YES 0.812
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
↳ BR
mainModule Main
| ((sum :: [Float] -> Float) :: [Float] -> Float) |
module Main where
Replaced joker patterns by fresh variables and removed binding patterns.
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
mainModule Main
| ((sum :: [Float] -> Float) :: [Float] -> Float) |
module Main where
Cond Reductions:
The following Function with conditions
is transformed to
undefined0 | True | = undefined |
undefined1 | | = undefined0 False |
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ NumRed
mainModule Main
| ((sum :: [Float] -> Float) :: [Float] -> Float) |
module Main where
Num Reduction: All numbers are transformed to thier corresponding representation with Pos, Neg, Succ and Zero.
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
mainModule Main
| (sum :: [Float] -> Float) |
module Main where
Haskell To QDPs
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primMinusNat(Succ(vx4000), Succ(vx30000)) → new_primMinusNat(vx4000, vx30000)
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(vx4000), Succ(vx30000)) → new_primMinusNat(vx4000, vx30000)
The graph contains the following edges 1 > 1, 2 > 2
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primPlusNat(Succ(vx4000), Succ(vx30000)) → new_primPlusNat(vx4000, vx30000)
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(vx4000), Succ(vx30000)) → new_primPlusNat(vx4000, vx30000)
The graph contains the following edges 1 > 1, 2 > 2
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primMulNat(Succ(vx4100), Succ(vx30100)) → new_primMulNat(vx4100, Succ(vx30100))
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(vx4100), Succ(vx30100)) → new_primMulNat(vx4100, Succ(vx30100))
The graph contains the following edges 1 > 1, 2 >= 2
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
Q DP problem:
The TRS P consists of the following rules:
new_enforceWHNF(Float(vx40, vx41), Float(vx300, vx301), vx5, :(vx310, vx311)) → new_enforceWHNF(Float(new_ps(vx40, vx300), new_sr(vx41, vx301)), vx310, Float(new_ps(vx40, vx300), new_sr(vx41, vx301)), vx311)
The TRS R consists of the following rules:
new_primMulNat0(Zero, Zero) → Zero
new_sr(Neg(vx410), Neg(vx3010)) → Pos(new_primMulNat0(vx410, vx3010))
new_ps(Pos(vx400), Pos(vx3000)) → Pos(new_primPlusNat0(vx400, vx3000))
new_primPlusNat0(Zero, Zero) → Zero
new_sr(Pos(vx410), Pos(vx3010)) → Pos(new_primMulNat0(vx410, vx3010))
new_sr(Pos(vx410), Neg(vx3010)) → Neg(new_primMulNat0(vx410, vx3010))
new_sr(Neg(vx410), Pos(vx3010)) → Neg(new_primMulNat0(vx410, vx3010))
new_primMinusNat0(Zero, Zero) → Pos(Zero)
new_primMinusNat0(Succ(vx4000), Zero) → Pos(Succ(vx4000))
new_primMulNat0(Zero, Succ(vx30100)) → Zero
new_primMulNat0(Succ(vx4100), Zero) → Zero
new_primMulNat0(Succ(vx4100), Succ(vx30100)) → new_primPlusNat0(new_primMulNat0(vx4100, Succ(vx30100)), Succ(vx30100))
new_primMinusNat0(Zero, Succ(vx30000)) → Neg(Succ(vx30000))
new_primPlusNat0(Zero, Succ(vx30000)) → Succ(vx30000)
new_primPlusNat0(Succ(vx4000), Zero) → Succ(vx4000)
new_primPlusNat0(Succ(vx4000), Succ(vx30000)) → Succ(Succ(new_primPlusNat0(vx4000, vx30000)))
new_primMinusNat0(Succ(vx4000), Succ(vx30000)) → new_primMinusNat0(vx4000, vx30000)
new_ps(Pos(vx400), Neg(vx3000)) → new_primMinusNat0(vx400, vx3000)
new_ps(Neg(vx400), Pos(vx3000)) → new_primMinusNat0(vx3000, vx400)
new_ps(Neg(vx400), Neg(vx3000)) → Neg(new_primPlusNat0(vx400, vx3000))
The set Q consists of the following terms:
new_ps(Pos(x0), Pos(x1))
new_sr(Neg(x0), Pos(x1))
new_sr(Pos(x0), Neg(x1))
new_primMulNat0(Zero, Succ(x0))
new_sr(Pos(x0), Pos(x1))
new_ps(Neg(x0), Neg(x1))
new_primMulNat0(Zero, Zero)
new_sr(Neg(x0), Neg(x1))
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Succ(x0), Succ(x1))
new_primPlusNat0(Zero, Zero)
new_primMinusNat0(Succ(x0), Zero)
new_primMulNat0(Succ(x0), Succ(x1))
new_primMinusNat0(Succ(x0), Succ(x1))
new_primMulNat0(Succ(x0), Zero)
new_primMinusNat0(Zero, Zero)
new_primPlusNat0(Zero, Succ(x0))
new_ps(Pos(x0), Neg(x1))
new_ps(Neg(x0), Pos(x1))
new_primMinusNat0(Zero, Succ(x0))
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_enforceWHNF(Float(vx40, vx41), Float(vx300, vx301), vx5, :(vx310, vx311)) → new_enforceWHNF(Float(new_ps(vx40, vx300), new_sr(vx41, vx301)), vx310, Float(new_ps(vx40, vx300), new_sr(vx41, vx301)), vx311)
The graph contains the following edges 4 > 2, 4 > 4