*************************************
Proof
{ system = (VAR x y
u v)
(STRATEGY INNERMOST)
(RULES f (true, x, y) -> f (gt (x, y), x, round (s (y)))
round (0) -> 0
round (s (0)) -> s (s (0))
round (s (s (x))) -> s (s (round (x)))
gt (0, v) -> false
gt (s (u), 0) -> true
gt (s (u), s (v)) -> gt (u, v))
, property = Termination , truth = Nothing , transform = Ignore_Strategy
, to = [ Proof
{ system = (VAR x y
u v)
(STRATEGY INNERMOST)
(RULES f (true, x, y) -> f (gt (x, y), x, round (s (y)))
round (0) -> 0
round (s (0)) -> s (s (0))
round (s (s (x))) -> s (s (round (x)))
gt (0, v) -> false
gt (s (u), 0) -> true
gt (s (u), s (v)) -> gt (u, v))
, property = Termination , truth = Nothing , transform = Dependency_Pair_Transformation
, to = [ Proof
{ system = (VAR x y
u v)
(STRATEGY INNERMOST)
(RULES fP (true, x, y) -> fP (gt (x, y), x, round (s (y)))
fP (true, x, y) -> gtP (x, y)
fP (true, x, y) -> roundP (s (y))
roundP (s (s (x))) -> roundP (x)
gtP (s (u), s (v)) -> gtP (u, v)
f (true, x, y) ->= f (gt (x, y), x, round (s (y)))
round (0) ->= 0
round (s (0)) ->= s (s (0))
round (s (s (x))) ->= s (s (round (x)))
gt (0, v) ->= false
gt (s (u), 0) ->= true
gt (s (u), s (v)) ->= gt (u, v))
, property = Top_Termination , truth = Nothing
, transform = Remove
{ interpretation = Interpretation
gtP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (2)
0 |-> () |-> E^0x1
f |-> (x , y , z) |-> E^0x0 * x + E^0x0 * y + E^0x0 * z + E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
false |-> () |-> E^0x1
fP |-> (x , y , z) |-> E^1x0 * x + E^1x0 * y + E^1x0 * z + (3)
gt |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
round |-> (x) |-> E^0x0 * x + E^0x1
true |-> () |-> E^0x1
roundP |-> (x) |-> E^1x0 * x + (0)
, removes_rules = [ fP (true, x, y) -> gtP (x, y) , fP (true, x, y) -> roundP (s (y)) ] , comment = size 0 , heights ( 3 , 7 ) , time
}
, to = [ Proof
{ system = (VAR x y
u v)
(STRATEGY INNERMOST)
(RULES fP (true, x, y) -> fP (gt (x, y), x, round (s (y)))
roundP (s (s (x))) -> roundP (x)
gtP (s (u), s (v)) -> gtP (u, v)
f (true, x, y) ->= f (gt (x, y), x, round (s (y)))
round (0) ->= 0
round (s (0)) ->= s (s (0))
round (s (s (x))) ->= s (s (round (x)))
gt (0, v) ->= false
gt (s (u), 0) ->= true
gt (s (u), s (v)) ->= gt (u, v))
, property = Top_Termination , truth = Nothing
, transform = Split
{ interpretation = Interpretation
gtP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (2)
0 |-> () |-> E^0x1
f |-> (x , y , z) |-> E^0x0 * x + E^0x0 * y + E^0x0 * z + E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
false |-> () |-> E^0x1
fP |-> (x , y , z) |-> E^1x0 * x + E^1x0 * y + E^1x0 * z + (3)
gt |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
round |-> (x) |-> E^0x0 * x + E^0x1
true |-> () |-> E^0x1
roundP |-> (x) |-> E^1x0 * x + (0)
, clusters = [ [ roundP (s (s (x))) -> roundP (x) ] , [ gtP (s (u), s (v)) -> gtP (u, v) ] , [ fP (true, x, y) -> fP (gt (x, y), x, round (s (y))) ] ]
, comment = size 0 , heights ( 3 , 7 ) , time
split_dimension: 0
}
, to = [ Proof
{ system = (VAR x y
u v)
(STRATEGY INNERMOST)
(RULES roundP (s (s (x))) -> roundP (x)
f (true, x, y) ->= f (gt (x, y), x, round (s (y)))
round (0) ->= 0
round (s (0)) ->= s (s (0))
round (s (s (x))) ->= s (s (round (x)))
gt (0, v) ->= false
gt (s (u), 0) ->= true
gt (s (u), s (v)) ->= gt (u, v))
, property = Top_Termination
, truth = Just
True
, transform = Remove
{ interpretation = Interpretation
0 |-> () |-> (2)
f |-> (x , y , z) |-> (0) * x + (0) * y + (0) * z + (0)
s |-> (x) |-> (1) * x + (1)
false |-> () |-> (4)
gt |-> (x , y) |-> (0) * x + (0) * y + (4)
round |-> (x) |-> (1) * x + (1)
true |-> () |-> (1)
roundP |-> (x) |-> (1) * x + (0)
, removes_rules = [ roundP (s (s (x))) -> roundP (x) ] , comment = size 1 , heights ( 7 , 15 ) , time 1 sec
}
, to = [ Proof
{ system = (VAR x y
u v)
(STRATEGY INNERMOST)
(RULES f (true, x, y) ->= f (gt (x, y), x, round (s (y)))
round (0) ->= 0
round (s (0)) ->= s (s (0))
round (s (s (x))) ->= s (s (round (x)))
gt (0, v) ->= false
gt (s (u), 0) ->= true
gt (s (u), s (v)) ->= gt (u, v))
, property = Top_Termination
, truth = Just
True
, transform = No_Strict_Rules , to = [ ]
}
]
}
, Proof
{ system = (VAR x y
u v)
(STRATEGY INNERMOST)
(RULES gtP (s (u), s (v)) -> gtP (u, v)
f (true, x, y) ->= f (gt (x, y), x, round (s (y)))
round (0) ->= 0
round (s (0)) ->= s (s (0))
round (s (s (x))) ->= s (s (round (x)))
gt (0, v) ->= false
gt (s (u), 0) ->= true
gt (s (u), s (v)) ->= gt (u, v))
, property = Top_Termination
, truth = Just
True
, transform = Remove
{ interpretation = Interpretation
gtP |-> (x , y) |-> (4) * x + (0) * y + (4)
0 |-> () |-> (0)
f |-> (x , y , z) |-> (0) * x + (0) * y + (0) * z + (0)
s |-> (x) |-> (2) * x + (1)
false |-> () |-> (1)
gt |-> (x , y) |-> (0) * x + (0) * y + (1)
round |-> (x) |-> (3) * x + (1)
true |-> () |-> (0)
, removes_rules = [ gtP (s (u), s (v)) -> gtP (u, v) ] , comment = size 1 , heights ( 7 , 15 ) , time 1 sec
}
, to = [ Proof
{ system = (VAR x y
u v)
(STRATEGY INNERMOST)
(RULES f (true, x, y) ->= f (gt (x, y), x, round (s (y)))
round (0) ->= 0
round (s (0)) ->= s (s (0))
round (s (s (x))) ->= s (s (round (x)))
gt (0, v) ->= false
gt (s (u), 0) ->= true
gt (s (u), s (v)) ->= gt (u, v))
, property = Top_Termination
, truth = Just
True
, transform = No_Strict_Rules , to = [ ]
}
]
}
, Claim
{ system = (VAR x y
u v)
(STRATEGY INNERMOST)
(RULES fP (true, x, y) -> fP (gt (x, y), x, round (s (y)))
f (true, x, y) ->= f (gt (x, y), x, round (s (y)))
round (0) ->= 0
round (s (0)) ->= s (s (0))
round (s (s (x))) ->= s (s (round (x)))
gt (0, v) ->= false
gt (s (u), 0) ->= true
gt (s (u), s (v)) ->= gt (u, v))
, property = Top_Termination
}
]
}
]
}
]
}
]
}
Proof summary:
value Nothing
for property Termination
for system with 7 strict rules and 0 non-strict rules
follows by transformation
Ignore_Strategy
from
value Nothing
for property Termination
for system with 7 strict rules and 0 non-strict rules
follows by transformation
Dependency_Pair_Transformation
from
value Nothing
for property Top_Termination
for system with 5 strict rules and 7 non-strict rules
follows by transformation
Remove
{ interpretation = Interpretation
gtP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (2)
0 |-> () |-> E^0x1
f |-> (x , y , z) |-> E^0x0 * x + E^0x0 * y + E^0x0 * z + E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
false |-> () |-> E^0x1
fP |-> (x , y , z) |-> E^1x0 * x + E^1x0 * y + E^1x0 * z + (3)
gt |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
round |-> (x) |-> E^0x0 * x + E^0x1
true |-> () |-> E^0x1
roundP |-> (x) |-> E^1x0 * x + (0)
, removes_rules = [ fP (true, x, y) -> gtP (x, y) , fP (true, x, y) -> roundP (s (y)) ] , comment = size 0 , heights ( 3 , 7 ) , time
}
from
value Nothing
for property Top_Termination
for system with 3 strict rules and 7 non-strict rules
follows by transformation
Split
{ interpretation = Interpretation
gtP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (2)
0 |-> () |-> E^0x1
f |-> (x , y , z) |-> E^0x0 * x + E^0x0 * y + E^0x0 * z + E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
false |-> () |-> E^0x1
fP |-> (x , y , z) |-> E^1x0 * x + E^1x0 * y + E^1x0 * z + (3)
gt |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
round |-> (x) |-> E^0x0 * x + E^0x1
true |-> () |-> E^0x1
roundP |-> (x) |-> E^1x0 * x + (0)
, clusters = [ [ roundP (s (s (x))) -> roundP (x) ] , [ gtP (s (u), s (v)) -> gtP (u, v) ] , [ fP (true, x, y) -> fP (gt (x, y), x, round (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 7 non-strict rules
follows by transformation
Remove
{ interpretation = Interpretation
0 |-> () |-> (2)
f |-> (x , y , z) |-> (0) * x + (0) * y + (0) * z + (0)
s |-> (x) |-> (1) * x + (1)
false |-> () |-> (4)
gt |-> (x , y) |-> (0) * x + (0) * y + (4)
round |-> (x) |-> (1) * x + (1)
true |-> () |-> (1)
roundP |-> (x) |-> (1) * x + (0)
, removes_rules = [ roundP (s (s (x))) -> roundP (x) ] , comment = size 1 , heights ( 7 , 15 ) , time 1 sec
}
from
value Just
True
for property Top_Termination
for system with 0 strict rules and 7 non-strict rules
follows by transformation
No_Strict_Rules
from
value Just
True
for property Top_Termination
for system with 1 strict rules and 7 non-strict rules
follows by transformation
Remove
{ interpretation = Interpretation
gtP |-> (x , y) |-> (4) * x + (0) * y + (4)
0 |-> () |-> (0)
f |-> (x , y , z) |-> (0) * x + (0) * y + (0) * z + (0)
s |-> (x) |-> (2) * x + (1)
false |-> () |-> (1)
gt |-> (x , y) |-> (0) * x + (0) * y + (1)
round |-> (x) |-> (3) * x + (1)
true |-> () |-> (0)
, removes_rules = [ gtP (s (u), s (v)) -> gtP (u, v) ] , comment = size 1 , heights ( 7 , 15 ) , time 1 sec
}
from
value Just
True
for property Top_Termination
for system with 0 strict rules and 7 non-strict rules
follows by transformation
No_Strict_Rules
from
Claim
{ system = (VAR x y
u v)
(STRATEGY INNERMOST)
(RULES fP (true, x, y) -> fP (gt (x, y), x, round (s (y)))
f (true, x, y) ->= f (gt (x, y), x, round (s (y)))
round (0) ->= 0
round (s (0)) ->= s (s (0))
round (s (s (x))) ->= s (s (round (x)))
gt (0, v) ->= false
gt (s (u), 0) ->= true
gt (s (u), s (v)) ->= gt (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/tmpQ-GIs4/ex09.trs
started : Thu Feb 22 16:40:21 CET 2007
finished : Thu Feb 22 16:41:11 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: 50 secs