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

YES 0.793 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/empty.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ BR

mainModule Main

((lookup :: Ordering -> [(Ordering,a)] -> Maybe a) :: Ordering -> [(Ordering,a)] -> Maybe a)

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

((lookup :: Ordering -> [(Ordering,a)] -> Maybe a) :: Ordering -> [(Ordering,a)] -> Maybe a)

module Main where

import qualified Prelude

Cond Reductions:
The following Function with conditions

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

The following Function with conditions

lookup k [] = Nothing

lookup k ((x,y) : xys)

| k == x

= Just y

| otherwise

= lookup k xys

is transformed to

lookup k [] = lookup3 k []

lookup k ((x,y) : xys) = lookup2 k ((x,y) : xys)

lookup0 k x y xys True = lookup k xys

lookup1 k x y xys True = Just y

lookup1 k x y xys False = lookup0 k x y xys otherwise

lookup2 k ((x,y) : xys) = lookup1 k x y xys (k == x)

lookup3 k [] = Nothing

lookup3 wu wv = lookup2 wu wv

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

mainModule Main

(lookup :: Ordering -> [(Ordering,a)] -> Maybe a)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="lookup",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="lookup ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="lookup ww3 ww4",fontsize=16,color="burlywood",shape="triangle"];66[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];4 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 5[label="",style="solid", color="burlywood", weight=3];
67[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="lookup ww3 (ww40 : ww41)",fontsize=16,color="burlywood",shape="box"];68[label="ww40/(ww400,ww401)",fontsize=10,color="white",style="solid",shape="box"];5 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="lookup ww3 []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="lookup ww3 ((ww400,ww401) : ww41)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="lookup3 ww3 []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="lookup2 ww3 ((ww400,ww401) : ww41)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="Nothing",fontsize=16,color="green",shape="box"];11[label="lookup1 ww3 ww400 ww401 ww41 (ww3 == ww400)",fontsize=16,color="burlywood",shape="box"];69[label="ww3/LT",fontsize=10,color="white",style="solid",shape="box"];11 -> 69[label="",style="solid", color="burlywood", weight=9];
69 -> 12[label="",style="solid", color="burlywood", weight=3];
70[label="ww3/EQ",fontsize=10,color="white",style="solid",shape="box"];11 -> 70[label="",style="solid", color="burlywood", weight=9];
70 -> 13[label="",style="solid", color="burlywood", weight=3];
71[label="ww3/GT",fontsize=10,color="white",style="solid",shape="box"];11 -> 71[label="",style="solid", color="burlywood", weight=9];
71 -> 14[label="",style="solid", color="burlywood", weight=3];
12[label="lookup1 LT ww400 ww401 ww41 (LT == ww400)",fontsize=16,color="burlywood",shape="box"];72[label="ww400/LT",fontsize=10,color="white",style="solid",shape="box"];12 -> 72[label="",style="solid", color="burlywood", weight=9];
72 -> 15[label="",style="solid", color="burlywood", weight=3];
73[label="ww400/EQ",fontsize=10,color="white",style="solid",shape="box"];12 -> 73[label="",style="solid", color="burlywood", weight=9];
73 -> 16[label="",style="solid", color="burlywood", weight=3];
74[label="ww400/GT",fontsize=10,color="white",style="solid",shape="box"];12 -> 74[label="",style="solid", color="burlywood", weight=9];
74 -> 17[label="",style="solid", color="burlywood", weight=3];
13[label="lookup1 EQ ww400 ww401 ww41 (EQ == ww400)",fontsize=16,color="burlywood",shape="box"];75[label="ww400/LT",fontsize=10,color="white",style="solid",shape="box"];13 -> 75[label="",style="solid", color="burlywood", weight=9];
75 -> 18[label="",style="solid", color="burlywood", weight=3];
76[label="ww400/EQ",fontsize=10,color="white",style="solid",shape="box"];13 -> 76[label="",style="solid", color="burlywood", weight=9];
76 -> 19[label="",style="solid", color="burlywood", weight=3];
77[label="ww400/GT",fontsize=10,color="white",style="solid",shape="box"];13 -> 77[label="",style="solid", color="burlywood", weight=9];
77 -> 20[label="",style="solid", color="burlywood", weight=3];
14[label="lookup1 GT ww400 ww401 ww41 (GT == ww400)",fontsize=16,color="burlywood",shape="box"];78[label="ww400/LT",fontsize=10,color="white",style="solid",shape="box"];14 -> 78[label="",style="solid", color="burlywood", weight=9];
78 -> 21[label="",style="solid", color="burlywood", weight=3];
79[label="ww400/EQ",fontsize=10,color="white",style="solid",shape="box"];14 -> 79[label="",style="solid", color="burlywood", weight=9];
79 -> 22[label="",style="solid", color="burlywood", weight=3];
80[label="ww400/GT",fontsize=10,color="white",style="solid",shape="box"];14 -> 80[label="",style="solid", color="burlywood", weight=9];
80 -> 23[label="",style="solid", color="burlywood", weight=3];
15[label="lookup1 LT LT ww401 ww41 (LT == LT)",fontsize=16,color="black",shape="box"];15 -> 24[label="",style="solid", color="black", weight=3];
16[label="lookup1 LT EQ ww401 ww41 (LT == EQ)",fontsize=16,color="black",shape="box"];16 -> 25[label="",style="solid", color="black", weight=3];
17[label="lookup1 LT GT ww401 ww41 (LT == GT)",fontsize=16,color="black",shape="box"];17 -> 26[label="",style="solid", color="black", weight=3];
18[label="lookup1 EQ LT ww401 ww41 (EQ == LT)",fontsize=16,color="black",shape="box"];18 -> 27[label="",style="solid", color="black", weight=3];
19[label="lookup1 EQ EQ ww401 ww41 (EQ == EQ)",fontsize=16,color="black",shape="box"];19 -> 28[label="",style="solid", color="black", weight=3];
20[label="lookup1 EQ GT ww401 ww41 (EQ == GT)",fontsize=16,color="black",shape="box"];20 -> 29[label="",style="solid", color="black", weight=3];
21[label="lookup1 GT LT ww401 ww41 (GT == LT)",fontsize=16,color="black",shape="box"];21 -> 30[label="",style="solid", color="black", weight=3];
22[label="lookup1 GT EQ ww401 ww41 (GT == EQ)",fontsize=16,color="black",shape="box"];22 -> 31[label="",style="solid", color="black", weight=3];
23[label="lookup1 GT GT ww401 ww41 (GT == GT)",fontsize=16,color="black",shape="box"];23 -> 32[label="",style="solid", color="black", weight=3];
24[label="lookup1 LT LT ww401 ww41 True",fontsize=16,color="black",shape="box"];24 -> 33[label="",style="solid", color="black", weight=3];
25[label="lookup1 LT EQ ww401 ww41 False",fontsize=16,color="black",shape="box"];25 -> 34[label="",style="solid", color="black", weight=3];
26[label="lookup1 LT GT ww401 ww41 False",fontsize=16,color="black",shape="box"];26 -> 35[label="",style="solid", color="black", weight=3];
27[label="lookup1 EQ LT ww401 ww41 False",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3];
28[label="lookup1 EQ EQ ww401 ww41 True",fontsize=16,color="black",shape="box"];28 -> 37[label="",style="solid", color="black", weight=3];
29[label="lookup1 EQ GT ww401 ww41 False",fontsize=16,color="black",shape="box"];29 -> 38[label="",style="solid", color="black", weight=3];
30[label="lookup1 GT LT ww401 ww41 False",fontsize=16,color="black",shape="box"];30 -> 39[label="",style="solid", color="black", weight=3];
31[label="lookup1 GT EQ ww401 ww41 False",fontsize=16,color="black",shape="box"];31 -> 40[label="",style="solid", color="black", weight=3];
32[label="lookup1 GT GT ww401 ww41 True",fontsize=16,color="black",shape="box"];32 -> 41[label="",style="solid", color="black", weight=3];
33[label="Just ww401",fontsize=16,color="green",shape="box"];34[label="lookup0 LT EQ ww401 ww41 otherwise",fontsize=16,color="black",shape="box"];34 -> 42[label="",style="solid", color="black", weight=3];
35[label="lookup0 LT GT ww401 ww41 otherwise",fontsize=16,color="black",shape="box"];35 -> 43[label="",style="solid", color="black", weight=3];
36[label="lookup0 EQ LT ww401 ww41 otherwise",fontsize=16,color="black",shape="box"];36 -> 44[label="",style="solid", color="black", weight=3];
37[label="Just ww401",fontsize=16,color="green",shape="box"];38[label="lookup0 EQ GT ww401 ww41 otherwise",fontsize=16,color="black",shape="box"];38 -> 45[label="",style="solid", color="black", weight=3];
39[label="lookup0 GT LT ww401 ww41 otherwise",fontsize=16,color="black",shape="box"];39 -> 46[label="",style="solid", color="black", weight=3];
40[label="lookup0 GT EQ ww401 ww41 otherwise",fontsize=16,color="black",shape="box"];40 -> 47[label="",style="solid", color="black", weight=3];
41[label="Just ww401",fontsize=16,color="green",shape="box"];42[label="lookup0 LT EQ ww401 ww41 True",fontsize=16,color="black",shape="box"];42 -> 48[label="",style="solid", color="black", weight=3];
43[label="lookup0 LT GT ww401 ww41 True",fontsize=16,color="black",shape="box"];43 -> 49[label="",style="solid", color="black", weight=3];
44[label="lookup0 EQ LT ww401 ww41 True",fontsize=16,color="black",shape="box"];44 -> 50[label="",style="solid", color="black", weight=3];
45[label="lookup0 EQ GT ww401 ww41 True",fontsize=16,color="black",shape="box"];45 -> 51[label="",style="solid", color="black", weight=3];
46[label="lookup0 GT LT ww401 ww41 True",fontsize=16,color="black",shape="box"];46 -> 52[label="",style="solid", color="black", weight=3];
47[label="lookup0 GT EQ ww401 ww41 True",fontsize=16,color="black",shape="box"];47 -> 53[label="",style="solid", color="black", weight=3];
48 -> 4[label="",style="dashed", color="red", weight=0];
48[label="lookup LT ww41",fontsize=16,color="magenta"];48 -> 54[label="",style="dashed", color="magenta", weight=3];
48 -> 55[label="",style="dashed", color="magenta", weight=3];
49 -> 4[label="",style="dashed", color="red", weight=0];
49[label="lookup LT ww41",fontsize=16,color="magenta"];49 -> 56[label="",style="dashed", color="magenta", weight=3];
49 -> 57[label="",style="dashed", color="magenta", weight=3];
50 -> 4[label="",style="dashed", color="red", weight=0];
50[label="lookup EQ ww41",fontsize=16,color="magenta"];50 -> 58[label="",style="dashed", color="magenta", weight=3];
50 -> 59[label="",style="dashed", color="magenta", weight=3];
51 -> 4[label="",style="dashed", color="red", weight=0];
51[label="lookup EQ ww41",fontsize=16,color="magenta"];51 -> 60[label="",style="dashed", color="magenta", weight=3];
51 -> 61[label="",style="dashed", color="magenta", weight=3];
52 -> 4[label="",style="dashed", color="red", weight=0];
52[label="lookup GT ww41",fontsize=16,color="magenta"];52 -> 62[label="",style="dashed", color="magenta", weight=3];
52 -> 63[label="",style="dashed", color="magenta", weight=3];
53 -> 4[label="",style="dashed", color="red", weight=0];
53[label="lookup GT ww41",fontsize=16,color="magenta"];53 -> 64[label="",style="dashed", color="magenta", weight=3];
53 -> 65[label="",style="dashed", color="magenta", weight=3];
54[label="LT",fontsize=16,color="green",shape="box"];55[label="ww41",fontsize=16,color="green",shape="box"];56[label="LT",fontsize=16,color="green",shape="box"];57[label="ww41",fontsize=16,color="green",shape="box"];58[label="EQ",fontsize=16,color="green",shape="box"];59[label="ww41",fontsize=16,color="green",shape="box"];60[label="EQ",fontsize=16,color="green",shape="box"];61[label="ww41",fontsize=16,color="green",shape="box"];62[label="GT",fontsize=16,color="green",shape="box"];63[label="ww41",fontsize=16,color="green",shape="box"];64[label="GT",fontsize=16,color="green",shape="box"];65[label="ww41",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ QDP

              ↳ DependencyGraphProof

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

new_lookup(LT, :(@2(GT, ww401), ww41), ba) → new_lookup(LT, ww41, ba)
new_lookup(GT, :(@2(EQ, ww401), ww41), ba) → new_lookup(GT, ww41, ba)
new_lookup(GT, :(@2(LT, ww401), ww41), ba) → new_lookup(GT, ww41, ba)
new_lookup(EQ, :(@2(LT, ww401), ww41), ba) → new_lookup(EQ, ww41, ba)
new_lookup(LT, :(@2(EQ, ww401), ww41), ba) → new_lookup(LT, ww41, ba)
new_lookup(EQ, :(@2(GT, ww401), ww41), ba) → new_lookup(EQ, ww41, ba)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

                  ↳ QDP

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

new_lookup(EQ, :(@2(LT, ww401), ww41), ba) → new_lookup(EQ, ww41, ba)
new_lookup(EQ, :(@2(GT, ww401), ww41), ba) → new_lookup(EQ, ww41, 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_lookup(EQ, :(@2(GT, ww401), ww41), ba) → new_lookup(EQ, ww41, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
new_lookup(EQ, :(@2(LT, ww401), ww41), ba) → new_lookup(EQ, ww41, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

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

new_lookup(GT, :(@2(EQ, ww401), ww41), ba) → new_lookup(GT, ww41, ba)
new_lookup(GT, :(@2(LT, ww401), ww41), ba) → new_lookup(GT, ww41, ba)

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

new_lookup(GT, :(@2(LT, ww401), ww41), ba) → new_lookup(GT, ww41, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
new_lookup(GT, :(@2(EQ, ww401), ww41), ba) → new_lookup(GT, ww41, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPSizeChangeProof

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

new_lookup(LT, :(@2(GT, ww401), ww41), ba) → new_lookup(LT, ww41, ba)
new_lookup(LT, :(@2(EQ, ww401), ww41), ba) → new_lookup(LT, ww41, ba)

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

new_lookup(LT, :(@2(EQ, ww401), ww41), ba) → new_lookup(LT, ww41, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
new_lookup(LT, :(@2(GT, ww401), ww41), ba) → new_lookup(LT, ww41, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3

lookup	k []	= lookup3 k []
lookup	k ((x,y) : xys)	= lookup2 k ((x,y) : xys)

lookup1	k x y xys True	= Just y
lookup1	k x y xys False	= lookup0 k x y xys otherwise