H-Termination proof of ../tpdb/qualif/reverse

YES H-Termination proof of ../tpdb/qualif/reverse_1.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ NumRed

mainModule Main

((reverse :: List a -> List a) :: List a -> List a)

module Main where

import qualified Prelude

data List a = Cons a (List a) | Nil

flip :: (c -> a -> b) -> a -> c -> b

flip

f x y

f y x

foldl :: (b -> a -> b) -> b -> List a -> b

foldl	f z Nil	=	z
foldl	f z (Cons x xs)	=	foldl f (f z x) xs

reverse :: List a -> List a

reverse

foldl (flip Cons) Nil

Num Reduction:All numbers are transformed to thier corresponding representation with Succ, Pred and Zero.

↳ HASKELL

  ↳ NumRed

    ↳ HASKELL

      ↳ Narrow

mainModule Main

(reverse :: List a -> List a)

module Main where

import qualified Prelude

data List a = Cons a (List a) | Nil

flip :: (a -> b -> c) -> b -> a -> c

flip

f x y

f y x

foldl :: (b -> a -> b) -> b -> List a -> b

foldl	f z Nil	=	z
foldl	f z (Cons x xs)	=	foldl f (f z x) xs

reverse :: List a -> List a

reverse

foldl (flip Cons) Nil

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="reverse\n",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="reverse u3\n",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="foldl (flip Cons) Nil u3\n",fontsize=16,color="burlywood",shape="box"];35[label="u3/Cons u30 u31",fontsize=10,color="white",style="solid",shape="box"];4 -> 35[label="",style="solid", color="burlywood", weight=9];
35 -> 5[label="",style="solid", color="burlywood", weight=3];
36[label="u3/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 36[label="",style="solid", color="burlywood", weight=9];
36 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="foldl (flip Cons) Nil (Cons u30 u31)\n",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="foldl (flip Cons) Nil Nil\n",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="foldl (flip Cons) (flip Cons Nil u30) u31\n",fontsize=16,color="burlywood",shape="box"];37[label="u31/Cons u310 u311",fontsize=10,color="white",style="solid",shape="box"];7 -> 37[label="",style="solid", color="burlywood", weight=9];
37 -> 9[label="",style="solid", color="burlywood", weight=3];
38[label="u31/Nil",fontsize=10,color="white",style="solid",shape="box"];7 -> 38[label="",style="solid", color="burlywood", weight=9];
38 -> 10[label="",style="solid", color="burlywood", weight=3];
8[label="Nil\n",fontsize=16,color="green",shape="box"];9[label="foldl (flip Cons) (flip Cons Nil u30) (Cons u310 u311)\n",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="foldl (flip Cons) (flip Cons Nil u30) Nil\n",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11 -> 18[label="",style="dashed", color="red", weight=0];
11[label="foldl (flip Cons) (flip Cons (flip Cons Nil u30) u310) u311\n",fontsize=16,color="magenta"];11 -> 19[label="",style="dashed", color="magenta", weight=3];
11 -> 20[label="",style="dashed", color="magenta", weight=3];
11 -> 21[label="",style="dashed", color="magenta", weight=3];
11 -> 22[label="",style="dashed", color="magenta", weight=3];
12[label="flip Cons Nil u30\n",fontsize=16,color="black",shape="box"];12 -> 15[label="",style="solid", color="black", weight=3];
19[label="u30\n",fontsize=16,color="green",shape="box"];20[label="u311\n",fontsize=16,color="green",shape="box"];21[label="u310\n",fontsize=16,color="green",shape="box"];22[label="Nil\n",fontsize=16,color="green",shape="box"];18[label="foldl (flip Cons) (flip Cons (flip Cons u4 u310) u3110) u3111\n",fontsize=16,color="burlywood",shape="triangle"];40[label="u3111/Cons u31110 u31111",fontsize=10,color="white",style="solid",shape="box"];18 -> 40[label="",style="solid", color="burlywood", weight=9];
40 -> 24[label="",style="solid", color="burlywood", weight=3];
41[label="u3111/Nil",fontsize=10,color="white",style="solid",shape="box"];18 -> 41[label="",style="solid", color="burlywood", weight=9];
41 -> 25[label="",style="solid", color="burlywood", weight=3];
15[label="Cons u30 Nil\n",fontsize=16,color="green",shape="box"];24[label="foldl (flip Cons) (flip Cons (flip Cons u4 u310) u3110) (Cons u31110 u31111)\n",fontsize=16,color="black",shape="box"];24 -> 26[label="",style="solid", color="black", weight=3];
25[label="foldl (flip Cons) (flip Cons (flip Cons u4 u310) u3110) Nil\n",fontsize=16,color="black",shape="box"];25 -> 27[label="",style="solid", color="black", weight=3];
26 -> 18[label="",style="dashed", color="red", weight=0];
26[label="foldl (flip Cons) (flip Cons (flip Cons (flip Cons u4 u310) u3110) u31110) u31111\n",fontsize=16,color="magenta"];26 -> 28[label="",style="dashed", color="magenta", weight=3];
26 -> 29[label="",style="dashed", color="magenta", weight=3];
26 -> 30[label="",style="dashed", color="magenta", weight=3];
26 -> 31[label="",style="dashed", color="magenta", weight=3];
27[label="flip Cons (flip Cons u4 u310) u3110\n",fontsize=16,color="black",shape="box"];27 -> 32[label="",style="solid", color="black", weight=3];
28[label="u3110\n",fontsize=16,color="green",shape="box"];29[label="u31111\n",fontsize=16,color="green",shape="box"];30[label="u31110\n",fontsize=16,color="green",shape="box"];31[label="flip Cons u4 u310\n",fontsize=16,color="black",shape="triangle"];31 -> 33[label="",style="solid", color="black", weight=3];
32[label="Cons u3110 (flip Cons u4 u310)\n",fontsize=16,color="green",shape="box"];32 -> 34[label="",style="dashed", color="green", weight=3];
33[label="Cons u310 u4\n",fontsize=16,color="green",shape="box"];34 -> 31[label="",style="dashed", color="red", weight=0];
34[label="flip Cons u4 u310\n",fontsize=16,color="magenta"];}

↳ HASKELL

  ↳ NumRed

    ↳ HASKELL

      ↳ Narrow

        ↳ QDP

          ↳ QDPSizeChangeProof

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

new_foldl(u4, u310, u3110, Cons(u31110, u31111), d) → new_foldl(new_flip(u4, u310, d), u3110, u31110, u31111, d)

The TRS R consists of the following rules:

new_flip(u4, u310, d) → Cons(u310, u4)

The set Q consists of the following terms:

new_flip(x0, x1, x2)

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_foldl(u4, u310, u3110, Cons(u31110, u31111), d) → new_foldl(new_flip(u4, u310, d), u3110, u31110, u31111, d)
The graph contains the following edges 3 >= 2, 4 > 3, 4 > 4, 5 >= 5