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