YES 15.424
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 :: Float -> Int) :: Float -> 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
↳ IFR
mainModule Main
| ((floor :: Float -> Int) :: Float -> 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
↳ IFR
↳ HASKELL
↳ BR
mainModule Main
| ((floor :: Float -> Int) :: Float -> 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 vz wu)
is replaced by the following term
Double vz wu
The bind variable of the following binding Pattern
frac@(Float wx wy)
is replaced by the following term
Float wx wy
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
mainModule Main
| ((floor :: Float -> Int) :: Float -> Int) |
module Main where
Cond Reductions:
The following Function with conditions
is transformed to
undefined0 | True | = undefined |
undefined1 | | = undefined0 False |
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
mainModule Main
| ((floor :: Float -> Int) :: Float -> 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
floorVu9 | xx | = properFraction xx |
floorR | xx | = floorR0 xx (floorVu9 xx) |
floorN | xx | = floorN0 xx (floorVu9 xx) |
floorFloor0 | xx True | = floorN xx - 1 |
floorFloor0 | xx False | = floorN xx |
The bindings of the following Let/Where expression
(fromIntegral q,r :% y) |
where | |
| |
| |
| |
| |
are unpacked to the following functions on top level
properFractionQ | xy xz | = properFractionQ1 xy xz (properFractionVu30 xy xz) |
properFractionQ1 | xy xz (q,xu) | = q |
properFractionVu30 | xy xz | = quotRem xy xz |
properFractionR | xy xz | = properFractionR1 xy xz (properFractionVu30 xy xz) |
properFractionR1 | xy xz (xv,r) | = r |
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
mainModule Main
| ((floor :: Float -> Int) :: Float -> Int) |
module Main where
Num Reduction: All numbers are transformed to thier corresponding representation with Pos, Neg, Succ and Zero.
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
mainModule Main
module Main where
Haskell To QDPs
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primPlusNat(Succ(yu30000), Succ(yu47500)) → new_primPlusNat(yu30000, yu47500)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- new_primPlusNat(Succ(yu30000), Succ(yu47500)) → new_primPlusNat(yu30000, yu47500)
The graph contains the following edges 1 > 1, 2 > 2
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primMinusNat(Succ(yu41400), Succ(yu43900)) → new_primMinusNat(yu41400, yu43900)
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_primMinusNat(Succ(yu41400), Succ(yu43900)) → new_primMinusNat(yu41400, yu43900)
The graph contains the following edges 1 > 1, 2 > 2
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(yu497, yu498, Zero, Zero) → new_primDivNatS00(yu497, yu498)
new_primDivNatS1(yu449, yu450) → new_primDivNatS01(yu449, yu450)
new_primDivNatS01(Succ(yu30000), Zero) → new_primDivNatS02(yu30000)
new_primDivNatS02(yu30000) → new_primDivNatS(Succ(yu30000), Zero, Zero)
new_primDivNatS0(yu497, yu498, Succ(yu4990), Succ(yu5000)) → new_primDivNatS0(yu497, yu498, yu4990, yu5000)
new_primDivNatS(Succ(yu5020), Zero, yu504) → new_primDivNatS1(yu5020, yu504)
new_primDivNatS(Succ(yu5020), Succ(yu5030), yu504) → new_primDivNatS(yu5020, yu5030, yu504)
new_primDivNatS01(Succ(yu30000), Succ(yu31000)) → new_primDivNatS0(yu30000, yu31000, yu30000, yu31000)
new_primDivNatS00(yu497, yu498) → new_primDivNatS(Succ(yu497), Succ(yu498), Succ(yu498))
new_primDivNatS0(yu497, yu498, Succ(yu4990), Zero) → new_primDivNatS(Succ(yu497), Succ(yu498), Succ(yu498))
new_primDivNatS01(Zero, Zero) → new_primDivNatS03
new_primDivNatS03 → new_primDivNatS(Zero, Zero, Zero)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(yu497, yu498, Zero, Zero) → new_primDivNatS00(yu497, yu498)
new_primDivNatS02(yu30000) → new_primDivNatS(Succ(yu30000), Zero, Zero)
new_primDivNatS01(Succ(yu30000), Zero) → new_primDivNatS02(yu30000)
new_primDivNatS1(yu449, yu450) → new_primDivNatS01(yu449, yu450)
new_primDivNatS0(yu497, yu498, Succ(yu4990), Succ(yu5000)) → new_primDivNatS0(yu497, yu498, yu4990, yu5000)
new_primDivNatS(Succ(yu5020), Zero, yu504) → new_primDivNatS1(yu5020, yu504)
new_primDivNatS(Succ(yu5020), Succ(yu5030), yu504) → new_primDivNatS(yu5020, yu5030, yu504)
new_primDivNatS01(Succ(yu30000), Succ(yu31000)) → new_primDivNatS0(yu30000, yu31000, yu30000, yu31000)
new_primDivNatS00(yu497, yu498) → new_primDivNatS(Succ(yu497), Succ(yu498), Succ(yu498))
new_primDivNatS0(yu497, yu498, Succ(yu4990), Zero) → new_primDivNatS(Succ(yu497), Succ(yu498), Succ(yu498))
R is empty.
Q is empty.
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(yu5020), Zero, yu504) → new_primDivNatS1(yu5020, yu504)
new_primDivNatS(Succ(yu5020), Succ(yu5030), yu504) → new_primDivNatS(yu5020, yu5030, yu504)
The remaining pairs can at least be oriented weakly.
new_primDivNatS0(yu497, yu498, Zero, Zero) → new_primDivNatS00(yu497, yu498)
new_primDivNatS02(yu30000) → new_primDivNatS(Succ(yu30000), Zero, Zero)
new_primDivNatS01(Succ(yu30000), Zero) → new_primDivNatS02(yu30000)
new_primDivNatS1(yu449, yu450) → new_primDivNatS01(yu449, yu450)
new_primDivNatS0(yu497, yu498, Succ(yu4990), Succ(yu5000)) → new_primDivNatS0(yu497, yu498, yu4990, yu5000)
new_primDivNatS01(Succ(yu30000), Succ(yu31000)) → new_primDivNatS0(yu30000, yu31000, yu30000, yu31000)
new_primDivNatS00(yu497, yu498) → new_primDivNatS(Succ(yu497), Succ(yu498), Succ(yu498))
new_primDivNatS0(yu497, yu498, Succ(yu4990), Zero) → new_primDivNatS(Succ(yu497), Succ(yu498), Succ(yu498))
Used ordering: Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 0
POL(new_primDivNatS(x1, x2, x3)) = x1
POL(new_primDivNatS0(x1, x2, x3, x4)) = 1 + x1
POL(new_primDivNatS00(x1, x2)) = 1 + x1
POL(new_primDivNatS01(x1, x2)) = x1
POL(new_primDivNatS02(x1)) = 1 + x1
POL(new_primDivNatS1(x1, x2)) = x1
The following usable rules [17] were oriented:
none
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(yu497, yu498, Zero, Zero) → new_primDivNatS00(yu497, yu498)
new_primDivNatS1(yu449, yu450) → new_primDivNatS01(yu449, yu450)
new_primDivNatS01(Succ(yu30000), Zero) → new_primDivNatS02(yu30000)
new_primDivNatS02(yu30000) → new_primDivNatS(Succ(yu30000), Zero, Zero)
new_primDivNatS0(yu497, yu498, Succ(yu4990), Succ(yu5000)) → new_primDivNatS0(yu497, yu498, yu4990, yu5000)
new_primDivNatS01(Succ(yu30000), Succ(yu31000)) → new_primDivNatS0(yu30000, yu31000, yu30000, yu31000)
new_primDivNatS00(yu497, yu498) → new_primDivNatS(Succ(yu497), Succ(yu498), Succ(yu498))
new_primDivNatS0(yu497, yu498, Succ(yu4990), Zero) → new_primDivNatS(Succ(yu497), Succ(yu498), Succ(yu498))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 7 less nodes.
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(yu497, yu498, Succ(yu4990), Succ(yu5000)) → new_primDivNatS0(yu497, yu498, yu4990, yu5000)
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(yu497, yu498, Succ(yu4990), Succ(yu5000)) → new_primDivNatS0(yu497, yu498, yu4990, yu5000)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
Q DP problem:
The TRS P consists of the following rules:
new_primMulNat(Succ(yu1120)) → new_primMulNat(yu1120)
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(yu1120)) → new_primMulNat(yu1120)
The graph contains the following edges 1 > 1