YES 5.682
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
| ((floor :: 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
| ((floor :: 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
| ((floor :: 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
floor0 | True | = n - 1 |
floor0 | 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
| ((floor :: 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
| ((floor :: 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 |
compare1 | x y True | = LT |
compare1 | x y False | = compare0 x y otherwise |
compare2 | x y True | = EQ |
compare2 | x y False | = compare1 x y (x <= y) |
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
absReal1 | x True | = x |
absReal1 | x False | = absReal0 x otherwise |
absReal0 | x True | = `negate` x |
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
| ((floor :: RealFrac a => a -> Int) :: RealFrac a => a -> Int) |
module Main where
Let/Where Reductions:
The bindings of the following Let/Where expression
floor0 (r < 0) |
where |
floor0 | True | = n - 1 |
floor0 | False | = n |
|
| |
| |
| |
| |
| |
are unpacked to the following functions on top level
floorN | vxx | = floorN0 vxx (floorVu9 vxx) |
floorVu9 | vxx | = properFraction vxx |
floorR | vxx | = floorR0 vxx (floorVu9 vxx) |
floorFloor0 | vxx True | = floorN vxx - 1 |
floorFloor0 | vxx False | = floorN 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
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 |
reduce2Reduce0 | vxy vxz x y True | = x `quot` reduce2D vxy vxz :% (y `quot` reduce2D 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'1 | True x vuz | = x |
gcd0Gcd'1 | vvu vvv vvw | = gcd0Gcd'0 vvv vvw |
gcd0Gcd'2 | x vuz | = gcd0Gcd'1 (vuz == 0) x vuz |
gcd0Gcd'2 | vvx vvy | = gcd0Gcd'0 vvx vvy |
gcd0Gcd' | x vuz | = gcd0Gcd'2 x vuz |
gcd0Gcd' | x y | = gcd0Gcd'0 x y |
The bindings of the following Let/Where expression
(fromIntegral q,r :% y) |
where | |
| |
| |
| |
| |
are unpacked to the following functions on top level
properFractionQ | vyu vyv | = properFractionQ1 vyu vyv (properFractionVu30 vyu vyv) |
properFractionR | vyu vyv | = properFractionR1 vyu vyv (properFractionVu30 vyu vyv) |
properFractionR1 | vyu vyv (zv,r) | = r |
properFractionQ1 | vyu vyv (q,zu) | = q |
properFractionVu30 | vyu vyv | = quotRem vyu vyv |
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
mainModule Main
| ((floor :: 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
| (floor :: 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
Q DP problem:
The TRS P consists of the following rules:
new_primMulNat(Succ(vyw101000)) → new_primMulNat(vyw101000)
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(vyw101000)) → new_primMulNat(vyw101000)
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
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_esEs(Succ(vyw190000), Succ(vyw300000), vyw31) → new_esEs(vyw190000, vyw300000, vyw31)
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(vyw190000), Succ(vyw300000), vyw31) → new_esEs(vyw190000, vyw300000, vyw31)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primMinusNatS(Succ(vyw1740), Succ(vyw1750)) → new_primMinusNatS(vyw1740, vyw1750)
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(vyw1740), Succ(vyw1750)) → new_primMinusNatS(vyw1740, vyw1750)
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
↳ DependencyGraphProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(vyw174, vyw175, Succ(vyw1760), Zero) → new_primModNatS(new_primMinusNatS2(vyw174, vyw175), Succ(vyw175))
new_primModNatS00(vyw174, vyw175) → new_primModNatS(new_primMinusNatS2(vyw174, vyw175), Succ(vyw175))
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(vyw174, vyw175, Succ(vyw1760), Succ(vyw1770)) → new_primModNatS0(vyw174, vyw175, vyw1760, vyw1770)
new_primModNatS0(vyw174, vyw175, Zero, Zero) → new_primModNatS00(vyw174, vyw175)
The TRS R consists of the following rules:
new_primMinusNatS3(Zero, Succ(vyw1750)) → Zero
new_primMinusNatS3(Succ(vyw1740), Succ(vyw1750)) → new_primMinusNatS3(vyw1740, vyw1750)
new_primMinusNatS1 → Zero
new_primMinusNatS2(vyw174, vyw175) → new_primMinusNatS3(vyw174, vyw175)
new_primMinusNatS0(vyw30000) → Succ(vyw30000)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vyw1740), Zero) → Succ(vyw1740)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS0(x0)
new_primMinusNatS1
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(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
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ 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_primMinusNatS3(Zero, Succ(vyw1750)) → Zero
new_primMinusNatS3(Succ(vyw1740), Succ(vyw1750)) → new_primMinusNatS3(vyw1740, vyw1750)
new_primMinusNatS1 → Zero
new_primMinusNatS2(vyw174, vyw175) → new_primMinusNatS3(vyw174, vyw175)
new_primMinusNatS0(vyw30000) → Succ(vyw30000)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vyw1740), Zero) → Succ(vyw1740)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS0(x0)
new_primMinusNatS1
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(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
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ 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_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS0(x0)
new_primMinusNatS1
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(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_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS1
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(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
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ 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
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ PisEmptyProof
↳ 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
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(vyw174, vyw175, Succ(vyw1760), Zero) → new_primModNatS(new_primMinusNatS2(vyw174, vyw175), Succ(vyw175))
new_primModNatS00(vyw174, vyw175) → new_primModNatS(new_primMinusNatS2(vyw174, vyw175), Succ(vyw175))
new_primModNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primModNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
new_primModNatS0(vyw174, vyw175, Succ(vyw1760), Succ(vyw1770)) → new_primModNatS0(vyw174, vyw175, vyw1760, vyw1770)
new_primModNatS0(vyw174, vyw175, Zero, Zero) → new_primModNatS00(vyw174, vyw175)
The TRS R consists of the following rules:
new_primMinusNatS3(Zero, Succ(vyw1750)) → Zero
new_primMinusNatS3(Succ(vyw1740), Succ(vyw1750)) → new_primMinusNatS3(vyw1740, vyw1750)
new_primMinusNatS1 → Zero
new_primMinusNatS2(vyw174, vyw175) → new_primMinusNatS3(vyw174, vyw175)
new_primMinusNatS0(vyw30000) → Succ(vyw30000)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vyw1740), Zero) → Succ(vyw1740)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS0(x0)
new_primMinusNatS1
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(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
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(vyw174, vyw175, Succ(vyw1760), Zero) → new_primModNatS(new_primMinusNatS2(vyw174, vyw175), Succ(vyw175))
new_primModNatS00(vyw174, vyw175) → new_primModNatS(new_primMinusNatS2(vyw174, vyw175), Succ(vyw175))
new_primModNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primModNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
new_primModNatS0(vyw174, vyw175, Succ(vyw1760), Succ(vyw1770)) → new_primModNatS0(vyw174, vyw175, vyw1760, vyw1770)
new_primModNatS0(vyw174, vyw175, Zero, Zero) → new_primModNatS00(vyw174, vyw175)
The TRS R consists of the following rules:
new_primMinusNatS2(vyw174, vyw175) → new_primMinusNatS3(vyw174, vyw175)
new_primMinusNatS3(Zero, Succ(vyw1750)) → Zero
new_primMinusNatS3(Succ(vyw1740), Succ(vyw1750)) → new_primMinusNatS3(vyw1740, vyw1750)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vyw1740), Zero) → Succ(vyw1740)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS0(x0)
new_primMinusNatS1
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(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(x0)
new_primMinusNatS1
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(vyw174, vyw175, Succ(vyw1760), Zero) → new_primModNatS(new_primMinusNatS2(vyw174, vyw175), Succ(vyw175))
new_primModNatS00(vyw174, vyw175) → new_primModNatS(new_primMinusNatS2(vyw174, vyw175), Succ(vyw175))
new_primModNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primModNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
new_primModNatS0(vyw174, vyw175, Zero, Zero) → new_primModNatS00(vyw174, vyw175)
new_primModNatS0(vyw174, vyw175, Succ(vyw1760), Succ(vyw1770)) → new_primModNatS0(vyw174, vyw175, vyw1760, vyw1770)
The TRS R consists of the following rules:
new_primMinusNatS2(vyw174, vyw175) → new_primMinusNatS3(vyw174, vyw175)
new_primMinusNatS3(Zero, Succ(vyw1750)) → Zero
new_primMinusNatS3(Succ(vyw1740), Succ(vyw1750)) → new_primMinusNatS3(vyw1740, vyw1750)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vyw1740), Zero) → Succ(vyw1740)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_primModNatS0(vyw174, vyw175, Succ(vyw1760), Zero) → new_primModNatS(new_primMinusNatS2(vyw174, vyw175), Succ(vyw175)) at position [0] we obtained the following new rules:
new_primModNatS0(vyw174, vyw175, Succ(vyw1760), Zero) → new_primModNatS(new_primMinusNatS3(vyw174, vyw175), Succ(vyw175))
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS00(vyw174, vyw175) → new_primModNatS(new_primMinusNatS2(vyw174, vyw175), Succ(vyw175))
new_primModNatS0(vyw174, vyw175, Succ(vyw1760), Zero) → new_primModNatS(new_primMinusNatS3(vyw174, vyw175), Succ(vyw175))
new_primModNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primModNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
new_primModNatS0(vyw174, vyw175, Succ(vyw1760), Succ(vyw1770)) → new_primModNatS0(vyw174, vyw175, vyw1760, vyw1770)
new_primModNatS0(vyw174, vyw175, Zero, Zero) → new_primModNatS00(vyw174, vyw175)
The TRS R consists of the following rules:
new_primMinusNatS2(vyw174, vyw175) → new_primMinusNatS3(vyw174, vyw175)
new_primMinusNatS3(Zero, Succ(vyw1750)) → Zero
new_primMinusNatS3(Succ(vyw1740), Succ(vyw1750)) → new_primMinusNatS3(vyw1740, vyw1750)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vyw1740), Zero) → Succ(vyw1740)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_primModNatS00(vyw174, vyw175) → new_primModNatS(new_primMinusNatS2(vyw174, vyw175), Succ(vyw175)) at position [0] we obtained the following new rules:
new_primModNatS00(vyw174, vyw175) → new_primModNatS(new_primMinusNatS3(vyw174, vyw175), Succ(vyw175))
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(vyw174, vyw175, Succ(vyw1760), Zero) → new_primModNatS(new_primMinusNatS3(vyw174, vyw175), Succ(vyw175))
new_primModNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primModNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
new_primModNatS00(vyw174, vyw175) → new_primModNatS(new_primMinusNatS3(vyw174, vyw175), Succ(vyw175))
new_primModNatS0(vyw174, vyw175, Zero, Zero) → new_primModNatS00(vyw174, vyw175)
new_primModNatS0(vyw174, vyw175, Succ(vyw1760), Succ(vyw1770)) → new_primModNatS0(vyw174, vyw175, vyw1760, vyw1770)
The TRS R consists of the following rules:
new_primMinusNatS2(vyw174, vyw175) → new_primMinusNatS3(vyw174, vyw175)
new_primMinusNatS3(Zero, Succ(vyw1750)) → Zero
new_primMinusNatS3(Succ(vyw1740), Succ(vyw1750)) → new_primMinusNatS3(vyw1740, vyw1750)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vyw1740), Zero) → Succ(vyw1740)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(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
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(vyw174, vyw175, Succ(vyw1760), Zero) → new_primModNatS(new_primMinusNatS3(vyw174, vyw175), Succ(vyw175))
new_primModNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primModNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
new_primModNatS00(vyw174, vyw175) → new_primModNatS(new_primMinusNatS3(vyw174, vyw175), Succ(vyw175))
new_primModNatS0(vyw174, vyw175, Zero, Zero) → new_primModNatS00(vyw174, vyw175)
new_primModNatS0(vyw174, vyw175, Succ(vyw1760), Succ(vyw1770)) → new_primModNatS0(vyw174, vyw175, vyw1760, vyw1770)
The TRS R consists of the following rules:
new_primMinusNatS3(Zero, Succ(vyw1750)) → Zero
new_primMinusNatS3(Succ(vyw1740), Succ(vyw1750)) → new_primMinusNatS3(vyw1740, vyw1750)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vyw1740), Zero) → Succ(vyw1740)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(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(x0, x1)
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(vyw174, vyw175, Succ(vyw1760), Zero) → new_primModNatS(new_primMinusNatS3(vyw174, vyw175), Succ(vyw175))
new_primModNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primModNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
new_primModNatS00(vyw174, vyw175) → new_primModNatS(new_primMinusNatS3(vyw174, vyw175), Succ(vyw175))
new_primModNatS0(vyw174, vyw175, Succ(vyw1760), Succ(vyw1770)) → new_primModNatS0(vyw174, vyw175, vyw1760, vyw1770)
new_primModNatS0(vyw174, vyw175, Zero, Zero) → new_primModNatS00(vyw174, vyw175)
The TRS R consists of the following rules:
new_primMinusNatS3(Zero, Succ(vyw1750)) → Zero
new_primMinusNatS3(Succ(vyw1740), Succ(vyw1750)) → new_primMinusNatS3(vyw1740, vyw1750)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vyw1740), Zero) → Succ(vyw1740)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(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_primModNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primModNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
The remaining pairs can at least be oriented weakly.
new_primModNatS0(vyw174, vyw175, Succ(vyw1760), Zero) → new_primModNatS(new_primMinusNatS3(vyw174, vyw175), Succ(vyw175))
new_primModNatS00(vyw174, vyw175) → new_primModNatS(new_primMinusNatS3(vyw174, vyw175), Succ(vyw175))
new_primModNatS0(vyw174, vyw175, Succ(vyw1760), Succ(vyw1770)) → new_primModNatS0(vyw174, vyw175, vyw1760, vyw1770)
new_primModNatS0(vyw174, vyw175, Zero, Zero) → new_primModNatS00(vyw174, vyw175)
Used ordering: Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 0
POL(new_primMinusNatS3(x1, x2)) = x1
POL(new_primModNatS(x1, x2)) = x1
POL(new_primModNatS0(x1, x2, x3, x4)) = x1
POL(new_primModNatS00(x1, x2)) = x1
The following usable rules [17] were oriented:
new_primMinusNatS3(Succ(vyw1740), Zero) → Succ(vyw1740)
new_primMinusNatS3(Succ(vyw1740), Succ(vyw1750)) → new_primMinusNatS3(vyw1740, vyw1750)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Zero, Succ(vyw1750)) → Zero
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(vyw174, vyw175, Succ(vyw1760), Zero) → new_primModNatS(new_primMinusNatS3(vyw174, vyw175), Succ(vyw175))
new_primModNatS00(vyw174, vyw175) → new_primModNatS(new_primMinusNatS3(vyw174, vyw175), Succ(vyw175))
new_primModNatS0(vyw174, vyw175, Zero, Zero) → new_primModNatS00(vyw174, vyw175)
new_primModNatS0(vyw174, vyw175, Succ(vyw1760), Succ(vyw1770)) → new_primModNatS0(vyw174, vyw175, vyw1760, vyw1770)
The TRS R consists of the following rules:
new_primMinusNatS3(Zero, Succ(vyw1750)) → Zero
new_primMinusNatS3(Succ(vyw1740), Succ(vyw1750)) → new_primMinusNatS3(vyw1740, vyw1750)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vyw1740), Zero) → Succ(vyw1740)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(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 3 less nodes.
↳ HASKELL
↳ LR
↳ HASKELL
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(vyw174, vyw175, Succ(vyw1760), Succ(vyw1770)) → new_primModNatS0(vyw174, vyw175, vyw1760, vyw1770)
The TRS R consists of the following rules:
new_primMinusNatS3(Zero, Succ(vyw1750)) → Zero
new_primMinusNatS3(Succ(vyw1740), Succ(vyw1750)) → new_primMinusNatS3(vyw1740, vyw1750)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vyw1740), Zero) → Succ(vyw1740)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(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
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(vyw174, vyw175, Succ(vyw1760), Succ(vyw1770)) → new_primModNatS0(vyw174, vyw175, vyw1760, vyw1770)
R is empty.
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(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_primMinusNatS3(Zero, Zero)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(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
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(vyw174, vyw175, Succ(vyw1760), Succ(vyw1770)) → new_primModNatS0(vyw174, vyw175, vyw1760, vyw1770)
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(vyw174, vyw175, Succ(vyw1760), Succ(vyw1770)) → new_primModNatS0(vyw174, vyw175, vyw1760, vyw1770)
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
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vyw169, vyw170, Zero, Zero) → new_primDivNatS00(vyw169, vyw170)
new_primDivNatS0(vyw169, vyw170, Succ(vyw1710), Succ(vyw1720)) → new_primDivNatS0(vyw169, vyw170, vyw1710, vyw1720)
new_primDivNatS0(vyw169, vyw170, Succ(vyw1710), Zero) → new_primDivNatS(new_primMinusNatS2(vyw169, vyw170), Succ(vyw170))
new_primDivNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primDivNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
new_primDivNatS00(vyw169, vyw170) → new_primDivNatS(new_primMinusNatS2(vyw169, vyw170), Succ(vyw170))
new_primDivNatS(Succ(Succ(vyw30000)), Zero) → new_primDivNatS(new_primMinusNatS0(vyw30000), Zero)
new_primDivNatS(Succ(Zero), Zero) → new_primDivNatS(new_primMinusNatS1, Zero)
The TRS R consists of the following rules:
new_primMinusNatS3(Zero, Succ(vyw1750)) → Zero
new_primMinusNatS3(Succ(vyw1740), Succ(vyw1750)) → new_primMinusNatS3(vyw1740, vyw1750)
new_primMinusNatS1 → Zero
new_primMinusNatS2(vyw174, vyw175) → new_primMinusNatS3(vyw174, vyw175)
new_primMinusNatS0(vyw30000) → Succ(vyw30000)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vyw1740), Zero) → Succ(vyw1740)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS0(x0)
new_primMinusNatS1
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(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
↳ CR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ 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_primMinusNatS3(Zero, Succ(vyw1750)) → Zero
new_primMinusNatS3(Succ(vyw1740), Succ(vyw1750)) → new_primMinusNatS3(vyw1740, vyw1750)
new_primMinusNatS1 → Zero
new_primMinusNatS2(vyw174, vyw175) → new_primMinusNatS3(vyw174, vyw175)
new_primMinusNatS0(vyw30000) → Succ(vyw30000)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vyw1740), Zero) → Succ(vyw1740)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS0(x0)
new_primMinusNatS1
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(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
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ 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_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS0(x0)
new_primMinusNatS1
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(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_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS1
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(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
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ 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
↳ 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_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
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vyw169, vyw170, Zero, Zero) → new_primDivNatS00(vyw169, vyw170)
new_primDivNatS0(vyw169, vyw170, Succ(vyw1710), Succ(vyw1720)) → new_primDivNatS0(vyw169, vyw170, vyw1710, vyw1720)
new_primDivNatS0(vyw169, vyw170, Succ(vyw1710), Zero) → new_primDivNatS(new_primMinusNatS2(vyw169, vyw170), Succ(vyw170))
new_primDivNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primDivNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
new_primDivNatS00(vyw169, vyw170) → new_primDivNatS(new_primMinusNatS2(vyw169, vyw170), Succ(vyw170))
The TRS R consists of the following rules:
new_primMinusNatS3(Zero, Succ(vyw1750)) → Zero
new_primMinusNatS3(Succ(vyw1740), Succ(vyw1750)) → new_primMinusNatS3(vyw1740, vyw1750)
new_primMinusNatS1 → Zero
new_primMinusNatS2(vyw174, vyw175) → new_primMinusNatS3(vyw174, vyw175)
new_primMinusNatS0(vyw30000) → Succ(vyw30000)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vyw1740), Zero) → Succ(vyw1740)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS0(x0)
new_primMinusNatS1
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(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
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vyw169, vyw170, Zero, Zero) → new_primDivNatS00(vyw169, vyw170)
new_primDivNatS0(vyw169, vyw170, Succ(vyw1710), Succ(vyw1720)) → new_primDivNatS0(vyw169, vyw170, vyw1710, vyw1720)
new_primDivNatS0(vyw169, vyw170, Succ(vyw1710), Zero) → new_primDivNatS(new_primMinusNatS2(vyw169, vyw170), Succ(vyw170))
new_primDivNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primDivNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
new_primDivNatS00(vyw169, vyw170) → new_primDivNatS(new_primMinusNatS2(vyw169, vyw170), Succ(vyw170))
The TRS R consists of the following rules:
new_primMinusNatS2(vyw174, vyw175) → new_primMinusNatS3(vyw174, vyw175)
new_primMinusNatS3(Zero, Succ(vyw1750)) → Zero
new_primMinusNatS3(Succ(vyw1740), Succ(vyw1750)) → new_primMinusNatS3(vyw1740, vyw1750)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vyw1740), Zero) → Succ(vyw1740)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS0(x0)
new_primMinusNatS1
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(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(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
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vyw169, vyw170, Zero, Zero) → new_primDivNatS00(vyw169, vyw170)
new_primDivNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primDivNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
new_primDivNatS0(vyw169, vyw170, Succ(vyw1710), Zero) → new_primDivNatS(new_primMinusNatS2(vyw169, vyw170), Succ(vyw170))
new_primDivNatS0(vyw169, vyw170, Succ(vyw1710), Succ(vyw1720)) → new_primDivNatS0(vyw169, vyw170, vyw1710, vyw1720)
new_primDivNatS00(vyw169, vyw170) → new_primDivNatS(new_primMinusNatS2(vyw169, vyw170), Succ(vyw170))
The TRS R consists of the following rules:
new_primMinusNatS2(vyw174, vyw175) → new_primMinusNatS3(vyw174, vyw175)
new_primMinusNatS3(Zero, Succ(vyw1750)) → Zero
new_primMinusNatS3(Succ(vyw1740), Succ(vyw1750)) → new_primMinusNatS3(vyw1740, vyw1750)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vyw1740), Zero) → Succ(vyw1740)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_primDivNatS0(vyw169, vyw170, Succ(vyw1710), Zero) → new_primDivNatS(new_primMinusNatS2(vyw169, vyw170), Succ(vyw170)) at position [0] we obtained the following new rules:
new_primDivNatS0(vyw169, vyw170, Succ(vyw1710), Zero) → new_primDivNatS(new_primMinusNatS3(vyw169, vyw170), Succ(vyw170))
↳ 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
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vyw169, vyw170, Zero, Zero) → new_primDivNatS00(vyw169, vyw170)
new_primDivNatS0(vyw169, vyw170, Succ(vyw1710), Succ(vyw1720)) → new_primDivNatS0(vyw169, vyw170, vyw1710, vyw1720)
new_primDivNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primDivNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
new_primDivNatS00(vyw169, vyw170) → new_primDivNatS(new_primMinusNatS2(vyw169, vyw170), Succ(vyw170))
new_primDivNatS0(vyw169, vyw170, Succ(vyw1710), Zero) → new_primDivNatS(new_primMinusNatS3(vyw169, vyw170), Succ(vyw170))
The TRS R consists of the following rules:
new_primMinusNatS2(vyw174, vyw175) → new_primMinusNatS3(vyw174, vyw175)
new_primMinusNatS3(Zero, Succ(vyw1750)) → Zero
new_primMinusNatS3(Succ(vyw1740), Succ(vyw1750)) → new_primMinusNatS3(vyw1740, vyw1750)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vyw1740), Zero) → Succ(vyw1740)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_primDivNatS00(vyw169, vyw170) → new_primDivNatS(new_primMinusNatS2(vyw169, vyw170), Succ(vyw170)) at position [0] we obtained the following new rules:
new_primDivNatS00(vyw169, vyw170) → new_primDivNatS(new_primMinusNatS3(vyw169, vyw170), Succ(vyw170))
↳ 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
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vyw169, vyw170, Zero, Zero) → new_primDivNatS00(vyw169, vyw170)
new_primDivNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primDivNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
new_primDivNatS0(vyw169, vyw170, Succ(vyw1710), Succ(vyw1720)) → new_primDivNatS0(vyw169, vyw170, vyw1710, vyw1720)
new_primDivNatS00(vyw169, vyw170) → new_primDivNatS(new_primMinusNatS3(vyw169, vyw170), Succ(vyw170))
new_primDivNatS0(vyw169, vyw170, Succ(vyw1710), Zero) → new_primDivNatS(new_primMinusNatS3(vyw169, vyw170), Succ(vyw170))
The TRS R consists of the following rules:
new_primMinusNatS2(vyw174, vyw175) → new_primMinusNatS3(vyw174, vyw175)
new_primMinusNatS3(Zero, Succ(vyw1750)) → Zero
new_primMinusNatS3(Succ(vyw1740), Succ(vyw1750)) → new_primMinusNatS3(vyw1740, vyw1750)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vyw1740), Zero) → Succ(vyw1740)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(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
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vyw169, vyw170, Zero, Zero) → new_primDivNatS00(vyw169, vyw170)
new_primDivNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primDivNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
new_primDivNatS0(vyw169, vyw170, Succ(vyw1710), Succ(vyw1720)) → new_primDivNatS0(vyw169, vyw170, vyw1710, vyw1720)
new_primDivNatS00(vyw169, vyw170) → new_primDivNatS(new_primMinusNatS3(vyw169, vyw170), Succ(vyw170))
new_primDivNatS0(vyw169, vyw170, Succ(vyw1710), Zero) → new_primDivNatS(new_primMinusNatS3(vyw169, vyw170), Succ(vyw170))
The TRS R consists of the following rules:
new_primMinusNatS3(Zero, Succ(vyw1750)) → Zero
new_primMinusNatS3(Succ(vyw1740), Succ(vyw1750)) → new_primMinusNatS3(vyw1740, vyw1750)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vyw1740), Zero) → Succ(vyw1740)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(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(x0, 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
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vyw169, vyw170, Zero, Zero) → new_primDivNatS00(vyw169, vyw170)
new_primDivNatS0(vyw169, vyw170, Succ(vyw1710), Succ(vyw1720)) → new_primDivNatS0(vyw169, vyw170, vyw1710, vyw1720)
new_primDivNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primDivNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
new_primDivNatS0(vyw169, vyw170, Succ(vyw1710), Zero) → new_primDivNatS(new_primMinusNatS3(vyw169, vyw170), Succ(vyw170))
new_primDivNatS00(vyw169, vyw170) → new_primDivNatS(new_primMinusNatS3(vyw169, vyw170), Succ(vyw170))
The TRS R consists of the following rules:
new_primMinusNatS3(Zero, Succ(vyw1750)) → Zero
new_primMinusNatS3(Succ(vyw1740), Succ(vyw1750)) → new_primMinusNatS3(vyw1740, vyw1750)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vyw1740), Zero) → Succ(vyw1740)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(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_primDivNatS(Succ(Succ(vyw30000)), Succ(vyw31000)) → new_primDivNatS0(vyw30000, vyw31000, vyw30000, vyw31000)
The remaining pairs can at least be oriented weakly.
new_primDivNatS0(vyw169, vyw170, Zero, Zero) → new_primDivNatS00(vyw169, vyw170)
new_primDivNatS0(vyw169, vyw170, Succ(vyw1710), Succ(vyw1720)) → new_primDivNatS0(vyw169, vyw170, vyw1710, vyw1720)
new_primDivNatS0(vyw169, vyw170, Succ(vyw1710), Zero) → new_primDivNatS(new_primMinusNatS3(vyw169, vyw170), Succ(vyw170))
new_primDivNatS00(vyw169, vyw170) → new_primDivNatS(new_primMinusNatS3(vyw169, vyw170), Succ(vyw170))
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)) = x1
POL(new_primDivNatS00(x1, x2)) = x1
POL(new_primMinusNatS3(x1, x2)) = x1
The following usable rules [17] were oriented:
new_primMinusNatS3(Succ(vyw1740), Succ(vyw1750)) → new_primMinusNatS3(vyw1740, vyw1750)
new_primMinusNatS3(Succ(vyw1740), Zero) → Succ(vyw1740)
new_primMinusNatS3(Zero, Succ(vyw1750)) → Zero
new_primMinusNatS3(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
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vyw169, vyw170, Zero, Zero) → new_primDivNatS00(vyw169, vyw170)
new_primDivNatS0(vyw169, vyw170, Succ(vyw1710), Succ(vyw1720)) → new_primDivNatS0(vyw169, vyw170, vyw1710, vyw1720)
new_primDivNatS00(vyw169, vyw170) → new_primDivNatS(new_primMinusNatS3(vyw169, vyw170), Succ(vyw170))
new_primDivNatS0(vyw169, vyw170, Succ(vyw1710), Zero) → new_primDivNatS(new_primMinusNatS3(vyw169, vyw170), Succ(vyw170))
The TRS R consists of the following rules:
new_primMinusNatS3(Zero, Succ(vyw1750)) → Zero
new_primMinusNatS3(Succ(vyw1740), Succ(vyw1750)) → new_primMinusNatS3(vyw1740, vyw1750)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vyw1740), Zero) → Succ(vyw1740)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(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 3 less nodes.
↳ 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
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vyw169, vyw170, Succ(vyw1710), Succ(vyw1720)) → new_primDivNatS0(vyw169, vyw170, vyw1710, vyw1720)
The TRS R consists of the following rules:
new_primMinusNatS3(Zero, Succ(vyw1750)) → Zero
new_primMinusNatS3(Succ(vyw1740), Succ(vyw1750)) → new_primMinusNatS3(vyw1740, vyw1750)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vyw1740), Zero) → Succ(vyw1740)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(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
↳ Rewriting
↳ QDP
↳ Rewriting
↳ 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(vyw169, vyw170, Succ(vyw1710), Succ(vyw1720)) → new_primDivNatS0(vyw169, vyw170, vyw1710, vyw1720)
R is empty.
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(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_primMinusNatS3(Zero, Zero)
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS3(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
↳ Rewriting
↳ QDP
↳ Rewriting
↳ 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(vyw169, vyw170, Succ(vyw1710), Succ(vyw1720)) → new_primDivNatS0(vyw169, vyw170, vyw1710, vyw1720)
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(vyw169, vyw170, Succ(vyw1710), Succ(vyw1720)) → new_primDivNatS0(vyw169, vyw170, vyw1710, vyw1720)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4