7.84/3.54	YES
9.07/3.97	proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
9.07/3.97	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.07/3.97	
9.07/3.97	
9.07/3.97	H-Termination with start terms of the given HASKELL could be proven:
9.07/3.97	
9.07/3.97	(0) HASKELL
9.07/3.97	(1) BR [EQUIVALENT, 0 ms]
9.07/3.97	(2) HASKELL
9.07/3.97	(3) COR [EQUIVALENT, 0 ms]
9.07/3.97	(4) HASKELL
9.07/3.97	(5) Narrow [SOUND, 0 ms]
9.07/3.97	(6) QDP
9.07/3.97	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.07/3.97	(8) YES
9.07/3.97	
9.07/3.97	
9.07/3.97	----------------------------------------
9.07/3.97	
9.07/3.97	(0)
9.07/3.97	Obligation:
9.07/3.97	mainModule Main 
9.07/3.97	module Main where {
9.07/3.97	import qualified Prelude;
9.07/3.97	}
9.07/3.97	
9.07/3.97	----------------------------------------
9.07/3.97	
9.07/3.97	(1) BR (EQUIVALENT)
9.07/3.97	Replaced joker patterns by fresh variables and removed binding patterns.
9.07/3.97	----------------------------------------
9.07/3.97	
9.07/3.97	(2)
9.07/3.97	Obligation:
9.07/3.97	mainModule Main 
9.07/3.97	module Main where {
9.07/3.97	import qualified Prelude;
9.07/3.97	}
9.07/3.97	
9.07/3.97	----------------------------------------
9.07/3.97	
9.07/3.97	(3) COR (EQUIVALENT)
9.07/3.97	Cond Reductions:
9.07/3.97	The following Function with conditions
9.07/3.97	"undefined |Falseundefined;
9.07/3.97	"
9.07/3.97	is transformed to
9.07/3.97	"undefined  = undefined1;
9.07/3.97	"
9.07/3.97	"undefined0 True = undefined;
9.07/3.97	"
9.07/3.97	"undefined1  = undefined0 False;
9.07/3.97	"
9.07/3.97	
9.07/3.97	----------------------------------------
9.07/3.97	
9.07/3.97	(4)
9.07/3.97	Obligation:
9.07/3.97	mainModule Main 
9.07/3.97	module Main where {
9.07/3.97	import qualified Prelude;
9.07/3.97	}
9.07/3.97	
9.07/3.97	----------------------------------------
9.07/3.97	
9.07/3.97	(5) Narrow (SOUND)
9.07/3.97	Haskell To QDPs
9.07/3.97	
9.07/3.97	digraph dp_graph {
9.07/3.97	node [outthreshold=100, inthreshold=100];1[label="foldr",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.07/3.97	3[label="foldr vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.07/3.97	4[label="foldr vx3 vx4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
9.07/3.97	5[label="foldr vx3 vx4 vx5",fontsize=16,color="burlywood",shape="triangle"];13[label="vx5/vx50 : vx51",fontsize=10,color="white",style="solid",shape="box"];5 -> 13[label="",style="solid", color="burlywood", weight=9];
9.07/3.97	13 -> 6[label="",style="solid", color="burlywood", weight=3];
9.07/3.97	14[label="vx5/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 14[label="",style="solid", color="burlywood", weight=9];
9.07/3.97	14 -> 7[label="",style="solid", color="burlywood", weight=3];
9.07/3.97	6[label="foldr vx3 vx4 (vx50 : vx51)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
9.07/3.97	7[label="foldr vx3 vx4 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
9.07/3.97	8[label="vx3 vx50 (foldr vx3 vx4 vx51)",fontsize=16,color="green",shape="box"];8 -> 10[label="",style="dashed", color="green", weight=3];
9.07/3.97	8 -> 11[label="",style="dashed", color="green", weight=3];
9.07/3.97	9[label="vx4",fontsize=16,color="green",shape="box"];10[label="vx50",fontsize=16,color="green",shape="box"];11 -> 5[label="",style="dashed", color="red", weight=0];
9.07/3.97	11[label="foldr vx3 vx4 vx51",fontsize=16,color="magenta"];11 -> 12[label="",style="dashed", color="magenta", weight=3];
9.07/3.97	12[label="vx51",fontsize=16,color="green",shape="box"];}
9.07/3.97	
9.07/3.97	----------------------------------------
9.07/3.97	
9.07/3.97	(6)
9.07/3.97	Obligation:
9.07/3.97	Q DP problem:
9.07/3.97	The TRS P consists of the following rules:
9.07/3.97	
9.07/3.97	   new_foldr(vx3, vx4, :(vx50, vx51), h, ba) -> new_foldr(vx3, vx4, vx51, h, ba)
9.07/3.97	
9.07/3.97	R is empty.
9.07/3.97	Q is empty.
9.07/3.97	We have to consider all minimal (P,Q,R)-chains.
9.07/3.97	----------------------------------------
9.07/3.97	
9.07/3.97	(7) QDPSizeChangeProof (EQUIVALENT)
9.07/3.97	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. 
9.07/3.97	
9.07/3.97	From the DPs we obtained the following set of size-change graphs:
9.07/3.97	*new_foldr(vx3, vx4, :(vx50, vx51), h, ba) -> new_foldr(vx3, vx4, vx51, h, ba)
9.07/3.97	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5
9.07/3.97	
9.07/3.97	
9.07/3.97	----------------------------------------
9.07/3.97	
9.07/3.97	(8)
9.07/3.97	YES
9.29/4.00	EOF