We consider the following Problem:
Strict Trs:
{ active(f(X, X)) -> mark(f(a(), b()))
, active(b()) -> mark(a())
, mark(f(X1, X2)) -> active(f(mark(X1), X2))
, mark(a()) -> active(a())
, mark(b()) -> active(b())
, f(mark(X1), X2) -> f(X1, X2)
, f(X1, mark(X2)) -> f(X1, X2)
, f(active(X1), X2) -> f(X1, X2)
, f(X1, active(X2)) -> f(X1, X2)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ active(f(X, X)) -> mark(f(a(), b()))
, active(b()) -> mark(a())
, mark(f(X1, X2)) -> active(f(mark(X1), X2))
, mark(a()) -> active(a())
, mark(b()) -> active(b())
, f(mark(X1), X2) -> f(X1, X2)
, f(X1, mark(X2)) -> f(X1, X2)
, f(active(X1), X2) -> f(X1, X2)
, f(X1, active(X2)) -> f(X1, X2)}
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(X1), X2) -> f(X1, X2)
, f(active(X1), X2) -> f(X1, X2)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(f) = {1}, Uargs(mark) = {}
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, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
mark(x1) = [1 0] x1 + [1]
[0 0] [1]
a() = [0]
[0]
b() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ active(f(X, X)) -> mark(f(a(), b()))
, active(b()) -> mark(a())
, mark(f(X1, X2)) -> active(f(mark(X1), X2))
, mark(a()) -> active(a())
, mark(b()) -> active(b())
, f(X1, mark(X2)) -> f(X1, X2)
, f(X1, active(X2)) -> f(X1, X2)}
Weak Trs:
{ f(mark(X1), X2) -> f(X1, X2)
, f(active(X1), X2) -> f(X1, X2)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {active(b()) -> mark(a())}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(f) = {1}, Uargs(mark) = {}
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, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
mark(x1) = [1 0] x1 + [1]
[0 0] [1]
a() = [0]
[0]
b() = [2]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ active(f(X, X)) -> mark(f(a(), b()))
, mark(f(X1, X2)) -> active(f(mark(X1), X2))
, mark(a()) -> active(a())
, mark(b()) -> active(b())
, f(X1, mark(X2)) -> f(X1, X2)
, f(X1, active(X2)) -> f(X1, X2)}
Weak Trs:
{ active(b()) -> mark(a())
, f(mark(X1), X2) -> f(X1, X2)
, f(active(X1), X2) -> f(X1, X2)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ mark(a()) -> active(a())
, mark(b()) -> active(b())}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(f) = {1}, Uargs(mark) = {}
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, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
mark(x1) = [1 0] x1 + [3]
[0 0] [1]
a() = [0]
[0]
b() = [2]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ active(f(X, X)) -> mark(f(a(), b()))
, mark(f(X1, X2)) -> active(f(mark(X1), X2))
, f(X1, mark(X2)) -> f(X1, X2)
, f(X1, active(X2)) -> f(X1, X2)}
Weak Trs:
{ mark(a()) -> active(a())
, mark(b()) -> active(b())
, active(b()) -> mark(a())
, f(mark(X1), X2) -> f(X1, X2)
, f(active(X1), X2) -> f(X1, X2)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {f(X1, mark(X2)) -> f(X1, X2)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(f) = {1}, Uargs(mark) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 0] x1 + [0]
[0 1] [0]
f(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [1]
mark(x1) = [1 0] x1 + [1]
[0 0] [1]
a() = [0]
[0]
b() = [2]
[1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ active(f(X, X)) -> mark(f(a(), b()))
, mark(f(X1, X2)) -> active(f(mark(X1), X2))
, f(X1, active(X2)) -> f(X1, X2)}
Weak Trs:
{ f(X1, mark(X2)) -> f(X1, X2)
, mark(a()) -> active(a())
, mark(b()) -> active(b())
, active(b()) -> mark(a())
, f(mark(X1), X2) -> f(X1, X2)
, f(active(X1), X2) -> f(X1, X2)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {f(X1, active(X2)) -> f(X1, X2)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(f) = {1}, Uargs(mark) = {}
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, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [1]
mark(x1) = [1 0] x1 + [1]
[0 0] [1]
a() = [0]
[0]
b() = [0]
[1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ active(f(X, X)) -> mark(f(a(), b()))
, mark(f(X1, X2)) -> active(f(mark(X1), X2))}
Weak Trs:
{ f(X1, active(X2)) -> f(X1, X2)
, f(X1, mark(X2)) -> f(X1, X2)
, mark(a()) -> active(a())
, mark(b()) -> active(b())
, active(b()) -> mark(a())
, f(mark(X1), X2) -> f(X1, X2)
, f(active(X1), X2) -> f(X1, X2)}
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(X, X)) -> mark(f(a(), b()))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(f) = {1}, Uargs(mark) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 2] x1 + [0]
[0 0] [0]
f(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [2]
mark(x1) = [1 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
b() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs: {mark(f(X1, X2)) -> active(f(mark(X1), X2))}
Weak Trs:
{ active(f(X, X)) -> mark(f(a(), b()))
, f(X1, active(X2)) -> f(X1, X2)
, f(X1, mark(X2)) -> f(X1, X2)
, mark(a()) -> active(a())
, mark(b()) -> active(b())
, active(b()) -> mark(a())
, f(mark(X1), X2) -> f(X1, X2)
, f(active(X1), X2) -> f(X1, X2)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs: {mark(f(X1, X2)) -> active(f(mark(X1), X2))}
Weak Trs:
{ active(f(X, X)) -> mark(f(a(), b()))
, f(X1, active(X2)) -> f(X1, X2)
, f(X1, mark(X2)) -> f(X1, X2)
, mark(a()) -> active(a())
, mark(b()) -> active(b())
, active(b()) -> mark(a())
, f(mark(X1), X2) -> f(X1, X2)
, f(active(X1), X2) -> f(X1, X2)}
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, 2) -> 1
, mark_0(2) -> 1
, a_0() -> 2
, b_0() -> 2}
Hurray, we answered YES(?,O(n^1))