7.78/3.52	YES
9.38/4.02	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.38/4.02	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.38/4.02	
9.38/4.02	
9.38/4.02	H-Termination with start terms of the given HASKELL could be proven:
9.38/4.02	
9.38/4.02	(0) HASKELL
9.38/4.02	(1) BR [EQUIVALENT, 0 ms]
9.38/4.02	(2) HASKELL
9.38/4.02	(3) COR [EQUIVALENT, 0 ms]
9.38/4.02	(4) HASKELL
9.38/4.02	(5) Narrow [SOUND, 0 ms]
9.38/4.02	(6) QDP
9.38/4.02	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.38/4.02	(8) YES
9.38/4.02	
9.38/4.02	
9.38/4.02	----------------------------------------
9.38/4.02	
9.38/4.02	(0)
9.38/4.02	Obligation:
9.38/4.02	mainModule Main 
9.38/4.02	module Main where {
9.38/4.02	import qualified Prelude;
9.38/4.02	data List a = Cons a (List a)  | Nil ;
9.38/4.02	
9.38/4.02	data MyBool = MyTrue  | MyFalse ;
9.38/4.02	
9.38/4.02	all :: (a  ->  MyBool)  ->  List a  ->  MyBool;
9.38/4.02	all p = pt and (map p);
9.38/4.02	
9.38/4.02	and :: List MyBool  ->  MyBool;
9.38/4.02	and = foldr asAs MyTrue;
9.38/4.02	
9.38/4.02	asAs :: MyBool  ->  MyBool  ->  MyBool;
9.38/4.02	asAs MyFalse x = MyFalse;
9.38/4.02	asAs MyTrue x = x;
9.38/4.02	
9.38/4.02	foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
9.38/4.02	foldr f z Nil = z;
9.38/4.02	foldr f z (Cons x xs) = f x (foldr f z xs);
9.38/4.02	
9.38/4.02	map :: (a  ->  b)  ->  List a  ->  List b;
9.38/4.02	map f Nil = Nil;
9.38/4.02	map f (Cons x xs) = Cons (f x) (map f xs);
9.38/4.02	
9.38/4.02	pt :: (a  ->  c)  ->  (b  ->  a)  ->  b  ->  c;
9.38/4.02	pt f g x = f (g x);
9.38/4.02	
9.38/4.02	}
9.38/4.02	
9.38/4.02	----------------------------------------
9.38/4.02	
9.38/4.02	(1) BR (EQUIVALENT)
9.38/4.02	Replaced joker patterns by fresh variables and removed binding patterns.
9.38/4.02	----------------------------------------
9.38/4.02	
9.38/4.02	(2)
9.38/4.02	Obligation:
9.38/4.02	mainModule Main 
9.38/4.02	module Main where {
9.38/4.02	import qualified Prelude;
9.38/4.02	data List a = Cons a (List a)  | Nil ;
9.38/4.02	
9.38/4.02	data MyBool = MyTrue  | MyFalse ;
9.38/4.02	
9.38/4.02	all :: (a  ->  MyBool)  ->  List a  ->  MyBool;
9.38/4.02	all p = pt and (map p);
9.38/4.02	
9.38/4.02	and :: List MyBool  ->  MyBool;
9.38/4.02	and = foldr asAs MyTrue;
9.38/4.02	
9.38/4.02	asAs :: MyBool  ->  MyBool  ->  MyBool;
9.38/4.02	asAs MyFalse x = MyFalse;
9.38/4.02	asAs MyTrue x = x;
9.38/4.02	
9.38/4.02	foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
9.38/4.02	foldr f z Nil = z;
9.38/4.02	foldr f z (Cons x xs) = f x (foldr f z xs);
9.38/4.02	
9.38/4.02	map :: (b  ->  a)  ->  List b  ->  List a;
9.38/4.02	map f Nil = Nil;
9.38/4.02	map f (Cons x xs) = Cons (f x) (map f xs);
9.38/4.02	
9.38/4.02	pt :: (b  ->  c)  ->  (a  ->  b)  ->  a  ->  c;
9.38/4.02	pt f g x = f (g x);
9.38/4.02	
9.38/4.02	}
9.38/4.02	
9.38/4.02	----------------------------------------
9.38/4.02	
9.38/4.02	(3) COR (EQUIVALENT)
9.38/4.02	Cond Reductions:
9.38/4.02	The following Function with conditions
9.38/4.02	"undefined |Falseundefined;
9.38/4.02	"
9.38/4.02	is transformed to
9.38/4.02	"undefined  = undefined1;
9.38/4.02	"
9.38/4.02	"undefined0 True = undefined;
9.38/4.02	"
9.38/4.02	"undefined1  = undefined0 False;
9.38/4.02	"
9.38/4.02	
9.38/4.02	----------------------------------------
9.38/4.02	
9.38/4.02	(4)
9.38/4.02	Obligation:
9.38/4.02	mainModule Main 
9.38/4.02	module Main where {
9.38/4.02	import qualified Prelude;
9.38/4.02	data List a = Cons a (List a)  | Nil ;
9.38/4.02	
9.38/4.02	data MyBool = MyTrue  | MyFalse ;
9.38/4.02	
9.38/4.02	all :: (a  ->  MyBool)  ->  List a  ->  MyBool;
9.38/4.02	all p = pt and (map p);
9.38/4.02	
9.38/4.02	and :: List MyBool  ->  MyBool;
9.38/4.02	and = foldr asAs MyTrue;
9.38/4.02	
9.38/4.02	asAs :: MyBool  ->  MyBool  ->  MyBool;
9.38/4.02	asAs MyFalse x = MyFalse;
9.38/4.02	asAs MyTrue x = x;
9.38/4.02	
9.38/4.02	foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
9.38/4.02	foldr f z Nil = z;
9.38/4.02	foldr f z (Cons x xs) = f x (foldr f z xs);
9.38/4.02	
9.38/4.02	map :: (b  ->  a)  ->  List b  ->  List a;
9.38/4.02	map f Nil = Nil;
9.38/4.02	map f (Cons x xs) = Cons (f x) (map f xs);
9.38/4.02	
9.38/4.02	pt :: (a  ->  b)  ->  (c  ->  a)  ->  c  ->  b;
9.38/4.02	pt f g x = f (g x);
9.38/4.02	
9.38/4.02	}
9.38/4.02	
9.38/4.02	----------------------------------------
9.38/4.02	
9.38/4.02	(5) Narrow (SOUND)
9.38/4.02	Haskell To QDPs
9.38/4.02	
9.38/4.02	digraph dp_graph {
9.38/4.02	node [outthreshold=100, inthreshold=100];1[label="all",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.38/4.02	3[label="all vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.38/4.02	4[label="all vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.38/4.02	5[label="pt and (map vx3) vx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.38/4.02	6[label="and (map vx3 vx4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
9.38/4.02	7[label="foldr asAs MyTrue (map vx3 vx4)",fontsize=16,color="burlywood",shape="triangle"];28[label="vx4/Cons vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];7 -> 28[label="",style="solid", color="burlywood", weight=9];
9.38/4.02	28 -> 8[label="",style="solid", color="burlywood", weight=3];
9.38/4.02	29[label="vx4/Nil",fontsize=10,color="white",style="solid",shape="box"];7 -> 29[label="",style="solid", color="burlywood", weight=9];
9.38/4.02	29 -> 9[label="",style="solid", color="burlywood", weight=3];
9.38/4.02	8[label="foldr asAs MyTrue (map vx3 (Cons vx40 vx41))",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9.38/4.02	9[label="foldr asAs MyTrue (map vx3 Nil)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
9.38/4.02	10[label="foldr asAs MyTrue (Cons (vx3 vx40) (map vx3 vx41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
9.38/4.02	11[label="foldr asAs MyTrue Nil",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
9.38/4.02	12 -> 17[label="",style="dashed", color="red", weight=0];
9.38/4.02	12[label="asAs (vx3 vx40) (foldr asAs MyTrue (map vx3 vx41))",fontsize=16,color="magenta"];12 -> 18[label="",style="dashed", color="magenta", weight=3];
9.38/4.02	12 -> 19[label="",style="dashed", color="magenta", weight=3];
9.38/4.02	13[label="MyTrue",fontsize=16,color="green",shape="box"];18 -> 7[label="",style="dashed", color="red", weight=0];
9.38/4.02	18[label="foldr asAs MyTrue (map vx3 vx41)",fontsize=16,color="magenta"];18 -> 21[label="",style="dashed", color="magenta", weight=3];
9.38/4.02	19[label="vx3 vx40",fontsize=16,color="green",shape="box"];19 -> 22[label="",style="dashed", color="green", weight=3];
9.38/4.02	17[label="asAs vx6 vx5",fontsize=16,color="burlywood",shape="triangle"];30[label="vx6/MyTrue",fontsize=10,color="white",style="solid",shape="box"];17 -> 30[label="",style="solid", color="burlywood", weight=9];
9.38/4.02	30 -> 23[label="",style="solid", color="burlywood", weight=3];
9.38/4.02	31[label="vx6/MyFalse",fontsize=10,color="white",style="solid",shape="box"];17 -> 31[label="",style="solid", color="burlywood", weight=9];
9.38/4.02	31 -> 24[label="",style="solid", color="burlywood", weight=3];
9.38/4.02	21[label="vx41",fontsize=16,color="green",shape="box"];22[label="vx40",fontsize=16,color="green",shape="box"];23[label="asAs MyTrue vx5",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
9.38/4.02	24[label="asAs MyFalse vx5",fontsize=16,color="black",shape="box"];24 -> 27[label="",style="solid", color="black", weight=3];
9.38/4.02	26[label="vx5",fontsize=16,color="green",shape="box"];27[label="MyFalse",fontsize=16,color="green",shape="box"];}
9.38/4.02	
9.38/4.02	----------------------------------------
9.38/4.02	
9.38/4.02	(6)
9.38/4.02	Obligation:
9.38/4.02	Q DP problem:
9.38/4.02	The TRS P consists of the following rules:
9.38/4.02	
9.38/4.02	   new_foldr(vx3, Cons(vx40, vx41), h) -> new_foldr(vx3, vx41, h)
9.38/4.02	
9.38/4.02	R is empty.
9.38/4.02	Q is empty.
9.38/4.02	We have to consider all minimal (P,Q,R)-chains.
9.38/4.02	----------------------------------------
9.38/4.02	
9.38/4.02	(7) QDPSizeChangeProof (EQUIVALENT)
9.38/4.02	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. 
9.38/4.02	
9.38/4.02	From the DPs we obtained the following set of size-change graphs:
9.38/4.02	*new_foldr(vx3, Cons(vx40, vx41), h) -> new_foldr(vx3, vx41, h)
9.38/4.02	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
9.38/4.02	
9.38/4.02	
9.38/4.02	----------------------------------------
9.38/4.02	
9.38/4.02	(8)
9.38/4.02	YES
9.55/4.05	EOF