YES 10.137
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
|
| ((realToFrac :: Int -> Ratio Int) :: Int -> Ratio Int) |
module Main where
If Reductions:
The following If expression
if primGEqNatS x y then Succ (primDivNatS (primMinusNatS x y) (Succ y)) else Zero
is transformed to
| primDivNatS0 | x y True | = Succ (primDivNatS (primMinusNatS x y) (Succ y)) |
| primDivNatS0 | x y False | = Zero |
The following If expression
if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x
is transformed to
| primModNatS0 | x y True | = primModNatS (primMinusNatS x y) (Succ y) |
| primModNatS0 | x y False | = Succ x |
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
mainModule Main
|
| ((realToFrac :: Int -> Ratio Int) :: Int -> Ratio Int) |
module Main where
Replaced joker patterns by fresh variables and removed binding patterns.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
mainModule Main
|
| ((realToFrac :: Int -> Ratio Int) :: Int -> Ratio Int) |
module Main where
Cond Reductions:
The following Function with conditions
| gcd' | x 0 | = x |
| gcd' | x y | = gcd' y (x `rem` y) |
is transformed to
| gcd' | x xz | = gcd'2 x xz |
| gcd' | x y | = gcd'0 x y |
| gcd'0 | x y | = gcd' y (x `rem` y) |
| gcd'1 | True x xz | = x |
| gcd'1 | yu yv yw | = gcd'0 yv yw |
| gcd'2 | x xz | = gcd'1 (xz == 0) x xz |
| gcd'2 | yx yy | = gcd'0 yx yy |
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 | yz zu | = gcd3 yz zu |
| gcd | x y | = gcd0 x y |
| gcd0 | x y | =
| gcd' (abs x) (abs y) |
| where |
| gcd' | x xz | = gcd'2 x xz |
| gcd' | x y | = gcd'0 x y |
|
|
| gcd'0 | x y | = gcd' y (x `rem` y) |
|
|
| gcd'1 | True x xz | = x |
| gcd'1 | yu yv yw | = gcd'0 yv yw |
|
|
| gcd'2 | x xz | = gcd'1 (xz == 0) x xz |
| gcd'2 | yx yy | = gcd'0 yx yy |
|
|
| gcd1 | True yz zu | = error [] |
| gcd1 | zv zw zx | = gcd0 zw zx |
| gcd2 | True yz zu | = gcd1 (zu == 0) yz zu |
| gcd2 | zy zz vuu | = gcd0 zz vuu |
| gcd3 | yz zu | = gcd2 (yz == 0) yz zu |
| gcd3 | vuv vuw | = gcd0 vuv vuw |
The following Function with conditions
is transformed to
| absReal0 | x True | = `negate` x |
| absReal1 | x True | = x |
| absReal1 | x False | = absReal0 x otherwise |
| absReal2 | x | = absReal1 x (x >= 0) |
The following Function with conditions
is transformed to
| undefined0 | True | = undefined |
| undefined1 | | = undefined0 False |
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
| signumReal | x |
| | | x == 0 | |
| | | x > 0 | |
| | | otherwise | |
|
is transformed to
| signumReal | x | = signumReal3 x |
| signumReal2 | x True | = 0 |
| signumReal2 | x False | = signumReal1 x (x > 0) |
| signumReal1 | x True | = 1 |
| signumReal1 | x False | = signumReal0 x otherwise |
| signumReal3 | x | = signumReal2 x (x == 0) |
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
mainModule Main
|
| ((realToFrac :: Int -> Ratio Int) :: Int -> Ratio Int) |
module Main where
Let/Where Reductions:
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 | vux vuy x y True | = error [] |
| reduce2Reduce1 | vux vuy x y False | = reduce2Reduce0 vux vuy x y otherwise |
| reduce2D | vux vuy | = gcd vux vuy |
| reduce2Reduce0 | vux vuy x y True | = x `quot` reduce2D vux vuy :% (y `quot` reduce2D vux vuy) |
The bindings of the following Let/Where expression
| gcd' (abs x) (abs y) |
| where |
| gcd' | x xz | = gcd'2 x xz |
| gcd' | x y | = gcd'0 x y |
|
|
| gcd'0 | x y | = gcd' y (x `rem` y) |
|
|
| gcd'1 | True x xz | = x |
| gcd'1 | yu yv yw | = gcd'0 yv yw |
|
|
| gcd'2 | x xz | = gcd'1 (xz == 0) x xz |
| gcd'2 | yx yy | = gcd'0 yx yy |
|
are unpacked to the following functions on top level
| gcd0Gcd' | x xz | = gcd0Gcd'2 x xz |
| gcd0Gcd' | x y | = gcd0Gcd'0 x y |
| gcd0Gcd'1 | True x xz | = x |
| gcd0Gcd'1 | yu yv yw | = gcd0Gcd'0 yv yw |
| gcd0Gcd'0 | x y | = gcd0Gcd' y (x `rem` y) |
| gcd0Gcd'2 | x xz | = gcd0Gcd'1 (xz == 0) x xz |
| gcd0Gcd'2 | yx yy | = gcd0Gcd'0 yx yy |
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
mainModule Main
|
| ((realToFrac :: Int -> Ratio Int) :: Int -> Ratio Int) |
module Main where
Num Reduction: All numbers are transformed to thier corresponding representation with Pos, Neg, Succ and Zero.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
mainModule Main
|
| (realToFrac :: Int -> Ratio Int) |
module Main where
Haskell To QDPs
↳ 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(vuz3000)) → new_primMulNat(vuz3000)
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(vuz3000)) → new_primMulNat(vuz3000)
The graph contains the following edges 1 > 1
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS(Succ(Zero)) → new_primDivNatS(new_primMinusNatS0)
new_primDivNatS(Succ(Succ(vuz750000))) → new_primDivNatS(new_primMinusNatS(vuz750000))
The TRS R consists of the following rules:
new_primMinusNatS0 → Zero
new_primMinusNatS(vuz750000) → Succ(vuz750000)
The set Q consists of the following terms:
new_primMinusNatS(x0)
new_primMinusNatS0
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
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS(Succ(Succ(vuz750000))) → new_primDivNatS(new_primMinusNatS(vuz750000))
The TRS R consists of the following rules:
new_primMinusNatS0 → Zero
new_primMinusNatS(vuz750000) → Succ(vuz750000)
The set Q consists of the following terms:
new_primMinusNatS(x0)
new_primMinusNatS0
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
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS(Succ(Succ(vuz750000))) → new_primDivNatS(new_primMinusNatS(vuz750000))
The TRS R consists of the following rules:
new_primMinusNatS(vuz750000) → Succ(vuz750000)
The set Q consists of the following terms:
new_primMinusNatS(x0)
new_primMinusNatS0
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
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS(Succ(Succ(vuz750000))) → new_primDivNatS(new_primMinusNatS(vuz750000))
The TRS R consists of the following rules:
new_primMinusNatS(vuz750000) → Succ(vuz750000)
The set Q consists of the following terms:
new_primMinusNatS(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(vuz750000))) → new_primDivNatS(new_primMinusNatS(vuz750000))
Strictly oriented rules of the TRS R:
new_primMinusNatS(vuz750000) → Succ(vuz750000)
Used ordering: POLO with Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + 2·x1
POL(new_primDivNatS(x1)) = x1
POL(new_primMinusNatS(x1)) = 2 + 2·x1
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
P is empty.
R is empty.
The set Q consists of the following terms:
new_primMinusNatS(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
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot(vuz70, Succ(Zero), vuz76) → new_quot(vuz70, new_primMinusNatS0, new_primMinusNatS0)
new_quot(vuz70, Succ(Succ(vuz7700)), vuz76) → new_quot(vuz70, new_primMinusNatS(vuz7700), new_primMinusNatS(vuz7700))
The TRS R consists of the following rules:
new_primMinusNatS0 → Zero
new_primMinusNatS(vuz750000) → Succ(vuz750000)
The set Q consists of the following terms:
new_primMinusNatS(x0)
new_primMinusNatS0
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
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot(vuz70, Succ(Succ(vuz7700)), vuz76) → new_quot(vuz70, new_primMinusNatS(vuz7700), new_primMinusNatS(vuz7700))
The TRS R consists of the following rules:
new_primMinusNatS0 → Zero
new_primMinusNatS(vuz750000) → Succ(vuz750000)
The set Q consists of the following terms:
new_primMinusNatS(x0)
new_primMinusNatS0
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
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot(vuz70, Succ(Succ(vuz7700)), vuz76) → new_quot(vuz70, new_primMinusNatS(vuz7700), new_primMinusNatS(vuz7700))
The TRS R consists of the following rules:
new_primMinusNatS(vuz750000) → Succ(vuz750000)
The set Q consists of the following terms:
new_primMinusNatS(x0)
new_primMinusNatS0
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
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot(vuz70, Succ(Succ(vuz7700)), vuz76) → new_quot(vuz70, new_primMinusNatS(vuz7700), new_primMinusNatS(vuz7700))
The TRS R consists of the following rules:
new_primMinusNatS(vuz750000) → Succ(vuz750000)
The set Q consists of the following terms:
new_primMinusNatS(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_quot(vuz70, Succ(Succ(vuz7700)), vuz76) → new_quot(vuz70, new_primMinusNatS(vuz7700), new_primMinusNatS(vuz7700))
Used ordering: POLO with Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + 2·x1
POL(new_primMinusNatS(x1)) = 1 + 2·x1
POL(new_quot(x1, x2, x3)) = x1 + x2 + x3
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
Q DP problem:
P is empty.
The TRS R consists of the following rules:
new_primMinusNatS(vuz750000) → Succ(vuz750000)
The set Q consists of the following terms:
new_primMinusNatS(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
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot0(Succ(vuz4200)) → new_quot0(vuz4200)
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_quot0(Succ(vuz4200)) → new_quot0(vuz4200)
The graph contains the following edges 1 > 1
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
Q DP problem:
The TRS P consists of the following rules:
new_quot1(vuz72, Succ(vuz730)) → new_quot1(vuz72, vuz730)
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_quot1(vuz72, Succ(vuz730)) → new_quot1(vuz72, vuz730)
The graph contains the following edges 1 >= 1, 2 > 2