/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.hs /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


H-Termination with start terms of the given HASKELL could be proven:

(0) HASKELL
(1) CR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 24 ms]
(6) HASKELL
(7) Narrow [SOUND, 0 ms]
(8) AND
    (9) QDP
        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (11) YES
    (12) QDP
        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (14) YES
    (15) QDP
        (16) DependencyGraphProof [EQUIVALENT, 0 ms]
        (17) QDP
        (18) TransformationProof [EQUIVALENT, 0 ms]
        (19) QDP
        (20) DependencyGraphProof [EQUIVALENT, 0 ms]
        (21) QDP
        (22) UsableRulesProof [EQUIVALENT, 0 ms]
        (23) QDP
        (24) QReductionProof [EQUIVALENT, 0 ms]
        (25) QDP
        (26) TransformationProof [EQUIVALENT, 0 ms]
        (27) QDP
        (28) UsableRulesProof [EQUIVALENT, 0 ms]
        (29) QDP
        (30) QReductionProof [EQUIVALENT, 0 ms]
        (31) QDP
        (32) TransformationProof [EQUIVALENT, 0 ms]
        (33) QDP
        (34) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (35) YES
    (36) QDP
        (37) DependencyGraphProof [EQUIVALENT, 0 ms]
        (38) QDP
        (39) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (40) YES


----------------------------------------

(0)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
merge :: (a  ->  a  ->  Ordering)  ->  [a]  ->  [a]  ->  [a];
merge cmp xs [] = xs;
merge cmp [] ys = ys;
merge cmp (x : xs) (y : ys) = case x `cmp` y of { 
GT-> y : merge cmp (x : xs) ys; 
_-> x : merge cmp xs (y : ys); 
 } ;

merge_pairs :: (a  ->  a  ->  Ordering)  ->  [[a]]  ->  [[a]];
merge_pairs cmp [] = [];
merge_pairs cmp (xs : []) = xs : [];
merge_pairs cmp (xs : ys : xss) = merge cmp xs ys : merge_pairs cmp xss;

mergesort :: (a  ->  a  ->  Ordering)  ->  [a]  ->  [a];
mergesort cmp = mergesort' cmp . map wrap;

mergesort' :: (a  ->  a  ->  Ordering)  ->  [[a]]  ->  [a];
mergesort' cmp [] = [];
mergesort' cmp (xs : []) = xs;
mergesort' cmp xss = mergesort' cmp (merge_pairs cmp xss);

sort :: Ord a => [a]  ->  [a];
sort l = mergesort compare l;

wrap :: a  ->  [a];
wrap x = x : [];

}
module Main where {
import qualified List;
import qualified Maybe;
import qualified Prelude;
}

----------------------------------------

(1) CR (EQUIVALENT)
Case Reductions:
The following Case expression
"case cmp x y of {
GT  -> y : merge cmp (x : xs) ys;
_  -> x : merge cmp xs (y : ys)}
"
is transformed to
"merge0 y cmp x xs ys GT = y : merge cmp (x : xs) ys;
merge0 y cmp x xs ys _ = x : merge cmp xs (y : ys);
"

----------------------------------------

(2)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
merge :: (a  ->  a  ->  Ordering)  ->  [a]  ->  [a]  ->  [a];
merge cmp xs [] = xs;
merge cmp [] ys = ys;
merge cmp (x : xs) (y : ys) = merge0 y cmp x xs ys (x `cmp` y);

merge0 y cmp x xs ys GT = y : merge cmp (x : xs) ys;
merge0 y cmp x xs ys _ = x : merge cmp xs (y : ys);

merge_pairs :: (a  ->  a  ->  Ordering)  ->  [[a]]  ->  [[a]];
merge_pairs cmp [] = [];
merge_pairs cmp (xs : []) = xs : [];
merge_pairs cmp (xs : ys : xss) = merge cmp xs ys : merge_pairs cmp xss;

mergesort :: (a  ->  a  ->  Ordering)  ->  [a]  ->  [a];
mergesort cmp = mergesort' cmp . map wrap;

mergesort' :: (a  ->  a  ->  Ordering)  ->  [[a]]  ->  [a];
mergesort' cmp [] = [];
mergesort' cmp (xs : []) = xs;
mergesort' cmp xss = mergesort' cmp (merge_pairs cmp xss);

sort :: Ord a => [a]  ->  [a];
sort l = mergesort compare l;

wrap :: a  ->  [a];
wrap x = x : [];

}
module Main where {
import qualified List;
import qualified Maybe;
import qualified Prelude;
}

----------------------------------------

(3) BR (EQUIVALENT)
Replaced joker patterns by fresh variables and removed binding patterns.
----------------------------------------

(4)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
merge :: (a  ->  a  ->  Ordering)  ->  [a]  ->  [a]  ->  [a];
merge cmp xs [] = xs;
merge cmp [] ys = ys;
merge cmp (x : xs) (y : ys) = merge0 y cmp x xs ys (x `cmp` y);

merge0 y cmp x xs ys GT = y : merge cmp (x : xs) ys;
merge0 y cmp x xs ys vy = x : merge cmp xs (y : ys);

merge_pairs :: (a  ->  a  ->  Ordering)  ->  [[a]]  ->  [[a]];
merge_pairs cmp [] = [];
merge_pairs cmp (xs : []) = xs : [];
merge_pairs cmp (xs : ys : xss) = merge cmp xs ys : merge_pairs cmp xss;

mergesort :: (a  ->  a  ->  Ordering)  ->  [a]  ->  [a];
mergesort cmp = mergesort' cmp . map wrap;

mergesort' :: (a  ->  a  ->  Ordering)  ->  [[a]]  ->  [a];
mergesort' cmp [] = [];
mergesort' cmp (xs : []) = xs;
mergesort' cmp xss = mergesort' cmp (merge_pairs cmp xss);

sort :: Ord a => [a]  ->  [a];
sort l = mergesort compare l;

wrap :: a  ->  [a];
wrap x = x : [];

}
module Main where {
import qualified List;
import qualified Maybe;
import qualified Prelude;
}

----------------------------------------

(5) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"

----------------------------------------

(6)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
merge :: (a  ->  a  ->  Ordering)  ->  [a]  ->  [a]  ->  [a];
merge cmp xs [] = xs;
merge cmp [] ys = ys;
merge cmp (x : xs) (y : ys) = merge0 y cmp x xs ys (x `cmp` y);

merge0 y cmp x xs ys GT = y : merge cmp (x : xs) ys;
merge0 y cmp x xs ys vy = x : merge cmp xs (y : ys);

merge_pairs :: (a  ->  a  ->  Ordering)  ->  [[a]]  ->  [[a]];
merge_pairs cmp [] = [];
merge_pairs cmp (xs : []) = xs : [];
merge_pairs cmp (xs : ys : xss) = merge cmp xs ys : merge_pairs cmp xss;

mergesort :: (a  ->  a  ->  Ordering)  ->  [a]  ->  [a];
mergesort cmp = mergesort' cmp . map wrap;

mergesort' :: (a  ->  a  ->  Ordering)  ->  [[a]]  ->  [a];
mergesort' cmp [] = [];
mergesort' cmp (xs : []) = xs;
mergesort' cmp xss = mergesort' cmp (merge_pairs cmp xss);

sort :: Ord a => [a]  ->  [a];
sort l = mergesort compare l;

wrap :: a  ->  [a];
wrap x = x : [];

}
module Main where {
import qualified List;
import qualified Maybe;
import qualified Prelude;
}

----------------------------------------

(7) Narrow (SOUND)
Haskell To QDPs

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522[label="compare vz3500 vz35100",fontsize=16,color="magenta"];522 -> 568[label="",style="dashed", color="magenta", weight=3];
522 -> 569[label="",style="dashed", color="magenta", weight=3];
527[label="error []",fontsize=16,color="red",shape="box"];528[label="error []",fontsize=16,color="red",shape="box"];529[label="error []",fontsize=16,color="red",shape="box"];530[label="error []",fontsize=16,color="red",shape="box"];531[label="error []",fontsize=16,color="red",shape="box"];532[label="error []",fontsize=16,color="red",shape="box"];533[label="vz340",fontsize=16,color="green",shape="box"];534[label="vz3500",fontsize=16,color="green",shape="box"];535[label="error []",fontsize=16,color="red",shape="box"];536[label="error []",fontsize=16,color="red",shape="box"];537[label="error []",fontsize=16,color="red",shape="box"];538[label="error []",fontsize=16,color="red",shape="box"];539[label="error []",fontsize=16,color="red",shape="box"];540[label="error []",fontsize=16,color="red",shape="box"];541[label="error []",fontsize=16,color="red",shape="box"];542[label="vz3500",fontsize=16,color="green",shape="box"];543[label="vz35100",fontsize=16,color="green",shape="box"];544[label="vz3500",fontsize=16,color="green",shape="box"];545[label="vz35100",fontsize=16,color="green",shape="box"];546[label="vz3500",fontsize=16,color="green",shape="box"];547[label="vz35100",fontsize=16,color="green",shape="box"];548[label="vz3500",fontsize=16,color="green",shape="box"];549[label="vz35100",fontsize=16,color="green",shape="box"];550[label="vz3500",fontsize=16,color="green",shape="box"];551[label="vz35100",fontsize=16,color="green",shape="box"];552[label="vz3500",fontsize=16,color="green",shape="box"];553[label="vz35100",fontsize=16,color="green",shape="box"];554[label="vz3500",fontsize=16,color="green",shape="box"];555[label="vz35100",fontsize=16,color="green",shape="box"];556[label="vz3500",fontsize=16,color="green",shape="box"];557[label="vz35100",fontsize=16,color="green",shape="box"];558[label="vz3500",fontsize=16,color="green",shape="box"];559[label="vz35100",fontsize=16,color="green",shape="box"];560[label="vz3500",fontsize=16,color="green",shape="box"];561[label="vz35100",fontsize=16,color="green",shape="box"];562[label="vz3500",fontsize=16,color="green",shape="box"];563[label="vz35100",fontsize=16,color="green",shape="box"];564[label="vz3500",fontsize=16,color="green",shape="box"];565[label="vz35100",fontsize=16,color="green",shape="box"];566[label="vz3500",fontsize=16,color="green",shape="box"];567[label="vz35100",fontsize=16,color="green",shape="box"];568[label="vz3500",fontsize=16,color="green",shape="box"];569[label="vz35100",fontsize=16,color="green",shape="box"];}

----------------------------------------

(8)
Complex Obligation (AND)

----------------------------------------

(9)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_merge_pairs(:(vz35110, :(vz351110, vz351111)), ba) -> new_merge_pairs(vz351111, ba)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(10) QDPSizeChangeProof (EQUIVALENT)
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. 

From the DPs we obtained the following set of size-change graphs:
*new_merge_pairs(:(vz35110, :(vz351110, vz351111)), ba) -> new_merge_pairs(vz351111, ba)
The graph contains the following edges 1 > 1, 2 >= 2


----------------------------------------

(11)
YES

----------------------------------------

(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_map(:(vz3110, vz3111)) -> new_map(vz3111)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(13) QDPSizeChangeProof (EQUIVALENT)
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. 

From the DPs we obtained the following set of size-change graphs:
*new_map(:(vz3110, vz3111)) -> new_map(vz3111)
The graph contains the following edges 1 > 1


----------------------------------------

(14)
YES

----------------------------------------

(15)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_merge0(vz44, vz45, vz46, vz47, EQ, ba) -> new_merge(vz46, :(vz44, vz47), ba)
   new_merge0(vz44, vz45, vz46, vz47, GT, ba) -> new_merge(:(vz45, vz46), vz47, ba)
   new_merge0(vz44, vz45, vz46, vz47, LT, ba) -> new_merge(vz46, :(vz44, vz47), ba)
   new_merge(:(vz340, vz341), :(vz3500, vz3501), bb) -> new_merge0(vz3500, vz340, vz341, vz3501, new_compare(vz340, vz3500, bb), bb)

The TRS R consists of the following rules:

   new_compare11(vz340, vz3500) -> error([])
   new_compare(vz340, vz3500, app(app(ty_Either, bh), ca)) -> new_compare9(vz340, vz3500, bh, ca)
   new_compare(vz340, vz3500, ty_Float) -> new_compare0(vz340, vz3500)
   new_compare(vz340, vz3500, ty_Double) -> new_compare7(vz340, vz3500)
   new_compare(vz340, vz3500, ty_Int) -> new_compare11(vz340, vz3500)
   new_compare(vz340, vz3500, app(app(ty_@2, cb), cc)) -> new_compare12(vz340, vz3500, cb, cc)
   new_compare(vz340, vz3500, ty_Bool) -> new_compare6(vz340, vz3500)
   new_compare12(vz340, vz3500, cb, cc) -> error([])
   new_compare(vz340, vz3500, ty_Ordering) -> new_compare3(vz340, vz3500)
   new_compare(vz340, vz3500, app(ty_[], cd)) -> new_compare13(vz340, vz3500, cd)
   new_compare(vz340, vz3500, ty_Integer) -> new_compare2(vz340, vz3500)
   new_compare13(vz340, vz3500, cd) -> error([])
   new_compare(vz340, vz3500, app(ty_Maybe, bf)) -> new_compare5(vz340, vz3500, bf)
   new_compare2(vz340, vz3500) -> error([])
   new_compare1(vz340, vz3500, bc, bd, be) -> error([])
   new_compare6(vz340, vz3500) -> error([])
   new_compare5(vz340, vz3500, bf) -> error([])
   new_compare9(vz340, vz3500, bh, ca) -> error([])
   new_compare(vz340, vz3500, ty_@0) -> new_compare4(vz340, vz3500)
   new_compare(vz340, vz3500, app(app(app(ty_@3, bc), bd), be)) -> new_compare1(vz340, vz3500, bc, bd, be)
   new_compare10(vz340, vz3500) -> error([])
   new_compare0(vz340, vz3500) -> error([])
   new_compare3(vz340, vz3500) -> error([])
   new_compare8(vz340, vz3500, bg) -> error([])
   new_compare(vz340, vz3500, ty_Char) -> new_compare10(vz340, vz3500)
   new_compare4(@0, @0) -> EQ
   new_compare(vz340, vz3500, app(ty_Ratio, bg)) -> new_compare8(vz340, vz3500, bg)
   new_compare7(vz340, vz3500) -> error([])

The set Q consists of the following terms:

   new_compare(x0, x1, app(app(ty_Either, x2), x3))
   new_compare(x0, x1, app(app(ty_@2, x2), x3))
   new_compare(x0, x1, app(ty_Ratio, x2))
   new_compare1(x0, x1, x2, x3, x4)
   new_compare10(x0, x1)
   new_compare11(x0, x1)
   new_compare(x0, x1, ty_Integer)
   new_compare12(x0, x1, x2, x3)
   new_compare6(x0, x1)
   new_compare(x0, x1, ty_Bool)
   new_compare4(@0, @0)
   new_compare2(x0, x1)
   new_compare(x0, x1, ty_Float)
   new_compare8(x0, x1, x2)
   new_compare(x0, x1, app(ty_Maybe, x2))
   new_compare(x0, x1, app(ty_[], x2))
   new_compare7(x0, x1)
   new_compare5(x0, x1, x2)
   new_compare(x0, x1, app(app(app(ty_@3, x2), x3), x4))
   new_compare(x0, x1, ty_Int)
   new_compare3(x0, x1)
   new_compare9(x0, x1, x2, x3)
   new_compare(x0, x1, ty_Char)
   new_compare13(x0, x1, x2)
   new_compare0(x0, x1)
   new_compare(x0, x1, ty_Double)
   new_compare(x0, x1, ty_@0)
   new_compare(x0, x1, ty_Ordering)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(16) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
----------------------------------------

(17)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_merge(:(vz340, vz341), :(vz3500, vz3501), bb) -> new_merge0(vz3500, vz340, vz341, vz3501, new_compare(vz340, vz3500, bb), bb)
   new_merge0(vz44, vz45, vz46, vz47, EQ, ba) -> new_merge(vz46, :(vz44, vz47), ba)

The TRS R consists of the following rules:

   new_compare11(vz340, vz3500) -> error([])
   new_compare(vz340, vz3500, app(app(ty_Either, bh), ca)) -> new_compare9(vz340, vz3500, bh, ca)
   new_compare(vz340, vz3500, ty_Float) -> new_compare0(vz340, vz3500)
   new_compare(vz340, vz3500, ty_Double) -> new_compare7(vz340, vz3500)
   new_compare(vz340, vz3500, ty_Int) -> new_compare11(vz340, vz3500)
   new_compare(vz340, vz3500, app(app(ty_@2, cb), cc)) -> new_compare12(vz340, vz3500, cb, cc)
   new_compare(vz340, vz3500, ty_Bool) -> new_compare6(vz340, vz3500)
   new_compare12(vz340, vz3500, cb, cc) -> error([])
   new_compare(vz340, vz3500, ty_Ordering) -> new_compare3(vz340, vz3500)
   new_compare(vz340, vz3500, app(ty_[], cd)) -> new_compare13(vz340, vz3500, cd)
   new_compare(vz340, vz3500, ty_Integer) -> new_compare2(vz340, vz3500)
   new_compare13(vz340, vz3500, cd) -> error([])
   new_compare(vz340, vz3500, app(ty_Maybe, bf)) -> new_compare5(vz340, vz3500, bf)
   new_compare2(vz340, vz3500) -> error([])
   new_compare1(vz340, vz3500, bc, bd, be) -> error([])
   new_compare6(vz340, vz3500) -> error([])
   new_compare5(vz340, vz3500, bf) -> error([])
   new_compare9(vz340, vz3500, bh, ca) -> error([])
   new_compare(vz340, vz3500, ty_@0) -> new_compare4(vz340, vz3500)
   new_compare(vz340, vz3500, app(app(app(ty_@3, bc), bd), be)) -> new_compare1(vz340, vz3500, bc, bd, be)
   new_compare10(vz340, vz3500) -> error([])
   new_compare0(vz340, vz3500) -> error([])
   new_compare3(vz340, vz3500) -> error([])
   new_compare8(vz340, vz3500, bg) -> error([])
   new_compare(vz340, vz3500, ty_Char) -> new_compare10(vz340, vz3500)
   new_compare4(@0, @0) -> EQ
   new_compare(vz340, vz3500, app(ty_Ratio, bg)) -> new_compare8(vz340, vz3500, bg)
   new_compare7(vz340, vz3500) -> error([])

The set Q consists of the following terms:

   new_compare(x0, x1, app(app(ty_Either, x2), x3))
   new_compare(x0, x1, app(app(ty_@2, x2), x3))
   new_compare(x0, x1, app(ty_Ratio, x2))
   new_compare1(x0, x1, x2, x3, x4)
   new_compare10(x0, x1)
   new_compare11(x0, x1)
   new_compare(x0, x1, ty_Integer)
   new_compare12(x0, x1, x2, x3)
   new_compare6(x0, x1)
   new_compare(x0, x1, ty_Bool)
   new_compare4(@0, @0)
   new_compare2(x0, x1)
   new_compare(x0, x1, ty_Float)
   new_compare8(x0, x1, x2)
   new_compare(x0, x1, app(ty_Maybe, x2))
   new_compare(x0, x1, app(ty_[], x2))
   new_compare7(x0, x1)
   new_compare5(x0, x1, x2)
   new_compare(x0, x1, app(app(app(ty_@3, x2), x3), x4))
   new_compare(x0, x1, ty_Int)
   new_compare3(x0, x1)
   new_compare9(x0, x1, x2, x3)
   new_compare(x0, x1, ty_Char)
   new_compare13(x0, x1, x2)
   new_compare0(x0, x1)
   new_compare(x0, x1, ty_Double)
   new_compare(x0, x1, ty_@0)
   new_compare(x0, x1, ty_Ordering)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(18) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_merge(:(vz340, vz341), :(vz3500, vz3501), bb) -> new_merge0(vz3500, vz340, vz341, vz3501, new_compare(vz340, vz3500, bb), bb) at position [4] we obtained the following new rules [LPAR04]:

   (new_merge(:(x0, y1), :(x1, y3), app(app(ty_Either, x2), x3)) -> new_merge0(x1, x0, y1, y3, new_compare9(x0, x1, x2, x3), app(app(ty_Either, x2), x3)),new_merge(:(x0, y1), :(x1, y3), app(app(ty_Either, x2), x3)) -> new_merge0(x1, x0, y1, y3, new_compare9(x0, x1, x2, x3), app(app(ty_Either, x2), x3)))
   (new_merge(:(x0, y1), :(x1, y3), ty_Float) -> new_merge0(x1, x0, y1, y3, new_compare0(x0, x1), ty_Float),new_merge(:(x0, y1), :(x1, y3), ty_Float) -> new_merge0(x1, x0, y1, y3, new_compare0(x0, x1), ty_Float))
   (new_merge(:(x0, y1), :(x1, y3), ty_Double) -> new_merge0(x1, x0, y1, y3, new_compare7(x0, x1), ty_Double),new_merge(:(x0, y1), :(x1, y3), ty_Double) -> new_merge0(x1, x0, y1, y3, new_compare7(x0, x1), ty_Double))
   (new_merge(:(x0, y1), :(x1, y3), ty_Int) -> new_merge0(x1, x0, y1, y3, new_compare11(x0, x1), ty_Int),new_merge(:(x0, y1), :(x1, y3), ty_Int) -> new_merge0(x1, x0, y1, y3, new_compare11(x0, x1), ty_Int))
   (new_merge(:(x0, y1), :(x1, y3), app(app(ty_@2, x2), x3)) -> new_merge0(x1, x0, y1, y3, new_compare12(x0, x1, x2, x3), app(app(ty_@2, x2), x3)),new_merge(:(x0, y1), :(x1, y3), app(app(ty_@2, x2), x3)) -> new_merge0(x1, x0, y1, y3, new_compare12(x0, x1, x2, x3), app(app(ty_@2, x2), x3)))
   (new_merge(:(x0, y1), :(x1, y3), ty_Bool) -> new_merge0(x1, x0, y1, y3, new_compare6(x0, x1), ty_Bool),new_merge(:(x0, y1), :(x1, y3), ty_Bool) -> new_merge0(x1, x0, y1, y3, new_compare6(x0, x1), ty_Bool))
   (new_merge(:(x0, y1), :(x1, y3), ty_Ordering) -> new_merge0(x1, x0, y1, y3, new_compare3(x0, x1), ty_Ordering),new_merge(:(x0, y1), :(x1, y3), ty_Ordering) -> new_merge0(x1, x0, y1, y3, new_compare3(x0, x1), ty_Ordering))
   (new_merge(:(x0, y1), :(x1, y3), app(ty_[], x2)) -> new_merge0(x1, x0, y1, y3, new_compare13(x0, x1, x2), app(ty_[], x2)),new_merge(:(x0, y1), :(x1, y3), app(ty_[], x2)) -> new_merge0(x1, x0, y1, y3, new_compare13(x0, x1, x2), app(ty_[], x2)))
   (new_merge(:(x0, y1), :(x1, y3), ty_Integer) -> new_merge0(x1, x0, y1, y3, new_compare2(x0, x1), ty_Integer),new_merge(:(x0, y1), :(x1, y3), ty_Integer) -> new_merge0(x1, x0, y1, y3, new_compare2(x0, x1), ty_Integer))
   (new_merge(:(x0, y1), :(x1, y3), app(ty_Maybe, x2)) -> new_merge0(x1, x0, y1, y3, new_compare5(x0, x1, x2), app(ty_Maybe, x2)),new_merge(:(x0, y1), :(x1, y3), app(ty_Maybe, x2)) -> new_merge0(x1, x0, y1, y3, new_compare5(x0, x1, x2), app(ty_Maybe, x2)))
   (new_merge(:(x0, y1), :(x1, y3), ty_@0) -> new_merge0(x1, x0, y1, y3, new_compare4(x0, x1), ty_@0),new_merge(:(x0, y1), :(x1, y3), ty_@0) -> new_merge0(x1, x0, y1, y3, new_compare4(x0, x1), ty_@0))
   (new_merge(:(x0, y1), :(x1, y3), app(app(app(ty_@3, x2), x3), x4)) -> new_merge0(x1, x0, y1, y3, new_compare1(x0, x1, x2, x3, x4), app(app(app(ty_@3, x2), x3), x4)),new_merge(:(x0, y1), :(x1, y3), app(app(app(ty_@3, x2), x3), x4)) -> new_merge0(x1, x0, y1, y3, new_compare1(x0, x1, x2, x3, x4), app(app(app(ty_@3, x2), x3), x4)))
   (new_merge(:(x0, y1), :(x1, y3), ty_Char) -> new_merge0(x1, x0, y1, y3, new_compare10(x0, x1), ty_Char),new_merge(:(x0, y1), :(x1, y3), ty_Char) -> new_merge0(x1, x0, y1, y3, new_compare10(x0, x1), ty_Char))
   (new_merge(:(x0, y1), :(x1, y3), app(ty_Ratio, x2)) -> new_merge0(x1, x0, y1, y3, new_compare8(x0, x1, x2), app(ty_Ratio, x2)),new_merge(:(x0, y1), :(x1, y3), app(ty_Ratio, x2)) -> new_merge0(x1, x0, y1, y3, new_compare8(x0, x1, x2), app(ty_Ratio, x2)))


----------------------------------------

(19)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_merge0(vz44, vz45, vz46, vz47, EQ, ba) -> new_merge(vz46, :(vz44, vz47), ba)
   new_merge(:(x0, y1), :(x1, y3), app(app(ty_Either, x2), x3)) -> new_merge0(x1, x0, y1, y3, new_compare9(x0, x1, x2, x3), app(app(ty_Either, x2), x3))
   new_merge(:(x0, y1), :(x1, y3), ty_Float) -> new_merge0(x1, x0, y1, y3, new_compare0(x0, x1), ty_Float)
   new_merge(:(x0, y1), :(x1, y3), ty_Double) -> new_merge0(x1, x0, y1, y3, new_compare7(x0, x1), ty_Double)
   new_merge(:(x0, y1), :(x1, y3), ty_Int) -> new_merge0(x1, x0, y1, y3, new_compare11(x0, x1), ty_Int)
   new_merge(:(x0, y1), :(x1, y3), app(app(ty_@2, x2), x3)) -> new_merge0(x1, x0, y1, y3, new_compare12(x0, x1, x2, x3), app(app(ty_@2, x2), x3))
   new_merge(:(x0, y1), :(x1, y3), ty_Bool) -> new_merge0(x1, x0, y1, y3, new_compare6(x0, x1), ty_Bool)
   new_merge(:(x0, y1), :(x1, y3), ty_Ordering) -> new_merge0(x1, x0, y1, y3, new_compare3(x0, x1), ty_Ordering)
   new_merge(:(x0, y1), :(x1, y3), app(ty_[], x2)) -> new_merge0(x1, x0, y1, y3, new_compare13(x0, x1, x2), app(ty_[], x2))
   new_merge(:(x0, y1), :(x1, y3), ty_Integer) -> new_merge0(x1, x0, y1, y3, new_compare2(x0, x1), ty_Integer)
   new_merge(:(x0, y1), :(x1, y3), app(ty_Maybe, x2)) -> new_merge0(x1, x0, y1, y3, new_compare5(x0, x1, x2), app(ty_Maybe, x2))
   new_merge(:(x0, y1), :(x1, y3), ty_@0) -> new_merge0(x1, x0, y1, y3, new_compare4(x0, x1), ty_@0)
   new_merge(:(x0, y1), :(x1, y3), app(app(app(ty_@3, x2), x3), x4)) -> new_merge0(x1, x0, y1, y3, new_compare1(x0, x1, x2, x3, x4), app(app(app(ty_@3, x2), x3), x4))
   new_merge(:(x0, y1), :(x1, y3), ty_Char) -> new_merge0(x1, x0, y1, y3, new_compare10(x0, x1), ty_Char)
   new_merge(:(x0, y1), :(x1, y3), app(ty_Ratio, x2)) -> new_merge0(x1, x0, y1, y3, new_compare8(x0, x1, x2), app(ty_Ratio, x2))

The TRS R consists of the following rules:

   new_compare11(vz340, vz3500) -> error([])
   new_compare(vz340, vz3500, app(app(ty_Either, bh), ca)) -> new_compare9(vz340, vz3500, bh, ca)
   new_compare(vz340, vz3500, ty_Float) -> new_compare0(vz340, vz3500)
   new_compare(vz340, vz3500, ty_Double) -> new_compare7(vz340, vz3500)
   new_compare(vz340, vz3500, ty_Int) -> new_compare11(vz340, vz3500)
   new_compare(vz340, vz3500, app(app(ty_@2, cb), cc)) -> new_compare12(vz340, vz3500, cb, cc)
   new_compare(vz340, vz3500, ty_Bool) -> new_compare6(vz340, vz3500)
   new_compare12(vz340, vz3500, cb, cc) -> error([])
   new_compare(vz340, vz3500, ty_Ordering) -> new_compare3(vz340, vz3500)
   new_compare(vz340, vz3500, app(ty_[], cd)) -> new_compare13(vz340, vz3500, cd)
   new_compare(vz340, vz3500, ty_Integer) -> new_compare2(vz340, vz3500)
   new_compare13(vz340, vz3500, cd) -> error([])
   new_compare(vz340, vz3500, app(ty_Maybe, bf)) -> new_compare5(vz340, vz3500, bf)
   new_compare2(vz340, vz3500) -> error([])
   new_compare1(vz340, vz3500, bc, bd, be) -> error([])
   new_compare6(vz340, vz3500) -> error([])
   new_compare5(vz340, vz3500, bf) -> error([])
   new_compare9(vz340, vz3500, bh, ca) -> error([])
   new_compare(vz340, vz3500, ty_@0) -> new_compare4(vz340, vz3500)
   new_compare(vz340, vz3500, app(app(app(ty_@3, bc), bd), be)) -> new_compare1(vz340, vz3500, bc, bd, be)
   new_compare10(vz340, vz3500) -> error([])
   new_compare0(vz340, vz3500) -> error([])
   new_compare3(vz340, vz3500) -> error([])
   new_compare8(vz340, vz3500, bg) -> error([])
   new_compare(vz340, vz3500, ty_Char) -> new_compare10(vz340, vz3500)
   new_compare4(@0, @0) -> EQ
   new_compare(vz340, vz3500, app(ty_Ratio, bg)) -> new_compare8(vz340, vz3500, bg)
   new_compare7(vz340, vz3500) -> error([])

The set Q consists of the following terms:

   new_compare(x0, x1, app(app(ty_Either, x2), x3))
   new_compare(x0, x1, app(app(ty_@2, x2), x3))
   new_compare(x0, x1, app(ty_Ratio, x2))
   new_compare1(x0, x1, x2, x3, x4)
   new_compare10(x0, x1)
   new_compare11(x0, x1)
   new_compare(x0, x1, ty_Integer)
   new_compare12(x0, x1, x2, x3)
   new_compare6(x0, x1)
   new_compare(x0, x1, ty_Bool)
   new_compare4(@0, @0)
   new_compare2(x0, x1)
   new_compare(x0, x1, ty_Float)
   new_compare8(x0, x1, x2)
   new_compare(x0, x1, app(ty_Maybe, x2))
   new_compare(x0, x1, app(ty_[], x2))
   new_compare7(x0, x1)
   new_compare5(x0, x1, x2)
   new_compare(x0, x1, app(app(app(ty_@3, x2), x3), x4))
   new_compare(x0, x1, ty_Int)
   new_compare3(x0, x1)
   new_compare9(x0, x1, x2, x3)
   new_compare(x0, x1, ty_Char)
   new_compare13(x0, x1, x2)
   new_compare0(x0, x1)
   new_compare(x0, x1, ty_Double)
   new_compare(x0, x1, ty_@0)
   new_compare(x0, x1, ty_Ordering)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(20) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 13 less nodes.
----------------------------------------

(21)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_merge(:(x0, y1), :(x1, y3), ty_@0) -> new_merge0(x1, x0, y1, y3, new_compare4(x0, x1), ty_@0)
   new_merge0(vz44, vz45, vz46, vz47, EQ, ba) -> new_merge(vz46, :(vz44, vz47), ba)

The TRS R consists of the following rules:

   new_compare11(vz340, vz3500) -> error([])
   new_compare(vz340, vz3500, app(app(ty_Either, bh), ca)) -> new_compare9(vz340, vz3500, bh, ca)
   new_compare(vz340, vz3500, ty_Float) -> new_compare0(vz340, vz3500)
   new_compare(vz340, vz3500, ty_Double) -> new_compare7(vz340, vz3500)
   new_compare(vz340, vz3500, ty_Int) -> new_compare11(vz340, vz3500)
   new_compare(vz340, vz3500, app(app(ty_@2, cb), cc)) -> new_compare12(vz340, vz3500, cb, cc)
   new_compare(vz340, vz3500, ty_Bool) -> new_compare6(vz340, vz3500)
   new_compare12(vz340, vz3500, cb, cc) -> error([])
   new_compare(vz340, vz3500, ty_Ordering) -> new_compare3(vz340, vz3500)
   new_compare(vz340, vz3500, app(ty_[], cd)) -> new_compare13(vz340, vz3500, cd)
   new_compare(vz340, vz3500, ty_Integer) -> new_compare2(vz340, vz3500)
   new_compare13(vz340, vz3500, cd) -> error([])
   new_compare(vz340, vz3500, app(ty_Maybe, bf)) -> new_compare5(vz340, vz3500, bf)
   new_compare2(vz340, vz3500) -> error([])
   new_compare1(vz340, vz3500, bc, bd, be) -> error([])
   new_compare6(vz340, vz3500) -> error([])
   new_compare5(vz340, vz3500, bf) -> error([])
   new_compare9(vz340, vz3500, bh, ca) -> error([])
   new_compare(vz340, vz3500, ty_@0) -> new_compare4(vz340, vz3500)
   new_compare(vz340, vz3500, app(app(app(ty_@3, bc), bd), be)) -> new_compare1(vz340, vz3500, bc, bd, be)
   new_compare10(vz340, vz3500) -> error([])
   new_compare0(vz340, vz3500) -> error([])
   new_compare3(vz340, vz3500) -> error([])
   new_compare8(vz340, vz3500, bg) -> error([])
   new_compare(vz340, vz3500, ty_Char) -> new_compare10(vz340, vz3500)
   new_compare4(@0, @0) -> EQ
   new_compare(vz340, vz3500, app(ty_Ratio, bg)) -> new_compare8(vz340, vz3500, bg)
   new_compare7(vz340, vz3500) -> error([])

The set Q consists of the following terms:

   new_compare(x0, x1, app(app(ty_Either, x2), x3))
   new_compare(x0, x1, app(app(ty_@2, x2), x3))
   new_compare(x0, x1, app(ty_Ratio, x2))
   new_compare1(x0, x1, x2, x3, x4)
   new_compare10(x0, x1)
   new_compare11(x0, x1)
   new_compare(x0, x1, ty_Integer)
   new_compare12(x0, x1, x2, x3)
   new_compare6(x0, x1)
   new_compare(x0, x1, ty_Bool)
   new_compare4(@0, @0)
   new_compare2(x0, x1)
   new_compare(x0, x1, ty_Float)
   new_compare8(x0, x1, x2)
   new_compare(x0, x1, app(ty_Maybe, x2))
   new_compare(x0, x1, app(ty_[], x2))
   new_compare7(x0, x1)
   new_compare5(x0, x1, x2)
   new_compare(x0, x1, app(app(app(ty_@3, x2), x3), x4))
   new_compare(x0, x1, ty_Int)
   new_compare3(x0, x1)
   new_compare9(x0, x1, x2, x3)
   new_compare(x0, x1, ty_Char)
   new_compare13(x0, x1, x2)
   new_compare0(x0, x1)
   new_compare(x0, x1, ty_Double)
   new_compare(x0, x1, ty_@0)
   new_compare(x0, x1, ty_Ordering)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(22) UsableRulesProof (EQUIVALENT)
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.
----------------------------------------

(23)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_merge(:(x0, y1), :(x1, y3), ty_@0) -> new_merge0(x1, x0, y1, y3, new_compare4(x0, x1), ty_@0)
   new_merge0(vz44, vz45, vz46, vz47, EQ, ba) -> new_merge(vz46, :(vz44, vz47), ba)

The TRS R consists of the following rules:

   new_compare4(@0, @0) -> EQ

The set Q consists of the following terms:

   new_compare(x0, x1, app(app(ty_Either, x2), x3))
   new_compare(x0, x1, app(app(ty_@2, x2), x3))
   new_compare(x0, x1, app(ty_Ratio, x2))
   new_compare1(x0, x1, x2, x3, x4)
   new_compare10(x0, x1)
   new_compare11(x0, x1)
   new_compare(x0, x1, ty_Integer)
   new_compare12(x0, x1, x2, x3)
   new_compare6(x0, x1)
   new_compare(x0, x1, ty_Bool)
   new_compare4(@0, @0)
   new_compare2(x0, x1)
   new_compare(x0, x1, ty_Float)
   new_compare8(x0, x1, x2)
   new_compare(x0, x1, app(ty_Maybe, x2))
   new_compare(x0, x1, app(ty_[], x2))
   new_compare7(x0, x1)
   new_compare5(x0, x1, x2)
   new_compare(x0, x1, app(app(app(ty_@3, x2), x3), x4))
   new_compare(x0, x1, ty_Int)
   new_compare3(x0, x1)
   new_compare9(x0, x1, x2, x3)
   new_compare(x0, x1, ty_Char)
   new_compare13(x0, x1, x2)
   new_compare0(x0, x1)
   new_compare(x0, x1, ty_Double)
   new_compare(x0, x1, ty_@0)
   new_compare(x0, x1, ty_Ordering)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(24) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_compare(x0, x1, app(app(ty_Either, x2), x3))
   new_compare(x0, x1, app(app(ty_@2, x2), x3))
   new_compare(x0, x1, app(ty_Ratio, x2))
   new_compare1(x0, x1, x2, x3, x4)
   new_compare10(x0, x1)
   new_compare11(x0, x1)
   new_compare(x0, x1, ty_Integer)
   new_compare12(x0, x1, x2, x3)
   new_compare6(x0, x1)
   new_compare(x0, x1, ty_Bool)
   new_compare2(x0, x1)
   new_compare(x0, x1, ty_Float)
   new_compare8(x0, x1, x2)
   new_compare(x0, x1, app(ty_Maybe, x2))
   new_compare(x0, x1, app(ty_[], x2))
   new_compare7(x0, x1)
   new_compare5(x0, x1, x2)
   new_compare(x0, x1, app(app(app(ty_@3, x2), x3), x4))
   new_compare(x0, x1, ty_Int)
   new_compare3(x0, x1)
   new_compare9(x0, x1, x2, x3)
   new_compare(x0, x1, ty_Char)
   new_compare13(x0, x1, x2)
   new_compare0(x0, x1)
   new_compare(x0, x1, ty_Double)
   new_compare(x0, x1, ty_@0)
   new_compare(x0, x1, ty_Ordering)


----------------------------------------

(25)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_merge(:(x0, y1), :(x1, y3), ty_@0) -> new_merge0(x1, x0, y1, y3, new_compare4(x0, x1), ty_@0)
   new_merge0(vz44, vz45, vz46, vz47, EQ, ba) -> new_merge(vz46, :(vz44, vz47), ba)

The TRS R consists of the following rules:

   new_compare4(@0, @0) -> EQ

The set Q consists of the following terms:

   new_compare4(@0, @0)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(26) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_merge(:(x0, y1), :(x1, y3), ty_@0) -> new_merge0(x1, x0, y1, y3, new_compare4(x0, x1), ty_@0) at position [4] we obtained the following new rules [LPAR04]:

   (new_merge(:(@0, y1), :(@0, y3), ty_@0) -> new_merge0(@0, @0, y1, y3, EQ, ty_@0),new_merge(:(@0, y1), :(@0, y3), ty_@0) -> new_merge0(@0, @0, y1, y3, EQ, ty_@0))


----------------------------------------

(27)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_merge0(vz44, vz45, vz46, vz47, EQ, ba) -> new_merge(vz46, :(vz44, vz47), ba)
   new_merge(:(@0, y1), :(@0, y3), ty_@0) -> new_merge0(@0, @0, y1, y3, EQ, ty_@0)

The TRS R consists of the following rules:

   new_compare4(@0, @0) -> EQ

The set Q consists of the following terms:

   new_compare4(@0, @0)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(28) UsableRulesProof (EQUIVALENT)
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.
----------------------------------------

(29)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_merge0(vz44, vz45, vz46, vz47, EQ, ba) -> new_merge(vz46, :(vz44, vz47), ba)
   new_merge(:(@0, y1), :(@0, y3), ty_@0) -> new_merge0(@0, @0, y1, y3, EQ, ty_@0)

R is empty.
The set Q consists of the following terms:

   new_compare4(@0, @0)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(30) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_compare4(@0, @0)


----------------------------------------

(31)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_merge0(vz44, vz45, vz46, vz47, EQ, ba) -> new_merge(vz46, :(vz44, vz47), ba)
   new_merge(:(@0, y1), :(@0, y3), ty_@0) -> new_merge0(@0, @0, y1, y3, EQ, ty_@0)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(32) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule new_merge0(vz44, vz45, vz46, vz47, EQ, ba) -> new_merge(vz46, :(vz44, vz47), ba) we obtained the following new rules [LPAR04]:

   (new_merge0(@0, @0, z0, z1, EQ, ty_@0) -> new_merge(z0, :(@0, z1), ty_@0),new_merge0(@0, @0, z0, z1, EQ, ty_@0) -> new_merge(z0, :(@0, z1), ty_@0))


----------------------------------------

(33)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_merge(:(@0, y1), :(@0, y3), ty_@0) -> new_merge0(@0, @0, y1, y3, EQ, ty_@0)
   new_merge0(@0, @0, z0, z1, EQ, ty_@0) -> new_merge(z0, :(@0, z1), ty_@0)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(34) QDPSizeChangeProof (EQUIVALENT)
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. 

From the DPs we obtained the following set of size-change graphs:
*new_merge0(@0, @0, z0, z1, EQ, ty_@0) -> new_merge(z0, :(@0, z1), ty_@0)
The graph contains the following edges 3 >= 1, 6 >= 3


*new_merge(:(@0, y1), :(@0, y3), ty_@0) -> new_merge0(@0, @0, y1, y3, EQ, ty_@0)
The graph contains the following edges 1 > 1, 2 > 1, 1 > 2, 2 > 2, 1 > 3, 2 > 4, 3 >= 6


----------------------------------------

(35)
YES

----------------------------------------

(36)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_mergesort'(vz34, :(vz350, []), ba) -> new_mergesort'(new_merge2(vz34, vz350, ba), [], ba)
   new_mergesort'(vz34, :(vz350, :(vz3510, vz3511)), ba) -> new_mergesort'(new_merge1(vz34, vz350, vz3510, ba), new_merge_pairs0(vz3511, ba), ba)

The TRS R consists of the following rules:

   new_compare11(vz340, vz3500) -> error([])
   new_compare(vz340, vz3500, ty_Float) -> new_compare0(vz340, vz3500)
   new_compare14(vz3500, vz35100, ty_Bool) -> new_compare6(vz3500, vz35100)
   new_compare14(vz3500, vz35100, ty_Double) -> new_compare7(vz3500, vz35100)
   new_compare(vz340, vz3500, ty_Int) -> new_compare11(vz340, vz3500)
   new_compare(vz340, vz3500, app(app(ty_@2, bg), bh)) -> new_compare12(vz340, vz3500, bg, bh)
   new_compare(vz340, vz3500, ty_Ordering) -> new_compare3(vz340, vz3500)
   new_compare12(vz340, vz3500, bg, bh) -> error([])
   new_compare(vz340, vz3500, app(ty_[], bf)) -> new_compare13(vz340, vz3500, bf)
   new_compare(vz340, vz3500, app(ty_Maybe, be)) -> new_compare5(vz340, vz3500, be)
   new_compare14(vz3500, vz35100, app(app(ty_Either, ca), cb)) -> new_compare9(vz3500, vz35100, ca, cb)
   new_merge_pairs0(:(vz35110, []), ba) -> :(vz35110, [])
   new_merge1(vz34, :(vz3500, vz3501), :(vz35100, vz35101), ba) -> new_merge2(vz34, new_merge00(vz35100, vz3500, vz3501, vz35101, new_compare14(vz3500, vz35100, ba), ba), ba)
   new_compare14(vz3500, vz35100, ty_Char) -> new_compare10(vz3500, vz35100)
   new_compare5(vz340, vz3500, be) -> error([])
   new_compare14(vz3500, vz35100, app(ty_Ratio, cc)) -> new_compare8(vz3500, vz35100, cc)
   new_merge2([], :(vz3500, vz3501), ba) -> :(vz3500, vz3501)
   new_compare0(vz340, vz3500) -> error([])
   new_compare14(vz3500, vz35100, app(app(ty_@2, bg), bh)) -> new_compare12(vz3500, vz35100, bg, bh)
   new_compare14(vz3500, vz35100, ty_Float) -> new_compare0(vz3500, vz35100)
   new_merge_pairs0(:(vz35110, :(vz351110, vz351111)), ba) -> :(new_merge2(vz35110, vz351110, ba), new_merge_pairs0(vz351111, ba))
   new_compare3(vz340, vz3500) -> error([])
   new_merge00(vz44, vz45, vz46, vz47, LT, cd) -> :(vz45, new_merge2(vz46, :(vz44, vz47), cd))
   new_compare(vz340, vz3500, ty_Char) -> new_compare10(vz340, vz3500)
   new_compare4(@0, @0) -> EQ
   new_compare7(vz340, vz3500) -> error([])
   new_compare14(vz3500, vz35100, ty_Ordering) -> new_compare3(vz3500, vz35100)
   new_compare(vz340, vz3500, app(app(ty_Either, ca), cb)) -> new_compare9(vz340, vz3500, ca, cb)
   new_merge2(:(vz340, vz341), :(vz3500, vz3501), ba) -> new_merge00(vz3500, vz340, vz341, vz3501, new_compare(vz340, vz3500, ba), ba)
   new_compare14(vz3500, vz35100, app(ty_Maybe, be)) -> new_compare5(vz3500, vz35100, be)
   new_compare(vz340, vz3500, ty_Double) -> new_compare7(vz340, vz3500)
   new_compare14(vz3500, vz35100, ty_Int) -> new_compare11(vz3500, vz35100)
   new_merge2(vz34, [], ba) -> vz34
   new_compare(vz340, vz3500, ty_Bool) -> new_compare6(vz340, vz3500)
   new_compare(vz340, vz3500, ty_Integer) -> new_compare2(vz340, vz3500)
   new_compare13(vz340, vz3500, bf) -> error([])
   new_merge_pairs0([], ba) -> []
   new_merge1(vz34, [], :(vz35100, vz35101), ba) -> new_merge2(vz34, :(vz35100, vz35101), ba)
   new_compare2(vz340, vz3500) -> error([])
   new_compare1(vz340, vz3500, bb, bc, bd) -> error([])
   new_compare6(vz340, vz3500) -> error([])
   new_merge00(vz44, vz45, vz46, vz47, GT, cd) -> :(vz44, new_merge2(:(vz45, vz46), vz47, cd))
   new_compare(vz340, vz3500, ty_@0) -> new_compare4(vz340, vz3500)
   new_compare9(vz340, vz3500, ca, cb) -> error([])
   new_compare(vz340, vz3500, app(app(app(ty_@3, bb), bc), bd)) -> new_compare1(vz340, vz3500, bb, bc, bd)
   new_compare14(vz3500, vz35100, ty_Integer) -> new_compare2(vz3500, vz35100)
   new_compare10(vz340, vz3500) -> error([])
   new_merge00(vz44, vz45, vz46, vz47, EQ, cd) -> :(vz45, new_merge2(vz46, :(vz44, vz47), cd))
   new_compare14(vz3500, vz35100, app(app(app(ty_@3, bb), bc), bd)) -> new_compare1(vz3500, vz35100, bb, bc, bd)
   new_compare8(vz340, vz3500, cc) -> error([])
   new_merge1(vz34, vz350, [], ba) -> new_merge2(vz34, vz350, ba)
   new_compare(vz340, vz3500, app(ty_Ratio, cc)) -> new_compare8(vz340, vz3500, cc)
   new_compare14(vz3500, vz35100, ty_@0) -> new_compare4(vz3500, vz35100)
   new_compare14(vz3500, vz35100, app(ty_[], bf)) -> new_compare13(vz3500, vz35100, bf)

The set Q consists of the following terms:

   new_compare14(x0, x1, ty_Double)
   new_merge1(x0, :(x1, x2), :(x3, x4), x5)
   new_compare10(x0, x1)
   new_compare(x0, x1, app(ty_[], x2))
   new_compare13(x0, x1, x2)
   new_compare(x0, x1, app(app(app(ty_@3, x2), x3), x4))
   new_compare14(x0, x1, ty_Bool)
   new_merge2([], :(x0, x1), x2)
   new_compare11(x0, x1)
   new_compare14(x0, x1, ty_@0)
   new_merge_pairs0(:(x0, :(x1, x2)), x3)
   new_compare(x0, x1, ty_Bool)
   new_compare(x0, x1, app(app(ty_Either, x2), x3))
   new_compare4(@0, @0)
   new_compare8(x0, x1, x2)
   new_compare14(x0, x1, ty_Char)
   new_merge00(x0, x1, x2, x3, EQ, x4)
   new_compare14(x0, x1, app(ty_[], x2))
   new_compare14(x0, x1, app(app(ty_Either, x2), x3))
   new_merge1(x0, [], :(x1, x2), x3)
   new_compare14(x0, x1, ty_Ordering)
   new_compare7(x0, x1)
   new_merge2(x0, [], x1)
   new_compare14(x0, x1, app(app(app(ty_@3, x2), x3), x4))
   new_compare(x0, x1, ty_Int)
   new_merge00(x0, x1, x2, x3, GT, x4)
   new_merge1(x0, x1, [], x2)
   new_compare14(x0, x1, app(app(ty_@2, x2), x3))
   new_compare12(x0, x1, x2, x3)
   new_compare14(x0, x1, ty_Int)
   new_compare(x0, x1, ty_@0)
   new_compare(x0, x1, ty_Ordering)
   new_compare(x0, x1, app(ty_Ratio, x2))
   new_compare(x0, x1, app(ty_Maybe, x2))
   new_compare1(x0, x1, x2, x3, x4)
   new_compare(x0, x1, ty_Integer)
   new_compare(x0, x1, app(app(ty_@2, x2), x3))
   new_compare6(x0, x1)
   new_compare2(x0, x1)
   new_compare(x0, x1, ty_Float)
   new_compare9(x0, x1, x2, x3)
   new_compare14(x0, x1, app(ty_Ratio, x2))
   new_compare3(x0, x1)
   new_compare5(x0, x1, x2)
   new_compare(x0, x1, ty_Char)
   new_compare14(x0, x1, app(ty_Maybe, x2))
   new_merge_pairs0(:(x0, []), x1)
   new_compare14(x0, x1, ty_Float)
   new_merge_pairs0([], x0)
   new_compare0(x0, x1)
   new_merge2(:(x0, x1), :(x2, x3), x4)
   new_compare(x0, x1, ty_Double)
   new_compare14(x0, x1, ty_Integer)
   new_merge00(x0, x1, x2, x3, LT, x4)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(37) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(38)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_mergesort'(vz34, :(vz350, :(vz3510, vz3511)), ba) -> new_mergesort'(new_merge1(vz34, vz350, vz3510, ba), new_merge_pairs0(vz3511, ba), ba)

The TRS R consists of the following rules:

   new_compare11(vz340, vz3500) -> error([])
   new_compare(vz340, vz3500, ty_Float) -> new_compare0(vz340, vz3500)
   new_compare14(vz3500, vz35100, ty_Bool) -> new_compare6(vz3500, vz35100)
   new_compare14(vz3500, vz35100, ty_Double) -> new_compare7(vz3500, vz35100)
   new_compare(vz340, vz3500, ty_Int) -> new_compare11(vz340, vz3500)
   new_compare(vz340, vz3500, app(app(ty_@2, bg), bh)) -> new_compare12(vz340, vz3500, bg, bh)
   new_compare(vz340, vz3500, ty_Ordering) -> new_compare3(vz340, vz3500)
   new_compare12(vz340, vz3500, bg, bh) -> error([])
   new_compare(vz340, vz3500, app(ty_[], bf)) -> new_compare13(vz340, vz3500, bf)
   new_compare(vz340, vz3500, app(ty_Maybe, be)) -> new_compare5(vz340, vz3500, be)
   new_compare14(vz3500, vz35100, app(app(ty_Either, ca), cb)) -> new_compare9(vz3500, vz35100, ca, cb)
   new_merge_pairs0(:(vz35110, []), ba) -> :(vz35110, [])
   new_merge1(vz34, :(vz3500, vz3501), :(vz35100, vz35101), ba) -> new_merge2(vz34, new_merge00(vz35100, vz3500, vz3501, vz35101, new_compare14(vz3500, vz35100, ba), ba), ba)
   new_compare14(vz3500, vz35100, ty_Char) -> new_compare10(vz3500, vz35100)
   new_compare5(vz340, vz3500, be) -> error([])
   new_compare14(vz3500, vz35100, app(ty_Ratio, cc)) -> new_compare8(vz3500, vz35100, cc)
   new_merge2([], :(vz3500, vz3501), ba) -> :(vz3500, vz3501)
   new_compare0(vz340, vz3500) -> error([])
   new_compare14(vz3500, vz35100, app(app(ty_@2, bg), bh)) -> new_compare12(vz3500, vz35100, bg, bh)
   new_compare14(vz3500, vz35100, ty_Float) -> new_compare0(vz3500, vz35100)
   new_merge_pairs0(:(vz35110, :(vz351110, vz351111)), ba) -> :(new_merge2(vz35110, vz351110, ba), new_merge_pairs0(vz351111, ba))
   new_compare3(vz340, vz3500) -> error([])
   new_merge00(vz44, vz45, vz46, vz47, LT, cd) -> :(vz45, new_merge2(vz46, :(vz44, vz47), cd))
   new_compare(vz340, vz3500, ty_Char) -> new_compare10(vz340, vz3500)
   new_compare4(@0, @0) -> EQ
   new_compare7(vz340, vz3500) -> error([])
   new_compare14(vz3500, vz35100, ty_Ordering) -> new_compare3(vz3500, vz35100)
   new_compare(vz340, vz3500, app(app(ty_Either, ca), cb)) -> new_compare9(vz340, vz3500, ca, cb)
   new_merge2(:(vz340, vz341), :(vz3500, vz3501), ba) -> new_merge00(vz3500, vz340, vz341, vz3501, new_compare(vz340, vz3500, ba), ba)
   new_compare14(vz3500, vz35100, app(ty_Maybe, be)) -> new_compare5(vz3500, vz35100, be)
   new_compare(vz340, vz3500, ty_Double) -> new_compare7(vz340, vz3500)
   new_compare14(vz3500, vz35100, ty_Int) -> new_compare11(vz3500, vz35100)
   new_merge2(vz34, [], ba) -> vz34
   new_compare(vz340, vz3500, ty_Bool) -> new_compare6(vz340, vz3500)
   new_compare(vz340, vz3500, ty_Integer) -> new_compare2(vz340, vz3500)
   new_compare13(vz340, vz3500, bf) -> error([])
   new_merge_pairs0([], ba) -> []
   new_merge1(vz34, [], :(vz35100, vz35101), ba) -> new_merge2(vz34, :(vz35100, vz35101), ba)
   new_compare2(vz340, vz3500) -> error([])
   new_compare1(vz340, vz3500, bb, bc, bd) -> error([])
   new_compare6(vz340, vz3500) -> error([])
   new_merge00(vz44, vz45, vz46, vz47, GT, cd) -> :(vz44, new_merge2(:(vz45, vz46), vz47, cd))
   new_compare(vz340, vz3500, ty_@0) -> new_compare4(vz340, vz3500)
   new_compare9(vz340, vz3500, ca, cb) -> error([])
   new_compare(vz340, vz3500, app(app(app(ty_@3, bb), bc), bd)) -> new_compare1(vz340, vz3500, bb, bc, bd)
   new_compare14(vz3500, vz35100, ty_Integer) -> new_compare2(vz3500, vz35100)
   new_compare10(vz340, vz3500) -> error([])
   new_merge00(vz44, vz45, vz46, vz47, EQ, cd) -> :(vz45, new_merge2(vz46, :(vz44, vz47), cd))
   new_compare14(vz3500, vz35100, app(app(app(ty_@3, bb), bc), bd)) -> new_compare1(vz3500, vz35100, bb, bc, bd)
   new_compare8(vz340, vz3500, cc) -> error([])
   new_merge1(vz34, vz350, [], ba) -> new_merge2(vz34, vz350, ba)
   new_compare(vz340, vz3500, app(ty_Ratio, cc)) -> new_compare8(vz340, vz3500, cc)
   new_compare14(vz3500, vz35100, ty_@0) -> new_compare4(vz3500, vz35100)
   new_compare14(vz3500, vz35100, app(ty_[], bf)) -> new_compare13(vz3500, vz35100, bf)

The set Q consists of the following terms:

   new_compare14(x0, x1, ty_Double)
   new_merge1(x0, :(x1, x2), :(x3, x4), x5)
   new_compare10(x0, x1)
   new_compare(x0, x1, app(ty_[], x2))
   new_compare13(x0, x1, x2)
   new_compare(x0, x1, app(app(app(ty_@3, x2), x3), x4))
   new_compare14(x0, x1, ty_Bool)
   new_merge2([], :(x0, x1), x2)
   new_compare11(x0, x1)
   new_compare14(x0, x1, ty_@0)
   new_merge_pairs0(:(x0, :(x1, x2)), x3)
   new_compare(x0, x1, ty_Bool)
   new_compare(x0, x1, app(app(ty_Either, x2), x3))
   new_compare4(@0, @0)
   new_compare8(x0, x1, x2)
   new_compare14(x0, x1, ty_Char)
   new_merge00(x0, x1, x2, x3, EQ, x4)
   new_compare14(x0, x1, app(ty_[], x2))
   new_compare14(x0, x1, app(app(ty_Either, x2), x3))
   new_merge1(x0, [], :(x1, x2), x3)
   new_compare14(x0, x1, ty_Ordering)
   new_compare7(x0, x1)
   new_merge2(x0, [], x1)
   new_compare14(x0, x1, app(app(app(ty_@3, x2), x3), x4))
   new_compare(x0, x1, ty_Int)
   new_merge00(x0, x1, x2, x3, GT, x4)
   new_merge1(x0, x1, [], x2)
   new_compare14(x0, x1, app(app(ty_@2, x2), x3))
   new_compare12(x0, x1, x2, x3)
   new_compare14(x0, x1, ty_Int)
   new_compare(x0, x1, ty_@0)
   new_compare(x0, x1, ty_Ordering)
   new_compare(x0, x1, app(ty_Ratio, x2))
   new_compare(x0, x1, app(ty_Maybe, x2))
   new_compare1(x0, x1, x2, x3, x4)
   new_compare(x0, x1, ty_Integer)
   new_compare(x0, x1, app(app(ty_@2, x2), x3))
   new_compare6(x0, x1)
   new_compare2(x0, x1)
   new_compare(x0, x1, ty_Float)
   new_compare9(x0, x1, x2, x3)
   new_compare14(x0, x1, app(ty_Ratio, x2))
   new_compare3(x0, x1)
   new_compare5(x0, x1, x2)
   new_compare(x0, x1, ty_Char)
   new_compare14(x0, x1, app(ty_Maybe, x2))
   new_merge_pairs0(:(x0, []), x1)
   new_compare14(x0, x1, ty_Float)
   new_merge_pairs0([], x0)
   new_compare0(x0, x1)
   new_merge2(:(x0, x1), :(x2, x3), x4)
   new_compare(x0, x1, ty_Double)
   new_compare14(x0, x1, ty_Integer)
   new_merge00(x0, x1, x2, x3, LT, x4)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(39) QDPSizeChangeProof (EQUIVALENT)
We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 

Order:Polynomial interpretation [POLO]:

   POL(:(x_1, x_2)) = 1 + x_2
   POL([]) = 0
   POL(new_merge2(x_1, x_2, x_3)) = 0
   POL(new_merge_pairs0(x_1, x_2)) = x_1




From the DPs we obtained the following set of size-change graphs:
*new_mergesort'(vz34, :(vz350, :(vz3510, vz3511)), ba) -> new_mergesort'(new_merge1(vz34, vz350, vz3510, ba), new_merge_pairs0(vz3511, ba), ba) (allowed arguments on rhs = {2, 3})
The graph contains the following edges 2 > 2, 3 >= 3



We oriented the following set of usable rules [AAECC05,FROCOS05].

   new_merge_pairs0([], ba) -> []
   new_merge_pairs0(:(vz35110, []), ba) -> :(vz35110, [])
   new_merge_pairs0(:(vz35110, :(vz351110, vz351111)), ba) -> :(new_merge2(vz35110, vz351110, ba), new_merge_pairs0(vz351111, ba))

----------------------------------------

(40)
YES