YES 2.865
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
↳ LR
mainModule Main
| ((succ :: Ratio Int -> Ratio Int) :: Ratio Int -> Ratio Int) |
module Main where
Lambda Reductions:
The following Lambda expression
\(m,_)→m
is transformed to
The following Lambda expression
\(_,r)→r
is transformed to
The following Lambda expression
\(q,_)→q
is transformed to
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
mainModule Main
| ((succ :: Ratio Int -> Ratio Int) :: Ratio Int -> Ratio Int) |
module Main where
If Reductions:
The following If expression
if primGEqNatS x y then Succ (primDivNatS (primMinusNatS x y) (Succ y)) else Zero
is transformed to
primDivNatS0 | x y True | = Succ (primDivNatS (primMinusNatS x y) (Succ y)) |
primDivNatS0 | x y False | = Zero |
The following If expression
if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x
is transformed to
primModNatS0 | x y True | = primModNatS (primMinusNatS x y) (Succ y) |
primModNatS0 | x y False | = Succ x |
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
mainModule Main
| ((succ :: Ratio Int -> Ratio Int) :: Ratio Int -> Ratio Int) |
module Main where
Replaced joker patterns by fresh variables and removed binding patterns.
Binding Reductions:
The bind variable of the following binding Pattern
frac@(Double vy vz)
is replaced by the following term
Double vy vz
The bind variable of the following binding Pattern
frac@(Float ww wx)
is replaced by the following term
Float ww wx
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
mainModule Main
| ((succ :: Ratio Int -> Ratio Int) :: Ratio Int -> Ratio Int) |
module Main where
Cond Reductions:
The following Function with conditions
is transformed to
undefined0 | True | = undefined |
undefined1 | | = undefined0 False |
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
mainModule Main
| ((succ :: Ratio Int -> Ratio Int) :: Ratio Int -> Ratio Int) |
module Main where
Let/Where Reductions:
The bindings of the following Let/Where expression
are unpacked to the following functions on top level
truncateM | xw | = truncateM0 xw (truncateVu6 xw) |
truncateVu6 | xw | = properFraction xw |
The bindings of the following Let/Where expression
(fromIntegral q,r :% y) |
where | |
| |
| |
| |
| |
are unpacked to the following functions on top level
properFractionQ1 | xx xy (q,wz) | = q |
properFractionR | xx xy | = properFractionR0 xx xy (properFractionVu30 xx xy) |
properFractionVu30 | xx xy | = quotRem xx xy |
properFractionR0 | xx xy (xu,r) | = r |
properFractionQ | xx xy | = properFractionQ1 xx xy (properFractionVu30 xx xy) |
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
mainModule Main
| ((succ :: Ratio Int -> Ratio Int) :: Ratio Int -> Ratio Int) |
module Main where
Num Reduction: All numbers are transformed to thier corresponding representation with Pos, Neg, Succ and Zero.
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
mainModule Main
| (succ :: Ratio Int -> Ratio Int) |
module Main where
Haskell To QDPs
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(xz134, xz135, Zero, Zero) → new_primDivNatS00(xz134, xz135)
new_primDivNatS0(xz134, xz135, Succ(xz1360), Succ(xz1370)) → new_primDivNatS0(xz134, xz135, xz1360, xz1370)
new_primDivNatS(Succ(xz1150), Succ(xz1160), xz117) → new_primDivNatS(xz1150, xz1160, xz117)
new_primDivNatS(Succ(Succ(xz11500)), Zero, Succ(xz1170)) → new_primDivNatS0(xz11500, xz1170, xz11500, xz1170)
new_primDivNatS(Succ(Zero), Zero, Zero) → new_primDivNatS(Zero, Zero, Zero)
new_primDivNatS(Succ(Succ(xz11500)), Zero, Zero) → new_primDivNatS(Succ(xz11500), Zero, Zero)
new_primDivNatS00(xz134, xz135) → new_primDivNatS(Succ(xz134), Succ(xz135), Succ(xz135))
new_primDivNatS0(xz134, xz135, Succ(xz1360), Zero) → new_primDivNatS(Succ(xz134), Succ(xz135), Succ(xz135))
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 with 1 less node.
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS(Succ(Succ(xz11500)), Zero, Zero) → new_primDivNatS(Succ(xz11500), Zero, Zero)
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_primDivNatS(Succ(Succ(xz11500)), Zero, Zero) → new_primDivNatS(Succ(xz11500), Zero, Zero)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 2, 2 >= 3, 3 >= 3
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(xz134, xz135, Zero, Zero) → new_primDivNatS00(xz134, xz135)
new_primDivNatS0(xz134, xz135, Succ(xz1360), Succ(xz1370)) → new_primDivNatS0(xz134, xz135, xz1360, xz1370)
new_primDivNatS(Succ(xz1150), Succ(xz1160), xz117) → new_primDivNatS(xz1150, xz1160, xz117)
new_primDivNatS(Succ(Succ(xz11500)), Zero, Succ(xz1170)) → new_primDivNatS0(xz11500, xz1170, xz11500, xz1170)
new_primDivNatS00(xz134, xz135) → new_primDivNatS(Succ(xz134), Succ(xz135), Succ(xz135))
new_primDivNatS0(xz134, xz135, Succ(xz1360), Zero) → new_primDivNatS(Succ(xz134), Succ(xz135), Succ(xz135))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
new_primDivNatS(Succ(xz1150), Succ(xz1160), xz117) → new_primDivNatS(xz1150, xz1160, xz117)
new_primDivNatS(Succ(Succ(xz11500)), Zero, Succ(xz1170)) → new_primDivNatS0(xz11500, xz1170, xz11500, xz1170)
The remaining pairs can at least be oriented weakly.
new_primDivNatS0(xz134, xz135, Zero, Zero) → new_primDivNatS00(xz134, xz135)
new_primDivNatS0(xz134, xz135, Succ(xz1360), Succ(xz1370)) → new_primDivNatS0(xz134, xz135, xz1360, xz1370)
new_primDivNatS00(xz134, xz135) → new_primDivNatS(Succ(xz134), Succ(xz135), Succ(xz135))
new_primDivNatS0(xz134, xz135, Succ(xz1360), Zero) → new_primDivNatS(Succ(xz134), Succ(xz135), Succ(xz135))
Used ordering: Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 0
POL(new_primDivNatS(x1, x2, x3)) = x1
POL(new_primDivNatS0(x1, x2, x3, x4)) = 1 + x1
POL(new_primDivNatS00(x1, x2)) = 1 + x1
The following usable rules [17] were oriented:
none
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(xz134, xz135, Zero, Zero) → new_primDivNatS00(xz134, xz135)
new_primDivNatS0(xz134, xz135, Succ(xz1360), Succ(xz1370)) → new_primDivNatS0(xz134, xz135, xz1360, xz1370)
new_primDivNatS00(xz134, xz135) → new_primDivNatS(Succ(xz134), Succ(xz135), Succ(xz135))
new_primDivNatS0(xz134, xz135, Succ(xz1360), Zero) → new_primDivNatS(Succ(xz134), Succ(xz135), Succ(xz135))
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 1 SCC with 3 less nodes.
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(xz134, xz135, Succ(xz1360), Succ(xz1370)) → new_primDivNatS0(xz134, xz135, xz1360, xz1370)
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_primDivNatS0(xz134, xz135, Succ(xz1360), Succ(xz1370)) → new_primDivNatS0(xz134, xz135, xz1360, xz1370)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primMinusNat(xz94, xz95, Succ(xz960), Succ(xz970)) → new_primMinusNat(xz94, xz95, xz960, xz970)
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_primMinusNat(xz94, xz95, Succ(xz960), Succ(xz970)) → new_primMinusNat(xz94, xz95, xz960, xz970)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primMinusNat0(Succ(xz570), Succ(xz580), xz59) → new_primMinusNat0(xz570, xz580, xz59)
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_primMinusNat0(Succ(xz570), Succ(xz580), xz59) → new_primMinusNat0(xz570, xz580, xz59)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primMinusNat1(xz45, xz46, Succ(xz470), Succ(xz480)) → new_primMinusNat1(xz45, xz46, xz470, xz480)
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_primMinusNat1(xz45, xz46, Succ(xz470), Succ(xz480)) → new_primMinusNat1(xz45, xz46, xz470, xz480)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primPlusNat(xz107, xz108, Succ(xz1090), Succ(xz1100)) → new_primPlusNat(xz107, xz108, xz1090, xz1100)
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(xz107, xz108, Succ(xz1090), Succ(xz1100)) → new_primPlusNat(xz107, xz108, xz1090, xz1100)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primPlusNat0(Succ(xz610), Succ(xz620), xz63) → new_primPlusNat0(xz610, xz620, xz63)
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_primPlusNat0(Succ(xz610), Succ(xz620), xz63) → new_primPlusNat0(xz610, xz620, xz63)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
Q DP problem:
The TRS P consists of the following rules:
new_primPlusNat1(xz50, xz51, Succ(xz520), Succ(xz530)) → new_primPlusNat1(xz50, xz51, xz520, xz530)
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_primPlusNat1(xz50, xz51, Succ(xz520), Succ(xz530)) → new_primPlusNat1(xz50, xz51, xz520, xz530)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4