7.73/3.57	YES
9.89/4.13	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.89/4.13	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.89/4.13	
9.89/4.13	
9.89/4.13	H-Termination with start terms of the given HASKELL could be proven:
9.89/4.13	
9.89/4.13	(0) HASKELL
9.89/4.13	(1) BR [EQUIVALENT, 0 ms]
9.89/4.13	(2) HASKELL
9.89/4.13	(3) COR [EQUIVALENT, 0 ms]
9.89/4.13	(4) HASKELL
9.89/4.13	(5) Narrow [SOUND, 0 ms]
9.89/4.13	(6) QDP
9.89/4.13	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.89/4.13	(8) YES
9.89/4.13	
9.89/4.13	
9.89/4.13	----------------------------------------
9.89/4.13	
9.89/4.13	(0)
9.89/4.13	Obligation:
9.89/4.13	mainModule Main 
9.89/4.13	module Main where {
9.89/4.13	import qualified Prelude;
9.89/4.13	data List a = Cons a (List a)  | Nil ;
9.89/4.13	
9.89/4.13	data MyBool = MyTrue  | MyFalse ;
9.89/4.13	
9.89/4.13	concat :: List (List a)  ->  List a;
9.89/4.13	concat = foldr psPs Nil;
9.89/4.13	
9.89/4.13	concatMap :: (a  ->  List b)  ->  List a  ->  List b;
9.89/4.13	concatMap f = pt concat (map f);
9.89/4.13	
9.89/4.13	filter :: (a  ->  MyBool)  ->  List a  ->  List a;
9.89/4.13	filter p xs = concatMap (filter0 p) xs;
9.89/4.13	
9.89/4.13	filter0 p vu39 = filter00 p vu39;
9.89/4.13	
9.89/4.13	filter00 p x = filter000 x (p x);
9.89/4.13	filter00 p vv = Nil;
9.89/4.13	
9.89/4.13	filter000 x MyTrue = Cons x Nil;
9.89/4.13	filter000 x MyFalse = Nil;
9.89/4.13	
9.89/4.13	foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
9.89/4.13	foldr f z Nil = z;
9.89/4.13	foldr f z (Cons x xs) = f x (foldr f z xs);
9.89/4.13	
9.89/4.13	map :: (a  ->  b)  ->  List a  ->  List b;
9.89/4.13	map f Nil = Nil;
9.89/4.13	map f (Cons x xs) = Cons (f x) (map f xs);
9.89/4.13	
9.89/4.13	psPs :: List a  ->  List a  ->  List a;
9.89/4.13	psPs Nil ys = ys;
9.89/4.13	psPs (Cons x xs) ys = Cons x (psPs xs ys);
9.89/4.13	
9.89/4.13	pt :: (c  ->  a)  ->  (b  ->  c)  ->  b  ->  a;
9.89/4.13	pt f g x = f (g x);
9.89/4.13	
9.89/4.13	}
9.89/4.13	
9.89/4.13	----------------------------------------
9.89/4.13	
9.89/4.13	(1) BR (EQUIVALENT)
9.89/4.13	Replaced joker patterns by fresh variables and removed binding patterns.
9.89/4.13	----------------------------------------
9.89/4.13	
9.89/4.13	(2)
9.89/4.13	Obligation:
9.89/4.13	mainModule Main 
9.89/4.13	module Main where {
9.89/4.13	import qualified Prelude;
9.89/4.13	data List a = Cons a (List a)  | Nil ;
9.89/4.13	
9.89/4.13	data MyBool = MyTrue  | MyFalse ;
9.89/4.13	
9.89/4.13	concat :: List (List a)  ->  List a;
9.89/4.13	concat = foldr psPs Nil;
9.89/4.13	
9.89/4.13	concatMap :: (a  ->  List b)  ->  List a  ->  List b;
9.89/4.13	concatMap f = pt concat (map f);
9.89/4.13	
9.89/4.13	filter :: (a  ->  MyBool)  ->  List a  ->  List a;
9.89/4.13	filter p xs = concatMap (filter0 p) xs;
9.89/4.13	
9.89/4.13	filter0 p vu39 = filter00 p vu39;
9.89/4.13	
9.89/4.13	filter00 p x = filter000 x (p x);
9.89/4.13	filter00 p vv = Nil;
9.89/4.13	
9.89/4.13	filter000 x MyTrue = Cons x Nil;
9.89/4.13	filter000 x MyFalse = Nil;
9.89/4.13	
9.89/4.13	foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
9.89/4.13	foldr f z Nil = z;
9.89/4.13	foldr f z (Cons x xs) = f x (foldr f z xs);
9.89/4.13	
9.89/4.13	map :: (a  ->  b)  ->  List a  ->  List b;
9.89/4.13	map f Nil = Nil;
9.89/4.13	map f (Cons x xs) = Cons (f x) (map f xs);
9.89/4.13	
9.89/4.13	psPs :: List a  ->  List a  ->  List a;
9.89/4.13	psPs Nil ys = ys;
9.89/4.13	psPs (Cons x xs) ys = Cons x (psPs xs ys);
9.89/4.13	
9.89/4.13	pt :: (b  ->  a)  ->  (c  ->  b)  ->  c  ->  a;
9.89/4.13	pt f g x = f (g x);
9.89/4.13	
9.89/4.13	}
9.89/4.13	
9.89/4.13	----------------------------------------
9.89/4.13	
9.89/4.13	(3) COR (EQUIVALENT)
9.89/4.13	Cond Reductions:
9.89/4.13	The following Function with conditions
9.89/4.13	"undefined |Falseundefined;
9.89/4.13	"
9.89/4.13	is transformed to
9.89/4.13	"undefined  = undefined1;
9.89/4.13	"
9.89/4.13	"undefined0 True = undefined;
9.89/4.13	"
9.89/4.13	"undefined1  = undefined0 False;
9.89/4.13	"
9.89/4.13	
9.89/4.13	----------------------------------------
9.89/4.13	
9.89/4.13	(4)
9.89/4.13	Obligation:
9.89/4.13	mainModule Main 
9.89/4.13	module Main where {
9.89/4.13	import qualified Prelude;
9.89/4.13	data List a = Cons a (List a)  | Nil ;
9.89/4.13	
9.89/4.13	data MyBool = MyTrue  | MyFalse ;
9.89/4.13	
9.89/4.13	concat :: List (List a)  ->  List a;
9.89/4.13	concat = foldr psPs Nil;
9.89/4.13	
9.89/4.13	concatMap :: (b  ->  List a)  ->  List b  ->  List a;
9.89/4.13	concatMap f = pt concat (map f);
9.89/4.13	
9.89/4.13	filter :: (a  ->  MyBool)  ->  List a  ->  List a;
9.89/4.13	filter p xs = concatMap (filter0 p) xs;
9.89/4.13	
9.89/4.13	filter0 p vu39 = filter00 p vu39;
9.89/4.13	
9.89/4.13	filter00 p x = filter000 x (p x);
9.89/4.13	filter00 p vv = Nil;
9.89/4.13	
9.89/4.13	filter000 x MyTrue = Cons x Nil;
9.89/4.13	filter000 x MyFalse = Nil;
9.89/4.13	
9.89/4.13	foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
9.89/4.13	foldr f z Nil = z;
9.89/4.13	foldr f z (Cons x xs) = f x (foldr f z xs);
9.89/4.13	
9.89/4.13	map :: (a  ->  b)  ->  List a  ->  List b;
9.89/4.13	map f Nil = Nil;
9.89/4.13	map f (Cons x xs) = Cons (f x) (map f xs);
9.89/4.13	
9.89/4.13	psPs :: List a  ->  List a  ->  List a;
9.89/4.13	psPs Nil ys = ys;
9.89/4.13	psPs (Cons x xs) ys = Cons x (psPs xs ys);
9.89/4.13	
9.89/4.13	pt :: (c  ->  b)  ->  (a  ->  c)  ->  a  ->  b;
9.89/4.13	pt f g x = f (g x);
9.89/4.13	
9.89/4.13	}
9.89/4.13	
9.89/4.13	----------------------------------------
9.89/4.13	
9.89/4.13	(5) Narrow (SOUND)
9.89/4.13	Haskell To QDPs
9.89/4.13	
9.89/4.13	digraph dp_graph {
9.89/4.13	node [outthreshold=100, inthreshold=100];1[label="filter",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.89/4.13	3[label="filter vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.89/4.13	4[label="filter vy3 vy4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.89/4.13	5[label="concatMap (filter0 vy3) vy4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.89/4.13	6[label="pt concat (map (filter0 vy3)) vy4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
9.89/4.13	7[label="concat (map (filter0 vy3) vy4)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
9.89/4.13	8[label="foldr psPs Nil (map (filter0 vy3) vy4)",fontsize=16,color="burlywood",shape="triangle"];31[label="vy4/Cons vy40 vy41",fontsize=10,color="white",style="solid",shape="box"];8 -> 31[label="",style="solid", color="burlywood", weight=9];
9.89/4.13	31 -> 9[label="",style="solid", color="burlywood", weight=3];
9.89/4.13	32[label="vy4/Nil",fontsize=10,color="white",style="solid",shape="box"];8 -> 32[label="",style="solid", color="burlywood", weight=9];
9.89/4.13	32 -> 10[label="",style="solid", color="burlywood", weight=3];
9.89/4.13	9[label="foldr psPs Nil (map (filter0 vy3) (Cons vy40 vy41))",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
9.89/4.13	10[label="foldr psPs Nil (map (filter0 vy3) Nil)",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
9.89/4.13	11[label="foldr psPs Nil (Cons (filter0 vy3 vy40) (map (filter0 vy3) vy41))",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
9.89/4.13	12[label="foldr psPs Nil Nil",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
9.89/4.13	13 -> 15[label="",style="dashed", color="red", weight=0];
9.89/4.13	13[label="psPs (filter0 vy3 vy40) (foldr psPs Nil (map (filter0 vy3) vy41))",fontsize=16,color="magenta"];13 -> 16[label="",style="dashed", color="magenta", weight=3];
9.89/4.13	14[label="Nil",fontsize=16,color="green",shape="box"];16 -> 8[label="",style="dashed", color="red", weight=0];
9.89/4.13	16[label="foldr psPs Nil (map (filter0 vy3) vy41)",fontsize=16,color="magenta"];16 -> 17[label="",style="dashed", color="magenta", weight=3];
9.89/4.13	15[label="psPs (filter0 vy3 vy40) vy5",fontsize=16,color="black",shape="triangle"];15 -> 18[label="",style="solid", color="black", weight=3];
9.89/4.13	17[label="vy41",fontsize=16,color="green",shape="box"];18[label="psPs (filter00 vy3 vy40) vy5",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
9.89/4.13	19 -> 20[label="",style="dashed", color="red", weight=0];
9.89/4.13	19[label="psPs (filter000 vy40 (vy3 vy40)) vy5",fontsize=16,color="magenta"];19 -> 21[label="",style="dashed", color="magenta", weight=3];
9.89/4.13	21[label="vy3 vy40",fontsize=16,color="green",shape="box"];21 -> 25[label="",style="dashed", color="green", weight=3];
9.89/4.13	20[label="psPs (filter000 vy40 vy6) vy5",fontsize=16,color="burlywood",shape="triangle"];33[label="vy6/MyTrue",fontsize=10,color="white",style="solid",shape="box"];20 -> 33[label="",style="solid", color="burlywood", weight=9];
9.89/4.13	33 -> 23[label="",style="solid", color="burlywood", weight=3];
9.89/4.13	34[label="vy6/MyFalse",fontsize=10,color="white",style="solid",shape="box"];20 -> 34[label="",style="solid", color="burlywood", weight=9];
9.89/4.13	34 -> 24[label="",style="solid", color="burlywood", weight=3];
9.89/4.13	25[label="vy40",fontsize=16,color="green",shape="box"];23[label="psPs (filter000 vy40 MyTrue) vy5",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
9.89/4.13	24[label="psPs (filter000 vy40 MyFalse) vy5",fontsize=16,color="black",shape="box"];24 -> 27[label="",style="solid", color="black", weight=3];
9.89/4.13	26[label="psPs (Cons vy40 Nil) vy5",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3];
9.89/4.13	27[label="psPs Nil vy5",fontsize=16,color="black",shape="triangle"];27 -> 29[label="",style="solid", color="black", weight=3];
9.89/4.13	28[label="Cons vy40 (psPs Nil vy5)",fontsize=16,color="green",shape="box"];28 -> 30[label="",style="dashed", color="green", weight=3];
9.89/4.13	29[label="vy5",fontsize=16,color="green",shape="box"];30 -> 27[label="",style="dashed", color="red", weight=0];
9.89/4.13	30[label="psPs Nil vy5",fontsize=16,color="magenta"];}
9.89/4.13	
9.89/4.13	----------------------------------------
9.89/4.13	
9.89/4.13	(6)
9.89/4.13	Obligation:
9.89/4.13	Q DP problem:
9.89/4.13	The TRS P consists of the following rules:
9.89/4.13	
9.89/4.13	   new_foldr(vy3, Cons(vy40, vy41), h) -> new_foldr(vy3, vy41, h)
9.89/4.13	
9.89/4.13	R is empty.
9.89/4.13	Q is empty.
9.89/4.13	We have to consider all minimal (P,Q,R)-chains.
9.89/4.13	----------------------------------------
9.89/4.13	
9.89/4.13	(7) QDPSizeChangeProof (EQUIVALENT)
9.89/4.13	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. 
9.89/4.13	
9.89/4.13	From the DPs we obtained the following set of size-change graphs:
9.89/4.13	*new_foldr(vy3, Cons(vy40, vy41), h) -> new_foldr(vy3, vy41, h)
9.89/4.13	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
9.89/4.13	
9.89/4.13	
9.89/4.13	----------------------------------------
9.89/4.13	
9.89/4.13	(8)
9.89/4.13	YES
9.99/4.21	EOF