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

YES 1.29 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

  ↳ BR

mainModule List

((isPrefixOf :: Eq a => [a] -> [a] -> Bool) :: Eq a => [a] -> [a] -> Bool)

module List where

import qualified Maybe
import qualified Prelude

isPrefixOf :: Eq a => [a] -> [a] -> Bool

isPrefixOf	[] _	=	True
isPrefixOf	_ []	=	False
isPrefixOf	(x : xs) (y : ys)	=	x == y && isPrefixOf xs ys

module Maybe where

import qualified List
import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule List

((isPrefixOf :: Eq a => [a] -> [a] -> Bool) :: Eq a => [a] -> [a] -> Bool)

module List where

import qualified Maybe
import qualified Prelude

isPrefixOf :: Eq a => [a] -> [a] -> Bool

isPrefixOf	[] vw	=	True
isPrefixOf	vx []	=	False
isPrefixOf	(x : xs) (y : ys)	=	x == y && isPrefixOf xs ys

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

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

mainModule List

(isPrefixOf :: Eq a => [a] -> [a] -> Bool)

module List where

import qualified Maybe
import qualified Prelude

isPrefixOf :: Eq a => [a] -> [a] -> Bool

isPrefixOf	[] vw	=	True
isPrefixOf	vx []	=	False
isPrefixOf	(x : xs) (y : ys)	=	x == y && isPrefixOf xs ys

module Maybe where

import qualified List
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="isPrefixOf\n  Eq a",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
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651 -> 5[label="",style="solid", color="burlywood", weight=3];
652[label="xy3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 652[label="",style="solid", color="burlywood", weight=9];
652 -> 6[label="",style="solid", color="burlywood", weight=3];
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653 -> 7[label="",style="solid", color="burlywood", weight=3];
654[label="xy4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 654[label="",style="solid", color="burlywood", weight=9];
654 -> 8[label="",style="solid", color="burlywood", weight=3];
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8[label="isPrefixOf (xy30 : xy31) []\n  Eq a",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9[label="True",fontsize=16,color="green",shape="box"];10 -> 16[label="",style="dashed", color="red", weight=0];
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10 -> 18[label="",style="dashed", color="magenta", weight=3];
11[label="False",fontsize=16,color="green",shape="box"];17 -> 4[label="",style="dashed", color="red", weight=0];
17[label="isPrefixOf xy31 xy41\n  Eq a",fontsize=16,color="magenta"];17 -> 21[label="",style="dashed", color="magenta", weight=3];
17 -> 22[label="",style="dashed", color="magenta", weight=3];
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657 -> 23[label="",style="solid", color="blue", weight=3];
658[label="== :: ((@3) a b c) -> ((@3) a b c) -> Bool",fontsize=10,color="white",style="solid",shape="box"];18 -> 658[label="",style="solid", color="blue", weight=9];
658 -> 24[label="",style="solid", color="blue", weight=3];
659[label="== :: Integer -> Integer -> Bool",fontsize=10,color="white",style="solid",shape="box"];18 -> 659[label="",style="solid", color="blue", weight=9];
659 -> 25[label="",style="solid", color="blue", weight=3];
660[label="== :: (Ratio a) -> (Ratio a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];18 -> 660[label="",style="solid", color="blue", weight=9];
660 -> 26[label="",style="solid", color="blue", weight=3];
661[label="== :: Double -> Double -> Bool",fontsize=10,color="white",style="solid",shape="box"];18 -> 661[label="",style="solid", color="blue", weight=9];
661 -> 27[label="",style="solid", color="blue", weight=3];
662[label="== :: Float -> Float -> Bool",fontsize=10,color="white",style="solid",shape="box"];18 -> 662[label="",style="solid", color="blue", weight=9];
662 -> 28[label="",style="solid", color="blue", weight=3];
663[label="== :: (Either a b) -> (Either a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];18 -> 663[label="",style="solid", color="blue", weight=9];
663 -> 29[label="",style="solid", color="blue", weight=3];
664[label="== :: Ordering -> Ordering -> Bool",fontsize=10,color="white",style="solid",shape="box"];18 -> 664[label="",style="solid", color="blue", weight=9];
664 -> 30[label="",style="solid", color="blue", weight=3];
665[label="== :: (Maybe a) -> (Maybe a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];18 -> 665[label="",style="solid", color="blue", weight=9];
665 -> 31[label="",style="solid", color="blue", weight=3];
666[label="== :: Bool -> Bool -> Bool",fontsize=10,color="white",style="solid",shape="box"];18 -> 666[label="",style="solid", color="blue", weight=9];
666 -> 32[label="",style="solid", color="blue", weight=3];
667[label="== :: Int -> Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];18 -> 667[label="",style="solid", color="blue", weight=9];
667 -> 33[label="",style="solid", color="blue", weight=3];
668[label="== :: Char -> Char -> Bool",fontsize=10,color="white",style="solid",shape="box"];18 -> 668[label="",style="solid", color="blue", weight=9];
668 -> 34[label="",style="solid", color="blue", weight=3];
669[label="== :: () -> () -> Bool",fontsize=10,color="white",style="solid",shape="box"];18 -> 669[label="",style="solid", color="blue", weight=9];
669 -> 35[label="",style="solid", color="blue", weight=3];
670[label="== :: ((@2) a b) -> ((@2) a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];18 -> 670[label="",style="solid", color="blue", weight=9];
670 -> 36[label="",style="solid", color="blue", weight=3];
16[label="xy9 && xy10",fontsize=16,color="burlywood",shape="triangle"];671[label="xy9/False",fontsize=10,color="white",style="solid",shape="box"];16 -> 671[label="",style="solid", color="burlywood", weight=9];
671 -> 37[label="",style="solid", color="burlywood", weight=3];
672[label="xy9/True",fontsize=10,color="white",style="solid",shape="box"];16 -> 672[label="",style="solid", color="burlywood", weight=9];
672 -> 38[label="",style="solid", color="burlywood", weight=3];
21[label="xy41\n  Eq a",fontsize=16,color="green",shape="box"];22[label="xy31\n  Eq a",fontsize=16,color="green",shape="box"];23[label="xy30 == xy40\n  Eq a",fontsize=16,color="burlywood",shape="triangle"];673[label="xy30/xy300 : xy301",fontsize=10,color="white",style="solid",shape="box"];23 -> 673[label="",style="solid", color="burlywood", weight=9];
673 -> 39[label="",style="solid", color="burlywood", weight=3];
674[label="xy30/[]",fontsize=10,color="white",style="solid",shape="box"];23 -> 674[label="",style="solid", color="burlywood", weight=9];
674 -> 40[label="",style="solid", color="burlywood", weight=3];
24[label="xy30 == xy40\n  Eq a  Eq b  Eq c",fontsize=16,color="burlywood",shape="triangle"];675[label="xy30/(xy300,xy301,xy302)",fontsize=10,color="white",style="solid",shape="box"];24 -> 675[label="",style="solid", color="burlywood", weight=9];
675 -> 41[label="",style="solid", color="burlywood", weight=3];
25[label="xy30 == xy40",fontsize=16,color="burlywood",shape="triangle"];676[label="xy30/Integer xy300",fontsize=10,color="white",style="solid",shape="box"];25 -> 676[label="",style="solid", color="burlywood", weight=9];
676 -> 42[label="",style="solid", color="burlywood", weight=3];
26[label="xy30 == xy40\n  Integral a",fontsize=16,color="burlywood",shape="triangle"];677[label="xy30/xy300 :% xy301",fontsize=10,color="white",style="solid",shape="box"];26 -> 677[label="",style="solid", color="burlywood", weight=9];
677 -> 43[label="",style="solid", color="burlywood", weight=3];
27[label="xy30 == xy40",fontsize=16,color="black",shape="triangle"];27 -> 44[label="",style="solid", color="black", weight=3];
28[label="xy30 == xy40",fontsize=16,color="black",shape="triangle"];28 -> 45[label="",style="solid", color="black", weight=3];
29[label="xy30 == xy40\n  Eq a  Eq b",fontsize=16,color="burlywood",shape="triangle"];678[label="xy30/Left xy300",fontsize=10,color="white",style="solid",shape="box"];29 -> 678[label="",style="solid", color="burlywood", weight=9];
678 -> 46[label="",style="solid", color="burlywood", weight=3];
679[label="xy30/Right xy300",fontsize=10,color="white",style="solid",shape="box"];29 -> 679[label="",style="solid", color="burlywood", weight=9];
679 -> 47[label="",style="solid", color="burlywood", weight=3];
30[label="xy30 == xy40",fontsize=16,color="burlywood",shape="triangle"];680[label="xy30/LT",fontsize=10,color="white",style="solid",shape="box"];30 -> 680[label="",style="solid", color="burlywood", weight=9];
680 -> 48[label="",style="solid", color="burlywood", weight=3];
681[label="xy30/EQ",fontsize=10,color="white",style="solid",shape="box"];30 -> 681[label="",style="solid", color="burlywood", weight=9];
681 -> 49[label="",style="solid", color="burlywood", weight=3];
682[label="xy30/GT",fontsize=10,color="white",style="solid",shape="box"];30 -> 682[label="",style="solid", color="burlywood", weight=9];
682 -> 50[label="",style="solid", color="burlywood", weight=3];
31[label="xy30 == xy40\n  Eq a",fontsize=16,color="burlywood",shape="triangle"];683[label="xy30/Nothing",fontsize=10,color="white",style="solid",shape="box"];31 -> 683[label="",style="solid", color="burlywood", weight=9];
683 -> 51[label="",style="solid", color="burlywood", weight=3];
684[label="xy30/Just xy300",fontsize=10,color="white",style="solid",shape="box"];31 -> 684[label="",style="solid", color="burlywood", weight=9];
684 -> 52[label="",style="solid", color="burlywood", weight=3];
32[label="xy30 == xy40",fontsize=16,color="burlywood",shape="triangle"];685[label="xy30/False",fontsize=10,color="white",style="solid",shape="box"];32 -> 685[label="",style="solid", color="burlywood", weight=9];
685 -> 53[label="",style="solid", color="burlywood", weight=3];
686[label="xy30/True",fontsize=10,color="white",style="solid",shape="box"];32 -> 686[label="",style="solid", color="burlywood", weight=9];
686 -> 54[label="",style="solid", color="burlywood", weight=3];
33[label="xy30 == xy40",fontsize=16,color="black",shape="triangle"];33 -> 55[label="",style="solid", color="black", weight=3];
34[label="xy30 == xy40",fontsize=16,color="black",shape="triangle"];34 -> 56[label="",style="solid", color="black", weight=3];
35[label="xy30 == xy40",fontsize=16,color="burlywood",shape="triangle"];687[label="xy30/()",fontsize=10,color="white",style="solid",shape="box"];35 -> 687[label="",style="solid", color="burlywood", weight=9];
687 -> 57[label="",style="solid", color="burlywood", weight=3];
36[label="xy30 == xy40\n  Eq a  Eq b",fontsize=16,color="burlywood",shape="triangle"];688[label="xy30/(xy300,xy301)",fontsize=10,color="white",style="solid",shape="box"];36 -> 688[label="",style="solid", color="burlywood", weight=9];
688 -> 58[label="",style="solid", color="burlywood", weight=3];
37[label="False && xy10",fontsize=16,color="black",shape="box"];37 -> 59[label="",style="solid", color="black", weight=3];
38[label="True && xy10",fontsize=16,color="black",shape="box"];38 -> 60[label="",style="solid", color="black", weight=3];
39[label="xy300 : xy301 == xy40\n  Eq a",fontsize=16,color="burlywood",shape="box"];689[label="xy40/xy400 : xy401",fontsize=10,color="white",style="solid",shape="box"];39 -> 689[label="",style="solid", color="burlywood", weight=9];
689 -> 61[label="",style="solid", color="burlywood", weight=3];
690[label="xy40/[]",fontsize=10,color="white",style="solid",shape="box"];39 -> 690[label="",style="solid", color="burlywood", weight=9];
690 -> 62[label="",style="solid", color="burlywood", weight=3];
40[label="[] == xy40\n  Eq a",fontsize=16,color="burlywood",shape="box"];691[label="xy40/xy400 : xy401",fontsize=10,color="white",style="solid",shape="box"];40 -> 691[label="",style="solid", color="burlywood", weight=9];
691 -> 63[label="",style="solid", color="burlywood", weight=3];
692[label="xy40/[]",fontsize=10,color="white",style="solid",shape="box"];40 -> 692[label="",style="solid", color="burlywood", weight=9];
692 -> 64[label="",style="solid", color="burlywood", weight=3];
41[label="(xy300,xy301,xy302) == xy40\n  Eq a  Eq b  Eq c",fontsize=16,color="burlywood",shape="box"];693[label="xy40/(xy400,xy401,xy402)",fontsize=10,color="white",style="solid",shape="box"];41 -> 693[label="",style="solid", color="burlywood", weight=9];
693 -> 65[label="",style="solid", color="burlywood", weight=3];
42[label="Integer xy300 == xy40",fontsize=16,color="burlywood",shape="box"];694[label="xy40/Integer xy400",fontsize=10,color="white",style="solid",shape="box"];42 -> 694[label="",style="solid", color="burlywood", weight=9];
694 -> 66[label="",style="solid", color="burlywood", weight=3];
43[label="xy300 :% xy301 == xy40\n  Integral a",fontsize=16,color="burlywood",shape="box"];695[label="xy40/xy400 :% xy401",fontsize=10,color="white",style="solid",shape="box"];43 -> 695[label="",style="solid", color="burlywood", weight=9];
695 -> 67[label="",style="solid", color="burlywood", weight=3];
44[label="primEqDouble xy30 xy40",fontsize=16,color="burlywood",shape="box"];696[label="xy30/Double xy300 xy301",fontsize=10,color="white",style="solid",shape="box"];44 -> 696[label="",style="solid", color="burlywood", weight=9];
696 -> 68[label="",style="solid", color="burlywood", weight=3];
45[label="primEqFloat xy30 xy40",fontsize=16,color="burlywood",shape="box"];697[label="xy30/Float xy300 xy301",fontsize=10,color="white",style="solid",shape="box"];45 -> 697[label="",style="solid", color="burlywood", weight=9];
697 -> 69[label="",style="solid", color="burlywood", weight=3];
46[label="Left xy300 == xy40\n  Eq a  Eq b",fontsize=16,color="burlywood",shape="box"];698[label="xy40/Left xy400",fontsize=10,color="white",style="solid",shape="box"];46 -> 698[label="",style="solid", color="burlywood", weight=9];
698 -> 70[label="",style="solid", color="burlywood", weight=3];
699[label="xy40/Right xy400",fontsize=10,color="white",style="solid",shape="box"];46 -> 699[label="",style="solid", color="burlywood", weight=9];
699 -> 71[label="",style="solid", color="burlywood", weight=3];
47[label="Right xy300 == xy40\n  Eq a  Eq b",fontsize=16,color="burlywood",shape="box"];700[label="xy40/Left xy400",fontsize=10,color="white",style="solid",shape="box"];47 -> 700[label="",style="solid", color="burlywood", weight=9];
700 -> 72[label="",style="solid", color="burlywood", weight=3];
701[label="xy40/Right xy400",fontsize=10,color="white",style="solid",shape="box"];47 -> 701[label="",style="solid", color="burlywood", weight=9];
701 -> 73[label="",style="solid", color="burlywood", weight=3];
48[label="LT == xy40",fontsize=16,color="burlywood",shape="box"];702[label="xy40/LT",fontsize=10,color="white",style="solid",shape="box"];48 -> 702[label="",style="solid", color="burlywood", weight=9];
702 -> 74[label="",style="solid", color="burlywood", weight=3];
703[label="xy40/EQ",fontsize=10,color="white",style="solid",shape="box"];48 -> 703[label="",style="solid", color="burlywood", weight=9];
703 -> 75[label="",style="solid", color="burlywood", weight=3];
704[label="xy40/GT",fontsize=10,color="white",style="solid",shape="box"];48 -> 704[label="",style="solid", color="burlywood", weight=9];
704 -> 76[label="",style="solid", color="burlywood", weight=3];
49[label="EQ == xy40",fontsize=16,color="burlywood",shape="box"];705[label="xy40/LT",fontsize=10,color="white",style="solid",shape="box"];49 -> 705[label="",style="solid", color="burlywood", weight=9];
705 -> 77[label="",style="solid", color="burlywood", weight=3];
706[label="xy40/EQ",fontsize=10,color="white",style="solid",shape="box"];49 -> 706[label="",style="solid", color="burlywood", weight=9];
706 -> 78[label="",style="solid", color="burlywood", weight=3];
707[label="xy40/GT",fontsize=10,color="white",style="solid",shape="box"];49 -> 707[label="",style="solid", color="burlywood", weight=9];
707 -> 79[label="",style="solid", color="burlywood", weight=3];
50[label="GT == xy40",fontsize=16,color="burlywood",shape="box"];708[label="xy40/LT",fontsize=10,color="white",style="solid",shape="box"];50 -> 708[label="",style="solid", color="burlywood", weight=9];
708 -> 80[label="",style="solid", color="burlywood", weight=3];
709[label="xy40/EQ",fontsize=10,color="white",style="solid",shape="box"];50 -> 709[label="",style="solid", color="burlywood", weight=9];
709 -> 81[label="",style="solid", color="burlywood", weight=3];
710[label="xy40/GT",fontsize=10,color="white",style="solid",shape="box"];50 -> 710[label="",style="solid", color="burlywood", weight=9];
710 -> 82[label="",style="solid", color="burlywood", weight=3];
51[label="Nothing == xy40\n  Eq a",fontsize=16,color="burlywood",shape="box"];711[label="xy40/Nothing",fontsize=10,color="white",style="solid",shape="box"];51 -> 711[label="",style="solid", color="burlywood", weight=9];
711 -> 83[label="",style="solid", color="burlywood", weight=3];
712[label="xy40/Just xy400",fontsize=10,color="white",style="solid",shape="box"];51 -> 712[label="",style="solid", color="burlywood", weight=9];
712 -> 84[label="",style="solid", color="burlywood", weight=3];
52[label="Just xy300 == xy40\n  Eq a",fontsize=16,color="burlywood",shape="box"];713[label="xy40/Nothing",fontsize=10,color="white",style="solid",shape="box"];52 -> 713[label="",style="solid", color="burlywood", weight=9];
713 -> 85[label="",style="solid", color="burlywood", weight=3];
714[label="xy40/Just xy400",fontsize=10,color="white",style="solid",shape="box"];52 -> 714[label="",style="solid", color="burlywood", weight=9];
714 -> 86[label="",style="solid", color="burlywood", weight=3];
53[label="False == xy40",fontsize=16,color="burlywood",shape="box"];715[label="xy40/False",fontsize=10,color="white",style="solid",shape="box"];53 -> 715[label="",style="solid", color="burlywood", weight=9];
715 -> 87[label="",style="solid", color="burlywood", weight=3];
716[label="xy40/True",fontsize=10,color="white",style="solid",shape="box"];53 -> 716[label="",style="solid", color="burlywood", weight=9];
716 -> 88[label="",style="solid", color="burlywood", weight=3];
54[label="True == xy40",fontsize=16,color="burlywood",shape="box"];717[label="xy40/False",fontsize=10,color="white",style="solid",shape="box"];54 -> 717[label="",style="solid", color="burlywood", weight=9];
717 -> 89[label="",style="solid", color="burlywood", weight=3];
718[label="xy40/True",fontsize=10,color="white",style="solid",shape="box"];54 -> 718[label="",style="solid", color="burlywood", weight=9];
718 -> 90[label="",style="solid", color="burlywood", weight=3];
55[label="primEqInt xy30 xy40",fontsize=16,color="burlywood",shape="triangle"];719[label="xy30/Pos xy300",fontsize=10,color="white",style="solid",shape="box"];55 -> 719[label="",style="solid", color="burlywood", weight=9];
719 -> 91[label="",style="solid", color="burlywood", weight=3];
720[label="xy30/Neg xy300",fontsize=10,color="white",style="solid",shape="box"];55 -> 720[label="",style="solid", color="burlywood", weight=9];
720 -> 92[label="",style="solid", color="burlywood", weight=3];
56[label="primEqChar xy30 xy40",fontsize=16,color="burlywood",shape="box"];721[label="xy30/Char xy300",fontsize=10,color="white",style="solid",shape="box"];56 -> 721[label="",style="solid", color="burlywood", weight=9];
721 -> 93[label="",style="solid", color="burlywood", weight=3];
57[label="() == xy40",fontsize=16,color="burlywood",shape="box"];722[label="xy40/()",fontsize=10,color="white",style="solid",shape="box"];57 -> 722[label="",style="solid", color="burlywood", weight=9];
722 -> 94[label="",style="solid", color="burlywood", weight=3];
58[label="(xy300,xy301) == xy40\n  Eq a  Eq b",fontsize=16,color="burlywood",shape="box"];723[label="xy40/(xy400,xy401)",fontsize=10,color="white",style="solid",shape="box"];58 -> 723[label="",style="solid", color="burlywood", weight=9];
723 -> 95[label="",style="solid", color="burlywood", weight=3];
59[label="False",fontsize=16,color="green",shape="box"];60[label="xy10",fontsize=16,color="green",shape="box"];61[label="xy300 : xy301 == xy400 : xy401\n  Eq a",fontsize=16,color="black",shape="box"];61 -> 96[label="",style="solid", color="black", weight=3];
62[label="xy300 : xy301 == []\n  Eq a",fontsize=16,color="black",shape="box"];62 -> 97[label="",style="solid", color="black", weight=3];
63[label="[] == xy400 : xy401\n  Eq a",fontsize=16,color="black",shape="box"];63 -> 98[label="",style="solid", color="black", weight=3];
64[label="[] == []\n  Eq a",fontsize=16,color="black",shape="box"];64 -> 99[label="",style="solid", color="black", weight=3];
65[label="(xy300,xy301,xy302) == (xy400,xy401,xy402)\n  Eq a  Eq b  Eq c",fontsize=16,color="black",shape="box"];65 -> 100[label="",style="solid", color="black", weight=3];
66[label="Integer xy300 == Integer xy400",fontsize=16,color="black",shape="box"];66 -> 101[label="",style="solid", color="black", weight=3];
67[label="xy300 :% xy301 == xy400 :% xy401\n  Integral a",fontsize=16,color="black",shape="box"];67 -> 102[label="",style="solid", color="black", weight=3];
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724 -> 103[label="",style="solid", color="burlywood", weight=3];
69[label="primEqFloat (Float xy300 xy301) xy40",fontsize=16,color="burlywood",shape="box"];725[label="xy40/Float xy400 xy401",fontsize=10,color="white",style="solid",shape="box"];69 -> 725[label="",style="solid", color="burlywood", weight=9];
725 -> 104[label="",style="solid", color="burlywood", weight=3];
70[label="Left xy300 == Left xy400\n  Eq a  Eq b",fontsize=16,color="black",shape="box"];70 -> 105[label="",style="solid", color="black", weight=3];
71[label="Left xy300 == Right xy400\n  Eq a  Eq b",fontsize=16,color="black",shape="box"];71 -> 106[label="",style="solid", color="black", weight=3];
72[label="Right xy300 == Left xy400\n  Eq a  Eq b",fontsize=16,color="black",shape="box"];72 -> 107[label="",style="solid", color="black", weight=3];
73[label="Right xy300 == Right xy400\n  Eq a  Eq b",fontsize=16,color="black",shape="box"];73 -> 108[label="",style="solid", color="black", weight=3];
74[label="LT == LT",fontsize=16,color="black",shape="box"];74 -> 109[label="",style="solid", color="black", weight=3];
75[label="LT == EQ",fontsize=16,color="black",shape="box"];75 -> 110[label="",style="solid", color="black", weight=3];
76[label="LT == GT",fontsize=16,color="black",shape="box"];76 -> 111[label="",style="solid", color="black", weight=3];
77[label="EQ == LT",fontsize=16,color="black",shape="box"];77 -> 112[label="",style="solid", color="black", weight=3];
78[label="EQ == EQ",fontsize=16,color="black",shape="box"];78 -> 113[label="",style="solid", color="black", weight=3];
79[label="EQ == GT",fontsize=16,color="black",shape="box"];79 -> 114[label="",style="solid", color="black", weight=3];
80[label="GT == LT",fontsize=16,color="black",shape="box"];80 -> 115[label="",style="solid", color="black", weight=3];
81[label="GT == EQ",fontsize=16,color="black",shape="box"];81 -> 116[label="",style="solid", color="black", weight=3];
82[label="GT == GT",fontsize=16,color="black",shape="box"];82 -> 117[label="",style="solid", color="black", weight=3];
83[label="Nothing == Nothing\n  Eq a",fontsize=16,color="black",shape="box"];83 -> 118[label="",style="solid", color="black", weight=3];
84[label="Nothing == Just xy400\n  Eq a",fontsize=16,color="black",shape="box"];84 -> 119[label="",style="solid", color="black", weight=3];
85[label="Just xy300 == Nothing\n  Eq a",fontsize=16,color="black",shape="box"];85 -> 120[label="",style="solid", color="black", weight=3];
86[label="Just xy300 == Just xy400\n  Eq a",fontsize=16,color="black",shape="box"];86 -> 121[label="",style="solid", color="black", weight=3];
87[label="False == False",fontsize=16,color="black",shape="box"];87 -> 122[label="",style="solid", color="black", weight=3];
88[label="False == True",fontsize=16,color="black",shape="box"];88 -> 123[label="",style="solid", color="black", weight=3];
89[label="True == False",fontsize=16,color="black",shape="box"];89 -> 124[label="",style="solid", color="black", weight=3];
90[label="True == True",fontsize=16,color="black",shape="box"];90 -> 125[label="",style="solid", color="black", weight=3];
91[label="primEqInt (Pos xy300) xy40",fontsize=16,color="burlywood",shape="box"];726[label="xy300/Succ xy3000",fontsize=10,color="white",style="solid",shape="box"];91 -> 726[label="",style="solid", color="burlywood", weight=9];
726 -> 126[label="",style="solid", color="burlywood", weight=3];
727[label="xy300/Zero",fontsize=10,color="white",style="solid",shape="box"];91 -> 727[label="",style="solid", color="burlywood", weight=9];
727 -> 127[label="",style="solid", color="burlywood", weight=3];
92[label="primEqInt (Neg xy300) xy40",fontsize=16,color="burlywood",shape="box"];728[label="xy300/Succ xy3000",fontsize=10,color="white",style="solid",shape="box"];92 -> 728[label="",style="solid", color="burlywood", weight=9];
728 -> 128[label="",style="solid", color="burlywood", weight=3];
729[label="xy300/Zero",fontsize=10,color="white",style="solid",shape="box"];92 -> 729[label="",style="solid", color="burlywood", weight=9];
729 -> 129[label="",style="solid", color="burlywood", weight=3];
93[label="primEqChar (Char xy300) xy40",fontsize=16,color="burlywood",shape="box"];730[label="xy40/Char xy400",fontsize=10,color="white",style="solid",shape="box"];93 -> 730[label="",style="solid", color="burlywood", weight=9];
730 -> 130[label="",style="solid", color="burlywood", weight=3];
94[label="() == ()",fontsize=16,color="black",shape="box"];94 -> 131[label="",style="solid", color="black", weight=3];
95[label="(xy300,xy301) == (xy400,xy401)\n  Eq a  Eq b",fontsize=16,color="black",shape="box"];95 -> 132[label="",style="solid", color="black", weight=3];
96 -> 16[label="",style="dashed", color="red", weight=0];
96[label="xy300 == xy400 && xy301 == xy401\n  Eq a",fontsize=16,color="magenta"];96 -> 133[label="",style="dashed", color="magenta", weight=3];
96 -> 134[label="",style="dashed", color="magenta", weight=3];
97[label="False",fontsize=16,color="green",shape="box"];98[label="False",fontsize=16,color="green",shape="box"];99[label="True",fontsize=16,color="green",shape="box"];100 -> 16[label="",style="dashed", color="red", weight=0];
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100 -> 136[label="",style="dashed", color="magenta", weight=3];
101 -> 55[label="",style="dashed", color="red", weight=0];
101[label="primEqInt xy300 xy400",fontsize=16,color="magenta"];101 -> 137[label="",style="dashed", color="magenta", weight=3];
101 -> 138[label="",style="dashed", color="magenta", weight=3];
102 -> 16[label="",style="dashed", color="red", weight=0];
102[label="xy300 == xy400 && xy301 == xy401\n  Integral a",fontsize=16,color="magenta"];102 -> 139[label="",style="dashed", color="magenta", weight=3];
102 -> 140[label="",style="dashed", color="magenta", weight=3];
103[label="primEqDouble (Double xy300 xy301) (Double xy400 xy401)",fontsize=16,color="black",shape="box"];103 -> 141[label="",style="solid", color="black", weight=3];
104[label="primEqFloat (Float xy300 xy301) (Float xy400 xy401)",fontsize=16,color="black",shape="box"];104 -> 142[label="",style="solid", color="black", weight=3];
105[label="xy300 == xy400\n  Eq a",fontsize=16,color="blue",shape="box"];735[label="== :: ([] a) -> ([] a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];105 -> 735[label="",style="solid", color="blue", weight=9];
735 -> 143[label="",style="solid", color="blue", weight=3];
736[label="== :: ((@3) a b c) -> ((@3) a b c) -> Bool",fontsize=10,color="white",style="solid",shape="box"];105 -> 736[label="",style="solid", color="blue", weight=9];
736 -> 144[label="",style="solid", color="blue", weight=3];
737[label="== :: Integer -> Integer -> Bool",fontsize=10,color="white",style="solid",shape="box"];105 -> 737[label="",style="solid", color="blue", weight=9];
737 -> 145[label="",style="solid", color="blue", weight=3];
738[label="== :: (Ratio a) -> (Ratio a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];105 -> 738[label="",style="solid", color="blue", weight=9];
738 -> 146[label="",style="solid", color="blue", weight=3];
739[label="== :: Double -> Double -> Bool",fontsize=10,color="white",style="solid",shape="box"];105 -> 739[label="",style="solid", color="blue", weight=9];
739 -> 147[label="",style="solid", color="blue", weight=3];
740[label="== :: Float -> Float -> Bool",fontsize=10,color="white",style="solid",shape="box"];105 -> 740[label="",style="solid", color="blue", weight=9];
740 -> 148[label="",style="solid", color="blue", weight=3];
741[label="== :: (Either a b) -> (Either a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];105 -> 741[label="",style="solid", color="blue", weight=9];
741 -> 149[label="",style="solid", color="blue", weight=3];
742[label="== :: Ordering -> Ordering -> Bool",fontsize=10,color="white",style="solid",shape="box"];105 -> 742[label="",style="solid", color="blue", weight=9];
742 -> 150[label="",style="solid", color="blue", weight=3];
743[label="== :: (Maybe a) -> (Maybe a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];105 -> 743[label="",style="solid", color="blue", weight=9];
743 -> 151[label="",style="solid", color="blue", weight=3];
744[label="== :: Bool -> Bool -> Bool",fontsize=10,color="white",style="solid",shape="box"];105 -> 744[label="",style="solid", color="blue", weight=9];
744 -> 152[label="",style="solid", color="blue", weight=3];
745[label="== :: Int -> Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];105 -> 745[label="",style="solid", color="blue", weight=9];
745 -> 153[label="",style="solid", color="blue", weight=3];
746[label="== :: Char -> Char -> Bool",fontsize=10,color="white",style="solid",shape="box"];105 -> 746[label="",style="solid", color="blue", weight=9];
746 -> 154[label="",style="solid", color="blue", weight=3];
747[label="== :: () -> () -> Bool",fontsize=10,color="white",style="solid",shape="box"];105 -> 747[label="",style="solid", color="blue", weight=9];
747 -> 155[label="",style="solid", color="blue", weight=3];
748[label="== :: ((@2) a b) -> ((@2) a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];105 -> 748[label="",style="solid", color="blue", weight=9];
748 -> 156[label="",style="solid", color="blue", weight=3];
106[label="False",fontsize=16,color="green",shape="box"];107[label="False",fontsize=16,color="green",shape="box"];108[label="xy300 == xy400\n  Eq a",fontsize=16,color="blue",shape="box"];749[label="== :: ([] a) -> ([] a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];108 -> 749[label="",style="solid", color="blue", weight=9];
749 -> 157[label="",style="solid", color="blue", weight=3];
750[label="== :: ((@3) a b c) -> ((@3) a b c) -> Bool",fontsize=10,color="white",style="solid",shape="box"];108 -> 750[label="",style="solid", color="blue", weight=9];
750 -> 158[label="",style="solid", color="blue", weight=3];
751[label="== :: Integer -> Integer -> Bool",fontsize=10,color="white",style="solid",shape="box"];108 -> 751[label="",style="solid", color="blue", weight=9];
751 -> 159[label="",style="solid", color="blue", weight=3];
752[label="== :: (Ratio a) -> (Ratio a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];108 -> 752[label="",style="solid", color="blue", weight=9];
752 -> 160[label="",style="solid", color="blue", weight=3];
753[label="== :: Double -> Double -> Bool",fontsize=10,color="white",style="solid",shape="box"];108 -> 753[label="",style="solid", color="blue", weight=9];
753 -> 161[label="",style="solid", color="blue", weight=3];
754[label="== :: Float -> Float -> Bool",fontsize=10,color="white",style="solid",shape="box"];108 -> 754[label="",style="solid", color="blue", weight=9];
754 -> 162[label="",style="solid", color="blue", weight=3];
755[label="== :: (Either a b) -> (Either a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];108 -> 755[label="",style="solid", color="blue", weight=9];
755 -> 163[label="",style="solid", color="blue", weight=3];
756[label="== :: Ordering -> Ordering -> Bool",fontsize=10,color="white",style="solid",shape="box"];108 -> 756[label="",style="solid", color="blue", weight=9];
756 -> 164[label="",style="solid", color="blue", weight=3];
757[label="== :: (Maybe a) -> (Maybe a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];108 -> 757[label="",style="solid", color="blue", weight=9];
757 -> 165[label="",style="solid", color="blue", weight=3];
758[label="== :: Bool -> Bool -> Bool",fontsize=10,color="white",style="solid",shape="box"];108 -> 758[label="",style="solid", color="blue", weight=9];
758 -> 166[label="",style="solid", color="blue", weight=3];
759[label="== :: Int -> Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];108 -> 759[label="",style="solid", color="blue", weight=9];
759 -> 167[label="",style="solid", color="blue", weight=3];
760[label="== :: Char -> Char -> Bool",fontsize=10,color="white",style="solid",shape="box"];108 -> 760[label="",style="solid", color="blue", weight=9];
760 -> 168[label="",style="solid", color="blue", weight=3];
761[label="== :: () -> () -> Bool",fontsize=10,color="white",style="solid",shape="box"];108 -> 761[label="",style="solid", color="blue", weight=9];
761 -> 169[label="",style="solid", color="blue", weight=3];
762[label="== :: ((@2) a b) -> ((@2) a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];108 -> 762[label="",style="solid", color="blue", weight=9];
762 -> 170[label="",style="solid", color="blue", weight=3];
109[label="True",fontsize=16,color="green",shape="box"];110[label="False",fontsize=16,color="green",shape="box"];111[label="False",fontsize=16,color="green",shape="box"];112[label="False",fontsize=16,color="green",shape="box"];113[label="True",fontsize=16,color="green",shape="box"];114[label="False",fontsize=16,color="green",shape="box"];115[label="False",fontsize=16,color="green",shape="box"];116[label="False",fontsize=16,color="green",shape="box"];117[label="True",fontsize=16,color="green",shape="box"];118[label="True",fontsize=16,color="green",shape="box"];119[label="False",fontsize=16,color="green",shape="box"];120[label="False",fontsize=16,color="green",shape="box"];121[label="xy300 == xy400\n  Eq a",fontsize=16,color="blue",shape="box"];763[label="== :: ([] a) -> ([] a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];121 -> 763[label="",style="solid", color="blue", weight=9];
763 -> 171[label="",style="solid", color="blue", weight=3];
764[label="== :: ((@3) a b c) -> ((@3) a b c) -> Bool",fontsize=10,color="white",style="solid",shape="box"];121 -> 764[label="",style="solid", color="blue", weight=9];
764 -> 172[label="",style="solid", color="blue", weight=3];
765[label="== :: Integer -> Integer -> Bool",fontsize=10,color="white",style="solid",shape="box"];121 -> 765[label="",style="solid", color="blue", weight=9];
765 -> 173[label="",style="solid", color="blue", weight=3];
766[label="== :: (Ratio a) -> (Ratio a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];121 -> 766[label="",style="solid", color="blue", weight=9];
766 -> 174[label="",style="solid", color="blue", weight=3];
767[label="== :: Double -> Double -> Bool",fontsize=10,color="white",style="solid",shape="box"];121 -> 767[label="",style="solid", color="blue", weight=9];
767 -> 175[label="",style="solid", color="blue", weight=3];
768[label="== :: Float -> Float -> Bool",fontsize=10,color="white",style="solid",shape="box"];121 -> 768[label="",style="solid", color="blue", weight=9];
768 -> 176[label="",style="solid", color="blue", weight=3];
769[label="== :: (Either a b) -> (Either a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];121 -> 769[label="",style="solid", color="blue", weight=9];
769 -> 177[label="",style="solid", color="blue", weight=3];
770[label="== :: Ordering -> Ordering -> Bool",fontsize=10,color="white",style="solid",shape="box"];121 -> 770[label="",style="solid", color="blue", weight=9];
770 -> 178[label="",style="solid", color="blue", weight=3];
771[label="== :: (Maybe a) -> (Maybe a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];121 -> 771[label="",style="solid", color="blue", weight=9];
771 -> 179[label="",style="solid", color="blue", weight=3];
772[label="== :: Bool -> Bool -> Bool",fontsize=10,color="white",style="solid",shape="box"];121 -> 772[label="",style="solid", color="blue", weight=9];
772 -> 180[label="",style="solid", color="blue", weight=3];
773[label="== :: Int -> Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];121 -> 773[label="",style="solid", color="blue", weight=9];
773 -> 181[label="",style="solid", color="blue", weight=3];
774[label="== :: Char -> Char -> Bool",fontsize=10,color="white",style="solid",shape="box"];121 -> 774[label="",style="solid", color="blue", weight=9];
774 -> 182[label="",style="solid", color="blue", weight=3];
775[label="== :: () -> () -> Bool",fontsize=10,color="white",style="solid",shape="box"];121 -> 775[label="",style="solid", color="blue", weight=9];
775 -> 183[label="",style="solid", color="blue", weight=3];
776[label="== :: ((@2) a b) -> ((@2) a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];121 -> 776[label="",style="solid", color="blue", weight=9];
776 -> 184[label="",style="solid", color="blue", weight=3];
122[label="True",fontsize=16,color="green",shape="box"];123[label="False",fontsize=16,color="green",shape="box"];124[label="False",fontsize=16,color="green",shape="box"];125[label="True",fontsize=16,color="green",shape="box"];126[label="primEqInt (Pos (Succ xy3000)) xy40",fontsize=16,color="burlywood",shape="box"];777[label="xy40/Pos xy400",fontsize=10,color="white",style="solid",shape="box"];126 -> 777[label="",style="solid", color="burlywood", weight=9];
777 -> 185[label="",style="solid", color="burlywood", weight=3];
778[label="xy40/Neg xy400",fontsize=10,color="white",style="solid",shape="box"];126 -> 778[label="",style="solid", color="burlywood", weight=9];
778 -> 186[label="",style="solid", color="burlywood", weight=3];
127[label="primEqInt (Pos Zero) xy40",fontsize=16,color="burlywood",shape="box"];779[label="xy40/Pos xy400",fontsize=10,color="white",style="solid",shape="box"];127 -> 779[label="",style="solid", color="burlywood", weight=9];
779 -> 187[label="",style="solid", color="burlywood", weight=3];
780[label="xy40/Neg xy400",fontsize=10,color="white",style="solid",shape="box"];127 -> 780[label="",style="solid", color="burlywood", weight=9];
780 -> 188[label="",style="solid", color="burlywood", weight=3];
128[label="primEqInt (Neg (Succ xy3000)) xy40",fontsize=16,color="burlywood",shape="box"];781[label="xy40/Pos xy400",fontsize=10,color="white",style="solid",shape="box"];128 -> 781[label="",style="solid", color="burlywood", weight=9];
781 -> 189[label="",style="solid", color="burlywood", weight=3];
782[label="xy40/Neg xy400",fontsize=10,color="white",style="solid",shape="box"];128 -> 782[label="",style="solid", color="burlywood", weight=9];
782 -> 190[label="",style="solid", color="burlywood", weight=3];
129[label="primEqInt (Neg Zero) xy40",fontsize=16,color="burlywood",shape="box"];783[label="xy40/Pos xy400",fontsize=10,color="white",style="solid",shape="box"];129 -> 783[label="",style="solid", color="burlywood", weight=9];
783 -> 191[label="",style="solid", color="burlywood", weight=3];
784[label="xy40/Neg xy400",fontsize=10,color="white",style="solid",shape="box"];129 -> 784[label="",style="solid", color="burlywood", weight=9];
784 -> 192[label="",style="solid", color="burlywood", weight=3];
130[label="primEqChar (Char xy300) (Char xy400)",fontsize=16,color="black",shape="box"];130 -> 193[label="",style="solid", color="black", weight=3];
131[label="True",fontsize=16,color="green",shape="box"];132 -> 16[label="",style="dashed", color="red", weight=0];
132[label="xy300 == xy400 && xy301 == xy401\n  Eq a  Eq b",fontsize=16,color="magenta"];132 -> 194[label="",style="dashed", color="magenta", weight=3];
132 -> 195[label="",style="dashed", color="magenta", weight=3];
133 -> 23[label="",style="dashed", color="red", weight=0];
133[label="xy301 == xy401\n  Eq a",fontsize=16,color="magenta"];133 -> 196[label="",style="dashed", color="magenta", weight=3];
133 -> 197[label="",style="dashed", color="magenta", weight=3];
134[label="xy300 == xy400\n  Eq a",fontsize=16,color="blue",shape="box"];787[label="== :: ([] a) -> ([] a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];134 -> 787[label="",style="solid", color="blue", weight=9];
787 -> 198[label="",style="solid", color="blue", weight=3];
788[label="== :: ((@3) a b c) -> ((@3) a b c) -> Bool",fontsize=10,color="white",style="solid",shape="box"];134 -> 788[label="",style="solid", color="blue", weight=9];
788 -> 199[label="",style="solid", color="blue", weight=3];
789[label="== :: Integer -> Integer -> Bool",fontsize=10,color="white",style="solid",shape="box"];134 -> 789[label="",style="solid", color="blue", weight=9];
789 -> 200[label="",style="solid", color="blue", weight=3];
790[label="== :: (Ratio a) -> (Ratio a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];134 -> 790[label="",style="solid", color="blue", weight=9];
790 -> 201[label="",style="solid", color="blue", weight=3];
791[label="== :: Double -> Double -> Bool",fontsize=10,color="white",style="solid",shape="box"];134 -> 791[label="",style="solid", color="blue", weight=9];
791 -> 202[label="",style="solid", color="blue", weight=3];
792[label="== :: Float -> Float -> Bool",fontsize=10,color="white",style="solid",shape="box"];134 -> 792[label="",style="solid", color="blue", weight=9];
792 -> 203[label="",style="solid", color="blue", weight=3];
793[label="== :: (Either a b) -> (Either a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];134 -> 793[label="",style="solid", color="blue", weight=9];
793 -> 204[label="",style="solid", color="blue", weight=3];
794[label="== :: Ordering -> Ordering -> Bool",fontsize=10,color="white",style="solid",shape="box"];134 -> 794[label="",style="solid", color="blue", weight=9];
794 -> 205[label="",style="solid", color="blue", weight=3];
795[label="== :: (Maybe a) -> (Maybe a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];134 -> 795[label="",style="solid", color="blue", weight=9];
795 -> 206[label="",style="solid", color="blue", weight=3];
796[label="== :: Bool -> Bool -> Bool",fontsize=10,color="white",style="solid",shape="box"];134 -> 796[label="",style="solid", color="blue", weight=9];
796 -> 207[label="",style="solid", color="blue", weight=3];
797[label="== :: Int -> Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];134 -> 797[label="",style="solid", color="blue", weight=9];
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640[label="primPlusNat (Succ xy1100) (Succ xy400000)",fontsize=16,color="black",shape="box"];640 -> 644[label="",style="solid", color="black", weight=3];
641[label="primPlusNat (Succ xy1100) Zero",fontsize=16,color="black",shape="box"];641 -> 645[label="",style="solid", color="black", weight=3];
642[label="primPlusNat Zero (Succ xy400000)",fontsize=16,color="black",shape="box"];642 -> 646[label="",style="solid", color="black", weight=3];
643[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];643 -> 647[label="",style="solid", color="black", weight=3];
644[label="Succ (Succ (primPlusNat xy1100 xy400000))",fontsize=16,color="green",shape="box"];644 -> 648[label="",style="dashed", color="green", weight=3];
645[label="Succ xy1100",fontsize=16,color="green",shape="box"];646[label="Succ xy400000",fontsize=16,color="green",shape="box"];647[label="Zero",fontsize=16,color="green",shape="box"];648 -> 637[label="",style="dashed", color="red", weight=0];
648[label="primPlusNat xy1100 xy400000",fontsize=16,color="magenta"];648 -> 649[label="",style="dashed", color="magenta", weight=3];
648 -> 650[label="",style="dashed", color="magenta", weight=3];
649[label="xy400000",fontsize=16,color="green",shape="box"];650[label="xy1100",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

new_primPlusNat(Succ(xy1100), Succ(xy400000)) → new_primPlusNat(xy1100, xy400000)

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_primPlusNat(Succ(xy1100), Succ(xy400000)) → new_primPlusNat(xy1100, xy400000)
The graph contains the following edges 1 > 1, 2 > 2

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

new_primMulNat(Succ(xy30000), Succ(xy40000)) → new_primMulNat(xy30000, Succ(xy40000))

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

new_primMulNat(Succ(xy30000), Succ(xy40000)) → new_primMulNat(xy30000, Succ(xy40000))
The graph contains the following edges 1 > 1, 2 >= 2

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

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

new_primEqNat(Succ(xy3000), Succ(xy4000)) → new_primEqNat(xy3000, xy4000)

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

new_primEqNat(Succ(xy3000), Succ(xy4000)) → new_primEqNat(xy3000, xy4000)
The graph contains the following edges 1 > 1, 2 > 2

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

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

new_esEs1(Left(xy300), Left(xy400), app(app(ty_@2, hc), hd), gd) → new_esEs3(xy300, xy400, hc, hd)
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), cc, cd, app(app(ty_Either, db), dc)) → new_esEs1(xy302, xy402, db, dc)
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), cc, cd, app(app(ty_@2, de), df)) → new_esEs3(xy302, xy402, de, df)
new_esEs2(Just(xy300), Just(xy400), app(ty_[], bag)) → new_esEs(xy300, xy400, bag)
new_esEs3(@2(xy300, xy301), @2(xy400, xy401), bbh, app(ty_Maybe, bcg)) → new_esEs2(xy301, xy401, bcg)
new_esEs3(@2(xy300, xy301), @2(xy400, xy401), app(app(ty_Either, bdg), bdh), bdc) → new_esEs1(xy300, xy400, bdg, bdh)
new_esEs1(Left(xy300), Left(xy400), app(app(app(ty_@3, ge), gf), gg), gd) → new_esEs0(xy300, xy400, ge, gf, gg)
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), app(ty_Maybe, fh), cd, dh) → new_esEs2(xy300, xy400, fh)
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), app(app(ty_Either, ff), fg), cd, dh) → new_esEs1(xy300, xy400, ff, fg)
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), cc, cd, app(ty_Maybe, dd)) → new_esEs2(xy302, xy402, dd)
new_esEs1(Right(xy300), Right(xy400), he, app(app(app(ty_@3, hg), hh), baa)) → new_esEs0(xy300, xy400, hg, hh, baa)
new_esEs3(@2(xy300, xy301), @2(xy400, xy401), app(ty_[], bdb), bdc) → new_esEs(xy300, xy400, bdb)
new_esEs3(@2(xy300, xy301), @2(xy400, xy401), app(app(app(ty_@3, bdd), bde), bdf), bdc) → new_esEs0(xy300, xy400, bdd, bde, bdf)
new_esEs1(Left(xy300), Left(xy400), app(app(ty_Either, gh), ha), gd) → new_esEs1(xy300, xy400, gh, ha)
new_esEs2(Just(xy300), Just(xy400), app(ty_Maybe, bbe)) → new_esEs2(xy300, xy400, bbe)
new_esEs1(Right(xy300), Right(xy400), he, app(app(ty_Either, bab), bac)) → new_esEs1(xy300, xy400, bab, bac)
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), cc, app(app(app(ty_@3, ea), eb), ec), dh) → new_esEs0(xy301, xy401, ea, eb, ec)
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), cc, cd, app(ty_[], ce)) → new_esEs(xy302, xy402, ce)
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), cc, app(app(ty_Either, ed), ee), dh) → new_esEs1(xy301, xy401, ed, ee)
new_esEs(:(xy300, xy301), :(xy400, xy401), app(ty_Maybe, bh)) → new_esEs2(xy300, xy400, bh)
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), cc, app(app(ty_@2, eg), eh), dh) → new_esEs3(xy301, xy401, eg, eh)
new_esEs3(@2(xy300, xy301), @2(xy400, xy401), bbh, app(ty_[], bca)) → new_esEs(xy301, xy401, bca)
new_esEs3(@2(xy300, xy301), @2(xy400, xy401), bbh, app(app(ty_@2, bch), bda)) → new_esEs3(xy301, xy401, bch, bda)
new_esEs2(Just(xy300), Just(xy400), app(app(app(ty_@3, bah), bba), bbb)) → new_esEs0(xy300, xy400, bah, bba, bbb)
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), app(app(ty_@2, ga), gb), cd, dh) → new_esEs3(xy300, xy400, ga, gb)
new_esEs2(Just(xy300), Just(xy400), app(app(ty_Either, bbc), bbd)) → new_esEs1(xy300, xy400, bbc, bbd)
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), app(ty_[], fa), cd, dh) → new_esEs(xy300, xy400, fa)
new_esEs(:(xy300, xy301), :(xy400, xy401), ba) → new_esEs(xy301, xy401, ba)
new_esEs3(@2(xy300, xy301), @2(xy400, xy401), app(ty_Maybe, bea), bdc) → new_esEs2(xy300, xy400, bea)
new_esEs3(@2(xy300, xy301), @2(xy400, xy401), bbh, app(app(ty_Either, bce), bcf)) → new_esEs1(xy301, xy401, bce, bcf)
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), cc, cd, app(app(app(ty_@3, cf), cg), da)) → new_esEs0(xy302, xy402, cf, cg, da)
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), cc, app(ty_[], dg), dh) → new_esEs(xy301, xy401, dg)
new_esEs1(Left(xy300), Left(xy400), app(ty_Maybe, hb), gd) → new_esEs2(xy300, xy400, hb)
new_esEs3(@2(xy300, xy301), @2(xy400, xy401), app(app(ty_@2, beb), bec), bdc) → new_esEs3(xy300, xy400, beb, bec)
new_esEs(:(xy300, xy301), :(xy400, xy401), app(app(app(ty_@3, bc), bd), be)) → new_esEs0(xy300, xy400, bc, bd, be)
new_esEs(:(xy300, xy301), :(xy400, xy401), app(app(ty_@2, ca), cb)) → new_esEs3(xy300, xy400, ca, cb)
new_esEs2(Just(xy300), Just(xy400), app(app(ty_@2, bbf), bbg)) → new_esEs3(xy300, xy400, bbf, bbg)
new_esEs3(@2(xy300, xy301), @2(xy400, xy401), bbh, app(app(app(ty_@3, bcb), bcc), bcd)) → new_esEs0(xy301, xy401, bcb, bcc, bcd)
new_esEs(:(xy300, xy301), :(xy400, xy401), app(app(ty_Either, bf), bg)) → new_esEs1(xy300, xy400, bf, bg)
new_esEs1(Right(xy300), Right(xy400), he, app(ty_[], hf)) → new_esEs(xy300, xy400, hf)
new_esEs1(Left(xy300), Left(xy400), app(ty_[], gc), gd) → new_esEs(xy300, xy400, gc)
new_esEs1(Right(xy300), Right(xy400), he, app(app(ty_@2, bae), baf)) → new_esEs3(xy300, xy400, bae, baf)
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), cc, app(ty_Maybe, ef), dh) → new_esEs2(xy301, xy401, ef)
new_esEs1(Right(xy300), Right(xy400), he, app(ty_Maybe, bad)) → new_esEs2(xy300, xy400, bad)
new_esEs(:(xy300, xy301), :(xy400, xy401), app(ty_[], bb)) → new_esEs(xy300, xy400, bb)
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), app(app(app(ty_@3, fb), fc), fd), cd, dh) → new_esEs0(xy300, xy400, fb, fc, fd)

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

new_esEs(:(xy300, xy301), :(xy400, xy401), app(app(ty_Either, bf), bg)) → new_esEs1(xy300, xy400, bf, bg)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3, 3 > 4
new_esEs(:(xy300, xy301), :(xy400, xy401), app(app(app(ty_@3, bc), bd), be)) → new_esEs0(xy300, xy400, bc, bd, be)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3, 3 > 4, 3 > 5
new_esEs(:(xy300, xy301), :(xy400, xy401), app(ty_Maybe, bh)) → new_esEs2(xy300, xy400, bh)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3
new_esEs(:(xy300, xy301), :(xy400, xy401), app(app(ty_@2, ca), cb)) → new_esEs3(xy300, xy400, ca, cb)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3, 3 > 4
new_esEs2(Just(xy300), Just(xy400), app(app(ty_Either, bbc), bbd)) → new_esEs1(xy300, xy400, bbc, bbd)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3, 3 > 4
new_esEs2(Just(xy300), Just(xy400), app(app(app(ty_@3, bah), bba), bbb)) → new_esEs0(xy300, xy400, bah, bba, bbb)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3, 3 > 4, 3 > 5
new_esEs2(Just(xy300), Just(xy400), app(ty_Maybe, bbe)) → new_esEs2(xy300, xy400, bbe)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3
new_esEs2(Just(xy300), Just(xy400), app(app(ty_@2, bbf), bbg)) → new_esEs3(xy300, xy400, bbf, bbg)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3, 3 > 4
new_esEs2(Just(xy300), Just(xy400), app(ty_[], bag)) → new_esEs(xy300, xy400, bag)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3
new_esEs3(@2(xy300, xy301), @2(xy400, xy401), app(app(ty_Either, bdg), bdh), bdc) → new_esEs1(xy300, xy400, bdg, bdh)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3, 3 > 4
new_esEs3(@2(xy300, xy301), @2(xy400, xy401), bbh, app(app(ty_Either, bce), bcf)) → new_esEs1(xy301, xy401, bce, bcf)
The graph contains the following edges 1 > 1, 2 > 2, 4 > 3, 4 > 4
new_esEs3(@2(xy300, xy301), @2(xy400, xy401), app(app(app(ty_@3, bdd), bde), bdf), bdc) → new_esEs0(xy300, xy400, bdd, bde, bdf)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3, 3 > 4, 3 > 5
new_esEs3(@2(xy300, xy301), @2(xy400, xy401), bbh, app(app(app(ty_@3, bcb), bcc), bcd)) → new_esEs0(xy301, xy401, bcb, bcc, bcd)
The graph contains the following edges 1 > 1, 2 > 2, 4 > 3, 4 > 4, 4 > 5
new_esEs3(@2(xy300, xy301), @2(xy400, xy401), bbh, app(ty_Maybe, bcg)) → new_esEs2(xy301, xy401, bcg)
The graph contains the following edges 1 > 1, 2 > 2, 4 > 3
new_esEs3(@2(xy300, xy301), @2(xy400, xy401), app(ty_Maybe, bea), bdc) → new_esEs2(xy300, xy400, bea)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3
new_esEs3(@2(xy300, xy301), @2(xy400, xy401), bbh, app(app(ty_@2, bch), bda)) → new_esEs3(xy301, xy401, bch, bda)
The graph contains the following edges 1 > 1, 2 > 2, 4 > 3, 4 > 4
new_esEs3(@2(xy300, xy301), @2(xy400, xy401), app(app(ty_@2, beb), bec), bdc) → new_esEs3(xy300, xy400, beb, bec)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3, 3 > 4
new_esEs3(@2(xy300, xy301), @2(xy400, xy401), app(ty_[], bdb), bdc) → new_esEs(xy300, xy400, bdb)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3
new_esEs3(@2(xy300, xy301), @2(xy400, xy401), bbh, app(ty_[], bca)) → new_esEs(xy301, xy401, bca)
The graph contains the following edges 1 > 1, 2 > 2, 4 > 3
new_esEs1(Left(xy300), Left(xy400), app(app(ty_Either, gh), ha), gd) → new_esEs1(xy300, xy400, gh, ha)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3, 3 > 4
new_esEs1(Right(xy300), Right(xy400), he, app(app(ty_Either, bab), bac)) → new_esEs1(xy300, xy400, bab, bac)
The graph contains the following edges 1 > 1, 2 > 2, 4 > 3, 4 > 4
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), cc, cd, app(app(ty_Either, db), dc)) → new_esEs1(xy302, xy402, db, dc)
The graph contains the following edges 1 > 1, 2 > 2, 5 > 3, 5 > 4
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), app(app(ty_Either, ff), fg), cd, dh) → new_esEs1(xy300, xy400, ff, fg)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3, 3 > 4
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), cc, app(app(ty_Either, ed), ee), dh) → new_esEs1(xy301, xy401, ed, ee)
The graph contains the following edges 1 > 1, 2 > 2, 4 > 3, 4 > 4
new_esEs1(Left(xy300), Left(xy400), app(app(app(ty_@3, ge), gf), gg), gd) → new_esEs0(xy300, xy400, ge, gf, gg)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3, 3 > 4, 3 > 5
new_esEs1(Right(xy300), Right(xy400), he, app(app(app(ty_@3, hg), hh), baa)) → new_esEs0(xy300, xy400, hg, hh, baa)
The graph contains the following edges 1 > 1, 2 > 2, 4 > 3, 4 > 4, 4 > 5
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), cc, app(app(app(ty_@3, ea), eb), ec), dh) → new_esEs0(xy301, xy401, ea, eb, ec)
The graph contains the following edges 1 > 1, 2 > 2, 4 > 3, 4 > 4, 4 > 5
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), cc, cd, app(app(app(ty_@3, cf), cg), da)) → new_esEs0(xy302, xy402, cf, cg, da)
The graph contains the following edges 1 > 1, 2 > 2, 5 > 3, 5 > 4, 5 > 5
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), app(app(app(ty_@3, fb), fc), fd), cd, dh) → new_esEs0(xy300, xy400, fb, fc, fd)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3, 3 > 4, 3 > 5
new_esEs1(Left(xy300), Left(xy400), app(ty_Maybe, hb), gd) → new_esEs2(xy300, xy400, hb)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3
new_esEs1(Right(xy300), Right(xy400), he, app(ty_Maybe, bad)) → new_esEs2(xy300, xy400, bad)
The graph contains the following edges 1 > 1, 2 > 2, 4 > 3
new_esEs1(Left(xy300), Left(xy400), app(app(ty_@2, hc), hd), gd) → new_esEs3(xy300, xy400, hc, hd)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3, 3 > 4
new_esEs1(Right(xy300), Right(xy400), he, app(app(ty_@2, bae), baf)) → new_esEs3(xy300, xy400, bae, baf)
The graph contains the following edges 1 > 1, 2 > 2, 4 > 3, 4 > 4
new_esEs1(Right(xy300), Right(xy400), he, app(ty_[], hf)) → new_esEs(xy300, xy400, hf)
The graph contains the following edges 1 > 1, 2 > 2, 4 > 3
new_esEs1(Left(xy300), Left(xy400), app(ty_[], gc), gd) → new_esEs(xy300, xy400, gc)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3
new_esEs(:(xy300, xy301), :(xy400, xy401), ba) → new_esEs(xy301, xy401, ba)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
new_esEs(:(xy300, xy301), :(xy400, xy401), app(ty_[], bb)) → new_esEs(xy300, xy400, bb)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), cc, cd, app(ty_Maybe, dd)) → new_esEs2(xy302, xy402, dd)
The graph contains the following edges 1 > 1, 2 > 2, 5 > 3
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), app(ty_Maybe, fh), cd, dh) → new_esEs2(xy300, xy400, fh)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), cc, app(ty_Maybe, ef), dh) → new_esEs2(xy301, xy401, ef)
The graph contains the following edges 1 > 1, 2 > 2, 4 > 3
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), cc, cd, app(app(ty_@2, de), df)) → new_esEs3(xy302, xy402, de, df)
The graph contains the following edges 1 > 1, 2 > 2, 5 > 3, 5 > 4
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), cc, app(app(ty_@2, eg), eh), dh) → new_esEs3(xy301, xy401, eg, eh)
The graph contains the following edges 1 > 1, 2 > 2, 4 > 3, 4 > 4
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), app(app(ty_@2, ga), gb), cd, dh) → new_esEs3(xy300, xy400, ga, gb)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3, 3 > 4
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), cc, cd, app(ty_[], ce)) → new_esEs(xy302, xy402, ce)
The graph contains the following edges 1 > 1, 2 > 2, 5 > 3
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), app(ty_[], fa), cd, dh) → new_esEs(xy300, xy400, fa)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3
new_esEs0(@3(xy300, xy301, xy302), @3(xy400, xy401, xy402), cc, app(ty_[], dg), dh) → new_esEs(xy301, xy401, dg)
The graph contains the following edges 1 > 1, 2 > 2, 4 > 3

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPSizeChangeProof

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

new_isPrefixOf(:(xy30, xy31), :(xy40, xy41), ba) → new_isPrefixOf(xy31, xy41, ba)

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

new_isPrefixOf(:(xy30, xy31), :(xy40, xy41), ba) → new_isPrefixOf(xy31, xy41, ba)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3