YES 0.887
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
| ((mapM_ :: Monad c => (b -> c a) -> [b] -> c ()) :: Monad c => (b -> c a) -> [b] -> c ()) |
module Main where
Lambda Reductions:
The following Lambda expression
\_→q
is transformed to
↳ HASKELL
↳ LR
↳ HASKELL
↳ BR
mainModule Main
| ((mapM_ :: Monad a => (c -> a b) -> [c] -> a ()) :: Monad a => (c -> a b) -> [c] -> a ()) |
module Main where
Replaced joker patterns by fresh variables and removed binding patterns.
↳ HASKELL
↳ LR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
mainModule Main
| ((mapM_ :: Monad c => (a -> c b) -> [a] -> c ()) :: Monad c => (a -> c b) -> [a] -> c ()) |
module Main where
Cond Reductions:
The following Function with conditions
is transformed to
undefined0 | True | = undefined |
undefined1 | | = undefined0 False |
↳ HASKELL
↳ LR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
mainModule Main
| (mapM_ :: Monad c => (a -> c b) -> [a] -> c ()) |
module Main where
Haskell To QDPs
↳ HASKELL
↳ LR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_psPs(:(vy50, vy51), vy8) → new_psPs(vy51, vy8)
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_psPs(:(vy50, vy51), vy8) → new_psPs(vy51, vy8)
The graph contains the following edges 1 > 1, 2 >= 2
↳ HASKELL
↳ LR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_gtGtEs(:(vy60, vy61), vy5, ba) → new_gtGtEs(vy61, vy5, ba)
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_gtGtEs(:(vy60, vy61), vy5, ba) → new_gtGtEs(vy61, vy5, ba)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
↳ HASKELL
↳ LR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
new_foldr(vy3, :(vy40, vy41), ty_IO, ba, bb) → new_foldr(vy3, vy41, ty_IO, ba, bb)
new_foldr(vy3, :(vy40, vy41), ty_Maybe, ba, bb) → new_foldr(vy3, vy41, ty_Maybe, ba, bb)
new_foldr(vy3, :(vy40, vy41), ty_[], ba, bb) → new_foldr(vy3, vy41, ty_[], ba, bb)
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 3 SCCs.
↳ HASKELL
↳ LR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_foldr(vy3, :(vy40, vy41), ty_[], ba, bb) → new_foldr(vy3, vy41, ty_[], ba, bb)
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_foldr(vy3, :(vy40, vy41), ty_[], ba, bb) → new_foldr(vy3, vy41, ty_[], ba, bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4, 5 >= 5
↳ HASKELL
↳ LR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_foldr(vy3, :(vy40, vy41), ty_Maybe, ba, bb) → new_foldr(vy3, vy41, ty_Maybe, ba, bb)
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_foldr(vy3, :(vy40, vy41), ty_Maybe, ba, bb) → new_foldr(vy3, vy41, ty_Maybe, ba, bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4, 5 >= 5
↳ HASKELL
↳ LR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
Q DP problem:
The TRS P consists of the following rules:
new_foldr(vy3, :(vy40, vy41), ty_IO, ba, bb) → new_foldr(vy3, vy41, ty_IO, ba, bb)
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_foldr(vy3, :(vy40, vy41), ty_IO, ba, bb) → new_foldr(vy3, vy41, ty_IO, ba, bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4, 5 >= 5