9.13/3.91	YES
10.93/4.43	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.93/4.43	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.93/4.43	
10.93/4.43	
10.93/4.43	H-Termination with start terms of the given HASKELL could be proven:
10.93/4.43	
10.93/4.43	(0) HASKELL
10.93/4.43	(1) LR [EQUIVALENT, 0 ms]
10.93/4.43	(2) HASKELL
10.93/4.43	(3) BR [EQUIVALENT, 0 ms]
10.93/4.43	(4) HASKELL
10.93/4.43	(5) COR [EQUIVALENT, 0 ms]
10.93/4.43	(6) HASKELL
10.93/4.43	(7) Narrow [SOUND, 0 ms]
10.93/4.43	(8) QDP
10.93/4.43	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.93/4.43	(10) YES
10.93/4.43	
10.93/4.43	
10.93/4.43	----------------------------------------
10.93/4.43	
10.93/4.43	(0)
10.93/4.43	Obligation:
10.93/4.43	mainModule Main 
10.93/4.43	module Maybe where {
10.93/4.43	import qualified Main;
10.93/4.43	import qualified Monad;
10.93/4.43	import qualified Prelude;
10.93/4.43	}
10.93/4.43	module Main where {
10.93/4.43	import qualified Maybe;
10.93/4.43	import qualified Monad;
10.93/4.43	import qualified Prelude;
10.93/4.43	}
10.93/4.43	module Monad where {
10.93/4.43	import qualified Main;
10.93/4.43	import qualified Maybe;
10.93/4.43	import qualified Prelude;
10.93/4.43	zipWithM :: Monad a => (b  ->  c  ->  a d)  ->  [b]  ->  [c]  ->  a [d];
10.93/4.43	zipWithM f xs ys = sequence (zipWith f xs ys);
10.93/4.43	
10.93/4.43	}
10.93/4.43	
10.93/4.43	----------------------------------------
10.93/4.43	
10.93/4.43	(1) LR (EQUIVALENT)
10.93/4.43	Lambda Reductions:
10.93/4.43	The following Lambda expression
10.93/4.43	"\xs->return (x : xs)"
10.93/4.43	is transformed to
10.93/4.43	"sequence0 x xs = return (x : xs);
10.93/4.43	"
10.93/4.43	The following Lambda expression
10.93/4.43	"\x->sequence cs >>= sequence0 x"
10.93/4.43	is transformed to
10.93/4.43	"sequence1 cs x = sequence cs >>= sequence0 x;
10.93/4.43	"
10.93/4.43	
10.93/4.43	----------------------------------------
10.93/4.43	
10.93/4.43	(2)
10.93/4.43	Obligation:
10.93/4.43	mainModule Main 
10.93/4.43	module Maybe where {
10.93/4.43	import qualified Main;
10.93/4.43	import qualified Monad;
10.93/4.43	import qualified Prelude;
10.93/4.43	}
10.93/4.43	module Main where {
10.93/4.43	import qualified Maybe;
10.93/4.43	import qualified Monad;
10.93/4.43	import qualified Prelude;
10.93/4.43	}
10.93/4.43	module Monad where {
10.93/4.43	import qualified Main;
10.93/4.43	import qualified Maybe;
10.93/4.43	import qualified Prelude;
10.93/4.43	zipWithM :: Monad b => (c  ->  d  ->  b a)  ->  [c]  ->  [d]  ->  b [a];
10.93/4.43	zipWithM f xs ys = sequence (zipWith f xs ys);
10.93/4.43	
10.93/4.43	}
10.93/4.43	
10.93/4.43	----------------------------------------
10.93/4.43	
10.93/4.43	(3) BR (EQUIVALENT)
10.93/4.43	Replaced joker patterns by fresh variables and removed binding patterns.
10.93/4.43	----------------------------------------
10.93/4.43	
10.93/4.43	(4)
10.93/4.43	Obligation:
10.93/4.43	mainModule Main 
10.93/4.43	module Maybe where {
10.93/4.43	import qualified Main;
10.93/4.43	import qualified Monad;
10.93/4.43	import qualified Prelude;
10.93/4.43	}
10.93/4.43	module Main where {
10.93/4.43	import qualified Maybe;
10.93/4.43	import qualified Monad;
10.93/4.43	import qualified Prelude;
10.93/4.43	}
10.93/4.43	module Monad where {
10.93/4.43	import qualified Main;
10.93/4.43	import qualified Maybe;
10.93/4.43	import qualified Prelude;
10.93/4.43	zipWithM :: Monad a => (b  ->  d  ->  a c)  ->  [b]  ->  [d]  ->  a [c];
10.93/4.43	zipWithM f xs ys = sequence (zipWith f xs ys);
10.93/4.43	
10.93/4.43	}
10.93/4.43	
10.93/4.43	----------------------------------------
10.93/4.43	
10.93/4.43	(5) COR (EQUIVALENT)
10.93/4.43	Cond Reductions:
10.93/4.43	The following Function with conditions
10.93/4.43	"undefined |Falseundefined;
10.93/4.43	"
10.93/4.43	is transformed to
10.93/4.43	"undefined  = undefined1;
10.93/4.43	"
10.93/4.43	"undefined0 True = undefined;
10.93/4.43	"
10.93/4.43	"undefined1  = undefined0 False;
10.93/4.43	"
10.93/4.43	
10.93/4.43	----------------------------------------
10.93/4.43	
10.93/4.43	(6)
10.93/4.43	Obligation:
10.93/4.43	mainModule Main 
10.93/4.43	module Maybe where {
10.93/4.43	import qualified Main;
10.93/4.43	import qualified Monad;
10.93/4.43	import qualified Prelude;
10.93/4.43	}
10.93/4.43	module Main where {
10.93/4.43	import qualified Maybe;
10.93/4.43	import qualified Monad;
10.93/4.43	import qualified Prelude;
10.93/4.43	}
10.93/4.43	module Monad where {
10.93/4.43	import qualified Main;
10.93/4.43	import qualified Maybe;
10.93/4.43	import qualified Prelude;
10.93/4.43	zipWithM :: Monad b => (c  ->  d  ->  b a)  ->  [c]  ->  [d]  ->  b [a];
10.93/4.43	zipWithM f xs ys = sequence (zipWith f xs ys);
10.93/4.43	
10.93/4.43	}
10.93/4.43	
10.93/4.43	----------------------------------------
10.93/4.43	
10.93/4.43	(7) Narrow (SOUND)
10.93/4.43	Haskell To QDPs
10.93/4.43	
10.93/4.43	digraph dp_graph {
10.93/4.43	node [outthreshold=100, inthreshold=100];1[label="Monad.zipWithM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.93/4.43	3[label="Monad.zipWithM wv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.93/4.43	4[label="Monad.zipWithM wv3 wv4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
10.93/4.43	5[label="Monad.zipWithM wv3 wv4 wv5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
10.93/4.43	6[label="sequence (zipWith wv3 wv4 wv5)",fontsize=16,color="burlywood",shape="triangle"];50[label="wv4/wv40 : wv41",fontsize=10,color="white",style="solid",shape="box"];6 -> 50[label="",style="solid", color="burlywood", weight=9];
10.93/4.43	50 -> 7[label="",style="solid", color="burlywood", weight=3];
10.93/4.43	51[label="wv4/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 51[label="",style="solid", color="burlywood", weight=9];
10.93/4.43	51 -> 8[label="",style="solid", color="burlywood", weight=3];
10.93/4.43	7[label="sequence (zipWith wv3 (wv40 : wv41) wv5)",fontsize=16,color="burlywood",shape="box"];52[label="wv5/wv50 : wv51",fontsize=10,color="white",style="solid",shape="box"];7 -> 52[label="",style="solid", color="burlywood", weight=9];
10.93/4.43	52 -> 9[label="",style="solid", color="burlywood", weight=3];
10.93/4.43	53[label="wv5/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 53[label="",style="solid", color="burlywood", weight=9];
10.93/4.43	53 -> 10[label="",style="solid", color="burlywood", weight=3];
10.93/4.43	8[label="sequence (zipWith wv3 [] wv5)",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
10.93/4.43	9[label="sequence (zipWith wv3 (wv40 : wv41) (wv50 : wv51))",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3];
10.93/4.43	10[label="sequence (zipWith wv3 (wv40 : wv41) [])",fontsize=16,color="black",shape="box"];10 -> 13[label="",style="solid", color="black", weight=3];
10.93/4.43	11[label="sequence []",fontsize=16,color="black",shape="triangle"];11 -> 14[label="",style="solid", color="black", weight=3];
10.93/4.43	12[label="sequence (wv3 wv40 wv50 : zipWith wv3 wv41 wv51)",fontsize=16,color="black",shape="box"];12 -> 15[label="",style="solid", color="black", weight=3];
10.93/4.43	13 -> 11[label="",style="dashed", color="red", weight=0];
10.93/4.43	13[label="sequence []",fontsize=16,color="magenta"];14[label="return []",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
10.93/4.43	15[label="wv3 wv40 wv50 >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
10.93/4.43	16[label="primretIO []",fontsize=16,color="black",shape="box"];16 -> 18[label="",style="solid", color="black", weight=3];
10.93/4.43	17 -> 19[label="",style="dashed", color="red", weight=0];
10.93/4.43	17[label="primbindIO (wv3 wv40 wv50) (sequence1 (zipWith wv3 wv41 wv51))",fontsize=16,color="magenta"];17 -> 20[label="",style="dashed", color="magenta", weight=3];
10.93/4.43	18[label="AProVE_IO []",fontsize=16,color="green",shape="box"];20[label="wv3 wv40 wv50",fontsize=16,color="green",shape="box"];20 -> 27[label="",style="dashed", color="green", weight=3];
10.93/4.43	20 -> 28[label="",style="dashed", color="green", weight=3];
10.93/4.43	19[label="primbindIO wv6 (sequence1 (zipWith wv3 wv41 wv51))",fontsize=16,color="burlywood",shape="triangle"];54[label="wv6/IO wv60",fontsize=10,color="white",style="solid",shape="box"];19 -> 54[label="",style="solid", color="burlywood", weight=9];
10.93/4.43	54 -> 23[label="",style="solid", color="burlywood", weight=3];
10.93/4.43	55[label="wv6/AProVE_IO wv60",fontsize=10,color="white",style="solid",shape="box"];19 -> 55[label="",style="solid", color="burlywood", weight=9];
10.93/4.43	55 -> 24[label="",style="solid", color="burlywood", weight=3];
10.93/4.43	56[label="wv6/AProVE_Exception wv60",fontsize=10,color="white",style="solid",shape="box"];19 -> 56[label="",style="solid", color="burlywood", weight=9];
10.93/4.43	56 -> 25[label="",style="solid", color="burlywood", weight=3];
10.93/4.43	57[label="wv6/AProVE_Error wv60",fontsize=10,color="white",style="solid",shape="box"];19 -> 57[label="",style="solid", color="burlywood", weight=9];
10.93/4.43	57 -> 26[label="",style="solid", color="burlywood", weight=3];
10.93/4.43	27[label="wv40",fontsize=16,color="green",shape="box"];28[label="wv50",fontsize=16,color="green",shape="box"];23[label="primbindIO (IO wv60) (sequence1 (zipWith wv3 wv41 wv51))",fontsize=16,color="black",shape="box"];23 -> 29[label="",style="solid", color="black", weight=3];
10.93/4.43	24[label="primbindIO (AProVE_IO wv60) (sequence1 (zipWith wv3 wv41 wv51))",fontsize=16,color="black",shape="box"];24 -> 30[label="",style="solid", color="black", weight=3];
10.93/4.43	25[label="primbindIO (AProVE_Exception wv60) (sequence1 (zipWith wv3 wv41 wv51))",fontsize=16,color="black",shape="box"];25 -> 31[label="",style="solid", color="black", weight=3];
10.93/4.43	26[label="primbindIO (AProVE_Error wv60) (sequence1 (zipWith wv3 wv41 wv51))",fontsize=16,color="black",shape="box"];26 -> 32[label="",style="solid", color="black", weight=3];
10.93/4.43	29[label="error []",fontsize=16,color="red",shape="box"];30[label="sequence1 (zipWith wv3 wv41 wv51) wv60",fontsize=16,color="black",shape="box"];30 -> 33[label="",style="solid", color="black", weight=3];
10.93/4.43	31[label="AProVE_Exception wv60",fontsize=16,color="green",shape="box"];32[label="AProVE_Error wv60",fontsize=16,color="green",shape="box"];33 -> 34[label="",style="dashed", color="red", weight=0];
10.93/4.43	33[label="sequence (zipWith wv3 wv41 wv51) >>= sequence0 wv60",fontsize=16,color="magenta"];33 -> 35[label="",style="dashed", color="magenta", weight=3];
10.93/4.43	35 -> 6[label="",style="dashed", color="red", weight=0];
10.93/4.43	35[label="sequence (zipWith wv3 wv41 wv51)",fontsize=16,color="magenta"];35 -> 36[label="",style="dashed", color="magenta", weight=3];
10.93/4.43	35 -> 37[label="",style="dashed", color="magenta", weight=3];
10.93/4.43	34[label="wv7 >>= sequence0 wv60",fontsize=16,color="black",shape="triangle"];34 -> 38[label="",style="solid", color="black", weight=3];
10.93/4.43	36[label="wv41",fontsize=16,color="green",shape="box"];37[label="wv51",fontsize=16,color="green",shape="box"];38[label="primbindIO wv7 (sequence0 wv60)",fontsize=16,color="burlywood",shape="box"];58[label="wv7/IO wv70",fontsize=10,color="white",style="solid",shape="box"];38 -> 58[label="",style="solid", color="burlywood", weight=9];
10.93/4.43	58 -> 39[label="",style="solid", color="burlywood", weight=3];
10.93/4.43	59[label="wv7/AProVE_IO wv70",fontsize=10,color="white",style="solid",shape="box"];38 -> 59[label="",style="solid", color="burlywood", weight=9];
10.93/4.43	59 -> 40[label="",style="solid", color="burlywood", weight=3];
10.93/4.43	60[label="wv7/AProVE_Exception wv70",fontsize=10,color="white",style="solid",shape="box"];38 -> 60[label="",style="solid", color="burlywood", weight=9];
10.93/4.43	60 -> 41[label="",style="solid", color="burlywood", weight=3];
10.93/4.43	61[label="wv7/AProVE_Error wv70",fontsize=10,color="white",style="solid",shape="box"];38 -> 61[label="",style="solid", color="burlywood", weight=9];
10.93/4.43	61 -> 42[label="",style="solid", color="burlywood", weight=3];
10.93/4.43	39[label="primbindIO (IO wv70) (sequence0 wv60)",fontsize=16,color="black",shape="box"];39 -> 43[label="",style="solid", color="black", weight=3];
10.93/4.43	40[label="primbindIO (AProVE_IO wv70) (sequence0 wv60)",fontsize=16,color="black",shape="box"];40 -> 44[label="",style="solid", color="black", weight=3];
10.93/4.43	41[label="primbindIO (AProVE_Exception wv70) (sequence0 wv60)",fontsize=16,color="black",shape="box"];41 -> 45[label="",style="solid", color="black", weight=3];
10.93/4.43	42[label="primbindIO (AProVE_Error wv70) (sequence0 wv60)",fontsize=16,color="black",shape="box"];42 -> 46[label="",style="solid", color="black", weight=3];
10.93/4.43	43[label="error []",fontsize=16,color="red",shape="box"];44[label="sequence0 wv60 wv70",fontsize=16,color="black",shape="box"];44 -> 47[label="",style="solid", color="black", weight=3];
10.93/4.43	45[label="AProVE_Exception wv70",fontsize=16,color="green",shape="box"];46[label="AProVE_Error wv70",fontsize=16,color="green",shape="box"];47[label="return (wv60 : wv70)",fontsize=16,color="black",shape="box"];47 -> 48[label="",style="solid", color="black", weight=3];
10.93/4.43	48[label="primretIO (wv60 : wv70)",fontsize=16,color="black",shape="box"];48 -> 49[label="",style="solid", color="black", weight=3];
10.93/4.43	49[label="AProVE_IO (wv60 : wv70)",fontsize=16,color="green",shape="box"];}
10.93/4.43	
10.93/4.43	----------------------------------------
10.93/4.43	
10.93/4.43	(8)
10.93/4.43	Obligation:
10.93/4.43	Q DP problem:
10.93/4.43	The TRS P consists of the following rules:
10.93/4.43	
10.93/4.43	   new_sequence(wv3, :(wv40, wv41), :(wv50, wv51), h, ba, bb) -> new_primbindIO(wv3, wv41, wv51, h, ba, bb)
10.93/4.43	   new_primbindIO(wv3, wv41, wv51, h, ba, bb) -> new_sequence(wv3, wv41, wv51, h, ba, bb)
10.93/4.43	
10.93/4.43	R is empty.
10.93/4.43	Q is empty.
10.93/4.43	We have to consider all minimal (P,Q,R)-chains.
10.93/4.43	----------------------------------------
10.93/4.43	
10.93/4.43	(9) QDPSizeChangeProof (EQUIVALENT)
10.93/4.43	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. 
10.93/4.43	
10.93/4.43	From the DPs we obtained the following set of size-change graphs:
10.93/4.43	*new_primbindIO(wv3, wv41, wv51, h, ba, bb) -> new_sequence(wv3, wv41, wv51, h, ba, bb)
10.93/4.43	The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6
10.93/4.43	
10.93/4.43	
10.93/4.43	*new_sequence(wv3, :(wv40, wv41), :(wv50, wv51), h, ba, bb) -> new_primbindIO(wv3, wv41, wv51, h, ba, bb)
10.93/4.43	The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6
10.93/4.43	
10.93/4.43	
10.93/4.43	----------------------------------------
10.93/4.43	
10.93/4.43	(10)
10.93/4.43	YES
11.12/4.47	EOF