*************************************
Proof
{ system = (VAR l x
y v
u n
m)
(STRATEGY INNERMOST)
(RULES log (x, s (s (y))) -> cond (le (x, s (s (y))), x, y)
cond (true, x, y) -> s (0)
cond (false, x, y) -> double (log (x, square (s (s (y)))))
le (0, v) -> true
le (s (u), 0) -> false
le (s (u), s (v)) -> le (u, v)
double (0) -> 0
double (s (x)) -> s (s (double (x)))
square (0) -> 0
square (s (x)) -> s (plus (square (x), double (x)))
plus (n, 0) -> n
plus (n, s (m)) -> s (plus (n, m)))
, property = Termination , truth = Nothing , transform = Ignore_Strategy
, to = [ Proof
{ system = (VAR l x
y v
u n
m)
(STRATEGY INNERMOST)
(RULES log (x, s (s (y))) -> cond (le (x, s (s (y))), x, y)
cond (true, x, y) -> s (0)
cond (false, x, y) -> double (log (x, square (s (s (y)))))
le (0, v) -> true
le (s (u), 0) -> false
le (s (u), s (v)) -> le (u, v)
double (0) -> 0
double (s (x)) -> s (s (double (x)))
square (0) -> 0
square (s (x)) -> s (plus (square (x), double (x)))
plus (n, 0) -> n
plus (n, s (m)) -> s (plus (n, m)))
, property = Termination , truth = Nothing , transform = Dependency_Pair_Transformation
, to = [ Proof
{ system = (VAR l x
y v
u n
m)
(STRATEGY INNERMOST)
(RULES logP (x, s (s (y))) -> condP (le (x, s (s (y))), x, y)
logP (x, s (s (y))) -> leP (x, s (s (y)))
condP (false, x, y) -> doubleP (log (x, square (s (s (y)))))
condP (false, x, y) -> logP (x, square (s (s (y))))
condP (false, x, y) -> squareP (s (s (y)))
leP (s (u), s (v)) -> leP (u, v)
doubleP (s (x)) -> doubleP (x)
squareP (s (x)) -> plusP (square (x), double (x))
squareP (s (x)) -> squareP (x)
squareP (s (x)) -> doubleP (x)
plusP (n, s (m)) -> plusP (n, m)
log (x, s (s (y))) ->= cond (le (x, s (s (y))), x, y)
cond (true, x, y) ->= s (0)
cond (false, x, y) ->= double (log (x, square (s (s (y)))))
le (0, v) ->= true
le (s (u), 0) ->= false
le (s (u), s (v)) ->= le (u, v)
double (0) ->= 0
double (s (x)) ->= s (s (double (x)))
square (0) ->= 0
square (s (x)) ->= s (plus (square (x), double (x)))
plus (n, 0) ->= n
plus (n, s (m)) ->= s (plus (n, m)))
, property = Top_Termination , truth = Nothing
, transform = Remove
{ interpretation = Interpretation
squareP |-> (x) |-> E^1x0 * x + (0)
leP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0)
doubleP |-> (x) |-> E^1x0 * x + (0)
plusP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0)
condP |-> (x , y , z) |-> E^1x0 * x + E^1x0 * y + E^1x0 * z + (1)
0 |-> () |-> E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
false |-> () |-> E^0x1
square |-> (x) |-> E^0x0 * x + E^0x1
le |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
logP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (1)
double |-> (x) |-> E^0x0 * x + E^0x1
true |-> () |-> E^0x1
log |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
plus |-> (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 = [ logP (x, s (s (y))) -> leP (x, s (s (y))) , condP (false, x, y) -> doubleP (log (x, square (s (s (y))))) , condP (false, x, y) -> squareP (s (s (y))) ] , comment = size 0 , heights ( 3 , 7 ) , time
}
, to = [ Proof
{ system = (VAR l x
y v
u n
m)
(STRATEGY INNERMOST)
(RULES logP (x, s (s (y))) -> condP (le (x, s (s (y))), x, y)
condP (false, x, y) -> logP (x, square (s (s (y))))
leP (s (u), s (v)) -> leP (u, v)
doubleP (s (x)) -> doubleP (x)
squareP (s (x)) -> plusP (square (x), double (x))
squareP (s (x)) -> squareP (x)
squareP (s (x)) -> doubleP (x)
plusP (n, s (m)) -> plusP (n, m)
log (x, s (s (y))) ->= cond (le (x, s (s (y))), x, y)
cond (true, x, y) ->= s (0)
cond (false, x, y) ->= double (log (x, square (s (s (y)))))
le (0, v) ->= true
le (s (u), 0) ->= false
le (s (u), s (v)) ->= le (u, v)
double (0) ->= 0
double (s (x)) ->= s (s (double (x)))
square (0) ->= 0
square (s (x)) ->= s (plus (square (x), double (x)))
plus (n, 0) ->= n
plus (n, s (m)) ->= s (plus (n, m)))
, property = Top_Termination , truth = Nothing
, transform = Split
{ interpretation = Interpretation
squareP |-> (x) |-> E^1x0 * x + (0)
leP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0)
doubleP |-> (x) |-> E^1x0 * x + (0)
plusP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0)
condP |-> (x , y , z) |-> E^1x0 * x + E^1x0 * y + E^1x0 * z + (1)
0 |-> () |-> E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
false |-> () |-> E^0x1
square |-> (x) |-> E^0x0 * x + E^0x1
le |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
logP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (1)
double |-> (x) |-> E^0x0 * x + E^0x1
true |-> () |-> E^0x1
log |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
plus |-> (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 = [ [ leP (s (u), s (v)) -> leP (u, v) , doubleP (s (x)) -> doubleP (x) , squareP (s (x)) -> plusP (square (x), double (x)) , squareP (s (x)) -> squareP (x) , squareP (s (x)) -> doubleP (x) , plusP (n, s (m)) -> plusP (n, m) ] , [ logP (x, s (s (y))) -> condP (le (x, s (s (y))), x, y) , condP (false, x, y) -> logP (x, square (s (s (y)))) ] ]
, comment = size 0 , heights ( 3 , 7 ) , time
split_dimension: 0
}
, to = [ Proof
{ system = (VAR l x
y v
u n
m)
(STRATEGY INNERMOST)
(RULES leP (s (u), s (v)) -> leP (u, v)
doubleP (s (x)) -> doubleP (x)
squareP (s (x)) -> plusP (square (x), double (x))
squareP (s (x)) -> squareP (x)
squareP (s (x)) -> doubleP (x)
plusP (n, s (m)) -> plusP (n, m)
log (x, s (s (y))) ->= cond (le (x, s (s (y))), x, y)
cond (true, x, y) ->= s (0)
cond (false, x, y) ->= double (log (x, square (s (s (y)))))
le (0, v) ->= true
le (s (u), 0) ->= false
le (s (u), s (v)) ->= le (u, v)
double (0) ->= 0
double (s (x)) ->= s (s (double (x)))
square (0) ->= 0
square (s (x)) ->= s (plus (square (x), double (x)))
plus (n, 0) ->= n
plus (n, s (m)) ->= s (plus (n, m)))
, property = Top_Termination , truth = Nothing
, transform = Remove
{ interpretation = Interpretation
squareP |-> (x) |-> E^1x0 * x + (2)
leP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0)
doubleP |-> (x) |-> E^1x0 * x + (0)
plusP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (1)
0 |-> () |-> E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
false |-> () |-> E^0x1
square |-> (x) |-> E^0x0 * x + E^0x1
le |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
double |-> (x) |-> E^0x0 * x + E^0x1
true |-> () |-> E^0x1
log |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
plus |-> (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 = [ squareP (s (x)) -> plusP (square (x), double (x)) , squareP (s (x)) -> doubleP (x) ] , comment = size 0 , heights ( 3 , 7 ) , time
}
, to = [ Proof
{ system = (VAR l x
y v
u n
m)
(STRATEGY INNERMOST)
(RULES leP (s (u), s (v)) -> leP (u, v)
doubleP (s (x)) -> doubleP (x)
squareP (s (x)) -> squareP (x)
plusP (n, s (m)) -> plusP (n, m)
log (x, s (s (y))) ->= cond (le (x, s (s (y))), x, y)
cond (true, x, y) ->= s (0)
cond (false, x, y) ->= double (log (x, square (s (s (y)))))
le (0, v) ->= true
le (s (u), 0) ->= false
le (s (u), s (v)) ->= le (u, v)
double (0) ->= 0
double (s (x)) ->= s (s (double (x)))
square (0) ->= 0
square (s (x)) ->= s (plus (square (x), double (x)))
plus (n, 0) ->= n
plus (n, s (m)) ->= s (plus (n, m)))
, property = Top_Termination , truth = Nothing
, transform = Split
{ interpretation = Interpretation
squareP |-> (x) |-> E^1x0 * x + (2)
leP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0)
doubleP |-> (x) |-> E^1x0 * x + (0)
plusP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (1)
0 |-> () |-> E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
false |-> () |-> E^0x1
square |-> (x) |-> E^0x0 * x + E^0x1
le |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
double |-> (x) |-> E^0x0 * x + E^0x1
true |-> () |-> E^0x1
log |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
plus |-> (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 = [ [ leP (s (u), s (v)) -> leP (u, v) , doubleP (s (x)) -> doubleP (x) ] , [ plusP (n, s (m)) -> plusP (n, m) ] , [ squareP (s (x)) -> squareP (x) ] ]
, comment = size 0 , heights ( 3 , 7 ) , time
split_dimension: 0
}
, to = [ Proof
{ system = (VAR l x
y v
u n
m)
(STRATEGY INNERMOST)
(RULES leP (s (u), s (v)) -> leP (u, v)
doubleP (s (x)) -> doubleP (x)
log (x, s (s (y))) ->= cond (le (x, s (s (y))), x, y)
cond (true, x, y) ->= s (0)
cond (false, x, y) ->= double (log (x, square (s (s (y)))))
le (0, v) ->= true
le (s (u), 0) ->= false
le (s (u), s (v)) ->= le (u, v)
double (0) ->= 0
double (s (x)) ->= s (s (double (x)))
square (0) ->= 0
square (s (x)) ->= s (plus (square (x), double (x)))
plus (n, 0) ->= n
plus (n, s (m)) ->= s (plus (n, m)))
, property = Top_Termination , truth = Nothing
, transform = Split
{ interpretation = Interpretation
leP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0)
doubleP |-> (x) |-> E^1x0 * x + (2)
0 |-> () |-> E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
false |-> () |-> E^0x1
square |-> (x) |-> E^0x0 * x + E^0x1
le |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
double |-> (x) |-> E^0x0 * x + E^0x1
true |-> () |-> E^0x1
log |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
plus |-> (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 = [ [ leP (s (u), s (v)) -> leP (u, v) ] , [ doubleP (s (x)) -> doubleP (x) ] ]
, comment = size 0 , heights ( 3 , 7 ) , time
split_dimension: 0
}
, to = [ Claim
{ system = (VAR l x
y v
u n
m)
(STRATEGY INNERMOST)
(RULES leP (s (u), s (v)) -> leP (u, v)
log (x, s (s (y))) ->= cond (le (x, s (s (y))), x, y)
cond (true, x, y) ->= s (0)
cond (false, x, y) ->= double (log (x, square (s (s (y)))))
le (0, v) ->= true
le (s (u), 0) ->= false
le (s (u), s (v)) ->= le (u, v)
double (0) ->= 0
double (s (x)) ->= s (s (double (x)))
square (0) ->= 0
square (s (x)) ->= s (plus (square (x), double (x)))
plus (n, 0) ->= n
plus (n, s (m)) ->= s (plus (n, m)))
, property = Top_Termination
}
, Claim
{ system = (VAR l x
y v
u n
m)
(STRATEGY INNERMOST)
(RULES doubleP (s (x)) -> doubleP (x)
log (x, s (s (y))) ->= cond (le (x, s (s (y))), x, y)
cond (true, x, y) ->= s (0)
cond (false, x, y) ->= double (log (x, square (s (s (y)))))
le (0, v) ->= true
le (s (u), 0) ->= false
le (s (u), s (v)) ->= le (u, v)
double (0) ->= 0
double (s (x)) ->= s (s (double (x)))
square (0) ->= 0
square (s (x)) ->= s (plus (square (x), double (x)))
plus (n, 0) ->= n
plus (n, s (m)) ->= s (plus (n, m)))
, property = Top_Termination
}
]
}
, Claim
{ system = (VAR l x
y v
u n
m)
(STRATEGY INNERMOST)
(RULES plusP (n, s (m)) -> plusP (n, m)
log (x, s (s (y))) ->= cond (le (x, s (s (y))), x, y)
cond (true, x, y) ->= s (0)
cond (false, x, y) ->= double (log (x, square (s (s (y)))))
le (0, v) ->= true
le (s (u), 0) ->= false
le (s (u), s (v)) ->= le (u, v)
double (0) ->= 0
double (s (x)) ->= s (s (double (x)))
square (0) ->= 0
square (s (x)) ->= s (plus (square (x), double (x)))
plus (n, 0) ->= n
plus (n, s (m)) ->= s (plus (n, m)))
, property = Top_Termination
}
, Claim
{ system = (VAR l x
y v
u n
m)
(STRATEGY INNERMOST)
(RULES squareP (s (x)) -> squareP (x)
log (x, s (s (y))) ->= cond (le (x, s (s (y))), x, y)
cond (true, x, y) ->= s (0)
cond (false, x, y) ->= double (log (x, square (s (s (y)))))
le (0, v) ->= true
le (s (u), 0) ->= false
le (s (u), s (v)) ->= le (u, v)
double (0) ->= 0
double (s (x)) ->= s (s (double (x)))
square (0) ->= 0
square (s (x)) ->= s (plus (square (x), double (x)))
plus (n, 0) ->= n
plus (n, s (m)) ->= s (plus (n, m)))
, property = Top_Termination
}
]
}
]
}
, Claim
{ system = (VAR l x
y v
u n
m)
(STRATEGY INNERMOST)
(RULES logP (x, s (s (y))) -> condP (le (x, s (s (y))), x, y)
condP (false, x, y) -> logP (x, square (s (s (y))))
log (x, s (s (y))) ->= cond (le (x, s (s (y))), x, y)
cond (true, x, y) ->= s (0)
cond (false, x, y) ->= double (log (x, square (s (s (y)))))
le (0, v) ->= true
le (s (u), 0) ->= false
le (s (u), s (v)) ->= le (u, v)
double (0) ->= 0
double (s (x)) ->= s (s (double (x)))
square (0) ->= 0
square (s (x)) ->= s (plus (square (x), double (x)))
plus (n, 0) ->= n
plus (n, s (m)) ->= s (plus (n, m)))
, property = Top_Termination
}
]
}
]
}
]
}
]
}
Proof summary:
value Nothing
for property Termination
for system with 12 strict rules and 0 non-strict rules
follows by transformation
Ignore_Strategy
from
value Nothing
for property Termination
for system with 12 strict rules and 0 non-strict rules
follows by transformation
Dependency_Pair_Transformation
from
value Nothing
for property Top_Termination
for system with 11 strict rules and 12 non-strict rules
follows by transformation
Remove
{ interpretation = Interpretation
squareP |-> (x) |-> E^1x0 * x + (0)
leP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0)
doubleP |-> (x) |-> E^1x0 * x + (0)
plusP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0)
condP |-> (x , y , z) |-> E^1x0 * x + E^1x0 * y + E^1x0 * z + (1)
0 |-> () |-> E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
false |-> () |-> E^0x1
square |-> (x) |-> E^0x0 * x + E^0x1
le |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
logP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (1)
double |-> (x) |-> E^0x0 * x + E^0x1
true |-> () |-> E^0x1
log |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
plus |-> (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 = [ logP (x, s (s (y))) -> leP (x, s (s (y))) , condP (false, x, y) -> doubleP (log (x, square (s (s (y))))) , condP (false, x, y) -> squareP (s (s (y))) ] , comment = size 0 , heights ( 3 , 7 ) , time
}
from
value Nothing
for property Top_Termination
for system with 8 strict rules and 12 non-strict rules
follows by transformation
Split
{ interpretation = Interpretation
squareP |-> (x) |-> E^1x0 * x + (0)
leP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0)
doubleP |-> (x) |-> E^1x0 * x + (0)
plusP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0)
condP |-> (x , y , z) |-> E^1x0 * x + E^1x0 * y + E^1x0 * z + (1)
0 |-> () |-> E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
false |-> () |-> E^0x1
square |-> (x) |-> E^0x0 * x + E^0x1
le |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
logP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (1)
double |-> (x) |-> E^0x0 * x + E^0x1
true |-> () |-> E^0x1
log |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
plus |-> (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 = [ [ leP (s (u), s (v)) -> leP (u, v) , doubleP (s (x)) -> doubleP (x) , squareP (s (x)) -> plusP (square (x), double (x)) , squareP (s (x)) -> squareP (x) , squareP (s (x)) -> doubleP (x) , plusP (n, s (m)) -> plusP (n, m) ] , [ logP (x, s (s (y))) -> condP (le (x, s (s (y))), x, y) , condP (false, x, y) -> logP (x, square (s (s (y)))) ] ]
, comment = size 0 , heights ( 3 , 7 ) , time
split_dimension: 0
}
from
value Nothing
for property Top_Termination
for system with 6 strict rules and 12 non-strict rules
follows by transformation
Remove
{ interpretation = Interpretation
squareP |-> (x) |-> E^1x0 * x + (2)
leP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0)
doubleP |-> (x) |-> E^1x0 * x + (0)
plusP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (1)
0 |-> () |-> E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
false |-> () |-> E^0x1
square |-> (x) |-> E^0x0 * x + E^0x1
le |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
double |-> (x) |-> E^0x0 * x + E^0x1
true |-> () |-> E^0x1
log |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
plus |-> (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 = [ squareP (s (x)) -> plusP (square (x), double (x)) , squareP (s (x)) -> doubleP (x) ] , comment = size 0 , heights ( 3 , 7 ) , time
}
from
value Nothing
for property Top_Termination
for system with 4 strict rules and 12 non-strict rules
follows by transformation
Split
{ interpretation = Interpretation
squareP |-> (x) |-> E^1x0 * x + (2)
leP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0)
doubleP |-> (x) |-> E^1x0 * x + (0)
plusP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (1)
0 |-> () |-> E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
false |-> () |-> E^0x1
square |-> (x) |-> E^0x0 * x + E^0x1
le |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
double |-> (x) |-> E^0x0 * x + E^0x1
true |-> () |-> E^0x1
log |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
plus |-> (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 = [ [ leP (s (u), s (v)) -> leP (u, v) , doubleP (s (x)) -> doubleP (x) ] , [ plusP (n, s (m)) -> plusP (n, m) ] , [ squareP (s (x)) -> squareP (x) ] ]
, comment = size 0 , heights ( 3 , 7 ) , time
split_dimension: 0
}
from
value Nothing
for property Top_Termination
for system with 2 strict rules and 12 non-strict rules
follows by transformation
Split
{ interpretation = Interpretation
leP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0)
doubleP |-> (x) |-> E^1x0 * x + (2)
0 |-> () |-> E^0x1
s |-> (x) |-> E^0x0 * x + E^0x1
false |-> () |-> E^0x1
square |-> (x) |-> E^0x0 * x + E^0x1
le |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
double |-> (x) |-> E^0x0 * x + E^0x1
true |-> () |-> E^0x1
log |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1
plus |-> (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 = [ [ leP (s (u), s (v)) -> leP (u, v) ] , [ doubleP (s (x)) -> doubleP (x) ] ]
, comment = size 0 , heights ( 3 , 7 ) , time
split_dimension: 0
}
from
Claim
{ system = (VAR l x
y v
u n
m)
(STRATEGY INNERMOST)
(RULES leP (s (u), s (v)) -> leP (u, v)
log (x, s (s (y))) ->= cond (le (x, s (s (y))), x, y)
cond (true, x, y) ->= s (0)
cond (false, x, y) ->= double (log (x, square (s (s (y)))))
le (0, v) ->= true
le (s (u), 0) ->= false
le (s (u), s (v)) ->= le (u, v)
double (0) ->= 0
double (s (x)) ->= s (s (double (x)))
square (0) ->= 0
square (s (x)) ->= s (plus (square (x), double (x)))
plus (n, 0) ->= n
plus (n, s (m)) ->= s (plus (n, m)))
, property = Top_Termination
}
Claim
{ system = (VAR l x
y v
u n
m)
(STRATEGY INNERMOST)
(RULES doubleP (s (x)) -> doubleP (x)
log (x, s (s (y))) ->= cond (le (x, s (s (y))), x, y)
cond (true, x, y) ->= s (0)
cond (false, x, y) ->= double (log (x, square (s (s (y)))))
le (0, v) ->= true
le (s (u), 0) ->= false
le (s (u), s (v)) ->= le (u, v)
double (0) ->= 0
double (s (x)) ->= s (s (double (x)))
square (0) ->= 0
square (s (x)) ->= s (plus (square (x), double (x)))
plus (n, 0) ->= n
plus (n, s (m)) ->= s (plus (n, m)))
, property = Top_Termination
}
Claim
{ system = (VAR l x
y v
u n
m)
(STRATEGY INNERMOST)
(RULES plusP (n, s (m)) -> plusP (n, m)
log (x, s (s (y))) ->= cond (le (x, s (s (y))), x, y)
cond (true, x, y) ->= s (0)
cond (false, x, y) ->= double (log (x, square (s (s (y)))))
le (0, v) ->= true
le (s (u), 0) ->= false
le (s (u), s (v)) ->= le (u, v)
double (0) ->= 0
double (s (x)) ->= s (s (double (x)))
square (0) ->= 0
square (s (x)) ->= s (plus (square (x), double (x)))
plus (n, 0) ->= n
plus (n, s (m)) ->= s (plus (n, m)))
, property = Top_Termination
}
Claim
{ system = (VAR l x
y v
u n
m)
(STRATEGY INNERMOST)
(RULES squareP (s (x)) -> squareP (x)
log (x, s (s (y))) ->= cond (le (x, s (s (y))), x, y)
cond (true, x, y) ->= s (0)
cond (false, x, y) ->= double (log (x, square (s (s (y)))))
le (0, v) ->= true
le (s (u), 0) ->= false
le (s (u), s (v)) ->= le (u, v)
double (0) ->= 0
double (s (x)) ->= s (s (double (x)))
square (0) ->= 0
square (s (x)) ->= s (plus (square (x), double (x)))
plus (n, 0) ->= n
plus (n, s (m)) ->= s (plus (n, m)))
, property = Top_Termination
}
Claim
{ system = (VAR l x
y v
u n
m)
(STRATEGY INNERMOST)
(RULES logP (x, s (s (y))) -> condP (le (x, s (s (y))), x, y)
condP (false, x, y) -> logP (x, square (s (s (y))))
log (x, s (s (y))) ->= cond (le (x, s (s (y))), x, y)
cond (true, x, y) ->= s (0)
cond (false, x, y) ->= double (log (x, square (s (s (y)))))
le (0, v) ->= true
le (s (u), 0) ->= false
le (s (u), s (v)) ->= le (u, v)
double (0) ->= 0
double (s (x)) ->= s (s (double (x)))
square (0) ->= 0
square (s (x)) ->= s (plus (square (x), double (x)))
plus (n, 0) ->= n
plus (n, s (m)) ->= s (plus (n, m)))
, 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/tmpqXca0t/ex18.trs
started : Thu Feb 22 16:52:32 CET 2007
finished : Thu Feb 22 16:53:23 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: 51 secs