10.79/4.50	YES
12.95/5.11	proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
12.95/5.11	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.95/5.11	
12.95/5.11	
12.95/5.11	H-Termination with start terms of the given HASKELL could be proven:
12.95/5.11	
12.95/5.11	(0) HASKELL
12.95/5.11	(1) LR [EQUIVALENT, 0 ms]
12.95/5.11	(2) HASKELL
12.95/5.11	(3) CR [EQUIVALENT, 0 ms]
12.95/5.11	(4) HASKELL
12.95/5.11	(5) BR [EQUIVALENT, 0 ms]
12.95/5.11	(6) HASKELL
12.95/5.11	(7) COR [EQUIVALENT, 0 ms]
12.95/5.11	(8) HASKELL
12.95/5.11	(9) Narrow [SOUND, 0 ms]
12.95/5.11	(10) AND
12.95/5.11	    (11) QDP
12.95/5.11	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.95/5.11	        (13) YES
12.95/5.11	    (14) QDP
12.95/5.11	        (15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.95/5.11	        (16) YES
12.95/5.11	    (17) QDP
12.95/5.11	        (18) QDPSizeChangeProof [EQUIVALENT, 13 ms]
12.95/5.11	        (19) YES
12.95/5.11	
12.95/5.11	
12.95/5.11	----------------------------------------
12.95/5.11	
12.95/5.11	(0)
12.95/5.11	Obligation:
12.95/5.11	mainModule Main 
12.95/5.11	module Maybe where {
12.95/5.11	import qualified List;
12.95/5.11	import qualified Main;
12.95/5.11	import qualified Prelude;
12.95/5.11	}
12.95/5.11	module List where {
12.95/5.11	import qualified Main;
12.95/5.11	import qualified Maybe;
12.95/5.11	import qualified Prelude;
12.95/5.11	transpose :: [[a]]  ->  [[a]];
12.95/5.11	transpose [] = [];
12.95/5.11	transpose ([] : xss) = transpose xss;
12.95/5.11	transpose ((x : xs) : xss) = (x : concatMap (\vv3 ->case vv3 of { 
12.95/5.11	h : t-> h : []; 
12.95/5.11	_-> []; 
12.95/5.11	 } ) xss) : transpose (xs : concatMap (\vv4 ->case vv4 of { 
12.95/5.11	h : t-> t : []; 
12.95/5.11	_-> []; 
12.95/5.11	 } ) xss);
12.95/5.11	
12.95/5.11	}
12.95/5.11	module Main where {
12.95/5.11	import qualified List;
12.95/5.11	import qualified Maybe;
12.95/5.11	import qualified Prelude;
12.95/5.11	}
12.95/5.11	
12.95/5.11	----------------------------------------
12.95/5.11	
12.95/5.11	(1) LR (EQUIVALENT)
12.95/5.11	Lambda Reductions:
12.95/5.11	The following Lambda expression
12.95/5.11	"\vv3->case vv3 of {
12.95/5.11	h : t  -> h : [];
12.95/5.11	_  -> []}
12.95/5.11	"
12.95/5.11	is transformed to
12.95/5.11	"transpose0 vv3 = case vv3 of {
12.95/5.11	h : t  -> h : [];
12.95/5.11	_  -> []}
12.95/5.11	;
12.95/5.11	"
12.95/5.11	The following Lambda expression
12.95/5.11	"\vv4->case vv4 of {
12.95/5.11	h : t  -> t : [];
12.95/5.11	_  -> []}
12.95/5.11	"
12.95/5.11	is transformed to
12.95/5.11	"transpose1 vv4 = case vv4 of {
12.95/5.11	h : t  -> t : [];
12.95/5.11	_  -> []}
12.95/5.11	;
12.95/5.11	"
12.95/5.11	
12.95/5.11	----------------------------------------
12.95/5.11	
12.95/5.11	(2)
12.95/5.11	Obligation:
12.95/5.11	mainModule Main 
12.95/5.11	module Maybe where {
12.95/5.11	import qualified List;
12.95/5.11	import qualified Main;
12.95/5.11	import qualified Prelude;
12.95/5.11	}
12.95/5.11	module List where {
12.95/5.11	import qualified Main;
12.95/5.11	import qualified Maybe;
12.95/5.11	import qualified Prelude;
12.95/5.11	transpose :: [[a]]  ->  [[a]];
12.95/5.11	transpose [] = [];
12.95/5.11	transpose ([] : xss) = transpose xss;
12.95/5.11	transpose ((x : xs) : xss) = (x : concatMap transpose0 xss) : transpose (xs : concatMap transpose1 xss);
12.95/5.11	
12.95/5.11	transpose0 vv3 = case vv3 of { 
12.95/5.11	h : t-> h : []; 
12.95/5.11	_-> []; 
12.95/5.11	 } ;
12.95/5.11	
12.95/5.11	transpose1 vv4 = case vv4 of { 
12.95/5.11	h : t-> t : []; 
12.95/5.11	_-> []; 
12.95/5.11	 } ;
12.95/5.11	
12.95/5.11	}
12.95/5.11	module Main where {
12.95/5.11	import qualified List;
12.95/5.11	import qualified Maybe;
12.95/5.11	import qualified Prelude;
12.95/5.11	}
12.95/5.11	
12.95/5.11	----------------------------------------
12.95/5.11	
12.95/5.11	(3) CR (EQUIVALENT)
12.95/5.11	Case Reductions:
12.95/5.11	The following Case expression
12.95/5.11	"case vv4 of {
12.95/5.11	h : t  -> t : [];
12.95/5.11	_  -> []}
12.95/5.11	"
12.95/5.11	is transformed to
12.95/5.11	"transpose10 (h : t) = t : [];
12.95/5.11	transpose10 _ = [];
12.95/5.11	"
12.95/5.11	The following Case expression
12.95/5.11	"case vv3 of {
12.95/5.11	h : t  -> h : [];
12.95/5.11	_  -> []}
12.95/5.11	"
12.95/5.11	is transformed to
12.95/5.11	"transpose00 (h : t) = h : [];
12.95/5.11	transpose00 _ = [];
12.95/5.11	"
12.95/5.11	
12.95/5.11	----------------------------------------
12.95/5.11	
12.95/5.11	(4)
12.95/5.11	Obligation:
12.95/5.11	mainModule Main 
12.95/5.11	module Maybe where {
12.95/5.11	import qualified List;
12.95/5.11	import qualified Main;
12.95/5.11	import qualified Prelude;
12.95/5.11	}
12.95/5.11	module List where {
12.95/5.11	import qualified Main;
12.95/5.11	import qualified Maybe;
12.95/5.11	import qualified Prelude;
12.95/5.11	transpose :: [[a]]  ->  [[a]];
12.95/5.11	transpose [] = [];
12.95/5.11	transpose ([] : xss) = transpose xss;
12.95/5.11	transpose ((x : xs) : xss) = (x : concatMap transpose0 xss) : transpose (xs : concatMap transpose1 xss);
12.95/5.11	
12.95/5.11	transpose0 vv3 = transpose00 vv3;
12.95/5.11	
12.95/5.11	transpose00 (h : t) = h : [];
12.95/5.11	transpose00 _ = [];
12.95/5.11	
12.95/5.11	transpose1 vv4 = transpose10 vv4;
12.95/5.11	
12.95/5.11	transpose10 (h : t) = t : [];
12.95/5.11	transpose10 _ = [];
12.95/5.11	
12.95/5.11	}
12.95/5.11	module Main where {
12.95/5.11	import qualified List;
12.95/5.11	import qualified Maybe;
12.95/5.11	import qualified Prelude;
12.95/5.11	}
12.95/5.11	
12.95/5.11	----------------------------------------
12.95/5.11	
12.95/5.11	(5) BR (EQUIVALENT)
12.95/5.11	Replaced joker patterns by fresh variables and removed binding patterns.
12.95/5.11	----------------------------------------
12.95/5.11	
12.95/5.11	(6)
12.95/5.11	Obligation:
12.95/5.11	mainModule Main 
12.95/5.11	module Maybe where {
12.95/5.11	import qualified List;
12.95/5.11	import qualified Main;
12.95/5.11	import qualified Prelude;
12.95/5.11	}
12.95/5.11	module List where {
12.95/5.11	import qualified Main;
12.95/5.11	import qualified Maybe;
12.95/5.11	import qualified Prelude;
12.95/5.11	transpose :: [[a]]  ->  [[a]];
12.95/5.11	transpose [] = [];
12.95/5.11	transpose ([] : xss) = transpose xss;
12.95/5.11	transpose ((x : xs) : xss) = (x : concatMap transpose0 xss) : transpose (xs : concatMap transpose1 xss);
12.95/5.11	
12.95/5.11	transpose0 vv3 = transpose00 vv3;
12.95/5.11	
12.95/5.11	transpose00 (h : t) = h : [];
12.95/5.11	transpose00 vz = [];
12.95/5.11	
12.95/5.11	transpose1 vv4 = transpose10 vv4;
12.95/5.11	
12.95/5.11	transpose10 (h : t) = t : [];
12.95/5.11	transpose10 vy = [];
12.95/5.11	
12.95/5.11	}
12.95/5.11	module Main where {
12.95/5.11	import qualified List;
12.95/5.11	import qualified Maybe;
12.95/5.11	import qualified Prelude;
12.95/5.11	}
12.95/5.11	
12.95/5.11	----------------------------------------
12.95/5.11	
12.95/5.11	(7) COR (EQUIVALENT)
12.95/5.11	Cond Reductions:
12.95/5.11	The following Function with conditions
12.95/5.11	"undefined |Falseundefined;
12.95/5.11	"
12.95/5.11	is transformed to
12.95/5.11	"undefined  = undefined1;
12.95/5.11	"
12.95/5.11	"undefined0 True = undefined;
12.95/5.11	"
12.95/5.11	"undefined1  = undefined0 False;
12.95/5.11	"
12.95/5.11	
12.95/5.11	----------------------------------------
12.95/5.11	
12.95/5.11	(8)
12.95/5.11	Obligation:
12.95/5.11	mainModule Main 
12.95/5.11	module Maybe where {
12.95/5.11	import qualified List;
12.95/5.11	import qualified Main;
12.95/5.11	import qualified Prelude;
12.95/5.11	}
12.95/5.11	module List where {
12.95/5.11	import qualified Main;
12.95/5.11	import qualified Maybe;
12.95/5.11	import qualified Prelude;
12.95/5.11	transpose :: [[a]]  ->  [[a]];
12.95/5.11	transpose [] = [];
12.95/5.11	transpose ([] : xss) = transpose xss;
12.95/5.11	transpose ((x : xs) : xss) = (x : concatMap transpose0 xss) : transpose (xs : concatMap transpose1 xss);
12.95/5.11	
12.95/5.11	transpose0 vv3 = transpose00 vv3;
12.95/5.11	
12.95/5.11	transpose00 (h : t) = h : [];
12.95/5.11	transpose00 vz = [];
12.95/5.11	
12.95/5.11	transpose1 vv4 = transpose10 vv4;
12.95/5.11	
12.95/5.11	transpose10 (h : t) = t : [];
12.95/5.11	transpose10 vy = [];
12.95/5.11	
12.95/5.11	}
12.95/5.11	module Main where {
12.95/5.11	import qualified List;
12.95/5.11	import qualified Maybe;
12.95/5.11	import qualified Prelude;
12.95/5.11	}
12.95/5.11	
12.95/5.11	----------------------------------------
12.95/5.11	
12.95/5.11	(9) Narrow (SOUND)
12.95/5.11	Haskell To QDPs
12.95/5.11	
12.95/5.11	digraph dp_graph {
12.95/5.11	node [outthreshold=100, inthreshold=100];1[label="List.transpose",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.95/5.11	3[label="List.transpose wu3",fontsize=16,color="burlywood",shape="triangle"];55[label="wu3/wu30 : wu31",fontsize=10,color="white",style="solid",shape="box"];3 -> 55[label="",style="solid", color="burlywood", weight=9];
12.95/5.11	55 -> 4[label="",style="solid", color="burlywood", weight=3];
12.95/5.11	56[label="wu3/[]",fontsize=10,color="white",style="solid",shape="box"];3 -> 56[label="",style="solid", color="burlywood", weight=9];
12.95/5.11	56 -> 5[label="",style="solid", color="burlywood", weight=3];
12.95/5.11	4[label="List.transpose (wu30 : wu31)",fontsize=16,color="burlywood",shape="box"];57[label="wu30/wu300 : wu301",fontsize=10,color="white",style="solid",shape="box"];4 -> 57[label="",style="solid", color="burlywood", weight=9];
12.95/5.11	57 -> 6[label="",style="solid", color="burlywood", weight=3];
12.95/5.11	58[label="wu30/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 58[label="",style="solid", color="burlywood", weight=9];
12.95/5.11	58 -> 7[label="",style="solid", color="burlywood", weight=3];
12.95/5.11	5[label="List.transpose []",fontsize=16,color="black",shape="box"];5 -> 8[label="",style="solid", color="black", weight=3];
12.95/5.11	6[label="List.transpose ((wu300 : wu301) : wu31)",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
12.95/5.11	7[label="List.transpose ([] : wu31)",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
12.95/5.11	8[label="[]",fontsize=16,color="green",shape="box"];9[label="(wu300 : concatMap List.transpose0 wu31) : List.transpose (wu301 : concatMap List.transpose1 wu31)",fontsize=16,color="green",shape="box"];9 -> 11[label="",style="dashed", color="green", weight=3];
12.95/5.11	9 -> 12[label="",style="dashed", color="green", weight=3];
12.95/5.11	10 -> 3[label="",style="dashed", color="red", weight=0];
12.95/5.11	10[label="List.transpose wu31",fontsize=16,color="magenta"];10 -> 13[label="",style="dashed", color="magenta", weight=3];
12.95/5.11	11[label="concatMap List.transpose0 wu31",fontsize=16,color="black",shape="box"];11 -> 14[label="",style="solid", color="black", weight=3];
12.95/5.11	12 -> 3[label="",style="dashed", color="red", weight=0];
12.95/5.11	12[label="List.transpose (wu301 : concatMap List.transpose1 wu31)",fontsize=16,color="magenta"];12 -> 15[label="",style="dashed", color="magenta", weight=3];
12.95/5.11	13[label="wu31",fontsize=16,color="green",shape="box"];14[label="concat . map List.transpose0",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
12.95/5.11	15[label="wu301 : concatMap List.transpose1 wu31",fontsize=16,color="green",shape="box"];15 -> 17[label="",style="dashed", color="green", weight=3];
12.95/5.11	16[label="concat (map List.transpose0 wu31)",fontsize=16,color="black",shape="box"];16 -> 18[label="",style="solid", color="black", weight=3];
12.95/5.11	17[label="concatMap List.transpose1 wu31",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
12.95/5.11	18[label="foldr (++) [] (map List.transpose0 wu31)",fontsize=16,color="burlywood",shape="triangle"];59[label="wu31/wu310 : wu311",fontsize=10,color="white",style="solid",shape="box"];18 -> 59[label="",style="solid", color="burlywood", weight=9];
12.95/5.11	59 -> 20[label="",style="solid", color="burlywood", weight=3];
12.95/5.11	60[label="wu31/[]",fontsize=10,color="white",style="solid",shape="box"];18 -> 60[label="",style="solid", color="burlywood", weight=9];
12.95/5.11	60 -> 21[label="",style="solid", color="burlywood", weight=3];
12.95/5.11	19[label="concat . map List.transpose1",fontsize=16,color="black",shape="box"];19 -> 22[label="",style="solid", color="black", weight=3];
12.95/5.11	20[label="foldr (++) [] (map List.transpose0 (wu310 : wu311))",fontsize=16,color="black",shape="box"];20 -> 23[label="",style="solid", color="black", weight=3];
12.95/5.11	21[label="foldr (++) [] (map List.transpose0 [])",fontsize=16,color="black",shape="box"];21 -> 24[label="",style="solid", color="black", weight=3];
12.95/5.11	22[label="concat (map List.transpose1 wu31)",fontsize=16,color="black",shape="box"];22 -> 25[label="",style="solid", color="black", weight=3];
12.95/5.11	23[label="foldr (++) [] (List.transpose0 wu310 : map List.transpose0 wu311)",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
12.95/5.11	24[label="foldr (++) [] []",fontsize=16,color="black",shape="triangle"];24 -> 27[label="",style="solid", color="black", weight=3];
12.95/5.11	25[label="foldr (++) [] (map List.transpose1 wu31)",fontsize=16,color="burlywood",shape="triangle"];61[label="wu31/wu310 : wu311",fontsize=10,color="white",style="solid",shape="box"];25 -> 61[label="",style="solid", color="burlywood", weight=9];
12.95/5.11	61 -> 28[label="",style="solid", color="burlywood", weight=3];
12.95/5.11	62[label="wu31/[]",fontsize=10,color="white",style="solid",shape="box"];25 -> 62[label="",style="solid", color="burlywood", weight=9];
12.95/5.11	62 -> 29[label="",style="solid", color="burlywood", weight=3];
12.95/5.11	26 -> 30[label="",style="dashed", color="red", weight=0];
12.95/5.11	26[label="(++) List.transpose0 wu310 foldr (++) [] (map List.transpose0 wu311)",fontsize=16,color="magenta"];26 -> 31[label="",style="dashed", color="magenta", weight=3];
12.95/5.11	27[label="[]",fontsize=16,color="green",shape="box"];28[label="foldr (++) [] (map List.transpose1 (wu310 : wu311))",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
12.95/5.11	29[label="foldr (++) [] (map List.transpose1 [])",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
12.95/5.11	31 -> 18[label="",style="dashed", color="red", weight=0];
12.95/5.11	31[label="foldr (++) [] (map List.transpose0 wu311)",fontsize=16,color="magenta"];31 -> 34[label="",style="dashed", color="magenta", weight=3];
12.95/5.11	30[label="(++) List.transpose0 wu310 wu4",fontsize=16,color="black",shape="triangle"];30 -> 35[label="",style="solid", color="black", weight=3];
12.95/5.11	32[label="foldr (++) [] (List.transpose1 wu310 : map List.transpose1 wu311)",fontsize=16,color="black",shape="box"];32 -> 36[label="",style="solid", color="black", weight=3];
12.95/5.11	33 -> 24[label="",style="dashed", color="red", weight=0];
12.95/5.11	33[label="foldr (++) [] []",fontsize=16,color="magenta"];34[label="wu311",fontsize=16,color="green",shape="box"];35[label="(++) List.transpose00 wu310 wu4",fontsize=16,color="burlywood",shape="box"];63[label="wu310/wu3100 : wu3101",fontsize=10,color="white",style="solid",shape="box"];35 -> 63[label="",style="solid", color="burlywood", weight=9];
12.95/5.11	63 -> 37[label="",style="solid", color="burlywood", weight=3];
12.95/5.11	64[label="wu310/[]",fontsize=10,color="white",style="solid",shape="box"];35 -> 64[label="",style="solid", color="burlywood", weight=9];
12.95/5.11	64 -> 38[label="",style="solid", color="burlywood", weight=3];
12.95/5.11	36 -> 39[label="",style="dashed", color="red", weight=0];
12.95/5.11	36[label="(++) List.transpose1 wu310 foldr (++) [] (map List.transpose1 wu311)",fontsize=16,color="magenta"];36 -> 40[label="",style="dashed", color="magenta", weight=3];
12.95/5.11	37[label="(++) List.transpose00 (wu3100 : wu3101) wu4",fontsize=16,color="black",shape="box"];37 -> 41[label="",style="solid", color="black", weight=3];
12.95/5.11	38[label="(++) List.transpose00 [] wu4",fontsize=16,color="black",shape="box"];38 -> 42[label="",style="solid", color="black", weight=3];
12.95/5.11	40 -> 25[label="",style="dashed", color="red", weight=0];
12.95/5.11	40[label="foldr (++) [] (map List.transpose1 wu311)",fontsize=16,color="magenta"];40 -> 43[label="",style="dashed", color="magenta", weight=3];
12.95/5.11	39[label="(++) List.transpose1 wu310 wu5",fontsize=16,color="black",shape="triangle"];39 -> 44[label="",style="solid", color="black", weight=3];
12.95/5.11	41[label="(++) (wu3100 : []) wu4",fontsize=16,color="black",shape="triangle"];41 -> 45[label="",style="solid", color="black", weight=3];
12.95/5.11	42[label="(++) [] wu4",fontsize=16,color="black",shape="triangle"];42 -> 46[label="",style="solid", color="black", weight=3];
12.95/5.11	43[label="wu311",fontsize=16,color="green",shape="box"];44[label="(++) List.transpose10 wu310 wu5",fontsize=16,color="burlywood",shape="box"];65[label="wu310/wu3100 : wu3101",fontsize=10,color="white",style="solid",shape="box"];44 -> 65[label="",style="solid", color="burlywood", weight=9];
12.95/5.11	65 -> 47[label="",style="solid", color="burlywood", weight=3];
12.95/5.11	66[label="wu310/[]",fontsize=10,color="white",style="solid",shape="box"];44 -> 66[label="",style="solid", color="burlywood", weight=9];
12.95/5.11	66 -> 48[label="",style="solid", color="burlywood", weight=3];
12.95/5.11	45[label="wu3100 : [] ++ wu4",fontsize=16,color="green",shape="box"];45 -> 49[label="",style="dashed", color="green", weight=3];
12.95/5.11	46[label="wu4",fontsize=16,color="green",shape="box"];47[label="(++) List.transpose10 (wu3100 : wu3101) wu5",fontsize=16,color="black",shape="box"];47 -> 50[label="",style="solid", color="black", weight=3];
12.95/5.11	48[label="(++) List.transpose10 [] wu5",fontsize=16,color="black",shape="box"];48 -> 51[label="",style="solid", color="black", weight=3];
12.95/5.11	49 -> 42[label="",style="dashed", color="red", weight=0];
12.95/5.11	49[label="[] ++ wu4",fontsize=16,color="magenta"];50 -> 41[label="",style="dashed", color="red", weight=0];
12.95/5.11	50[label="(++) (wu3101 : []) wu5",fontsize=16,color="magenta"];50 -> 52[label="",style="dashed", color="magenta", weight=3];
12.95/5.11	50 -> 53[label="",style="dashed", color="magenta", weight=3];
12.95/5.11	51 -> 42[label="",style="dashed", color="red", weight=0];
12.95/5.11	51[label="(++) [] wu5",fontsize=16,color="magenta"];51 -> 54[label="",style="dashed", color="magenta", weight=3];
12.95/5.11	52[label="wu5",fontsize=16,color="green",shape="box"];53[label="wu3101",fontsize=16,color="green",shape="box"];54[label="wu5",fontsize=16,color="green",shape="box"];}
12.95/5.11	
12.95/5.11	----------------------------------------
12.95/5.11	
12.95/5.11	(10)
12.95/5.11	Complex Obligation (AND)
12.95/5.11	
12.95/5.11	----------------------------------------
12.95/5.11	
12.95/5.11	(11)
12.95/5.11	Obligation:
12.95/5.11	Q DP problem:
12.95/5.11	The TRS P consists of the following rules:
12.95/5.11	
12.95/5.11	   new_foldr0(:(wu310, wu311), ba) -> new_foldr0(wu311, ba)
12.95/5.11	
12.95/5.11	R is empty.
12.95/5.11	Q is empty.
12.95/5.11	We have to consider all minimal (P,Q,R)-chains.
12.95/5.11	----------------------------------------
12.95/5.11	
12.95/5.11	(12) QDPSizeChangeProof (EQUIVALENT)
12.95/5.11	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. 
12.95/5.11	
12.95/5.11	From the DPs we obtained the following set of size-change graphs:
12.95/5.11	*new_foldr0(:(wu310, wu311), ba) -> new_foldr0(wu311, ba)
12.95/5.11	The graph contains the following edges 1 > 1, 2 >= 2
12.95/5.11	
12.95/5.11	
12.95/5.11	----------------------------------------
12.95/5.11	
12.95/5.11	(13)
12.95/5.11	YES
12.95/5.11	
12.95/5.11	----------------------------------------
12.95/5.11	
12.95/5.11	(14)
12.95/5.11	Obligation:
12.95/5.11	Q DP problem:
12.95/5.11	The TRS P consists of the following rules:
12.95/5.11	
12.95/5.11	   new_foldr(:(wu310, wu311), ba) -> new_foldr(wu311, ba)
12.95/5.11	
12.95/5.11	R is empty.
12.95/5.11	Q is empty.
12.95/5.11	We have to consider all minimal (P,Q,R)-chains.
12.95/5.11	----------------------------------------
12.95/5.11	
12.95/5.11	(15) QDPSizeChangeProof (EQUIVALENT)
12.95/5.11	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. 
12.95/5.11	
12.95/5.11	From the DPs we obtained the following set of size-change graphs:
12.95/5.11	*new_foldr(:(wu310, wu311), ba) -> new_foldr(wu311, ba)
12.95/5.11	The graph contains the following edges 1 > 1, 2 >= 2
12.95/5.11	
12.95/5.11	
12.95/5.11	----------------------------------------
12.95/5.11	
12.95/5.11	(16)
12.95/5.11	YES
12.95/5.11	
12.95/5.11	----------------------------------------
12.95/5.11	
12.95/5.11	(17)
12.95/5.11	Obligation:
12.95/5.11	Q DP problem:
12.95/5.11	The TRS P consists of the following rules:
12.95/5.11	
12.95/5.11	   new_transpose(:([], wu31), ba) -> new_transpose(wu31, ba)
12.95/5.11	   new_transpose(:(:(wu300, wu301), wu31), ba) -> new_transpose(:(wu301, new_foldr1(wu31, ba)), ba)
12.95/5.11	
12.95/5.11	The TRS R consists of the following rules:
12.95/5.11	
12.95/5.11	   new_foldr1([], ba) -> new_foldr2(app(ty_[], ba))
12.95/5.11	   new_psPs0(wu4, ba) -> wu4
12.95/5.11	   new_foldr2(ba) -> []
12.95/5.11	   new_foldr1(:(wu310, wu311), ba) -> new_psPs1(wu310, new_foldr1(wu311, ba), ba)
12.95/5.11	   new_psPs1(:(wu3100, wu3101), wu5, ba) -> new_psPs(wu3101, wu5, app(ty_[], ba))
12.95/5.11	   new_psPs(wu3100, wu4, ba) -> :(wu3100, new_psPs0(wu4, ba))
12.95/5.11	   new_psPs1([], wu5, ba) -> new_psPs0(wu5, app(ty_[], ba))
12.95/5.11	
12.95/5.11	The set Q consists of the following terms:
12.95/5.11	
12.95/5.11	   new_psPs0(x0, x1)
12.95/5.11	   new_foldr1([], x0)
12.95/5.11	   new_foldr1(:(x0, x1), x2)
12.95/5.11	   new_psPs1([], x0, x1)
12.95/5.11	   new_psPs1(:(x0, x1), x2, x3)
12.95/5.11	   new_psPs(x0, x1, x2)
12.95/5.11	   new_foldr2(x0)
12.95/5.11	
12.95/5.11	We have to consider all minimal (P,Q,R)-chains.
12.95/5.11	----------------------------------------
12.95/5.11	
12.95/5.11	(18) QDPSizeChangeProof (EQUIVALENT)
12.95/5.11	We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 
12.95/5.11	
12.95/5.11	Order:Polynomial interpretation [POLO]:
12.95/5.11	
12.95/5.11	   POL(:(x_1, x_2)) = 1 + x_1 + x_2
12.95/5.11	   POL([]) = 1
12.95/5.11	   POL(app(x_1, x_2)) = x_1
12.95/5.11	   POL(new_foldr1(x_1, x_2)) = x_1
12.95/5.11	   POL(new_foldr2(x_1)) = 1
12.95/5.11	   POL(new_psPs(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
12.95/5.11	   POL(new_psPs0(x_1, x_2)) = x_1 + x_2
12.95/5.11	   POL(new_psPs1(x_1, x_2, x_3)) = 1 + x_1 + x_2
12.95/5.11	   POL(ty_[]) = 0
12.95/5.11	
12.95/5.11	
12.95/5.11	
12.95/5.11	
12.95/5.11	From the DPs we obtained the following set of size-change graphs:
12.95/5.11	*new_transpose(:([], wu31), ba) -> new_transpose(wu31, ba) (allowed arguments on rhs = {1, 2})
12.95/5.11	The graph contains the following edges 1 > 1, 2 >= 2
12.95/5.11	
12.95/5.11	
12.95/5.11	*new_transpose(:(:(wu300, wu301), wu31), ba) -> new_transpose(:(wu301, new_foldr1(wu31, ba)), ba) (allowed arguments on rhs = {1, 2})
12.95/5.11	The graph contains the following edges 1 > 1, 2 >= 2
12.95/5.11	
12.95/5.11	
12.95/5.11	
12.95/5.11	We oriented the following set of usable rules [AAECC05,FROCOS05].
12.95/5.11	
12.95/5.11	   new_psPs1([], wu5, ba) -> new_psPs0(wu5, app(ty_[], ba))
12.95/5.11	   new_psPs1(:(wu3100, wu3101), wu5, ba) -> new_psPs(wu3101, wu5, app(ty_[], ba))
12.95/5.11	   new_psPs0(wu4, ba) -> wu4
12.95/5.11	   new_psPs(wu3100, wu4, ba) -> :(wu3100, new_psPs0(wu4, ba))
12.95/5.11	   new_foldr2(ba) -> []
12.95/5.11	   new_foldr1([], ba) -> new_foldr2(app(ty_[], ba))
12.95/5.11	   new_foldr1(:(wu310, wu311), ba) -> new_psPs1(wu310, new_foldr1(wu311, ba), ba)
12.95/5.11	
12.95/5.11	----------------------------------------
12.95/5.11	
12.95/5.11	(19)
12.95/5.11	YES
12.95/5.14	EOF