7.64/3.51	YES
9.39/4.03	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.39/4.03	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.39/4.03	
9.39/4.03	
9.39/4.03	H-Termination with start terms of the given HASKELL could be proven:
9.39/4.03	
9.39/4.03	(0) HASKELL
9.39/4.03	(1) BR [EQUIVALENT, 0 ms]
9.39/4.03	(2) HASKELL
9.39/4.03	(3) COR [EQUIVALENT, 0 ms]
9.39/4.03	(4) HASKELL
9.39/4.03	(5) Narrow [SOUND, 0 ms]
9.39/4.03	(6) AND
9.39/4.03	    (7) QDP
9.39/4.03	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.39/4.03	        (9) YES
9.39/4.03	    (10) QDP
9.39/4.03	        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.39/4.03	        (12) YES
9.39/4.03	
9.39/4.03	
9.39/4.03	----------------------------------------
9.39/4.03	
9.39/4.03	(0)
9.39/4.03	Obligation:
9.39/4.03	mainModule Main 
9.39/4.03	module Main where {
9.39/4.03	import qualified Prelude;
9.39/4.03	data List a = Cons a (List a)  | Nil ;
9.39/4.03	
9.39/4.03	concat :: List (List a)  ->  List a;
9.39/4.03	concat = foldr psPs Nil;
9.39/4.03	
9.39/4.03	concatMap :: (a  ->  List b)  ->  List a  ->  List b;
9.39/4.03	concatMap f = pt concat (map f);
9.39/4.03	
9.39/4.03	foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
9.39/4.03	foldr f z Nil = z;
9.39/4.03	foldr f z (Cons x xs) = f x (foldr f z xs);
9.39/4.03	
9.39/4.03	map :: (b  ->  a)  ->  List b  ->  List a;
9.39/4.03	map f Nil = Nil;
9.39/4.03	map f (Cons x xs) = Cons (f x) (map f xs);
9.39/4.03	
9.39/4.03	psPs :: List a  ->  List a  ->  List a;
9.39/4.03	psPs Nil ys = ys;
9.39/4.03	psPs (Cons x xs) ys = Cons x (psPs xs ys);
9.39/4.03	
9.39/4.03	pt :: (c  ->  b)  ->  (a  ->  c)  ->  a  ->  b;
9.39/4.03	pt f g x = f (g x);
9.39/4.03	
9.39/4.03	}
9.39/4.03	
9.39/4.03	----------------------------------------
9.39/4.03	
9.39/4.03	(1) BR (EQUIVALENT)
9.39/4.03	Replaced joker patterns by fresh variables and removed binding patterns.
9.39/4.03	----------------------------------------
9.39/4.03	
9.39/4.03	(2)
9.39/4.03	Obligation:
9.39/4.03	mainModule Main 
9.39/4.03	module Main where {
9.39/4.03	import qualified Prelude;
9.39/4.03	data List a = Cons a (List a)  | Nil ;
9.39/4.03	
9.39/4.03	concat :: List (List a)  ->  List a;
9.39/4.03	concat = foldr psPs Nil;
9.39/4.03	
9.39/4.03	concatMap :: (b  ->  List a)  ->  List b  ->  List a;
9.39/4.03	concatMap f = pt concat (map f);
9.39/4.03	
9.39/4.03	foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
9.39/4.03	foldr f z Nil = z;
9.39/4.03	foldr f z (Cons x xs) = f x (foldr f z xs);
9.39/4.03	
9.39/4.03	map :: (b  ->  a)  ->  List b  ->  List a;
9.39/4.03	map f Nil = Nil;
9.39/4.03	map f (Cons x xs) = Cons (f x) (map f xs);
9.39/4.03	
9.39/4.03	psPs :: List a  ->  List a  ->  List a;
9.39/4.03	psPs Nil ys = ys;
9.39/4.03	psPs (Cons x xs) ys = Cons x (psPs xs ys);
9.39/4.03	
9.39/4.03	pt :: (a  ->  b)  ->  (c  ->  a)  ->  c  ->  b;
9.39/4.03	pt f g x = f (g x);
9.39/4.03	
9.39/4.03	}
9.39/4.03	
9.39/4.03	----------------------------------------
9.39/4.03	
9.39/4.03	(3) COR (EQUIVALENT)
9.39/4.03	Cond Reductions:
9.39/4.03	The following Function with conditions
9.39/4.03	"undefined |Falseundefined;
9.39/4.03	"
9.39/4.03	is transformed to
9.39/4.03	"undefined  = undefined1;
9.39/4.03	"
9.39/4.03	"undefined0 True = undefined;
9.39/4.03	"
9.39/4.03	"undefined1  = undefined0 False;
9.39/4.03	"
9.39/4.03	
9.39/4.03	----------------------------------------
9.39/4.03	
9.39/4.03	(4)
9.39/4.03	Obligation:
9.39/4.03	mainModule Main 
9.39/4.03	module Main where {
9.39/4.03	import qualified Prelude;
9.39/4.03	data List a = Cons a (List a)  | Nil ;
9.39/4.03	
9.39/4.03	concat :: List (List a)  ->  List a;
9.39/4.03	concat = foldr psPs Nil;
9.39/4.03	
9.39/4.03	concatMap :: (a  ->  List b)  ->  List a  ->  List b;
9.39/4.03	concatMap f = pt concat (map f);
9.39/4.03	
9.39/4.03	foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
9.39/4.03	foldr f z Nil = z;
9.39/4.03	foldr f z (Cons x xs) = f x (foldr f z xs);
9.39/4.03	
9.39/4.03	map :: (b  ->  a)  ->  List b  ->  List a;
9.39/4.03	map f Nil = Nil;
9.39/4.03	map f (Cons x xs) = Cons (f x) (map f xs);
9.39/4.03	
9.39/4.03	psPs :: List a  ->  List a  ->  List a;
9.39/4.03	psPs Nil ys = ys;
9.39/4.03	psPs (Cons x xs) ys = Cons x (psPs xs ys);
9.39/4.03	
9.39/4.03	pt :: (b  ->  a)  ->  (c  ->  b)  ->  c  ->  a;
9.39/4.03	pt f g x = f (g x);
9.39/4.03	
9.39/4.03	}
9.39/4.03	
9.39/4.03	----------------------------------------
9.39/4.03	
9.39/4.03	(5) Narrow (SOUND)
9.39/4.03	Haskell To QDPs
9.39/4.03	
9.39/4.03	digraph dp_graph {
9.39/4.03	node [outthreshold=100, inthreshold=100];1[label="concatMap",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.39/4.03	3[label="concatMap vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.39/4.03	4[label="concatMap vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.39/4.03	5[label="pt concat (map vx3) vx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.39/4.03	6[label="concat (map vx3 vx4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
9.39/4.03	7[label="foldr psPs Nil (map vx3 vx4)",fontsize=16,color="burlywood",shape="triangle"];30[label="vx4/Cons vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];7 -> 30[label="",style="solid", color="burlywood", weight=9];
9.39/4.03	30 -> 8[label="",style="solid", color="burlywood", weight=3];
9.39/4.03	31[label="vx4/Nil",fontsize=10,color="white",style="solid",shape="box"];7 -> 31[label="",style="solid", color="burlywood", weight=9];
9.39/4.03	31 -> 9[label="",style="solid", color="burlywood", weight=3];
9.39/4.03	8[label="foldr psPs Nil (map vx3 (Cons vx40 vx41))",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9.39/4.03	9[label="foldr psPs Nil (map vx3 Nil)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
9.39/4.03	10[label="foldr psPs Nil (Cons (vx3 vx40) (map vx3 vx41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
9.39/4.03	11[label="foldr psPs Nil Nil",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
9.39/4.03	12 -> 17[label="",style="dashed", color="red", weight=0];
9.39/4.03	12[label="psPs (vx3 vx40) (foldr psPs Nil (map vx3 vx41))",fontsize=16,color="magenta"];12 -> 18[label="",style="dashed", color="magenta", weight=3];
9.39/4.03	12 -> 19[label="",style="dashed", color="magenta", weight=3];
9.39/4.03	13[label="Nil",fontsize=16,color="green",shape="box"];18[label="vx3 vx40",fontsize=16,color="green",shape="box"];18 -> 21[label="",style="dashed", color="green", weight=3];
9.39/4.03	19 -> 7[label="",style="dashed", color="red", weight=0];
9.39/4.03	19[label="foldr psPs Nil (map vx3 vx41)",fontsize=16,color="magenta"];19 -> 22[label="",style="dashed", color="magenta", weight=3];
9.39/4.03	17[label="psPs vx6 vx5",fontsize=16,color="burlywood",shape="triangle"];32[label="vx6/Cons vx60 vx61",fontsize=10,color="white",style="solid",shape="box"];17 -> 32[label="",style="solid", color="burlywood", weight=9];
9.39/4.03	32 -> 23[label="",style="solid", color="burlywood", weight=3];
9.39/4.03	33[label="vx6/Nil",fontsize=10,color="white",style="solid",shape="box"];17 -> 33[label="",style="solid", color="burlywood", weight=9];
9.39/4.03	33 -> 24[label="",style="solid", color="burlywood", weight=3];
9.39/4.03	21[label="vx40",fontsize=16,color="green",shape="box"];22[label="vx41",fontsize=16,color="green",shape="box"];23[label="psPs (Cons vx60 vx61) vx5",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
9.39/4.03	24[label="psPs Nil vx5",fontsize=16,color="black",shape="box"];24 -> 27[label="",style="solid", color="black", weight=3];
9.39/4.03	26[label="Cons vx60 (psPs vx61 vx5)",fontsize=16,color="green",shape="box"];26 -> 28[label="",style="dashed", color="green", weight=3];
9.39/4.03	27[label="vx5",fontsize=16,color="green",shape="box"];28 -> 17[label="",style="dashed", color="red", weight=0];
9.39/4.03	28[label="psPs vx61 vx5",fontsize=16,color="magenta"];28 -> 29[label="",style="dashed", color="magenta", weight=3];
9.39/4.03	29[label="vx61",fontsize=16,color="green",shape="box"];}
9.39/4.03	
9.39/4.03	----------------------------------------
9.39/4.03	
9.39/4.03	(6)
9.39/4.03	Complex Obligation (AND)
9.39/4.03	
9.39/4.03	----------------------------------------
9.39/4.03	
9.39/4.03	(7)
9.39/4.03	Obligation:
9.39/4.03	Q DP problem:
9.39/4.03	The TRS P consists of the following rules:
9.39/4.03	
9.39/4.03	   new_psPs(Cons(vx60, vx61), vx5, h) -> new_psPs(vx61, vx5, h)
9.39/4.03	
9.39/4.03	R is empty.
9.39/4.03	Q is empty.
9.39/4.03	We have to consider all minimal (P,Q,R)-chains.
9.39/4.03	----------------------------------------
9.39/4.03	
9.39/4.03	(8) QDPSizeChangeProof (EQUIVALENT)
9.39/4.03	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. 
9.39/4.03	
9.39/4.03	From the DPs we obtained the following set of size-change graphs:
9.39/4.03	*new_psPs(Cons(vx60, vx61), vx5, h) -> new_psPs(vx61, vx5, h)
9.39/4.03	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
9.39/4.03	
9.39/4.03	
9.39/4.03	----------------------------------------
9.39/4.03	
9.39/4.03	(9)
9.39/4.03	YES
9.39/4.03	
9.39/4.03	----------------------------------------
9.39/4.03	
9.39/4.03	(10)
9.39/4.03	Obligation:
9.39/4.03	Q DP problem:
9.39/4.03	The TRS P consists of the following rules:
9.39/4.03	
9.39/4.03	   new_foldr(vx3, Cons(vx40, vx41), h, ba) -> new_foldr(vx3, vx41, h, ba)
9.39/4.03	
9.39/4.03	R is empty.
9.39/4.03	Q is empty.
9.39/4.03	We have to consider all minimal (P,Q,R)-chains.
9.39/4.03	----------------------------------------
9.39/4.03	
9.39/4.03	(11) QDPSizeChangeProof (EQUIVALENT)
9.39/4.03	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. 
9.39/4.03	
9.39/4.03	From the DPs we obtained the following set of size-change graphs:
9.39/4.03	*new_foldr(vx3, Cons(vx40, vx41), h, ba) -> new_foldr(vx3, vx41, h, ba)
9.39/4.03	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
9.39/4.03	
9.39/4.03	
9.39/4.03	----------------------------------------
9.39/4.03	
9.39/4.03	(12)
9.39/4.03	YES
9.71/4.12	EOF