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

YES H-Termination proof of ../tpdb/qualif/abs_1.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ NumRed

mainModule Main

((absRatio :: Ratio MyInt -> Ratio MyInt) :: Ratio MyInt -> Ratio MyInt)

module Main where

import qualified Prelude

absMyInt :: MyInt -> MyInt

absMyInt

absReal

absRatio :: Ratio MyInt -> Ratio MyInt

absRatio

(CnPc x y)

CnPc (absMyInt x) y

absReal

absReal2 x

absReal0

x MyTrue

negateMyInt x

absReal1	x MyTrue	=	x
absReal1	x MyFalse	=	absReal0 x otherwise

absReal2

absReal1 x (gtEsMyInt x (fromIntMyInt (Pos Zero)))

compareMyInt :: MyInt -> MyInt -> Ordering

compareMyInt

primCmpInt

esEsOrdering :: Ordering -> Ordering -> MyBool

esEsOrdering	LT LT	=	MyTrue
esEsOrdering	LT EQ	=	MyFalse
esEsOrdering	LT GT	=	MyFalse
esEsOrdering	EQ LT	=	MyFalse
esEsOrdering	EQ EQ	=	MyTrue
esEsOrdering	EQ GT	=	MyFalse
esEsOrdering	GT LT	=	MyFalse
esEsOrdering	GT EQ	=	MyFalse
esEsOrdering	GT GT	=	MyTrue

fromIntMyInt :: MyInt -> MyInt

fromIntMyInt

fsEsOrdering :: Ordering -> Ordering -> MyBool

fsEsOrdering

x y

not (esEsOrdering x y)

gtEsMyInt :: MyInt -> MyInt -> MyBool

gtEsMyInt

x y

fsEsOrdering (compareMyInt x y) LT

negateMyInt :: MyInt -> MyInt

negateMyInt

primNegInt

not :: MyBool -> MyBool

not	MyTrue	=	MyFalse
not	MyFalse	=	MyTrue

otherwise :: MyBool

otherwise

MyTrue

primCmpInt :: MyInt -> MyInt -> Ordering

primCmpInt	(Pos Zero) (Pos Zero)	=	EQ
primCmpInt	(Pos Zero) (Neg Zero)	=	EQ
primCmpInt	(Neg Zero) (Pos Zero)	=	EQ
primCmpInt	(Neg Zero) (Neg Zero)	=	EQ
primCmpInt	(Pos x) (Pos y)	=	primCmpNat x y
primCmpInt	(Pos x) (Neg y)	=	GT
primCmpInt	(Neg x) (Pos y)	=	LT
primCmpInt	(Neg x) (Neg y)	=	primCmpNat y x

primCmpNat :: Nat -> Nat -> Ordering

primCmpNat	Zero Zero	=	EQ
primCmpNat	Zero (Succ y)	=	LT
primCmpNat	(Succ x) Zero	=	GT
primCmpNat	(Succ x) (Succ y)	=	primCmpNat x y

primNegInt :: MyInt -> MyInt

primNegInt	(Pos x)	=	Neg x
primNegInt	(Neg x)	=	Pos x

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

↳ HASKELL

  ↳ NumRed

    ↳ HASKELL

      ↳ Narrow

mainModule Main

(absRatio :: Ratio MyInt -> Ratio MyInt)

module Main where

import qualified Prelude

absMyInt :: MyInt -> MyInt

absMyInt

absReal

absRatio :: Ratio MyInt -> Ratio MyInt

absRatio

(CnPc x y)

CnPc (absMyInt x) y

absReal

absReal2 x

absReal0

x MyTrue

negateMyInt x

absReal1	x MyTrue	=	x
absReal1	x MyFalse	=	absReal0 x otherwise

absReal2

absReal1 x (gtEsMyInt x (fromIntMyInt (Pos Zero)))

compareMyInt :: MyInt -> MyInt -> Ordering

compareMyInt

primCmpInt

esEsOrdering :: Ordering -> Ordering -> MyBool

esEsOrdering	LT LT	=	MyTrue
esEsOrdering	LT EQ	=	MyFalse
esEsOrdering	LT GT	=	MyFalse
esEsOrdering	EQ LT	=	MyFalse
esEsOrdering	EQ EQ	=	MyTrue
esEsOrdering	EQ GT	=	MyFalse
esEsOrdering	GT LT	=	MyFalse
esEsOrdering	GT EQ	=	MyFalse
esEsOrdering	GT GT	=	MyTrue

fromIntMyInt :: MyInt -> MyInt

fromIntMyInt

fsEsOrdering :: Ordering -> Ordering -> MyBool

fsEsOrdering

x y

not (esEsOrdering x y)

gtEsMyInt :: MyInt -> MyInt -> MyBool

gtEsMyInt

x y

fsEsOrdering (compareMyInt x y) LT

negateMyInt :: MyInt -> MyInt

negateMyInt

primNegInt

not :: MyBool -> MyBool

not	MyTrue	=	MyFalse
not	MyFalse	=	MyTrue

otherwise :: MyBool

otherwise

MyTrue

primCmpInt :: MyInt -> MyInt -> Ordering

primCmpInt	(Pos Zero) (Pos Zero)	=	EQ
primCmpInt	(Pos Zero) (Neg Zero)	=	EQ
primCmpInt	(Neg Zero) (Pos Zero)	=	EQ
primCmpInt	(Neg Zero) (Neg Zero)	=	EQ
primCmpInt	(Pos x) (Pos y)	=	primCmpNat x y
primCmpInt	(Pos x) (Neg y)	=	GT
primCmpInt	(Neg x) (Pos y)	=	LT
primCmpInt	(Neg x) (Neg y)	=	primCmpNat y x

primCmpNat :: Nat -> Nat -> Ordering

primCmpNat	Zero Zero	=	EQ
primCmpNat	Zero (Succ y)	=	LT
primCmpNat	(Succ x) Zero	=	GT
primCmpNat	(Succ x) (Succ y)	=	primCmpNat x y

primNegInt :: MyInt -> MyInt

primNegInt	(Pos x)	=	Neg x
primNegInt	(Neg x)	=	Pos x

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="absRatio\n",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="absRatio u3\n",fontsize=16,color="burlywood",shape="triangle"];44[label="u3/CnPc u30 u31",fontsize=10,color="white",style="solid",shape="box"];3 -> 44[label="",style="solid", color="burlywood", weight=9];
44 -> 4[label="",style="solid", color="burlywood", weight=3];
4[label="absRatio (CnPc u30 u31)\n",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="CnPc (absMyInt u30) u31\n",fontsize=16,color="green",shape="box"];5 -> 6[label="",style="dashed", color="green", weight=3];
6[label="absMyInt u30\n",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="absReal u30\n",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="absReal2 u30\n",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="absReal1 u30 (gtEsMyInt u30 (fromIntMyInt (Pos Zero)))\n",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10[label="absReal1 u30 (fsEsOrdering (compareMyInt u30 (fromIntMyInt (Pos Zero))) LT)\n",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11[label="absReal1 u30 (not (esEsOrdering (compareMyInt u30 (fromIntMyInt (Pos Zero))) LT))\n",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12[label="absReal1 u30 (not (esEsOrdering (primCmpInt u30 (fromIntMyInt (Pos Zero))) LT))\n",fontsize=16,color="burlywood",shape="box"];45[label="u30/Pos u300",fontsize=10,color="white",style="solid",shape="box"];12 -> 45[label="",style="solid", color="burlywood", weight=9];
45 -> 13[label="",style="solid", color="burlywood", weight=3];
46[label="u30/Neg u300",fontsize=10,color="white",style="solid",shape="box"];12 -> 46[label="",style="solid", color="burlywood", weight=9];
46 -> 14[label="",style="solid", color="burlywood", weight=3];
13[label="absReal1 (Pos u300) (not (esEsOrdering (primCmpInt (Pos u300) (fromIntMyInt (Pos Zero))) LT))\n",fontsize=16,color="burlywood",shape="box"];47[label="u300/Succ u3000",fontsize=10,color="white",style="solid",shape="box"];13 -> 47[label="",style="solid", color="burlywood", weight=9];
47 -> 15[label="",style="solid", color="burlywood", weight=3];
48[label="u300/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 48[label="",style="solid", color="burlywood", weight=9];
48 -> 16[label="",style="solid", color="burlywood", weight=3];
14[label="absReal1 (Neg u300) (not (esEsOrdering (primCmpInt (Neg u300) (fromIntMyInt (Pos Zero))) LT))\n",fontsize=16,color="burlywood",shape="box"];49[label="u300/Succ u3000",fontsize=10,color="white",style="solid",shape="box"];14 -> 49[label="",style="solid", color="burlywood", weight=9];
49 -> 17[label="",style="solid", color="burlywood", weight=3];
50[label="u300/Zero",fontsize=10,color="white",style="solid",shape="box"];14 -> 50[label="",style="solid", color="burlywood", weight=9];
50 -> 18[label="",style="solid", color="burlywood", weight=3];
15[label="absReal1 (Pos (Succ u3000)) (not (esEsOrdering (primCmpInt (Pos (Succ u3000)) (fromIntMyInt (Pos Zero))) LT))\n",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
16[label="absReal1 (Pos Zero) (not (esEsOrdering (primCmpInt (Pos Zero) (fromIntMyInt (Pos Zero))) LT))\n",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="absReal1 (Neg (Succ u3000)) (not (esEsOrdering (primCmpInt (Neg (Succ u3000)) (fromIntMyInt (Pos Zero))) LT))\n",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
18[label="absReal1 (Neg Zero) (not (esEsOrdering (primCmpInt (Neg Zero) (fromIntMyInt (Pos Zero))) LT))\n",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="absReal1 (Pos (Succ u3000)) (not (esEsOrdering (primCmpInt (Pos (Succ u3000)) (Pos Zero)) LT))\n",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="absReal1 (Pos Zero) (not (esEsOrdering (primCmpInt (Pos Zero) (Pos Zero)) LT))\n",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="absReal1 (Neg (Succ u3000)) (not (esEsOrdering (primCmpInt (Neg (Succ u3000)) (Pos Zero)) LT))\n",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="absReal1 (Neg Zero) (not (esEsOrdering (primCmpInt (Neg Zero) (Pos Zero)) LT))\n",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="absReal1 (Pos (Succ u3000)) (not (esEsOrdering (primCmpNat (Succ u3000) Zero) LT))\n",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="absReal1 (Pos Zero) (not (esEsOrdering EQ LT))\n",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="absReal1 (Neg (Succ u3000)) (not (esEsOrdering LT LT))\n",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="absReal1 (Neg Zero) (not (esEsOrdering EQ LT))\n",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="absReal1 (Pos (Succ u3000)) (not (esEsOrdering GT LT))\n",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="absReal1 (Pos Zero) (not MyFalse)\n",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="absReal1 (Neg (Succ u3000)) (not MyTrue)\n",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
30[label="absReal1 (Neg Zero) (not MyFalse)\n",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3];
31[label="absReal1 (Pos (Succ u3000)) (not MyFalse)\n",fontsize=16,color="black",shape="box"];31 -> 35[label="",style="solid", color="black", weight=3];
32[label="absReal1 (Pos Zero) MyTrue\n",fontsize=16,color="black",shape="box"];32 -> 36[label="",style="solid", color="black", weight=3];
33[label="absReal1 (Neg (Succ u3000)) MyFalse\n",fontsize=16,color="black",shape="box"];33 -> 37[label="",style="solid", color="black", weight=3];
34[label="absReal1 (Neg Zero) MyTrue\n",fontsize=16,color="black",shape="box"];34 -> 38[label="",style="solid", color="black", weight=3];
35[label="absReal1 (Pos (Succ u3000)) MyTrue\n",fontsize=16,color="black",shape="box"];35 -> 39[label="",style="solid", color="black", weight=3];
36[label="Pos Zero\n",fontsize=16,color="green",shape="box"];37[label="absReal0 (Neg (Succ u3000)) otherwise\n",fontsize=16,color="black",shape="box"];37 -> 40[label="",style="solid", color="black", weight=3];
38[label="Neg Zero\n",fontsize=16,color="green",shape="box"];39[label="Pos (Succ u3000)\n",fontsize=16,color="green",shape="box"];40[label="absReal0 (Neg (Succ u3000)) MyTrue\n",fontsize=16,color="black",shape="box"];40 -> 41[label="",style="solid", color="black", weight=3];
41[label="negateMyInt (Neg (Succ u3000))\n",fontsize=16,color="black",shape="box"];41 -> 42[label="",style="solid", color="black", weight=3];
42[label="primNegInt (Neg (Succ u3000))\n",fontsize=16,color="black",shape="box"];42 -> 43[label="",style="solid", color="black", weight=3];
43[label="Pos (Succ u3000)\n",fontsize=16,color="green",shape="box"];}