We consider the following Problem:
Strict Trs:
{ active(f(X)) -> mark(g(h(f(X))))
, mark(f(X)) -> active(f(mark(X)))
, mark(g(X)) -> active(g(X))
, mark(h(X)) -> active(h(mark(X)))
, f(mark(X)) -> f(X)
, f(active(X)) -> f(X)
, 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(f(X)) -> mark(g(h(f(X))))
, mark(f(X)) -> active(f(mark(X)))
, mark(g(X)) -> active(g(X))
, mark(h(X)) -> active(h(mark(X)))
, f(mark(X)) -> f(X)
, f(active(X)) -> f(X)
, 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:
{ f(mark(X)) -> f(X)
, f(active(X)) -> f(X)
, h(mark(X)) -> h(X)
, h(active(X)) -> h(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(f) = {1}, Uargs(mark) = {},
Uargs(g) = {}, Uargs(h) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 0] x1 + [1]
[0 1] [1]
f(x1) = [1 0] x1 + [0]
[0 0] [0]
mark(x1) = [1 0] x1 + [1]
[0 0] [1]
g(x1) = [0 0] x1 + [0]
[0 1] [0]
h(x1) = [1 0] x1 + [0]
[0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ active(f(X)) -> mark(g(h(f(X))))
, mark(f(X)) -> active(f(mark(X)))
, mark(g(X)) -> active(g(X))
, mark(h(X)) -> active(h(mark(X)))
, g(mark(X)) -> g(X)
, g(active(X)) -> g(X)}
Weak Trs:
{ f(mark(X)) -> f(X)
, f(active(X)) -> f(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(g(X)) -> active(g(X))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(f) = {1}, Uargs(mark) = {},
Uargs(g) = {}, Uargs(h) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 0] x1 + [1]
[0 0] [1]
f(x1) = [1 0] x1 + [0]
[0 0] [1]
mark(x1) = [1 0] x1 + [3]
[0 0] [1]
g(x1) = [0 0] x1 + [0]
[0 0] [1]
h(x1) = [1 0] x1 + [0]
[0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ active(f(X)) -> mark(g(h(f(X))))
, mark(f(X)) -> active(f(mark(X)))
, mark(h(X)) -> active(h(mark(X)))
, g(mark(X)) -> g(X)
, g(active(X)) -> g(X)}
Weak Trs:
{ mark(g(X)) -> active(g(X))
, f(mark(X)) -> f(X)
, f(active(X)) -> f(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: {active(f(X)) -> mark(g(h(f(X))))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(f) = {1}, Uargs(mark) = {},
Uargs(g) = {}, Uargs(h) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 0] x1 + [1]
[0 0] [1]
f(x1) = [1 0] x1 + [2]
[0 0] [1]
mark(x1) = [1 0] x1 + [1]
[0 1] [1]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
h(x1) = [1 0] x1 + [0]
[0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ mark(f(X)) -> active(f(mark(X)))
, mark(h(X)) -> active(h(mark(X)))
, g(mark(X)) -> g(X)
, g(active(X)) -> g(X)}
Weak Trs:
{ active(f(X)) -> mark(g(h(f(X))))
, mark(g(X)) -> active(g(X))
, f(mark(X)) -> f(X)
, f(active(X)) -> f(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:
{ g(mark(X)) -> g(X)
, g(active(X)) -> g(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(f) = {1}, Uargs(mark) = {},
Uargs(g) = {}, Uargs(h) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 0] x1 + [1]
[0 0] [1]
f(x1) = [1 0] x1 + [1]
[0 0] [1]
mark(x1) = [1 0] x1 + [1]
[0 0] [1]
g(x1) = [1 0] x1 + [0]
[0 0] [1]
h(x1) = [1 0] x1 + [0]
[0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ mark(f(X)) -> active(f(mark(X)))
, mark(h(X)) -> active(h(mark(X)))}
Weak Trs:
{ g(mark(X)) -> g(X)
, g(active(X)) -> g(X)
, active(f(X)) -> mark(g(h(f(X))))
, mark(g(X)) -> active(g(X))
, f(mark(X)) -> f(X)
, f(active(X)) -> f(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:
{ mark(f(X)) -> active(f(mark(X)))
, mark(h(X)) -> active(h(mark(X)))}
Weak Trs:
{ g(mark(X)) -> g(X)
, g(active(X)) -> g(X)
, active(f(X)) -> mark(g(h(f(X))))
, mark(g(X)) -> active(g(X))
, f(mark(X)) -> f(X)
, f(active(X)) -> f(X)
, h(mark(X)) -> h(X)
, h(active(X)) -> h(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The problem is match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ active_0(2) -> 1
, f_0(2) -> 1
, mark_0(2) -> 1
, g_0(2) -> 1
, h_0(2) -> 1}
Hurray, we answered YES(?,O(n^1))