11.13/4.59	YES
14.10/5.37	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
14.10/5.37	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
14.10/5.37	
14.10/5.37	
14.10/5.37	H-Termination with start terms of the given HASKELL could be proven:
14.10/5.37	
14.10/5.37	(0) HASKELL
14.10/5.37	(1) LR [EQUIVALENT, 0 ms]
14.10/5.37	(2) HASKELL
14.10/5.37	(3) CR [EQUIVALENT, 0 ms]
14.10/5.37	(4) HASKELL
14.10/5.37	(5) IFR [EQUIVALENT, 0 ms]
14.10/5.37	(6) HASKELL
14.10/5.37	(7) BR [EQUIVALENT, 0 ms]
14.10/5.37	(8) HASKELL
14.10/5.37	(9) COR [EQUIVALENT, 0 ms]
14.10/5.37	(10) HASKELL
14.10/5.37	(11) LetRed [EQUIVALENT, 0 ms]
14.10/5.37	(12) HASKELL
14.10/5.37	(13) NumRed [SOUND, 0 ms]
14.10/5.37	(14) HASKELL
14.10/5.37	(15) Narrow [SOUND, 0 ms]
14.10/5.37	(16) QDP
14.10/5.37	(17) TransformationProof [EQUIVALENT, 0 ms]
14.10/5.37	(18) QDP
14.10/5.37	(19) DependencyGraphProof [EQUIVALENT, 0 ms]
14.10/5.37	(20) QDP
14.10/5.37	(21) TransformationProof [EQUIVALENT, 0 ms]
14.10/5.37	(22) QDP
14.10/5.37	(23) DependencyGraphProof [EQUIVALENT, 0 ms]
14.10/5.37	(24) QDP
14.10/5.37	(25) UsableRulesProof [EQUIVALENT, 0 ms]
14.10/5.37	(26) QDP
14.10/5.37	(27) QReductionProof [EQUIVALENT, 0 ms]
14.10/5.37	(28) QDP
14.10/5.37	(29) QDPSizeChangeProof [EQUIVALENT, 0 ms]
14.10/5.37	(30) YES
14.10/5.37	
14.10/5.37	
14.10/5.37	----------------------------------------
14.10/5.37	
14.10/5.37	(0)
14.10/5.37	Obligation:
14.10/5.37	mainModule Main 
14.10/5.37	module Maybe where {
14.10/5.37	import qualified List;
14.10/5.38	import qualified Main;
14.10/5.38	import qualified Prelude;
14.10/5.38	}
14.10/5.38	module List where {
14.10/5.38	import qualified Main;
14.10/5.38	import qualified Maybe;
14.10/5.38	import qualified Prelude;
14.10/5.38	genericSplitAt :: Integral a => a  ->  [b]  ->  ([b],[b]);
14.10/5.38	genericSplitAt 0 xs = ([],xs);
14.10/5.38	genericSplitAt _ [] = ([],[]);
14.10/5.38	genericSplitAt n (x : xs)  | n > 0 = (x : xs',xs'') where { 
14.10/5.38	vv9 = genericSplitAt (n - 1) xs;
14.10/5.38	xs' = (\(xs',_) ->xs') vv9;
14.10/5.38	xs'' = (\(_,xs'') ->xs'') vv9;
14.10/5.38	 };
14.10/5.38	genericSplitAt _ _ = error [];
14.10/5.38	
14.10/5.38	}
14.10/5.38	module Main where {
14.10/5.38	import qualified List;
14.10/5.38	import qualified Maybe;
14.10/5.38	import qualified Prelude;
14.10/5.38	}
14.10/5.38	
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(1) LR (EQUIVALENT)
14.10/5.38	Lambda Reductions:
14.10/5.38	The following Lambda expression
14.10/5.38	"\(_,xs'')->xs''"
14.10/5.38	is transformed to
14.10/5.38	"xs''0 (_,xs'') = xs'';
14.10/5.38	"
14.10/5.38	The following Lambda expression
14.10/5.38	"\(xs',_)->xs'"
14.10/5.38	is transformed to
14.10/5.38	"xs'0 (xs',_) = xs';
14.10/5.38	"
14.10/5.38	
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(2)
14.10/5.38	Obligation:
14.10/5.38	mainModule Main 
14.10/5.38	module Maybe where {
14.10/5.38	import qualified List;
14.10/5.38	import qualified Main;
14.10/5.38	import qualified Prelude;
14.10/5.38	}
14.10/5.38	module List where {
14.10/5.38	import qualified Main;
14.10/5.38	import qualified Maybe;
14.10/5.38	import qualified Prelude;
14.10/5.38	genericSplitAt :: Integral a => a  ->  [b]  ->  ([b],[b]);
14.10/5.38	genericSplitAt 0 xs = ([],xs);
14.10/5.38	genericSplitAt _ [] = ([],[]);
14.10/5.38	genericSplitAt n (x : xs)  | n > 0 = (x : xs',xs'') where { 
14.10/5.38	vv9 = genericSplitAt (n - 1) xs;
14.10/5.38	xs' = xs'0 vv9;
14.10/5.38	xs'' = xs''0 vv9;
14.10/5.38	xs''0 (_,xs'') = xs'';
14.10/5.38	xs'0 (xs',_) = xs';
14.10/5.38	 };
14.10/5.38	genericSplitAt _ _ = error [];
14.10/5.38	
14.10/5.38	}
14.10/5.38	module Main where {
14.10/5.38	import qualified List;
14.10/5.38	import qualified Maybe;
14.10/5.38	import qualified Prelude;
14.10/5.38	}
14.10/5.38	
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(3) CR (EQUIVALENT)
14.10/5.38	Case Reductions:
14.10/5.38	The following Case expression
14.10/5.38	"case compare x y of {
14.10/5.38	EQ  -> o;
14.10/5.38	LT  -> LT;
14.10/5.38	GT  -> GT}
14.10/5.38	"
14.10/5.38	is transformed to
14.10/5.38	"primCompAux0 o EQ = o;
14.10/5.38	primCompAux0 o LT = LT;
14.10/5.38	primCompAux0 o GT = GT;
14.10/5.38	"
14.10/5.38	
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(4)
14.10/5.38	Obligation:
14.10/5.38	mainModule Main 
14.10/5.38	module Maybe where {
14.10/5.38	import qualified List;
14.10/5.38	import qualified Main;
14.10/5.38	import qualified Prelude;
14.10/5.38	}
14.10/5.38	module List where {
14.10/5.38	import qualified Main;
14.10/5.38	import qualified Maybe;
14.10/5.38	import qualified Prelude;
14.10/5.38	genericSplitAt :: Integral a => a  ->  [b]  ->  ([b],[b]);
14.10/5.38	genericSplitAt 0 xs = ([],xs);
14.10/5.38	genericSplitAt _ [] = ([],[]);
14.10/5.38	genericSplitAt n (x : xs)  | n > 0 = (x : xs',xs'') where { 
14.10/5.38	vv9 = genericSplitAt (n - 1) xs;
14.10/5.38	xs' = xs'0 vv9;
14.10/5.38	xs'' = xs''0 vv9;
14.10/5.38	xs''0 (_,xs'') = xs'';
14.10/5.38	xs'0 (xs',_) = xs';
14.10/5.38	 };
14.10/5.38	genericSplitAt _ _ = error [];
14.10/5.38	
14.10/5.38	}
14.10/5.38	module Main where {
14.10/5.38	import qualified List;
14.10/5.38	import qualified Maybe;
14.10/5.38	import qualified Prelude;
14.10/5.38	}
14.10/5.38	
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(5) IFR (EQUIVALENT)
14.10/5.38	If Reductions:
14.10/5.38	The following If expression
14.10/5.38	"if primGEqNatS x y then Succ (primDivNatS (primMinusNatS x y) (Succ y)) else Zero"
14.10/5.38	is transformed to
14.10/5.38	"primDivNatS0 x y True = Succ (primDivNatS (primMinusNatS x y) (Succ y));
14.10/5.38	primDivNatS0 x y False = Zero;
14.10/5.38	"
14.10/5.38	The following If expression
14.10/5.38	"if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x"
14.10/5.38	is transformed to
14.10/5.38	"primModNatS0 x y True = primModNatS (primMinusNatS x y) (Succ y);
14.10/5.38	primModNatS0 x y False = Succ x;
14.10/5.38	"
14.10/5.38	
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(6)
14.10/5.38	Obligation:
14.10/5.38	mainModule Main 
14.10/5.38	module Maybe where {
14.10/5.38	import qualified List;
14.10/5.38	import qualified Main;
14.10/5.38	import qualified Prelude;
14.10/5.38	}
14.10/5.38	module List where {
14.10/5.38	import qualified Main;
14.10/5.38	import qualified Maybe;
14.10/5.38	import qualified Prelude;
14.10/5.38	genericSplitAt :: Integral b => b  ->  [a]  ->  ([a],[a]);
14.10/5.38	genericSplitAt 0 xs = ([],xs);
14.10/5.38	genericSplitAt _ [] = ([],[]);
14.10/5.38	genericSplitAt n (x : xs)  | n > 0 = (x : xs',xs'') where { 
14.10/5.38	vv9 = genericSplitAt (n - 1) xs;
14.10/5.38	xs' = xs'0 vv9;
14.10/5.38	xs'' = xs''0 vv9;
14.10/5.38	xs''0 (_,xs'') = xs'';
14.10/5.38	xs'0 (xs',_) = xs';
14.10/5.38	 };
14.10/5.38	genericSplitAt _ _ = error [];
14.10/5.38	
14.10/5.38	}
14.10/5.38	module Main where {
14.10/5.38	import qualified List;
14.10/5.38	import qualified Maybe;
14.10/5.38	import qualified Prelude;
14.10/5.38	}
14.10/5.38	
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(7) BR (EQUIVALENT)
14.10/5.38	Replaced joker patterns by fresh variables and removed binding patterns.
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(8)
14.10/5.38	Obligation:
14.10/5.38	mainModule Main 
14.10/5.38	module Maybe where {
14.10/5.38	import qualified List;
14.10/5.38	import qualified Main;
14.10/5.38	import qualified Prelude;
14.10/5.38	}
14.10/5.38	module List where {
14.10/5.38	import qualified Main;
14.10/5.38	import qualified Maybe;
14.10/5.38	import qualified Prelude;
14.10/5.38	genericSplitAt :: Integral a => a  ->  [b]  ->  ([b],[b]);
14.10/5.38	genericSplitAt 0 xs = ([],xs);
14.10/5.38	genericSplitAt zy [] = ([],[]);
14.10/5.38	genericSplitAt n (x : xs)  | n > 0 = (x : xs',xs'') where { 
14.10/5.38	vv9 = genericSplitAt (n - 1) xs;
14.10/5.38	xs' = xs'0 vv9;
14.10/5.38	xs'' = xs''0 vv9;
14.10/5.38	xs''0 (zz,xs'') = xs'';
14.10/5.38	xs'0 (xs',vuu) = xs';
14.10/5.38	 };
14.10/5.38	genericSplitAt vuv vuw = error [];
14.10/5.38	
14.10/5.38	}
14.10/5.38	module Main where {
14.10/5.38	import qualified List;
14.10/5.38	import qualified Maybe;
14.10/5.38	import qualified Prelude;
14.10/5.38	}
14.10/5.38	
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(9) COR (EQUIVALENT)
14.10/5.38	Cond Reductions:
14.10/5.38	The following Function with conditions
14.10/5.38	"compare x y|x == yEQ|x <= yLT|otherwiseGT;
14.10/5.38	"
14.10/5.38	is transformed to
14.10/5.38	"compare x y = compare3 x y;
14.10/5.38	"
14.10/5.38	"compare0 x y True = GT;
14.10/5.38	"
14.10/5.38	"compare1 x y True = LT;
14.10/5.38	compare1 x y False = compare0 x y otherwise;
14.10/5.38	"
14.10/5.38	"compare2 x y True = EQ;
14.10/5.38	compare2 x y False = compare1 x y (x <= y);
14.10/5.38	"
14.10/5.38	"compare3 x y = compare2 x y (x == y);
14.10/5.38	"
14.10/5.38	The following Function with conditions
14.10/5.38	"gcd' x 0 = x;
14.10/5.38	gcd' x y = gcd' y (x `rem` y);
14.10/5.38	"
14.10/5.38	is transformed to
14.10/5.38	"gcd' x vux = gcd'2 x vux;
14.10/5.38	gcd' x y = gcd'0 x y;
14.10/5.38	"
14.10/5.38	"gcd'0 x y = gcd' y (x `rem` y);
14.10/5.38	"
14.10/5.38	"gcd'1 True x vux = x;
14.10/5.38	gcd'1 vuy vuz vvu = gcd'0 vuz vvu;
14.10/5.38	"
14.10/5.38	"gcd'2 x vux = gcd'1 (vux == 0) x vux;
14.10/5.38	gcd'2 vvv vvw = gcd'0 vvv vvw;
14.10/5.38	"
14.10/5.38	The following Function with conditions
14.10/5.38	"gcd 0 0 = error [];
14.10/5.38	gcd x y = gcd' (abs x) (abs y) where {
14.10/5.38	gcd' x 0 = x;
14.10/5.38	gcd' x y = gcd' y (x `rem` y);
14.10/5.38	} 
14.10/5.38	;
14.10/5.38	"
14.10/5.38	is transformed to
14.10/5.38	"gcd vvx vvy = gcd3 vvx vvy;
14.10/5.38	gcd x y = gcd0 x y;
14.10/5.38	"
14.10/5.38	"gcd0 x y = gcd' (abs x) (abs y) where {
14.10/5.38	gcd' x vux = gcd'2 x vux;
14.10/5.38	gcd' x y = gcd'0 x y;
14.10/5.38	;
14.10/5.38	gcd'0 x y = gcd' y (x `rem` y);
14.10/5.38	;
14.10/5.38	gcd'1 True x vux = x;
14.10/5.38	gcd'1 vuy vuz vvu = gcd'0 vuz vvu;
14.10/5.38	;
14.10/5.38	gcd'2 x vux = gcd'1 (vux == 0) x vux;
14.10/5.38	gcd'2 vvv vvw = gcd'0 vvv vvw;
14.10/5.38	} 
14.10/5.38	;
14.10/5.38	"
14.10/5.38	"gcd1 True vvx vvy = error [];
14.10/5.38	gcd1 vvz vwu vwv = gcd0 vwu vwv;
14.10/5.38	"
14.10/5.38	"gcd2 True vvx vvy = gcd1 (vvy == 0) vvx vvy;
14.10/5.38	gcd2 vww vwx vwy = gcd0 vwx vwy;
14.10/5.38	"
14.10/5.38	"gcd3 vvx vvy = gcd2 (vvx == 0) vvx vvy;
14.10/5.38	gcd3 vwz vxu = gcd0 vwz vxu;
14.10/5.38	"
14.10/5.38	The following Function with conditions
14.10/5.38	"reduce x y|y == 0error []|otherwisex `quot` d :% (y `quot` d) where {
14.10/5.38	d  = gcd x y;
14.10/5.38	} 
14.10/5.38	;
14.10/5.38	"
14.10/5.38	is transformed to
14.10/5.38	"reduce x y = reduce2 x y;
14.10/5.38	"
14.10/5.38	"reduce2 x y = reduce1 x y (y == 0) where {
14.10/5.38	d  = gcd x y;
14.10/5.38	;
14.10/5.38	reduce0 x y True = x `quot` d :% (y `quot` d);
14.10/5.38	;
14.10/5.38	reduce1 x y True = error [];
14.10/5.38	reduce1 x y False = reduce0 x y otherwise;
14.10/5.38	} 
14.10/5.38	;
14.10/5.38	"
14.10/5.38	The following Function with conditions
14.10/5.38	"absReal x|x >= 0x|otherwise`negate` x;
14.10/5.38	"
14.10/5.38	is transformed to
14.10/5.38	"absReal x = absReal2 x;
14.10/5.38	"
14.10/5.38	"absReal0 x True = `negate` x;
14.10/5.38	"
14.10/5.38	"absReal1 x True = x;
14.10/5.38	absReal1 x False = absReal0 x otherwise;
14.10/5.38	"
14.10/5.38	"absReal2 x = absReal1 x (x >= 0);
14.10/5.38	"
14.10/5.38	The following Function with conditions
14.10/5.38	"undefined |Falseundefined;
14.10/5.38	"
14.10/5.38	is transformed to
14.10/5.38	"undefined  = undefined1;
14.10/5.38	"
14.10/5.38	"undefined0 True = undefined;
14.10/5.38	"
14.10/5.38	"undefined1  = undefined0 False;
14.10/5.38	"
14.10/5.38	The following Function with conditions
14.10/5.38	"genericSplitAt 0 xs = ([],xs);
14.10/5.38	genericSplitAt zy [] = ([],[]);
14.10/5.38	genericSplitAt n (x : xs)|n > 0(x : xs',xs'') where {
14.10/5.38	vv9  = genericSplitAt (n - 1) xs;
14.10/5.38	;
14.10/5.38	xs'  = xs'0 vv9;
14.10/5.38	;
14.10/5.38	xs''  = xs''0 vv9;
14.10/5.38	;
14.10/5.38	xs''0 (zz,xs'') = xs'';
14.10/5.38	;
14.10/5.38	xs'0 (xs',vuu) = xs';
14.10/5.38	} 
14.10/5.38	;
14.10/5.38	genericSplitAt vuv vuw = error [];
14.10/5.38	"
14.10/5.38	is transformed to
14.10/5.38	"genericSplitAt vyv xs = genericSplitAt5 vyv xs;
14.10/5.38	genericSplitAt zy [] = genericSplitAt3 zy [];
14.10/5.38	genericSplitAt n (x : xs) = genericSplitAt2 n (x : xs);
14.10/5.38	genericSplitAt vuv vuw = genericSplitAt0 vuv vuw;
14.10/5.38	"
14.10/5.38	"genericSplitAt0 vuv vuw = error [];
14.10/5.38	"
14.10/5.38	"genericSplitAt2 n (x : xs) = genericSplitAt1 n x xs (n > 0) where {
14.10/5.38	genericSplitAt1 n x xs True = (x : xs',xs'');
14.10/5.38	genericSplitAt1 n x xs False = genericSplitAt0 n (x : xs);
14.10/5.38	;
14.10/5.38	vv9  = genericSplitAt (n - 1) xs;
14.10/5.38	;
14.10/5.38	xs'  = xs'0 vv9;
14.10/5.38	;
14.10/5.38	xs''  = xs''0 vv9;
14.10/5.38	;
14.10/5.38	xs''0 (zz,xs'') = xs'';
14.10/5.38	;
14.10/5.38	xs'0 (xs',vuu) = xs';
14.10/5.38	} 
14.10/5.38	;
14.10/5.38	genericSplitAt2 vxw vxx = genericSplitAt0 vxw vxx;
14.10/5.38	"
14.10/5.38	"genericSplitAt3 zy [] = ([],[]);
14.10/5.38	genericSplitAt3 vxz vyu = genericSplitAt2 vxz vyu;
14.10/5.38	"
14.10/5.38	"genericSplitAt4 True vyv xs = ([],xs);
14.10/5.38	genericSplitAt4 vyw vyx vyy = genericSplitAt3 vyx vyy;
14.10/5.38	"
14.10/5.38	"genericSplitAt5 vyv xs = genericSplitAt4 (vyv == 0) vyv xs;
14.10/5.38	genericSplitAt5 vyz vzu = genericSplitAt3 vyz vzu;
14.10/5.38	"
14.10/5.38	
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(10)
14.10/5.38	Obligation:
14.10/5.38	mainModule Main 
14.10/5.38	module Maybe where {
14.10/5.38	import qualified List;
14.10/5.38	import qualified Main;
14.10/5.38	import qualified Prelude;
14.10/5.38	}
14.10/5.38	module List where {
14.10/5.38	import qualified Main;
14.10/5.38	import qualified Maybe;
14.10/5.38	import qualified Prelude;
14.10/5.38	genericSplitAt :: Integral a => a  ->  [b]  ->  ([b],[b]);
14.10/5.38	genericSplitAt vyv xs = genericSplitAt5 vyv xs;
14.10/5.38	genericSplitAt zy [] = genericSplitAt3 zy [];
14.10/5.38	genericSplitAt n (x : xs) = genericSplitAt2 n (x : xs);
14.10/5.38	genericSplitAt vuv vuw = genericSplitAt0 vuv vuw;
14.10/5.38	
14.10/5.38	genericSplitAt0 vuv vuw = error [];
14.10/5.38	
14.10/5.38	genericSplitAt2 n (x : xs) = genericSplitAt1 n x xs (n > 0) where { 
14.10/5.38	genericSplitAt1 n x xs True = (x : xs',xs'');
14.10/5.38	genericSplitAt1 n x xs False = genericSplitAt0 n (x : xs);
14.10/5.38	vv9 = genericSplitAt (n - 1) xs;
14.10/5.38	xs' = xs'0 vv9;
14.10/5.38	xs'' = xs''0 vv9;
14.10/5.38	xs''0 (zz,xs'') = xs'';
14.10/5.38	xs'0 (xs',vuu) = xs';
14.10/5.38	 };
14.10/5.38	genericSplitAt2 vxw vxx = genericSplitAt0 vxw vxx;
14.10/5.38	
14.10/5.38	genericSplitAt3 zy [] = ([],[]);
14.10/5.38	genericSplitAt3 vxz vyu = genericSplitAt2 vxz vyu;
14.10/5.38	
14.10/5.38	genericSplitAt4 True vyv xs = ([],xs);
14.10/5.38	genericSplitAt4 vyw vyx vyy = genericSplitAt3 vyx vyy;
14.10/5.38	
14.10/5.38	genericSplitAt5 vyv xs = genericSplitAt4 (vyv == 0) vyv xs;
14.10/5.38	genericSplitAt5 vyz vzu = genericSplitAt3 vyz vzu;
14.10/5.38	
14.10/5.38	}
14.10/5.38	module Main where {
14.10/5.38	import qualified List;
14.10/5.38	import qualified Maybe;
14.10/5.38	import qualified Prelude;
14.10/5.38	}
14.10/5.38	
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(11) LetRed (EQUIVALENT)
14.10/5.38	Let/Where Reductions:
14.10/5.38	The bindings of the following Let/Where expression
14.10/5.38	"gcd' (abs x) (abs y) where {
14.10/5.38	gcd' x vux = gcd'2 x vux;
14.10/5.38	gcd' x y = gcd'0 x y;
14.10/5.38	;
14.10/5.38	gcd'0 x y = gcd' y (x `rem` y);
14.10/5.38	;
14.10/5.38	gcd'1 True x vux = x;
14.10/5.38	gcd'1 vuy vuz vvu = gcd'0 vuz vvu;
14.10/5.38	;
14.10/5.38	gcd'2 x vux = gcd'1 (vux == 0) x vux;
14.10/5.38	gcd'2 vvv vvw = gcd'0 vvv vvw;
14.10/5.38	} 
14.10/5.38	"
14.10/5.38	are unpacked to the following functions on top level
14.10/5.38	"gcd0Gcd'1 True x vux = x;
14.10/5.38	gcd0Gcd'1 vuy vuz vvu = gcd0Gcd'0 vuz vvu;
14.10/5.38	"
14.10/5.38	"gcd0Gcd'2 x vux = gcd0Gcd'1 (vux == 0) x vux;
14.10/5.38	gcd0Gcd'2 vvv vvw = gcd0Gcd'0 vvv vvw;
14.10/5.38	"
14.10/5.38	"gcd0Gcd' x vux = gcd0Gcd'2 x vux;
14.10/5.38	gcd0Gcd' x y = gcd0Gcd'0 x y;
14.10/5.38	"
14.10/5.38	"gcd0Gcd'0 x y = gcd0Gcd' y (x `rem` y);
14.10/5.38	"
14.10/5.38	The bindings of the following Let/Where expression
14.10/5.38	"reduce1 x y (y == 0) where {
14.10/5.38	d  = gcd x y;
14.10/5.38	;
14.10/5.38	reduce0 x y True = x `quot` d :% (y `quot` d);
14.10/5.38	;
14.10/5.38	reduce1 x y True = error [];
14.10/5.38	reduce1 x y False = reduce0 x y otherwise;
14.10/5.38	} 
14.10/5.38	"
14.10/5.38	are unpacked to the following functions on top level
14.10/5.38	"reduce2D vzv vzw = gcd vzv vzw;
14.10/5.38	"
14.10/5.38	"reduce2Reduce1 vzv vzw x y True = error [];
14.10/5.38	reduce2Reduce1 vzv vzw x y False = reduce2Reduce0 vzv vzw x y otherwise;
14.10/5.38	"
14.10/5.38	"reduce2Reduce0 vzv vzw x y True = x `quot` reduce2D vzv vzw :% (y `quot` reduce2D vzv vzw);
14.10/5.38	"
14.10/5.38	The bindings of the following Let/Where expression
14.10/5.38	"genericSplitAt1 n x xs (n > 0) where {
14.10/5.38	genericSplitAt1 n x xs True = (x : xs',xs'');
14.10/5.38	genericSplitAt1 n x xs False = genericSplitAt0 n (x : xs);
14.10/5.38	;
14.10/5.38	vv9  = genericSplitAt (n - 1) xs;
14.10/5.38	;
14.10/5.38	xs'  = xs'0 vv9;
14.10/5.38	;
14.10/5.38	xs''  = xs''0 vv9;
14.10/5.38	;
14.10/5.38	xs''0 (zz,xs'') = xs'';
14.10/5.38	;
14.10/5.38	xs'0 (xs',vuu) = xs';
14.10/5.38	} 
14.10/5.38	"
14.10/5.38	are unpacked to the following functions on top level
14.10/5.38	"genericSplitAt2Vv9 vzx vzy = genericSplitAt (vzx - 1) vzy;
14.10/5.38	"
14.10/5.38	"genericSplitAt2Xs' vzx vzy = genericSplitAt2Xs'0 vzx vzy (genericSplitAt2Vv9 vzx vzy);
14.10/5.38	"
14.10/5.38	"genericSplitAt2Xs'' vzx vzy = genericSplitAt2Xs''0 vzx vzy (genericSplitAt2Vv9 vzx vzy);
14.10/5.38	"
14.10/5.38	"genericSplitAt2GenericSplitAt1 vzx vzy n x xs True = (x : genericSplitAt2Xs' vzx vzy,genericSplitAt2Xs'' vzx vzy);
14.10/5.38	genericSplitAt2GenericSplitAt1 vzx vzy n x xs False = genericSplitAt0 n (x : xs);
14.10/5.38	"
14.10/5.38	"genericSplitAt2Xs''0 vzx vzy (zz,xs'') = xs'';
14.10/5.38	"
14.10/5.38	"genericSplitAt2Xs'0 vzx vzy (xs',vuu) = xs';
14.10/5.38	"
14.10/5.38	
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(12)
14.10/5.38	Obligation:
14.10/5.38	mainModule Main 
14.10/5.38	module Maybe where {
14.10/5.38	import qualified List;
14.10/5.38	import qualified Main;
14.10/5.38	import qualified Prelude;
14.10/5.38	}
14.10/5.38	module List where {
14.10/5.38	import qualified Main;
14.10/5.38	import qualified Maybe;
14.10/5.38	import qualified Prelude;
14.10/5.38	genericSplitAt :: Integral a => a  ->  [b]  ->  ([b],[b]);
14.10/5.38	genericSplitAt vyv xs = genericSplitAt5 vyv xs;
14.10/5.38	genericSplitAt zy [] = genericSplitAt3 zy [];
14.10/5.38	genericSplitAt n (x : xs) = genericSplitAt2 n (x : xs);
14.10/5.38	genericSplitAt vuv vuw = genericSplitAt0 vuv vuw;
14.10/5.38	
14.10/5.38	genericSplitAt0 vuv vuw = error [];
14.10/5.38	
14.10/5.38	genericSplitAt2 n (x : xs) = genericSplitAt2GenericSplitAt1 n xs n x xs (n > 0);
14.10/5.38	genericSplitAt2 vxw vxx = genericSplitAt0 vxw vxx;
14.10/5.38	
14.10/5.38	genericSplitAt2GenericSplitAt1 vzx vzy n x xs True = (x : genericSplitAt2Xs' vzx vzy,genericSplitAt2Xs'' vzx vzy);
14.10/5.38	genericSplitAt2GenericSplitAt1 vzx vzy n x xs False = genericSplitAt0 n (x : xs);
14.10/5.38	
14.10/5.38	genericSplitAt2Vv9 vzx vzy = genericSplitAt (vzx - 1) vzy;
14.10/5.38	
14.10/5.38	genericSplitAt2Xs' vzx vzy = genericSplitAt2Xs'0 vzx vzy (genericSplitAt2Vv9 vzx vzy);
14.10/5.38	
14.10/5.38	genericSplitAt2Xs'' vzx vzy = genericSplitAt2Xs''0 vzx vzy (genericSplitAt2Vv9 vzx vzy);
14.10/5.38	
14.10/5.38	genericSplitAt2Xs''0 vzx vzy (zz,xs'') = xs'';
14.10/5.38	
14.10/5.38	genericSplitAt2Xs'0 vzx vzy (xs',vuu) = xs';
14.10/5.38	
14.10/5.38	genericSplitAt3 zy [] = ([],[]);
14.10/5.38	genericSplitAt3 vxz vyu = genericSplitAt2 vxz vyu;
14.10/5.38	
14.10/5.38	genericSplitAt4 True vyv xs = ([],xs);
14.10/5.38	genericSplitAt4 vyw vyx vyy = genericSplitAt3 vyx vyy;
14.10/5.38	
14.10/5.38	genericSplitAt5 vyv xs = genericSplitAt4 (vyv == 0) vyv xs;
14.10/5.38	genericSplitAt5 vyz vzu = genericSplitAt3 vyz vzu;
14.10/5.38	
14.10/5.38	}
14.10/5.38	module Main where {
14.10/5.38	import qualified List;
14.10/5.38	import qualified Maybe;
14.10/5.38	import qualified Prelude;
14.10/5.38	}
14.10/5.38	
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(13) NumRed (SOUND)
14.10/5.38	Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(14)
14.10/5.38	Obligation:
14.10/5.38	mainModule Main 
14.10/5.38	module Maybe where {
14.10/5.38	import qualified List;
14.10/5.38	import qualified Main;
14.10/5.38	import qualified Prelude;
14.10/5.38	}
14.10/5.38	module List where {
14.10/5.38	import qualified Main;
14.10/5.38	import qualified Maybe;
14.10/5.38	import qualified Prelude;
14.10/5.38	genericSplitAt :: Integral a => a  ->  [b]  ->  ([b],[b]);
14.10/5.38	genericSplitAt vyv xs = genericSplitAt5 vyv xs;
14.10/5.38	genericSplitAt zy [] = genericSplitAt3 zy [];
14.10/5.38	genericSplitAt n (x : xs) = genericSplitAt2 n (x : xs);
14.10/5.38	genericSplitAt vuv vuw = genericSplitAt0 vuv vuw;
14.10/5.38	
14.10/5.38	genericSplitAt0 vuv vuw = error [];
14.10/5.38	
14.10/5.38	genericSplitAt2 n (x : xs) = genericSplitAt2GenericSplitAt1 n xs n x xs (n > fromInt (Pos Zero));
14.10/5.38	genericSplitAt2 vxw vxx = genericSplitAt0 vxw vxx;
14.10/5.38	
14.10/5.38	genericSplitAt2GenericSplitAt1 vzx vzy n x xs True = (x : genericSplitAt2Xs' vzx vzy,genericSplitAt2Xs'' vzx vzy);
14.10/5.38	genericSplitAt2GenericSplitAt1 vzx vzy n x xs False = genericSplitAt0 n (x : xs);
14.10/5.38	
14.10/5.38	genericSplitAt2Vv9 vzx vzy = genericSplitAt (vzx - fromInt (Pos (Succ Zero))) vzy;
14.10/5.38	
14.10/5.38	genericSplitAt2Xs' vzx vzy = genericSplitAt2Xs'0 vzx vzy (genericSplitAt2Vv9 vzx vzy);
14.10/5.38	
14.10/5.38	genericSplitAt2Xs'' vzx vzy = genericSplitAt2Xs''0 vzx vzy (genericSplitAt2Vv9 vzx vzy);
14.10/5.38	
14.10/5.38	genericSplitAt2Xs''0 vzx vzy (zz,xs'') = xs'';
14.10/5.38	
14.10/5.38	genericSplitAt2Xs'0 vzx vzy (xs',vuu) = xs';
14.10/5.38	
14.10/5.38	genericSplitAt3 zy [] = ([],[]);
14.10/5.38	genericSplitAt3 vxz vyu = genericSplitAt2 vxz vyu;
14.10/5.38	
14.10/5.38	genericSplitAt4 True vyv xs = ([],xs);
14.10/5.38	genericSplitAt4 vyw vyx vyy = genericSplitAt3 vyx vyy;
14.10/5.38	
14.10/5.38	genericSplitAt5 vyv xs = genericSplitAt4 (vyv == fromInt (Pos Zero)) vyv xs;
14.10/5.38	genericSplitAt5 vyz vzu = genericSplitAt3 vyz vzu;
14.10/5.38	
14.10/5.38	}
14.10/5.38	module Main where {
14.10/5.38	import qualified List;
14.10/5.38	import qualified Maybe;
14.10/5.38	import qualified Prelude;
14.10/5.38	}
14.10/5.38	
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(15) Narrow (SOUND)
14.10/5.38	Haskell To QDPs
14.10/5.38	
14.10/5.38	digraph dp_graph {
14.10/5.38	node [outthreshold=100, inthreshold=100];1[label="List.genericSplitAt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
14.10/5.38	3[label="List.genericSplitAt vzz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
14.10/5.38	4[label="List.genericSplitAt vzz3 vzz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
14.10/5.38	5[label="List.genericSplitAt5 vzz3 vzz4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
14.10/5.38	6[label="List.genericSplitAt4 (vzz3 == fromInt (Pos Zero)) vzz3 vzz4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
14.10/5.38	7[label="List.genericSplitAt4 (primEqInt vzz3 (fromInt (Pos Zero))) vzz3 vzz4",fontsize=16,color="burlywood",shape="box"];78[label="vzz3/Pos vzz30",fontsize=10,color="white",style="solid",shape="box"];7 -> 78[label="",style="solid", color="burlywood", weight=9];
14.10/5.38	78 -> 8[label="",style="solid", color="burlywood", weight=3];
14.10/5.38	79[label="vzz3/Neg vzz30",fontsize=10,color="white",style="solid",shape="box"];7 -> 79[label="",style="solid", color="burlywood", weight=9];
14.10/5.38	79 -> 9[label="",style="solid", color="burlywood", weight=3];
14.10/5.38	8[label="List.genericSplitAt4 (primEqInt (Pos vzz30) (fromInt (Pos Zero))) (Pos vzz30) vzz4",fontsize=16,color="burlywood",shape="box"];80[label="vzz30/Succ vzz300",fontsize=10,color="white",style="solid",shape="box"];8 -> 80[label="",style="solid", color="burlywood", weight=9];
14.10/5.38	80 -> 10[label="",style="solid", color="burlywood", weight=3];
14.10/5.38	81[label="vzz30/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 81[label="",style="solid", color="burlywood", weight=9];
14.10/5.38	81 -> 11[label="",style="solid", color="burlywood", weight=3];
14.10/5.38	9[label="List.genericSplitAt4 (primEqInt (Neg vzz30) (fromInt (Pos Zero))) (Neg vzz30) vzz4",fontsize=16,color="burlywood",shape="box"];82[label="vzz30/Succ vzz300",fontsize=10,color="white",style="solid",shape="box"];9 -> 82[label="",style="solid", color="burlywood", weight=9];
14.10/5.38	82 -> 12[label="",style="solid", color="burlywood", weight=3];
14.10/5.38	83[label="vzz30/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 83[label="",style="solid", color="burlywood", weight=9];
14.10/5.38	83 -> 13[label="",style="solid", color="burlywood", weight=3];
14.10/5.38	10[label="List.genericSplitAt4 (primEqInt (Pos (Succ vzz300)) (fromInt (Pos Zero))) (Pos (Succ vzz300)) vzz4",fontsize=16,color="black",shape="box"];10 -> 14[label="",style="solid", color="black", weight=3];
14.10/5.38	11[label="List.genericSplitAt4 (primEqInt (Pos Zero) (fromInt (Pos Zero))) (Pos Zero) vzz4",fontsize=16,color="black",shape="box"];11 -> 15[label="",style="solid", color="black", weight=3];
14.10/5.38	12[label="List.genericSplitAt4 (primEqInt (Neg (Succ vzz300)) (fromInt (Pos Zero))) (Neg (Succ vzz300)) vzz4",fontsize=16,color="black",shape="box"];12 -> 16[label="",style="solid", color="black", weight=3];
14.10/5.38	13[label="List.genericSplitAt4 (primEqInt (Neg Zero) (fromInt (Pos Zero))) (Neg Zero) vzz4",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3];
14.10/5.38	14[label="List.genericSplitAt4 (primEqInt (Pos (Succ vzz300)) (Pos Zero)) (Pos (Succ vzz300)) vzz4",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3];
14.10/5.38	15[label="List.genericSplitAt4 (primEqInt (Pos Zero) (Pos Zero)) (Pos Zero) vzz4",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
14.10/5.38	16[label="List.genericSplitAt4 (primEqInt (Neg (Succ vzz300)) (Pos Zero)) (Neg (Succ vzz300)) vzz4",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
14.10/5.38	17[label="List.genericSplitAt4 (primEqInt (Neg Zero) (Pos Zero)) (Neg Zero) vzz4",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
14.10/5.38	18[label="List.genericSplitAt4 False (Pos (Succ vzz300)) vzz4",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
14.10/5.38	19[label="List.genericSplitAt4 True (Pos Zero) vzz4",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
14.10/5.38	20[label="List.genericSplitAt4 False (Neg (Succ vzz300)) vzz4",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
14.10/5.38	21[label="List.genericSplitAt4 True (Neg Zero) vzz4",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
14.10/5.38	22[label="List.genericSplitAt3 (Pos (Succ vzz300)) vzz4",fontsize=16,color="burlywood",shape="box"];84[label="vzz4/vzz40 : vzz41",fontsize=10,color="white",style="solid",shape="box"];22 -> 84[label="",style="solid", color="burlywood", weight=9];
14.10/5.38	84 -> 26[label="",style="solid", color="burlywood", weight=3];
14.10/5.38	85[label="vzz4/[]",fontsize=10,color="white",style="solid",shape="box"];22 -> 85[label="",style="solid", color="burlywood", weight=9];
14.10/5.38	85 -> 27[label="",style="solid", color="burlywood", weight=3];
14.10/5.38	23[label="([],vzz4)",fontsize=16,color="green",shape="box"];24[label="List.genericSplitAt3 (Neg (Succ vzz300)) vzz4",fontsize=16,color="burlywood",shape="box"];86[label="vzz4/vzz40 : vzz41",fontsize=10,color="white",style="solid",shape="box"];24 -> 86[label="",style="solid", color="burlywood", weight=9];
14.10/5.38	86 -> 28[label="",style="solid", color="burlywood", weight=3];
14.10/5.38	87[label="vzz4/[]",fontsize=10,color="white",style="solid",shape="box"];24 -> 87[label="",style="solid", color="burlywood", weight=9];
14.10/5.38	87 -> 29[label="",style="solid", color="burlywood", weight=3];
14.10/5.38	25[label="([],vzz4)",fontsize=16,color="green",shape="box"];26[label="List.genericSplitAt3 (Pos (Succ vzz300)) (vzz40 : vzz41)",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
14.10/5.38	27[label="List.genericSplitAt3 (Pos (Succ vzz300)) []",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
14.10/5.38	28[label="List.genericSplitAt3 (Neg (Succ vzz300)) (vzz40 : vzz41)",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
14.10/5.38	29[label="List.genericSplitAt3 (Neg (Succ vzz300)) []",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
14.10/5.38	30[label="List.genericSplitAt2 (Pos (Succ vzz300)) (vzz40 : vzz41)",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3];
14.10/5.38	31[label="([],[])",fontsize=16,color="green",shape="box"];32[label="List.genericSplitAt2 (Neg (Succ vzz300)) (vzz40 : vzz41)",fontsize=16,color="black",shape="box"];32 -> 35[label="",style="solid", color="black", weight=3];
14.10/5.38	33[label="([],[])",fontsize=16,color="green",shape="box"];34[label="List.genericSplitAt2GenericSplitAt1 (Pos (Succ vzz300)) vzz41 (Pos (Succ vzz300)) vzz40 vzz41 (Pos (Succ vzz300) > fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3];
14.10/5.38	35[label="List.genericSplitAt2GenericSplitAt1 (Neg (Succ vzz300)) vzz41 (Neg (Succ vzz300)) vzz40 vzz41 (Neg (Succ vzz300) > fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
14.10/5.38	36[label="List.genericSplitAt2GenericSplitAt1 (Pos (Succ vzz300)) vzz41 (Pos (Succ vzz300)) vzz40 vzz41 (compare (Pos (Succ vzz300)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];36 -> 38[label="",style="solid", color="black", weight=3];
14.10/5.38	37[label="List.genericSplitAt2GenericSplitAt1 (Neg (Succ vzz300)) vzz41 (Neg (Succ vzz300)) vzz40 vzz41 (compare (Neg (Succ vzz300)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];37 -> 39[label="",style="solid", color="black", weight=3];
14.10/5.38	38[label="List.genericSplitAt2GenericSplitAt1 (Pos (Succ vzz300)) vzz41 (Pos (Succ vzz300)) vzz40 vzz41 (primCmpInt (Pos (Succ vzz300)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];38 -> 40[label="",style="solid", color="black", weight=3];
14.10/5.38	39[label="List.genericSplitAt2GenericSplitAt1 (Neg (Succ vzz300)) vzz41 (Neg (Succ vzz300)) vzz40 vzz41 (primCmpInt (Neg (Succ vzz300)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];39 -> 41[label="",style="solid", color="black", weight=3];
14.10/5.38	40[label="List.genericSplitAt2GenericSplitAt1 (Pos (Succ vzz300)) vzz41 (Pos (Succ vzz300)) vzz40 vzz41 (primCmpInt (Pos (Succ vzz300)) (Pos Zero) == GT)",fontsize=16,color="black",shape="box"];40 -> 42[label="",style="solid", color="black", weight=3];
14.10/5.38	41[label="List.genericSplitAt2GenericSplitAt1 (Neg (Succ vzz300)) vzz41 (Neg (Succ vzz300)) vzz40 vzz41 (primCmpInt (Neg (Succ vzz300)) (Pos Zero) == GT)",fontsize=16,color="black",shape="box"];41 -> 43[label="",style="solid", color="black", weight=3];
14.10/5.38	42[label="List.genericSplitAt2GenericSplitAt1 (Pos (Succ vzz300)) vzz41 (Pos (Succ vzz300)) vzz40 vzz41 (primCmpNat (Succ vzz300) Zero == GT)",fontsize=16,color="black",shape="box"];42 -> 44[label="",style="solid", color="black", weight=3];
14.10/5.38	43[label="List.genericSplitAt2GenericSplitAt1 (Neg (Succ vzz300)) vzz41 (Neg (Succ vzz300)) vzz40 vzz41 (LT == GT)",fontsize=16,color="black",shape="box"];43 -> 45[label="",style="solid", color="black", weight=3];
14.10/5.38	44[label="List.genericSplitAt2GenericSplitAt1 (Pos (Succ vzz300)) vzz41 (Pos (Succ vzz300)) vzz40 vzz41 (GT == GT)",fontsize=16,color="black",shape="box"];44 -> 46[label="",style="solid", color="black", weight=3];
14.10/5.38	45[label="List.genericSplitAt2GenericSplitAt1 (Neg (Succ vzz300)) vzz41 (Neg (Succ vzz300)) vzz40 vzz41 False",fontsize=16,color="black",shape="box"];45 -> 47[label="",style="solid", color="black", weight=3];
14.10/5.38	46[label="List.genericSplitAt2GenericSplitAt1 (Pos (Succ vzz300)) vzz41 (Pos (Succ vzz300)) vzz40 vzz41 True",fontsize=16,color="black",shape="box"];46 -> 48[label="",style="solid", color="black", weight=3];
14.10/5.38	47[label="List.genericSplitAt0 (Neg (Succ vzz300)) (vzz40 : vzz41)",fontsize=16,color="black",shape="box"];47 -> 49[label="",style="solid", color="black", weight=3];
14.10/5.38	48[label="(vzz40 : List.genericSplitAt2Xs' (Pos (Succ vzz300)) vzz41,List.genericSplitAt2Xs'' (Pos (Succ vzz300)) vzz41)",fontsize=16,color="green",shape="box"];48 -> 50[label="",style="dashed", color="green", weight=3];
14.10/5.38	48 -> 51[label="",style="dashed", color="green", weight=3];
14.10/5.38	49[label="error []",fontsize=16,color="black",shape="box"];49 -> 52[label="",style="solid", color="black", weight=3];
14.10/5.38	50[label="List.genericSplitAt2Xs' (Pos (Succ vzz300)) vzz41",fontsize=16,color="black",shape="box"];50 -> 53[label="",style="solid", color="black", weight=3];
14.10/5.38	51[label="List.genericSplitAt2Xs'' (Pos (Succ vzz300)) vzz41",fontsize=16,color="black",shape="box"];51 -> 54[label="",style="solid", color="black", weight=3];
14.10/5.38	52[label="error []",fontsize=16,color="red",shape="box"];53 -> 57[label="",style="dashed", color="red", weight=0];
14.10/5.38	53[label="List.genericSplitAt2Xs'0 (Pos (Succ vzz300)) vzz41 (List.genericSplitAt2Vv9 (Pos (Succ vzz300)) vzz41)",fontsize=16,color="magenta"];53 -> 58[label="",style="dashed", color="magenta", weight=3];
14.10/5.38	54 -> 62[label="",style="dashed", color="red", weight=0];
14.10/5.38	54[label="List.genericSplitAt2Xs''0 (Pos (Succ vzz300)) vzz41 (List.genericSplitAt2Vv9 (Pos (Succ vzz300)) vzz41)",fontsize=16,color="magenta"];54 -> 63[label="",style="dashed", color="magenta", weight=3];
14.10/5.38	58[label="List.genericSplitAt2Vv9 (Pos (Succ vzz300)) vzz41",fontsize=16,color="black",shape="triangle"];58 -> 60[label="",style="solid", color="black", weight=3];
14.10/5.38	57[label="List.genericSplitAt2Xs'0 (Pos (Succ vzz300)) vzz41 vzz5",fontsize=16,color="burlywood",shape="triangle"];88[label="vzz5/(vzz50,vzz51)",fontsize=10,color="white",style="solid",shape="box"];57 -> 88[label="",style="solid", color="burlywood", weight=9];
14.10/5.38	88 -> 61[label="",style="solid", color="burlywood", weight=3];
14.10/5.38	63 -> 58[label="",style="dashed", color="red", weight=0];
14.10/5.38	63[label="List.genericSplitAt2Vv9 (Pos (Succ vzz300)) vzz41",fontsize=16,color="magenta"];62[label="List.genericSplitAt2Xs''0 (Pos (Succ vzz300)) vzz41 vzz6",fontsize=16,color="burlywood",shape="triangle"];89[label="vzz6/(vzz60,vzz61)",fontsize=10,color="white",style="solid",shape="box"];62 -> 89[label="",style="solid", color="burlywood", weight=9];
14.10/5.38	89 -> 65[label="",style="solid", color="burlywood", weight=3];
14.10/5.38	60 -> 4[label="",style="dashed", color="red", weight=0];
14.10/5.38	60[label="List.genericSplitAt (Pos (Succ vzz300) - fromInt (Pos (Succ Zero))) vzz41",fontsize=16,color="magenta"];60 -> 66[label="",style="dashed", color="magenta", weight=3];
14.10/5.38	60 -> 67[label="",style="dashed", color="magenta", weight=3];
14.10/5.38	61[label="List.genericSplitAt2Xs'0 (Pos (Succ vzz300)) vzz41 (vzz50,vzz51)",fontsize=16,color="black",shape="box"];61 -> 68[label="",style="solid", color="black", weight=3];
14.10/5.38	65[label="List.genericSplitAt2Xs''0 (Pos (Succ vzz300)) vzz41 (vzz60,vzz61)",fontsize=16,color="black",shape="box"];65 -> 69[label="",style="solid", color="black", weight=3];
14.10/5.38	66[label="vzz41",fontsize=16,color="green",shape="box"];67[label="Pos (Succ vzz300) - fromInt (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];67 -> 70[label="",style="solid", color="black", weight=3];
14.10/5.38	68[label="vzz50",fontsize=16,color="green",shape="box"];69[label="vzz61",fontsize=16,color="green",shape="box"];70[label="primMinusInt (Pos (Succ vzz300)) (fromInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];70 -> 71[label="",style="solid", color="black", weight=3];
14.10/5.38	71[label="primMinusInt (Pos (Succ vzz300)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];71 -> 72[label="",style="solid", color="black", weight=3];
14.10/5.38	72[label="primMinusNat (Succ vzz300) (Succ Zero)",fontsize=16,color="black",shape="box"];72 -> 73[label="",style="solid", color="black", weight=3];
14.10/5.38	73[label="primMinusNat vzz300 Zero",fontsize=16,color="burlywood",shape="box"];90[label="vzz300/Succ vzz3000",fontsize=10,color="white",style="solid",shape="box"];73 -> 90[label="",style="solid", color="burlywood", weight=9];
14.10/5.38	90 -> 74[label="",style="solid", color="burlywood", weight=3];
14.10/5.38	91[label="vzz300/Zero",fontsize=10,color="white",style="solid",shape="box"];73 -> 91[label="",style="solid", color="burlywood", weight=9];
14.10/5.38	91 -> 75[label="",style="solid", color="burlywood", weight=3];
14.10/5.38	74[label="primMinusNat (Succ vzz3000) Zero",fontsize=16,color="black",shape="box"];74 -> 76[label="",style="solid", color="black", weight=3];
14.10/5.38	75[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];75 -> 77[label="",style="solid", color="black", weight=3];
14.10/5.38	76[label="Pos (Succ vzz3000)",fontsize=16,color="green",shape="box"];77[label="Pos Zero",fontsize=16,color="green",shape="box"];}
14.10/5.38	
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(16)
14.10/5.38	Obligation:
14.10/5.38	Q DP problem:
14.10/5.38	The TRS P consists of the following rules:
14.10/5.38	
14.10/5.38	   new_genericSplitAt2Vv9(vzz300, vzz41, ba) -> new_genericSplitAt(new_primMinusNat(vzz300), vzz41, ba)
14.10/5.38	   new_genericSplitAt(Pos(Succ(vzz300)), :(vzz40, vzz41), ba) -> new_genericSplitAt2Vv9(vzz300, vzz41, ba)
14.10/5.38	   new_genericSplitAt(Pos(Succ(vzz300)), :(vzz40, vzz41), ba) -> new_genericSplitAt(new_primMinusNat(vzz300), vzz41, ba)
14.10/5.38	
14.10/5.38	The TRS R consists of the following rules:
14.10/5.38	
14.10/5.38	   new_primMinusNat(Succ(vzz3000)) -> Pos(Succ(vzz3000))
14.10/5.38	   new_primMinusNat(Zero) -> Pos(Zero)
14.10/5.38	
14.10/5.38	The set Q consists of the following terms:
14.10/5.38	
14.10/5.38	   new_primMinusNat(Succ(x0))
14.10/5.38	   new_primMinusNat(Zero)
14.10/5.38	
14.10/5.38	We have to consider all minimal (P,Q,R)-chains.
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(17) TransformationProof (EQUIVALENT)
14.10/5.38	By narrowing [LPAR04] the rule new_genericSplitAt2Vv9(vzz300, vzz41, ba) -> new_genericSplitAt(new_primMinusNat(vzz300), vzz41, ba) at position [0] we obtained the following new rules [LPAR04]:
14.10/5.38	
14.10/5.38	   (new_genericSplitAt2Vv9(Succ(x0), y1, y2) -> new_genericSplitAt(Pos(Succ(x0)), y1, y2),new_genericSplitAt2Vv9(Succ(x0), y1, y2) -> new_genericSplitAt(Pos(Succ(x0)), y1, y2))
14.10/5.38	   (new_genericSplitAt2Vv9(Zero, y1, y2) -> new_genericSplitAt(Pos(Zero), y1, y2),new_genericSplitAt2Vv9(Zero, y1, y2) -> new_genericSplitAt(Pos(Zero), y1, y2))
14.10/5.38	
14.10/5.38	
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(18)
14.10/5.38	Obligation:
14.10/5.38	Q DP problem:
14.10/5.38	The TRS P consists of the following rules:
14.10/5.38	
14.10/5.38	   new_genericSplitAt(Pos(Succ(vzz300)), :(vzz40, vzz41), ba) -> new_genericSplitAt2Vv9(vzz300, vzz41, ba)
14.10/5.38	   new_genericSplitAt(Pos(Succ(vzz300)), :(vzz40, vzz41), ba) -> new_genericSplitAt(new_primMinusNat(vzz300), vzz41, ba)
14.10/5.38	   new_genericSplitAt2Vv9(Succ(x0), y1, y2) -> new_genericSplitAt(Pos(Succ(x0)), y1, y2)
14.10/5.38	   new_genericSplitAt2Vv9(Zero, y1, y2) -> new_genericSplitAt(Pos(Zero), y1, y2)
14.10/5.38	
14.10/5.38	The TRS R consists of the following rules:
14.10/5.38	
14.10/5.38	   new_primMinusNat(Succ(vzz3000)) -> Pos(Succ(vzz3000))
14.10/5.38	   new_primMinusNat(Zero) -> Pos(Zero)
14.10/5.38	
14.10/5.38	The set Q consists of the following terms:
14.10/5.38	
14.10/5.38	   new_primMinusNat(Succ(x0))
14.10/5.38	   new_primMinusNat(Zero)
14.10/5.38	
14.10/5.38	We have to consider all minimal (P,Q,R)-chains.
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(19) DependencyGraphProof (EQUIVALENT)
14.10/5.38	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(20)
14.10/5.38	Obligation:
14.10/5.38	Q DP problem:
14.10/5.38	The TRS P consists of the following rules:
14.10/5.38	
14.10/5.38	   new_genericSplitAt2Vv9(Succ(x0), y1, y2) -> new_genericSplitAt(Pos(Succ(x0)), y1, y2)
14.10/5.38	   new_genericSplitAt(Pos(Succ(vzz300)), :(vzz40, vzz41), ba) -> new_genericSplitAt2Vv9(vzz300, vzz41, ba)
14.10/5.38	   new_genericSplitAt(Pos(Succ(vzz300)), :(vzz40, vzz41), ba) -> new_genericSplitAt(new_primMinusNat(vzz300), vzz41, ba)
14.10/5.38	
14.10/5.38	The TRS R consists of the following rules:
14.10/5.38	
14.10/5.38	   new_primMinusNat(Succ(vzz3000)) -> Pos(Succ(vzz3000))
14.10/5.38	   new_primMinusNat(Zero) -> Pos(Zero)
14.10/5.38	
14.10/5.38	The set Q consists of the following terms:
14.10/5.38	
14.10/5.38	   new_primMinusNat(Succ(x0))
14.10/5.38	   new_primMinusNat(Zero)
14.10/5.38	
14.10/5.38	We have to consider all minimal (P,Q,R)-chains.
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(21) TransformationProof (EQUIVALENT)
14.10/5.38	By narrowing [LPAR04] the rule new_genericSplitAt(Pos(Succ(vzz300)), :(vzz40, vzz41), ba) -> new_genericSplitAt(new_primMinusNat(vzz300), vzz41, ba) at position [0] we obtained the following new rules [LPAR04]:
14.10/5.38	
14.10/5.38	   (new_genericSplitAt(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_genericSplitAt(Pos(Succ(x0)), y2, y3),new_genericSplitAt(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_genericSplitAt(Pos(Succ(x0)), y2, y3))
14.10/5.38	   (new_genericSplitAt(Pos(Succ(Zero)), :(y1, y2), y3) -> new_genericSplitAt(Pos(Zero), y2, y3),new_genericSplitAt(Pos(Succ(Zero)), :(y1, y2), y3) -> new_genericSplitAt(Pos(Zero), y2, y3))
14.10/5.38	
14.10/5.38	
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(22)
14.10/5.38	Obligation:
14.10/5.38	Q DP problem:
14.10/5.38	The TRS P consists of the following rules:
14.10/5.38	
14.10/5.38	   new_genericSplitAt2Vv9(Succ(x0), y1, y2) -> new_genericSplitAt(Pos(Succ(x0)), y1, y2)
14.10/5.38	   new_genericSplitAt(Pos(Succ(vzz300)), :(vzz40, vzz41), ba) -> new_genericSplitAt2Vv9(vzz300, vzz41, ba)
14.10/5.38	   new_genericSplitAt(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_genericSplitAt(Pos(Succ(x0)), y2, y3)
14.10/5.38	   new_genericSplitAt(Pos(Succ(Zero)), :(y1, y2), y3) -> new_genericSplitAt(Pos(Zero), y2, y3)
14.10/5.38	
14.10/5.38	The TRS R consists of the following rules:
14.10/5.38	
14.10/5.38	   new_primMinusNat(Succ(vzz3000)) -> Pos(Succ(vzz3000))
14.10/5.38	   new_primMinusNat(Zero) -> Pos(Zero)
14.10/5.38	
14.10/5.38	The set Q consists of the following terms:
14.10/5.38	
14.10/5.38	   new_primMinusNat(Succ(x0))
14.10/5.38	   new_primMinusNat(Zero)
14.10/5.38	
14.10/5.38	We have to consider all minimal (P,Q,R)-chains.
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(23) DependencyGraphProof (EQUIVALENT)
14.10/5.38	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(24)
14.10/5.38	Obligation:
14.10/5.38	Q DP problem:
14.10/5.38	The TRS P consists of the following rules:
14.10/5.38	
14.10/5.38	   new_genericSplitAt(Pos(Succ(vzz300)), :(vzz40, vzz41), ba) -> new_genericSplitAt2Vv9(vzz300, vzz41, ba)
14.10/5.38	   new_genericSplitAt2Vv9(Succ(x0), y1, y2) -> new_genericSplitAt(Pos(Succ(x0)), y1, y2)
14.10/5.38	   new_genericSplitAt(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_genericSplitAt(Pos(Succ(x0)), y2, y3)
14.10/5.38	
14.10/5.38	The TRS R consists of the following rules:
14.10/5.38	
14.10/5.38	   new_primMinusNat(Succ(vzz3000)) -> Pos(Succ(vzz3000))
14.10/5.38	   new_primMinusNat(Zero) -> Pos(Zero)
14.10/5.38	
14.10/5.38	The set Q consists of the following terms:
14.10/5.38	
14.10/5.38	   new_primMinusNat(Succ(x0))
14.10/5.38	   new_primMinusNat(Zero)
14.10/5.38	
14.10/5.38	We have to consider all minimal (P,Q,R)-chains.
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(25) UsableRulesProof (EQUIVALENT)
14.10/5.38	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(26)
14.10/5.38	Obligation:
14.10/5.38	Q DP problem:
14.10/5.38	The TRS P consists of the following rules:
14.10/5.38	
14.10/5.38	   new_genericSplitAt(Pos(Succ(vzz300)), :(vzz40, vzz41), ba) -> new_genericSplitAt2Vv9(vzz300, vzz41, ba)
14.10/5.38	   new_genericSplitAt2Vv9(Succ(x0), y1, y2) -> new_genericSplitAt(Pos(Succ(x0)), y1, y2)
14.10/5.38	   new_genericSplitAt(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_genericSplitAt(Pos(Succ(x0)), y2, y3)
14.10/5.38	
14.10/5.38	R is empty.
14.10/5.38	The set Q consists of the following terms:
14.10/5.38	
14.10/5.38	   new_primMinusNat(Succ(x0))
14.10/5.38	   new_primMinusNat(Zero)
14.10/5.38	
14.10/5.38	We have to consider all minimal (P,Q,R)-chains.
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(27) QReductionProof (EQUIVALENT)
14.10/5.38	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
14.10/5.38	
14.10/5.38	   new_primMinusNat(Succ(x0))
14.10/5.38	   new_primMinusNat(Zero)
14.10/5.38	
14.10/5.38	
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(28)
14.10/5.38	Obligation:
14.10/5.38	Q DP problem:
14.10/5.38	The TRS P consists of the following rules:
14.10/5.38	
14.10/5.38	   new_genericSplitAt(Pos(Succ(vzz300)), :(vzz40, vzz41), ba) -> new_genericSplitAt2Vv9(vzz300, vzz41, ba)
14.10/5.38	   new_genericSplitAt2Vv9(Succ(x0), y1, y2) -> new_genericSplitAt(Pos(Succ(x0)), y1, y2)
14.10/5.38	   new_genericSplitAt(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_genericSplitAt(Pos(Succ(x0)), y2, y3)
14.10/5.38	
14.10/5.38	R is empty.
14.10/5.38	Q is empty.
14.10/5.38	We have to consider all minimal (P,Q,R)-chains.
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(29) QDPSizeChangeProof (EQUIVALENT)
14.10/5.38	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. 
14.10/5.38	
14.10/5.38	From the DPs we obtained the following set of size-change graphs:
14.10/5.38	*new_genericSplitAt2Vv9(Succ(x0), y1, y2) -> new_genericSplitAt(Pos(Succ(x0)), y1, y2)
14.10/5.38	The graph contains the following edges 2 >= 2, 3 >= 3
14.10/5.38	
14.10/5.38	
14.10/5.38	*new_genericSplitAt(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_genericSplitAt(Pos(Succ(x0)), y2, y3)
14.10/5.38	The graph contains the following edges 2 > 2, 3 >= 3
14.10/5.38	
14.10/5.38	
14.10/5.38	*new_genericSplitAt(Pos(Succ(vzz300)), :(vzz40, vzz41), ba) -> new_genericSplitAt2Vv9(vzz300, vzz41, ba)
14.10/5.38	The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
14.10/5.38	
14.10/5.38	
14.10/5.38	----------------------------------------
14.10/5.38	
14.10/5.38	(30)
14.10/5.38	YES
14.22/5.42	EOF