*************************************
Proof
{ system = (VAR x y
v u)
(STRATEGY INNERMOST)
(RULES minus (x, x) -> 0
minus (x, y) -> cond (min (x, y), x, y)
cond (y, x, y) -> s (minus (x, s (y)))
min (0, v) -> 0
min (u, 0) -> 0
min (s (u), s (v)) -> s (min (u, v)))
, property = Termination , truth = Nothing , transform = Ignore_Strategy
, to = [ Proof
{ system = (VAR x y
v u)
(STRATEGY INNERMOST)
(RULES minus (x, x) -> 0
minus (x, y) -> cond (min (x, y), x, y)
cond (y, x, y) -> s (minus (x, s (y)))
min (0, v) -> 0
min (u, 0) -> 0
min (s (u), s (v)) -> s (min (u, v)))
, property = Termination , truth = Nothing , transform = Dependency_Pair_Transformation
, to = [ Proof
{ system = (VAR x y
v u)
(STRATEGY INNERMOST)
(RULES minusP (x, y) -> condP (min (x, y), x, y)
minusP (x, y) -> minP (x, y)
condP (y, x, y) -> minusP (x, s (y))
minP (s (u), s (v)) -> minP (u, v)
minus (x, x) ->= 0
minus (x, y) ->= cond (min (x, y), x, y)
cond (y, x, y) ->= s (minus (x, s (y)))
min (0, v) ->= 0
min (u, 0) ->= 0
min (s (u), s (v)) ->= s (min (u, v)))
, property = Top_Termination , truth = Nothing
, transform = Remove
{ interpretation = Interpretation
minusP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (2)
condP |-> (x , y , z) |-> E^1x0 * x + E^1x0 * y + E^1x0 * z + (2)
0 |-> () |-> E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
minP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (1)
minus |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
min |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
cond |-> (x , y , z) |-> E^0x0 * x + E^0x0 * y + E^0x0 * z + E^0x1
, removes_rules = [ minusP (x, y) -> minP (x, y) ] , comment = size 0 , heights ( 3 , 7 ) , time
}
, to = [ Proof
{ system = (VAR x y
v u)
(STRATEGY INNERMOST)
(RULES minusP (x, y) -> condP (min (x, y), x, y)
condP (y, x, y) -> minusP (x, s (y))
minP (s (u), s (v)) -> minP (u, v)
minus (x, x) ->= 0
minus (x, y) ->= cond (min (x, y), x, y)
cond (y, x, y) ->= s (minus (x, s (y)))
min (0, v) ->= 0
min (u, 0) ->= 0
min (s (u), s (v)) ->= s (min (u, v)))
, property = Top_Termination , truth = Nothing
, transform = Split
{ interpretation = Interpretation
minusP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (2)
condP |-> (x , y , z) |-> E^1x0 * x + E^1x0 * y + E^1x0 * z + (2)
0 |-> () |-> E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
minP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (1)
minus |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
min |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
cond |-> (x , y , z) |-> E^0x0 * x + E^0x0 * y + E^0x0 * z + E^0x1
, clusters = [ [ minP (s (u), s (v)) -> minP (u, v) ] , [ minusP (x, y) -> condP (min (x, y), x, y) , condP (y, x, y) -> minusP (x, s (y)) ] ]
, comment = size 0 , heights ( 3 , 7 ) , time
split_dimension: 0
}
, to = [ Claim
{ system = (VAR x y
v u)
(STRATEGY INNERMOST)
(RULES minP (s (u), s (v)) -> minP (u, v)
minus (x, x) ->= 0
minus (x, y) ->= cond (min (x, y), x, y)
cond (y, x, y) ->= s (minus (x, s (y)))
min (0, v) ->= 0
min (u, 0) ->= 0
min (s (u), s (v)) ->= s (min (u, v)))
, property = Top_Termination
}
, Claim
{ system = (VAR x y
v u)
(STRATEGY INNERMOST)
(RULES minusP (x, y) -> condP (min (x, y), x, y)
condP (y, x, y) -> minusP (x, s (y))
minus (x, x) ->= 0
minus (x, y) ->= cond (min (x, y), x, y)
cond (y, x, y) ->= s (minus (x, s (y)))
min (0, v) ->= 0
min (u, 0) ->= 0
min (s (u), s (v)) ->= s (min (u, v)))
, property = Top_Termination
}
]
}
]
}
]
}
]
}
Proof summary:
value Nothing
for property Termination
for system with 6 strict rules and 0 non-strict rules
follows by transformation
Ignore_Strategy
from
value Nothing
for property Termination
for system with 6 strict rules and 0 non-strict rules
follows by transformation
Dependency_Pair_Transformation
from
value Nothing
for property Top_Termination
for system with 4 strict rules and 6 non-strict rules
follows by transformation
Remove
{ interpretation = Interpretation
minusP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (2)
condP |-> (x , y , z) |-> E^1x0 * x + E^1x0 * y + E^1x0 * z + (2)
0 |-> () |-> E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
minP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (1)
minus |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
min |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
cond |-> (x , y , z) |-> E^0x0 * x + E^0x0 * y + E^0x0 * z + E^0x1
, removes_rules = [ minusP (x, y) -> minP (x, y) ] , comment = size 0 , heights ( 3 , 7 ) , time
}
from
value Nothing
for property Top_Termination
for system with 3 strict rules and 6 non-strict rules
follows by transformation
Split
{ interpretation = Interpretation
minusP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (2)
condP |-> (x , y , z) |-> E^1x0 * x + E^1x0 * y + E^1x0 * z + (2)
0 |-> () |-> E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
minP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (1)
minus |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
min |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
cond |-> (x , y , z) |-> E^0x0 * x + E^0x0 * y + E^0x0 * z + E^0x1
, clusters = [ [ minP (s (u), s (v)) -> minP (u, v) ] , [ minusP (x, y) -> condP (min (x, y), x, y) , condP (y, x, y) -> minusP (x, s (y)) ] ]
, comment = size 0 , heights ( 3 , 7 ) , time
split_dimension: 0
}
from
Claim
{ system = (VAR x y
v u)
(STRATEGY INNERMOST)
(RULES minP (s (u), s (v)) -> minP (u, v)
minus (x, x) ->= 0
minus (x, y) ->= cond (min (x, y), x, y)
cond (y, x, y) ->= s (minus (x, s (y)))
min (0, v) ->= 0
min (u, 0) ->= 0
min (s (u), s (v)) ->= s (min (u, v)))
, property = Top_Termination
}
Claim
{ system = (VAR x y
v u)
(STRATEGY INNERMOST)
(RULES minusP (x, y) -> condP (min (x, y), x, y)
condP (y, x, y) -> minusP (x, s (y))
minus (x, x) ->= 0
minus (x, y) ->= cond (min (x, y), x, y)
cond (y, x, y) ->= s (minus (x, s (y)))
min (0, v) ->= 0
min (u, 0) ->= 0
min (s (u), s (v)) ->= s (min (u, v)))
, 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/tmpZM05RT/ex04.trs
started : Thu Feb 22 16:26:27 CET 2007
finished : Thu Feb 22 16:27:19 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: 52 secs