
10.95/4.44	(10) QDPSizeChangeProof (EQUIVALENT)
10.95/4.44	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. 
10.95/4.44	
10.95/4.44	From the DPs we obtained the following set of size-change graphs:
10.95/4.44	*new_gtGtEs(:(ww80, ww81), ww6, h) -> new_gtGtEs(ww81, ww6, h)
10.95/4.44	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
10.95/4.44	
10.95/4.44	
10.95/4.44	----------------------------------------
10.95/4.44	
10.95/4.44	(11)
10.95/4.44	YES
10.95/4.44	
10.95/4.44	----------------------------------------
10.95/4.44	
10.95/4.44	(12)
10.95/4.44	Obligation:
10.95/4.44	Q DP problem:
10.95/4.44	The TRS P consists of the following rules:
10.95/4.44	
10.95/4.44	   new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), h, ba, bb) -> new_foldr(ww3, ww41, ww51, h, ba, bb)
10.95/4.44	
10.95/4.44	R is empty.
10.95/4.44	Q is empty.
10.95/4.44	We have to consider all minimal (P,Q,R)-chains.
10.95/4.44	----------------------------------------
10.95/4.44	
10.95/4.44	(13) QDPSizeChangeProof (EQUIVALENT)
10.95/4.44	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. 
10.95/4.44	
10.95/4.44	From the DPs we obtained the following set of size-change graphs:
10.95/4.44	*new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), h, ba, bb) -> new_foldr(ww3, ww41, ww51, h, ba, bb)
10.95/4.44	The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6
10.95/4.44	
10.95/4.44	
10.95/4.44	----------------------------------------
10.95/4.44	
10.95/4.44	(14)
10.95/4.44	YES
10.95/4.44	
10.95/4.44	----------------------------------------
10.95/4.44	
10.95/4.44	(15)
10.95/4.44	Obligation:
10.95/4.44	Q DP problem:
10.95/4.44	The TRS P consists of the following rules:
10.95/4.44	
10.95/4.44	   new_psPs(:(ww60, ww61), ww9) -> new_psPs(ww61, ww9)
10.95/4.44	
10.95/4.44	R is empty.
10.95/4.44	Q is empty.
10.95/4.44	We have to consider all minimal (P,Q,R)-chains.
10.95/4.44	----------------------------------------
10.95/4.44	
10.95/4.44	(16) QDPSizeChangeProof (EQUIVALENT)
10.95/4.44	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. 
10.95/4.44	
10.95/4.44	From the DPs we obtained the following set of size-change graphs:
10.95/4.44	*new_psPs(:(ww60, ww61), ww9) -> new_psPs(ww61, ww9)
10.95/4.44	The graph contains the following edges 1 > 1, 2 >= 2
10.95/4.44	
10.95/4.44	
10.95/4.44	----------------------------------------
10.95/4.44	
10.95/4.44	(17)
10.95/4.44	YES
11.12/4.53	EOF