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