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