We consider the following Problem:
Strict Trs:
{ a__f(f(a())) -> c(f(g(f(a()))))
, mark(f(X)) -> a__f(mark(X))
, mark(a()) -> a()
, mark(c(X)) -> c(X)
, mark(g(X)) -> g(mark(X))
, a__f(X) -> f(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ a__f(f(a())) -> c(f(g(f(a()))))
, mark(f(X)) -> a__f(mark(X))
, mark(a()) -> a()
, mark(c(X)) -> c(X)
, mark(g(X)) -> g(mark(X))
, a__f(X) -> 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:
{ mark(a()) -> a()
, a__f(X) -> f(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__f) = {1}, Uargs(f) = {}, Uargs(c) = {}, Uargs(g) = {1},
Uargs(mark) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__f(x1) = [1 0] x1 + [1]
[0 0] [1]
f(x1) = [0 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
c(x1) = [0 0] x1 + [1]
[0 0] [1]
g(x1) = [1 0] x1 + [0]
[0 0] [1]
mark(x1) = [0 0] x1 + [1]
[0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ a__f(f(a())) -> c(f(g(f(a()))))
, mark(f(X)) -> a__f(mark(X))
, mark(c(X)) -> c(X)
, mark(g(X)) -> g(mark(X))}
Weak Trs:
{ mark(a()) -> a()
, a__f(X) -> 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: {mark(c(X)) -> c(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__f) = {1}, Uargs(f) = {}, Uargs(c) = {}, Uargs(g) = {1},
Uargs(mark) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__f(x1) = [1 0] x1 + [1]
[0 1] [1]
f(x1) = [0 0] x1 + [0]
[0 1] [0]
a() = [0]
[0]
c(x1) = [0 0] x1 + [1]
[0 0] [1]
g(x1) = [1 0] x1 + [0]
[0 1] [3]
mark(x1) = [0 3] x1 + [1]
[0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ a__f(f(a())) -> c(f(g(f(a()))))
, mark(f(X)) -> a__f(mark(X))
, mark(g(X)) -> g(mark(X))}
Weak Trs:
{ mark(c(X)) -> c(X)
, mark(a()) -> a()
, a__f(X) -> 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: {a__f(f(a())) -> c(f(g(f(a()))))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__f) = {1}, Uargs(f) = {}, Uargs(c) = {}, Uargs(g) = {1},
Uargs(mark) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__f(x1) = [1 1] x1 + [1]
[0 0] [1]
f(x1) = [0 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
c(x1) = [0 0] x1 + [0]
[0 0] [0]
g(x1) = [1 1] x1 + [2]
[0 0] [1]
mark(x1) = [0 0] x1 + [0]
[0 0] [3]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ mark(f(X)) -> a__f(mark(X))
, mark(g(X)) -> g(mark(X))}
Weak Trs:
{ a__f(f(a())) -> c(f(g(f(a()))))
, mark(c(X)) -> c(X)
, mark(a()) -> a()
, a__f(X) -> 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: {mark(g(X)) -> g(mark(X))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__f) = {1}, Uargs(f) = {}, Uargs(c) = {}, Uargs(g) = {1},
Uargs(mark) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__f(x1) = [1 0] x1 + [1]
[0 1] [0]
f(x1) = [0 0] x1 + [0]
[0 1] [0]
a() = [0]
[2]
c(x1) = [0 0] x1 + [1]
[0 0] [1]
g(x1) = [1 0] x1 + [1]
[0 1] [2]
mark(x1) = [0 1] x1 + [0]
[0 1] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs: {mark(f(X)) -> a__f(mark(X))}
Weak Trs:
{ mark(g(X)) -> g(mark(X))
, a__f(f(a())) -> c(f(g(f(a()))))
, mark(c(X)) -> c(X)
, mark(a()) -> a()
, a__f(X) -> 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: {mark(f(X)) -> a__f(mark(X))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__f) = {1}, Uargs(f) = {}, Uargs(c) = {}, Uargs(g) = {1},
Uargs(mark) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__f(x1) = [1 0] x1 + [1]
[0 1] [2]
f(x1) = [0 0] x1 + [0]
[0 1] [2]
a() = [0]
[0]
c(x1) = [0 0] x1 + [1]
[1 0] [2]
g(x1) = [1 0] x1 + [0]
[0 1] [0]
mark(x1) = [0 1] x1 + [0]
[0 1] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Weak Trs:
{ mark(f(X)) -> a__f(mark(X))
, mark(g(X)) -> g(mark(X))
, a__f(f(a())) -> c(f(g(f(a()))))
, mark(c(X)) -> c(X)
, mark(a()) -> a()
, a__f(X) -> f(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ mark(f(X)) -> a__f(mark(X))
, mark(g(X)) -> g(mark(X))
, a__f(f(a())) -> c(f(g(f(a()))))
, mark(c(X)) -> c(X)
, mark(a()) -> a()
, a__f(X) -> 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))