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
  import qualified Prelude



Lambda Reductions:
The following Lambda expression
\(_,r)→r

is transformed to
r0 (_,r) = r

The following Lambda expression
\(n,_)→n

is transformed to
n0 (n,_) = n

The following Lambda expression
\(_,r)→r

is transformed to
r1 (_,r) = r

The following Lambda expression
\(q,_)→q

is transformed to
q1 (q,_) = q



↳ HASKELL
  ↳ LR
HASKELL
      ↳ IFR

mainModule Main
  ((floor :: Float  ->  Int) :: Float  ->  Int)

module Main where
  import qualified Prelude



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
  import qualified Prelude



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
  import qualified Prelude



Cond Reductions:
The following Function with conditions
undefined 
 | False
 = undefined

is transformed to
undefined  = undefined1

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
  import qualified Prelude



Let/Where Reductions:
The bindings of the following Let/Where expression
floor0 (r < 0)
where 
floor0 True = n - 1
floor0 False = n
n  = n0 vu9
n0 (n,vw) = n
r  = r0 vu9
r0 (vv,r) = r
vu9  = properFraction x

are unpacked to the following functions on top level
floorVu9 xx = properFraction xx

floorR xx = floorR0 xx (floorVu9 xx)

floorN0 xx (n,vw) = n

floorR0 xx (vv,r) = r

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 
q  = q1 vu30
q1 (q,xu) = q
r  = r1 vu30
r1 (xv,r) = r
vu30  = quotRem x y

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
  import qualified Prelude



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
  (floor :: Float  ->  Int)

module Main where
  import qualified Prelude



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:



↳ 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:



↳ 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_primDivNatS03new_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:



↳ 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: