Tool CaT
stdout:
MAYBE
Problem:
and(tt(),tt()) -> tt()
is_nat(0()) -> tt()
is_nat(s(x)) -> is_nat(x)
is_natlist(nil()) -> tt()
is_natlist(cons(x,xs)) -> and(is_nat(x),is_natlist(xs))
from(x) -> fromCond(is_natlist(x),x)
fromCond(tt(),cons(x,xs)) -> from(cons(s(x),cons(x,xs)))
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ and(tt(), tt()) -> tt()
, is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, from(x) -> fromCond(is_natlist(x), x)
, fromCond(tt(), cons(x, xs)) -> from(cons(s(x), cons(x, xs)))}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: and^#(tt(), tt()) -> c_0()
, 2: is_nat^#(0()) -> c_1()
, 3: is_nat^#(s(x)) -> c_2(is_nat^#(x))
, 4: is_natlist^#(nil()) -> c_3()
, 5: is_natlist^#(cons(x, xs)) ->
c_4(and^#(is_nat(x), is_natlist(xs)))
, 6: from^#(x) -> c_5(fromCond^#(is_natlist(x), x))
, 7: fromCond^#(tt(), cons(x, xs)) ->
c_6(from^#(cons(s(x), cons(x, xs))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6,7} [ NA ]
->{5} [ YES(?,O(n^3)) ]
|
`->{1} [ YES(?,O(n^3)) ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(n^2)) ]
|
`->{2} [ YES(?,O(n^2)) ]
Sub-problems:
-------------
* Path {3}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {}, Uargs(is_nat^#) = {},
Uargs(c_2) = {1}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
tt() = [0]
[0]
[0]
is_nat(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
is_natlist(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fromCond(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
is_nat^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
is_natlist^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fromCond^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {is_nat^#(s(x)) -> c_2(is_nat^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(is_nat^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 1 2] [2]
[0 0 0] [0]
is_nat^#(x1) = [0 1 0] x1 + [2]
[6 0 0] [0]
[2 3 0] [2]
c_2(x1) = [1 0 0] x1 + [1]
[2 0 2] [0]
[0 0 0] [0]
* Path {3}->{2}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {}, Uargs(is_nat^#) = {},
Uargs(c_2) = {1}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
tt() = [0]
[0]
[0]
is_nat(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
is_natlist(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fromCond(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
is_nat^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
is_natlist^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fromCond^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {is_nat^#(0()) -> c_1()}
Weak Rules: {is_nat^#(s(x)) -> c_2(is_nat^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(is_nat^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 1 0] x1 + [0]
[0 1 1] [1]
[0 0 0] [0]
is_nat^#(x1) = [2 2 2] x1 + [0]
[0 6 0] [0]
[0 0 2] [0]
c_1() = [1]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [2]
[0 0 0] [3]
[0 0 0] [0]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {}, Uargs(is_nat^#) = {},
Uargs(c_2) = {}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
tt() = [0]
[0]
[0]
is_nat(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
is_natlist(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fromCond(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
is_nat^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
is_natlist^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fromCond^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {is_natlist^#(nil()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(is_natlist^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
[2]
is_natlist^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_3() = [0]
[1]
[1]
* Path {5}: YES(?,O(n^3))
-----------------------
The usable rules for this path are:
{ is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {1, 2}, Uargs(is_nat^#) = {},
Uargs(c_2) = {}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {1},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [2 0 0] x1 + [1 2 1] x2 + [0]
[0 0 0] [0 1 0] [0]
[0 0 0] [0 0 0] [0]
tt() = [0]
[1]
[0]
is_nat(x1) = [1 0 0] x1 + [1]
[0 0 0] [1]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [2]
[0 0 0] [0]
[0 0 0] [0]
is_natlist(x1) = [2 0 2] x1 + [0]
[0 1 0] [0]
[0 1 2] [0]
nil() = [1]
[2]
[0]
cons(x1, x2) = [1 0 0] x1 + [1 2 2] x2 + [0]
[0 0 0] [0 1 0] [0]
[0 0 0] [0 0 0] [2]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fromCond(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
and^#(x1, x2) = [1 0 0] x1 + [1 0 1] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_0() = [0]
[0]
[0]
is_nat^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
is_natlist^#(x1) = [2 1 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fromCond^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{is_natlist^#(cons(x, xs)) ->
c_4(and^#(is_nat(x), is_natlist(xs)))}
Weak Rules:
{ is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()}
Proof Output:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(and^#) = {},
Uargs(is_natlist^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [4]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
tt() = [0]
[0]
[0]
is_nat(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 2 0] [0]
0() = [0]
[2]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
is_natlist(x1) = [1 2 2] x1 + [6]
[0 0 0] [4]
[1 0 2] [6]
nil() = [0]
[2]
[2]
cons(x1, x2) = [1 2 0] x1 + [1 2 0] x2 + [2]
[0 1 2] [0 0 0] [0]
[0 0 0] [0 0 1] [0]
and^#(x1, x2) = [2 2 0] x1 + [0 0 0] x2 + [2]
[0 2 3] [0 0 1] [0]
[0 0 0] [0 0 0] [0]
is_natlist^#(x1) = [4 0 4] x1 + [7]
[4 4 4] [7]
[2 4 0] [3]
c_4(x1) = [2 0 0] x1 + [5]
[0 2 0] [3]
[0 0 0] [3]
* Path {5}->{1}: YES(?,O(n^3))
----------------------------
The usable rules for this path are:
{ is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {1, 2}, Uargs(is_nat^#) = {},
Uargs(c_2) = {}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {1},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [2 0 0] x1 + [1 0 0] x2 + [1]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
tt() = [0]
[0]
[0]
is_nat(x1) = [1 0 0] x1 + [1]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [2]
[0 0 0] [0]
[0 0 0] [0]
is_natlist(x1) = [2 3 1] x1 + [0]
[3 1 0] [0]
[2 2 0] [0]
nil() = [2]
[0]
[0]
cons(x1, x2) = [1 2 2] x1 + [1 3 0] x2 + [2]
[0 0 2] [0 1 0] [3]
[0 0 0] [0 0 1] [3]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fromCond(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
and^#(x1, x2) = [3 0 0] x1 + [3 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
is_nat^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
is_natlist^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fromCond^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^3))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {and^#(tt(), tt()) -> c_0()}
Weak Rules:
{ is_natlist^#(cons(x, xs)) ->
c_4(and^#(is_nat(x), is_natlist(xs)))
, is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()}
Proof Output:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(and^#) = {},
Uargs(is_natlist^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 2 0] x1 + [0 2 0] x2 + [0]
[2 0 2] [0 0 0] [0]
[2 2 2] [0 0 0] [0]
tt() = [2]
[2]
[2]
is_nat(x1) = [0 0 1] x1 + [0]
[0 0 0] [4]
[0 0 1] [0]
0() = [0]
[0]
[4]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 1] [0]
is_natlist(x1) = [6 1 0] x1 + [0]
[0 4 0] [0]
[4 0 4] [0]
nil() = [2]
[2]
[0]
cons(x1, x2) = [1 0 1] x1 + [1 2 0] x2 + [2]
[0 1 2] [0 1 2] [0]
[0 0 1] [0 0 1] [1]
and^#(x1, x2) = [2 2 2] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[2 0 1] [0 0 0] [0]
c_0() = [1]
[0]
[0]
is_natlist^#(x1) = [4 1 6] x1 + [0]
[0 4 4] [6]
[4 1 6] [1]
c_4(x1) = [1 0 2] x1 + [3]
[0 0 0] [3]
[0 0 0] [7]
* Path {6,7}: NA
--------------
The usable rules for this path are:
{ is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()
, is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: and^#(tt(), tt()) -> c_0()
, 2: is_nat^#(0()) -> c_1()
, 3: is_nat^#(s(x)) -> c_2(is_nat^#(x))
, 4: is_natlist^#(nil()) -> c_3()
, 5: is_natlist^#(cons(x, xs)) ->
c_4(and^#(is_nat(x), is_natlist(xs)))
, 6: from^#(x) -> c_5(fromCond^#(is_natlist(x), x))
, 7: fromCond^#(tt(), cons(x, xs)) ->
c_6(from^#(cons(s(x), cons(x, xs))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6,7} [ MAYBE ]
->{5} [ YES(?,O(n^2)) ]
|
`->{1} [ YES(?,O(n^2)) ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(n^1)) ]
|
`->{2} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {3}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {}, Uargs(is_nat^#) = {},
Uargs(c_2) = {1}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
is_nat(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
is_natlist(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
fromCond(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
is_nat^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
is_natlist^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
fromCond^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {is_nat^#(s(x)) -> c_2(is_nat^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(is_nat^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
is_nat^#(x1) = [0 1] x1 + [1]
[0 0] [0]
c_2(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {3}->{2}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {}, Uargs(is_nat^#) = {},
Uargs(c_2) = {1}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
is_nat(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
is_natlist(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
fromCond(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
is_nat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
is_natlist^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
fromCond^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {is_nat^#(0()) -> c_1()}
Weak Rules: {is_nat^#(s(x)) -> c_2(is_nat^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(is_nat^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [1]
[0 0] [3]
is_nat^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_1() = [1]
[0]
c_2(x1) = [1 0] x1 + [5]
[2 0] [3]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {}, Uargs(is_nat^#) = {},
Uargs(c_2) = {}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
is_nat(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
is_natlist(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
fromCond(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
is_nat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
is_natlist^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
fromCond^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {is_natlist^#(nil()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(is_natlist^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
is_natlist^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_3() = [0]
[1]
* Path {5}: YES(?,O(n^2))
-----------------------
The usable rules for this path are:
{ is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {1, 2}, Uargs(is_nat^#) = {},
Uargs(c_2) = {}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {1},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 2] x2 + [0]
[0 0] [0 0] [1]
tt() = [0]
[1]
is_nat(x1) = [2 0] x1 + [1]
[0 0] [2]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
is_natlist(x1) = [3 2] x1 + [0]
[0 1] [0]
nil() = [0]
[2]
cons(x1, x2) = [1 2] x1 + [1 1] x2 + [0]
[0 1] [0 1] [2]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
fromCond(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
is_nat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
is_natlist^#(x1) = [3 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
fromCond^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{is_natlist^#(cons(x, xs)) ->
c_4(and^#(is_nat(x), is_natlist(xs)))}
Weak Rules:
{ is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()}
Proof Output:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(and^#) = {},
Uargs(is_natlist^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
is_nat(x1) = [0 0] x1 + [0]
[2 0] [0]
0() = [4]
[0]
s(x1) = [1 2] x1 + [2]
[0 0] [0]
is_natlist(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 0] [0 1] [2]
and^#(x1, x2) = [2 0] x1 + [0 2] x2 + [2]
[0 0] [0 0] [0]
is_natlist^#(x1) = [0 4] x1 + [7]
[2 2] [7]
c_4(x1) = [2 0] x1 + [7]
[2 0] [7]
* Path {5}->{1}: YES(?,O(n^2))
----------------------------
The usable rules for this path are:
{ is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {1, 2}, Uargs(is_nat^#) = {},
Uargs(c_2) = {}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {1},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 1] x2 + [0]
[0 0] [0 1] [0]
tt() = [0]
[2]
is_nat(x1) = [2 0] x1 + [1]
[0 0] [2]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
is_natlist(x1) = [3 0] x1 + [0]
[0 1] [0]
nil() = [3]
[2]
cons(x1, x2) = [1 0] x1 + [1 3] x2 + [2]
[0 1] [0 1] [0]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
fromCond(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
is_nat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
is_natlist^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
fromCond^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {and^#(tt(), tt()) -> c_0()}
Weak Rules:
{ is_natlist^#(cons(x, xs)) ->
c_4(and^#(is_nat(x), is_natlist(xs)))
, is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()}
Proof Output:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(and^#) = {},
Uargs(is_natlist^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [2 2] x1 + [0 0] x2 + [0]
[0 4] [0 0] [0]
tt() = [2]
[2]
is_nat(x1) = [0 2] x1 + [0]
[0 2] [0]
0() = [0]
[2]
s(x1) = [1 0] x1 + [0]
[0 1] [2]
is_natlist(x1) = [4 2] x1 + [0]
[4 0] [2]
nil() = [0]
[2]
cons(x1, x2) = [1 2] x1 + [1 1] x2 + [2]
[0 0] [0 0] [2]
and^#(x1, x2) = [2 0] x1 + [0 0] x2 + [0]
[0 2] [2 0] [0]
c_0() = [1]
[0]
is_natlist^#(x1) = [4 0] x1 + [6]
[2 0] [3]
c_4(x1) = [2 0] x1 + [7]
[0 0] [3]
* Path {6,7}: MAYBE
-----------------
The usable rules for this path are:
{ is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()
, is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ from^#(x) -> c_5(fromCond^#(is_natlist(x), x))
, fromCond^#(tt(), cons(x, xs)) ->
c_6(from^#(cons(s(x), cons(x, xs))))
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()
, is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)}
Proof Output:
The input cannot be shown compatible
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: and^#(tt(), tt()) -> c_0()
, 2: is_nat^#(0()) -> c_1()
, 3: is_nat^#(s(x)) -> c_2(is_nat^#(x))
, 4: is_natlist^#(nil()) -> c_3()
, 5: is_natlist^#(cons(x, xs)) ->
c_4(and^#(is_nat(x), is_natlist(xs)))
, 6: from^#(x) -> c_5(fromCond^#(is_natlist(x), x))
, 7: fromCond^#(tt(), cons(x, xs)) ->
c_6(from^#(cons(s(x), cons(x, xs))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6,7} [ MAYBE ]
->{5} [ YES(?,O(n^1)) ]
|
`->{1} [ NA ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(n^1)) ]
|
`->{2} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {3}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {}, Uargs(is_nat^#) = {},
Uargs(c_2) = {1}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
is_nat(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
is_natlist(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
fromCond(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
is_nat^#(x1) = [3] x1 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
is_natlist^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
fromCond^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {is_nat^#(s(x)) -> c_2(is_nat^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(is_nat^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
is_nat^#(x1) = [2] x1 + [0]
c_2(x1) = [1] x1 + [7]
* Path {3}->{2}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {}, Uargs(is_nat^#) = {},
Uargs(c_2) = {1}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
is_nat(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
is_natlist(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
fromCond(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
is_nat^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
is_natlist^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
fromCond^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {is_nat^#(0()) -> c_1()}
Weak Rules: {is_nat^#(s(x)) -> c_2(is_nat^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(is_nat^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [0]
is_nat^#(x1) = [2] x1 + [0]
c_1() = [1]
c_2(x1) = [1] x1 + [0]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {}, Uargs(is_nat^#) = {},
Uargs(c_2) = {}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
is_nat(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
is_natlist(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
fromCond(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
is_nat^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
is_natlist^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
fromCond^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {is_natlist^#(nil()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(is_natlist^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [7]
is_natlist^#(x1) = [1] x1 + [7]
c_3() = [1]
* Path {5}: YES(?,O(n^1))
-----------------------
The usable rules for this path are:
{ is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {1, 2}, Uargs(is_nat^#) = {},
Uargs(c_2) = {}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {1},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [2] x1 + [1] x2 + [0]
tt() = [3]
is_nat(x1) = [1] x1 + [1]
0() = [3]
s(x1) = [1] x1 + [3]
is_natlist(x1) = [2] x1 + [0]
nil() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
from(x1) = [0] x1 + [0]
fromCond(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [3] x1 + [1] x2 + [0]
c_0() = [0]
is_nat^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
is_natlist^#(x1) = [3] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
fromCond^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{is_natlist^#(cons(x, xs)) ->
c_4(and^#(is_nat(x), is_natlist(xs)))}
Weak Rules:
{ is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()}
Proof Output:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(and^#) = {},
Uargs(is_natlist^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [2] x1 + [0] x2 + [4]
tt() = [2]
is_nat(x1) = [0] x1 + [2]
0() = [0]
s(x1) = [1] x1 + [0]
is_natlist(x1) = [1] x1 + [4]
nil() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [4]
and^#(x1, x2) = [2] x1 + [0] x2 + [0]
is_natlist^#(x1) = [2] x1 + [7]
c_4(x1) = [2] x1 + [3]
* Path {5}->{1}: NA
-----------------
The usable rules for this path are:
{ is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {1, 2}, Uargs(is_nat^#) = {},
Uargs(c_2) = {}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {1},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [2]
is_nat(x1) = [1] x1 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
is_natlist(x1) = [1] x1 + [2]
nil() = [1]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
from(x1) = [0] x1 + [0]
fromCond(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
is_nat^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
is_natlist^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
fromCond^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {6,7}: MAYBE
-----------------
The usable rules for this path are:
{ is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()
, is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ from^#(x) -> c_5(fromCond^#(is_natlist(x), x))
, fromCond^#(tt(), cons(x, xs)) ->
c_6(from^#(cons(s(x), cons(x, xs))))
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()
, is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)}
Proof Output:
The input cannot be shown compatible
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool RC1
stdout:
MAYBE
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ and(tt(), tt()) -> tt()
, is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, from(x) -> fromCond(is_natlist(x), x)
, fromCond(tt(), cons(x, xs)) -> from(cons(s(x), cons(x, xs)))}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: and^#(tt(), tt()) -> c_0()
, 2: is_nat^#(0()) -> c_1()
, 3: is_nat^#(s(x)) -> c_2(is_nat^#(x))
, 4: is_natlist^#(nil()) -> c_3()
, 5: is_natlist^#(cons(x, xs)) ->
c_4(and^#(is_nat(x), is_natlist(xs)))
, 6: from^#(x) -> c_5(fromCond^#(is_natlist(x), x))
, 7: fromCond^#(tt(), cons(x, xs)) ->
c_6(from^#(cons(s(x), cons(x, xs))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6,7} [ NA ]
->{5} [ YES(?,O(n^3)) ]
|
`->{1} [ YES(?,O(n^3)) ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(n^2)) ]
|
`->{2} [ YES(?,O(n^2)) ]
Sub-problems:
-------------
* Path {3}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {}, Uargs(is_nat^#) = {},
Uargs(c_2) = {1}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
tt() = [0]
[0]
[0]
is_nat(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
is_natlist(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fromCond(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
is_nat^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
is_natlist^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fromCond^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {is_nat^#(s(x)) -> c_2(is_nat^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(is_nat^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 1 2] [2]
[0 0 0] [0]
is_nat^#(x1) = [0 1 0] x1 + [2]
[6 0 0] [0]
[2 3 0] [2]
c_2(x1) = [1 0 0] x1 + [1]
[2 0 2] [0]
[0 0 0] [0]
* Path {3}->{2}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {}, Uargs(is_nat^#) = {},
Uargs(c_2) = {1}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
tt() = [0]
[0]
[0]
is_nat(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
is_natlist(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fromCond(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
is_nat^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
is_natlist^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fromCond^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {is_nat^#(0()) -> c_1()}
Weak Rules: {is_nat^#(s(x)) -> c_2(is_nat^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(is_nat^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 1 0] x1 + [0]
[0 1 1] [1]
[0 0 0] [0]
is_nat^#(x1) = [2 2 2] x1 + [0]
[0 6 0] [0]
[0 0 2] [0]
c_1() = [1]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [2]
[0 0 0] [3]
[0 0 0] [0]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {}, Uargs(is_nat^#) = {},
Uargs(c_2) = {}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
tt() = [0]
[0]
[0]
is_nat(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
is_natlist(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fromCond(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
is_nat^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
is_natlist^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fromCond^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {is_natlist^#(nil()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(is_natlist^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
[2]
is_natlist^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_3() = [0]
[1]
[1]
* Path {5}: YES(?,O(n^3))
-----------------------
The usable rules for this path are:
{ is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {1, 2}, Uargs(is_nat^#) = {},
Uargs(c_2) = {}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {1},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [2 0 0] x1 + [1 2 1] x2 + [0]
[0 0 0] [0 1 0] [0]
[0 0 0] [0 0 0] [0]
tt() = [0]
[1]
[0]
is_nat(x1) = [1 0 0] x1 + [1]
[0 0 0] [1]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [2]
[0 0 0] [0]
[0 0 0] [0]
is_natlist(x1) = [2 0 2] x1 + [0]
[0 1 0] [0]
[0 1 2] [0]
nil() = [1]
[2]
[0]
cons(x1, x2) = [1 0 0] x1 + [1 2 2] x2 + [0]
[0 0 0] [0 1 0] [0]
[0 0 0] [0 0 0] [2]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fromCond(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
and^#(x1, x2) = [1 0 0] x1 + [1 0 1] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_0() = [0]
[0]
[0]
is_nat^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
is_natlist^#(x1) = [2 1 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fromCond^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{is_natlist^#(cons(x, xs)) ->
c_4(and^#(is_nat(x), is_natlist(xs)))}
Weak Rules:
{ is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()}
Proof Output:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(and^#) = {},
Uargs(is_natlist^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [4]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
tt() = [0]
[0]
[0]
is_nat(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 2 0] [0]
0() = [0]
[2]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
is_natlist(x1) = [1 2 2] x1 + [6]
[0 0 0] [4]
[1 0 2] [6]
nil() = [0]
[2]
[2]
cons(x1, x2) = [1 2 0] x1 + [1 2 0] x2 + [2]
[0 1 2] [0 0 0] [0]
[0 0 0] [0 0 1] [0]
and^#(x1, x2) = [2 2 0] x1 + [0 0 0] x2 + [2]
[0 2 3] [0 0 1] [0]
[0 0 0] [0 0 0] [0]
is_natlist^#(x1) = [4 0 4] x1 + [7]
[4 4 4] [7]
[2 4 0] [3]
c_4(x1) = [2 0 0] x1 + [5]
[0 2 0] [3]
[0 0 0] [3]
* Path {5}->{1}: YES(?,O(n^3))
----------------------------
The usable rules for this path are:
{ is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {1, 2}, Uargs(is_nat^#) = {},
Uargs(c_2) = {}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {1},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [2 0 0] x1 + [1 0 0] x2 + [1]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
tt() = [0]
[0]
[0]
is_nat(x1) = [1 0 0] x1 + [1]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [2]
[0 0 0] [0]
[0 0 0] [0]
is_natlist(x1) = [2 3 1] x1 + [0]
[3 1 0] [0]
[2 2 0] [0]
nil() = [2]
[0]
[0]
cons(x1, x2) = [1 2 2] x1 + [1 3 0] x2 + [2]
[0 0 2] [0 1 0] [3]
[0 0 0] [0 0 1] [3]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fromCond(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
and^#(x1, x2) = [3 0 0] x1 + [3 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
is_nat^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
is_natlist^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fromCond^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^3))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {and^#(tt(), tt()) -> c_0()}
Weak Rules:
{ is_natlist^#(cons(x, xs)) ->
c_4(and^#(is_nat(x), is_natlist(xs)))
, is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()}
Proof Output:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(and^#) = {},
Uargs(is_natlist^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 2 0] x1 + [0 2 0] x2 + [0]
[2 0 2] [0 0 0] [0]
[2 2 2] [0 0 0] [0]
tt() = [2]
[2]
[2]
is_nat(x1) = [0 0 1] x1 + [0]
[0 0 0] [4]
[0 0 1] [0]
0() = [0]
[0]
[4]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 1] [0]
is_natlist(x1) = [6 1 0] x1 + [0]
[0 4 0] [0]
[4 0 4] [0]
nil() = [2]
[2]
[0]
cons(x1, x2) = [1 0 1] x1 + [1 2 0] x2 + [2]
[0 1 2] [0 1 2] [0]
[0 0 1] [0 0 1] [1]
and^#(x1, x2) = [2 2 2] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[2 0 1] [0 0 0] [0]
c_0() = [1]
[0]
[0]
is_natlist^#(x1) = [4 1 6] x1 + [0]
[0 4 4] [6]
[4 1 6] [1]
c_4(x1) = [1 0 2] x1 + [3]
[0 0 0] [3]
[0 0 0] [7]
* Path {6,7}: NA
--------------
The usable rules for this path are:
{ is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()
, is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: and^#(tt(), tt()) -> c_0()
, 2: is_nat^#(0()) -> c_1()
, 3: is_nat^#(s(x)) -> c_2(is_nat^#(x))
, 4: is_natlist^#(nil()) -> c_3()
, 5: is_natlist^#(cons(x, xs)) ->
c_4(and^#(is_nat(x), is_natlist(xs)))
, 6: from^#(x) -> c_5(fromCond^#(is_natlist(x), x))
, 7: fromCond^#(tt(), cons(x, xs)) ->
c_6(from^#(cons(s(x), cons(x, xs))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6,7} [ MAYBE ]
->{5} [ YES(?,O(n^2)) ]
|
`->{1} [ YES(?,O(n^2)) ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(n^1)) ]
|
`->{2} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {3}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {}, Uargs(is_nat^#) = {},
Uargs(c_2) = {1}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
is_nat(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
is_natlist(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
fromCond(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
is_nat^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
is_natlist^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
fromCond^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {is_nat^#(s(x)) -> c_2(is_nat^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(is_nat^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
is_nat^#(x1) = [0 1] x1 + [1]
[0 0] [0]
c_2(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {3}->{2}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {}, Uargs(is_nat^#) = {},
Uargs(c_2) = {1}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
is_nat(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
is_natlist(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
fromCond(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
is_nat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
is_natlist^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
fromCond^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {is_nat^#(0()) -> c_1()}
Weak Rules: {is_nat^#(s(x)) -> c_2(is_nat^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(is_nat^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [1]
[0 0] [3]
is_nat^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_1() = [1]
[0]
c_2(x1) = [1 0] x1 + [5]
[2 0] [3]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {}, Uargs(is_nat^#) = {},
Uargs(c_2) = {}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
is_nat(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
is_natlist(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
fromCond(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
is_nat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
is_natlist^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
fromCond^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {is_natlist^#(nil()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(is_natlist^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
is_natlist^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_3() = [0]
[1]
* Path {5}: YES(?,O(n^2))
-----------------------
The usable rules for this path are:
{ is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {1, 2}, Uargs(is_nat^#) = {},
Uargs(c_2) = {}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {1},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 2] x2 + [0]
[0 0] [0 0] [1]
tt() = [0]
[1]
is_nat(x1) = [2 0] x1 + [1]
[0 0] [2]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
is_natlist(x1) = [3 2] x1 + [0]
[0 1] [0]
nil() = [0]
[2]
cons(x1, x2) = [1 2] x1 + [1 1] x2 + [0]
[0 1] [0 1] [2]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
fromCond(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
is_nat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
is_natlist^#(x1) = [3 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
fromCond^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{is_natlist^#(cons(x, xs)) ->
c_4(and^#(is_nat(x), is_natlist(xs)))}
Weak Rules:
{ is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()}
Proof Output:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(and^#) = {},
Uargs(is_natlist^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
is_nat(x1) = [0 0] x1 + [0]
[2 0] [0]
0() = [4]
[0]
s(x1) = [1 2] x1 + [2]
[0 0] [0]
is_natlist(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 0] [0 1] [2]
and^#(x1, x2) = [2 0] x1 + [0 2] x2 + [2]
[0 0] [0 0] [0]
is_natlist^#(x1) = [0 4] x1 + [7]
[2 2] [7]
c_4(x1) = [2 0] x1 + [7]
[2 0] [7]
* Path {5}->{1}: YES(?,O(n^2))
----------------------------
The usable rules for this path are:
{ is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {1, 2}, Uargs(is_nat^#) = {},
Uargs(c_2) = {}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {1},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1 0] x1 + [1 1] x2 + [0]
[0 0] [0 1] [0]
tt() = [0]
[2]
is_nat(x1) = [2 0] x1 + [1]
[0 0] [2]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
is_natlist(x1) = [3 0] x1 + [0]
[0 1] [0]
nil() = [3]
[2]
cons(x1, x2) = [1 0] x1 + [1 3] x2 + [2]
[0 1] [0 1] [0]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
fromCond(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
is_nat^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
is_natlist^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
fromCond^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {and^#(tt(), tt()) -> c_0()}
Weak Rules:
{ is_natlist^#(cons(x, xs)) ->
c_4(and^#(is_nat(x), is_natlist(xs)))
, is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()}
Proof Output:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(and^#) = {},
Uargs(is_natlist^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [2 2] x1 + [0 0] x2 + [0]
[0 4] [0 0] [0]
tt() = [2]
[2]
is_nat(x1) = [0 2] x1 + [0]
[0 2] [0]
0() = [0]
[2]
s(x1) = [1 0] x1 + [0]
[0 1] [2]
is_natlist(x1) = [4 2] x1 + [0]
[4 0] [2]
nil() = [0]
[2]
cons(x1, x2) = [1 2] x1 + [1 1] x2 + [2]
[0 0] [0 0] [2]
and^#(x1, x2) = [2 0] x1 + [0 0] x2 + [0]
[0 2] [2 0] [0]
c_0() = [1]
[0]
is_natlist^#(x1) = [4 0] x1 + [6]
[2 0] [3]
c_4(x1) = [2 0] x1 + [7]
[0 0] [3]
* Path {6,7}: MAYBE
-----------------
The usable rules for this path are:
{ is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()
, is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ from^#(x) -> c_5(fromCond^#(is_natlist(x), x))
, fromCond^#(tt(), cons(x, xs)) ->
c_6(from^#(cons(s(x), cons(x, xs))))
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()
, is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)}
Proof Output:
The input cannot be shown compatible
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: and^#(tt(), tt()) -> c_0()
, 2: is_nat^#(0()) -> c_1()
, 3: is_nat^#(s(x)) -> c_2(is_nat^#(x))
, 4: is_natlist^#(nil()) -> c_3()
, 5: is_natlist^#(cons(x, xs)) ->
c_4(and^#(is_nat(x), is_natlist(xs)))
, 6: from^#(x) -> c_5(fromCond^#(is_natlist(x), x))
, 7: fromCond^#(tt(), cons(x, xs)) ->
c_6(from^#(cons(s(x), cons(x, xs))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6,7} [ MAYBE ]
->{5} [ YES(?,O(n^1)) ]
|
`->{1} [ NA ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(n^1)) ]
|
`->{2} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {3}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {}, Uargs(is_nat^#) = {},
Uargs(c_2) = {1}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
is_nat(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
is_natlist(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
fromCond(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
is_nat^#(x1) = [3] x1 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
is_natlist^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
fromCond^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {is_nat^#(s(x)) -> c_2(is_nat^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(is_nat^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
is_nat^#(x1) = [2] x1 + [0]
c_2(x1) = [1] x1 + [7]
* Path {3}->{2}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {}, Uargs(is_nat^#) = {},
Uargs(c_2) = {1}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
is_nat(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
is_natlist(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
fromCond(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
is_nat^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
is_natlist^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
fromCond^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {is_nat^#(0()) -> c_1()}
Weak Rules: {is_nat^#(s(x)) -> c_2(is_nat^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(is_nat^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [0]
is_nat^#(x1) = [2] x1 + [0]
c_1() = [1]
c_2(x1) = [1] x1 + [0]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {}, Uargs(is_nat^#) = {},
Uargs(c_2) = {}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
is_nat(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
is_natlist(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
fromCond(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
is_nat^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
is_natlist^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
fromCond^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {is_natlist^#(nil()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(is_natlist^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [7]
is_natlist^#(x1) = [1] x1 + [7]
c_3() = [1]
* Path {5}: YES(?,O(n^1))
-----------------------
The usable rules for this path are:
{ is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {1, 2}, Uargs(is_nat^#) = {},
Uargs(c_2) = {}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {1},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [2] x1 + [1] x2 + [0]
tt() = [3]
is_nat(x1) = [1] x1 + [1]
0() = [3]
s(x1) = [1] x1 + [3]
is_natlist(x1) = [2] x1 + [0]
nil() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
from(x1) = [0] x1 + [0]
fromCond(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [3] x1 + [1] x2 + [0]
c_0() = [0]
is_nat^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
is_natlist^#(x1) = [3] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
fromCond^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{is_natlist^#(cons(x, xs)) ->
c_4(and^#(is_nat(x), is_natlist(xs)))}
Weak Rules:
{ is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()}
Proof Output:
The following argument positions are usable:
Uargs(and) = {}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(and^#) = {},
Uargs(is_natlist^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [2] x1 + [0] x2 + [4]
tt() = [2]
is_nat(x1) = [0] x1 + [2]
0() = [0]
s(x1) = [1] x1 + [0]
is_natlist(x1) = [1] x1 + [4]
nil() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [4]
and^#(x1, x2) = [2] x1 + [0] x2 + [0]
is_natlist^#(x1) = [2] x1 + [7]
c_4(x1) = [2] x1 + [3]
* Path {5}->{1}: NA
-----------------
The usable rules for this path are:
{ is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(is_nat) = {}, Uargs(s) = {},
Uargs(is_natlist) = {}, Uargs(cons) = {}, Uargs(from) = {},
Uargs(fromCond) = {}, Uargs(and^#) = {1, 2}, Uargs(is_nat^#) = {},
Uargs(c_2) = {}, Uargs(is_natlist^#) = {}, Uargs(c_4) = {1},
Uargs(from^#) = {}, Uargs(c_5) = {}, Uargs(fromCond^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [2]
is_nat(x1) = [1] x1 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
is_natlist(x1) = [1] x1 + [2]
nil() = [1]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
from(x1) = [0] x1 + [0]
fromCond(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
is_nat^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
is_natlist^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
fromCond^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {6,7}: MAYBE
-----------------
The usable rules for this path are:
{ is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()
, is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ from^#(x) -> c_5(fromCond^#(is_natlist(x), x))
, fromCond^#(tt(), cons(x, xs)) ->
c_6(from^#(cons(s(x), cons(x, xs))))
, is_natlist(nil()) -> tt()
, is_natlist(cons(x, xs)) -> and(is_nat(x), is_natlist(xs))
, and(tt(), tt()) -> tt()
, is_nat(0()) -> tt()
, is_nat(s(x)) -> is_nat(x)}
Proof Output:
The input cannot be shown compatible
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.