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