9.47/3.92	YES
11.34/4.47	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
11.34/4.47	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.34/4.47	
11.34/4.47	
11.34/4.47	H-Termination with start terms of the given HASKELL could be proven:
11.34/4.47	
11.34/4.47	(0) HASKELL
11.34/4.47	(1) LR [EQUIVALENT, 0 ms]
11.34/4.47	(2) HASKELL
11.34/4.47	(3) BR [EQUIVALENT, 0 ms]
11.34/4.47	(4) HASKELL
11.34/4.47	(5) COR [EQUIVALENT, 0 ms]
11.34/4.47	(6) HASKELL
11.34/4.47	(7) Narrow [SOUND, 0 ms]
11.34/4.47	(8) QDP
11.34/4.47	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.34/4.47	(10) YES
11.34/4.47	
11.34/4.47	
11.34/4.47	----------------------------------------
11.34/4.47	
11.34/4.47	(0)
11.34/4.47	Obligation:
11.34/4.47	mainModule Main 
11.34/4.47	module Maybe where {
11.34/4.47	import qualified Main;
11.34/4.47	import qualified Monad;
11.34/4.47	import qualified Prelude;
11.34/4.47	}
11.34/4.47	module Main where {
11.34/4.47	import qualified Maybe;
11.34/4.47	import qualified Monad;
11.34/4.47	import qualified Prelude;
11.34/4.47	}
11.34/4.47	module Monad where {
11.34/4.47	import qualified Main;
11.34/4.47	import qualified Maybe;
11.34/4.47	import qualified Prelude;
11.34/4.47	foldM :: Monad b => (c  ->  a  ->  b c)  ->  c  ->  [a]  ->  b c;
11.34/4.47	foldM _ a [] = return a;
11.34/4.47	foldM f a (x : xs) = f a x >>= (\fax ->foldM f fax xs);
11.34/4.47	
11.34/4.47	}
11.34/4.47	
11.34/4.47	----------------------------------------
11.34/4.47	
11.34/4.47	(1) LR (EQUIVALENT)
11.34/4.47	Lambda Reductions:
11.34/4.47	The following Lambda expression
11.34/4.47	"\fax->foldM f fax xs"
11.34/4.47	is transformed to
11.34/4.47	"foldM0 f xs fax = foldM f fax xs;
11.34/4.47	"
11.34/4.47	
11.34/4.47	----------------------------------------
11.34/4.47	
11.34/4.47	(2)
11.34/4.47	Obligation:
11.34/4.47	mainModule Main 
11.34/4.47	module Maybe where {
11.34/4.47	import qualified Main;
11.34/4.47	import qualified Monad;
11.34/4.47	import qualified Prelude;
11.34/4.47	}
11.34/4.47	module Main where {
11.34/4.47	import qualified Maybe;
11.34/4.47	import qualified Monad;
11.34/4.47	import qualified Prelude;
11.34/4.47	}
11.34/4.47	module Monad where {
11.34/4.47	import qualified Main;
11.34/4.47	import qualified Maybe;
11.34/4.47	import qualified Prelude;
11.34/4.47	foldM :: Monad b => (a  ->  c  ->  b a)  ->  a  ->  [c]  ->  b a;
11.34/4.47	foldM _ a [] = return a;
11.34/4.47	foldM f a (x : xs) = f a x >>= foldM0 f xs;
11.34/4.47	
11.34/4.47	foldM0 f xs fax = foldM f fax xs;
11.34/4.47	
11.34/4.47	}
11.34/4.47	
11.34/4.47	----------------------------------------
11.34/4.47	
11.34/4.47	(3) BR (EQUIVALENT)
11.34/4.47	Replaced joker patterns by fresh variables and removed binding patterns.
11.34/4.47	----------------------------------------
11.34/4.47	
11.34/4.47	(4)
11.34/4.47	Obligation:
11.34/4.47	mainModule Main 
11.34/4.47	module Maybe where {
11.34/4.47	import qualified Main;
11.34/4.47	import qualified Monad;
11.34/4.47	import qualified Prelude;
11.34/4.47	}
11.34/4.47	module Main where {
11.34/4.47	import qualified Maybe;
11.34/4.47	import qualified Monad;
11.34/4.47	import qualified Prelude;
11.34/4.47	}
11.34/4.47	module Monad where {
11.34/4.47	import qualified Main;
11.34/4.47	import qualified Maybe;
11.34/4.47	import qualified Prelude;
11.34/4.47	foldM :: Monad a => (c  ->  b  ->  a c)  ->  c  ->  [b]  ->  a c;
11.34/4.47	foldM vy a [] = return a;
11.34/4.47	foldM f a (x : xs) = f a x >>= foldM0 f xs;
11.34/4.47	
11.34/4.47	foldM0 f xs fax = foldM f fax xs;
11.34/4.47	
11.34/4.47	}
11.34/4.47	
11.34/4.47	----------------------------------------
11.34/4.47	
11.34/4.47	(5) COR (EQUIVALENT)
11.34/4.47	Cond Reductions:
11.34/4.47	The following Function with conditions
11.34/4.47	"undefined |Falseundefined;
11.34/4.47	"
11.34/4.47	is transformed to
11.34/4.47	"undefined  = undefined1;
11.34/4.47	"
11.34/4.47	"undefined0 True = undefined;
11.34/4.47	"
11.34/4.47	"undefined1  = undefined0 False;
11.34/4.47	"
11.34/4.47	
11.34/4.47	----------------------------------------
11.34/4.47	
11.34/4.47	(6)
11.34/4.47	Obligation:
11.34/4.47	mainModule Main 
11.34/4.47	module Maybe where {
11.34/4.47	import qualified Main;
11.34/4.47	import qualified Monad;
11.34/4.47	import qualified Prelude;
11.34/4.47	}
11.34/4.47	module Main where {
11.34/4.47	import qualified Maybe;
11.34/4.47	import qualified Monad;
11.34/4.47	import qualified Prelude;
11.34/4.47	}
11.34/4.47	module Monad where {
11.34/4.47	import qualified Main;
11.34/4.47	import qualified Maybe;
11.34/4.47	import qualified Prelude;
11.34/4.47	foldM :: Monad c => (b  ->  a  ->  c b)  ->  b  ->  [a]  ->  c b;
11.34/4.47	foldM vy a [] = return a;
11.34/4.47	foldM f a (x : xs) = f a x >>= foldM0 f xs;
11.34/4.47	
11.34/4.47	foldM0 f xs fax = foldM f fax xs;
11.34/4.47	
11.34/4.47	}
11.34/4.47	
11.34/4.47	----------------------------------------
11.34/4.47	
11.34/4.47	(7) Narrow (SOUND)
11.34/4.47	Haskell To QDPs
11.34/4.47	
11.34/4.47	digraph dp_graph {
11.34/4.47	node [outthreshold=100, inthreshold=100];1[label="Monad.foldM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.34/4.47	3[label="Monad.foldM vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
11.34/4.47	4[label="Monad.foldM vz3 vz4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
11.34/4.47	5[label="Monad.foldM vz3 vz4 vz5",fontsize=16,color="burlywood",shape="triangle"];24[label="vz5/vz50 : vz51",fontsize=10,color="white",style="solid",shape="box"];5 -> 24[label="",style="solid", color="burlywood", weight=9];
11.34/4.47	24 -> 6[label="",style="solid", color="burlywood", weight=3];
11.34/4.47	25[label="vz5/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 25[label="",style="solid", color="burlywood", weight=9];
11.34/4.47	25 -> 7[label="",style="solid", color="burlywood", weight=3];
11.34/4.47	6[label="Monad.foldM vz3 vz4 (vz50 : vz51)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
11.34/4.47	7[label="Monad.foldM vz3 vz4 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
11.34/4.47	8 -> 10[label="",style="dashed", color="red", weight=0];
11.34/4.47	8[label="vz3 vz4 vz50 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="magenta"];8 -> 11[label="",style="dashed", color="magenta", weight=3];
11.34/4.47	9[label="return vz4",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3];
11.34/4.47	11[label="vz3 vz4 vz50",fontsize=16,color="green",shape="box"];11 -> 17[label="",style="dashed", color="green", weight=3];
11.34/4.47	11 -> 18[label="",style="dashed", color="green", weight=3];
11.34/4.47	10[label="vz6 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="burlywood",shape="triangle"];26[label="vz6/Nothing",fontsize=10,color="white",style="solid",shape="box"];10 -> 26[label="",style="solid", color="burlywood", weight=9];
11.34/4.47	26 -> 15[label="",style="solid", color="burlywood", weight=3];
11.34/4.47	27[label="vz6/Just vz60",fontsize=10,color="white",style="solid",shape="box"];10 -> 27[label="",style="solid", color="burlywood", weight=9];
11.34/4.47	27 -> 16[label="",style="solid", color="burlywood", weight=3];
11.34/4.47	12[label="Just vz4",fontsize=16,color="green",shape="box"];17[label="vz4",fontsize=16,color="green",shape="box"];18[label="vz50",fontsize=16,color="green",shape="box"];15[label="Nothing >>= Monad.foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
11.34/4.47	16[label="Just vz60 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
11.34/4.47	19[label="Nothing",fontsize=16,color="green",shape="box"];20[label="Monad.foldM0 vz3 vz51 vz60",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3];
11.34/4.47	21 -> 5[label="",style="dashed", color="red", weight=0];
11.34/4.47	21[label="Monad.foldM vz3 vz60 vz51",fontsize=16,color="magenta"];21 -> 22[label="",style="dashed", color="magenta", weight=3];
11.34/4.47	21 -> 23[label="",style="dashed", color="magenta", weight=3];
11.34/4.47	22[label="vz60",fontsize=16,color="green",shape="box"];23[label="vz51",fontsize=16,color="green",shape="box"];}
11.34/4.47	
11.34/4.47	----------------------------------------
11.34/4.47	
11.34/4.47	(8)
11.34/4.47	Obligation:
11.34/4.47	Q DP problem:
11.34/4.47	The TRS P consists of the following rules:
11.34/4.47	
11.34/4.47	   new_foldM(vz3, :(vz50, vz51), h, ba) -> new_gtGtEs(vz3, vz51, h, ba)
11.34/4.47	   new_gtGtEs(vz3, vz51, h, ba) -> new_foldM(vz3, vz51, h, ba)
11.34/4.47	
11.34/4.47	R is empty.
11.34/4.47	Q is empty.
11.34/4.47	We have to consider all minimal (P,Q,R)-chains.
11.34/4.47	----------------------------------------
11.34/4.47	
11.34/4.47	(9) QDPSizeChangeProof (EQUIVALENT)
11.34/4.47	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. 
11.34/4.47	
11.34/4.47	From the DPs we obtained the following set of size-change graphs:
11.34/4.47	*new_gtGtEs(vz3, vz51, h, ba) -> new_foldM(vz3, vz51, h, ba)
11.34/4.47	The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4
11.34/4.47	
11.34/4.47	
11.34/4.47	*new_foldM(vz3, :(vz50, vz51), h, ba) -> new_gtGtEs(vz3, vz51, h, ba)
11.34/4.47	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
11.34/4.47	
11.34/4.47	
11.34/4.47	----------------------------------------
11.34/4.47	
11.34/4.47	(10)
11.34/4.47	YES
11.34/4.51	EOF