YES 19.696
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
| ((ceiling :: RealFrac a => a -> Int) :: RealFrac a => a -> Int) |
module Main where
Lambda Reductions:
The following Lambda expression
\(_,r)→r
is transformed to
The following Lambda expression
\(n,_)→n
is transformed to
The following Lambda expression
\(_,r)→r
is transformed to
The following Lambda expression
\(q,_)→q
is transformed to
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
mainModule Main
| ((ceiling :: RealFrac a => a -> Int) :: RealFrac a => a -> Int) |
module Main where
Case Reductions:
The following Case expression
case | compare x y of |
| EQ | → o |
| LT | → LT |
| GT | → GT |
is transformed to
primCompAux0 | o EQ | = o |
primCompAux0 | o LT | = LT |
primCompAux0 | o GT | = GT |
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
mainModule Main
| ((ceiling :: RealFrac a => a -> Int) :: RealFrac a => a -> Int) |
module Main where
If Reductions:
The following If expression
if r > 0 then n + 1 else n
is transformed to
ceiling0 | True | = n + 1 |
ceiling0 | False | = n |
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
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
mainModule Main
| ((ceiling :: RealFrac a => a -> Int) :: RealFrac 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 wy wz)
is replaced by the following term
Double wy wz
The bind variable of the following binding Pattern
frac@(Float xu xv)
is replaced by the following term
Float xu xv
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
mainModule Main
| ((ceiling :: RealFrac a => a -> Int) :: RealFrac a => a -> Int) |
module Main where
Cond Reductions:
The following Function with conditions
compare | x y |
| | x == y | |
| | x <= y | |
| | otherwise | |
|
is transformed to
compare | x y | = compare3 x y |
compare2 | x y True | = EQ |
compare2 | x y False | = compare1 x y (x <= y) |
compare1 | x y True | = LT |
compare1 | x y False | = compare0 x y otherwise |
compare3 | x y | = compare2 x y (x == y) |
The following Function with conditions
gcd' | x 0 | = x |
gcd' | x y | = gcd' y (x `rem` y) |
is transformed to
gcd' | x vuz | = gcd'2 x vuz |
gcd' | x y | = gcd'0 x y |
gcd'0 | x y | = gcd' y (x `rem` y) |
gcd'1 | True x vuz | = x |
gcd'1 | vvu vvv vvw | = gcd'0 vvv vvw |
gcd'2 | x vuz | = gcd'1 (vuz == 0) x vuz |
gcd'2 | vvx vvy | = gcd'0 vvx vvy |
The following Function with conditions
gcd | 0 0 | = error [] |
gcd | x y | =
gcd' (abs x) (abs y) |
where |
gcd' | x 0 | = x |
gcd' | x y | = gcd' y (x `rem` y) |
|
|
is transformed to
gcd | vvz vwu | = gcd3 vvz vwu |
gcd | x y | = gcd0 x y |
gcd0 | x y | =
gcd' (abs x) (abs y) |
where |
gcd' | x vuz | = gcd'2 x vuz |
gcd' | x y | = gcd'0 x y |
|
|
gcd'0 | x y | = gcd' y (x `rem` y) |
|
|
gcd'1 | True x vuz | = x |
gcd'1 | vvu vvv vvw | = gcd'0 vvv vvw |
|
|
gcd'2 | x vuz | = gcd'1 (vuz == 0) x vuz |
gcd'2 | vvx vvy | = gcd'0 vvx vvy |
|
|
gcd1 | True vvz vwu | = error [] |
gcd1 | vwv vww vwx | = gcd0 vww vwx |
gcd2 | True vvz vwu | = gcd1 (vwu == 0) vvz vwu |
gcd2 | vwy vwz vxu | = gcd0 vwz vxu |
gcd3 | vvz vwu | = gcd2 (vvz == 0) vvz vwu |
gcd3 | vxv vxw | = gcd0 vxv vxw |
The following Function with conditions
reduce | x y |
| | y == 0 | |
| | otherwise |
= | x `quot` d :% (y `quot` d) |
|
|
where | |
|
is transformed to
reduce2 | x y | =
reduce1 x y (y == 0) |
where | |
|
reduce0 | x y True | = x `quot` d :% (y `quot` d) |
|
|
reduce1 | x y True | = error [] |
reduce1 | x y False | = reduce0 x y otherwise |
|
|
The following Function with conditions
is transformed to
absReal0 | x True | = `negate` x |
absReal1 | x True | = x |
absReal1 | x False | = absReal0 x otherwise |
absReal2 | x | = absReal1 x (x >= 0) |
The following Function with conditions
is transformed to
undefined0 | True | = undefined |
undefined1 | | = undefined0 False |
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
mainModule Main
| ((ceiling :: RealFrac a => a -> Int) :: RealFrac a => a -> Int) |
module Main where
Let/Where Reductions:
The bindings of the following Let/Where expression
ceiling0 (r > 0) |
where |
ceiling0 | True | = n + 1 |
ceiling0 | False | = n |
|
| |
| |
| |
| |
| |
are unpacked to the following functions on top level
ceilingN | vxx | = ceilingN0 vxx (ceilingVu8 vxx) |
ceilingCeiling0 | vxx True | = ceilingN vxx + 1 |
ceilingCeiling0 | vxx False | = ceilingN vxx |
ceilingVu8 | vxx | = properFraction vxx |
ceilingR | vxx | = ceilingR0 vxx (ceilingVu8 vxx) |
The bindings of the following Let/Where expression
reduce1 x y (y == 0) |
where | |
|
reduce0 | x y True | = x `quot` d :% (y `quot` d) |
|
|
reduce1 | x y True | = error [] |
reduce1 | x y False | = reduce0 x y otherwise |
|
are unpacked to the following functions on top level
reduce2Reduce0 | vxy vxz x y True | = x `quot` reduce2D vxy vxz :% (y `quot` reduce2D vxy vxz) |
reduce2Reduce1 | vxy vxz x y True | = error [] |
reduce2Reduce1 | vxy vxz x y False | = reduce2Reduce0 vxy vxz x y otherwise |
reduce2D | vxy vxz | = gcd vxy vxz |
The bindings of the following Let/Where expression
gcd' (abs x) (abs y) |
where |
gcd' | x vuz | = gcd'2 x vuz |
gcd' | x y | = gcd'0 x y |
|
|
gcd'0 | x y | = gcd' y (x `rem` y) |
|
|
gcd'1 | True x vuz | = x |
gcd'1 | vvu vvv vvw | = gcd'0 vvv vvw |
|
|
gcd'2 | x vuz | = gcd'1 (vuz == 0) x vuz |
gcd'2 | vvx vvy | = gcd'0 vvx vvy |
|
are unpacked to the following functions on top level
gcd0Gcd'0 | x y | = gcd0Gcd' y (x `rem` y) |
gcd0Gcd' | x vuz | = gcd0Gcd'2 x vuz |
gcd0Gcd' | x y | = gcd0Gcd'0 x y |
gcd0Gcd'2 | x vuz | = gcd0Gcd'1 (vuz == 0) x vuz |
gcd0Gcd'2 | vvx vvy | = gcd0Gcd'0 vvx vvy |
gcd0Gcd'1 | True x vuz | = x |
gcd0Gcd'1 | vvu vvv vvw | = gcd0Gcd'0 vvv vvw |
The bindings of the following Let/Where expression
(fromIntegral q,r :% y) |
where | |
| |
| |
| |
| |
are unpacked to the following functions on top level
properFractionR1 | vyu vyv (zv,r) | = r |
properFractionQ | vyu vyv | = properFractionQ1 vyu vyv (properFractionVu30 vyu vyv) |
properFractionVu30 | vyu vyv | = quotRem vyu vyv |
properFractionR | vyu vyv | = properFractionR1 vyu vyv (properFractionVu30 vyu vyv) |
properFractionQ1 | vyu vyv (q,zu) | = q |
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
mainModule Main
| ((ceiling :: RealFrac a => a -> Int) :: RealFrac 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
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
mainModule Main
| (ceiling :: RealFrac a => a -> Int) |
module Main where
Haskell To QDPs
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primPlusNat(Succ(vyw30000), Succ(vyw18000)) → new_primPlusNat(vyw30000, vyw18000)
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(vyw30000), Succ(vyw18000)) → new_primPlusNat(vyw30000, vyw18000)
The graph contains the following edges 1 > 1, 2 > 2
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_esEs(Succ(vyw42300), Succ(vyw44000)) → new_esEs(vyw42300, vyw44000)
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_esEs(Succ(vyw42300), Succ(vyw44000)) → new_esEs(vyw42300, vyw44000)
The graph contains the following edges 1 > 1, 2 > 2
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primMulNat(Succ(vyw31100)) → new_primMulNat(vyw31100)
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_primMulNat(Succ(vyw31100)) → new_primMulNat(vyw31100)
The graph contains the following edges 1 > 1
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primMinusNatS(Succ(vyw5620), Succ(vyw5630)) → new_primMinusNatS(vyw5620, vyw5630)
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(vyw5620), Succ(vyw5630)) → new_primMinusNatS(vyw5620, vyw5630)
The graph contains the following edges 1 > 1, 2 > 2
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(vyw562, vyw563, Succ(vyw5640), Zero) → new_primModNatS(new_primMinusNatS2(vyw562, vyw563), Succ(vyw563))
new_primModNatS00(vyw562, vyw563) → new_primModNatS(new_primMinusNatS2(vyw562, vyw563), Succ(vyw563))
new_primModNatS(Succ(Zero), Zero) → new_primModNatS(new_primMinusNatS1, Zero)
new_primModNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primModNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
new_primModNatS(Succ(Succ(vyw30000)), Zero) → new_primModNatS(new_primMinusNatS0(vyw30000), Zero)
new_primModNatS0(vyw562, vyw563, Succ(vyw5640), Succ(vyw5650)) → new_primModNatS0(vyw562, vyw563, vyw5640, vyw5650)
new_primModNatS0(vyw562, vyw563, Zero, Zero) → new_primModNatS00(vyw562, vyw563)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vyw5620), Succ(vyw5630)) → new_primMinusNatS2(vyw5620, vyw5630)
new_primMinusNatS2(Zero, Succ(vyw5630)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vyw5620), Zero) → Succ(vyw5620)
new_primMinusNatS1 → Zero
new_primMinusNatS0(vyw30000) → Succ(vyw30000)
The set Q consists of the following terms:
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS0(x0)
new_primMinusNatS1
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
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS(Succ(Succ(vyw30000)), Zero) → new_primModNatS(new_primMinusNatS0(vyw30000), Zero)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vyw5620), Succ(vyw5630)) → new_primMinusNatS2(vyw5620, vyw5630)
new_primMinusNatS2(Zero, Succ(vyw5630)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vyw5620), Zero) → Succ(vyw5620)
new_primMinusNatS1 → Zero
new_primMinusNatS0(vyw30000) → Succ(vyw30000)
The set Q consists of the following terms:
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS0(x0)
new_primMinusNatS1
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
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS(Succ(Succ(vyw30000)), Zero) → new_primModNatS(new_primMinusNatS0(vyw30000), Zero)
The TRS R consists of the following rules:
new_primMinusNatS0(vyw30000) → Succ(vyw30000)
The set Q consists of the following terms:
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS0(x0)
new_primMinusNatS1
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(Succ(x0), Zero)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS1
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS(Succ(Succ(vyw30000)), Zero) → new_primModNatS(new_primMinusNatS0(vyw30000), Zero)
The TRS R consists of the following rules:
new_primMinusNatS0(vyw30000) → Succ(vyw30000)
The set Q consists of the following terms:
new_primMinusNatS0(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_primModNatS(Succ(Succ(vyw30000)), Zero) → new_primModNatS(new_primMinusNatS0(vyw30000), Zero)
Strictly oriented rules of the TRS R:
new_primMinusNatS0(vyw30000) → Succ(vyw30000)
Used ordering: POLO with Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + 2·x1
POL(Zero) = 0
POL(new_primMinusNatS0(x1)) = 2 + 2·x1
POL(new_primModNatS(x1, x2)) = x1 + x2
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
P is empty.
R is empty.
The set Q consists of the following terms:
new_primMinusNatS0(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
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(vyw562, vyw563, Succ(vyw5640), Zero) → new_primModNatS(new_primMinusNatS2(vyw562, vyw563), Succ(vyw563))
new_primModNatS00(vyw562, vyw563) → new_primModNatS(new_primMinusNatS2(vyw562, vyw563), Succ(vyw563))
new_primModNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primModNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
new_primModNatS0(vyw562, vyw563, Succ(vyw5640), Succ(vyw5650)) → new_primModNatS0(vyw562, vyw563, vyw5640, vyw5650)
new_primModNatS0(vyw562, vyw563, Zero, Zero) → new_primModNatS00(vyw562, vyw563)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vyw5620), Succ(vyw5630)) → new_primMinusNatS2(vyw5620, vyw5630)
new_primMinusNatS2(Zero, Succ(vyw5630)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vyw5620), Zero) → Succ(vyw5620)
new_primMinusNatS1 → Zero
new_primMinusNatS0(vyw30000) → Succ(vyw30000)
The set Q consists of the following terms:
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS0(x0)
new_primMinusNatS1
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
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(vyw562, vyw563, Succ(vyw5640), Zero) → new_primModNatS(new_primMinusNatS2(vyw562, vyw563), Succ(vyw563))
new_primModNatS00(vyw562, vyw563) → new_primModNatS(new_primMinusNatS2(vyw562, vyw563), Succ(vyw563))
new_primModNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primModNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
new_primModNatS0(vyw562, vyw563, Succ(vyw5640), Succ(vyw5650)) → new_primModNatS0(vyw562, vyw563, vyw5640, vyw5650)
new_primModNatS0(vyw562, vyw563, Zero, Zero) → new_primModNatS00(vyw562, vyw563)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vyw5620), Succ(vyw5630)) → new_primMinusNatS2(vyw5620, vyw5630)
new_primMinusNatS2(Zero, Succ(vyw5630)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vyw5620), Zero) → Succ(vyw5620)
The set Q consists of the following terms:
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS0(x0)
new_primMinusNatS1
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(x0)
new_primMinusNatS1
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(vyw562, vyw563, Succ(vyw5640), Zero) → new_primModNatS(new_primMinusNatS2(vyw562, vyw563), Succ(vyw563))
new_primModNatS00(vyw562, vyw563) → new_primModNatS(new_primMinusNatS2(vyw562, vyw563), Succ(vyw563))
new_primModNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primModNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
new_primModNatS0(vyw562, vyw563, Zero, Zero) → new_primModNatS00(vyw562, vyw563)
new_primModNatS0(vyw562, vyw563, Succ(vyw5640), Succ(vyw5650)) → new_primModNatS0(vyw562, vyw563, vyw5640, vyw5650)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vyw5620), Succ(vyw5630)) → new_primMinusNatS2(vyw5620, vyw5630)
new_primMinusNatS2(Zero, Succ(vyw5630)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vyw5620), Zero) → Succ(vyw5620)
The set Q consists of the following terms:
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(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_primModNatS0(vyw562, vyw563, Succ(vyw5640), Zero) → new_primModNatS(new_primMinusNatS2(vyw562, vyw563), Succ(vyw563))
new_primModNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primModNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
new_primModNatS0(vyw562, vyw563, Zero, Zero) → new_primModNatS00(vyw562, vyw563)
The remaining pairs can at least be oriented weakly.
new_primModNatS00(vyw562, vyw563) → new_primModNatS(new_primMinusNatS2(vyw562, vyw563), Succ(vyw563))
new_primModNatS0(vyw562, vyw563, Succ(vyw5640), Succ(vyw5650)) → new_primModNatS0(vyw562, vyw563, vyw5640, vyw5650)
Used ordering: Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 0
POL(new_primMinusNatS2(x1, x2)) = x1
POL(new_primModNatS(x1, x2)) = x1
POL(new_primModNatS0(x1, x2, x3, x4)) = 1 + x1
POL(new_primModNatS00(x1, x2)) = x1
The following usable rules [17] were oriented:
new_primMinusNatS2(Succ(vyw5620), Succ(vyw5630)) → new_primMinusNatS2(vyw5620, vyw5630)
new_primMinusNatS2(Succ(vyw5620), Zero) → Succ(vyw5620)
new_primMinusNatS2(Zero, Succ(vyw5630)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS00(vyw562, vyw563) → new_primModNatS(new_primMinusNatS2(vyw562, vyw563), Succ(vyw563))
new_primModNatS0(vyw562, vyw563, Succ(vyw5640), Succ(vyw5650)) → new_primModNatS0(vyw562, vyw563, vyw5640, vyw5650)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vyw5620), Succ(vyw5630)) → new_primMinusNatS2(vyw5620, vyw5630)
new_primMinusNatS2(Zero, Succ(vyw5630)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vyw5620), Zero) → Succ(vyw5620)
The set Q consists of the following terms:
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(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
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(vyw562, vyw563, Succ(vyw5640), Succ(vyw5650)) → new_primModNatS0(vyw562, vyw563, vyw5640, vyw5650)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vyw5620), Succ(vyw5630)) → new_primMinusNatS2(vyw5620, vyw5630)
new_primMinusNatS2(Zero, Succ(vyw5630)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vyw5620), Zero) → Succ(vyw5620)
The set Q consists of the following terms:
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(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
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(vyw562, vyw563, Succ(vyw5640), Succ(vyw5650)) → new_primModNatS0(vyw562, vyw563, vyw5640, vyw5650)
R is empty.
The set Q consists of the following terms:
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(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(Succ(x0), Zero)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(vyw562, vyw563, Succ(vyw5640), Succ(vyw5650)) → new_primModNatS0(vyw562, vyw563, vyw5640, vyw5650)
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_primModNatS0(vyw562, vyw563, Succ(vyw5640), Succ(vyw5650)) → new_primModNatS0(vyw562, vyw563, vyw5640, vyw5650)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vyw577, vyw578, Zero, Zero) → new_primDivNatS00(vyw577, vyw578)
new_primDivNatS0(vyw577, vyw578, Succ(vyw5790), Succ(vyw5800)) → new_primDivNatS0(vyw577, vyw578, vyw5790, vyw5800)
new_primDivNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primDivNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
new_primDivNatS00(vyw577, vyw578) → new_primDivNatS(new_primMinusNatS2(Succ(vyw577), Succ(vyw578)), Succ(vyw578))
new_primDivNatS(Succ(Succ(vyw30000)), Zero) → new_primDivNatS(new_primMinusNatS0(vyw30000), Zero)
new_primDivNatS0(vyw577, vyw578, Succ(vyw5790), Zero) → new_primDivNatS(new_primMinusNatS2(Succ(vyw577), Succ(vyw578)), Succ(vyw578))
new_primDivNatS(Succ(Zero), Zero) → new_primDivNatS(new_primMinusNatS1, Zero)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vyw5620), Succ(vyw5630)) → new_primMinusNatS2(vyw5620, vyw5630)
new_primMinusNatS2(Zero, Succ(vyw5630)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vyw5620), Zero) → Succ(vyw5620)
new_primMinusNatS1 → Zero
new_primMinusNatS0(vyw30000) → Succ(vyw30000)
The set Q consists of the following terms:
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS0(x0)
new_primMinusNatS1
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
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS(Succ(Succ(vyw30000)), Zero) → new_primDivNatS(new_primMinusNatS0(vyw30000), Zero)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vyw5620), Succ(vyw5630)) → new_primMinusNatS2(vyw5620, vyw5630)
new_primMinusNatS2(Zero, Succ(vyw5630)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vyw5620), Zero) → Succ(vyw5620)
new_primMinusNatS1 → Zero
new_primMinusNatS0(vyw30000) → Succ(vyw30000)
The set Q consists of the following terms:
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS0(x0)
new_primMinusNatS1
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
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS(Succ(Succ(vyw30000)), Zero) → new_primDivNatS(new_primMinusNatS0(vyw30000), Zero)
The TRS R consists of the following rules:
new_primMinusNatS0(vyw30000) → Succ(vyw30000)
The set Q consists of the following terms:
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS0(x0)
new_primMinusNatS1
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(Succ(x0), Zero)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS1
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS(Succ(Succ(vyw30000)), Zero) → new_primDivNatS(new_primMinusNatS0(vyw30000), Zero)
The TRS R consists of the following rules:
new_primMinusNatS0(vyw30000) → Succ(vyw30000)
The set Q consists of the following terms:
new_primMinusNatS0(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(vyw30000)), Zero) → new_primDivNatS(new_primMinusNatS0(vyw30000), Zero)
Strictly oriented rules of the TRS R:
new_primMinusNatS0(vyw30000) → Succ(vyw30000)
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_primMinusNatS0(x1)) = 2 + 2·x1
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
P is empty.
R is empty.
The set Q consists of the following terms:
new_primMinusNatS0(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
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vyw577, vyw578, Zero, Zero) → new_primDivNatS00(vyw577, vyw578)
new_primDivNatS0(vyw577, vyw578, Succ(vyw5790), Succ(vyw5800)) → new_primDivNatS0(vyw577, vyw578, vyw5790, vyw5800)
new_primDivNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primDivNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
new_primDivNatS00(vyw577, vyw578) → new_primDivNatS(new_primMinusNatS2(Succ(vyw577), Succ(vyw578)), Succ(vyw578))
new_primDivNatS0(vyw577, vyw578, Succ(vyw5790), Zero) → new_primDivNatS(new_primMinusNatS2(Succ(vyw577), Succ(vyw578)), Succ(vyw578))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vyw5620), Succ(vyw5630)) → new_primMinusNatS2(vyw5620, vyw5630)
new_primMinusNatS2(Zero, Succ(vyw5630)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vyw5620), Zero) → Succ(vyw5620)
new_primMinusNatS1 → Zero
new_primMinusNatS0(vyw30000) → Succ(vyw30000)
The set Q consists of the following terms:
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS0(x0)
new_primMinusNatS1
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
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vyw577, vyw578, Zero, Zero) → new_primDivNatS00(vyw577, vyw578)
new_primDivNatS0(vyw577, vyw578, Succ(vyw5790), Succ(vyw5800)) → new_primDivNatS0(vyw577, vyw578, vyw5790, vyw5800)
new_primDivNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primDivNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
new_primDivNatS00(vyw577, vyw578) → new_primDivNatS(new_primMinusNatS2(Succ(vyw577), Succ(vyw578)), Succ(vyw578))
new_primDivNatS0(vyw577, vyw578, Succ(vyw5790), Zero) → new_primDivNatS(new_primMinusNatS2(Succ(vyw577), Succ(vyw578)), Succ(vyw578))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vyw5620), Succ(vyw5630)) → new_primMinusNatS2(vyw5620, vyw5630)
new_primMinusNatS2(Zero, Succ(vyw5630)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vyw5620), Zero) → Succ(vyw5620)
The set Q consists of the following terms:
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS0(x0)
new_primMinusNatS1
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(x0)
new_primMinusNatS1
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vyw577, vyw578, Zero, Zero) → new_primDivNatS00(vyw577, vyw578)
new_primDivNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primDivNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
new_primDivNatS0(vyw577, vyw578, Succ(vyw5790), Succ(vyw5800)) → new_primDivNatS0(vyw577, vyw578, vyw5790, vyw5800)
new_primDivNatS00(vyw577, vyw578) → new_primDivNatS(new_primMinusNatS2(Succ(vyw577), Succ(vyw578)), Succ(vyw578))
new_primDivNatS0(vyw577, vyw578, Succ(vyw5790), Zero) → new_primDivNatS(new_primMinusNatS2(Succ(vyw577), Succ(vyw578)), Succ(vyw578))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vyw5620), Succ(vyw5630)) → new_primMinusNatS2(vyw5620, vyw5630)
new_primMinusNatS2(Zero, Succ(vyw5630)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vyw5620), Zero) → Succ(vyw5620)
The set Q consists of the following terms:
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_primDivNatS00(vyw577, vyw578) → new_primDivNatS(new_primMinusNatS2(Succ(vyw577), Succ(vyw578)), Succ(vyw578)) at position [0] we obtained the following new rules:
new_primDivNatS00(vyw577, vyw578) → new_primDivNatS(new_primMinusNatS2(vyw577, vyw578), Succ(vyw578))
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vyw577, vyw578, Zero, Zero) → new_primDivNatS00(vyw577, vyw578)
new_primDivNatS0(vyw577, vyw578, Succ(vyw5790), Succ(vyw5800)) → new_primDivNatS0(vyw577, vyw578, vyw5790, vyw5800)
new_primDivNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primDivNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
new_primDivNatS00(vyw577, vyw578) → new_primDivNatS(new_primMinusNatS2(vyw577, vyw578), Succ(vyw578))
new_primDivNatS0(vyw577, vyw578, Succ(vyw5790), Zero) → new_primDivNatS(new_primMinusNatS2(Succ(vyw577), Succ(vyw578)), Succ(vyw578))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vyw5620), Succ(vyw5630)) → new_primMinusNatS2(vyw5620, vyw5630)
new_primMinusNatS2(Zero, Succ(vyw5630)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vyw5620), Zero) → Succ(vyw5620)
The set Q consists of the following terms:
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_primDivNatS0(vyw577, vyw578, Succ(vyw5790), Zero) → new_primDivNatS(new_primMinusNatS2(Succ(vyw577), Succ(vyw578)), Succ(vyw578)) at position [0] we obtained the following new rules:
new_primDivNatS0(vyw577, vyw578, Succ(vyw5790), Zero) → new_primDivNatS(new_primMinusNatS2(vyw577, vyw578), Succ(vyw578))
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vyw577, vyw578, Zero, Zero) → new_primDivNatS00(vyw577, vyw578)
new_primDivNatS0(vyw577, vyw578, Succ(vyw5790), Zero) → new_primDivNatS(new_primMinusNatS2(vyw577, vyw578), Succ(vyw578))
new_primDivNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primDivNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
new_primDivNatS0(vyw577, vyw578, Succ(vyw5790), Succ(vyw5800)) → new_primDivNatS0(vyw577, vyw578, vyw5790, vyw5800)
new_primDivNatS00(vyw577, vyw578) → new_primDivNatS(new_primMinusNatS2(vyw577, vyw578), Succ(vyw578))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vyw5620), Succ(vyw5630)) → new_primMinusNatS2(vyw5620, vyw5630)
new_primMinusNatS2(Zero, Succ(vyw5630)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vyw5620), Zero) → Succ(vyw5620)
The set Q consists of the following terms:
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(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(vyw577, vyw578, Succ(vyw5790), Zero) → new_primDivNatS(new_primMinusNatS2(vyw577, vyw578), Succ(vyw578))
new_primDivNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primDivNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
new_primDivNatS00(vyw577, vyw578) → new_primDivNatS(new_primMinusNatS2(vyw577, vyw578), Succ(vyw578))
The remaining pairs can at least be oriented weakly.
new_primDivNatS0(vyw577, vyw578, Zero, Zero) → new_primDivNatS00(vyw577, vyw578)
new_primDivNatS0(vyw577, vyw578, Succ(vyw5790), Succ(vyw5800)) → new_primDivNatS0(vyw577, vyw578, vyw5790, vyw5800)
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)) = 1 + x1
POL(new_primMinusNatS2(x1, x2)) = x1
The following usable rules [17] were oriented:
new_primMinusNatS2(Zero, Succ(vyw5630)) → Zero
new_primMinusNatS2(Succ(vyw5620), Succ(vyw5630)) → new_primMinusNatS2(vyw5620, vyw5630)
new_primMinusNatS2(Succ(vyw5620), Zero) → Succ(vyw5620)
new_primMinusNatS2(Zero, Zero) → Zero
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vyw577, vyw578, Zero, Zero) → new_primDivNatS00(vyw577, vyw578)
new_primDivNatS0(vyw577, vyw578, Succ(vyw5790), Succ(vyw5800)) → new_primDivNatS0(vyw577, vyw578, vyw5790, vyw5800)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vyw5620), Succ(vyw5630)) → new_primMinusNatS2(vyw5620, vyw5630)
new_primMinusNatS2(Zero, Succ(vyw5630)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vyw5620), Zero) → Succ(vyw5620)
The set Q consists of the following terms:
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(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
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vyw577, vyw578, Succ(vyw5790), Succ(vyw5800)) → new_primDivNatS0(vyw577, vyw578, vyw5790, vyw5800)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vyw5620), Succ(vyw5630)) → new_primMinusNatS2(vyw5620, vyw5630)
new_primMinusNatS2(Zero, Succ(vyw5630)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vyw5620), Zero) → Succ(vyw5620)
The set Q consists of the following terms:
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(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
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vyw577, vyw578, Succ(vyw5790), Succ(vyw5800)) → new_primDivNatS0(vyw577, vyw578, vyw5790, vyw5800)
R is empty.
The set Q consists of the following terms:
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(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(Succ(x0), Zero)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vyw577, vyw578, Succ(vyw5790), Succ(vyw5800)) → new_primDivNatS0(vyw577, vyw578, vyw5790, vyw5800)
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(vyw577, vyw578, Succ(vyw5790), Succ(vyw5800)) → new_primDivNatS0(vyw577, vyw578, vyw5790, vyw5800)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_ceilingCeiling0(vyw587, vyw588, Succ(vyw5890), Succ(vyw5900), ba) → new_ceilingCeiling0(vyw587, vyw588, vyw5890, vyw5900, ba)
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_ceilingCeiling0(vyw587, vyw588, Succ(vyw5890), Succ(vyw5900), ba) → new_ceilingCeiling0(vyw587, vyw588, vyw5890, vyw5900, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4, 5 >= 5
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_ceilingCeiling00(vyw572, vyw573, Succ(vyw5740), Succ(vyw5750), ba) → new_ceilingCeiling00(vyw572, vyw573, vyw5740, vyw5750, ba)
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_ceilingCeiling00(vyw572, vyw573, Succ(vyw5740), Succ(vyw5750), ba) → new_ceilingCeiling00(vyw572, vyw573, vyw5740, vyw5750, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4, 5 >= 5
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_ceilingCeiling01(vyw582, vyw583, Succ(vyw5840), Succ(vyw5850), ba) → new_ceilingCeiling01(vyw582, vyw583, vyw5840, vyw5850, ba)
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_ceilingCeiling01(vyw582, vyw583, Succ(vyw5840), Succ(vyw5850), ba) → new_ceilingCeiling01(vyw582, vyw583, vyw5840, vyw5850, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4, 5 >= 5
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
Q DP problem:
The TRS P consists of the following rules:
new_ceilingCeiling02(vyw567, vyw568, Succ(vyw5690), Succ(vyw5700), ba) → new_ceilingCeiling02(vyw567, vyw568, vyw5690, vyw5700, ba)
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_ceilingCeiling02(vyw567, vyw568, Succ(vyw5690), Succ(vyw5700), ba) → new_ceilingCeiling02(vyw567, vyw568, vyw5690, vyw5700, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4, 5 >= 5