We consider the following Problem:
Strict Trs:
{ f(n__f(n__a())) -> f(n__g(n__f(n__a())))
, f(X) -> n__f(X)
, a() -> n__a()
, g(X) -> n__g(X)
, activate(n__f(X)) -> f(X)
, activate(n__a()) -> a()
, activate(n__g(X)) -> g(activate(X))
, activate(X) -> X}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ f(n__f(n__a())) -> f(n__g(n__f(n__a())))
, f(X) -> n__f(X)
, a() -> n__a()
, g(X) -> n__g(X)
, activate(n__f(X)) -> f(X)
, activate(n__a()) -> a()
, activate(n__g(X)) -> g(activate(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)
, activate(n__a()) -> a()}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {}, Uargs(n__f) = {}, Uargs(n__g) = {}, Uargs(g) = {1},
Uargs(activate) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [1]
[0 0] [1]
n__f(x1) = [0 0] x1 + [0]
[0 0] [0]
n__a() = [0]
[0]
n__g(x1) = [1 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
g(x1) = [1 0] x1 + [0]
[0 0] [1]
activate(x1) = [1 0] x1 + [1]
[0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ f(n__f(n__a())) -> f(n__g(n__f(n__a())))
, a() -> n__a()
, g(X) -> n__g(X)
, activate(n__f(X)) -> f(X)
, activate(n__g(X)) -> g(activate(X))
, activate(X) -> X}
Weak Trs:
{ f(X) -> n__f(X)
, activate(n__a()) -> a()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {a() -> n__a()}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {}, Uargs(n__f) = {}, Uargs(n__g) = {}, Uargs(g) = {1},
Uargs(activate) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [1]
[0 0] [1]
n__f(x1) = [0 0] x1 + [0]
[0 0] [0]
n__a() = [0]
[0]
n__g(x1) = [1 0] x1 + [0]
[0 0] [0]
a() = [1]
[0]
g(x1) = [1 0] x1 + [0]
[0 0] [1]
activate(x1) = [1 0] x1 + [1]
[0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ f(n__f(n__a())) -> f(n__g(n__f(n__a())))
, g(X) -> n__g(X)
, activate(n__f(X)) -> f(X)
, activate(n__g(X)) -> g(activate(X))
, activate(X) -> X}
Weak Trs:
{ a() -> n__a()
, f(X) -> n__f(X)
, activate(n__a()) -> a()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {g(X) -> n__g(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {}, Uargs(n__f) = {}, Uargs(n__g) = {}, Uargs(g) = {1},
Uargs(activate) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [1]
[0 0] [1]
n__f(x1) = [0 0] x1 + [0]
[0 0] [0]
n__a() = [0]
[0]
n__g(x1) = [1 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
g(x1) = [1 0] x1 + [2]
[0 0] [1]
activate(x1) = [1 0] x1 + [1]
[0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ f(n__f(n__a())) -> f(n__g(n__f(n__a())))
, activate(n__f(X)) -> f(X)
, activate(n__g(X)) -> g(activate(X))
, activate(X) -> X}
Weak Trs:
{ g(X) -> n__g(X)
, a() -> n__a()
, f(X) -> n__f(X)
, activate(n__a()) -> a()}
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) = {}, Uargs(n__f) = {}, Uargs(n__g) = {}, Uargs(g) = {1},
Uargs(activate) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [1]
[0 0] [1]
n__f(x1) = [0 0] x1 + [0]
[0 0] [0]
n__a() = [0]
[0]
n__g(x1) = [1 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
g(x1) = [1 2] x1 + [0]
[0 0] [1]
activate(x1) = [1 0] x1 + [3]
[0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ f(n__f(n__a())) -> f(n__g(n__f(n__a())))
, activate(n__g(X)) -> g(activate(X))
, activate(X) -> X}
Weak Trs:
{ activate(n__f(X)) -> f(X)
, g(X) -> n__g(X)
, a() -> n__a()
, f(X) -> n__f(X)
, activate(n__a()) -> a()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {f(n__f(n__a())) -> f(n__g(n__f(n__a())))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {}, Uargs(n__f) = {}, Uargs(n__g) = {}, Uargs(g) = {1},
Uargs(activate) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1) = [0 1] x1 + [0]
[0 0] [2]
n__f(x1) = [0 1] x1 + [0]
[0 0] [2]
n__a() = [0]
[0]
n__g(x1) = [1 0] x1 + [1]
[0 0] [0]
a() = [0]
[0]
g(x1) = [1 0] x1 + [1]
[1 0] [1]
activate(x1) = [1 0] x1 + [0]
[0 0] [2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ activate(n__g(X)) -> g(activate(X))
, activate(X) -> X}
Weak Trs:
{ f(n__f(n__a())) -> f(n__g(n__f(n__a())))
, activate(n__f(X)) -> f(X)
, g(X) -> n__g(X)
, a() -> n__a()
, f(X) -> n__f(X)
, activate(n__a()) -> a()}
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) = {}, Uargs(n__f) = {}, Uargs(n__g) = {}, Uargs(g) = {1},
Uargs(activate) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [1]
[0 0] [1]
n__f(x1) = [0 0] x1 + [0]
[0 0] [0]
n__a() = [0]
[0]
n__g(x1) = [1 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
g(x1) = [1 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: {activate(n__g(X)) -> g(activate(X))}
Weak Trs:
{ activate(X) -> X
, f(n__f(n__a())) -> f(n__g(n__f(n__a())))
, activate(n__f(X)) -> f(X)
, g(X) -> n__g(X)
, a() -> n__a()
, f(X) -> n__f(X)
, activate(n__a()) -> a()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs: {activate(n__g(X)) -> g(activate(X))}
Weak Trs:
{ activate(X) -> X
, f(n__f(n__a())) -> f(n__g(n__f(n__a())))
, activate(n__f(X)) -> f(X)
, g(X) -> n__g(X)
, a() -> n__a()
, f(X) -> n__f(X)
, activate(n__a()) -> a()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ f_0(2) -> 1
, f_1(2) -> 3
, f_1(4) -> 1
, f_1(4) -> 3
, n__f_0(2) -> 1
, n__f_0(2) -> 2
, n__f_0(2) -> 3
, n__f_1(2) -> 3
, n__f_1(4) -> 1
, n__f_1(4) -> 3
, n__f_1(6) -> 5
, n__a_0() -> 1
, n__a_0() -> 2
, n__a_0() -> 3
, n__a_1() -> 3
, n__a_1() -> 6
, n__g_0(2) -> 1
, n__g_0(2) -> 2
, n__g_0(2) -> 3
, n__g_1(3) -> 1
, n__g_1(3) -> 3
, n__g_1(5) -> 4
, a_0() -> 1
, a_1() -> 3
, g_0(2) -> 1
, g_1(3) -> 1
, g_1(3) -> 3
, activate_0(2) -> 1
, activate_1(2) -> 3}
Hurray, we answered YES(?,O(n^1))