11.21/4.63	YES
13.57/5.32	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
13.57/5.32	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
13.57/5.32	
13.57/5.32	
13.57/5.32	H-Termination with start terms of the given HASKELL could be proven:
13.57/5.32	
13.57/5.32	(0) HASKELL
13.57/5.32	(1) LR [EQUIVALENT, 0 ms]
13.57/5.32	(2) HASKELL
13.57/5.32	(3) CR [EQUIVALENT, 0 ms]
13.57/5.32	(4) HASKELL
13.57/5.32	(5) IFR [EQUIVALENT, 0 ms]
13.57/5.32	(6) HASKELL
13.57/5.32	(7) BR [EQUIVALENT, 0 ms]
13.57/5.32	(8) HASKELL
13.57/5.32	(9) COR [EQUIVALENT, 0 ms]
13.57/5.32	(10) HASKELL
13.57/5.32	(11) NumRed [SOUND, 0 ms]
13.57/5.32	(12) HASKELL
13.57/5.32	(13) Narrow [SOUND, 0 ms]
13.57/5.32	(14) QDP
13.57/5.32	(15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
13.57/5.32	(16) YES
13.57/5.32	
13.57/5.32	
13.57/5.32	----------------------------------------
13.57/5.32	
13.57/5.32	(0)
13.57/5.32	Obligation:
13.57/5.32	mainModule Main 
13.57/5.32	module Maybe where {
13.57/5.32	import qualified List;
13.57/5.32	import qualified Main;
13.57/5.32	import qualified Prelude;
13.57/5.32	}
13.57/5.32	module List where {
13.57/5.32	import qualified Main;
13.57/5.32	import qualified Maybe;
13.57/5.32	import qualified Prelude;
13.57/5.32	findIndices :: (a  ->  Bool)  ->  [a]  ->  [Int];
13.57/5.32	findIndices p xs = concatMap (\vv1 ->case vv1 of { 
13.57/5.32	(x,i)->  if p x then i : [] else []; 
13.57/5.32	_-> []; 
13.57/5.32	 } ) (zip xs (enumFrom 0));
13.57/5.32	
13.57/5.32	}
13.57/5.32	module Main where {
13.57/5.32	import qualified List;
13.57/5.32	import qualified Maybe;
13.57/5.32	import qualified Prelude;
13.57/5.32	}
13.57/5.32	
13.57/5.32	----------------------------------------
13.57/5.32	
13.57/5.32	(1) LR (EQUIVALENT)
13.57/5.32	Lambda Reductions:
13.57/5.32	The following Lambda expression
13.57/5.32	"\ab->(a,b)"
13.57/5.32	is transformed to
13.57/5.32	"zip0 a b = (a,b);
13.57/5.32	"
13.57/5.32	The following Lambda expression
13.57/5.32	"\vv1->case vv1 of {
13.57/5.32	(x,i)  -> if p x then i : [] else [];
13.57/5.32	_  -> []}
13.57/5.32	"
13.57/5.32	is transformed to
13.57/5.32	"findIndices0 p vv1 = case vv1 of {
13.57/5.32	(x,i)  -> if p x then i : [] else [];
13.57/5.32	_  -> []}
13.57/5.32	;
13.57/5.32	"
13.57/5.32	
13.57/5.32	----------------------------------------
13.57/5.32	
13.57/5.32	(2)
13.57/5.32	Obligation:
13.57/5.32	mainModule Main 
13.57/5.32	module Maybe where {
13.57/5.32	import qualified List;
13.57/5.32	import qualified Main;
13.57/5.32	import qualified Prelude;
13.57/5.32	}
13.57/5.32	module List where {
13.57/5.32	import qualified Main;
13.57/5.32	import qualified Maybe;
13.57/5.32	import qualified Prelude;
13.57/5.32	findIndices :: (a  ->  Bool)  ->  [a]  ->  [Int];
13.57/5.32	findIndices p xs = concatMap (findIndices0 p) (zip xs (enumFrom 0));
13.57/5.32	
13.57/5.32	findIndices0 p vv1 = case vv1 of { 
13.57/5.32	(x,i)->  if p x then i : [] else []; 
13.57/5.32	_-> []; 
13.57/5.32	 } ;
13.57/5.32	
13.57/5.32	}
13.57/5.32	module Main where {
13.57/5.32	import qualified List;
13.57/5.32	import qualified Maybe;
13.57/5.32	import qualified Prelude;
13.57/5.32	}
13.57/5.32	
13.57/5.32	----------------------------------------
13.57/5.32	
13.57/5.32	(3) CR (EQUIVALENT)
13.57/5.32	Case Reductions:
13.57/5.32	The following Case expression
13.57/5.32	"case vv1 of {
13.57/5.32	(x,i)  -> if p x then i : [] else [];
13.57/5.32	_  -> []}
13.57/5.32	"
13.57/5.32	is transformed to
13.57/5.32	"findIndices00 p (x,i) = if p x then i : [] else [];
13.57/5.32	findIndices00 p _ = [];
13.57/5.32	"
13.57/5.32	
13.57/5.32	----------------------------------------
13.57/5.32	
13.57/5.32	(4)
13.57/5.32	Obligation:
13.57/5.32	mainModule Main 
13.57/5.32	module Maybe where {
13.57/5.32	import qualified List;
13.57/5.32	import qualified Main;
13.57/5.32	import qualified Prelude;
13.57/5.32	}
13.57/5.32	module List where {
13.57/5.32	import qualified Main;
13.57/5.32	import qualified Maybe;
13.57/5.32	import qualified Prelude;
13.57/5.32	findIndices :: (a  ->  Bool)  ->  [a]  ->  [Int];
13.57/5.32	findIndices p xs = concatMap (findIndices0 p) (zip xs (enumFrom 0));
13.57/5.32	
13.57/5.32	findIndices0 p vv1 = findIndices00 p vv1;
13.57/5.32	
13.57/5.32	findIndices00 p (x,i) =  if p x then i : [] else [];
13.57/5.32	findIndices00 p _ = [];
13.57/5.32	
13.57/5.32	}
13.57/5.32	module Main where {
13.57/5.32	import qualified List;
13.57/5.32	import qualified Maybe;
13.57/5.32	import qualified Prelude;
13.57/5.32	}
13.57/5.32	
13.57/5.32	----------------------------------------
13.57/5.32	
13.57/5.32	(5) IFR (EQUIVALENT)
13.57/5.32	If Reductions:
13.57/5.32	The following If expression
13.57/5.32	"if p x then i : [] else []"
13.57/5.32	is transformed to
13.57/5.32	"findIndices000 i True = i : [];
13.57/5.32	findIndices000 i False = [];
13.57/5.32	"
13.57/5.32	
13.57/5.32	----------------------------------------
13.57/5.32	
13.57/5.32	(6)
13.57/5.32	Obligation:
13.57/5.32	mainModule Main 
13.57/5.32	module Maybe where {
13.57/5.32	import qualified List;
13.57/5.32	import qualified Main;
13.57/5.32	import qualified Prelude;
13.57/5.32	}
13.57/5.32	module List where {
13.57/5.32	import qualified Main;
13.57/5.32	import qualified Maybe;
13.57/5.32	import qualified Prelude;
13.57/5.32	findIndices :: (a  ->  Bool)  ->  [a]  ->  [Int];
13.57/5.32	findIndices p xs = concatMap (findIndices0 p) (zip xs (enumFrom 0));
13.57/5.32	
13.57/5.32	findIndices0 p vv1 = findIndices00 p vv1;
13.57/5.32	
13.57/5.32	findIndices00 p (x,i) = findIndices000 i (p x);
13.57/5.32	findIndices00 p _ = [];
13.57/5.32	
13.57/5.32	findIndices000 i True = i : [];
13.57/5.32	findIndices000 i False = [];
13.57/5.32	
13.57/5.32	}
13.57/5.32	module Main where {
13.57/5.32	import qualified List;
13.57/5.32	import qualified Maybe;
13.57/5.32	import qualified Prelude;
13.57/5.32	}
13.57/5.32	
13.57/5.32	----------------------------------------
13.57/5.32	
13.57/5.32	(7) BR (EQUIVALENT)
13.57/5.32	Replaced joker patterns by fresh variables and removed binding patterns.
13.57/5.32	----------------------------------------
13.57/5.32	
13.57/5.32	(8)
13.57/5.32	Obligation:
13.57/5.32	mainModule Main 
13.57/5.32	module Maybe where {
13.57/5.32	import qualified List;
13.57/5.32	import qualified Main;
13.57/5.32	import qualified Prelude;
13.57/5.32	}
13.57/5.32	module List where {
13.57/5.32	import qualified Main;
13.57/5.32	import qualified Maybe;
13.57/5.32	import qualified Prelude;
13.57/5.32	findIndices :: (a  ->  Bool)  ->  [a]  ->  [Int];
13.57/5.32	findIndices p xs = concatMap (findIndices0 p) (zip xs (enumFrom 0));
13.57/5.32	
13.57/5.32	findIndices0 p vv1 = findIndices00 p vv1;
13.57/5.32	
13.57/5.32	findIndices00 p (x,i) = findIndices000 i (p x);
13.57/5.32	findIndices00 p wv = [];
13.57/5.32	
13.57/5.32	findIndices000 i True = i : [];
13.57/5.32	findIndices000 i False = [];
13.57/5.32	
13.57/5.32	}
13.57/5.32	module Main where {
13.57/5.32	import qualified List;
13.57/5.32	import qualified Maybe;
13.57/5.32	import qualified Prelude;
13.57/5.32	}
13.57/5.32	
13.57/5.32	----------------------------------------
13.57/5.32	
13.57/5.32	(9) COR (EQUIVALENT)
13.57/5.32	Cond Reductions:
13.57/5.32	The following Function with conditions
13.57/5.32	"undefined |Falseundefined;
13.57/5.32	"
13.57/5.32	is transformed to
13.57/5.32	"undefined  = undefined1;
13.57/5.32	"
13.57/5.32	"undefined0 True = undefined;
13.57/5.32	"
13.57/5.32	"undefined1  = undefined0 False;
13.57/5.32	"
13.57/5.32	
13.57/5.32	----------------------------------------
13.57/5.32	
13.57/5.32	(10)
13.57/5.32	Obligation:
13.57/5.32	mainModule Main 
13.57/5.32	module Maybe where {
13.57/5.32	import qualified List;
13.57/5.32	import qualified Main;
13.57/5.32	import qualified Prelude;
13.57/5.32	}
13.57/5.32	module List where {
13.57/5.32	import qualified Main;
13.57/5.32	import qualified Maybe;
13.57/5.32	import qualified Prelude;
13.57/5.32	findIndices :: (a  ->  Bool)  ->  [a]  ->  [Int];
13.57/5.32	findIndices p xs = concatMap (findIndices0 p) (zip xs (enumFrom 0));
13.57/5.32	
13.57/5.32	findIndices0 p vv1 = findIndices00 p vv1;
13.57/5.32	
13.57/5.32	findIndices00 p (x,i) = findIndices000 i (p x);
13.57/5.32	findIndices00 p wv = [];
13.57/5.32	
13.57/5.32	findIndices000 i True = i : [];
13.57/5.32	findIndices000 i False = [];
13.57/5.32	
13.57/5.32	}
13.57/5.32	module Main where {
13.57/5.32	import qualified List;
13.57/5.32	import qualified Maybe;
13.57/5.32	import qualified Prelude;
13.57/5.32	}
13.57/5.32	
13.57/5.32	----------------------------------------
13.57/5.32	
13.57/5.32	(11) NumRed (SOUND)
13.57/5.32	Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
13.57/5.32	----------------------------------------
13.57/5.32	
13.57/5.32	(12)
13.57/5.32	Obligation:
13.57/5.32	mainModule Main 
13.57/5.32	module Maybe where {
13.57/5.32	import qualified List;
13.57/5.32	import qualified Main;
13.57/5.32	import qualified Prelude;
13.57/5.32	}
13.57/5.32	module List where {
13.57/5.32	import qualified Main;
13.57/5.32	import qualified Maybe;
13.57/5.32	import qualified Prelude;
13.57/5.32	findIndices :: (a  ->  Bool)  ->  [a]  ->  [Int];
13.57/5.32	findIndices p xs = concatMap (findIndices0 p) (zip xs (enumFrom (Pos Zero)));
13.57/5.32	
13.57/5.32	findIndices0 p vv1 = findIndices00 p vv1;
13.57/5.32	
13.57/5.32	findIndices00 p (x,i) = findIndices000 i (p x);
13.57/5.32	findIndices00 p wv = [];
13.57/5.32	
13.57/5.32	findIndices000 i True = i : [];
13.57/5.32	findIndices000 i False = [];
13.57/5.32	
13.57/5.32	}
13.57/5.32	module Main where {
13.57/5.32	import qualified List;
13.57/5.32	import qualified Maybe;
13.57/5.32	import qualified Prelude;
13.57/5.32	}
13.57/5.32	
13.57/5.32	----------------------------------------
13.57/5.32	
13.57/5.32	(13) Narrow (SOUND)
13.57/5.32	Haskell To QDPs
13.57/5.32	
13.57/5.32	digraph dp_graph {
13.57/5.32	node [outthreshold=100, inthreshold=100];1[label="List.findIndices",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
13.57/5.32	3[label="List.findIndices ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
13.57/5.32	4[label="List.findIndices ww3 ww4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
13.57/5.32	5[label="concatMap (List.findIndices0 ww3) (zip ww4 (enumFrom (Pos Zero)))",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
13.57/5.32	6[label="concat . map (List.findIndices0 ww3)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
13.57/5.32	7[label="concat (map (List.findIndices0 ww3) (zip ww4 (enumFrom (Pos Zero))))",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
13.57/5.32	8[label="foldr (++) [] (map (List.findIndices0 ww3) (zip ww4 (enumFrom (Pos Zero))))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
13.57/5.32	9[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww4 (enumFrom (Pos Zero))))",fontsize=16,color="burlywood",shape="box"];115[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];9 -> 115[label="",style="solid", color="burlywood", weight=9];
13.57/5.32	115 -> 10[label="",style="solid", color="burlywood", weight=3];
13.57/5.32	116[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 116[label="",style="solid", color="burlywood", weight=9];
13.57/5.32	116 -> 11[label="",style="solid", color="burlywood", weight=3];
13.57/5.32	10[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 (ww40 : ww41) (enumFrom (Pos Zero))))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
13.57/5.32	11[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 [] (enumFrom (Pos Zero))))",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
13.57/5.32	12[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 (ww40 : ww41) (numericEnumFrom (Pos Zero))))",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13.57/5.32	13[label="foldr (++) [] (map (List.findIndices0 ww3) [])",fontsize=16,color="black",shape="triangle"];13 -> 15[label="",style="solid", color="black", weight=3];
13.57/5.32	14[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 (ww40 : ww41) (Pos Zero : (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
13.57/5.32	15[label="foldr (++) [] []",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
13.57/5.32	16[label="foldr (++) [] (map (List.findIndices0 ww3) (zip0 ww40 (Pos Zero) : zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];16 -> 18[label="",style="solid", color="black", weight=3];
13.57/5.32	17[label="[]",fontsize=16,color="green",shape="box"];18[label="foldr (++) [] (List.findIndices0 ww3 (zip0 ww40 (Pos Zero)) : map (List.findIndices0 ww3) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
13.57/5.32	19[label="(++) List.findIndices0 ww3 (zip0 ww40 (Pos Zero)) foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3];
13.57/5.32	20[label="(++) List.findIndices00 ww3 (zip0 ww40 (Pos Zero)) foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3];
13.57/5.32	21[label="(++) List.findIndices00 ww3 (ww40,Pos Zero) foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
13.57/5.32	22 -> 23[label="",style="dashed", color="red", weight=0];
13.57/5.32	22[label="(++) List.findIndices000 (Pos Zero) (ww3 ww40) foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];22 -> 24[label="",style="dashed", color="magenta", weight=3];
13.57/5.32	24[label="ww3 ww40",fontsize=16,color="green",shape="box"];24 -> 28[label="",style="dashed", color="green", weight=3];
13.57/5.32	23[label="(++) List.findIndices000 (Pos Zero) ww5 foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="burlywood",shape="triangle"];117[label="ww5/False",fontsize=10,color="white",style="solid",shape="box"];23 -> 117[label="",style="solid", color="burlywood", weight=9];
13.57/5.32	117 -> 26[label="",style="solid", color="burlywood", weight=3];
13.57/5.32	118[label="ww5/True",fontsize=10,color="white",style="solid",shape="box"];23 -> 118[label="",style="solid", color="burlywood", weight=9];
13.57/5.32	118 -> 27[label="",style="solid", color="burlywood", weight=3];
13.57/5.32	28[label="ww40",fontsize=16,color="green",shape="box"];26[label="(++) List.findIndices000 (Pos Zero) False foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];26 -> 29[label="",style="solid", color="black", weight=3];
13.57/5.32	27[label="(++) List.findIndices000 (Pos Zero) True foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];27 -> 30[label="",style="solid", color="black", weight=3];
13.57/5.32	29[label="(++) [] foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="triangle"];29 -> 31[label="",style="solid", color="black", weight=3];
13.57/5.32	30[label="(++) (Pos Zero : []) foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];30 -> 32[label="",style="solid", color="black", weight=3];
13.57/5.32	31[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="burlywood",shape="box"];119[label="ww41/ww410 : ww411",fontsize=10,color="white",style="solid",shape="box"];31 -> 119[label="",style="solid", color="burlywood", weight=9];
13.57/5.32	119 -> 33[label="",style="solid", color="burlywood", weight=3];
13.57/5.32	120[label="ww41/[]",fontsize=10,color="white",style="solid",shape="box"];31 -> 120[label="",style="solid", color="burlywood", weight=9];
13.57/5.32	120 -> 34[label="",style="solid", color="burlywood", weight=3];
13.57/5.32	32[label="Pos Zero : [] ++ foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="green",shape="box"];32 -> 35[label="",style="dashed", color="green", weight=3];
13.57/5.32	33[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 (ww410 : ww411) (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];33 -> 36[label="",style="solid", color="black", weight=3];
13.57/5.32	34[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 [] (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];34 -> 37[label="",style="solid", color="black", weight=3];
13.57/5.32	35 -> 29[label="",style="dashed", color="red", weight=0];
13.57/5.32	35[label="[] ++ foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];36 -> 67[label="",style="dashed", color="red", weight=0];
13.57/5.32	36[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 (ww410 : ww411) (Pos Zero + fromInt (Pos (Succ Zero)) `seq` numericEnumFrom (Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="magenta"];36 -> 68[label="",style="dashed", color="magenta", weight=3];
13.57/5.32	36 -> 69[label="",style="dashed", color="magenta", weight=3];
13.57/5.32	36 -> 70[label="",style="dashed", color="magenta", weight=3];
13.57/5.32	36 -> 71[label="",style="dashed", color="magenta", weight=3];
13.57/5.32	37 -> 13[label="",style="dashed", color="red", weight=0];
13.57/5.32	37[label="foldr (++) [] (map (List.findIndices0 ww3) [])",fontsize=16,color="magenta"];68[label="Zero",fontsize=16,color="green",shape="box"];69[label="ww410",fontsize=16,color="green",shape="box"];70[label="Zero",fontsize=16,color="green",shape="box"];71[label="ww411",fontsize=16,color="green",shape="box"];67[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 (ww4110 : ww4111) (Pos ww7 + fromInt (Pos (Succ Zero)) `seq` numericEnumFrom (Pos ww8 + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="triangle"];67 -> 74[label="",style="solid", color="black", weight=3];
13.57/5.32	74[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 (ww4110 : ww4111) (enforceWHNF (WHNF (Pos ww7 + fromInt (Pos (Succ Zero)))) (numericEnumFrom (Pos ww8 + fromInt (Pos (Succ Zero)))))))",fontsize=16,color="black",shape="box"];74 -> 75[label="",style="solid", color="black", weight=3];
13.57/5.32	75[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 (ww4110 : ww4111) (enforceWHNF (WHNF (primPlusInt (Pos ww7) (fromInt (Pos (Succ Zero))))) (numericEnumFrom (primPlusInt (Pos ww8) (fromInt (Pos (Succ Zero))))))))",fontsize=16,color="black",shape="box"];75 -> 76[label="",style="solid", color="black", weight=3];
13.57/5.32	76[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 (ww4110 : ww4111) (enforceWHNF (WHNF (primPlusInt (Pos ww7) (Pos (Succ Zero)))) (numericEnumFrom (primPlusInt (Pos ww8) (Pos (Succ Zero)))))))",fontsize=16,color="black",shape="box"];76 -> 77[label="",style="solid", color="black", weight=3];
13.57/5.32	77[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 (ww4110 : ww4111) (enforceWHNF (WHNF (Pos (primPlusNat ww7 (Succ Zero)))) (numericEnumFrom (Pos (primPlusNat ww7 (Succ Zero)))))))",fontsize=16,color="black",shape="box"];77 -> 78[label="",style="solid", color="black", weight=3];
13.57/5.32	78[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 (ww4110 : ww4111) (numericEnumFrom (Pos (primPlusNat ww7 (Succ Zero))))))",fontsize=16,color="black",shape="box"];78 -> 79[label="",style="solid", color="black", weight=3];
13.57/5.32	79[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 (ww4110 : ww4111) (Pos (primPlusNat ww7 (Succ Zero)) : (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];79 -> 80[label="",style="solid", color="black", weight=3];
13.57/5.32	80[label="foldr (++) [] (map (List.findIndices0 ww3) (zip0 ww4110 (Pos (primPlusNat ww7 (Succ Zero))) : zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];80 -> 81[label="",style="solid", color="black", weight=3];
13.57/5.32	81[label="foldr (++) [] (List.findIndices0 ww3 (zip0 ww4110 (Pos (primPlusNat ww7 (Succ Zero)))) : map (List.findIndices0 ww3) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];81 -> 82[label="",style="solid", color="black", weight=3];
13.57/5.32	82[label="(++) List.findIndices0 ww3 (zip0 ww4110 (Pos (primPlusNat ww7 (Succ Zero)))) foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];82 -> 83[label="",style="solid", color="black", weight=3];
13.57/5.32	83[label="(++) List.findIndices00 ww3 (zip0 ww4110 (Pos (primPlusNat ww7 (Succ Zero)))) foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];83 -> 84[label="",style="solid", color="black", weight=3];
13.57/5.32	84[label="(++) List.findIndices00 ww3 (ww4110,Pos (primPlusNat ww7 (Succ Zero))) foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];84 -> 85[label="",style="solid", color="black", weight=3];
13.57/5.32	85 -> 86[label="",style="dashed", color="red", weight=0];
13.57/5.32	85[label="(++) List.findIndices000 (Pos (primPlusNat ww7 (Succ Zero))) (ww3 ww4110) foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];85 -> 87[label="",style="dashed", color="magenta", weight=3];
13.57/5.32	87[label="ww3 ww4110",fontsize=16,color="green",shape="box"];87 -> 91[label="",style="dashed", color="green", weight=3];
13.57/5.32	86[label="(++) List.findIndices000 (Pos (primPlusNat ww7 (Succ Zero))) ww9 foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="burlywood",shape="triangle"];121[label="ww9/False",fontsize=10,color="white",style="solid",shape="box"];86 -> 121[label="",style="solid", color="burlywood", weight=9];
13.57/5.32	121 -> 89[label="",style="solid", color="burlywood", weight=3];
13.57/5.32	122[label="ww9/True",fontsize=10,color="white",style="solid",shape="box"];86 -> 122[label="",style="solid", color="burlywood", weight=9];
13.57/5.32	122 -> 90[label="",style="solid", color="burlywood", weight=3];
13.57/5.32	91[label="ww4110",fontsize=16,color="green",shape="box"];89[label="(++) List.findIndices000 (Pos (primPlusNat ww7 (Succ Zero))) False foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];89 -> 92[label="",style="solid", color="black", weight=3];
13.57/5.32	90[label="(++) List.findIndices000 (Pos (primPlusNat ww7 (Succ Zero))) True foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];90 -> 93[label="",style="solid", color="black", weight=3];
13.57/5.32	92[label="(++) [] foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="triangle"];92 -> 94[label="",style="solid", color="black", weight=3];
13.57/5.32	93[label="(++) (Pos (primPlusNat ww7 (Succ Zero)) : []) foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];93 -> 95[label="",style="solid", color="black", weight=3];
13.57/5.32	94[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="burlywood",shape="box"];123[label="ww4111/ww41110 : ww41111",fontsize=10,color="white",style="solid",shape="box"];94 -> 123[label="",style="solid", color="burlywood", weight=9];
13.57/5.32	123 -> 96[label="",style="solid", color="burlywood", weight=3];
13.57/5.32	124[label="ww4111/[]",fontsize=10,color="white",style="solid",shape="box"];94 -> 124[label="",style="solid", color="burlywood", weight=9];
13.57/5.32	124 -> 97[label="",style="solid", color="burlywood", weight=3];
13.57/5.32	95[label="Pos (primPlusNat ww7 (Succ Zero)) : [] ++ foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="green",shape="box"];95 -> 98[label="",style="dashed", color="green", weight=3];
13.57/5.32	95 -> 99[label="",style="dashed", color="green", weight=3];
13.57/5.32	96[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 (ww41110 : ww41111) (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];96 -> 100[label="",style="solid", color="black", weight=3];
13.57/5.32	97[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 [] (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];97 -> 101[label="",style="solid", color="black", weight=3];
13.57/5.32	98[label="primPlusNat ww7 (Succ Zero)",fontsize=16,color="burlywood",shape="triangle"];125[label="ww7/Succ ww70",fontsize=10,color="white",style="solid",shape="box"];98 -> 125[label="",style="solid", color="burlywood", weight=9];
13.57/5.32	125 -> 102[label="",style="solid", color="burlywood", weight=3];
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13.57/5.32	100[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 (ww41110 : ww41111) (Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)) `seq` numericEnumFrom (Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="magenta"];100 -> 104[label="",style="dashed", color="magenta", weight=3];
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13.57/5.32	100 -> 107[label="",style="dashed", color="magenta", weight=3];
13.57/5.32	101 -> 13[label="",style="dashed", color="red", weight=0];
13.57/5.32	101[label="foldr (++) [] (map (List.findIndices0 ww3) [])",fontsize=16,color="magenta"];102[label="primPlusNat (Succ ww70) (Succ Zero)",fontsize=16,color="black",shape="box"];102 -> 108[label="",style="solid", color="black", weight=3];
13.57/5.32	103[label="primPlusNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];103 -> 109[label="",style="solid", color="black", weight=3];
13.57/5.32	104 -> 98[label="",style="dashed", color="red", weight=0];
13.57/5.32	104[label="primPlusNat ww7 (Succ Zero)",fontsize=16,color="magenta"];105[label="ww41110",fontsize=16,color="green",shape="box"];106 -> 98[label="",style="dashed", color="red", weight=0];
13.57/5.32	106[label="primPlusNat ww7 (Succ Zero)",fontsize=16,color="magenta"];107[label="ww41111",fontsize=16,color="green",shape="box"];108[label="Succ (Succ (primPlusNat ww70 Zero))",fontsize=16,color="green",shape="box"];108 -> 110[label="",style="dashed", color="green", weight=3];
13.57/5.32	109[label="Succ Zero",fontsize=16,color="green",shape="box"];110[label="primPlusNat ww70 Zero",fontsize=16,color="burlywood",shape="box"];127[label="ww70/Succ ww700",fontsize=10,color="white",style="solid",shape="box"];110 -> 127[label="",style="solid", color="burlywood", weight=9];
13.57/5.32	127 -> 111[label="",style="solid", color="burlywood", weight=3];
13.57/5.32	128[label="ww70/Zero",fontsize=10,color="white",style="solid",shape="box"];110 -> 128[label="",style="solid", color="burlywood", weight=9];
13.57/5.32	128 -> 112[label="",style="solid", color="burlywood", weight=3];
13.57/5.32	111[label="primPlusNat (Succ ww700) Zero",fontsize=16,color="black",shape="box"];111 -> 113[label="",style="solid", color="black", weight=3];
13.57/5.32	112[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];112 -> 114[label="",style="solid", color="black", weight=3];
13.57/5.32	113[label="Succ ww700",fontsize=16,color="green",shape="box"];114[label="Zero",fontsize=16,color="green",shape="box"];}
13.57/5.32	
13.57/5.32	----------------------------------------
13.57/5.32	
13.57/5.32	(14)
13.57/5.32	Obligation:
13.57/5.32	Q DP problem:
13.57/5.32	The TRS P consists of the following rules:
13.57/5.32	
13.57/5.32	   new_psPs(ww7, ww3, :(ww41110, ww41111), ba) -> new_foldr(ww3, ww41110, ww41111, new_primPlusNat(ww7), new_primPlusNat(ww7), ba)
13.57/5.32	   new_psPs0(ww3, :(ww41110, ww41111), ww7, ba) -> new_foldr(ww3, ww41110, ww41111, new_primPlusNat(ww7), new_primPlusNat(ww7), ba)
13.57/5.32	   new_foldr(ww3, ww4110, ww4111, ww7, ww8, ba) -> new_psPs(ww7, ww3, ww4111, ba)
13.57/5.32	   new_psPs(ww7, ww3, ww4111, ba) -> new_psPs0(ww3, ww4111, ww7, ba)
13.57/5.32	
13.57/5.32	The TRS R consists of the following rules:
13.57/5.32	
13.57/5.32	   new_primPlusNat(Succ(ww70)) -> Succ(Succ(new_primPlusNat0(ww70)))
13.57/5.32	   new_primPlusNat(Zero) -> Succ(Zero)
13.57/5.32	   new_primPlusNat0(Succ(ww700)) -> Succ(ww700)
13.57/5.32	   new_primPlusNat0(Zero) -> Zero
13.57/5.32	
13.57/5.32	The set Q consists of the following terms:
13.57/5.32	
13.57/5.32	   new_primPlusNat(Succ(x0))
13.57/5.32	   new_primPlusNat0(Zero)
13.57/5.32	   new_primPlusNat0(Succ(x0))
13.57/5.32	   new_primPlusNat(Zero)
13.57/5.32	
13.57/5.32	We have to consider all minimal (P,Q,R)-chains.
13.57/5.32	----------------------------------------
13.57/5.32	
13.57/5.32	(15) QDPSizeChangeProof (EQUIVALENT)
13.57/5.32	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. 
13.57/5.32	
13.57/5.32	From the DPs we obtained the following set of size-change graphs:
13.57/5.32	*new_foldr(ww3, ww4110, ww4111, ww7, ww8, ba) -> new_psPs(ww7, ww3, ww4111, ba)
13.57/5.32	The graph contains the following edges 4 >= 1, 1 >= 2, 3 >= 3, 6 >= 4
13.57/5.32	
13.57/5.32	
13.57/5.32	*new_psPs(ww7, ww3, ww4111, ba) -> new_psPs0(ww3, ww4111, ww7, ba)
13.57/5.32	The graph contains the following edges 2 >= 1, 3 >= 2, 1 >= 3, 4 >= 4
13.57/5.32	
13.57/5.32	
13.57/5.32	*new_psPs(ww7, ww3, :(ww41110, ww41111), ba) -> new_foldr(ww3, ww41110, ww41111, new_primPlusNat(ww7), new_primPlusNat(ww7), ba)
13.57/5.32	The graph contains the following edges 2 >= 1, 3 > 2, 3 > 3, 4 >= 6
13.57/5.32	
13.57/5.32	
13.57/5.32	*new_psPs0(ww3, :(ww41110, ww41111), ww7, ba) -> new_foldr(ww3, ww41110, ww41111, new_primPlusNat(ww7), new_primPlusNat(ww7), ba)
13.57/5.32	The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 4 >= 6
13.57/5.32	
13.57/5.32	
13.57/5.32	----------------------------------------
13.57/5.32	
13.57/5.32	(16)
13.57/5.32	YES
13.77/5.36	EOF