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

YES 0.676 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/Queue.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ BR

mainModule Queue

((queueToList :: Queue a -> [a]) :: Queue a -> [a])

module Queue where

import qualified Prelude

data Queue a = Q [a] [a] [a]

queueToList :: Queue a -> [a]

queueToList

(Q xs ys _)

xs ++ reverse ys

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Queue

((queueToList :: Queue a -> [a]) :: Queue a -> [a])

module Queue where

import qualified Prelude

data Queue a = Q [a] [a] [a]

queueToList :: Queue a -> [a]

queueToList

(Q xs ys vv)

xs ++ reverse ys

Cond Reductions:
The following Function with conditions

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

mainModule Queue

(queueToList :: Queue a -> [a])

module Queue where

import qualified Prelude

data Queue a = Q [a] [a] [a]

queueToList :: Queue a -> [a]

queueToList

(Q xs ys vv)

xs ++ reverse ys

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="queueToList",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="queueToList vy3",fontsize=16,color="burlywood",shape="triangle"];43[label="vy3/Q vy30 vy31 vy32",fontsize=10,color="white",style="solid",shape="box"];3 -> 43[label="",style="solid", color="burlywood", weight=9];
43 -> 4[label="",style="solid", color="burlywood", weight=3];
4[label="queueToList (Q vy30 vy31 vy32)",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="vy30 ++ reverse vy31",fontsize=16,color="burlywood",shape="triangle"];44[label="vy30/vy300 : vy301",fontsize=10,color="white",style="solid",shape="box"];5 -> 44[label="",style="solid", color="burlywood", weight=9];
44 -> 6[label="",style="solid", color="burlywood", weight=3];
45[label="vy30/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 45[label="",style="solid", color="burlywood", weight=9];
45 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="(vy300 : vy301) ++ reverse vy31",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="[] ++ reverse vy31",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="vy300 : vy301 ++ reverse vy31",fontsize=16,color="green",shape="box"];8 -> 10[label="",style="dashed", color="green", weight=3];
9[label="reverse vy31",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10 -> 5[label="",style="dashed", color="red", weight=0];
10[label="vy301 ++ reverse vy31",fontsize=16,color="magenta"];10 -> 12[label="",style="dashed", color="magenta", weight=3];
11[label="foldl (flip (:)) [] vy31",fontsize=16,color="burlywood",shape="box"];47[label="vy31/vy310 : vy311",fontsize=10,color="white",style="solid",shape="box"];11 -> 47[label="",style="solid", color="burlywood", weight=9];
47 -> 13[label="",style="solid", color="burlywood", weight=3];
48[label="vy31/[]",fontsize=10,color="white",style="solid",shape="box"];11 -> 48[label="",style="solid", color="burlywood", weight=9];
48 -> 14[label="",style="solid", color="burlywood", weight=3];
12[label="vy301",fontsize=16,color="green",shape="box"];13[label="foldl (flip (:)) [] (vy310 : vy311)",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
14[label="foldl (flip (:)) [] []",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
15[label="foldl (flip (:)) (flip (:) [] vy310) vy311",fontsize=16,color="burlywood",shape="box"];49[label="vy311/vy3110 : vy3111",fontsize=10,color="white",style="solid",shape="box"];15 -> 49[label="",style="solid", color="burlywood", weight=9];
49 -> 17[label="",style="solid", color="burlywood", weight=3];
50[label="vy311/[]",fontsize=10,color="white",style="solid",shape="box"];15 -> 50[label="",style="solid", color="burlywood", weight=9];
50 -> 18[label="",style="solid", color="burlywood", weight=3];
16[label="[]",fontsize=16,color="green",shape="box"];17[label="foldl (flip (:)) (flip (:) [] vy310) (vy3110 : vy3111)",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
18[label="foldl (flip (:)) (flip (:) [] vy310) []",fontsize=16,color="black",shape="box"];18 -> 20[label="",style="solid", color="black", weight=3];
19 -> 26[label="",style="dashed", color="red", weight=0];
19[label="foldl (flip (:)) (flip (:) (flip (:) [] vy310) vy3110) vy3111",fontsize=16,color="magenta"];19 -> 27[label="",style="dashed", color="magenta", weight=3];
19 -> 28[label="",style="dashed", color="magenta", weight=3];
19 -> 29[label="",style="dashed", color="magenta", weight=3];
19 -> 30[label="",style="dashed", color="magenta", weight=3];
20[label="flip (:) [] vy310",fontsize=16,color="black",shape="box"];20 -> 23[label="",style="solid", color="black", weight=3];
27[label="[]",fontsize=16,color="green",shape="box"];28[label="vy3111",fontsize=16,color="green",shape="box"];29[label="vy310",fontsize=16,color="green",shape="box"];30[label="vy3110",fontsize=16,color="green",shape="box"];26[label="foldl (flip (:)) (flip (:) (flip (:) vy4 vy3110) vy31110) vy31111",fontsize=16,color="burlywood",shape="triangle"];52[label="vy31111/vy311110 : vy311111",fontsize=10,color="white",style="solid",shape="box"];26 -> 52[label="",style="solid", color="burlywood", weight=9];
52 -> 32[label="",style="solid", color="burlywood", weight=3];
53[label="vy31111/[]",fontsize=10,color="white",style="solid",shape="box"];26 -> 53[label="",style="solid", color="burlywood", weight=9];
53 -> 33[label="",style="solid", color="burlywood", weight=3];
23[label="(:) vy310 []",fontsize=16,color="green",shape="box"];32[label="foldl (flip (:)) (flip (:) (flip (:) vy4 vy3110) vy31110) (vy311110 : vy311111)",fontsize=16,color="black",shape="box"];32 -> 34[label="",style="solid", color="black", weight=3];
33[label="foldl (flip (:)) (flip (:) (flip (:) vy4 vy3110) vy31110) []",fontsize=16,color="black",shape="box"];33 -> 35[label="",style="solid", color="black", weight=3];
34 -> 26[label="",style="dashed", color="red", weight=0];
34[label="foldl (flip (:)) (flip (:) (flip (:) (flip (:) vy4 vy3110) vy31110) vy311110) vy311111",fontsize=16,color="magenta"];34 -> 36[label="",style="dashed", color="magenta", weight=3];
34 -> 37[label="",style="dashed", color="magenta", weight=3];
34 -> 38[label="",style="dashed", color="magenta", weight=3];
34 -> 39[label="",style="dashed", color="magenta", weight=3];
35[label="flip (:) (flip (:) vy4 vy3110) vy31110",fontsize=16,color="black",shape="box"];35 -> 40[label="",style="solid", color="black", weight=3];
36[label="flip (:) vy4 vy3110",fontsize=16,color="black",shape="triangle"];36 -> 41[label="",style="solid", color="black", weight=3];
37[label="vy311111",fontsize=16,color="green",shape="box"];38[label="vy31110",fontsize=16,color="green",shape="box"];39[label="vy311110",fontsize=16,color="green",shape="box"];40[label="(:) vy31110 flip (:) vy4 vy3110",fontsize=16,color="green",shape="box"];40 -> 42[label="",style="dashed", color="green", weight=3];
41[label="(:) vy3110 vy4",fontsize=16,color="green",shape="box"];42 -> 36[label="",style="dashed", color="red", weight=0];
42[label="flip (:) vy4 vy3110",fontsize=16,color="magenta"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

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

new_foldl(vy4, vy3110, vy31110, :(vy311110, vy311111), h) → new_foldl(new_flip(vy4, vy3110, h), vy31110, vy311110, vy311111, h)

The TRS R consists of the following rules:

new_flip(vy4, vy3110, h) → :(vy3110, vy4)

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(vy4, vy3110, vy31110, :(vy311110, vy311111), h) → new_foldl(new_flip(vy4, vy3110, h), vy31110, vy311110, vy311111, h)
The graph contains the following edges 3 >= 2, 4 > 3, 4 > 4, 5 >= 5

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ QDPSizeChangeProof

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

new_psPs(:(vy300, vy301), vy31, h) → new_psPs(vy301, vy31, h)

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_psPs(:(vy300, vy301), vy31, h) → new_psPs(vy301, vy31, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3