*************************************
Proof
{ system = (VAR x y)
(STRATEGY INNERMOST)
(RULES f (t, x, y) -> f (g (x, y), x, s (y))
g (s (x), 0) -> t
g (s (x), s (y)) -> g (x, y))
, property = Termination , truth = Nothing , transform = Ignore_Strategy
, to = [ Proof
{ system = (VAR x y)
(STRATEGY INNERMOST)
(RULES f (t, x, y) -> f (g (x, y), x, s (y))
g (s (x), 0) -> t
g (s (x), s (y)) -> g (x, y))
, property = Termination , truth = Nothing , transform = Dependency_Pair_Transformation
, to = [ Proof
{ system = (VAR x y)
(STRATEGY INNERMOST)
(RULES fP (t, x, y) -> fP (g (x, y), x, s (y))
fP (t, x, y) -> gP (x, y)
gP (s (x), s (y)) -> gP (x, y)
f (t, x, y) ->= f (g (x, y), x, s (y))
g (s (x), 0) ->= t
g (s (x), s (y)) ->= g (x, y))
, property = Top_Termination , truth = Nothing
, transform = Remove
{ interpretation = Interpretation
0 |-> () |-> E^0x1
f |-> (x , y , z) |-> E^0x0 * x + E^0x0 * y + E^0x0 * z + E^0x1
g |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
t |-> () |-> E^0x1
fP |-> (x , y , z) |-> E^1x0 * x + E^1x0 * y + E^1x0 * z + (2)
gP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0)
, removes_rules = [ fP (t, x, y) -> gP (x, y) ] , comment = size 0 , heights ( 3 , 7 ) , time
}
, to = [ Proof
{ system = (VAR x y)
(STRATEGY INNERMOST)
(RULES fP (t, x, y) -> fP (g (x, y), x, s (y))
gP (s (x), s (y)) -> gP (x, y)
f (t, x, y) ->= f (g (x, y), x, s (y))
g (s (x), 0) ->= t
g (s (x), s (y)) ->= g (x, y))
, property = Top_Termination , truth = Nothing
, transform = Split
{ interpretation = Interpretation
0 |-> () |-> E^0x1
f |-> (x , y , z) |-> E^0x0 * x + E^0x0 * y + E^0x0 * z + E^0x1
g |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
t |-> () |-> E^0x1
fP |-> (x , y , z) |-> E^1x0 * x + E^1x0 * y + E^1x0 * z + (2)
gP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0)
, clusters = [ [ gP (s (x), s (y)) -> gP (x, y) ] , [ fP (t, x, y) -> fP (g (x, y), x, s (y)) ] ]
, comment = size 0 , heights ( 3 , 7 ) , time
split_dimension: 0
}
, to = [ Proof
{ system = (VAR x y)
(STRATEGY INNERMOST)
(RULES gP (s (x), s (y)) -> gP (x, y)
f (t, x, y) ->= f (g (x, y), x, s (y))
g (s (x), 0) ->= t
g (s (x), s (y)) ->= g (x, y))
, property = Top_Termination
, truth = Just
True
, transform = Remove
{ interpretation = Interpretation
0 |-> () |-> (0)
f |-> (x , y , z) |-> (0) * x + (0) * y + (0) * z + (0)
g |-> (x , y) |-> (0) * x + (1) * y + (0)
s |-> (x) |-> (2) * x + (1)
t |-> () |-> (0)
gP |-> (x , y) |-> (4) * x + (4) * y + (0)
, removes_rules = [ gP (s (x), s (y)) -> gP (x, y) ] , comment = size 1 , heights ( 7 , 15 ) , time
}
, to = [ Proof
{ system = (VAR x y)
(STRATEGY INNERMOST)
(RULES f (t, x, y) ->= f (g (x, y), x, s (y))
g (s (x), 0) ->= t
g (s (x), s (y)) ->= g (x, y))
, property = Top_Termination
, truth = Just
True
, transform = No_Strict_Rules , to = [ ]
}
]
}
, Claim
{ system = (VAR x y)
(STRATEGY INNERMOST)
(RULES fP (t, x, y) -> fP (g (x, y), x, s (y))
f (t, x, y) ->= f (g (x, y), x, s (y))
g (s (x), 0) ->= t
g (s (x), s (y)) ->= g (x, y))
, property = Top_Termination
}
]
}
]
}
]
}
]
}
Proof summary:
value Nothing
for property Termination
for system with 3 strict rules and 0 non-strict rules
follows by transformation
Ignore_Strategy
from
value Nothing
for property Termination
for system with 3 strict rules and 0 non-strict rules
follows by transformation
Dependency_Pair_Transformation
from
value Nothing
for property Top_Termination
for system with 3 strict rules and 3 non-strict rules
follows by transformation
Remove
{ interpretation = Interpretation
0 |-> () |-> E^0x1
f |-> (x , y , z) |-> E^0x0 * x + E^0x0 * y + E^0x0 * z + E^0x1
g |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
t |-> () |-> E^0x1
fP |-> (x , y , z) |-> E^1x0 * x + E^1x0 * y + E^1x0 * z + (2)
gP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0)
, removes_rules = [ fP (t, x, y) -> gP (x, y) ] , comment = size 0 , heights ( 3 , 7 ) , time
}
from
value Nothing
for property Top_Termination
for system with 2 strict rules and 3 non-strict rules
follows by transformation
Split
{ interpretation = Interpretation
0 |-> () |-> E^0x1
f |-> (x , y , z) |-> E^0x0 * x + E^0x0 * y + E^0x0 * z + E^0x1
g |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
t |-> () |-> E^0x1
fP |-> (x , y , z) |-> E^1x0 * x + E^1x0 * y + E^1x0 * z + (2)
gP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0)
, clusters = [ [ gP (s (x), s (y)) -> gP (x, y) ] , [ fP (t, x, y) -> fP (g (x, y), x, s (y)) ] ]
, comment = size 0 , heights ( 3 , 7 ) , time
split_dimension: 0
}
from
value Just
True
for property Top_Termination
for system with 1 strict rules and 3 non-strict rules
follows by transformation
Remove
{ interpretation = Interpretation
0 |-> () |-> (0)
f |-> (x , y , z) |-> (0) * x + (0) * y + (0) * z + (0)
g |-> (x , y) |-> (0) * x + (1) * y + (0)
s |-> (x) |-> (2) * x + (1)
t |-> () |-> (0)
gP |-> (x , y) |-> (4) * x + (4) * y + (0)
, removes_rules = [ gP (s (x), s (y)) -> gP (x, y) ] , comment = size 1 , heights ( 7 , 15 ) , time
}
from
value Just
True
for property Top_Termination
for system with 0 strict rules and 3 non-strict rules
follows by transformation
No_Strict_Rules
from
Claim
{ system = (VAR x y)
(STRATEGY INNERMOST)
(RULES fP (t, x, y) -> fP (g (x, y), x, s (y))
f (t, x, y) ->= f (g (x, y), x, s (y))
g (s (x), 0) ->= t
g (s (x), s (y)) ->= g (x, y))
, property = Top_Termination
}
------------------------------------------------------------------
matchbox general information (including details on proof methods):
http://dfa.imn.htwk-leipzig.de/matchbox/
this matchbox implementation uses the SAT solver
SatELite/MiniSat by Niklas Een and Niklas Sörensson
http://www.cs.chalmers.se/Cs/Research/FormalMethods/MiniSat/
matchbox process information
arguments : --solver=/home/nowonder/forschung/increasing/wst06/matchbox/SatELiteGTI --timeout-command=/home/nowonder/forschung/increasing/wst06/matchbox/timeout --tmpdir=/home/nowonder/forschung/increasing/wst06/matchbox --timeout=60 /tmp/tmpf72jp5/ex02.trs
started : Thu Feb 22 16:24:42 CET 2007
finished : Thu Feb 22 16:25:36 CET 2007
run system : Linux aprove 2.6.14-gentoo-r5 #1 SMP Sun Dec 25 15:42:02 CET 2005 x86_64
release date : Thu Jun 8 23:18:07 CEST 2006
build date : Thu Jun 8 23:18:07 CEST 2006
build system : Linux dfa 2.6.8-2-k7 #1 Tue Aug 16 14:00:15 UTC 2005 i686 GNU/Linux
used clock time: 54 secs