We consider the following Problem:
Strict Trs:
{ f(0()) -> cons(0(), n__f(s(0())))
, f(s(0())) -> f(p(s(0())))
, p(s(0())) -> 0()
, f(X) -> n__f(X)
, activate(n__f(X)) -> f(X)
, activate(X) -> X}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ f(0()) -> cons(0(), n__f(s(0())))
, f(s(0())) -> f(p(s(0())))
, p(s(0())) -> 0()
, f(X) -> n__f(X)
, activate(n__f(X)) -> f(X)
, activate(X) -> X}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {f(X) -> n__f(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {1}, Uargs(cons) = {}, Uargs(n__f) = {}, Uargs(s) = {},
Uargs(p) = {}, Uargs(activate) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1) = [1 0] x1 + [1]
[1 0] [1]
0() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
n__f(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [1]
activate(x1) = [1 0] x1 + [1]
[1 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ f(0()) -> cons(0(), n__f(s(0())))
, f(s(0())) -> f(p(s(0())))
, p(s(0())) -> 0()
, activate(n__f(X)) -> f(X)
, activate(X) -> X}
Weak Trs: {f(X) -> n__f(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {activate(n__f(X)) -> f(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {1}, Uargs(cons) = {}, Uargs(n__f) = {}, Uargs(s) = {},
Uargs(p) = {}, Uargs(activate) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1) = [1 0] x1 + [1]
[1 0] [1]
0() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
n__f(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [1]
activate(x1) = [1 0] x1 + [3]
[1 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ f(0()) -> cons(0(), n__f(s(0())))
, f(s(0())) -> f(p(s(0())))
, p(s(0())) -> 0()
, activate(X) -> X}
Weak Trs:
{ activate(n__f(X)) -> f(X)
, f(X) -> n__f(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {activate(X) -> X}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {1}, Uargs(cons) = {}, Uargs(n__f) = {}, Uargs(s) = {},
Uargs(p) = {}, Uargs(activate) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1) = [1 0] x1 + [1]
[0 0] [1]
0() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
[0 1] [0 0] [1]
n__f(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [1]
activate(x1) = [1 0] x1 + [1]
[0 1] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ f(0()) -> cons(0(), n__f(s(0())))
, f(s(0())) -> f(p(s(0())))
, p(s(0())) -> 0()}
Weak Trs:
{ activate(X) -> X
, activate(n__f(X)) -> f(X)
, f(X) -> n__f(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {f(0()) -> cons(0(), n__f(s(0())))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {1}, Uargs(cons) = {}, Uargs(n__f) = {}, Uargs(s) = {},
Uargs(p) = {}, Uargs(activate) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1) = [1 2] x1 + [1]
[0 0] [1]
0() = [0]
[2]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
n__f(x1) = [1 2] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
activate(x1) = [1 0] x1 + [1]
[0 1] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ f(s(0())) -> f(p(s(0())))
, p(s(0())) -> 0()}
Weak Trs:
{ f(0()) -> cons(0(), n__f(s(0())))
, activate(X) -> X
, activate(n__f(X)) -> f(X)
, f(X) -> n__f(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {p(s(0())) -> 0()}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {1}, Uargs(cons) = {}, Uargs(n__f) = {}, Uargs(s) = {},
Uargs(p) = {}, Uargs(activate) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1) = [1 0] x1 + [1]
[0 0] [1]
0() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
[0 1] [0 0] [1]
n__f(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [2]
p(x1) = [0 2] x1 + [0]
[0 0] [1]
activate(x1) = [1 0] x1 + [1]
[0 1] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs: {f(s(0())) -> f(p(s(0())))}
Weak Trs:
{ p(s(0())) -> 0()
, f(0()) -> cons(0(), n__f(s(0())))
, activate(X) -> X
, activate(n__f(X)) -> f(X)
, f(X) -> n__f(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {f(s(0())) -> f(p(s(0())))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {1}, Uargs(cons) = {}, Uargs(n__f) = {}, Uargs(s) = {},
Uargs(p) = {}, Uargs(activate) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1) = [1 1] x1 + [0]
[0 0] [0]
0() = [2]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [0]
n__f(x1) = [1 1] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [3]
[0 0] [3]
p(x1) = [0 1] x1 + [0]
[0 0] [1]
activate(x1) = [1 0] x1 + [1]
[0 1] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Weak Trs:
{ f(s(0())) -> f(p(s(0())))
, p(s(0())) -> 0()
, f(0()) -> cons(0(), n__f(s(0())))
, activate(X) -> X
, activate(n__f(X)) -> f(X)
, f(X) -> n__f(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ f(s(0())) -> f(p(s(0())))
, p(s(0())) -> 0()
, f(0()) -> cons(0(), n__f(s(0())))
, activate(X) -> X
, activate(n__f(X)) -> f(X)
, f(X) -> n__f(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
Hurray, we answered YES(?,O(n^1))