YES 0.719
H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/Monad.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:
↳ HASKELL
↳ LR
mainModule Monad
| ((liftM4 :: (b -> c -> e -> a -> d) -> [b] -> [c] -> [e] -> [a] -> [d]) :: (b -> c -> e -> a -> d) -> [b] -> [c] -> [e] -> [a] -> [d]) |
module Monad where
| import qualified Maybe import qualified Prelude
|
| liftM4 :: Monad b => (e -> c -> f -> a -> d) -> b e -> b c -> b f -> b a -> b d
liftM4 | f m1 m2 m3 m4 | = | m1 >>= (\x1 ->m2 >>= (\x2 ->m3 >>= (\x3 ->m4 >>= (\x4 ->return (f x1 x2 x3 x4))))) |
|
module Maybe where
| import qualified Monad import qualified Prelude
|
Lambda Reductions:
The following Lambda expression
\x4→return (f x1 x2 x3 x4)
is transformed to
liftM40 | f x1 x2 x3 x4 | = return (f x1 x2 x3 x4) |
The following Lambda expression
\x3→m4 >>= liftM40 f x1 x2 x3
is transformed to
liftM41 | m4 f x1 x2 x3 | = m4 >>= liftM40 f x1 x2 x3 |
The following Lambda expression
\x2→m3 >>= liftM41 m4 f x1 x2
is transformed to
liftM42 | m3 m4 f x1 x2 | = m3 >>= liftM41 m4 f x1 x2 |
The following Lambda expression
\x1→m2 >>= liftM42 m3 m4 f x1
is transformed to
liftM43 | m2 m3 m4 f x1 | = m2 >>= liftM42 m3 m4 f x1 |
↳ HASKELL
↳ LR
↳ HASKELL
↳ BR
mainModule Monad
| ((liftM4 :: (e -> a -> d -> c -> b) -> [e] -> [a] -> [d] -> [c] -> [b]) :: (e -> a -> d -> c -> b) -> [e] -> [a] -> [d] -> [c] -> [b]) |
module Maybe where
| import qualified Monad import qualified Prelude
|
module Monad where
| import qualified Maybe import qualified Prelude
|
| liftM4 :: Monad d => (c -> b -> a -> e -> f) -> d c -> d b -> d a -> d e -> d f
liftM4 | f m1 m2 m3 m4 | = | m1 >>= liftM43 m2 m3 m4 f |
|
|
liftM40 | f x1 x2 x3 x4 | = | return (f x1 x2 x3 x4) |
|
|
liftM41 | m4 f x1 x2 x3 | = | m4 >>= liftM40 f x1 x2 x3 |
|
|
liftM42 | m3 m4 f x1 x2 | = | m3 >>= liftM41 m4 f x1 x2 |
|
|
liftM43 | m2 m3 m4 f x1 | = | m2 >>= liftM42 m3 m4 f x1 |
|
Replaced joker patterns by fresh variables and removed binding patterns.
↳ HASKELL
↳ LR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
mainModule Monad
| ((liftM4 :: (a -> d -> b -> e -> c) -> [a] -> [d] -> [b] -> [e] -> [c]) :: (a -> d -> b -> e -> c) -> [a] -> [d] -> [b] -> [e] -> [c]) |
module Monad where
| import qualified Maybe import qualified Prelude
|
| liftM4 :: Monad c => (a -> e -> b -> f -> d) -> c a -> c e -> c b -> c f -> c d
liftM4 | f m1 m2 m3 m4 | = | m1 >>= liftM43 m2 m3 m4 f |
|
|
liftM40 | f x1 x2 x3 x4 | = | return (f x1 x2 x3 x4) |
|
|
liftM41 | m4 f x1 x2 x3 | = | m4 >>= liftM40 f x1 x2 x3 |
|
|
liftM42 | m3 m4 f x1 x2 | = | m3 >>= liftM41 m4 f x1 x2 |
|
|
liftM43 | m2 m3 m4 f x1 | = | m2 >>= liftM42 m3 m4 f x1 |
|
module Maybe where
| import qualified Monad import qualified Prelude
|
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 Monad
| (liftM4 :: (d -> a -> b -> c -> e) -> [d] -> [a] -> [b] -> [c] -> [e]) |
module Maybe where
| import qualified Monad import qualified Prelude
|
module Monad where
| import qualified Maybe import qualified Prelude
|
| liftM4 :: Monad e => (a -> b -> f -> d -> c) -> e a -> e b -> e f -> e d -> e c
liftM4 | f m1 m2 m3 m4 | = | m1 >>= liftM43 m2 m3 m4 f |
|
|
liftM40 | f x1 x2 x3 x4 | = | return (f x1 x2 x3 x4) |
|
|
liftM41 | m4 f x1 x2 x3 | = | m4 >>= liftM40 f x1 x2 x3 |
|
|
liftM42 | m3 m4 f x1 x2 | = | m3 >>= liftM41 m4 f x1 x2 |
|
|
liftM43 | m2 m3 m4 f x1 | = | m2 >>= liftM42 m3 m4 f x1 |
|
Haskell To QDPs
↳ HASKELL
↳ LR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_psPs(:(vy90, vy91), vy8, h) → new_psPs(vy91, vy8, h)
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(:(vy90, vy91), vy8, h) → new_psPs(vy91, vy8, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
↳ HASKELL
↳ LR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_gtGtEs(:(vy70, vy71), vy3, vy40, vy50, vy60, h, ba, bb, bc, bd) → new_gtGtEs(vy71, vy3, vy40, vy50, vy60, h, ba, bb, bc, bd)
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(:(vy70, vy71), vy3, vy40, vy50, vy60, h, ba, bb, bc, bd) → new_gtGtEs(vy71, vy3, vy40, vy50, vy60, h, ba, bb, bc, bd)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 8 >= 8, 9 >= 9, 10 >= 10
↳ HASKELL
↳ LR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_gtGtEs0(:(vy60, vy61), vy7, vy3, vy40, vy50, h, ba, bb, bc, bd) → new_gtGtEs0(vy61, vy7, vy3, vy40, vy50, h, ba, bb, bc, bd)
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_gtGtEs0(:(vy60, vy61), vy7, vy3, vy40, vy50, h, ba, bb, bc, bd) → new_gtGtEs0(vy61, vy7, vy3, vy40, vy50, h, ba, bb, bc, bd)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 8 >= 8, 9 >= 9, 10 >= 10
↳ HASKELL
↳ LR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_gtGtEs1(:(vy50, vy51), vy6, vy7, vy3, vy40, h, ba, bb, bc, bd) → new_gtGtEs1(vy51, vy6, vy7, vy3, vy40, h, ba, bb, bc, bd)
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_gtGtEs1(:(vy50, vy51), vy6, vy7, vy3, vy40, h, ba, bb, bc, bd) → new_gtGtEs1(vy51, vy6, vy7, vy3, vy40, h, ba, bb, bc, bd)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 8 >= 8, 9 >= 9, 10 >= 10
↳ HASKELL
↳ LR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
Q DP problem:
The TRS P consists of the following rules:
new_gtGtEs2(:(vy40, vy41), vy5, vy6, vy7, vy3, h, ba, bb, bc, bd) → new_gtGtEs2(vy41, vy5, vy6, vy7, vy3, h, ba, bb, bc, bd)
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_gtGtEs2(:(vy40, vy41), vy5, vy6, vy7, vy3, h, ba, bb, bc, bd) → new_gtGtEs2(vy41, vy5, vy6, vy7, vy3, h, ba, bb, bc, bd)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 8 >= 8, 9 >= 9, 10 >= 10