We consider the following Problem:
Strict Trs:
{ active(f(f(a()))) -> mark(f(g(f(a()))))
, mark(f(X)) -> active(f(X))
, mark(a()) -> active(a())
, mark(g(X)) -> active(g(mark(X)))
, f(mark(X)) -> f(X)
, f(active(X)) -> f(X)
, g(mark(X)) -> g(X)
, g(active(X)) -> g(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ active(f(f(a()))) -> mark(f(g(f(a()))))
, mark(f(X)) -> active(f(X))
, mark(a()) -> active(a())
, mark(g(X)) -> active(g(mark(X)))
, f(mark(X)) -> f(X)
, f(active(X)) -> f(X)
, g(mark(X)) -> g(X)
, g(active(X)) -> g(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ g(mark(X)) -> g(X)
, g(active(X)) -> g(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(f) = {}, Uargs(mark) = {},
Uargs(g) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 0] x1 + [1]
[0 0] [1]
f(x1) = [0 0] x1 + [0]
[0 0] [1]
a() = [0]
[0]
mark(x1) = [1 0] x1 + [1]
[0 0] [1]
g(x1) = [1 0] x1 + [0]
[0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ active(f(f(a()))) -> mark(f(g(f(a()))))
, mark(f(X)) -> active(f(X))
, mark(a()) -> active(a())
, mark(g(X)) -> active(g(mark(X)))
, f(mark(X)) -> f(X)
, f(active(X)) -> f(X)}
Weak Trs:
{ g(mark(X)) -> g(X)
, g(active(X)) -> g(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ mark(f(X)) -> active(f(X))
, mark(a()) -> active(a())}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(f) = {}, Uargs(mark) = {},
Uargs(g) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 0] x1 + [1]
[0 1] [0]
f(x1) = [0 0] x1 + [0]
[0 0] [1]
a() = [0]
[0]
mark(x1) = [1 0] x1 + [3]
[0 0] [1]
g(x1) = [1 0] x1 + [0]
[0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ active(f(f(a()))) -> mark(f(g(f(a()))))
, mark(g(X)) -> active(g(mark(X)))
, f(mark(X)) -> f(X)
, f(active(X)) -> f(X)}
Weak Trs:
{ mark(f(X)) -> active(f(X))
, mark(a()) -> active(a())
, g(mark(X)) -> g(X)
, g(active(X)) -> g(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(mark(X)) -> f(X)
, f(active(X)) -> f(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(f) = {}, Uargs(mark) = {},
Uargs(g) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 0] x1 + [0]
[0 1] [1]
f(x1) = [0 1] x1 + [0]
[0 0] [0]
a() = [0]
[3]
mark(x1) = [1 0] x1 + [0]
[0 1] [1]
g(x1) = [1 0] x1 + [0]
[0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ active(f(f(a()))) -> mark(f(g(f(a()))))
, mark(g(X)) -> active(g(mark(X)))}
Weak Trs:
{ f(mark(X)) -> f(X)
, f(active(X)) -> f(X)
, mark(f(X)) -> active(f(X))
, mark(a()) -> active(a())
, g(mark(X)) -> g(X)
, g(active(X)) -> g(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {active(f(f(a()))) -> mark(f(g(f(a()))))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(f) = {}, Uargs(mark) = {},
Uargs(g) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 0] x1 + [0]
[0 1] [1]
f(x1) = [0 2] x1 + [0]
[0 0] [2]
a() = [0]
[0]
mark(x1) = [1 0] x1 + [0]
[0 1] [1]
g(x1) = [1 0] x1 + [0]
[0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs: {mark(g(X)) -> active(g(mark(X)))}
Weak Trs:
{ active(f(f(a()))) -> mark(f(g(f(a()))))
, f(mark(X)) -> f(X)
, f(active(X)) -> f(X)
, mark(f(X)) -> active(f(X))
, mark(a()) -> active(a())
, g(mark(X)) -> g(X)
, g(active(X)) -> g(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs: {mark(g(X)) -> active(g(mark(X)))}
Weak Trs:
{ active(f(f(a()))) -> mark(f(g(f(a()))))
, f(mark(X)) -> f(X)
, f(active(X)) -> f(X)
, mark(f(X)) -> active(f(X))
, mark(a()) -> active(a())
, g(mark(X)) -> g(X)
, g(active(X)) -> g(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The problem is match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ active_0(2) -> 1
, f_0(2) -> 1
, a_0() -> 2
, mark_0(2) -> 1
, g_0(2) -> 1}
Hurray, we answered YES(?,O(n^1))