9.32/3.96	YES
11.03/4.44	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
11.03/4.44	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.03/4.44	
11.03/4.44	
11.03/4.44	H-Termination with start terms of the given HASKELL could be proven:
11.03/4.44	
11.03/4.44	(0) HASKELL
11.03/4.44	(1) BR [EQUIVALENT, 0 ms]
11.03/4.44	(2) HASKELL
11.03/4.44	(3) COR [EQUIVALENT, 0 ms]
11.03/4.44	(4) HASKELL
11.03/4.44	(5) Narrow [SOUND, 0 ms]
11.03/4.44	(6) AND
11.03/4.44	    (7) QDP
11.03/4.44	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.03/4.44	        (9) YES
11.03/4.44	    (10) QDP
11.03/4.44	        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.03/4.44	        (12) YES
11.03/4.44	
11.03/4.44	
11.03/4.44	----------------------------------------
11.03/4.44	
11.03/4.44	(0)
11.03/4.44	Obligation:
11.03/4.44	mainModule Main 
11.03/4.44	module Maybe where {
11.03/4.44	import qualified Main;
11.03/4.44	import qualified Monad;
11.03/4.44	import qualified Prelude;
11.03/4.44	}
11.03/4.44	module Main where {
11.03/4.44	import qualified Maybe;
11.03/4.44	import qualified Monad;
11.03/4.44	import qualified Prelude;
11.03/4.44	}
11.03/4.44	module Monad where {
11.03/4.44	import qualified Main;
11.03/4.44	import qualified Maybe;
11.03/4.44	import qualified Prelude;
11.03/4.44	join :: Monad a => a (a b)  ->  a b;
11.03/4.44	join x = x >>= id;
11.03/4.44	
11.03/4.44	}
11.03/4.44	
11.03/4.44	----------------------------------------
11.03/4.44	
11.03/4.44	(1) BR (EQUIVALENT)
11.03/4.44	Replaced joker patterns by fresh variables and removed binding patterns.
11.03/4.44	----------------------------------------
11.03/4.44	
11.03/4.44	(2)
11.03/4.44	Obligation:
11.03/4.44	mainModule Main 
11.03/4.44	module Maybe where {
11.03/4.44	import qualified Main;
11.03/4.44	import qualified Monad;
11.03/4.44	import qualified Prelude;
11.03/4.44	}
11.03/4.44	module Main where {
11.03/4.44	import qualified Maybe;
11.03/4.44	import qualified Monad;
11.03/4.44	import qualified Prelude;
11.03/4.44	}
11.03/4.44	module Monad where {
11.03/4.44	import qualified Main;
11.03/4.44	import qualified Maybe;
11.03/4.44	import qualified Prelude;
11.03/4.44	join :: Monad b => b (b a)  ->  b a;
11.03/4.44	join x = x >>= id;
11.03/4.44	
11.03/4.44	}
11.03/4.44	
11.03/4.44	----------------------------------------
11.03/4.44	
11.03/4.44	(3) COR (EQUIVALENT)
11.03/4.44	Cond Reductions:
11.03/4.44	The following Function with conditions
11.03/4.44	"undefined |Falseundefined;
11.03/4.44	"
11.03/4.44	is transformed to
11.03/4.44	"undefined  = undefined1;
11.03/4.44	"
11.03/4.44	"undefined0 True = undefined;
11.03/4.44	"
11.03/4.44	"undefined1  = undefined0 False;
11.03/4.44	"
11.03/4.44	
11.03/4.44	----------------------------------------
11.03/4.44	
11.03/4.44	(4)
11.03/4.44	Obligation:
11.03/4.44	mainModule Main 
11.03/4.44	module Maybe where {
11.03/4.44	import qualified Main;
11.03/4.44	import qualified Monad;
11.03/4.44	import qualified Prelude;
11.03/4.44	}
11.03/4.44	module Main where {
11.03/4.44	import qualified Maybe;
11.03/4.44	import qualified Monad;
11.03/4.44	import qualified Prelude;
11.03/4.44	}
11.03/4.44	module Monad where {
11.03/4.44	import qualified Main;
11.03/4.44	import qualified Maybe;
11.03/4.44	import qualified Prelude;
11.03/4.44	join :: Monad b => b (b a)  ->  b a;
11.03/4.44	join x = x >>= id;
11.03/4.44	
11.03/4.44	}
11.03/4.44	
11.03/4.44	----------------------------------------
11.03/4.44	
11.03/4.44	(5) Narrow (SOUND)
11.03/4.44	Haskell To QDPs
11.03/4.44	
11.03/4.44	digraph dp_graph {
11.03/4.44	node [outthreshold=100, inthreshold=100];1[label="Monad.join",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.03/4.44	3[label="Monad.join vy3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
11.03/4.44	4[label="vy3 >>= id",fontsize=16,color="burlywood",shape="triangle"];19[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];4 -> 19[label="",style="solid", color="burlywood", weight=9];
11.03/4.44	19 -> 5[label="",style="solid", color="burlywood", weight=3];
11.03/4.44	20[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 20[label="",style="solid", color="burlywood", weight=9];
11.03/4.44	20 -> 6[label="",style="solid", color="burlywood", weight=3];
11.03/4.44	5[label="vy30 : vy31 >>= id",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
11.03/4.44	6[label="[] >>= id",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
11.03/4.44	7 -> 9[label="",style="dashed", color="red", weight=0];
11.03/4.44	7[label="id vy30 ++ (vy31 >>= id)",fontsize=16,color="magenta"];7 -> 10[label="",style="dashed", color="magenta", weight=3];
11.03/4.44	8[label="[]",fontsize=16,color="green",shape="box"];10 -> 4[label="",style="dashed", color="red", weight=0];
11.03/4.44	10[label="vy31 >>= id",fontsize=16,color="magenta"];10 -> 11[label="",style="dashed", color="magenta", weight=3];
11.03/4.44	9[label="id vy30 ++ vy4",fontsize=16,color="black",shape="triangle"];9 -> 12[label="",style="solid", color="black", weight=3];
11.03/4.44	11[label="vy31",fontsize=16,color="green",shape="box"];12[label="vy30 ++ vy4",fontsize=16,color="burlywood",shape="triangle"];21[label="vy30/vy300 : vy301",fontsize=10,color="white",style="solid",shape="box"];12 -> 21[label="",style="solid", color="burlywood", weight=9];
11.03/4.44	21 -> 13[label="",style="solid", color="burlywood", weight=3];
11.03/4.44	22[label="vy30/[]",fontsize=10,color="white",style="solid",shape="box"];12 -> 22[label="",style="solid", color="burlywood", weight=9];
11.03/4.44	22 -> 14[label="",style="solid", color="burlywood", weight=3];
11.03/4.44	13[label="(vy300 : vy301) ++ vy4",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
11.03/4.44	14[label="[] ++ vy4",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
11.03/4.44	15[label="vy300 : vy301 ++ vy4",fontsize=16,color="green",shape="box"];15 -> 17[label="",style="dashed", color="green", weight=3];
11.03/4.44	16[label="vy4",fontsize=16,color="green",shape="box"];17 -> 12[label="",style="dashed", color="red", weight=0];
11.03/4.44	17[label="vy301 ++ vy4",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
11.03/4.44	18[label="vy301",fontsize=16,color="green",shape="box"];}
11.03/4.44	
11.03/4.44	----------------------------------------
11.03/4.44	
11.03/4.44	(6)
11.03/4.44	Complex Obligation (AND)
11.03/4.44	
11.03/4.44	----------------------------------------
11.03/4.44	
11.03/4.44	(7)
11.03/4.44	Obligation:
11.03/4.44	Q DP problem:
11.03/4.44	The TRS P consists of the following rules:
11.03/4.44	
11.03/4.44	   new_gtGtEs(:(vy30, vy31), h) -> new_gtGtEs(vy31, h)
11.03/4.44	
11.03/4.44	R is empty.
11.03/4.44	Q is empty.
11.03/4.44	We have to consider all minimal (P,Q,R)-chains.
11.03/4.44	----------------------------------------
11.03/4.44	
11.03/4.44	(8) QDPSizeChangeProof (EQUIVALENT)
11.03/4.44	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. 
11.03/4.44	
11.03/4.44	From the DPs we obtained the following set of size-change graphs:
11.03/4.44	*new_gtGtEs(:(vy30, vy31), h) -> new_gtGtEs(vy31, h)
11.03/4.44	The graph contains the following edges 1 > 1, 2 >= 2
11.03/4.44	
11.03/4.44	
11.03/4.44	----------------------------------------
11.03/4.44	
11.03/4.44	(9)
11.03/4.44	YES
11.03/4.44	
11.03/4.44	----------------------------------------
11.03/4.44	
11.03/4.44	(10)
11.03/4.44	Obligation:
11.03/4.44	Q DP problem:
11.03/4.44	The TRS P consists of the following rules:
11.03/4.44	
11.03/4.44	   new_psPs(:(vy300, vy301), vy4, h) -> new_psPs(vy301, vy4, h)
11.03/4.44	
11.03/4.44	R is empty.
11.03/4.44	Q is empty.
11.03/4.44	We have to consider all minimal (P,Q,R)-chains.
11.03/4.44	----------------------------------------
11.03/4.44	
11.03/4.44	(11) QDPSizeChangeProof (EQUIVALENT)
11.03/4.44	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. 
11.03/4.44	
11.03/4.44	From the DPs we obtained the following set of size-change graphs:
11.03/4.44	*new_psPs(:(vy300, vy301), vy4, h) -> new_psPs(vy301, vy4, h)
11.03/4.44	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
11.03/4.44	
11.03/4.44	
11.03/4.44	----------------------------------------
11.03/4.44	
11.03/4.44	(12)
11.03/4.44	YES
11.36/4.49	EOF