*************************************
Proof
{ system = (VAR x y
z u
v n
m)
(STRATEGY INNERMOST)
(RULES div (x, s (y)) -> d (x, s (y), 0)
d (x, s (y), z) -> cond (ge (x, z), x, y, z)
cond (true, x, y, z) -> s (d (x, s (y), plus (s (y), z)))
cond (false, x, y, z) -> 0
ge (u, 0) -> true
ge (0, s (v)) -> false
ge (s (u), s (v)) -> ge (u, v)
plus (n, 0) -> n
plus (n, s (m)) -> s (plus (n, m))
plus (plus (n, m), u) -> plus (n, plus (m, u)))
, property = Termination , truth = Nothing , transform = Ignore_Strategy
, to = [ Proof
{ system = (VAR x y
z u
v n
m)
(STRATEGY INNERMOST)
(RULES div (x, s (y)) -> d (x, s (y), 0)
d (x, s (y), z) -> cond (ge (x, z), x, y, z)
cond (true, x, y, z) -> s (d (x, s (y), plus (s (y), z)))
cond (false, x, y, z) -> 0
ge (u, 0) -> true
ge (0, s (v)) -> false
ge (s (u), s (v)) -> ge (u, v)
plus (n, 0) -> n
plus (n, s (m)) -> s (plus (n, m))
plus (plus (n, m), u) -> plus (n, plus (m, u)))
, property = Termination , truth = Nothing , transform = Dependency_Pair_Transformation
, to = [ Proof
{ system = (VAR x y
z u
v n
m)
(STRATEGY INNERMOST)
(RULES divP (x, s (y)) -> dP (x, s (y), 0)
dP (x, s (y), z) -> condP (ge (x, z), x, y, z)
dP (x, s (y), z) -> geP (x, z)
condP (true, x, y, z) -> dP (x, s (y), plus (s (y), z))
condP (true, x, y, z) -> plusP (s (y), z)
geP (s (u), s (v)) -> geP (u, v)
plusP (n, s (m)) -> plusP (n, m)
plusP (plus (n, m), u) -> plusP (n, plus (m, u))
plusP (plus (n, m), u) -> plusP (m, u)
div (x, s (y)) ->= d (x, s (y), 0)
d (x, s (y), z) ->= cond (ge (x, z), x, y, z)
cond (true, x, y, z) ->= s (d (x, s (y), plus (s (y), z)))
cond (false, x, y, z) ->= 0
ge (u, 0) ->= true
ge (0, s (v)) ->= false
ge (s (u), s (v)) ->= ge (u, v)
plus (n, 0) ->= n
plus (n, s (m)) ->= s (plus (n, m))
plus (plus (n, m), u) ->= plus (n, plus (m, u)))
, property = Top_Termination , truth = Nothing
, transform = Remove
{ interpretation = Interpretation
geP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0)
plusP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0)
condP |-> (p , q , r , s) |-> E^1x0 * p + E^1x0 * q + E^1x0 * r + E^1x0 * s + (0)
0 |-> () |-> E^0x1
d |-> (x , y , z) |-> E^0x0 * x + E^0x0 * y + E^0x0 * z + E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
false |-> () |-> E^0x1
dP |-> (x , y , z) |-> E^1x0 * x + E^1x0 * y + E^1x0 * z + (0)
ge |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
true |-> () |-> E^0x1
divP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (1)
div |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
plus |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
cond |-> (p , q , r , s) |-> E^0x0 * p + E^0x0 * q + E^0x0 * r + E^0x0 * s + E^0x1
, removes_rules = [ divP (x, s (y)) -> dP (x, s (y), 0) ] , comment = size 0 , heights ( 3 , 7 ) , time 1 sec
}
, to = [ Proof
{ system = (VAR x y
z u
v n
m)
(STRATEGY INNERMOST)
(RULES dP (x, s (y), z) -> condP (ge (x, z), x, y, z)
dP (x, s (y), z) -> geP (x, z)
condP (true, x, y, z) -> dP (x, s (y), plus (s (y), z))
condP (true, x, y, z) -> plusP (s (y), z)
geP (s (u), s (v)) -> geP (u, v)
plusP (n, s (m)) -> plusP (n, m)
plusP (plus (n, m), u) -> plusP (n, plus (m, u))
plusP (plus (n, m), u) -> plusP (m, u)
div (x, s (y)) ->= d (x, s (y), 0)
d (x, s (y), z) ->= cond (ge (x, z), x, y, z)
cond (true, x, y, z) ->= s (d (x, s (y), plus (s (y), z)))
cond (false, x, y, z) ->= 0
ge (u, 0) ->= true
ge (0, s (v)) ->= false
ge (s (u), s (v)) ->= ge (u, v)
plus (n, 0) ->= n
plus (n, s (m)) ->= s (plus (n, m))
plus (plus (n, m), u) ->= plus (n, plus (m, u)))
, property = Top_Termination , truth = Nothing
, transform = Remove
{ interpretation = Interpretation
geP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0)
plusP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (1)
condP |-> (p , q , r , s) |-> E^1x0 * p + E^1x0 * q + E^1x0 * r + E^1x0 * s + (2)
0 |-> () |-> E^0x1
d |-> (x , y , z) |-> E^0x0 * x + E^0x0 * y + E^0x0 * z + E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
false |-> () |-> E^0x1
dP |-> (x , y , z) |-> E^1x0 * x + E^1x0 * y + E^1x0 * z + (2)
ge |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
true |-> () |-> E^0x1
div |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
plus |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
cond |-> (p , q , r , s) |-> E^0x0 * p + E^0x0 * q + E^0x0 * r + E^0x0 * s + E^0x1
, removes_rules = [ dP (x, s (y), z) -> geP (x, z) , condP (true, x, y, z) -> plusP (s (y), z) ] , comment = size 0 , heights ( 3 , 7 ) , time
}
, to = [ Proof
{ system = (VAR x y
z u
v n
m)
(STRATEGY INNERMOST)
(RULES dP (x, s (y), z) -> condP (ge (x, z), x, y, z)
condP (true, x, y, z) -> dP (x, s (y), plus (s (y), z))
geP (s (u), s (v)) -> geP (u, v)
plusP (n, s (m)) -> plusP (n, m)
plusP (plus (n, m), u) -> plusP (n, plus (m, u))
plusP (plus (n, m), u) -> plusP (m, u)
div (x, s (y)) ->= d (x, s (y), 0)
d (x, s (y), z) ->= cond (ge (x, z), x, y, z)
cond (true, x, y, z) ->= s (d (x, s (y), plus (s (y), z)))
cond (false, x, y, z) ->= 0
ge (u, 0) ->= true
ge (0, s (v)) ->= false
ge (s (u), s (v)) ->= ge (u, v)
plus (n, 0) ->= n
plus (n, s (m)) ->= s (plus (n, m))
plus (plus (n, m), u) ->= plus (n, plus (m, u)))
, property = Top_Termination , truth = Nothing
, transform = Split
{ interpretation = Interpretation
geP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0)
plusP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (1)
condP |-> (p , q , r , s) |-> E^1x0 * p + E^1x0 * q + E^1x0 * r + E^1x0 * s + (2)
0 |-> () |-> E^0x1
d |-> (x , y , z) |-> E^0x0 * x + E^0x0 * y + E^0x0 * z + E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
false |-> () |-> E^0x1
dP |-> (x , y , z) |-> E^1x0 * x + E^1x0 * y + E^1x0 * z + (2)
ge |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
true |-> () |-> E^0x1
div |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
plus |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
cond |-> (p , q , r , s) |-> E^0x0 * p + E^0x0 * q + E^0x0 * r + E^0x0 * s + E^0x1
, clusters = [ [ geP (s (u), s (v)) -> geP (u, v) ] , [ plusP (n, s (m)) -> plusP (n, m) , plusP (plus (n, m), u) -> plusP (n, plus (m, u)) , plusP (plus (n, m), u) -> plusP (m, u) ] , [ dP (x, s (y), z) -> condP (ge (x, z), x, y, z) , condP (true, x, y, z) -> dP (x, s (y), plus (s (y), z)) ] ]
, comment = size 0 , heights ( 3 , 7 ) , time
split_dimension: 0
}
, to = [ Claim
{ system = (VAR x y
z u
v n
m)
(STRATEGY INNERMOST)
(RULES geP (s (u), s (v)) -> geP (u, v)
div (x, s (y)) ->= d (x, s (y), 0)
d (x, s (y), z) ->= cond (ge (x, z), x, y, z)
cond (true, x, y, z) ->= s (d (x, s (y), plus (s (y), z)))
cond (false, x, y, z) ->= 0
ge (u, 0) ->= true
ge (0, s (v)) ->= false
ge (s (u), s (v)) ->= ge (u, v)
plus (n, 0) ->= n
plus (n, s (m)) ->= s (plus (n, m))
plus (plus (n, m), u) ->= plus (n, plus (m, u)))
, property = Top_Termination
}
, Claim
{ system = (VAR x y
z u
v n
m)
(STRATEGY INNERMOST)
(RULES plusP (n, s (m)) -> plusP (n, m)
plusP (plus (n, m), u) -> plusP (n, plus (m, u))
plusP (plus (n, m), u) -> plusP (m, u)
div (x, s (y)) ->= d (x, s (y), 0)
d (x, s (y), z) ->= cond (ge (x, z), x, y, z)
cond (true, x, y, z) ->= s (d (x, s (y), plus (s (y), z)))
cond (false, x, y, z) ->= 0
ge (u, 0) ->= true
ge (0, s (v)) ->= false
ge (s (u), s (v)) ->= ge (u, v)
plus (n, 0) ->= n
plus (n, s (m)) ->= s (plus (n, m))
plus (plus (n, m), u) ->= plus (n, plus (m, u)))
, property = Top_Termination
}
, Claim
{ system = (VAR x y
z u
v n
m)
(STRATEGY INNERMOST)
(RULES dP (x, s (y), z) -> condP (ge (x, z), x, y, z)
condP (true, x, y, z) -> dP (x, s (y), plus (s (y), z))
div (x, s (y)) ->= d (x, s (y), 0)
d (x, s (y), z) ->= cond (ge (x, z), x, y, z)
cond (true, x, y, z) ->= s (d (x, s (y), plus (s (y), z)))
cond (false, x, y, z) ->= 0
ge (u, 0) ->= true
ge (0, s (v)) ->= false
ge (s (u), s (v)) ->= ge (u, v)
plus (n, 0) ->= n
plus (n, s (m)) ->= s (plus (n, m))
plus (plus (n, m), u) ->= plus (n, plus (m, u)))
, property = Top_Termination
}
]
}
]
}
]
}
]
}
]
}
Proof summary:
value Nothing
for property Termination
for system with 10 strict rules and 0 non-strict rules
follows by transformation
Ignore_Strategy
from
value Nothing
for property Termination
for system with 10 strict rules and 0 non-strict rules
follows by transformation
Dependency_Pair_Transformation
from
value Nothing
for property Top_Termination
for system with 9 strict rules and 10 non-strict rules
follows by transformation
Remove
{ interpretation = Interpretation
geP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0)
plusP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0)
condP |-> (p , q , r , s) |-> E^1x0 * p + E^1x0 * q + E^1x0 * r + E^1x0 * s + (0)
0 |-> () |-> E^0x1
d |-> (x , y , z) |-> E^0x0 * x + E^0x0 * y + E^0x0 * z + E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
false |-> () |-> E^0x1
dP |-> (x , y , z) |-> E^1x0 * x + E^1x0 * y + E^1x0 * z + (0)
ge |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
true |-> () |-> E^0x1
divP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (1)
div |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
plus |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
cond |-> (p , q , r , s) |-> E^0x0 * p + E^0x0 * q + E^0x0 * r + E^0x0 * s + E^0x1
, removes_rules = [ divP (x, s (y)) -> dP (x, s (y), 0) ] , comment = size 0 , heights ( 3 , 7 ) , time 1 sec
}
from
value Nothing
for property Top_Termination
for system with 8 strict rules and 10 non-strict rules
follows by transformation
Remove
{ interpretation = Interpretation
geP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0)
plusP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (1)
condP |-> (p , q , r , s) |-> E^1x0 * p + E^1x0 * q + E^1x0 * r + E^1x0 * s + (2)
0 |-> () |-> E^0x1
d |-> (x , y , z) |-> E^0x0 * x + E^0x0 * y + E^0x0 * z + E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
false |-> () |-> E^0x1
dP |-> (x , y , z) |-> E^1x0 * x + E^1x0 * y + E^1x0 * z + (2)
ge |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
true |-> () |-> E^0x1
div |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
plus |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
cond |-> (p , q , r , s) |-> E^0x0 * p + E^0x0 * q + E^0x0 * r + E^0x0 * s + E^0x1
, removes_rules = [ dP (x, s (y), z) -> geP (x, z) , condP (true, x, y, z) -> plusP (s (y), z) ] , comment = size 0 , heights ( 3 , 7 ) , time
}
from
value Nothing
for property Top_Termination
for system with 6 strict rules and 10 non-strict rules
follows by transformation
Split
{ interpretation = Interpretation
geP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0)
plusP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (1)
condP |-> (p , q , r , s) |-> E^1x0 * p + E^1x0 * q + E^1x0 * r + E^1x0 * s + (2)
0 |-> () |-> E^0x1
d |-> (x , y , z) |-> E^0x0 * x + E^0x0 * y + E^0x0 * z + E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
false |-> () |-> E^0x1
dP |-> (x , y , z) |-> E^1x0 * x + E^1x0 * y + E^1x0 * z + (2)
ge |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
true |-> () |-> E^0x1
div |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
plus |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
cond |-> (p , q , r , s) |-> E^0x0 * p + E^0x0 * q + E^0x0 * r + E^0x0 * s + E^0x1
, clusters = [ [ geP (s (u), s (v)) -> geP (u, v) ] , [ plusP (n, s (m)) -> plusP (n, m) , plusP (plus (n, m), u) -> plusP (n, plus (m, u)) , plusP (plus (n, m), u) -> plusP (m, u) ] , [ dP (x, s (y), z) -> condP (ge (x, z), x, y, z) , condP (true, x, y, z) -> dP (x, s (y), plus (s (y), z)) ] ]
, comment = size 0 , heights ( 3 , 7 ) , time
split_dimension: 0
}
from
Claim
{ system = (VAR x y
z u
v n
m)
(STRATEGY INNERMOST)
(RULES geP (s (u), s (v)) -> geP (u, v)
div (x, s (y)) ->= d (x, s (y), 0)
d (x, s (y), z) ->= cond (ge (x, z), x, y, z)
cond (true, x, y, z) ->= s (d (x, s (y), plus (s (y), z)))
cond (false, x, y, z) ->= 0
ge (u, 0) ->= true
ge (0, s (v)) ->= false
ge (s (u), s (v)) ->= ge (u, v)
plus (n, 0) ->= n
plus (n, s (m)) ->= s (plus (n, m))
plus (plus (n, m), u) ->= plus (n, plus (m, u)))
, property = Top_Termination
}
Claim
{ system = (VAR x y
z u
v n
m)
(STRATEGY INNERMOST)
(RULES plusP (n, s (m)) -> plusP (n, m)
plusP (plus (n, m), u) -> plusP (n, plus (m, u))
plusP (plus (n, m), u) -> plusP (m, u)
div (x, s (y)) ->= d (x, s (y), 0)
d (x, s (y), z) ->= cond (ge (x, z), x, y, z)
cond (true, x, y, z) ->= s (d (x, s (y), plus (s (y), z)))
cond (false, x, y, z) ->= 0
ge (u, 0) ->= true
ge (0, s (v)) ->= false
ge (s (u), s (v)) ->= ge (u, v)
plus (n, 0) ->= n
plus (n, s (m)) ->= s (plus (n, m))
plus (plus (n, m), u) ->= plus (n, plus (m, u)))
, property = Top_Termination
}
Claim
{ system = (VAR x y
z u
v n
m)
(STRATEGY INNERMOST)
(RULES dP (x, s (y), z) -> condP (ge (x, z), x, y, z)
condP (true, x, y, z) -> dP (x, s (y), plus (s (y), z))
div (x, s (y)) ->= d (x, s (y), 0)
d (x, s (y), z) ->= cond (ge (x, z), x, y, z)
cond (true, x, y, z) ->= s (d (x, s (y), plus (s (y), z)))
cond (false, x, y, z) ->= 0
ge (u, 0) ->= true
ge (0, s (v)) ->= false
ge (s (u), s (v)) ->= ge (u, v)
plus (n, 0) ->= n
plus (n, s (m)) ->= s (plus (n, m))
plus (plus (n, m), u) ->= plus (n, plus (m, u)))
, 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/tmpxLiSZI/ex13.trs
started : Thu Feb 22 16:43:50 CET 2007
finished : Thu Feb 22 16:44:44 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