8.52/3.75	YES
10.48/4.28	proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
10.48/4.28	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.48/4.28	
10.48/4.28	
10.48/4.28	H-Termination with start terms of the given HASKELL could be proven:
10.48/4.28	
10.48/4.28	(0) HASKELL
10.48/4.28	(1) LR [EQUIVALENT, 0 ms]
10.48/4.28	(2) HASKELL
10.48/4.28	(3) BR [EQUIVALENT, 0 ms]
10.48/4.28	(4) HASKELL
10.48/4.28	(5) COR [EQUIVALENT, 0 ms]
10.48/4.28	(6) HASKELL
10.48/4.28	(7) Narrow [SOUND, 0 ms]
10.48/4.28	(8) QDP
10.48/4.28	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.48/4.28	(10) YES
10.48/4.28	
10.48/4.28	
10.48/4.28	----------------------------------------
10.48/4.28	
10.48/4.28	(0)
10.48/4.28	Obligation:
10.48/4.28	mainModule Main 
10.48/4.28	module Maybe where {
10.48/4.28	import qualified Main;
10.48/4.28	import qualified Monad;
10.48/4.28	import qualified Prelude;
10.48/4.28	}
10.48/4.28	module Main where {
10.48/4.28	import qualified Maybe;
10.48/4.28	import qualified Monad;
10.48/4.28	import qualified Prelude;
10.48/4.28	}
10.48/4.28	module Monad where {
10.48/4.28	import qualified Main;
10.48/4.28	import qualified Maybe;
10.48/4.28	import qualified Prelude;
10.48/4.28	foldM :: Monad c => (a  ->  b  ->  c a)  ->  a  ->  [b]  ->  c a;
10.48/4.28	foldM _ a [] = return a;
10.48/4.28	foldM f a (x : xs) = f a x >>= (\fax ->foldM f fax xs);
10.48/4.28	
10.48/4.28	}
10.48/4.28	
10.48/4.28	----------------------------------------
10.48/4.28	
10.48/4.28	(1) LR (EQUIVALENT)
10.48/4.28	Lambda Reductions:
10.48/4.28	The following Lambda expression
10.48/4.28	"\fax->foldM f fax xs"
10.48/4.28	is transformed to
10.48/4.28	"foldM0 f xs fax = foldM f fax xs;
10.48/4.28	"
10.48/4.28	
10.48/4.28	----------------------------------------
10.48/4.28	
10.48/4.28	(2)
10.48/4.28	Obligation:
10.48/4.28	mainModule Main 
10.48/4.28	module Maybe where {
10.48/4.28	import qualified Main;
10.48/4.28	import qualified Monad;
10.48/4.28	import qualified Prelude;
10.48/4.28	}
10.48/4.28	module Main where {
10.48/4.28	import qualified Maybe;
10.48/4.28	import qualified Monad;
10.48/4.28	import qualified Prelude;
10.48/4.28	}
10.48/4.28	module Monad where {
10.48/4.28	import qualified Main;
10.48/4.28	import qualified Maybe;
10.48/4.28	import qualified Prelude;
10.48/4.28	foldM :: Monad c => (a  ->  b  ->  c a)  ->  a  ->  [b]  ->  c a;
10.48/4.28	foldM _ a [] = return a;
10.48/4.28	foldM f a (x : xs) = f a x >>= foldM0 f xs;
10.48/4.28	
10.48/4.28	foldM0 f xs fax = foldM f fax xs;
10.48/4.28	
10.48/4.28	}
10.48/4.28	
10.48/4.28	----------------------------------------
10.48/4.28	
10.48/4.28	(3) BR (EQUIVALENT)
10.48/4.28	Replaced joker patterns by fresh variables and removed binding patterns.
10.48/4.28	----------------------------------------
10.48/4.28	
10.48/4.28	(4)
10.48/4.28	Obligation:
10.48/4.28	mainModule Main 
10.48/4.28	module Maybe where {
10.48/4.28	import qualified Main;
10.48/4.28	import qualified Monad;
10.48/4.28	import qualified Prelude;
10.48/4.28	}
10.48/4.28	module Main where {
10.48/4.28	import qualified Maybe;
10.48/4.28	import qualified Monad;
10.48/4.28	import qualified Prelude;
10.48/4.28	}
10.48/4.28	module Monad where {
10.48/4.28	import qualified Main;
10.48/4.28	import qualified Maybe;
10.48/4.28	import qualified Prelude;
10.48/4.28	foldM :: Monad a => (b  ->  c  ->  a b)  ->  b  ->  [c]  ->  a b;
10.48/4.28	foldM vy a [] = return a;
10.48/4.28	foldM f a (x : xs) = f a x >>= foldM0 f xs;
10.48/4.28	
10.48/4.28	foldM0 f xs fax = foldM f fax xs;
10.48/4.28	
10.48/4.28	}
10.48/4.28	
10.48/4.28	----------------------------------------
10.48/4.28	
10.48/4.28	(5) COR (EQUIVALENT)
10.48/4.28	Cond Reductions:
10.48/4.28	The following Function with conditions
10.48/4.28	"undefined |Falseundefined;
10.48/4.28	"
10.48/4.28	is transformed to
10.48/4.28	"undefined  = undefined1;
10.48/4.28	"
10.48/4.28	"undefined0 True = undefined;
10.48/4.28	"
10.48/4.28	"undefined1  = undefined0 False;
10.48/4.28	"
10.48/4.28	
10.48/4.28	----------------------------------------
10.48/4.28	
10.48/4.28	(6)
10.48/4.28	Obligation:
10.48/4.28	mainModule Main 
10.48/4.28	module Maybe where {
10.48/4.28	import qualified Main;
10.48/4.28	import qualified Monad;
10.48/4.28	import qualified Prelude;
10.48/4.28	}
10.48/4.28	module Main where {
10.48/4.28	import qualified Maybe;
10.48/4.28	import qualified Monad;
10.48/4.28	import qualified Prelude;
10.48/4.28	}
10.48/4.28	module Monad where {
10.48/4.28	import qualified Main;
10.48/4.28	import qualified Maybe;
10.48/4.28	import qualified Prelude;
10.48/4.28	foldM :: Monad b => (a  ->  c  ->  b a)  ->  a  ->  [c]  ->  b a;
10.48/4.28	foldM vy a [] = return a;
10.48/4.28	foldM f a (x : xs) = f a x >>= foldM0 f xs;
10.48/4.28	
10.48/4.28	foldM0 f xs fax = foldM f fax xs;
10.48/4.28	
10.48/4.28	}
10.48/4.28	
10.48/4.28	----------------------------------------
10.48/4.28	
10.48/4.28	(7) Narrow (SOUND)
10.48/4.28	Haskell To QDPs
10.48/4.28	
10.48/4.28	digraph dp_graph {
10.48/4.28	node [outthreshold=100, inthreshold=100];1[label="Monad.foldM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.48/4.28	3[label="Monad.foldM vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.48/4.28	4[label="Monad.foldM vz3 vz4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
10.48/4.28	5[label="Monad.foldM vz3 vz4 vz5",fontsize=16,color="burlywood",shape="triangle"];30[label="vz5/vz50 : vz51",fontsize=10,color="white",style="solid",shape="box"];5 -> 30[label="",style="solid", color="burlywood", weight=9];
10.48/4.28	30 -> 6[label="",style="solid", color="burlywood", weight=3];
10.48/4.28	31[label="vz5/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 31[label="",style="solid", color="burlywood", weight=9];
10.48/4.28	31 -> 7[label="",style="solid", color="burlywood", weight=3];
10.48/4.28	6[label="Monad.foldM vz3 vz4 (vz50 : vz51)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
10.48/4.28	7[label="Monad.foldM vz3 vz4 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
10.48/4.28	8[label="vz3 vz4 vz50 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
10.48/4.28	9[label="return vz4",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10.48/4.28	10 -> 12[label="",style="dashed", color="red", weight=0];
10.48/4.28	10[label="primbindIO (vz3 vz4 vz50) (Monad.foldM0 vz3 vz51)",fontsize=16,color="magenta"];10 -> 13[label="",style="dashed", color="magenta", weight=3];
10.48/4.28	11[label="primretIO vz4",fontsize=16,color="black",shape="box"];11 -> 14[label="",style="solid", color="black", weight=3];
10.48/4.28	13[label="vz3 vz4 vz50",fontsize=16,color="green",shape="box"];13 -> 21[label="",style="dashed", color="green", weight=3];
10.48/4.28	13 -> 22[label="",style="dashed", color="green", weight=3];
10.48/4.28	12[label="primbindIO vz6 (Monad.foldM0 vz3 vz51)",fontsize=16,color="burlywood",shape="triangle"];32[label="vz6/IO vz60",fontsize=10,color="white",style="solid",shape="box"];12 -> 32[label="",style="solid", color="burlywood", weight=9];
10.48/4.28	32 -> 17[label="",style="solid", color="burlywood", weight=3];
10.48/4.28	33[label="vz6/AProVE_IO vz60",fontsize=10,color="white",style="solid",shape="box"];12 -> 33[label="",style="solid", color="burlywood", weight=9];
10.48/4.28	33 -> 18[label="",style="solid", color="burlywood", weight=3];
10.48/4.28	34[label="vz6/AProVE_Exception vz60",fontsize=10,color="white",style="solid",shape="box"];12 -> 34[label="",style="solid", color="burlywood", weight=9];
10.48/4.28	34 -> 19[label="",style="solid", color="burlywood", weight=3];
10.48/4.28	35[label="vz6/AProVE_Error vz60",fontsize=10,color="white",style="solid",shape="box"];12 -> 35[label="",style="solid", color="burlywood", weight=9];
10.48/4.28	35 -> 20[label="",style="solid", color="burlywood", weight=3];
10.48/4.28	14[label="AProVE_IO vz4",fontsize=16,color="green",shape="box"];21[label="vz4",fontsize=16,color="green",shape="box"];22[label="vz50",fontsize=16,color="green",shape="box"];17[label="primbindIO (IO vz60) (Monad.foldM0 vz3 vz51)",fontsize=16,color="black",shape="box"];17 -> 23[label="",style="solid", color="black", weight=3];
10.48/4.28	18[label="primbindIO (AProVE_IO vz60) (Monad.foldM0 vz3 vz51)",fontsize=16,color="black",shape="box"];18 -> 24[label="",style="solid", color="black", weight=3];
10.48/4.28	19[label="primbindIO (AProVE_Exception vz60) (Monad.foldM0 vz3 vz51)",fontsize=16,color="black",shape="box"];19 -> 25[label="",style="solid", color="black", weight=3];
10.48/4.28	20[label="primbindIO (AProVE_Error vz60) (Monad.foldM0 vz3 vz51)",fontsize=16,color="black",shape="box"];20 -> 26[label="",style="solid", color="black", weight=3];
10.48/4.28	23[label="error []",fontsize=16,color="red",shape="box"];24[label="Monad.foldM0 vz3 vz51 vz60",fontsize=16,color="black",shape="box"];24 -> 27[label="",style="solid", color="black", weight=3];
10.48/4.28	25[label="AProVE_Exception vz60",fontsize=16,color="green",shape="box"];26[label="AProVE_Error vz60",fontsize=16,color="green",shape="box"];27 -> 5[label="",style="dashed", color="red", weight=0];
10.48/4.28	27[label="Monad.foldM vz3 vz60 vz51",fontsize=16,color="magenta"];27 -> 28[label="",style="dashed", color="magenta", weight=3];
10.48/4.28	27 -> 29[label="",style="dashed", color="magenta", weight=3];
10.48/4.28	28[label="vz51",fontsize=16,color="green",shape="box"];29[label="vz60",fontsize=16,color="green",shape="box"];}
10.48/4.28	
10.48/4.28	----------------------------------------
10.48/4.28	
10.48/4.28	(8)
10.48/4.28	Obligation:
10.48/4.28	Q DP problem:
10.48/4.28	The TRS P consists of the following rules:
10.48/4.28	
10.48/4.28	   new_primbindIO(vz3, vz51, h, ba) -> new_foldM(vz3, vz51, h, ba)
10.48/4.28	   new_foldM(vz3, :(vz50, vz51), h, ba) -> new_primbindIO(vz3, vz51, h, ba)
10.48/4.28	
10.48/4.28	R is empty.
10.48/4.28	Q is empty.
10.48/4.28	We have to consider all minimal (P,Q,R)-chains.
10.48/4.28	----------------------------------------
10.48/4.28	
10.48/4.28	(9) QDPSizeChangeProof (EQUIVALENT)
10.48/4.28	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. 
10.48/4.28	
10.48/4.28	From the DPs we obtained the following set of size-change graphs:
10.48/4.28	*new_foldM(vz3, :(vz50, vz51), h, ba) -> new_primbindIO(vz3, vz51, h, ba)
10.48/4.28	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
10.48/4.28	
10.48/4.28	
10.48/4.28	*new_primbindIO(vz3, vz51, h, ba) -> new_foldM(vz3, vz51, h, ba)
10.48/4.28	The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4
10.48/4.28	
10.48/4.28	
10.48/4.28	----------------------------------------
10.48/4.28	
10.48/4.28	(10)
10.48/4.28	YES
10.48/4.32	EOF