YES 2.004
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
| ((fromEnum :: Enum a => a -> Int) :: Enum a => a -> 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
| ((fromEnum :: Enum a => a -> Int) :: Enum a => a -> 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
| ((fromEnum :: Enum a => a -> Int) :: Enum a => a -> 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
| ((fromEnum :: Enum a => a -> Int) :: Enum a => a -> 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
| ((fromEnum :: Enum a => a -> Int) :: Enum a => a -> Int) |
module Main where
Let/Where Reductions:
The bindings of the following Let/Where expression
are unpacked to the following functions on top level
truncateVu6 | xw | = properFraction xw |
truncateM | xw | = truncateM0 xw (truncateVu6 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 |
properFractionR0 | xx xy (xu,r) | = r |
properFractionQ | xx xy | = properFractionQ1 xx xy (properFractionVu30 xx xy) |
properFractionR | xx xy | = properFractionR0 xx xy (properFractionVu30 xx xy) |
properFractionVu30 | xx xy | = quotRem xx xy |
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
mainModule Main
| ((fromEnum :: Enum a => a -> Int) :: Enum a => a -> 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
| (fromEnum :: Enum a => a -> Int) |
module Main where
Haskell To QDPs
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primMinusNatS(Succ(xz240), Succ(xz250)) → new_primMinusNatS(xz240, xz250)
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_primMinusNatS(Succ(xz240), Succ(xz250)) → new_primMinusNatS(xz240, xz250)
The graph contains the following edges 1 > 1, 2 > 2
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(xz24, xz25, Zero, Zero) → new_primDivNatS00(xz24, xz25)
new_primDivNatS0(xz24, xz25, Succ(xz260), Succ(xz270)) → new_primDivNatS0(xz24, xz25, xz260, xz270)
new_primDivNatS00(xz24, xz25) → new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25))
new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) → new_primDivNatS0(xz30000, xz31000, xz30000, xz31000)
new_primDivNatS0(xz24, xz25, Succ(xz260), Zero) → new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25))
new_primDivNatS(Succ(Zero), Zero) → new_primDivNatS(new_primMinusNatS2, Zero)
new_primDivNatS(Succ(Succ(xz30000)), Zero) → new_primDivNatS(new_primMinusNatS1(xz30000), Zero)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(xz250)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS1(xz30000) → Succ(xz30000)
new_primMinusNatS0(Succ(xz240), Succ(xz250)) → new_primMinusNatS0(xz240, xz250)
new_primMinusNatS0(Succ(xz240), Zero) → Succ(xz240)
new_primMinusNatS2 → Zero
The set Q consists of the following terms:
new_primMinusNatS2
new_primMinusNatS1(x0)
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
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
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS(Succ(Succ(xz30000)), Zero) → new_primDivNatS(new_primMinusNatS1(xz30000), Zero)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(xz250)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS1(xz30000) → Succ(xz30000)
new_primMinusNatS0(Succ(xz240), Succ(xz250)) → new_primMinusNatS0(xz240, xz250)
new_primMinusNatS0(Succ(xz240), Zero) → Succ(xz240)
new_primMinusNatS2 → Zero
The set Q consists of the following terms:
new_primMinusNatS2
new_primMinusNatS1(x0)
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS(Succ(Succ(xz30000)), Zero) → new_primDivNatS(new_primMinusNatS1(xz30000), Zero)
The TRS R consists of the following rules:
new_primMinusNatS1(xz30000) → Succ(xz30000)
The set Q consists of the following terms:
new_primMinusNatS2
new_primMinusNatS1(x0)
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primMinusNatS2
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS(Succ(Succ(xz30000)), Zero) → new_primDivNatS(new_primMinusNatS1(xz30000), Zero)
The TRS R consists of the following rules:
new_primMinusNatS1(xz30000) → Succ(xz30000)
The set Q consists of the following terms:
new_primMinusNatS1(x0)
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
new_primDivNatS(Succ(Succ(xz30000)), Zero) → new_primDivNatS(new_primMinusNatS1(xz30000), Zero)
Strictly oriented rules of the TRS R:
new_primMinusNatS1(xz30000) → Succ(xz30000)
Used ordering: POLO with Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + 2·x1
POL(Zero) = 0
POL(new_primDivNatS(x1, x2)) = x1 + x2
POL(new_primMinusNatS1(x1)) = 2 + 2·x1
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ PisEmptyProof
↳ QDP
Q DP problem:
P is empty.
R is empty.
The set Q consists of the following terms:
new_primMinusNatS1(x0)
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(xz24, xz25, Zero, Zero) → new_primDivNatS00(xz24, xz25)
new_primDivNatS0(xz24, xz25, Succ(xz260), Succ(xz270)) → new_primDivNatS0(xz24, xz25, xz260, xz270)
new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) → new_primDivNatS0(xz30000, xz31000, xz30000, xz31000)
new_primDivNatS00(xz24, xz25) → new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25))
new_primDivNatS0(xz24, xz25, Succ(xz260), Zero) → new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25))
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(xz250)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS1(xz30000) → Succ(xz30000)
new_primMinusNatS0(Succ(xz240), Succ(xz250)) → new_primMinusNatS0(xz240, xz250)
new_primMinusNatS0(Succ(xz240), Zero) → Succ(xz240)
new_primMinusNatS2 → Zero
The set Q consists of the following terms:
new_primMinusNatS2
new_primMinusNatS1(x0)
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(xz24, xz25, Zero, Zero) → new_primDivNatS00(xz24, xz25)
new_primDivNatS0(xz24, xz25, Succ(xz260), Succ(xz270)) → new_primDivNatS0(xz24, xz25, xz260, xz270)
new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) → new_primDivNatS0(xz30000, xz31000, xz30000, xz31000)
new_primDivNatS00(xz24, xz25) → new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25))
new_primDivNatS0(xz24, xz25, Succ(xz260), Zero) → new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25))
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(xz250)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(xz240), Succ(xz250)) → new_primMinusNatS0(xz240, xz250)
new_primMinusNatS0(Succ(xz240), Zero) → Succ(xz240)
The set Q consists of the following terms:
new_primMinusNatS2
new_primMinusNatS1(x0)
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primMinusNatS2
new_primMinusNatS1(x0)
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(xz24, xz25, Zero, Zero) → new_primDivNatS00(xz24, xz25)
new_primDivNatS00(xz24, xz25) → new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25))
new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) → new_primDivNatS0(xz30000, xz31000, xz30000, xz31000)
new_primDivNatS0(xz24, xz25, Succ(xz260), Succ(xz270)) → new_primDivNatS0(xz24, xz25, xz260, xz270)
new_primDivNatS0(xz24, xz25, Succ(xz260), Zero) → new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25))
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(xz250)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(xz240), Succ(xz250)) → new_primMinusNatS0(xz240, xz250)
new_primMinusNatS0(Succ(xz240), Zero) → Succ(xz240)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
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_primDivNatS0(xz24, xz25, Zero, Zero) → new_primDivNatS00(xz24, xz25)
new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) → new_primDivNatS0(xz30000, xz31000, xz30000, xz31000)
new_primDivNatS0(xz24, xz25, Succ(xz260), Zero) → new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25))
The remaining pairs can at least be oriented weakly.
new_primDivNatS00(xz24, xz25) → new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25))
new_primDivNatS0(xz24, xz25, Succ(xz260), Succ(xz270)) → new_primDivNatS0(xz24, xz25, xz260, xz270)
Used ordering: Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 0
POL(new_primDivNatS(x1, x2)) = x1
POL(new_primDivNatS0(x1, x2, x3, x4)) = 1 + x1
POL(new_primDivNatS00(x1, x2)) = x1
POL(new_primMinusNatS0(x1, x2)) = x1
The following usable rules [17] were oriented:
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(xz240), Zero) → Succ(xz240)
new_primMinusNatS0(Succ(xz240), Succ(xz250)) → new_primMinusNatS0(xz240, xz250)
new_primMinusNatS0(Zero, Succ(xz250)) → Zero
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(xz24, xz25, Succ(xz260), Succ(xz270)) → new_primDivNatS0(xz24, xz25, xz260, xz270)
new_primDivNatS00(xz24, xz25) → new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25))
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(xz250)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(xz240), Succ(xz250)) → new_primMinusNatS0(xz240, xz250)
new_primMinusNatS0(Succ(xz240), Zero) → Succ(xz240)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(xz24, xz25, Succ(xz260), Succ(xz270)) → new_primDivNatS0(xz24, xz25, xz260, xz270)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(xz250)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(xz240), Succ(xz250)) → new_primMinusNatS0(xz240, xz250)
new_primMinusNatS0(Succ(xz240), Zero) → Succ(xz240)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(xz24, xz25, Succ(xz260), Succ(xz270)) → new_primDivNatS0(xz24, xz25, xz260, xz270)
R is empty.
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(xz24, xz25, Succ(xz260), Succ(xz270)) → new_primDivNatS0(xz24, xz25, xz260, xz270)
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(xz24, xz25, Succ(xz260), Succ(xz270)) → new_primDivNatS0(xz24, xz25, xz260, xz270)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4