Tool CaT
stdout:
MAYBE
Problem:
is_empty(nil()) -> true()
is_empty(cons(x,l)) -> false()
hd(cons(x,l)) -> x
tl(cons(x,l)) -> cons(x,l)
append(l1,l2) -> ifappend(l1,l2,is_empty(l1))
ifappend(l1,l2,true()) -> l2
ifappend(l1,l2,false()) -> cons(hd(l1),append(tl(l1),l2))
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
TIMEOUT
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: TIMEOUT
Input Problem: innermost runtime-complexity with respect to
Rules:
{ is_empty(nil()) -> true()
, is_empty(cons(x, l)) -> false()
, hd(cons(x, l)) -> x
, tl(cons(x, l)) -> cons(x, l)
, append(l1, l2) -> ifappend(l1, l2, is_empty(l1))
, ifappend(l1, l2, true()) -> l2
, ifappend(l1, l2, false()) -> cons(hd(l1), append(tl(l1), l2))}
Proof Output:
Computation stopped due to timeout after 60.0 secondsTool RC1
stdout:
MAYBE
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ is_empty(nil()) -> true()
, is_empty(cons(x, l)) -> false()
, hd(cons(x, l)) -> x
, tl(cons(x, l)) -> cons(x, l)
, append(l1, l2) -> ifappend(l1, l2, is_empty(l1))
, ifappend(l1, l2, true()) -> l2
, ifappend(l1, l2, false()) -> cons(hd(l1), append(tl(l1), l2))}
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: is_empty^#(nil()) -> c_0()
, 2: is_empty^#(cons(x, l)) -> c_1()
, 3: hd^#(cons(x, l)) -> c_2(x)
, 4: tl^#(cons(x, l)) -> c_3(x, l)
, 5: append^#(l1, l2) -> c_4(ifappend^#(l1, l2, is_empty(l1)))
, 6: ifappend^#(l1, l2, true()) -> c_5(l2)
, 7: ifappend^#(l1, l2, false()) ->
c_6(hd^#(l1), append^#(tl(l1), l2))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5,7} [ MAYBE ]
|
|->{3} [ NA ]
|
`->{6} [ NA ]
->{4} [ YES(?,O(n^3)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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(is_empty) = {}, Uargs(cons) = {}, Uargs(hd) = {},
Uargs(tl) = {}, Uargs(append) = {}, Uargs(ifappend) = {},
Uargs(is_empty^#) = {}, Uargs(hd^#) = {}, Uargs(c_2) = {},
Uargs(tl^#) = {}, Uargs(c_3) = {}, Uargs(append^#) = {},
Uargs(c_4) = {}, Uargs(ifappend^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
is_empty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
true() = [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]
false() = [0]
[0]
[0]
hd(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tl(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
append(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]
ifappend(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
is_empty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
hd^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tl^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(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]
append^#(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_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifappend^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 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]
c_6(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]
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_empty^#(nil()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(is_empty^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
[2]
is_empty^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_0() = [0]
[1]
[1]
* Path {2}: 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(is_empty) = {}, Uargs(cons) = {}, Uargs(hd) = {},
Uargs(tl) = {}, Uargs(append) = {}, Uargs(ifappend) = {},
Uargs(is_empty^#) = {}, Uargs(hd^#) = {}, Uargs(c_2) = {},
Uargs(tl^#) = {}, Uargs(c_3) = {}, Uargs(append^#) = {},
Uargs(c_4) = {}, Uargs(ifappend^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
is_empty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
true() = [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]
false() = [0]
[0]
[0]
hd(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tl(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
append(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]
ifappend(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
is_empty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
hd^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tl^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(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]
append^#(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_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifappend^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 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]
c_6(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]
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_empty^#(cons(x, l)) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(is_empty^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [2]
[0 0 0] [0 0 0] [2]
[0 0 0] [0 0 0] [2]
is_empty^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_1() = [0]
[1]
[1]
* Path {4}: YES(?,O(n^3))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(is_empty) = {}, Uargs(cons) = {}, Uargs(hd) = {},
Uargs(tl) = {}, Uargs(append) = {}, Uargs(ifappend) = {},
Uargs(is_empty^#) = {}, Uargs(hd^#) = {}, Uargs(c_2) = {},
Uargs(tl^#) = {}, Uargs(c_3) = {}, Uargs(append^#) = {},
Uargs(c_4) = {}, Uargs(ifappend^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
is_empty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
true() = [0]
[0]
[0]
cons(x1, x2) = [1 3 3] x1 + [1 3 3] x2 + [0]
[0 1 1] [0 1 1] [0]
[0 0 1] [0 0 1] [0]
false() = [0]
[0]
[0]
hd(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tl(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
append(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]
ifappend(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
is_empty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
hd^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tl^#(x1) = [1 3 3] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1, x2) = [1 0 1] x1 + [1 0 1] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
append^#(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_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifappend^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 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]
c_6(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]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {tl^#(cons(x, l)) -> c_3(x, l)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(tl^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 2 2] x1 + [1 7 2] x2 + [2]
[0 0 2] [0 0 2] [2]
[0 0 0] [0 0 0] [2]
tl^#(x1) = [2 2 2] x1 + [3]
[2 2 0] [7]
[2 2 0] [7]
c_3(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [1]
[0 0 0] [0 0 0] [1]
* Path {5,7}: MAYBE
-----------------
The usable rules for this path are:
{ is_empty(nil()) -> true()
, is_empty(cons(x, l)) -> false()
, tl(cons(x, l)) -> cons(x, l)}
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 3'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ append^#(l1, l2) -> c_4(ifappend^#(l1, l2, is_empty(l1)))
, ifappend^#(l1, l2, false()) ->
c_6(hd^#(l1), append^#(tl(l1), l2))
, is_empty(nil()) -> true()
, is_empty(cons(x, l)) -> false()
, tl(cons(x, l)) -> cons(x, l)}
Proof Output:
The input cannot be shown compatible
* Path {5,7}->{3}: NA
-------------------
The usable rules for this path are:
{ is_empty(nil()) -> true()
, is_empty(cons(x, l)) -> false()
, tl(cons(x, l)) -> cons(x, l)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(is_empty) = {1}, Uargs(cons) = {}, Uargs(hd) = {},
Uargs(tl) = {1}, Uargs(append) = {}, Uargs(ifappend) = {},
Uargs(is_empty^#) = {}, Uargs(hd^#) = {1}, Uargs(c_2) = {},
Uargs(tl^#) = {}, Uargs(c_3) = {}, Uargs(append^#) = {1},
Uargs(c_4) = {1}, Uargs(ifappend^#) = {1, 3}, Uargs(c_5) = {},
Uargs(c_6) = {1, 2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
is_empty(x1) = [2 0 0] x1 + [2]
[0 0 0] [3]
[0 0 0] [3]
nil() = [0]
[0]
[0]
true() = [1]
[1]
[1]
cons(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
false() = [1]
[1]
[1]
hd(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tl(x1) = [2 0 0] x1 + [3]
[0 0 0] [3]
[0 0 0] [3]
append(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]
ifappend(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
is_empty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
hd^#(x1) = [3 3 3] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tl^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(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]
append^#(x1, x2) = [3 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
ifappend^#(x1, x2, x3) = [3 0 0] x1 + [0 0 0] x2 + [3 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 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]
c_6(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
[0 1 0] [0 1 0] [0]
[0 0 1] [0 0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {5,7}->{6}: NA
-------------------
The usable rules for this path are:
{ is_empty(nil()) -> true()
, is_empty(cons(x, l)) -> false()
, tl(cons(x, l)) -> cons(x, l)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(is_empty) = {1}, Uargs(cons) = {}, Uargs(hd) = {},
Uargs(tl) = {1}, Uargs(append) = {}, Uargs(ifappend) = {},
Uargs(is_empty^#) = {}, Uargs(hd^#) = {1}, Uargs(c_2) = {},
Uargs(tl^#) = {}, Uargs(c_3) = {}, Uargs(append^#) = {1},
Uargs(c_4) = {1}, Uargs(ifappend^#) = {1, 3}, Uargs(c_5) = {},
Uargs(c_6) = {1, 2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
is_empty(x1) = [2 0 0] x1 + [2]
[0 0 0] [3]
[0 0 0] [3]
nil() = [0]
[0]
[0]
true() = [1]
[1]
[1]
cons(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
false() = [1]
[1]
[1]
hd(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tl(x1) = [2 0 0] x1 + [3]
[0 0 0] [3]
[0 0 0] [3]
append(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]
ifappend(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
is_empty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
hd^#(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tl^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(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]
append^#(x1, x2) = [3 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
ifappend^#(x1, x2, x3) = [3 3 3] x1 + [0 0 0] x2 + [3 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_5(x1) = [1 1 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
[0 1 0] [0 1 0] [0]
[0 0 1] [0 0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
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: is_empty^#(nil()) -> c_0()
, 2: is_empty^#(cons(x, l)) -> c_1()
, 3: hd^#(cons(x, l)) -> c_2(x)
, 4: tl^#(cons(x, l)) -> c_3(x, l)
, 5: append^#(l1, l2) -> c_4(ifappend^#(l1, l2, is_empty(l1)))
, 6: ifappend^#(l1, l2, true()) -> c_5(l2)
, 7: ifappend^#(l1, l2, false()) ->
c_6(hd^#(l1), append^#(tl(l1), l2))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5,7} [ MAYBE ]
|
|->{3} [ NA ]
|
`->{6} [ NA ]
->{4} [ YES(?,O(n^2)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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(is_empty) = {}, Uargs(cons) = {}, Uargs(hd) = {},
Uargs(tl) = {}, Uargs(append) = {}, Uargs(ifappend) = {},
Uargs(is_empty^#) = {}, Uargs(hd^#) = {}, Uargs(c_2) = {},
Uargs(tl^#) = {}, Uargs(c_3) = {}, Uargs(append^#) = {},
Uargs(c_4) = {}, Uargs(ifappend^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
is_empty(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
true() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
hd(x1) = [0 0] x1 + [0]
[0 0] [0]
tl(x1) = [0 0] x1 + [0]
[0 0] [0]
append(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifappend(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
is_empty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
hd^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
tl^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
append^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
ifappend^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 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_empty^#(nil()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(is_empty^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
is_empty^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_0() = [0]
[1]
* Path {2}: 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(is_empty) = {}, Uargs(cons) = {}, Uargs(hd) = {},
Uargs(tl) = {}, Uargs(append) = {}, Uargs(ifappend) = {},
Uargs(is_empty^#) = {}, Uargs(hd^#) = {}, Uargs(c_2) = {},
Uargs(tl^#) = {}, Uargs(c_3) = {}, Uargs(append^#) = {},
Uargs(c_4) = {}, Uargs(ifappend^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
is_empty(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
true() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
hd(x1) = [0 0] x1 + [0]
[0 0] [0]
tl(x1) = [0 0] x1 + [0]
[0 0] [0]
append(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifappend(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
is_empty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
hd^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
tl^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
append^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
ifappend^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 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_empty^#(cons(x, l)) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(is_empty^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [2]
is_empty^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_1() = [0]
[1]
* Path {4}: 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(is_empty) = {}, Uargs(cons) = {}, Uargs(hd) = {},
Uargs(tl) = {}, Uargs(append) = {}, Uargs(ifappend) = {},
Uargs(is_empty^#) = {}, Uargs(hd^#) = {}, Uargs(c_2) = {},
Uargs(tl^#) = {}, Uargs(c_3) = {}, Uargs(append^#) = {},
Uargs(c_4) = {}, Uargs(ifappend^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
is_empty(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
true() = [0]
[0]
cons(x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 1] [0 1] [0]
false() = [0]
[0]
hd(x1) = [0 0] x1 + [0]
[0 0] [0]
tl(x1) = [0 0] x1 + [0]
[0 0] [0]
append(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifappend(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
is_empty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
hd^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
tl^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_3(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
append^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
ifappend^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 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: {tl^#(cons(x, l)) -> c_3(x, l)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(tl^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 2] x1 + [1 2] x2 + [0]
[0 0] [0 0] [0]
tl^#(x1) = [2 0] x1 + [7]
[0 0] [7]
c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
* Path {5,7}: MAYBE
-----------------
The usable rules for this path are:
{ is_empty(nil()) -> true()
, is_empty(cons(x, l)) -> false()
, tl(cons(x, l)) -> cons(x, l)}
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:
{ append^#(l1, l2) -> c_4(ifappend^#(l1, l2, is_empty(l1)))
, ifappend^#(l1, l2, false()) ->
c_6(hd^#(l1), append^#(tl(l1), l2))
, is_empty(nil()) -> true()
, is_empty(cons(x, l)) -> false()
, tl(cons(x, l)) -> cons(x, l)}
Proof Output:
The input cannot be shown compatible
* Path {5,7}->{3}: NA
-------------------
The usable rules for this path are:
{ is_empty(nil()) -> true()
, is_empty(cons(x, l)) -> false()
, tl(cons(x, l)) -> cons(x, l)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(is_empty) = {1}, Uargs(cons) = {}, Uargs(hd) = {},
Uargs(tl) = {1}, Uargs(append) = {}, Uargs(ifappend) = {},
Uargs(is_empty^#) = {}, Uargs(hd^#) = {1}, Uargs(c_2) = {},
Uargs(tl^#) = {}, Uargs(c_3) = {}, Uargs(append^#) = {1},
Uargs(c_4) = {1}, Uargs(ifappend^#) = {1, 3}, Uargs(c_5) = {},
Uargs(c_6) = {1, 2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
is_empty(x1) = [2 0] x1 + [3]
[3 3] [3]
nil() = [0]
[0]
true() = [0]
[1]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[1]
hd(x1) = [0 0] x1 + [0]
[0 0] [0]
tl(x1) = [2 0] x1 + [3]
[0 0] [3]
append(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifappend(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
is_empty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
hd^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_2(x1) = [1 0] x1 + [0]
[0 0] [0]
tl^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
append^#(x1, x2) = [3 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
ifappend^#(x1, x2, x3) = [3 0] x1 + [0 0] x2 + [3 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {5,7}->{6}: NA
-------------------
The usable rules for this path are:
{ is_empty(nil()) -> true()
, is_empty(cons(x, l)) -> false()
, tl(cons(x, l)) -> cons(x, l)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(is_empty) = {1}, Uargs(cons) = {}, Uargs(hd) = {},
Uargs(tl) = {1}, Uargs(append) = {}, Uargs(ifappend) = {},
Uargs(is_empty^#) = {}, Uargs(hd^#) = {1}, Uargs(c_2) = {},
Uargs(tl^#) = {}, Uargs(c_3) = {}, Uargs(append^#) = {1},
Uargs(c_4) = {1}, Uargs(ifappend^#) = {1, 3}, Uargs(c_5) = {},
Uargs(c_6) = {1, 2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
is_empty(x1) = [2 0] x1 + [3]
[3 3] [3]
nil() = [0]
[0]
true() = [0]
[1]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[1]
hd(x1) = [0 0] x1 + [0]
[0 0] [0]
tl(x1) = [2 0] x1 + [3]
[0 0] [3]
append(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifappend(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
is_empty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
hd^#(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
tl^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
append^#(x1, x2) = [3 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
ifappend^#(x1, x2, x3) = [3 3] x1 + [0 0] x2 + [3 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_5(x1) = [1 1] x1 + [0]
[0 0] [0]
c_6(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: is_empty^#(nil()) -> c_0()
, 2: is_empty^#(cons(x, l)) -> c_1()
, 3: hd^#(cons(x, l)) -> c_2(x)
, 4: tl^#(cons(x, l)) -> c_3(x, l)
, 5: append^#(l1, l2) -> c_4(ifappend^#(l1, l2, is_empty(l1)))
, 6: ifappend^#(l1, l2, true()) -> c_5(l2)
, 7: ifappend^#(l1, l2, false()) ->
c_6(hd^#(l1), append^#(tl(l1), l2))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5,7} [ MAYBE ]
|
|->{3} [ NA ]
|
`->{6} [ NA ]
->{4} [ YES(?,O(n^1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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(is_empty) = {}, Uargs(cons) = {}, Uargs(hd) = {},
Uargs(tl) = {}, Uargs(append) = {}, Uargs(ifappend) = {},
Uargs(is_empty^#) = {}, Uargs(hd^#) = {}, Uargs(c_2) = {},
Uargs(tl^#) = {}, Uargs(c_3) = {}, Uargs(append^#) = {},
Uargs(c_4) = {}, Uargs(ifappend^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
is_empty(x1) = [0] x1 + [0]
nil() = [0]
true() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
hd(x1) = [0] x1 + [0]
tl(x1) = [0] x1 + [0]
append(x1, x2) = [0] x1 + [0] x2 + [0]
ifappend(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
is_empty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
hd^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
tl^#(x1) = [0] x1 + [0]
c_3(x1, x2) = [0] x1 + [0] x2 + [0]
append^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
ifappend^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1, x2) = [0] x1 + [0] x2 + [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_empty^#(nil()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(is_empty^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [7]
is_empty^#(x1) = [1] x1 + [7]
c_0() = [1]
* Path {2}: 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(is_empty) = {}, Uargs(cons) = {}, Uargs(hd) = {},
Uargs(tl) = {}, Uargs(append) = {}, Uargs(ifappend) = {},
Uargs(is_empty^#) = {}, Uargs(hd^#) = {}, Uargs(c_2) = {},
Uargs(tl^#) = {}, Uargs(c_3) = {}, Uargs(append^#) = {},
Uargs(c_4) = {}, Uargs(ifappend^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
is_empty(x1) = [0] x1 + [0]
nil() = [0]
true() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
hd(x1) = [0] x1 + [0]
tl(x1) = [0] x1 + [0]
append(x1, x2) = [0] x1 + [0] x2 + [0]
ifappend(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
is_empty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
hd^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
tl^#(x1) = [0] x1 + [0]
c_3(x1, x2) = [0] x1 + [0] x2 + [0]
append^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
ifappend^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1, x2) = [0] x1 + [0] x2 + [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_empty^#(cons(x, l)) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(is_empty^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [0] x2 + [7]
is_empty^#(x1) = [1] x1 + [7]
c_1() = [1]
* Path {4}: 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(is_empty) = {}, Uargs(cons) = {}, Uargs(hd) = {},
Uargs(tl) = {}, Uargs(append) = {}, Uargs(ifappend) = {},
Uargs(is_empty^#) = {}, Uargs(hd^#) = {}, Uargs(c_2) = {},
Uargs(tl^#) = {}, Uargs(c_3) = {}, Uargs(append^#) = {},
Uargs(c_4) = {}, Uargs(ifappend^#) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
is_empty(x1) = [0] x1 + [0]
nil() = [0]
true() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
false() = [0]
hd(x1) = [0] x1 + [0]
tl(x1) = [0] x1 + [0]
append(x1, x2) = [0] x1 + [0] x2 + [0]
ifappend(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
is_empty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
hd^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
tl^#(x1) = [3] x1 + [0]
c_3(x1, x2) = [1] x1 + [1] x2 + [0]
append^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
ifappend^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1, x2) = [0] x1 + [0] x2 + [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: {tl^#(cons(x, l)) -> c_3(x, l)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(tl^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1] x1 + [1] x2 + [5]
tl^#(x1) = [3] x1 + [0]
c_3(x1, x2) = [1] x1 + [1] x2 + [0]
* Path {5,7}: MAYBE
-----------------
The usable rules for this path are:
{ is_empty(nil()) -> true()
, is_empty(cons(x, l)) -> false()
, tl(cons(x, l)) -> cons(x, l)}
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:
{ append^#(l1, l2) -> c_4(ifappend^#(l1, l2, is_empty(l1)))
, ifappend^#(l1, l2, false()) ->
c_6(hd^#(l1), append^#(tl(l1), l2))
, is_empty(nil()) -> true()
, is_empty(cons(x, l)) -> false()
, tl(cons(x, l)) -> cons(x, l)}
Proof Output:
The input cannot be shown compatible
* Path {5,7}->{3}: NA
-------------------
The usable rules for this path are:
{ is_empty(nil()) -> true()
, is_empty(cons(x, l)) -> false()
, tl(cons(x, l)) -> cons(x, l)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(is_empty) = {1}, Uargs(cons) = {}, Uargs(hd) = {},
Uargs(tl) = {1}, Uargs(append) = {}, Uargs(ifappend) = {},
Uargs(is_empty^#) = {}, Uargs(hd^#) = {1}, Uargs(c_2) = {},
Uargs(tl^#) = {}, Uargs(c_3) = {}, Uargs(append^#) = {1},
Uargs(c_4) = {1}, Uargs(ifappend^#) = {1, 3}, Uargs(c_5) = {},
Uargs(c_6) = {1, 2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
is_empty(x1) = [3] x1 + [3]
nil() = [1]
true() = [1]
cons(x1, x2) = [1] x1 + [1] x2 + [1]
false() = [1]
hd(x1) = [0] x1 + [0]
tl(x1) = [3] x1 + [3]
append(x1, x2) = [0] x1 + [0] x2 + [0]
ifappend(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
is_empty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
hd^#(x1) = [3] x1 + [0]
c_2(x1) = [1] x1 + [0]
tl^#(x1) = [0] x1 + [0]
c_3(x1, x2) = [0] x1 + [0] x2 + [0]
append^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_4(x1) = [1] x1 + [0]
ifappend^#(x1, x2, x3) = [3] x1 + [0] x2 + [3] x3 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1, x2) = [1] x1 + [1] x2 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {5,7}->{6}: NA
-------------------
The usable rules for this path are:
{ is_empty(nil()) -> true()
, is_empty(cons(x, l)) -> false()
, tl(cons(x, l)) -> cons(x, l)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(is_empty) = {1}, Uargs(cons) = {}, Uargs(hd) = {},
Uargs(tl) = {1}, Uargs(append) = {}, Uargs(ifappend) = {},
Uargs(is_empty^#) = {}, Uargs(hd^#) = {1}, Uargs(c_2) = {},
Uargs(tl^#) = {}, Uargs(c_3) = {}, Uargs(append^#) = {1},
Uargs(c_4) = {1}, Uargs(ifappend^#) = {1, 3}, Uargs(c_5) = {},
Uargs(c_6) = {1, 2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
is_empty(x1) = [1] x1 + [3]
nil() = [3]
true() = [1]
cons(x1, x2) = [1] x1 + [1] x2 + [3]
false() = [1]
hd(x1) = [0] x1 + [0]
tl(x1) = [1] x1 + [3]
append(x1, x2) = [0] x1 + [0] x2 + [0]
ifappend(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
is_empty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
hd^#(x1) = [3] x1 + [0]
c_2(x1) = [0] x1 + [0]
tl^#(x1) = [0] x1 + [0]
c_3(x1, x2) = [0] x1 + [0] x2 + [0]
append^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_4(x1) = [1] x1 + [0]
ifappend^#(x1, x2, x3) = [3] x1 + [0] x2 + [3] x3 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1, x2) = [1] x1 + [1] x2 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
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.