YES 0.769
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 :: [Int] -> Int) :: [Int] -> Int) |
module Main where
Replaced joker patterns by fresh variables and removed binding patterns.
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
mainModule Main
| ((sum :: [Int] -> Int) :: [Int] -> Int) |
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 :: [Int] -> Int) :: [Int] -> Int) |
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
module Main where
Haskell To QDPs
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primPlusNat(Succ(vx400), Succ(vx3000)) → new_primPlusNat(vx400, vx3000)
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(vx400), Succ(vx3000)) → new_primPlusNat(vx400, vx3000)
The graph contains the following edges 1 > 1, 2 > 2
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
Q DP problem:
The TRS P consists of the following rules:
new_enforceWHNF(Pos(Zero), Neg(Zero), vx5, :(vx310, vx311)) → new_enforceWHNF(Pos(Zero), vx310, Pos(Zero), vx311)
new_enforceWHNF(Pos(Succ(vx400)), Neg(Zero), vx5, :(vx310, vx311)) → new_enforceWHNF(Pos(Succ(vx400)), vx310, Pos(Succ(vx400)), vx311)
new_enforceWHNF0(Zero, Zero, :(vx310, vx311)) → new_enforceWHNF(Pos(Zero), vx310, Pos(Zero), vx311)
new_enforceWHNF(Pos(vx40), Pos(vx300), vx5, :(vx310, vx311)) → new_enforceWHNF(Pos(new_primPlusNat0(vx40, vx300)), vx310, Pos(new_primPlusNat0(vx40, vx300)), vx311)
new_enforceWHNF0(Succ(vx400), Succ(vx3000), vx31) → new_enforceWHNF0(vx400, vx3000, vx31)
new_seq(vx6, vx310, vx7, vx311) → new_enforceWHNF(Neg(vx6), vx310, Neg(vx7), vx311)
new_enforceWHNF(Pos(Succ(vx400)), Neg(Succ(vx3000)), vx5, vx31) → new_enforceWHNF0(vx400, vx3000, vx31)
new_enforceWHNF(Neg(vx40), Neg(vx300), vx5, :(vx310, vx311)) → new_seq(new_primPlusNat0(vx40, vx300), vx310, new_primPlusNat0(vx40, vx300), vx311)
new_enforceWHNF(Pos(Zero), Neg(Succ(vx3000)), vx5, :(vx310, vx311)) → new_seq(Succ(vx3000), vx310, Succ(vx3000), vx311)
new_enforceWHNF(Neg(vx40), Pos(vx300), vx5, vx31) → new_enforceWHNF0(vx300, vx40, vx31)
new_enforceWHNF0(Succ(vx400), Zero, :(vx310, vx311)) → new_enforceWHNF(Pos(Succ(vx400)), vx310, Pos(Succ(vx400)), vx311)
new_enforceWHNF0(Zero, Succ(vx3000), :(vx310, vx311)) → new_seq(Succ(vx3000), vx310, Succ(vx3000), vx311)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(vx400), Zero) → Succ(vx400)
new_primPlusNat0(Zero, Succ(vx3000)) → Succ(vx3000)
new_primPlusNat0(Succ(vx400), Succ(vx3000)) → Succ(Succ(new_primPlusNat0(vx400, vx3000)))
The set Q consists of the following terms:
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Succ(x1))
new_primPlusNat0(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(Pos(Zero), Neg(Succ(vx3000)), vx5, :(vx310, vx311)) → new_seq(Succ(vx3000), vx310, Succ(vx3000), vx311)
The graph contains the following edges 2 > 1, 4 > 2, 2 > 3, 4 > 4
- new_enforceWHNF0(Zero, Zero, :(vx310, vx311)) → new_enforceWHNF(Pos(Zero), vx310, Pos(Zero), vx311)
The graph contains the following edges 3 > 2, 3 > 4
- new_enforceWHNF(Pos(vx40), Pos(vx300), vx5, :(vx310, vx311)) → new_enforceWHNF(Pos(new_primPlusNat0(vx40, vx300)), vx310, Pos(new_primPlusNat0(vx40, vx300)), vx311)
The graph contains the following edges 4 > 2, 4 > 4
- new_enforceWHNF(Pos(Zero), Neg(Zero), vx5, :(vx310, vx311)) → new_enforceWHNF(Pos(Zero), vx310, Pos(Zero), vx311)
The graph contains the following edges 1 >= 1, 4 > 2, 1 >= 3, 4 > 4
- new_enforceWHNF(Pos(Succ(vx400)), Neg(Zero), vx5, :(vx310, vx311)) → new_enforceWHNF(Pos(Succ(vx400)), vx310, Pos(Succ(vx400)), vx311)
The graph contains the following edges 1 >= 1, 4 > 2, 1 >= 3, 4 > 4
- new_enforceWHNF(Pos(Succ(vx400)), Neg(Succ(vx3000)), vx5, vx31) → new_enforceWHNF0(vx400, vx3000, vx31)
The graph contains the following edges 1 > 1, 2 > 2, 4 >= 3
- new_enforceWHNF0(Succ(vx400), Zero, :(vx310, vx311)) → new_enforceWHNF(Pos(Succ(vx400)), vx310, Pos(Succ(vx400)), vx311)
The graph contains the following edges 3 > 2, 3 > 4
- new_enforceWHNF0(Succ(vx400), Succ(vx3000), vx31) → new_enforceWHNF0(vx400, vx3000, vx31)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
- new_enforceWHNF(Neg(vx40), Pos(vx300), vx5, vx31) → new_enforceWHNF0(vx300, vx40, vx31)
The graph contains the following edges 2 > 1, 1 > 2, 4 >= 3
- new_enforceWHNF0(Zero, Succ(vx3000), :(vx310, vx311)) → new_seq(Succ(vx3000), vx310, Succ(vx3000), vx311)
The graph contains the following edges 2 >= 1, 3 > 2, 2 >= 3, 3 > 4
- new_enforceWHNF(Neg(vx40), Neg(vx300), vx5, :(vx310, vx311)) → new_seq(new_primPlusNat0(vx40, vx300), vx310, new_primPlusNat0(vx40, vx300), vx311)
The graph contains the following edges 4 > 2, 4 > 4
- new_seq(vx6, vx310, vx7, vx311) → new_enforceWHNF(Neg(vx6), vx310, Neg(vx7), vx311)
The graph contains the following edges 2 >= 2, 4 >= 4