We consider the following Problem:
Strict Trs:
{ active(f(f(X))) -> mark(c(f(g(f(X)))))
, active(c(X)) -> mark(d(X))
, active(h(X)) -> mark(c(d(X)))
, mark(f(X)) -> active(f(mark(X)))
, mark(c(X)) -> active(c(X))
, mark(g(X)) -> active(g(X))
, mark(d(X)) -> active(d(X))
, mark(h(X)) -> active(h(mark(X)))
, f(mark(X)) -> f(X)
, f(active(X)) -> f(X)
, c(mark(X)) -> c(X)
, c(active(X)) -> c(X)
, g(mark(X)) -> g(X)
, g(active(X)) -> g(X)
, d(mark(X)) -> d(X)
, d(active(X)) -> d(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(f(X))) -> mark(c(f(g(f(X)))))
, active(c(X)) -> mark(d(X))
, active(h(X)) -> mark(c(d(X)))
, mark(f(X)) -> active(f(mark(X)))
, mark(c(X)) -> active(c(X))
, mark(g(X)) -> active(g(X))
, mark(d(X)) -> active(d(X))
, mark(h(X)) -> active(h(mark(X)))
, f(mark(X)) -> f(X)
, f(active(X)) -> f(X)
, c(mark(X)) -> c(X)
, c(active(X)) -> c(X)
, g(mark(X)) -> g(X)
, g(active(X)) -> g(X)
, d(mark(X)) -> d(X)
, d(active(X)) -> d(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)
, c(mark(X)) -> c(X)
, c(active(X)) -> c(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) = {1},
Uargs(c) = {1}, Uargs(g) = {}, Uargs(d) = {}, 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 + [1]
[0 0] [1]
c(x1) = [1 0] x1 + [0]
[0 0] [1]
g(x1) = [0 0] x1 + [0]
[0 0] [1]
d(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(f(X))) -> mark(c(f(g(f(X)))))
, active(c(X)) -> mark(d(X))
, active(h(X)) -> mark(c(d(X)))
, mark(f(X)) -> active(f(mark(X)))
, mark(c(X)) -> active(c(X))
, mark(g(X)) -> active(g(X))
, mark(d(X)) -> active(d(X))
, mark(h(X)) -> active(h(mark(X)))
, g(mark(X)) -> g(X)
, g(active(X)) -> g(X)
, d(mark(X)) -> d(X)
, d(active(X)) -> d(X)}
Weak Trs:
{ f(mark(X)) -> f(X)
, f(active(X)) -> f(X)
, c(mark(X)) -> c(X)
, c(active(X)) -> c(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(h(X)) -> mark(c(d(X)))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(f) = {1}, Uargs(mark) = {1},
Uargs(c) = {1}, Uargs(g) = {}, Uargs(d) = {}, Uargs(h) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 3] x1 + [1]
[0 0] [1]
f(x1) = [1 0] x1 + [0]
[0 0] [0]
mark(x1) = [1 0] x1 + [1]
[0 0] [1]
c(x1) = [1 0] x1 + [0]
[0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
d(x1) = [0 0] x1 + [0]
[0 0] [0]
h(x1) = [1 0] x1 + [0]
[0 0] [3]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ active(f(f(X))) -> mark(c(f(g(f(X)))))
, active(c(X)) -> mark(d(X))
, mark(f(X)) -> active(f(mark(X)))
, mark(c(X)) -> active(c(X))
, mark(g(X)) -> active(g(X))
, mark(d(X)) -> active(d(X))
, mark(h(X)) -> active(h(mark(X)))
, g(mark(X)) -> g(X)
, g(active(X)) -> g(X)
, d(mark(X)) -> d(X)
, d(active(X)) -> d(X)}
Weak Trs:
{ active(h(X)) -> mark(c(d(X)))
, f(mark(X)) -> f(X)
, f(active(X)) -> f(X)
, c(mark(X)) -> c(X)
, c(active(X)) -> c(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(c(X)) -> mark(d(X))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(f) = {1}, Uargs(mark) = {1},
Uargs(c) = {1}, Uargs(g) = {}, Uargs(d) = {}, Uargs(h) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 3] x1 + [1]
[0 0] [1]
f(x1) = [1 0] x1 + [0]
[0 0] [0]
mark(x1) = [1 0] x1 + [1]
[0 0] [1]
c(x1) = [1 0] x1 + [0]
[0 0] [2]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
d(x1) = [0 0] x1 + [0]
[0 0] [0]
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(f(X))) -> mark(c(f(g(f(X)))))
, mark(f(X)) -> active(f(mark(X)))
, mark(c(X)) -> active(c(X))
, mark(g(X)) -> active(g(X))
, mark(d(X)) -> active(d(X))
, mark(h(X)) -> active(h(mark(X)))
, g(mark(X)) -> g(X)
, g(active(X)) -> g(X)
, d(mark(X)) -> d(X)
, d(active(X)) -> d(X)}
Weak Trs:
{ active(c(X)) -> mark(d(X))
, active(h(X)) -> mark(c(d(X)))
, f(mark(X)) -> f(X)
, f(active(X)) -> f(X)
, c(mark(X)) -> c(X)
, c(active(X)) -> c(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(f(X))) -> mark(c(f(g(f(X)))))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(f) = {1}, Uargs(mark) = {1},
Uargs(c) = {1}, Uargs(g) = {}, Uargs(d) = {}, 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 1] [2]
mark(x1) = [1 0] x1 + [0]
[0 1] [1]
c(x1) = [1 0] x1 + [0]
[0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
d(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(c(X)) -> active(c(X))
, mark(g(X)) -> active(g(X))
, mark(d(X)) -> active(d(X))
, mark(h(X)) -> active(h(mark(X)))
, g(mark(X)) -> g(X)
, g(active(X)) -> g(X)
, d(mark(X)) -> d(X)
, d(active(X)) -> d(X)}
Weak Trs:
{ active(f(f(X))) -> mark(c(f(g(f(X)))))
, active(c(X)) -> mark(d(X))
, active(h(X)) -> mark(c(d(X)))
, f(mark(X)) -> f(X)
, f(active(X)) -> f(X)
, c(mark(X)) -> c(X)
, c(active(X)) -> c(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(active(X)) -> g(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(f) = {1}, Uargs(mark) = {1},
Uargs(c) = {1}, Uargs(g) = {}, Uargs(d) = {}, Uargs(h) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 0] x1 + [0]
[0 1] [1]
f(x1) = [1 0] x1 + [0]
[0 0] [0]
mark(x1) = [1 0] x1 + [0]
[0 1] [0]
c(x1) = [1 0] x1 + [0]
[0 0] [0]
g(x1) = [0 2] x1 + [0]
[0 0] [1]
d(x1) = [0 0] x1 + [0]
[0 0] [0]
h(x1) = [1 0] x1 + [1]
[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(c(X)) -> active(c(X))
, mark(g(X)) -> active(g(X))
, mark(d(X)) -> active(d(X))
, mark(h(X)) -> active(h(mark(X)))
, g(mark(X)) -> g(X)
, d(mark(X)) -> d(X)
, d(active(X)) -> d(X)}
Weak Trs:
{ g(active(X)) -> g(X)
, active(f(f(X))) -> mark(c(f(g(f(X)))))
, active(c(X)) -> mark(d(X))
, active(h(X)) -> mark(c(d(X)))
, f(mark(X)) -> f(X)
, f(active(X)) -> f(X)
, c(mark(X)) -> c(X)
, c(active(X)) -> c(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(X)) -> active(c(X))
, mark(g(X)) -> active(g(X))
, mark(d(X)) -> active(d(X))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(f) = {1}, Uargs(mark) = {1},
Uargs(c) = {1}, Uargs(g) = {}, Uargs(d) = {}, Uargs(h) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 0] x1 + [0]
[0 0] [0]
f(x1) = [1 0] x1 + [2]
[0 0] [0]
mark(x1) = [1 0] x1 + [1]
[0 0] [0]
c(x1) = [1 0] x1 + [1]
[0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
d(x1) = [0 0] x1 + [0]
[0 0] [1]
h(x1) = [1 0] x1 + [2]
[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)
, d(mark(X)) -> d(X)
, d(active(X)) -> d(X)}
Weak Trs:
{ mark(c(X)) -> active(c(X))
, mark(g(X)) -> active(g(X))
, mark(d(X)) -> active(d(X))
, g(active(X)) -> g(X)
, active(f(f(X))) -> mark(c(f(g(f(X)))))
, active(c(X)) -> mark(d(X))
, active(h(X)) -> mark(c(d(X)))
, f(mark(X)) -> f(X)
, f(active(X)) -> f(X)
, c(mark(X)) -> c(X)
, c(active(X)) -> c(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:
{ d(mark(X)) -> d(X)
, d(active(X)) -> d(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(f) = {1}, Uargs(mark) = {1},
Uargs(c) = {1}, Uargs(g) = {}, Uargs(d) = {}, Uargs(h) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 0] x1 + [0]
[0 1] [2]
f(x1) = [1 0] x1 + [0]
[0 0] [0]
mark(x1) = [1 0] x1 + [0]
[0 1] [2]
c(x1) = [1 2] x1 + [0]
[0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [1]
d(x1) = [0 2] x1 + [0]
[0 0] [0]
h(x1) = [1 2] x1 + [1]
[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)}
Weak Trs:
{ d(mark(X)) -> d(X)
, d(active(X)) -> d(X)
, mark(c(X)) -> active(c(X))
, mark(g(X)) -> active(g(X))
, mark(d(X)) -> active(d(X))
, g(active(X)) -> g(X)
, active(f(f(X))) -> mark(c(f(g(f(X)))))
, active(c(X)) -> mark(d(X))
, active(h(X)) -> mark(c(d(X)))
, f(mark(X)) -> f(X)
, f(active(X)) -> f(X)
, c(mark(X)) -> c(X)
, c(active(X)) -> c(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)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(f) = {1}, Uargs(mark) = {1},
Uargs(c) = {1}, Uargs(g) = {}, Uargs(d) = {}, Uargs(h) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 2] x1 + [0]
[0 0] [2]
f(x1) = [1 0] x1 + [2]
[0 0] [0]
mark(x1) = [1 0] x1 + [0]
[0 1] [2]
c(x1) = [1 0] x1 + [0]
[0 0] [0]
g(x1) = [1 2] x1 + [0]
[0 0] [0]
d(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)))}
Weak Trs:
{ g(mark(X)) -> g(X)
, d(mark(X)) -> d(X)
, d(active(X)) -> d(X)
, mark(c(X)) -> active(c(X))
, mark(g(X)) -> active(g(X))
, mark(d(X)) -> active(d(X))
, g(active(X)) -> g(X)
, active(f(f(X))) -> mark(c(f(g(f(X)))))
, active(c(X)) -> mark(d(X))
, active(h(X)) -> mark(c(d(X)))
, f(mark(X)) -> f(X)
, f(active(X)) -> f(X)
, c(mark(X)) -> c(X)
, c(active(X)) -> c(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)
, d(mark(X)) -> d(X)
, d(active(X)) -> d(X)
, mark(c(X)) -> active(c(X))
, mark(g(X)) -> active(g(X))
, mark(d(X)) -> active(d(X))
, g(active(X)) -> g(X)
, active(f(f(X))) -> mark(c(f(g(f(X)))))
, active(c(X)) -> mark(d(X))
, active(h(X)) -> mark(c(d(X)))
, f(mark(X)) -> f(X)
, f(active(X)) -> f(X)
, c(mark(X)) -> c(X)
, c(active(X)) -> c(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
, c_0(2) -> 1
, g_0(2) -> 1
, d_0(2) -> 1
, h_0(2) -> 1}
Hurray, we answered YES(?,O(n^1))