We consider the following Problem:
Strict Trs:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__b() -> a()
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__f(X1, X2, X3) -> f(X1, X2, X3)
, a__b() -> b()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__b() -> a()
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__f(X1, X2, X3) -> f(X1, X2, X3)
, a__b() -> b()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ mark(b()) -> a__b()
, mark(a()) -> a()
, a__f(X1, X2, X3) -> f(X1, X2, X3)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__f) = {2}, Uargs(mark) = {}, Uargs(f) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__f(x1, x2, x3) = [0 0] x1 + [1 1] x2 + [0 0] x3 + [1]
[1 1] [0 0] [1 1] [1]
a() = [0]
[0]
a__b() = [0]
[0]
b() = [0]
[0]
mark(x1) = [0 0] x1 + [1]
[0 0] [0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__b() -> a()
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__b() -> b()}
Weak Trs:
{ mark(b()) -> a__b()
, mark(a()) -> a()
, a__f(X1, X2, X3) -> f(X1, X2, X3)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ a__b() -> a()
, a__b() -> b()}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__f) = {2}, Uargs(mark) = {}, Uargs(f) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__f(x1, x2, x3) = [0 0] x1 + [1 1] x2 + [0 0] x3 + [1]
[1 1] [0 0] [1 1] [1]
a() = [0]
[0]
a__b() = [2]
[2]
b() = [0]
[0]
mark(x1) = [0 0] x1 + [3]
[0 0] [2]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)}
Weak Trs:
{ a__b() -> a()
, a__b() -> b()
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__f(X1, X2, X3) -> f(X1, X2, X3)}
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(X1, X2, X3)) -> a__f(X1, mark(X2), X3)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__f) = {2}, Uargs(mark) = {}, Uargs(f) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__f(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [0 1] x3 + [0]
[0 0] [0 1] [0 1] [3]
a() = [0]
[0]
a__b() = [0]
[3]
b() = [0]
[1]
mark(x1) = [0 1] x1 + [1]
[0 1] [2]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 1] [0 1] [3]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs: {a__f(a(), X, X) -> a__f(X, a__b(), b())}
Weak Trs:
{ mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__b() -> a()
, a__b() -> b()
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__f(X1, X2, X3) -> f(X1, X2, X3)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {a__f(a(), X, X) -> a__f(X, a__b(), b())}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__f) = {2}, Uargs(mark) = {}, Uargs(f) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__f(x1, x2, x3) = [0 1] x1 + [1 0] x2 + [0 1] x3 + [0]
[0 0] [1 0] [0 1] [1]
a() = [0]
[1]
a__b() = [0]
[2]
b() = [0]
[0]
mark(x1) = [1 0] x1 + [1]
[1 0] [3]
f(x1, x2, x3) = [0 1] x1 + [1 0] x2 + [0 1] x3 + [0]
[0 0] [0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Weak Trs:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__b() -> a()
, a__b() -> b()
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__f(X1, X2, X3) -> f(X1, X2, X3)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__b() -> a()
, a__b() -> b()
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__f(X1, X2, X3) -> f(X1, X2, X3)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
Hurray, we answered YES(?,O(n^1))