YES 2.257
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
↳ IFR
mainModule Main
| ((mod :: Int -> Int -> Int) :: Int -> Int -> Int) |
module Main where
If Reductions:
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 |
The following If expression
if primGEqNatS x y then primModNatP (primMinusNatS x y) (Succ y) else primMinusNatS y x
is transformed to
primModNatP0 | x y True | = primModNatP (primMinusNatS x y) (Succ y) |
primModNatP0 | x y False | = primMinusNatS y x |
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
mainModule Main
| ((mod :: Int -> Int -> Int) :: Int -> Int -> Int) |
module Main where
Replaced joker patterns by fresh variables and removed binding patterns.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
mainModule Main
| ((mod :: Int -> Int -> Int) :: Int -> Int -> Int) |
module Main where
Cond Reductions:
The following Function with conditions
is transformed to
undefined0 | True | = undefined |
undefined1 | | = undefined0 False |
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
mainModule Main
| (mod :: Int -> Int -> Int) |
module Main where
Haskell To QDPs
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primMinusNatS(Succ(vz560), Succ(vz550)) → new_primMinusNatS(vz560, vz550)
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(vz560), Succ(vz550)) → new_primMinusNatS(vz560, vz550)
The graph contains the following edges 1 > 1, 2 > 2
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatP0(vz55, vz56, Zero, Zero) → new_primModNatP00(vz55, vz56)
new_primModNatP(Succ(Succ(vz3000)), Zero) → new_primModNatP(new_primMinusNatS0(vz3000), Zero)
new_primModNatP0(vz55, vz56, Succ(vz570), Zero) → new_primModNatP(new_primMinusNatS2(vz55, vz56), Succ(vz56))
new_primModNatP0(vz55, vz56, Succ(vz570), Succ(vz580)) → new_primModNatP0(vz55, vz56, vz570, vz580)
new_primModNatP(Succ(Zero), Zero) → new_primModNatP(new_primMinusNatS1, Zero)
new_primModNatP00(vz55, vz56) → new_primModNatP(new_primMinusNatS2(vz55, vz56), Succ(vz56))
new_primModNatP(Succ(Succ(vz3000)), Succ(vz4000)) → new_primModNatP0(vz3000, vz4000, vz3000, vz4000)
The TRS R consists of the following rules:
new_primMinusNatS3(Zero, Succ(vz550)) → Zero
new_primMinusNatS3(Succ(vz560), Succ(vz550)) → new_primMinusNatS3(vz560, vz550)
new_primMinusNatS1 → Zero
new_primMinusNatS2(vz56, vz55) → new_primMinusNatS3(vz56, vz55)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS0(vz4000) → Succ(vz4000)
new_primMinusNatS3(Succ(vz560), Zero) → Succ(vz560)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS1
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS0(x0)
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
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatP(Succ(Succ(vz3000)), Zero) → new_primModNatP(new_primMinusNatS0(vz3000), Zero)
The TRS R consists of the following rules:
new_primMinusNatS3(Zero, Succ(vz550)) → Zero
new_primMinusNatS3(Succ(vz560), Succ(vz550)) → new_primMinusNatS3(vz560, vz550)
new_primMinusNatS1 → Zero
new_primMinusNatS2(vz56, vz55) → new_primMinusNatS3(vz56, vz55)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS0(vz4000) → Succ(vz4000)
new_primMinusNatS3(Succ(vz560), Zero) → Succ(vz560)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS1
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS0(x0)
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
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatP(Succ(Succ(vz3000)), Zero) → new_primModNatP(new_primMinusNatS0(vz3000), Zero)
The TRS R consists of the following rules:
new_primMinusNatS0(vz4000) → Succ(vz4000)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS1
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS0(x0)
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(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS1
new_primMinusNatS3(Succ(x0), Zero)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatP(Succ(Succ(vz3000)), Zero) → new_primModNatP(new_primMinusNatS0(vz3000), Zero)
The TRS R consists of the following rules:
new_primMinusNatS0(vz4000) → Succ(vz4000)
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_primModNatP(Succ(Succ(vz3000)), Zero) → new_primModNatP(new_primMinusNatS0(vz3000), Zero)
Strictly oriented rules of the TRS R:
new_primMinusNatS0(vz4000) → Succ(vz4000)
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_primModNatP(x1, x2)) = x1 + x2
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ 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
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatP0(vz55, vz56, Zero, Zero) → new_primModNatP00(vz55, vz56)
new_primModNatP0(vz55, vz56, Succ(vz570), Zero) → new_primModNatP(new_primMinusNatS2(vz55, vz56), Succ(vz56))
new_primModNatP0(vz55, vz56, Succ(vz570), Succ(vz580)) → new_primModNatP0(vz55, vz56, vz570, vz580)
new_primModNatP00(vz55, vz56) → new_primModNatP(new_primMinusNatS2(vz55, vz56), Succ(vz56))
new_primModNatP(Succ(Succ(vz3000)), Succ(vz4000)) → new_primModNatP0(vz3000, vz4000, vz3000, vz4000)
The TRS R consists of the following rules:
new_primMinusNatS3(Zero, Succ(vz550)) → Zero
new_primMinusNatS3(Succ(vz560), Succ(vz550)) → new_primMinusNatS3(vz560, vz550)
new_primMinusNatS1 → Zero
new_primMinusNatS2(vz56, vz55) → new_primMinusNatS3(vz56, vz55)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS0(vz4000) → Succ(vz4000)
new_primMinusNatS3(Succ(vz560), Zero) → Succ(vz560)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS1
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS0(x0)
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
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatP0(vz55, vz56, Zero, Zero) → new_primModNatP00(vz55, vz56)
new_primModNatP0(vz55, vz56, Succ(vz570), Zero) → new_primModNatP(new_primMinusNatS2(vz55, vz56), Succ(vz56))
new_primModNatP0(vz55, vz56, Succ(vz570), Succ(vz580)) → new_primModNatP0(vz55, vz56, vz570, vz580)
new_primModNatP00(vz55, vz56) → new_primModNatP(new_primMinusNatS2(vz55, vz56), Succ(vz56))
new_primModNatP(Succ(Succ(vz3000)), Succ(vz4000)) → new_primModNatP0(vz3000, vz4000, vz3000, vz4000)
The TRS R consists of the following rules:
new_primMinusNatS2(vz56, vz55) → new_primMinusNatS3(vz56, vz55)
new_primMinusNatS3(Zero, Succ(vz550)) → Zero
new_primMinusNatS3(Succ(vz560), Succ(vz550)) → new_primMinusNatS3(vz560, vz550)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vz560), Zero) → Succ(vz560)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS1
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS0(x0)
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_primMinusNatS1
new_primMinusNatS0(x0)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatP0(vz55, vz56, Zero, Zero) → new_primModNatP00(vz55, vz56)
new_primModNatP0(vz55, vz56, Succ(vz570), Zero) → new_primModNatP(new_primMinusNatS2(vz55, vz56), Succ(vz56))
new_primModNatP0(vz55, vz56, Succ(vz570), Succ(vz580)) → new_primModNatP0(vz55, vz56, vz570, vz580)
new_primModNatP00(vz55, vz56) → new_primModNatP(new_primMinusNatS2(vz55, vz56), Succ(vz56))
new_primModNatP(Succ(Succ(vz3000)), Succ(vz4000)) → new_primModNatP0(vz3000, vz4000, vz3000, vz4000)
The TRS R consists of the following rules:
new_primMinusNatS2(vz56, vz55) → new_primMinusNatS3(vz56, vz55)
new_primMinusNatS3(Zero, Succ(vz550)) → Zero
new_primMinusNatS3(Succ(vz560), Succ(vz550)) → new_primMinusNatS3(vz560, vz550)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vz560), Zero) → Succ(vz560)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_primModNatP0(vz55, vz56, Succ(vz570), Zero) → new_primModNatP(new_primMinusNatS2(vz55, vz56), Succ(vz56)) at position [0] we obtained the following new rules:
new_primModNatP0(vz55, vz56, Succ(vz570), Zero) → new_primModNatP(new_primMinusNatS3(vz55, vz56), Succ(vz56))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ 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_primModNatP0(vz55, vz56, Succ(vz570), Zero) → new_primModNatP(new_primMinusNatS3(vz55, vz56), Succ(vz56))
new_primModNatP0(vz55, vz56, Zero, Zero) → new_primModNatP00(vz55, vz56)
new_primModNatP0(vz55, vz56, Succ(vz570), Succ(vz580)) → new_primModNatP0(vz55, vz56, vz570, vz580)
new_primModNatP00(vz55, vz56) → new_primModNatP(new_primMinusNatS2(vz55, vz56), Succ(vz56))
new_primModNatP(Succ(Succ(vz3000)), Succ(vz4000)) → new_primModNatP0(vz3000, vz4000, vz3000, vz4000)
The TRS R consists of the following rules:
new_primMinusNatS2(vz56, vz55) → new_primMinusNatS3(vz56, vz55)
new_primMinusNatS3(Zero, Succ(vz550)) → Zero
new_primMinusNatS3(Succ(vz560), Succ(vz550)) → new_primMinusNatS3(vz560, vz550)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vz560), Zero) → Succ(vz560)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_primModNatP00(vz55, vz56) → new_primModNatP(new_primMinusNatS2(vz55, vz56), Succ(vz56)) at position [0] we obtained the following new rules:
new_primModNatP00(vz55, vz56) → new_primModNatP(new_primMinusNatS3(vz55, vz56), Succ(vz56))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ 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_primModNatP0(vz55, vz56, Zero, Zero) → new_primModNatP00(vz55, vz56)
new_primModNatP0(vz55, vz56, Succ(vz570), Zero) → new_primModNatP(new_primMinusNatS3(vz55, vz56), Succ(vz56))
new_primModNatP0(vz55, vz56, Succ(vz570), Succ(vz580)) → new_primModNatP0(vz55, vz56, vz570, vz580)
new_primModNatP00(vz55, vz56) → new_primModNatP(new_primMinusNatS3(vz55, vz56), Succ(vz56))
new_primModNatP(Succ(Succ(vz3000)), Succ(vz4000)) → new_primModNatP0(vz3000, vz4000, vz3000, vz4000)
The TRS R consists of the following rules:
new_primMinusNatS2(vz56, vz55) → new_primMinusNatS3(vz56, vz55)
new_primMinusNatS3(Zero, Succ(vz550)) → Zero
new_primMinusNatS3(Succ(vz560), Succ(vz550)) → new_primMinusNatS3(vz560, vz550)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vz560), Zero) → Succ(vz560)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
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
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ 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_primModNatP0(vz55, vz56, Succ(vz570), Zero) → new_primModNatP(new_primMinusNatS3(vz55, vz56), Succ(vz56))
new_primModNatP0(vz55, vz56, Zero, Zero) → new_primModNatP00(vz55, vz56)
new_primModNatP0(vz55, vz56, Succ(vz570), Succ(vz580)) → new_primModNatP0(vz55, vz56, vz570, vz580)
new_primModNatP00(vz55, vz56) → new_primModNatP(new_primMinusNatS3(vz55, vz56), Succ(vz56))
new_primModNatP(Succ(Succ(vz3000)), Succ(vz4000)) → new_primModNatP0(vz3000, vz4000, vz3000, vz4000)
The TRS R consists of the following rules:
new_primMinusNatS3(Zero, Succ(vz550)) → Zero
new_primMinusNatS3(Succ(vz560), Succ(vz550)) → new_primMinusNatS3(vz560, vz550)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vz560), Zero) → Succ(vz560)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
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
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ 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_primModNatP0(vz55, vz56, Zero, Zero) → new_primModNatP00(vz55, vz56)
new_primModNatP0(vz55, vz56, Succ(vz570), Zero) → new_primModNatP(new_primMinusNatS3(vz55, vz56), Succ(vz56))
new_primModNatP0(vz55, vz56, Succ(vz570), Succ(vz580)) → new_primModNatP0(vz55, vz56, vz570, vz580)
new_primModNatP00(vz55, vz56) → new_primModNatP(new_primMinusNatS3(vz55, vz56), Succ(vz56))
new_primModNatP(Succ(Succ(vz3000)), Succ(vz4000)) → new_primModNatP0(vz3000, vz4000, vz3000, vz4000)
The TRS R consists of the following rules:
new_primMinusNatS3(Zero, Succ(vz550)) → Zero
new_primMinusNatS3(Succ(vz560), Succ(vz550)) → new_primMinusNatS3(vz560, vz550)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vz560), Zero) → Succ(vz560)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS3(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
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_primModNatP0(vz55, vz56, Succ(vz570), Zero) → new_primModNatP(new_primMinusNatS3(vz55, vz56), Succ(vz56))
new_primModNatP00(vz55, vz56) → new_primModNatP(new_primMinusNatS3(vz55, vz56), Succ(vz56))
new_primModNatP(Succ(Succ(vz3000)), Succ(vz4000)) → new_primModNatP0(vz3000, vz4000, vz3000, vz4000)
The remaining pairs can at least be oriented weakly.
new_primModNatP0(vz55, vz56, Zero, Zero) → new_primModNatP00(vz55, vz56)
new_primModNatP0(vz55, vz56, Succ(vz570), Succ(vz580)) → new_primModNatP0(vz55, vz56, vz570, vz580)
Used ordering: Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 0
POL(new_primMinusNatS3(x1, x2)) = x1
POL(new_primModNatP(x1, x2)) = x1
POL(new_primModNatP0(x1, x2, x3, x4)) = 1 + x1
POL(new_primModNatP00(x1, x2)) = 1 + x1
The following usable rules [17] were oriented:
new_primMinusNatS3(Succ(vz560), Zero) → Succ(vz560)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Zero, Succ(vz550)) → Zero
new_primMinusNatS3(Succ(vz560), Succ(vz550)) → new_primMinusNatS3(vz560, vz550)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ 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_primModNatP0(vz55, vz56, Zero, Zero) → new_primModNatP00(vz55, vz56)
new_primModNatP0(vz55, vz56, Succ(vz570), Succ(vz580)) → new_primModNatP0(vz55, vz56, vz570, vz580)
The TRS R consists of the following rules:
new_primMinusNatS3(Zero, Succ(vz550)) → Zero
new_primMinusNatS3(Succ(vz560), Succ(vz550)) → new_primMinusNatS3(vz560, vz550)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vz560), Zero) → Succ(vz560)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS3(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
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
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ 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_primModNatP0(vz55, vz56, Succ(vz570), Succ(vz580)) → new_primModNatP0(vz55, vz56, vz570, vz580)
The TRS R consists of the following rules:
new_primMinusNatS3(Zero, Succ(vz550)) → Zero
new_primMinusNatS3(Succ(vz560), Succ(vz550)) → new_primMinusNatS3(vz560, vz550)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vz560), Zero) → Succ(vz560)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS3(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
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
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ 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_primModNatP0(vz55, vz56, Succ(vz570), Succ(vz580)) → new_primModNatP0(vz55, vz56, vz570, vz580)
R is empty.
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS3(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
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(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ 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_primModNatP0(vz55, vz56, Succ(vz570), Succ(vz580)) → new_primModNatP0(vz55, vz56, vz570, vz580)
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_primModNatP0(vz55, vz56, Succ(vz570), Succ(vz580)) → new_primModNatP0(vz55, vz56, vz570, vz580)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(vz50, vz51, Succ(vz520), Zero) → new_primModNatS(new_primMinusNatS2(vz50, vz51), Succ(vz51))
new_primModNatS00(vz50, vz51) → new_primModNatS(new_primMinusNatS2(vz50, vz51), Succ(vz51))
new_primModNatS(Succ(Zero), Zero) → new_primModNatS(new_primMinusNatS1, Zero)
new_primModNatS(Succ(Succ(vz3000)), Succ(vz4000)) → new_primModNatS0(vz3000, vz4000, vz3000, vz4000)
new_primModNatS(Succ(Succ(vz3000)), Zero) → new_primModNatS(new_primMinusNatS0(vz3000), Zero)
new_primModNatS0(vz50, vz51, Succ(vz520), Succ(vz530)) → new_primModNatS0(vz50, vz51, vz520, vz530)
new_primModNatS0(vz50, vz51, Zero, Zero) → new_primModNatS00(vz50, vz51)
The TRS R consists of the following rules:
new_primMinusNatS3(Zero, Succ(vz550)) → Zero
new_primMinusNatS3(Succ(vz560), Succ(vz550)) → new_primMinusNatS3(vz560, vz550)
new_primMinusNatS1 → Zero
new_primMinusNatS2(vz56, vz55) → new_primMinusNatS3(vz56, vz55)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS0(vz4000) → Succ(vz4000)
new_primMinusNatS3(Succ(vz560), Zero) → Succ(vz560)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS1
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS0(x0)
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
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS(Succ(Succ(vz3000)), Zero) → new_primModNatS(new_primMinusNatS0(vz3000), Zero)
The TRS R consists of the following rules:
new_primMinusNatS3(Zero, Succ(vz550)) → Zero
new_primMinusNatS3(Succ(vz560), Succ(vz550)) → new_primMinusNatS3(vz560, vz550)
new_primMinusNatS1 → Zero
new_primMinusNatS2(vz56, vz55) → new_primMinusNatS3(vz56, vz55)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS0(vz4000) → Succ(vz4000)
new_primMinusNatS3(Succ(vz560), Zero) → Succ(vz560)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS1
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS0(x0)
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
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS(Succ(Succ(vz3000)), Zero) → new_primModNatS(new_primMinusNatS0(vz3000), Zero)
The TRS R consists of the following rules:
new_primMinusNatS0(vz4000) → Succ(vz4000)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS1
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS0(x0)
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(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS1
new_primMinusNatS3(Succ(x0), Zero)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS(Succ(Succ(vz3000)), Zero) → new_primModNatS(new_primMinusNatS0(vz3000), Zero)
The TRS R consists of the following rules:
new_primMinusNatS0(vz4000) → Succ(vz4000)
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(vz3000)), Zero) → new_primModNatS(new_primMinusNatS0(vz3000), Zero)
Strictly oriented rules of the TRS R:
new_primMinusNatS0(vz4000) → Succ(vz4000)
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
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ 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
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(vz50, vz51, Succ(vz520), Zero) → new_primModNatS(new_primMinusNatS2(vz50, vz51), Succ(vz51))
new_primModNatS00(vz50, vz51) → new_primModNatS(new_primMinusNatS2(vz50, vz51), Succ(vz51))
new_primModNatS(Succ(Succ(vz3000)), Succ(vz4000)) → new_primModNatS0(vz3000, vz4000, vz3000, vz4000)
new_primModNatS0(vz50, vz51, Succ(vz520), Succ(vz530)) → new_primModNatS0(vz50, vz51, vz520, vz530)
new_primModNatS0(vz50, vz51, Zero, Zero) → new_primModNatS00(vz50, vz51)
The TRS R consists of the following rules:
new_primMinusNatS3(Zero, Succ(vz550)) → Zero
new_primMinusNatS3(Succ(vz560), Succ(vz550)) → new_primMinusNatS3(vz560, vz550)
new_primMinusNatS1 → Zero
new_primMinusNatS2(vz56, vz55) → new_primMinusNatS3(vz56, vz55)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS0(vz4000) → Succ(vz4000)
new_primMinusNatS3(Succ(vz560), Zero) → Succ(vz560)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS1
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS0(x0)
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
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(vz50, vz51, Succ(vz520), Zero) → new_primModNatS(new_primMinusNatS2(vz50, vz51), Succ(vz51))
new_primModNatS00(vz50, vz51) → new_primModNatS(new_primMinusNatS2(vz50, vz51), Succ(vz51))
new_primModNatS(Succ(Succ(vz3000)), Succ(vz4000)) → new_primModNatS0(vz3000, vz4000, vz3000, vz4000)
new_primModNatS0(vz50, vz51, Succ(vz520), Succ(vz530)) → new_primModNatS0(vz50, vz51, vz520, vz530)
new_primModNatS0(vz50, vz51, Zero, Zero) → new_primModNatS00(vz50, vz51)
The TRS R consists of the following rules:
new_primMinusNatS2(vz56, vz55) → new_primMinusNatS3(vz56, vz55)
new_primMinusNatS3(Zero, Succ(vz550)) → Zero
new_primMinusNatS3(Succ(vz560), Succ(vz550)) → new_primMinusNatS3(vz560, vz550)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vz560), Zero) → Succ(vz560)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS1
new_primMinusNatS3(Succ(x0), Zero)
new_primMinusNatS0(x0)
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_primMinusNatS1
new_primMinusNatS0(x0)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(vz50, vz51, Succ(vz520), Zero) → new_primModNatS(new_primMinusNatS2(vz50, vz51), Succ(vz51))
new_primModNatS00(vz50, vz51) → new_primModNatS(new_primMinusNatS2(vz50, vz51), Succ(vz51))
new_primModNatS(Succ(Succ(vz3000)), Succ(vz4000)) → new_primModNatS0(vz3000, vz4000, vz3000, vz4000)
new_primModNatS0(vz50, vz51, Zero, Zero) → new_primModNatS00(vz50, vz51)
new_primModNatS0(vz50, vz51, Succ(vz520), Succ(vz530)) → new_primModNatS0(vz50, vz51, vz520, vz530)
The TRS R consists of the following rules:
new_primMinusNatS2(vz56, vz55) → new_primMinusNatS3(vz56, vz55)
new_primMinusNatS3(Zero, Succ(vz550)) → Zero
new_primMinusNatS3(Succ(vz560), Succ(vz550)) → new_primMinusNatS3(vz560, vz550)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vz560), Zero) → Succ(vz560)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_primModNatS0(vz50, vz51, Succ(vz520), Zero) → new_primModNatS(new_primMinusNatS2(vz50, vz51), Succ(vz51)) at position [0] we obtained the following new rules:
new_primModNatS0(vz50, vz51, Succ(vz520), Zero) → new_primModNatS(new_primMinusNatS3(vz50, vz51), Succ(vz51))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ 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_primModNatS00(vz50, vz51) → new_primModNatS(new_primMinusNatS2(vz50, vz51), Succ(vz51))
new_primModNatS0(vz50, vz51, Succ(vz520), Zero) → new_primModNatS(new_primMinusNatS3(vz50, vz51), Succ(vz51))
new_primModNatS(Succ(Succ(vz3000)), Succ(vz4000)) → new_primModNatS0(vz3000, vz4000, vz3000, vz4000)
new_primModNatS0(vz50, vz51, Succ(vz520), Succ(vz530)) → new_primModNatS0(vz50, vz51, vz520, vz530)
new_primModNatS0(vz50, vz51, Zero, Zero) → new_primModNatS00(vz50, vz51)
The TRS R consists of the following rules:
new_primMinusNatS2(vz56, vz55) → new_primMinusNatS3(vz56, vz55)
new_primMinusNatS3(Zero, Succ(vz550)) → Zero
new_primMinusNatS3(Succ(vz560), Succ(vz550)) → new_primMinusNatS3(vz560, vz550)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vz560), Zero) → Succ(vz560)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_primModNatS00(vz50, vz51) → new_primModNatS(new_primMinusNatS2(vz50, vz51), Succ(vz51)) at position [0] we obtained the following new rules:
new_primModNatS00(vz50, vz51) → new_primModNatS(new_primMinusNatS3(vz50, vz51), Succ(vz51))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ 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_primModNatS0(vz50, vz51, Succ(vz520), Zero) → new_primModNatS(new_primMinusNatS3(vz50, vz51), Succ(vz51))
new_primModNatS(Succ(Succ(vz3000)), Succ(vz4000)) → new_primModNatS0(vz3000, vz4000, vz3000, vz4000)
new_primModNatS00(vz50, vz51) → new_primModNatS(new_primMinusNatS3(vz50, vz51), Succ(vz51))
new_primModNatS0(vz50, vz51, Zero, Zero) → new_primModNatS00(vz50, vz51)
new_primModNatS0(vz50, vz51, Succ(vz520), Succ(vz530)) → new_primModNatS0(vz50, vz51, vz520, vz530)
The TRS R consists of the following rules:
new_primMinusNatS2(vz56, vz55) → new_primMinusNatS3(vz56, vz55)
new_primMinusNatS3(Zero, Succ(vz550)) → Zero
new_primMinusNatS3(Succ(vz560), Succ(vz550)) → new_primMinusNatS3(vz560, vz550)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vz560), Zero) → Succ(vz560)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
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
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ 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_primModNatS0(vz50, vz51, Succ(vz520), Zero) → new_primModNatS(new_primMinusNatS3(vz50, vz51), Succ(vz51))
new_primModNatS(Succ(Succ(vz3000)), Succ(vz4000)) → new_primModNatS0(vz3000, vz4000, vz3000, vz4000)
new_primModNatS00(vz50, vz51) → new_primModNatS(new_primMinusNatS3(vz50, vz51), Succ(vz51))
new_primModNatS0(vz50, vz51, Zero, Zero) → new_primModNatS00(vz50, vz51)
new_primModNatS0(vz50, vz51, Succ(vz520), Succ(vz530)) → new_primModNatS0(vz50, vz51, vz520, vz530)
The TRS R consists of the following rules:
new_primMinusNatS3(Zero, Succ(vz550)) → Zero
new_primMinusNatS3(Succ(vz560), Succ(vz550)) → new_primMinusNatS3(vz560, vz550)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vz560), Zero) → Succ(vz560)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS2(x0, x1)
new_primMinusNatS3(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
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
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ 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_primModNatS0(vz50, vz51, Succ(vz520), Zero) → new_primModNatS(new_primMinusNatS3(vz50, vz51), Succ(vz51))
new_primModNatS(Succ(Succ(vz3000)), Succ(vz4000)) → new_primModNatS0(vz3000, vz4000, vz3000, vz4000)
new_primModNatS00(vz50, vz51) → new_primModNatS(new_primMinusNatS3(vz50, vz51), Succ(vz51))
new_primModNatS0(vz50, vz51, Succ(vz520), Succ(vz530)) → new_primModNatS0(vz50, vz51, vz520, vz530)
new_primModNatS0(vz50, vz51, Zero, Zero) → new_primModNatS00(vz50, vz51)
The TRS R consists of the following rules:
new_primMinusNatS3(Zero, Succ(vz550)) → Zero
new_primMinusNatS3(Succ(vz560), Succ(vz550)) → new_primMinusNatS3(vz560, vz550)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vz560), Zero) → Succ(vz560)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS3(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
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(vz3000)), Succ(vz4000)) → new_primModNatS0(vz3000, vz4000, vz3000, vz4000)
The remaining pairs can at least be oriented weakly.
new_primModNatS0(vz50, vz51, Succ(vz520), Zero) → new_primModNatS(new_primMinusNatS3(vz50, vz51), Succ(vz51))
new_primModNatS00(vz50, vz51) → new_primModNatS(new_primMinusNatS3(vz50, vz51), Succ(vz51))
new_primModNatS0(vz50, vz51, Succ(vz520), Succ(vz530)) → new_primModNatS0(vz50, vz51, vz520, vz530)
new_primModNatS0(vz50, vz51, Zero, Zero) → new_primModNatS00(vz50, vz51)
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(Zero, Succ(vz550)) → Zero
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vz560), Succ(vz550)) → new_primMinusNatS3(vz560, vz550)
new_primMinusNatS3(Succ(vz560), Zero) → Succ(vz560)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ 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_primModNatS0(vz50, vz51, Succ(vz520), Zero) → new_primModNatS(new_primMinusNatS3(vz50, vz51), Succ(vz51))
new_primModNatS00(vz50, vz51) → new_primModNatS(new_primMinusNatS3(vz50, vz51), Succ(vz51))
new_primModNatS0(vz50, vz51, Zero, Zero) → new_primModNatS00(vz50, vz51)
new_primModNatS0(vz50, vz51, Succ(vz520), Succ(vz530)) → new_primModNatS0(vz50, vz51, vz520, vz530)
The TRS R consists of the following rules:
new_primMinusNatS3(Zero, Succ(vz550)) → Zero
new_primMinusNatS3(Succ(vz560), Succ(vz550)) → new_primMinusNatS3(vz560, vz550)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vz560), Zero) → Succ(vz560)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS3(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
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
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ 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_primModNatS0(vz50, vz51, Succ(vz520), Succ(vz530)) → new_primModNatS0(vz50, vz51, vz520, vz530)
The TRS R consists of the following rules:
new_primMinusNatS3(Zero, Succ(vz550)) → Zero
new_primMinusNatS3(Succ(vz560), Succ(vz550)) → new_primMinusNatS3(vz560, vz550)
new_primMinusNatS3(Zero, Zero) → Zero
new_primMinusNatS3(Succ(vz560), Zero) → Succ(vz560)
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS3(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
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
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ 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_primModNatS0(vz50, vz51, Succ(vz520), Succ(vz530)) → new_primModNatS0(vz50, vz51, vz520, vz530)
R is empty.
The set Q consists of the following terms:
new_primMinusNatS3(Zero, Zero)
new_primMinusNatS3(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
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(Succ(x0), Succ(x1))
new_primMinusNatS3(Zero, Succ(x0))
new_primMinusNatS3(Succ(x0), Zero)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ 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_primModNatS0(vz50, vz51, Succ(vz520), Succ(vz530)) → new_primModNatS0(vz50, vz51, vz520, vz530)
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(vz50, vz51, Succ(vz520), Succ(vz530)) → new_primModNatS0(vz50, vz51, vz520, vz530)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4