We consider the following Problem:
Strict Trs:
{ active(g(X)) -> mark(h(X))
, active(c()) -> mark(d())
, active(h(d())) -> mark(g(c()))
, proper(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))
, proper(c()) -> ok(c())
, proper(d()) -> ok(d())
, g(ok(X)) -> ok(g(X))
, h(ok(X)) -> ok(h(X))
, top(mark(X)) -> top(proper(X))
, top(ok(X)) -> top(active(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()))
, proper(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))
, proper(c()) -> ok(c())
, proper(d()) -> ok(d())
, g(ok(X)) -> ok(g(X))
, h(ok(X)) -> ok(h(X))
, top(mark(X)) -> top(proper(X))
, top(ok(X)) -> top(active(X))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {top(ok(X)) -> top(active(X))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {}, Uargs(g) = {1}, Uargs(mark) = {1},
Uargs(h) = {1}, Uargs(proper) = {}, Uargs(ok) = {1},
Uargs(top) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 0] x1 + [1]
[1 0] [1]
g(x1) = [1 0] x1 + [0]
[0 0] [1]
mark(x1) = [1 0] x1 + [1]
[1 0] [1]
h(x1) = [1 0] x1 + [0]
[0 0] [1]
c() = [0]
[0]
d() = [0]
[0]
proper(x1) = [0 0] x1 + [1]
[0 0] [0]
ok(x1) = [1 0] x1 + [3]
[0 1] [3]
top(x1) = [1 0] x1 + [0]
[1 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()))
, proper(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))
, proper(c()) -> ok(c())
, proper(d()) -> ok(d())
, g(ok(X)) -> ok(g(X))
, h(ok(X)) -> ok(h(X))
, top(mark(X)) -> top(proper(X))}
Weak Trs: {top(ok(X)) -> top(active(X))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ proper(c()) -> ok(c())
, proper(d()) -> ok(d())}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {}, Uargs(g) = {1}, Uargs(mark) = {1},
Uargs(h) = {1}, Uargs(proper) = {}, Uargs(ok) = {1},
Uargs(top) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 0] x1 + [1]
[1 0] [1]
g(x1) = [1 0] x1 + [0]
[0 0] [0]
mark(x1) = [1 0] x1 + [1]
[1 0] [1]
h(x1) = [1 0] x1 + [0]
[0 0] [0]
c() = [0]
[0]
d() = [0]
[0]
proper(x1) = [0 0] x1 + [3]
[0 0] [2]
ok(x1) = [1 0] x1 + [1]
[0 0] [1]
top(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(g(X)) -> mark(h(X))
, active(c()) -> mark(d())
, active(h(d())) -> mark(g(c()))
, proper(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))
, g(ok(X)) -> ok(g(X))
, h(ok(X)) -> ok(h(X))
, top(mark(X)) -> top(proper(X))}
Weak Trs:
{ proper(c()) -> ok(c())
, proper(d()) -> ok(d())
, top(ok(X)) -> top(active(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(d())) -> mark(g(c()))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {}, Uargs(g) = {1}, Uargs(mark) = {1},
Uargs(h) = {1}, Uargs(proper) = {}, Uargs(ok) = {1},
Uargs(top) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 0] x1 + [1]
[1 0] [1]
g(x1) = [1 0] x1 + [0]
[0 0] [0]
mark(x1) = [1 0] x1 + [1]
[1 0] [1]
h(x1) = [1 0] x1 + [0]
[0 0] [1]
c() = [0]
[0]
d() = [1]
[0]
proper(x1) = [0 0] x1 + [2]
[0 0] [1]
ok(x1) = [1 0] x1 + [1]
[0 0] [1]
top(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(g(X)) -> mark(h(X))
, active(c()) -> mark(d())
, proper(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))
, g(ok(X)) -> ok(g(X))
, h(ok(X)) -> ok(h(X))
, top(mark(X)) -> top(proper(X))}
Weak Trs:
{ active(h(d())) -> mark(g(c()))
, proper(c()) -> ok(c())
, proper(d()) -> ok(d())
, top(ok(X)) -> top(active(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(g(X)) -> mark(h(X))
, active(c()) -> mark(d())}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {}, Uargs(g) = {1}, Uargs(mark) = {1},
Uargs(h) = {1}, Uargs(proper) = {}, Uargs(ok) = {1},
Uargs(top) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 0] x1 + [1]
[1 0] [0]
g(x1) = [1 0] x1 + [0]
[0 0] [1]
mark(x1) = [1 0] x1 + [0]
[1 0] [0]
h(x1) = [1 0] x1 + [0]
[0 0] [0]
c() = [0]
[0]
d() = [0]
[0]
proper(x1) = [0 0] x1 + [2]
[0 1] [0]
ok(x1) = [1 0] x1 + [2]
[0 1] [0]
top(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:
{ proper(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))
, g(ok(X)) -> ok(g(X))
, h(ok(X)) -> ok(h(X))
, top(mark(X)) -> top(proper(X))}
Weak Trs:
{ active(g(X)) -> mark(h(X))
, active(c()) -> mark(d())
, active(h(d())) -> mark(g(c()))
, proper(c()) -> ok(c())
, proper(d()) -> ok(d())
, top(ok(X)) -> top(active(X))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {top(mark(X)) -> top(proper(X))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {}, Uargs(g) = {1}, Uargs(mark) = {1},
Uargs(h) = {1}, Uargs(proper) = {}, Uargs(ok) = {1},
Uargs(top) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 0] x1 + [3]
[0 0] [0]
g(x1) = [1 0] x1 + [1]
[0 0] [0]
mark(x1) = [1 2] x1 + [1]
[0 0] [0]
h(x1) = [1 0] x1 + [1]
[0 0] [0]
c() = [2]
[2]
d() = [0]
[2]
proper(x1) = [1 0] x1 + [0]
[0 1] [0]
ok(x1) = [1 0] x1 + [0]
[0 0] [2]
top(x1) = [1 2] x1 + [0]
[0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ proper(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))
, g(ok(X)) -> ok(g(X))
, h(ok(X)) -> ok(h(X))}
Weak Trs:
{ top(mark(X)) -> top(proper(X))
, active(g(X)) -> mark(h(X))
, active(c()) -> mark(d())
, active(h(d())) -> mark(g(c()))
, proper(c()) -> ok(c())
, proper(d()) -> ok(d())
, top(ok(X)) -> top(active(X))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ proper(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {}, Uargs(g) = {1}, Uargs(mark) = {1},
Uargs(h) = {1}, Uargs(proper) = {}, Uargs(ok) = {1},
Uargs(top) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 1] x1 + [0]
[0 0] [1]
g(x1) = [1 0] x1 + [0]
[0 1] [1]
mark(x1) = [1 1] x1 + [0]
[0 0] [1]
h(x1) = [1 0] x1 + [0]
[0 1] [1]
c() = [0]
[0]
d() = [0]
[0]
proper(x1) = [0 1] x1 + [0]
[0 1] [0]
ok(x1) = [1 2] x1 + [0]
[0 0] [0]
top(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:
{ g(ok(X)) -> ok(g(X))
, h(ok(X)) -> ok(h(X))}
Weak Trs:
{ proper(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))
, top(mark(X)) -> top(proper(X))
, active(g(X)) -> mark(h(X))
, active(c()) -> mark(d())
, active(h(d())) -> mark(g(c()))
, proper(c()) -> ok(c())
, proper(d()) -> ok(d())
, top(ok(X)) -> top(active(X))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ g(ok(X)) -> ok(g(X))
, h(ok(X)) -> ok(h(X))}
Weak Trs:
{ proper(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))
, top(mark(X)) -> top(proper(X))
, active(g(X)) -> mark(h(X))
, active(c()) -> mark(d())
, active(h(d())) -> mark(g(c()))
, proper(c()) -> ok(c())
, proper(d()) -> ok(d())
, top(ok(X)) -> top(active(X))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ active_0(3) -> 1
, active_0(5) -> 1
, active_0(6) -> 1
, active_0(8) -> 1
, g_0(3) -> 2
, g_0(5) -> 2
, g_0(6) -> 2
, g_0(8) -> 2
, g_1(3) -> 10
, g_1(5) -> 10
, g_1(6) -> 10
, g_1(8) -> 10
, mark_0(3) -> 3
, mark_0(5) -> 3
, mark_0(6) -> 1
, mark_0(6) -> 3
, mark_0(8) -> 3
, h_0(3) -> 4
, h_0(5) -> 4
, h_0(6) -> 4
, h_0(8) -> 4
, h_1(3) -> 11
, h_1(5) -> 11
, h_1(6) -> 11
, h_1(8) -> 11
, c_0() -> 5
, d_0() -> 6
, proper_0(3) -> 7
, proper_0(5) -> 7
, proper_0(6) -> 7
, proper_0(8) -> 7
, ok_0(3) -> 8
, ok_0(5) -> 7
, ok_0(5) -> 8
, ok_0(6) -> 7
, ok_0(6) -> 8
, ok_0(8) -> 8
, ok_1(10) -> 2
, ok_1(10) -> 10
, ok_1(11) -> 4
, ok_1(11) -> 11
, top_0(1) -> 9
, top_0(3) -> 9
, top_0(5) -> 9
, top_0(6) -> 9
, top_0(7) -> 9
, top_0(8) -> 9}
Hurray, we answered YES(?,O(n^1))