H-Termination proof of /home/matraf/haskell/eval_FullyBlown

YES 1.7870000000000001 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/List.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ LR

mainModule List

((transpose :: [[a]] -> [[a]]) :: [[a]] -> [[a]])

module List where

import qualified Maybe
import qualified Prelude

transpose :: [[a]] -> [[a]]

transpose

[]

transpose

([] : xss)

transpose xss

transpose

((x : xs) : xss)

(x : concatMap (\vv3 ->

case	vv3 of
	h : t	->	h : []
	_	->	[]

) xss) : transpose (xs : concatMap (\vv4 ->

case	vv4 of
	h : t	->	t : []
	_	->	[]

) xss)

module Maybe where

import qualified List
import qualified Prelude

Lambda Reductions:
The following Lambda expression

\vv3→

case vv3 of
h : t → h : []

_ → []

is transformed to

transpose0 vv3 =

case vv3 of
h : t → h : []

_ → []

The following Lambda expression

\vv4→

case vv4 of
h : t → t : []

_ → []

is transformed to

transpose1 vv4 =

case vv4 of
h : t → t : []

_ → []

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

mainModule List

((transpose :: [[a]] -> [[a]]) :: [[a]] -> [[a]])

module List where

import qualified Maybe
import qualified Prelude

transpose :: [[a]] -> [[a]]

transpose	[]	=	[]
transpose	([] : xss)	=	transpose xss
transpose	((x : xs) : xss)	=	(x : concatMap transpose0 xss) : transpose (xs : concatMap transpose1 xss)

transpose0

vv3

case	vv3 of
	h : t	->	h : []
	_	->	[]

transpose1

vv4

case	vv4 of
	h : t	->	t : []
	_	->	[]

module Maybe where

import qualified List
import qualified Prelude

Case Reductions:
The following Case expression

case vv3 of
h : t → h : []

_ → []

is transformed to

transpose00 (h : t) = h : []

transpose00 _ = []

The following Case expression

case vv4 of
h : t → t : []

_ → []

is transformed to

transpose10 (h : t) = t : []

transpose10 _ = []

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ BR

mainModule List

((transpose :: [[a]] -> [[a]]) :: [[a]] -> [[a]])

module List where

import qualified Maybe
import qualified Prelude

transpose :: [[a]] -> [[a]]

transpose	[]	=	[]
transpose	([] : xss)	=	transpose xss
transpose	((x : xs) : xss)	=	(x : concatMap transpose0 xss) : transpose (xs : concatMap transpose1 xss)

transpose0

vv3

transpose00 vv3

transpose00	(h : t)	=	h : []
transpose00	_	=	[]

transpose1

vv4

transpose10 vv4

transpose10	(h : t)	=	t : []
transpose10	_	=	[]

module Maybe where

import qualified List
import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

mainModule List

((transpose :: [[a]] -> [[a]]) :: [[a]] -> [[a]])

module List where

import qualified Maybe
import qualified Prelude

transpose :: [[a]] -> [[a]]

transpose	[]	=	[]
transpose	([] : xss)	=	transpose xss
transpose	((x : xs) : xss)	=	(x : concatMap transpose0 xss) : transpose (xs : concatMap transpose1 xss)

transpose0

vv3

transpose00 vv3

transpose00	(h : t)	=	h : []
transpose00	vx	=	[]

transpose1

vv4

transpose10 vv4

transpose10	(h : t)	=	t : []
transpose10	vw	=	[]

module Maybe where

import qualified List
import qualified Prelude

Cond Reductions:
The following Function with conditions

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ Narrow

mainModule List

(transpose :: [[a]] -> [[a]])

module List where

import qualified Maybe
import qualified Prelude

transpose :: [[a]] -> [[a]]

transpose	[]	=	[]
transpose	([] : xss)	=	transpose xss
transpose	((x : xs) : xss)	=	(x : concatMap transpose0 xss) : transpose (xs : concatMap transpose1 xss)

transpose0

vv3

transpose00 vv3

transpose00	(h : t)	=	h : []
transpose00	vx	=	[]

transpose1

vv4

transpose10 vv4

transpose10	(h : t)	=	t : []
transpose10	vw	=	[]

module Maybe where

import qualified List
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="transpose",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="transpose wu3",fontsize=16,color="burlywood",shape="triangle"];55[label="wu3/wu30 : wu31",fontsize=10,color="white",style="solid",shape="box"];3 -> 55[label="",style="solid", color="burlywood", weight=9];
55 -> 4[label="",style="solid", color="burlywood", weight=3];
56[label="wu3/[]",fontsize=10,color="white",style="solid",shape="box"];3 -> 56[label="",style="solid", color="burlywood", weight=9];
56 -> 5[label="",style="solid", color="burlywood", weight=3];
4[label="transpose (wu30 : wu31)",fontsize=16,color="burlywood",shape="box"];57[label="wu30/wu300 : wu301",fontsize=10,color="white",style="solid",shape="box"];4 -> 57[label="",style="solid", color="burlywood", weight=9];
57 -> 6[label="",style="solid", color="burlywood", weight=3];
58[label="wu30/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 58[label="",style="solid", color="burlywood", weight=9];
58 -> 7[label="",style="solid", color="burlywood", weight=3];
5[label="transpose []",fontsize=16,color="black",shape="box"];5 -> 8[label="",style="solid", color="black", weight=3];
6[label="transpose ((wu300 : wu301) : wu31)",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
7[label="transpose ([] : wu31)",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
8[label="[]",fontsize=16,color="green",shape="box"];9[label="(wu300 : concatMap transpose0 wu31) : transpose (wu301 : concatMap transpose1 wu31)",fontsize=16,color="green",shape="box"];9 -> 11[label="",style="dashed", color="green", weight=3];
9 -> 12[label="",style="dashed", color="green", weight=3];
10 -> 3[label="",style="dashed", color="red", weight=0];
10[label="transpose wu31",fontsize=16,color="magenta"];10 -> 13[label="",style="dashed", color="magenta", weight=3];
11[label="concatMap transpose0 wu31",fontsize=16,color="black",shape="box"];11 -> 14[label="",style="solid", color="black", weight=3];
12 -> 3[label="",style="dashed", color="red", weight=0];
12[label="transpose (wu301 : concatMap transpose1 wu31)",fontsize=16,color="magenta"];12 -> 15[label="",style="dashed", color="magenta", weight=3];
13[label="wu31",fontsize=16,color="green",shape="box"];14[label="concat . map transpose0",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
15[label="wu301 : concatMap transpose1 wu31",fontsize=16,color="green",shape="box"];15 -> 17[label="",style="dashed", color="green", weight=3];
16[label="concat (map transpose0 wu31)",fontsize=16,color="black",shape="box"];16 -> 18[label="",style="solid", color="black", weight=3];
17[label="concatMap transpose1 wu31",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
18[label="foldr (++) [] (map transpose0 wu31)",fontsize=16,color="burlywood",shape="triangle"];61[label="wu31/wu310 : wu311",fontsize=10,color="white",style="solid",shape="box"];18 -> 61[label="",style="solid", color="burlywood", weight=9];
61 -> 20[label="",style="solid", color="burlywood", weight=3];
62[label="wu31/[]",fontsize=10,color="white",style="solid",shape="box"];18 -> 62[label="",style="solid", color="burlywood", weight=9];
62 -> 21[label="",style="solid", color="burlywood", weight=3];
19[label="concat . map transpose1",fontsize=16,color="black",shape="box"];19 -> 22[label="",style="solid", color="black", weight=3];
20[label="foldr (++) [] (map transpose0 (wu310 : wu311))",fontsize=16,color="black",shape="box"];20 -> 23[label="",style="solid", color="black", weight=3];
21[label="foldr (++) [] (map transpose0 [])",fontsize=16,color="black",shape="box"];21 -> 24[label="",style="solid", color="black", weight=3];
22[label="concat (map transpose1 wu31)",fontsize=16,color="black",shape="box"];22 -> 25[label="",style="solid", color="black", weight=3];
23[label="foldr (++) [] (transpose0 wu310 : map transpose0 wu311)",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
24[label="foldr (++) [] []",fontsize=16,color="black",shape="triangle"];24 -> 27[label="",style="solid", color="black", weight=3];
25[label="foldr (++) [] (map transpose1 wu31)",fontsize=16,color="burlywood",shape="triangle"];63[label="wu31/wu310 : wu311",fontsize=10,color="white",style="solid",shape="box"];25 -> 63[label="",style="solid", color="burlywood", weight=9];
63 -> 28[label="",style="solid", color="burlywood", weight=3];
64[label="wu31/[]",fontsize=10,color="white",style="solid",shape="box"];25 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 29[label="",style="solid", color="burlywood", weight=3];
26 -> 30[label="",style="dashed", color="red", weight=0];
26[label="(++) transpose0 wu310 foldr (++) [] (map transpose0 wu311)",fontsize=16,color="magenta"];26 -> 31[label="",style="dashed", color="magenta", weight=3];
27[label="[]",fontsize=16,color="green",shape="box"];28[label="foldr (++) [] (map transpose1 (wu310 : wu311))",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="foldr (++) [] (map transpose1 [])",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
31 -> 18[label="",style="dashed", color="red", weight=0];
31[label="foldr (++) [] (map transpose0 wu311)",fontsize=16,color="magenta"];31 -> 34[label="",style="dashed", color="magenta", weight=3];
30[label="(++) transpose0 wu310 wu4",fontsize=16,color="black",shape="triangle"];30 -> 35[label="",style="solid", color="black", weight=3];
32[label="foldr (++) [] (transpose1 wu310 : map transpose1 wu311)",fontsize=16,color="black",shape="box"];32 -> 36[label="",style="solid", color="black", weight=3];
33 -> 24[label="",style="dashed", color="red", weight=0];
33[label="foldr (++) [] []",fontsize=16,color="magenta"];34[label="wu311",fontsize=16,color="green",shape="box"];35[label="(++) transpose00 wu310 wu4",fontsize=16,color="burlywood",shape="box"];68[label="wu310/wu3100 : wu3101",fontsize=10,color="white",style="solid",shape="box"];35 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 37[label="",style="solid", color="burlywood", weight=3];
69[label="wu310/[]",fontsize=10,color="white",style="solid",shape="box"];35 -> 69[label="",style="solid", color="burlywood", weight=9];
69 -> 38[label="",style="solid", color="burlywood", weight=3];
36 -> 39[label="",style="dashed", color="red", weight=0];
36[label="(++) transpose1 wu310 foldr (++) [] (map transpose1 wu311)",fontsize=16,color="magenta"];36 -> 40[label="",style="dashed", color="magenta", weight=3];
37[label="(++) transpose00 (wu3100 : wu3101) wu4",fontsize=16,color="black",shape="box"];37 -> 41[label="",style="solid", color="black", weight=3];
38[label="(++) transpose00 [] wu4",fontsize=16,color="black",shape="box"];38 -> 42[label="",style="solid", color="black", weight=3];
40 -> 25[label="",style="dashed", color="red", weight=0];
40[label="foldr (++) [] (map transpose1 wu311)",fontsize=16,color="magenta"];40 -> 43[label="",style="dashed", color="magenta", weight=3];
39[label="(++) transpose1 wu310 wu5",fontsize=16,color="black",shape="triangle"];39 -> 44[label="",style="solid", color="black", weight=3];
41[label="(++) (wu3100 : []) wu4",fontsize=16,color="black",shape="triangle"];41 -> 45[label="",style="solid", color="black", weight=3];
42[label="(++) [] wu4",fontsize=16,color="black",shape="triangle"];42 -> 46[label="",style="solid", color="black", weight=3];
43[label="wu311",fontsize=16,color="green",shape="box"];44[label="(++) transpose10 wu310 wu5",fontsize=16,color="burlywood",shape="box"];72[label="wu310/wu3100 : wu3101",fontsize=10,color="white",style="solid",shape="box"];44 -> 72[label="",style="solid", color="burlywood", weight=9];
72 -> 47[label="",style="solid", color="burlywood", weight=3];
73[label="wu310/[]",fontsize=10,color="white",style="solid",shape="box"];44 -> 73[label="",style="solid", color="burlywood", weight=9];
73 -> 48[label="",style="solid", color="burlywood", weight=3];
45[label="wu3100 : [] ++ wu4",fontsize=16,color="green",shape="box"];45 -> 49[label="",style="dashed", color="green", weight=3];
46[label="wu4",fontsize=16,color="green",shape="box"];47[label="(++) transpose10 (wu3100 : wu3101) wu5",fontsize=16,color="black",shape="box"];47 -> 50[label="",style="solid", color="black", weight=3];
48[label="(++) transpose10 [] wu5",fontsize=16,color="black",shape="box"];48 -> 51[label="",style="solid", color="black", weight=3];
49 -> 42[label="",style="dashed", color="red", weight=0];
49[label="[] ++ wu4",fontsize=16,color="magenta"];50 -> 41[label="",style="dashed", color="red", weight=0];
50[label="(++) (wu3101 : []) wu5",fontsize=16,color="magenta"];50 -> 52[label="",style="dashed", color="magenta", weight=3];
50 -> 53[label="",style="dashed", color="magenta", weight=3];
51 -> 42[label="",style="dashed", color="red", weight=0];
51[label="(++) [] wu5",fontsize=16,color="magenta"];51 -> 54[label="",style="dashed", color="magenta", weight=3];
52[label="wu3101",fontsize=16,color="green",shape="box"];53[label="wu5",fontsize=16,color="green",shape="box"];54[label="wu5",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                        ↳ QDPSizeChangeProof

                      ↳ QDP

                      ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

new_foldr(:(wu310, wu311), ba) → new_foldr(wu311, ba)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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_foldr(:(wu310, wu311), ba) → new_foldr(wu311, ba)
The graph contains the following edges 1 > 1, 2 >= 2

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPSizeChangeProof

                      ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

new_foldr0(:(wu310, wu311), ba) → new_foldr0(wu311, ba)

From the DPs we obtained the following set of size-change graphs:

new_foldr0(:(wu310, wu311), ba) → new_foldr0(wu311, ba)
The graph contains the following edges 1 > 1, 2 >= 2

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                        ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

new_transpose(:([], wu31), ba) → new_transpose(wu31, ba)
new_transpose(:(:(wu300, wu301), wu31), ba) → new_transpose(:(wu301, new_foldr1(wu31, ba)), ba)

The TRS R consists of the following rules:

new_psPs1(:(wu3100, wu3101), wu5, ba) → new_psPs0(wu3101, wu5, app(ty_[], ba))
new_psPs(wu4, ba) → wu4
new_psPs0(wu3100, wu4, ba) → :(wu3100, new_psPs(wu4, ba))
new_foldr1([], ba) → new_foldr2(app(ty_[], ba))
new_psPs1([], wu5, ba) → new_psPs(wu5, app(ty_[], ba))
new_foldr1(:(wu310, wu311), ba) → new_psPs1(wu310, new_foldr1(wu311, ba), ba)
new_foldr2(ba) → []

The set Q consists of the following terms:

new_foldr1([], x0)
new_psPs1([], x0, x1)
new_foldr2(x0)
new_psPs0(x0, x1, x2)
new_psPs(x0, x1)
new_psPs1(:(x0, x1), x2, x3)
new_foldr1(:(x0, x1), x2)

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

new_transpose(:([], wu31), ba) → new_transpose(wu31, ba)
new_transpose(:(:(wu300, wu301), wu31), ba) → new_transpose(:(wu301, new_foldr1(wu31, ba)), ba)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25]:

POL(:(x₁, x₂)) = 1 + x₁ + x₂
POL([]) = 0
POL(app(x₁, x₂)) = 0
POL(new_foldr1(x₁, x₂)) = x₁
POL(new_foldr2(x₁)) = 0
POL(new_psPs(x₁, x₂)) = x₁
POL(new_psPs0(x₁, x₂, x₃)) = 1 + x₁ + x₂
POL(new_psPs1(x₁, x₂, x₃)) = 1 + x₁ + x₂
POL(new_transpose(x₁, x₂)) = x₁
POL(ty_[]) = 0

The following usable rules [17] were oriented:

new_psPs0(wu3100, wu4, ba) → :(wu3100, new_psPs(wu4, ba))
new_foldr2(ba) → []
new_psPs1([], wu5, ba) → new_psPs(wu5, app(ty_[], ba))
new_foldr1(:(wu310, wu311), ba) → new_psPs1(wu310, new_foldr1(wu311, ba), ba)
new_psPs(wu4, ba) → wu4
new_psPs1(:(wu3100, wu3101), wu5, ba) → new_psPs0(wu3101, wu5, app(ty_[], ba))
new_foldr1([], ba) → new_foldr2(app(ty_[], ba))

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ PisEmptyProof

                        ↳ QDPOrderProof

Q DP problem:
P is empty.
The TRS R consists of the following rules:

new_psPs1(:(wu3100, wu3101), wu5, ba) → new_psPs0(wu3101, wu5, app(ty_[], ba))
new_psPs(wu4, ba) → wu4
new_psPs0(wu3100, wu4, ba) → :(wu3100, new_psPs(wu4, ba))
new_foldr1([], ba) → new_foldr2(app(ty_[], ba))
new_psPs1([], wu5, ba) → new_psPs(wu5, app(ty_[], ba))
new_foldr1(:(wu310, wu311), ba) → new_psPs1(wu310, new_foldr1(wu311, ba), ba)
new_foldr2(ba) → []

The set Q consists of the following terms:

new_foldr1([], x0)
new_psPs1([], x0, x1)
new_foldr2(x0)
new_psPs0(x0, x1, x2)
new_psPs(x0, x1)
new_psPs1(:(x0, x1), x2, x3)
new_foldr1(:(x0, x1), x2)

We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

new_transpose(:([], wu31), ba) → new_transpose(wu31, ba)
new_transpose(:(:(wu300, wu301), wu31), ba) → new_transpose(:(wu301, new_foldr1(wu31, ba)), ba)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( new_foldr1(x₁, x₂) ) =

/	1	\
\	0	/

/	0	1	\
\	0	1	/

x₁

/	0	0	\
\	0	0	/

x₂

M( new_foldr2(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( [] ) =

/	0	\
\	0	/

M( app(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( new_psPs1(x₁, ..., x₃) ) =

/	0	\
\	1	/

/	0	0	\
\	0	1	/

x₁

/	1	0	\
\	0	1	/

x₂

/	0	0	\
\	0	0	/

x₃

M( new_psPs(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

/	0	0	\
\	0	0	/

x₂

M( new_psPs0(x₁, ..., x₃) ) =

/	0	\
\	1	/

/	0	0	\
\	0	1	/

x₁

/	0	0	\
\	0	1	/

x₂

/	0	0	\
\	0	0	/

x₃

M( :(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	1	/

x₁

/	0	0	\
\	0	1	/

x₂

M( ty_[] ) =

/	0	\
\	0	/

Tuple symbols:

M( new_transpose(x₁, x₂) ) =

[

]

x₁

[

]

x₂

Matrix type:
We used a basic matrix type which is not further parametrizeable.

As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:

new_psPs0(wu3100, wu4, ba) → :(wu3100, new_psPs(wu4, ba))
new_foldr2(ba) → []
new_psPs1([], wu5, ba) → new_psPs(wu5, app(ty_[], ba))
new_foldr1(:(wu310, wu311), ba) → new_psPs1(wu310, new_foldr1(wu311, ba), ba)
new_psPs(wu4, ba) → wu4
new_psPs1(:(wu3100, wu3101), wu5, ba) → new_psPs0(wu3101, wu5, app(ty_[], ba))
new_foldr1([], ba) → new_foldr2(app(ty_[], ba))

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                        ↳ QDPOrderProof

                          ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

new_psPs1(:(wu3100, wu3101), wu5, ba) → new_psPs0(wu3101, wu5, app(ty_[], ba))
new_psPs(wu4, ba) → wu4
new_psPs0(wu3100, wu4, ba) → :(wu3100, new_psPs(wu4, ba))
new_foldr1([], ba) → new_foldr2(app(ty_[], ba))
new_psPs1([], wu5, ba) → new_psPs(wu5, app(ty_[], ba))
new_foldr1(:(wu310, wu311), ba) → new_psPs1(wu310, new_foldr1(wu311, ba), ba)
new_foldr2(ba) → []

The set Q consists of the following terms:

new_foldr1([], x0)
new_psPs1([], x0, x1)
new_foldr2(x0)
new_psPs0(x0, x1, x2)
new_psPs(x0, x1)
new_psPs1(:(x0, x1), x2, x3)
new_foldr1(:(x0, x1), x2)

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