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

YES 0.924 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 :: Char -> [(Char,a)] -> Maybe a) :: Char -> [(Char,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 :: Char -> [(Char,a)] -> Maybe a) :: Char -> [(Char,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)

lookup1 k x y xys True = Just y

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

lookup0 k x y xys True = lookup k xys

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 :: Char -> [(Char,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"];365[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];4 -> 365[label="",style="solid", color="burlywood", weight=9];
365 -> 5[label="",style="solid", color="burlywood", weight=3];
366[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 366[label="",style="solid", color="burlywood", weight=9];
366 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="lookup ww3 (ww40 : ww41)",fontsize=16,color="burlywood",shape="box"];367[label="ww40/(ww400,ww401)",fontsize=10,color="white",style="solid",shape="box"];5 -> 367[label="",style="solid", color="burlywood", weight=9];
367 -> 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="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12[label="lookup1 ww3 ww400 ww401 ww41 (primEqChar ww3 ww400)",fontsize=16,color="burlywood",shape="box"];368[label="ww3/Char ww30",fontsize=10,color="white",style="solid",shape="box"];12 -> 368[label="",style="solid", color="burlywood", weight=9];
368 -> 13[label="",style="solid", color="burlywood", weight=3];
13[label="lookup1 (Char ww30) ww400 ww401 ww41 (primEqChar (Char ww30) ww400)",fontsize=16,color="burlywood",shape="box"];369[label="ww400/Char ww4000",fontsize=10,color="white",style="solid",shape="box"];13 -> 369[label="",style="solid", color="burlywood", weight=9];
369 -> 14[label="",style="solid", color="burlywood", weight=3];
14[label="lookup1 (Char ww30) (Char ww4000) ww401 ww41 (primEqChar (Char ww30) (Char ww4000))",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
15[label="lookup1 (Char ww30) (Char ww4000) ww401 ww41 (primEqNat ww30 ww4000)",fontsize=16,color="burlywood",shape="box"];370[label="ww30/Succ ww300",fontsize=10,color="white",style="solid",shape="box"];15 -> 370[label="",style="solid", color="burlywood", weight=9];
370 -> 16[label="",style="solid", color="burlywood", weight=3];
371[label="ww30/Zero",fontsize=10,color="white",style="solid",shape="box"];15 -> 371[label="",style="solid", color="burlywood", weight=9];
371 -> 17[label="",style="solid", color="burlywood", weight=3];
16[label="lookup1 (Char (Succ ww300)) (Char ww4000) ww401 ww41 (primEqNat (Succ ww300) ww4000)",fontsize=16,color="burlywood",shape="box"];372[label="ww4000/Succ ww40000",fontsize=10,color="white",style="solid",shape="box"];16 -> 372[label="",style="solid", color="burlywood", weight=9];
372 -> 18[label="",style="solid", color="burlywood", weight=3];
373[label="ww4000/Zero",fontsize=10,color="white",style="solid",shape="box"];16 -> 373[label="",style="solid", color="burlywood", weight=9];
373 -> 19[label="",style="solid", color="burlywood", weight=3];
17[label="lookup1 (Char Zero) (Char ww4000) ww401 ww41 (primEqNat Zero ww4000)",fontsize=16,color="burlywood",shape="box"];374[label="ww4000/Succ ww40000",fontsize=10,color="white",style="solid",shape="box"];17 -> 374[label="",style="solid", color="burlywood", weight=9];
374 -> 20[label="",style="solid", color="burlywood", weight=3];
375[label="ww4000/Zero",fontsize=10,color="white",style="solid",shape="box"];17 -> 375[label="",style="solid", color="burlywood", weight=9];
375 -> 21[label="",style="solid", color="burlywood", weight=3];
18[label="lookup1 (Char (Succ ww300)) (Char (Succ ww40000)) ww401 ww41 (primEqNat (Succ ww300) (Succ ww40000))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="lookup1 (Char (Succ ww300)) (Char Zero) ww401 ww41 (primEqNat (Succ ww300) Zero)",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="lookup1 (Char Zero) (Char (Succ ww40000)) ww401 ww41 (primEqNat Zero (Succ ww40000))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="lookup1 (Char Zero) (Char Zero) ww401 ww41 (primEqNat Zero Zero)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22 -> 286[label="",style="dashed", color="red", weight=0];
22[label="lookup1 (Char (Succ ww300)) (Char (Succ ww40000)) ww401 ww41 (primEqNat ww300 ww40000)",fontsize=16,color="magenta"];22 -> 287[label="",style="dashed", color="magenta", weight=3];
22 -> 288[label="",style="dashed", color="magenta", weight=3];
22 -> 289[label="",style="dashed", color="magenta", weight=3];
22 -> 290[label="",style="dashed", color="magenta", weight=3];
22 -> 291[label="",style="dashed", color="magenta", weight=3];
22 -> 292[label="",style="dashed", color="magenta", weight=3];
23[label="lookup1 (Char (Succ ww300)) (Char Zero) ww401 ww41 False",fontsize=16,color="black",shape="box"];23 -> 28[label="",style="solid", color="black", weight=3];
24[label="lookup1 (Char Zero) (Char (Succ ww40000)) ww401 ww41 False",fontsize=16,color="black",shape="box"];24 -> 29[label="",style="solid", color="black", weight=3];
25[label="lookup1 (Char Zero) (Char Zero) ww401 ww41 True",fontsize=16,color="black",shape="box"];25 -> 30[label="",style="solid", color="black", weight=3];
287[label="ww40000",fontsize=16,color="green",shape="box"];288[label="ww300",fontsize=16,color="green",shape="box"];289[label="ww300",fontsize=16,color="green",shape="box"];290[label="ww41",fontsize=16,color="green",shape="box"];291[label="ww401",fontsize=16,color="green",shape="box"];292[label="ww40000",fontsize=16,color="green",shape="box"];286[label="lookup1 (Char (Succ ww36)) (Char (Succ ww37)) ww38 ww39 (primEqNat ww40 ww41)",fontsize=16,color="burlywood",shape="triangle"];377[label="ww40/Succ ww400",fontsize=10,color="white",style="solid",shape="box"];286 -> 377[label="",style="solid", color="burlywood", weight=9];
377 -> 347[label="",style="solid", color="burlywood", weight=3];
378[label="ww40/Zero",fontsize=10,color="white",style="solid",shape="box"];286 -> 378[label="",style="solid", color="burlywood", weight=9];
378 -> 348[label="",style="solid", color="burlywood", weight=3];
28[label="lookup0 (Char (Succ ww300)) (Char Zero) ww401 ww41 otherwise",fontsize=16,color="black",shape="box"];28 -> 35[label="",style="solid", color="black", weight=3];
29[label="lookup0 (Char Zero) (Char (Succ ww40000)) ww401 ww41 otherwise",fontsize=16,color="black",shape="box"];29 -> 36[label="",style="solid", color="black", weight=3];
30[label="Just ww401",fontsize=16,color="green",shape="box"];347[label="lookup1 (Char (Succ ww36)) (Char (Succ ww37)) ww38 ww39 (primEqNat (Succ ww400) ww41)",fontsize=16,color="burlywood",shape="box"];379[label="ww41/Succ ww410",fontsize=10,color="white",style="solid",shape="box"];347 -> 379[label="",style="solid", color="burlywood", weight=9];
379 -> 349[label="",style="solid", color="burlywood", weight=3];
380[label="ww41/Zero",fontsize=10,color="white",style="solid",shape="box"];347 -> 380[label="",style="solid", color="burlywood", weight=9];
380 -> 350[label="",style="solid", color="burlywood", weight=3];
348[label="lookup1 (Char (Succ ww36)) (Char (Succ ww37)) ww38 ww39 (primEqNat Zero ww41)",fontsize=16,color="burlywood",shape="box"];381[label="ww41/Succ ww410",fontsize=10,color="white",style="solid",shape="box"];348 -> 381[label="",style="solid", color="burlywood", weight=9];
381 -> 351[label="",style="solid", color="burlywood", weight=3];
382[label="ww41/Zero",fontsize=10,color="white",style="solid",shape="box"];348 -> 382[label="",style="solid", color="burlywood", weight=9];
382 -> 352[label="",style="solid", color="burlywood", weight=3];
35[label="lookup0 (Char (Succ ww300)) (Char Zero) ww401 ww41 True",fontsize=16,color="black",shape="box"];35 -> 41[label="",style="solid", color="black", weight=3];
36[label="lookup0 (Char Zero) (Char (Succ ww40000)) ww401 ww41 True",fontsize=16,color="black",shape="box"];36 -> 42[label="",style="solid", color="black", weight=3];
349[label="lookup1 (Char (Succ ww36)) (Char (Succ ww37)) ww38 ww39 (primEqNat (Succ ww400) (Succ ww410))",fontsize=16,color="black",shape="box"];349 -> 353[label="",style="solid", color="black", weight=3];
350[label="lookup1 (Char (Succ ww36)) (Char (Succ ww37)) ww38 ww39 (primEqNat (Succ ww400) Zero)",fontsize=16,color="black",shape="box"];350 -> 354[label="",style="solid", color="black", weight=3];
351[label="lookup1 (Char (Succ ww36)) (Char (Succ ww37)) ww38 ww39 (primEqNat Zero (Succ ww410))",fontsize=16,color="black",shape="box"];351 -> 355[label="",style="solid", color="black", weight=3];
352[label="lookup1 (Char (Succ ww36)) (Char (Succ ww37)) ww38 ww39 (primEqNat Zero Zero)",fontsize=16,color="black",shape="box"];352 -> 356[label="",style="solid", color="black", weight=3];
41 -> 4[label="",style="dashed", color="red", weight=0];
41[label="lookup (Char (Succ ww300)) ww41",fontsize=16,color="magenta"];41 -> 48[label="",style="dashed", color="magenta", weight=3];
41 -> 49[label="",style="dashed", color="magenta", weight=3];
42 -> 4[label="",style="dashed", color="red", weight=0];
42[label="lookup (Char Zero) ww41",fontsize=16,color="magenta"];42 -> 50[label="",style="dashed", color="magenta", weight=3];
42 -> 51[label="",style="dashed", color="magenta", weight=3];
353 -> 286[label="",style="dashed", color="red", weight=0];
353[label="lookup1 (Char (Succ ww36)) (Char (Succ ww37)) ww38 ww39 (primEqNat ww400 ww410)",fontsize=16,color="magenta"];353 -> 357[label="",style="dashed", color="magenta", weight=3];
353 -> 358[label="",style="dashed", color="magenta", weight=3];
354[label="lookup1 (Char (Succ ww36)) (Char (Succ ww37)) ww38 ww39 False",fontsize=16,color="black",shape="triangle"];354 -> 359[label="",style="solid", color="black", weight=3];
355 -> 354[label="",style="dashed", color="red", weight=0];
355[label="lookup1 (Char (Succ ww36)) (Char (Succ ww37)) ww38 ww39 False",fontsize=16,color="magenta"];356[label="lookup1 (Char (Succ ww36)) (Char (Succ ww37)) ww38 ww39 True",fontsize=16,color="black",shape="box"];356 -> 360[label="",style="solid", color="black", weight=3];
48[label="ww41",fontsize=16,color="green",shape="box"];49[label="Char (Succ ww300)",fontsize=16,color="green",shape="box"];50[label="ww41",fontsize=16,color="green",shape="box"];51[label="Char Zero",fontsize=16,color="green",shape="box"];357[label="ww400",fontsize=16,color="green",shape="box"];358[label="ww410",fontsize=16,color="green",shape="box"];359[label="lookup0 (Char (Succ ww36)) (Char (Succ ww37)) ww38 ww39 otherwise",fontsize=16,color="black",shape="box"];359 -> 361[label="",style="solid", color="black", weight=3];
360[label="Just ww38",fontsize=16,color="green",shape="box"];361[label="lookup0 (Char (Succ ww36)) (Char (Succ ww37)) ww38 ww39 True",fontsize=16,color="black",shape="box"];361 -> 362[label="",style="solid", color="black", weight=3];
362 -> 4[label="",style="dashed", color="red", weight=0];
362[label="lookup (Char (Succ ww36)) ww39",fontsize=16,color="magenta"];362 -> 363[label="",style="dashed", color="magenta", weight=3];
362 -> 364[label="",style="dashed", color="magenta", weight=3];
363[label="ww39",fontsize=16,color="green",shape="box"];364[label="Char (Succ ww36)",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_lookup10(ww36, ww37, ww38, ww39, ba) → new_lookup(Char(Succ(ww36)), ww39, ba)
new_lookup1(ww36, ww37, ww38, ww39, Zero, Succ(ww410), ba) → new_lookup10(ww36, ww37, ww38, ww39, ba)
new_lookup(Char(Succ(ww300)), :(@2(Char(Succ(ww40000)), ww401), ww41), bb) → new_lookup1(ww300, ww40000, ww401, ww41, ww300, ww40000, bb)
new_lookup(Char(Zero), :(@2(Char(Succ(ww40000)), ww401), ww41), bb) → new_lookup(Char(Zero), ww41, bb)
new_lookup1(ww36, ww37, ww38, ww39, Succ(ww400), Succ(ww410), ba) → new_lookup1(ww36, ww37, ww38, ww39, ww400, ww410, ba)
new_lookup1(ww36, ww37, ww38, ww39, Succ(ww400), Zero, ba) → new_lookup(Char(Succ(ww36)), ww39, ba)
new_lookup(Char(Succ(ww300)), :(@2(Char(Zero), ww401), ww41), bb) → new_lookup(Char(Succ(ww300)), ww41, bb)

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 2 SCCs.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

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

new_lookup(Char(Zero), :(@2(Char(Succ(ww40000)), ww401), ww41), bb) → new_lookup(Char(Zero), ww41, bb)

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(Char(Zero), :(@2(Char(Succ(ww40000)), ww401), ww41), bb) → new_lookup(Char(Zero), ww41, bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPSizeChangeProof

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

new_lookup10(ww36, ww37, ww38, ww39, ba) → new_lookup(Char(Succ(ww36)), ww39, ba)
new_lookup1(ww36, ww37, ww38, ww39, Zero, Succ(ww410), ba) → new_lookup10(ww36, ww37, ww38, ww39, ba)
new_lookup(Char(Succ(ww300)), :(@2(Char(Succ(ww40000)), ww401), ww41), bb) → new_lookup1(ww300, ww40000, ww401, ww41, ww300, ww40000, bb)
new_lookup1(ww36, ww37, ww38, ww39, Succ(ww400), Succ(ww410), ba) → new_lookup1(ww36, ww37, ww38, ww39, ww400, ww410, ba)
new_lookup1(ww36, ww37, ww38, ww39, Succ(ww400), Zero, ba) → new_lookup(Char(Succ(ww36)), ww39, ba)
new_lookup(Char(Succ(ww300)), :(@2(Char(Zero), ww401), ww41), bb) → new_lookup(Char(Succ(ww300)), ww41, bb)

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

new_lookup1(ww36, ww37, ww38, ww39, Succ(ww400), Zero, ba) → new_lookup(Char(Succ(ww36)), ww39, ba)
The graph contains the following edges 4 >= 2, 7 >= 3
new_lookup(Char(Succ(ww300)), :(@2(Char(Succ(ww40000)), ww401), ww41), bb) → new_lookup1(ww300, ww40000, ww401, ww41, ww300, ww40000, bb)
The graph contains the following edges 1 > 1, 2 > 2, 2 > 3, 2 > 4, 1 > 5, 2 > 6, 3 >= 7
new_lookup1(ww36, ww37, ww38, ww39, Succ(ww400), Succ(ww410), ba) → new_lookup1(ww36, ww37, ww38, ww39, ww400, ww410, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 > 5, 6 > 6, 7 >= 7
new_lookup1(ww36, ww37, ww38, ww39, Zero, Succ(ww410), ba) → new_lookup10(ww36, ww37, ww38, ww39, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 7 >= 5
new_lookup10(ww36, ww37, ww38, ww39, ba) → new_lookup(Char(Succ(ww36)), ww39, ba)
The graph contains the following edges 4 >= 2, 5 >= 3
new_lookup(Char(Succ(ww300)), :(@2(Char(Zero), ww401), ww41), bb) → new_lookup(Char(Succ(ww300)), ww41, bb)
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